METHOD AND SYSTEM FOR TRACKING

Inventor: Manus Henry University of Oxford

Patent Application Attorney Docket: 34513-4001

METHOD AND SYSTEM FOR TRACKING SINUSOIDAL WAVE PARAMETERS FROM A RECEIVED SIGNAL THAT INCLUDES NOISE

FIELD [0001] The disclosed embodiments relate generally to systems and methods for tracking

sinusoidal wave parameters in the presence of noise. BACKGROUND [0002] Sine waves are a widely occurring natural phenomenon. At the most basic level of

physics, all waveforms across the electromagnetic spectrum, from radio waves through visible light to gamma rays, are sinusoidal. Many mechanical systems, whether natural or artificial, have sinusoidal characteristics, typically arising from resonant vibration or circular motion.

For example, motors and alternators usually operate using a form of circular

motion: in the case of an alternator the resulting AC voltage is an inherently sinusoidal waveform. [0003] Accordingly, given the prevalence of sine waves in physical processes, sinusoidal

signals occur very widely in sensing and data processing systems. They may arise either because the underlying physical process has sinusoidal characteristics, or because a sensing or data processing mechanism employs another process with sinusoidal characteristics. An example of the first case is the use of an accelerometer to monitor an engine. As the engine vibrates, the accelerometer generates generally sinusoidal signals. As an example of the latter case, in a Coriolis mass flow meter, generally non-sinusoidal characteristics of a process fluid, for example its mass flow rate and density, may be sensed to high accuracy using a suitably designed, controlled and monitored, resonating flowtube. The motion of the resonant flowtube is detected using sensors which generate generally sinusoidal signals.

These sinusoidal signals are processed using suitable

algorithms to extract the desired non-sinusoidal characteristics of mass flow rate and density for the process fluid. A further class of applications is where sinusoidal waveforms

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are used as a communication means, for example by optical, electrical or radio signals, where information is coded into the transmitted, generally sinusoidal, signal and decoded from the received signal. More broadly, and as is well understood by those familiar with the art, an arbitrary signal may be decomposed into an equivalent sum of sinusoids, via Fourier analysis, and so aspects of these disclosures may be usefully applied to a wide range of signal processing tasks.

[0004] A pure sine wave is a time varying signal, usually represented as taking the form: s(t) =A sin(2;ift + ¢0

Equation (1)

)

where: s(t) is the time varying sinusoid, in units appropriate for the signal, for example volts (V)

tis time, typically in seconds (s)

A is the amplitude of the sinusoid, in the same units as s(t), for example volts (V) f is the frequency of the sinusoid, in hertz (Hz)

¢0 is the phase of the sinusoid at time t

= 0,

where phase is in radians restricted to the

range -TT < ¢o < TT, wrapping around any value outside the range using modulo 2n arithmetic. Those skilled in the art will recognize that for each of these parameters alternative but equivalent units and ranges may be used. sin is the sine function, a widely-used and well-understood mathematical function.

Equivalently, the cosine function could be used throughout this disclosure, where the phase shift relationship between sine and cosine is well-understood by those familiar with the art. Other related mathematical functions, such as the complex exponent, could also be used to represent a sinusoidal function, subject to suitable adjustments made to the mathematical treatment in these disclosures, as would be understood by those familiar with the art.

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[0005] f, A and ¢o are the parameters of the sine wave. There is a well-known relationship

between phase and frequency, namely, that frequency is the rate of change of phase with respect to time. Accordingly, Equation 1 can be rewritten as Equation (2)

s(t) =A sin ¢(t) where the time varying phase ¢(t) is given by

¢(t) = 2eft+ rflo

Equation (3)

and where ¢(t) is restricted to the range -TT
3) of Gs, combined in a similar fashion to that demonstrated in Figures 48 to 51. [00930] One potential implementation issue is the possible loss of precision where

consecutive low (un-normalised) gain values are generated by Prisms in the chain. This can be ameliorated by applying a fixed multiplier (for example, a power of two) between each Prism or Prism stage: such multipliers can be accounted for when correcting for the gain at any particular frequency during subsequent calculations. [00931] Figures 53 to 64 demonstrate, through a series of exemplary computer

simulations, differences in the properties of conventional versus Prism method-based FIR filters. In each simulation, a signal consisting solely of white noise is passed through a filter,

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and it is demonstrated that the spectrum of the filtered signal broadly matches the expected, theoretical frequency response of the corresponding filter. Other performance aspects evaluated in these examples are, for varying filter orders, the filter design time and filter computational time, as well as the previously discussed GMACs (i.e., billion multiply and accumulate operations per second) performance metric. [00932] Given the wide variety of computing devices available, with vastly different

computational power, the most important aspects of the simulation studies are the relative changes in performance with increasing filter order, rather than the absolute values. All simulations have been carried out on a laptop with an Intel Core i?-2860 QM CPU operating at 2.5 GHz, with 16 Gbytes of memory, and running a 64 bit version of Windows 7. All filters are implemented in C++ code. The Equiripple filters are coded as a straightforward implementation of Equation 3, where raw data, filter coefficients and accumulator are all double precision values. The Prism filters are also implemented using conventional double precision arithmetic, without using the 128 bit totalizers as described above. Given the presence of noise in the input signal, a single stage of Romberg Integration is applied during Prism filter operation. [00933] In each example, 200s of sampled white noise is generated in MATLAB® and

passed to the C++ routine for storage. Once all the data has been passed across, the C++ code is instructed to filter the entire dataset as a single continuous operation. Time stamps are recorded at the start and the end of the filtering process. After filtering is complete, the filtered data is passed back to MATLAB for subsequent analysis and plotting. This arrangement is intended to provide a reasonably continuous filtering operation on the CPU, whereby little processing power is diverted to other tasks, so that reasonably representative performance statistics can be generated for the filtering calculation. Those skilled in the art will recognize that additional and/or alternative steps could be taken to provide equivalent or improved computational efficiency of this implementation, without undermining the basic findings described here.

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[00934] The Equiripple FIR filters were designed using the MATLAB® Filter Design and

Analysis Tool. The filter coefficients, once calculated, were transferred manually into the C++ code for execution. Those familiar with the art will recognize that these implementation details are in keeping with common practice, but many alternative implementations could equally be used. Similarly, alternative Equiripple FIR design parameters might result in different filter characteristics, but the resulting relationship between filter length and execution time would be broadly similar to the results shown here. [00935] Figures 53 to 55 show the performance of the Equiripple filter for filter orders 96,

960, and 9600 (graph 530, graph 540 and graph 550). In each case the sampling frequency is fixed at 48 kHz, while the filter stop frequencies (as required by the MATLAB filter design tool) are 1.5 kHz, 150 Hz, and 15 Hz respectively. The pass frequency is in each case one third of the stop frequency. Other parameter values are the defaults offered by the MATLAB tool. Thus the three filters are scaled by an order of magnitude relative to one another in terms of cutoff frequency and filter order. Figures 53 to 55 show that the resulting frequency responses are broadly similar in character, allowing for the reduction by a factor of ten of the displayed frequency axis in each case, and that the white noise filtering performance broadly matches the theoretical frequency response as calculated by the MATLAB filter design tool. [00936] The simulations in Figures 53 to 55 demonstrate that the computational effort

required per sample is, for a conventional Equiripple FIR filter, approximately linear with the order of the filter. Thus in graph 530, with a 961h order filter, the time required to filter each sample is approximately 97 nanoseconds (ns); in graph 540, with a 9601h order filter, the time required to filter each sample is 884 ns; and in graph 550, with a 96001h order filter, the time required to filter each sample is 8710 ns. Thus an increase in filter order by a factor of ten, results in an approximate tenfold increase in the time required to filter each sample. Accordingly the estimated GMACs performance in each case is similar at around 1.1 billion MAC operations per second.

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[00937] The other performance criterion observed for the Equiripple filters is the time required to perform filter design. This performance metric may be sensitive to a wide range of influencing variables, including the toolset used and the full set of parameters defining the filter specification, and so only broad conclusions may be drawn from the results. The filter design times for the three experiments were 0.027 s, 0.671 s, and 55.91 s, respectively, showing a faster than linear increase in the required design time against filter order. Furthermore attempts to design longer Equiripple filters, for example of 960001h order, were, for the equipment and toolset used, not successful, with the MATLAB filter design toolset registering an error. These results suggest that conventional FIR filter design is a non-trivial, resource-intensive task, particularly for long filters. This may provide an impediment to performing new filter design in real time within field devices with limited resources, for example if attempting to design a new filter to match the observed or changing characteristics of the signal being monitored.

[00938] Considering now the Prism filter performance, the first observation to be made is that, in contrast with the Equiripple filter, Prism method or system filter designs are very simple, and capable of being carried out by computationally limited devices. The equivalent of the filter coefficients for the Prism is the set of values for the sine and cosine modulation functions in, for example, Equation 12. These are simply calculated as linearly-spaced sine and cosine values, based upon the harmonic number h of the Prism and the window length (itself encapsulating the ratio between the sample rate fs and the modulation frequency m, as discussed above). The following exemplary C code, readily understood by one familiar with the art, is sufficient to generate the required Prism filter coefficients (or equivalently the modulation function values): delta_phase for (i

=2*pi*h/window_length;

= O; i < window_length; i++) { ms(i)

=sin(i*delta_phase);

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mc(i)


= cos(i*delta_phase);

} Here ms and me are vectors to store the filter coefficients for subsequent Prism operation. Thus Prism filter design is simple, requires low computational resources (both in terms of computational power and coding complexity), and is therefore suited to being carried out in the field even by low power devices, for example designing a new filter to match the observed or changing characteristics of the signal being monitored. [00939] Figures 56 to 60 (graph 560, graph 570, graph 580, graph 590 and graph 600)

show equivalent simulations to those of Figures 53 to 55, using single Prism filters. With the common sample rate of 48 kHz, the values of mare selected to provide the equivalent filter order (or window length in samples) to the Equiripple filters. Thus, in graph 560, mis 1 kHz, each integral window contains 48 data points, leading to a filter order equivalent of 96, so that the Prism filter in Figure 56 is roughly equivalent to the Equiripple filter in Figure 53. The time required for the Prism filter, at 26 ns, is less than the 97 ns required by the Equiripple filter of Figure 53. This single performance comparison may however be strongly influenced by implementation detail. [00940] For the Prism simulations, a 'GMAC equivalent' performance has been calculated,

based on the following premise: assume the Prism is a conventional FIR filter, calculate the filter length (or filter order+ 1) divided by the time required to filter each sample. In the case of the m

=1

kHz Prism of Figure 56, the value obtained is 3.72 billion multiply and

accumulate operations per second (GMAC). [00941] Graph 570 shows the results of a simulation with m

= 100

Hz. Here each window

length is 480 samples, leading to an equivalent filter order of 960, so that the Prism filter in Graph 570is roughly equivalent to the Equiripple filter in Figure 54. In this case the time required for the Prism filter, only 27 ns, is roughly comparable with that of Figure 56, while the equivalent Equiripple filter of Figure 54 requires 884 ns. Thus while for the Equiripple

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filter, the required computational time per sample increases linearly with the order of the filter, for the Prism, the calculation time per sample remains approximately constant. Correspondingly, the GMAC equivalent performance for the Prism of Figure 57 increases to 35.65 billion MACs. [00942] This pattern of behavior is repeated in Figure 58 (graph 580), where m is set to 10

Hz, so that the equivalent filter order is 9600, comparable with the Equiripple filter of Figure 55. The time required to filter each sample remains roughly constant at 26 ns, so that the GMAC equivalent performance rises to 365. This contrasts with the Equiripple filter which requires 8710 ns per filtered sample. [00943] Figures 59 and 60 (graph 590 and graph 600) show Prism simulations for which no

Equiripple equivalents have been generated using the MATLAB filter design tool. In Figure 59, m

= 1 Hz

is used, thus creating a filter with an equivalent order of 96,000. Again the

time required to filter each sample remains roughly unchanged, at 27 ns per sample, so that the GMAC performance indicator increases to around 3567. Finally, in Figure 60, the sample rate fs is increased to 196 kHz while m remains at 1 Hz. This creates the equivalent of a 384,0001h order filter. Despite this high order, the simulated filter output broadly matches the theoretical prediction, and the filter design process remains the same as before. The time required to filter each sample has increased marginally to 32 ns. This is perhaps attributable to operating system issues, such as memory caching with large data vectors, as the algorithm itself remains unchanged. Nevertheless, the GMAC equivalent performance has increased to 12069. [00944] Figures 61 to 65 (graph 610, graph 620, graph 630, graph 640 and graph 650)

demonstrate that the same performance trends are observed when a Prism chain is used to implement a filter. The 3 stage bandpass filter, described above and illustrated in Figures 48 - 52, has been used to filter white noise with a variety of sample rates and central frequencies. As the filter comprises six Prisms in a sequence, the time required to filter each sample is approximately six times the processing time for an individual Prism. Thus,

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for example, in Figure 61, 143 ns are required to process each sample, compared with 27 ns for a typical single Prism experiment. [00945] In each of the figures, the values of m1 and m2, the equivalent orders of each

Prism type, and the total filter length in samples are given, as well as the previously described performance parameters. The same basic performance trends are observed, whereby as the total filter length increases from 1729 in Figure 61, to 6,887, 137 in Figure 65, an increase of approximately 4,000 times, the time per sample required by each filter increases from 143 ns to 208 ns, an increase of less than a factor of two, and one that may be attributable in part to operating system issues rather than the filtering algorithm itself. The GMACs equivalent performance increases from 12.13 to 33, 165 billion multiply and accumulate operations per second (GMACs), an improvement by a factor of approximately 3,000. [00946] It is useful to compare this final GMACs performance with the best claimed FIR

performance for the Xilinx FPGA family. The Z-7100 device, with up to 2,020 Programmable DSP slices operating in parallel, can achieve a peak FIR performance of 2,622 GMACs. This is an order of magnitude less than the performance shown in Figure 65, achieved on a single thread of C++ on a conventional laptop. Note further that, while the FPGA performance is likely to be based upon limited, fixed integer arithmetic (for example 25 bits - see Figure 12), the Prism performance has been achieved using full 64 bit double precision. Overall these results suggest that the implementation of Prism techniques within FPGA hardware or other specialized DSP hardware could lead to significant performance improvements over conventional FIR filtering methods. [00947] In summary, the Prism methods and systems facilitate FIR filtering with two

significant advantages: the filter design is very simple, and the calculation is recursive, so that the computational cost is essentially independent of the length or order of the filter. Those familiar with the art will recognize that a wide variety of other filters may be constructed using appropriate combinations of Prisms to deliver a desired filtering

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performance. Furthermore, banks of Prism-based filters may be constructed whereby, for example, the central frequency of each bandpass filter is different, and where the properties of the Prisms in each filter are selected to deliver the desired passband range for that filter. Such an arrangement may be used to provide comprehensive, simultaneous tracking of all frequencies components in a signal, whereby consecutive, approximately adjacent, narrow bands of potential component frequencies have corresponding bandpass filters and trackers. Each such bandpass filter and tracker might advantageously track the frequency, phase and amplitude of any signal component passing through the bandpass filter and reject any component with a frequency falling outside the corresponding bandpass filter range, thus reducing the influence of frequency leakage, a phenomenon well-known to those familiar with the art. Such an arrangement could provide, whether via on-line or off-line analysis, comprehensive tracking of the time-varying frequency, amplitude and phase behavior of all frequency components in a signal, with the advantages of low computational cost due to recursive Prism calculation, and reduced frequency leakage. Prism-based signal trackers [00948] In addition to its application as a type of FIR filter, the Prism can be used for

tracking sine waves (i.e., generating estimates of the frequency, phase and/or amplitude values of one or more frequency components in a possibly noisy signal). [00949] Initially, the problem of a single component will be considered. Later, additional

Prism-based techniques for dealing with multiple components in a single signal will be disclosed. Hereafter the term 'signal tracker' (or simply 'tracker') is used to describe a Prism-based signal processing block which generates estimates of frequency, phase and/or amplitude from a potentially noisy signal containing typically, but not exclusively, a single sinusoidal component. A signal tracker comprises a Prism network, itself comprising one or more Prisms, together with a calculation block which uses the Prism outputs and possibly other information to calculate the estimates of the sinusoid parameters. In addition,

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two classes of signal tracker are disclosed, which may be designated as FIR and llR (or recursive). These terms are used in a consistent manner compared with earlier texts in this disclosure for example with regard to filter characteristics, but in the current context the terms refer to the nature of the tracker calculation. In an FIR tracker, there is no recursive element in the calculation. For example, in the two FIR trackers described in these disclosures, the Fast Signal Tracker and the Deep Signal Tracker, the tracking calculations are based entirely on the current values of the Prism outputs, with no reference to previously calculated values. As the Prism outputs themselves are FIR, this approach ensures the trackers are also FIR. In an llR or recursive tracker, information from previous tracker calculations is used in the current calculation. One example of an llR tracker, the Recursive Signal Tracker, is disclosed here. [00950] The trackers can be used in combination with other Prisms, Prism chains or Prism

networks, for example to provide signal pre-filtering such as low-pass, band-pass, static notch filtering and/or dynamic notch filtering. More generally, trackers can be used to track individual signal components, as part of a larger Prism signal processing network tracking a multi-component signal, where individual components are first isolated using Prism filtering or other techniques. [00951] Figure 66 shows an embodiment of the structure of the Fast Signal Tracker (FST),

which is a simple signal processing object for tracking a single, possibly noisy, sinusoid input 660 in real time. This embodiment of the FST comprises two Prisms (Prism 662 and Prism 664) in parallel, both with characteristic frequency m Hz, but with harmonic numbers 1 and 2 respectively. The outputs of Prism 662 and Prism 664, labelled Gs, Ge, Gz, and Gk, are combined in a block 666 using a simple set of equations to obtain estimates of the frequency, phase and amplitude of the sinusoid being tracked. The equations shown in block 666 and discussed below are for illustration of one embodiment and do not limit the disclosure.

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[00952] Given sample-by-sample numerical values of the Prism outputs, the following

equations (derived from Equations 25 and 26 above) are used to calculate estimates of the sinusoid component properties.

X=

4G; +GI G2s +G2c

Equation (41)

Note that this square root, itself formed from squared values, is always real. The estimated value of r,

r,

is derived using

r= ~4x-2'

Equation (42)

x-2

where taking the positive root in Equation 42 yields a correct value for between 0 and 1. Given

r,

r

when it falls

the frequency of the sinusoid f is estimated using

]=rm.

Equation (43)

The amplitude and phase of the sinusoid are estimated using

Equation (44)

and

¢=

atan2

(-Gs' rGC )+ 2nr'

Equation (45)

where ¢ is adjusted modulo 2rr to fall within ±rr. In Equation 44 the sine function has been expanded into its component terms (Equation 18) to simplify the final calculation. [00953] These estimates can be calculated once per sample (or less frequently if desired),

based on the outputs of the Prisms (which must themselves be updated every sample to remain valid). Note that there is no requirement for prior estimates off, A, or ¢, nor are any

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estimated values carried over between samples: the entire calculation is FIR, with each estimate based solely on the instantaneous values of the Prism outputs, which are themselves FIR filters. [00954] The FST tracking performance has been evaluated for a range of noise-free

sinusoids in simulation. m was fixed at 200 Hz, so that 1/m is 5ms. The sample rate fs was set at 51.2 kHz to provide 256 (+1) samples in each integral, and hence a total data window length of 512 samples. Four stages of Romberg Integration were applied to provide high precision. The amplitude of the sinusoid was kept constant for each simulation at a nominal 1 Volt, while the frequency was varied from 0.5 Hz to 199.5 Hz in steps of 0.5 Hz. In terms of the frequency ratio r, this range is equivalent to 0.0025 to 0.9975 in steps of 0.0025. For each value of r, a 5 second simulation was carried out, with the FST estimates of frequency, amplitude and phase recorded and compared with their true values. An overall root sum square error for each parameter was calculated, using every sample over the last four seconds of the simulation, as follows: l

k=5fs

4fs

k=lfs

rss error= -

2:[valest(k)-valtrue(k)]

2

Equation (46)

where, for clarity, the true frequency and amplitude values were constant during each simulation, while the phase advanced linearly each sample, modulo ±rr, at a rate determined by the corresponding frequency and sample rate. This root sum square error has the units of the corresponding parameter. A relative error is further calculated as follows: relative rss error = ( rss error] .

l

Equation (47)

valtrue

[00955] Here val1rue is the constant value of frequency and amplitude over each simulation,

while it is assigned the value rr for phase. Figures 67-69 (graph 670, graph 680 and graph 690) show the results generated by the FST in noise-free (i.e. subject only to the errors

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inherent in using double precision representation of numerical values) simulations, for frequency, amplitude and phase respectively. Figure 70 (graph 700) shows the 'tracked signal' error, i.e. the difference between a reconstructed signal based on the calculated values Aand

¢ (i.e.

Asin(J) ),

and the original sinusoid. All plots show broadly similar trends.

Even for low frequencies (r < 0.1 ), tracking is good, with relative errors for all parameters below 1o-8 . For frequencies above

r = 0.1, errors are consistently below 10-12 for all

parameters. While most parameters show a general lowering of the error as towards 1, the amplitude error has a relatively sharp minimum between

r increases

r = 0.5 and 0.6,

where the error drops below 10-14 .

[00956] A second set of simulations have been carried out in which white noise was added to the signal s(t). The sample rate fs and detection frequency m was kept at 51.2 kHz and 200 Hz respectively. Simulations have been carried out over a range of noise levels and frequencies. The signal/noise ratio (SNR) in decibels is defined above in Equation 11, and in the simulation this was varied from 140 dB down to 40dB, with the frequency ratio

r

varying from 0.2 to 0.9. As the additive white noise is defined in terms of decibels, the dimensionless relative rss error of Equation 46 is converted to decibels using the following scaling: relative rss error (dB)= 20 log10 (relative rss error)

Equation (48)

[00957] The results are shown in Figures 71 to 74 (graph 710, graph 720, graph 730 and graph 740), which show the relative error in decibels for the given frequency and SNR, for the calculated values of frequency, amplitude, phase and tracked signal. To a first approximation, all parameters show a roughly linear relationship between the error and the level of noise. In other words, if the level of white noise were to increase by (say) a factor of ten, then the observed error in each parameter would also increase by a factor of ten. The results further show generally better error reduction for mid-range values of r. The most distinctive feature of these results is in graph 720, where the amplitude error is markedly

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lower at mid-range values of

r than for low or high values, reflecting the noise-free

behaviour displayed in graph 680. [00958] Figures 75 to 78 (graph 750, graph 760, graph 770 and graph 780) compare the

FST performance to the CRLB for each parameter, defined above in Equations 10 - 12, in both absolute and relative terms. Given the approximately linear performance with noise exhibited in Figures 71 to 74, a single SNR ratio of 40 dB was selected, and a fixed nominal amplitude of 1 Volt. The previously applied values of sample rate fs

= 51.2

kHz and m

=

200 Hz were reused. Given these experimental parameters, Figures 75 to 77 show the frequency, amplitude and phase errors generated by the FST in hertz, volts and radians respectively, alongside the corresponding CRLB values. Note that here each CRLB is expressed as a standard deviation, in the same linear units as the parameter being measured, by taking the square root of each value calculated in Equations 10 - 12. Figure 78 shows the ratio of the FST errors to the CRLB for each parameter. The phase and frequency error ratios track each other closely, achieving roughly 5: 1 or better for the approximate interval

r E[0.6, 0.7], while the FST amplitude achieves a 2:1 ratio compared

with the CRLB for the approximate interval approximately

r E[0.5, 0.6], with a minimum of around 1.4 at

r = 0.55.

[00959] For SNR values below 40 dB, the simple implementation of the FST algorithm

described here may generate large errors. Similarly, for the noise/signal ranges considered here, frequency values outside the interval

r E[0.25, 0.9] may result in large errors. One

familiar with the art will appreciate that it remains feasible to use a wider range of

r values

(for example, to use a high m value, and hence a lower r value, in order to benefit from a fast response time) if the SNR is sufficiently high, or if additional checks and rules are applied to prevent extreme errors from being generated in Equations 42 to 45. In addition, other Prism-based signal trackers with greater noise tolerance and better noise reduction behaviour can be created, as described below.

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[00960] The FIR nature of the Fast Signal Tracker suggests that it should exhibit a delay of

half the duration of its data window. The Prism double integral has a duration of 2/m, implying a delay of 1/m. This 1/m delay is already implied in Equations 25 and 26, where the phase delay term 2rrr corresponds to a time delay of 1/m seconds for the sinusoid of frequency f

= rm.

[00961] Two simulation examples are given to demonstrate the dynamic response of the

FST. In the first, the sinusoid amplitude is changed linearly with time, demonstrating that the delay in the FST tracking of amplitude is indeed 1/m over a wide range of r, and hence independent of the frequency of the sinusoid being tracked. [00962] The second example demonstrates the FIR-style robustness of the FST. The

frequency of a tracked (noise-free) sinusoid jumps instantly from 190 Hz down to 10 Hz, and the FST is able to track the new frequency with high accuracy as soon as the new data has completely filled its double integrals, i.e. after a period of 2/m seconds. [00963] Figures 79A to 83 (graph 790, graph 792, graph 800, graph 802, graph 810, graph

812, graph 820, graph 822, and graph 830) show the results of a simulation in which the amplitude of the input sinusoid is varied with time. As previously, the sample rate fs

= 51.2

kHz, and m is 200 Hz. Graph 790 shows the input sinusoid time series, while Graph 792 shows its power spectrum. The frequency of the input sinusoid is fixed at 133.33 Hz, so

r = 0.666. The amplitude initially takes a value of 1 V, but after t = 1 s the amplitude is reduced by 0.25 V/s until t = 3 s; thereafter the amplitude remains constant at 0.5 V.

that

[00964] Figures 80A and 808 show how the FST tracks the signal (based on the tracked

amplitude and phase parameter values as discussed above) over the entire time series. Figure 80A shows the tracked signal together with the original sinusoidal input while Figure 808 shows the error, i.e. the difference between the true and tracked values. The errors are small during the periods of steady amplitude, but between

t = 1 s and t = 3 s when the

amplitude is dropping, the tracked signal error range increases to approximately

± 1.25e-3 V. - 178 -



[00965] Figure 81A shows the true and tracked amplitude, while Figure 818 shows the

corresponding error. The error has an essentially constant value of 1.25e-3 V during the period of amplitude reduction. For the amplitude rate of change of 0.25 V/s, and assuming delay is the only source of error, the amplitude error implies a delay of 5ms between input and output. [00966] Figure 82A and 828 shows the amplitude and its error in more detail, around

t=

2.0 s. The difference between the true and FST calculated value are shown in Figure 82A, where the time delay is estimated directly. The true amplitude takes the value 0.75 V at

t = 2.0 s (left arrow), while the FST calculated amplitude reaches the value 0. 75 V at exactly t = 2.005 s (right arrow). This indicates a delay of 5 ms, corresponding to 1/m exactly

where mis 200 Hz. Figure 828 shows the amplitude error, which averages 1.25e-3 V, but which shows a small level of modulation. This modulation is at a minimum at

r = 0.666, and

hence the selection off= 133.33 Hz in this example. [00967] To demonstrate that the delay is 1/m over a range of

r values, the simulation of

Figures 79A to 82 has been repeated using different input frequencies. As already noted, the average amplitude error divided by the rate of change of amplitude gives the average delay. Figure 83 shows the results of this calculation for r between zero and 1. The delay is approximately 1/m for frequency ratio

r E [0.1, 0.9].

[00968] The second simulation demonstrates the robustness of the FST to sudden

parameter change, reflecting the FIR structure of the Prism. The same FST structure was used as previously, with a sample rate of 51.2 kHz and m = 200 Hz. The input signal is noise free, and, as shown in Figures 84A and 848, has the following behaviour: the amplitude is constant throughout the 2s simulation at 1 V, but at t = 1s, the frequency steps from 190 Hz down to 10 Hz instantly. Thus Figure 84A shows the time series of input signal, where there is a clear step change in behaviour at

t = 1.0 s, when the sinusoidal

frequency changes instantly from 190 Hz down to 10 Hz. Figure 848 shows the corresponding power spectrum of the entire input time series, which shows peaks at the

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two frequencies 190Hz and 1OHz. The purpose of the simulation is to demonstrate the FST's ability to adapt rapidly to sudden changes in a sinusoid parameter value. As the results demonstrate, as soon as data at the new frequency has worked through the Prism data windows (i.e. after 10 ms, or 2/m), the FST reports the new frequency, amplitude and phase parameters to high accuracy. [00969] Figures 85A to 928 (graph 850, graph 852, graph 860, graph 862, graph 870,

graph 872, graph 880, graph 882, graph 890, graph 892, graph 900, graph 902, graph 910, graph 912, ,graph 920 and graph 922) show the parameter outputs from the FST, over two timescales. Figures 85A to 888 show frequency, amplitude, phase and signal values respectively over the full simulation period, while Figures 89A to 928 show the same parameter values during the transition period around

t = 1.0 s. In each case, the true value

of each parameter is shown in the upper plot (e.g. Figure 85A) alongside the value calculated by the FST. In the lower plot (e.g. Figure 858) the logarithm of the (absolute value of) the FST parameter error is shown. Note that absolute error, rather than relative error in decibels, has been used here because of the very large step change in frequency value during the simulation, rendering the relative error difficult to interpret. [00970] Figures 85A to 888 show a similar pattern: there is good agreement between the

true and calculated parameter values both before and after the frequency step change, but large deviations at the point of transition itself. Errors before and after the transition are roughly 10-12 for all parameters. [00971] Figures 89A to 928 show the transition behaviour of each of the FST parameter

t = 0.995 sand t = 1.015 s. While there is some variation in behaviour, all parameters get back 'on track' at t = 1.01 s, with error levels reverting to around 10-12 .

outputs between

[00972] Thus the calculated frequency (Figures 89A and 898) swings in value between

1.0s and 1

= 1.01 s,

t=

but stays within the range of 10 Hz and 190 Hz. Similarly, the

calculated phase (Figures 91A and 918) stays within its nominal range of ±n radians, though of course it is constrained to do so: its behaviour is more erratic during the transition

- 180 -



than that of frequency. However, the calculated amplitude (Figures 90A and 908), and hence also the calculated signal value (Figures 92A and 928), shows wide variation, with values up to an order of magnitude higher than the true amplitude during the transition period. Those familiar with the art will recognise that the extreme values generated during the transition could be suppressed using jacketing code around the basic calculation, for example by detecting an unreasonably large calculated amplitude value: the purpose of this example is simply to demonstrate that as soon as the post-transition data fills the FST integrals, at around

t = 1.01 s, the tracking becomes accurate again.

[00973] In summary, the FST simulations have demonstrated the dynamic response of the

FST, arising from the FIR properties of the Prism itself: •

Where continuous changes occur in the parameter values of the input sinusoid, the FST output will be delayed by 1/m seconds in the reporting of these parameter changes;

•

When a disruptively large step change occurs in the input sinusoid, the period of transition for the disruption to pass through the FST and normal tracking performance to be restored is 2/m seconds.

•

The FST delay is independent of the frequency of the input signal, and may, by careful design, be a small fraction of the period of the input signal - hence the designation 'Fast'.

The Deep Signal Tracker [00974] Figure 93 shows an embodiment of the structure of the Deep Signal Tracker (DST)

930. Input signals 931 arrive at the disclosed DST 930, which comprises five Prisms (Prism 932, Prism 933, Prism 934, Prism 935, Prism 936) arranged in a cascade of three layers, together with a processing block 931. Processing block 937 uses the outputs of the five Prisms to calculate an estimate of the frequency ratio r, and then estimates of the

- 181 -



frequency, amplitude and phase of the input signal, j, Aand ¢respectively. Equations depicted in block 937 illustrate one embodiment and do not limit the disclosure. [00975] In this embodiment of the DST 930, all Prisms are configured to have the same

value of m and harmonic number h

= 1,

so notation related to h will be excluded for

discussion of the DST. Prism outputs are labelled based on the number of sine or cosine output stages they have passed through. Thus

Gccc

(t) is the output of the Prism network

which has passed s(t) through three cosine output stages, and so on. [00976] Simple analytic expressions describe the outputs of the Prism network, based on

the basic first harmonic Equations 21 and 24. As stated previously, the gain of the output signal is the product of the each of the Prisms the signal has passed through, while the phase delay is simply the sum of the delays of each Prism the signal has passed through. The phase calculation is particularly straightforward as the delay at each stage is simply 2rrr. Applying these principles to the DST, the following equations are derived to describe the outputs of the Prism network. Defining 2

k

=

2

Equation (49)

sinc (r)--f-r -1

Then: Gsss(t)

=

Equation (50)

3

Ak sin(¢(t)-6nr)

-A

3

Equation (51)

3

Equation (52)

Gcss(t)=-k cos(¢(t)-6nr) r

Gscc (t)

-A

= -

r

2

k sin(¢(t)- 6nr)

and Equation (53)

- 182 -



[00977] Figures 94A - 978 (graph 940, graph 942, graph 950, graph 952, graph 960, graph

962, graph 970, graph 972) show the results of comparing the values from Equations 49 to 53 with numerically generated integral values (i.e., numerically generating s(t) and passing its value through the Prism network shown in Figure 93) for a particular set of parameter values. [00978] The upper graph in each instance (e.g. Figure 94A) plots the analytical values and

those obtained by numerical integration, while the lower graph in each Figure (e.g. Figure 948) shows the difference between them. In each case the sampling rate fs is set at 51.2

= 200 Hz, and four stages of Romberg Integration are applied, while the input to the is a noise-free sinusoid with amplitude A = 1 V and frequency of 82 Hz. The results

kHz, m DST

shown in each instance demonstrate a good match between numerical and analytical results even when signals are propagated through several Prisms. The relative differences are typically of the order of 10-13 of the amplitude of the corresponding function value. [00979] Creating a Prism chain three deep as shown in Figure 93 may cause powerful

filtering. Figures 98A to 1018 (graph 980, graph 982, graph 990, graph 992, graph 1000, graph 1002, graph 1010 and graph 1012) plot the gains of the functions in Equations 50 to 53, which act as strong low pass filters. In each case the upper plot (e.g. Figure 98A) shows the variation of relative gain (i.e., the gain is scaled by its maximum value in order to show the relative variation with frequency from a maximum nominal value of (-)1 ). The lower plot in each case (e.g. Figure 988) shows the magnitude of the relative gain on a decibel scale. For all the functions shown, all signal content above the frequency 2m is significantly attenuated. [00980] Given the analytic values of the functions in Equations 50 to 53 it is straightforward

to construct a combination of these function values to provide an estimate of r, denoted r:

Equation (54)

- 183 -



This particular form of Equation ensures that the denominator is never zero, irrespective of the phase of the input signal. From the value of

r,

f may be calculated using:

]=rm.

Equation (55)

The amplitude and phase of the sinusoid can be estimated from any sin/cos pair of functions; for example, using Gccc (t) and Gscc (t):

Equation (56)

and Equation (57) where

¢ is adjusted modulo 2rr to fall within ±rr.

[00981] These estimates can be calculated once per sample (or less frequently if desired), based on the current values of Gsss(t), Gcss(t), Gscc(t) and Gccc(t). Thus, like the FST, the DST is an FIR calculation, with no carry-over of information between samples beyond the operation of the Prisms themselves. [00982] The basic DST measurement performance is presented via three simulation studies. In the first, noise free performance is examined. In the second, white noise is introduced. In the third, performance against the Cramer-Rao Lower Bound (CRLB) is assessed. [00983] The performance of the DST in noise-free conditions demonstrates the correctness of the calculation method and the best possible performance from the algorithm. For ease of comparison, the parameter values and procedure used for the FST, and shown in Figures 67 to 70, are repeated. fs, the sample rate, is 51.2 kHz while m is 200 Hz. These - 184 -



choices result in 256 sample intervals within each Prism integral. Each Prism has two layers of integrals, and the DST has three layers of Prisms, so the total window length of the DST is 3 x 2 x 256

= 1536 samples in this example.

[00984] The relative root sum square error (Equations 46 and 47) for each of the

measurement parameters is calculated for values of the frequency ratio r between 0 and 1; the results are shown in Figures 102 to 105 (graph 1020, graph 1030, graph 1040 and graph 1050). Figures 102 to 104 show frequency, amplitude and phase relative errors, respectively. Figure 105 shows the tracked signal error (i.e., the difference between a reconstructed waveform based on the Aand ¢ (i.e.

Asin(J) ),

and the original signal, relative

to the constant amplitude of the original signal). [00985] The noise free performance is good for all parameters, and shows an overall

improvement of approximately a factor of 10 compared with the FST (Figures 67 to 70). For

r E [0.3, 0.9], the phase, amplitude and signal relative errors are all close to 10-15 , while the frequency error is nearer 10-14 . The somewhat higher errors observed for all parameters when r < 0.3 may be attributable to the small gain values of the Prism functions for low values of r (Figures 98A to 101 B). [00986] To evaluate the DST for the purposes of real-time measurement, white noise is

added to the signal s(t) to observe the resulting measurement errors. A simulations study has examined DST performance for r

E

[0.2, 0.8] and a signal-to-noise ratio (SNR)

between 20 dB and 140 dB. This simulation study is similar to the one carried out for the FST, and discussed in the context of Figures 71 to 74. Figures 106 - 109 (graph 1060, graph 1070, graph 1080, and graph 1090) show the variation in root sum square error (in decibels) for the frequency, amplitude and phase parameters, as well as the calculated signal error. All parameter errors are essentially linear with SNR over the range shown, while the errors are lowest for mid-range values of r (approximately r

E

[0.4, 0.6]). As

shown later, the DST can generate stable measurement results at 0 dB over a more limited range of r values.

- 185 -



[00987] Figures 110 to 112 (graph 1100, graph 1110, and graph 1120) show how the

performance of the DST compares with the CRLB in both relative and absolute terms, for a specific SNR, 40 dB. Note that the relative performance vis-a-vis the CRLB is likely to be independent of the particular noise level selected, given the approximate linearity with noise exhibited in Figures 106 to 108. [00988] Every DST parameter value shows an order of magnitude or greater difference in

its error over the range of minimum error is around

r values used in the simulation. For phase and frequency, the

r = 0.5, while the amplitude error exhibits a plateau for r E [0.4,

0.6], approaching the CRLB. Figure 113 (graph 1130) shows the ratio of each DST parameter error to the corresponding CRLB. The amplitude plateau is at around 1.3 x CRLB. The frequency and phase CRLB ratios follow parallel trajectories against r, with both falling between 2 - 3 x CRLB for r E [0.4, 0.6]. [00989] It is important to note that the CRLB is a function of the number of sample points

N, as well as the SNR. As the DST uses three times as many points as the FST, its CRLB is correspondingly lower. Table 3 compares the CRLB limits and actual performance of the FST and the DST for the specific case A

= 1V, SNR = 40 dB and r = 0.5 (i.e., f = 100 Hz).

[00990] In Table 3, the Frequency CLRB is 2.4e-2 Hz for the 512 point FST, but 4.?e-3 Hz

for the 1536 point DST. Thus, while the CLRB ratio improves from 6.3 to 2.0, between the FST and the DST, the actual rss error performance is over fifteen times better, from 1.5e-1 Hz down to 9.2e-3 Hz. Similarly, there is a five-fold improvement in the rss error for phase between FST and DST. The improved performance of the DST is obtained at the cost of the additional computation associated with its extra Prisms, and also in its slower dynamic response, as discussed below.

- 186 -


Table 3


Comparison of FST and DST Errors,

Parameter

FST, N = 512 FST 1.5e-1

r = 0.5

Ratio 6.3

CLRB 4.7e-3

DST, N = 1536 DST 9.2e-3

Ratio 2.0

Freq. (Hz)

CLRB 2.4e-2

Phase (rad.)

8.8e-4

4.8e-3

5.5

5.1e-4

9.2e-4

1.8

Amplitude (V)

4.4e-4

8.4e-4

1.9

2.6e-4

3.5e-4

1.4

[00991] The dynamic response of the DST is presented from two perspectives. Firstly, its

ability to track parameter change in the absence of noise will be demonstrated. As may be expected for an FIR object with window length 6/m, the DST shows a delay of 3/m seconds in responding to parameter change. Secondly, the tracking performance of the DST will be demonstrated in a simulation of extreme conditions, where both the amplitude and frequency of the input sinusoid vary, while the SNR decreases to 0 dB. [00992] In Figures 114A to 1178 (graph 1140, graph 1142, graph 1150, graph 1152, graph

1160, graph 1162, graph 1170 and graph 1172), the results are shown of a DST simulation in which the amplitude of a noise-free input sinusoid drops from 1 V down to 0.5V from 1s to

t=

t = 3s. Here the previously employed values for the sample rate fs = 51.2 kHz and m

= 200 Hz are maintained. This simulation study is similar to the one carried out for the FST, and discussed in the context of Figures 79A - 828. [00993] Figure 114A shows the sinusoid input as a time series, while Figure 1148 shows

the corresponding power spectrum. The sinusoid frequency is fixed at 104 Hz throughout the simulation. Figure 115A shows the corresponding tracked signal generated by the DST, while Figure 1158 shows its error with respect to the original signal of Figure 114A. Prior to

t = 1s and after t = 3s, this error is small, but during the period of amplitude reduction the tracked signal error is significantly greater, with peak values of around ± 4 mV. In Figure 116A the true and DST calculated value of the signal amplitude is shown, while Figure 1168 shows the corresponding error. Here the error jumps to around 3. 75 mV

- 187 -



during the change in signal amplitude between

t = 1 s and t = 3 s. Figure 117A shows the

amplitude tracking in more detail. Here the dashed line is the true, time varying amplitude, while the continuous line is the amplitude estimate from the DST. Figure 117B shows the corresponding error. It is reasonable to assert that the fairly constant amplitude error with a mean value of around 3. 75 mV during this period is attributable to a simple time delay between the true amplitude and the corresponding value from the DST. This delay is shown

t = 2.0 s (left arrow) the true amplitude is 0. 75 V, while the DST reported amplitude reached 0. 75 V at t = 2.015 s (right arrow). Form = 200 Hz, this corresponds to a

to be 15 ms: at

delay of 3/m seconds. This compares with the broadly similar simulation on the FST shown in Figures 79A to 82B where the delay was 5ms, corresponding to 1/m. [00994] A series of such experiments has been carried out using the DST, where only the

frequency of the input signal was varied, and the calculated delay is plotted in Figure 118 (graph 1180). Over the ranger

E

[0.1, 0.9], the observed delay is close to 3/m. Thus, these

simulations confirm the expected result that the delay of the DST is typically around 3/m seconds. [00995] One benefit of using the DST is its ability to operate in the presence of moderate

levels of white noise. Figures 119A to 128B (graph 1190, graph 1192, graph 1200, graph 1202, graph 1210, graph 1212, graph 1220, graph 1222, graph 1230, graph 1232, graph 1240, graph 1242, graph 1250, graph 1252, graph 1260, graph 1262, graph 1270, graph 1272, graph 1280 and graph 1282) illustrate this behaviour in a simulation with the following characteristics. As previously, the sample rate fs

= 51.2 kHz and m = 200Hz. The

input sinusoid has steady amplitude and frequency values for

t

E

[0.0, 1.0] s and

t

E

[2.0,

3.0] s. However, between these steady state conditions there are simultaneous linear changes in amplitude and frequency. The amplitude is reduced from 1.0 V to 0.5 V, while the frequency increases from 80 Hz to 120 Hz. White noise of constant standard deviation 0.5 I ~2 V is added to the signal. This results in a SNR of 6 dB at the start of simulation, and of 0 dB after the reduction in amplitude.

- 188 -



[00996] Figures 119A, 119B, 120A and 120B provide an overview of the simulation. Figure

119A shows the whole time sequence. The noisy input signal is shown in grey, while the true and DST calculated sinusoidal signal values (i.e. Asin(¢)) are shown as the dashed and continuous lines respectively. Figure 119B shows the noise time series (i.e. the true sinusoidal signal subtracted from the noisy signal), and the DST calculated signal error. The noise standard deviation remains constant through the simulation, while the true signal amplitude reduces. The error in the tracked signal, while remaining small compared with the original signal noise, is higher after

t = 3.0s, reflecting the lower SNR ratio.

[00997] Figures 120A and 120B show a plot of the same parameters over a more limited

time period, i.e. during a 100 ms period after the parameter changes have been completed, during which the SNR ratio is 0 dB. In Figure 120A the difference between the true and calculated sinusoidal signal can be perceived more clearly, along with the relative magnitudes of signal and noise. In Figure 120B, the most noticeable feature is the low frequency characteristic of the DST error, which contrasts with the high frequency content of the noise, and is a consequence of the low pass filtering characteristic of the DST Prism network. [00998] This low frequency error characteristic is shared by the all the DST parameter

estimates, as illustrated in Figures 121A to 128B. Figures 121A to 126B show the frequency, amplitude and phase estimates on both full and detailed timescales, while Figures 127A, 127B, 128A and 128B show the tracked signal of Figures 119A, 119B, 120A and 120B with the noise signal excluded. In each case the upper graph (e.g. Figure 121A) shows the true and DST calculated value of the corresponding parameter, while the lower graph (e.g. Figure 121B) shows the corresponding error i.e. the difference between the true and DST calculated value. [00999] Figures 121A and 122A show the frequency results on different timescales, while

Figures 121 B and 122B show the corresponding errors. The DST successfully tracks the rise in frequency. Before and during the rise in frequency, the errors are approximately

- 189 -



within ± 2 Hz; after the rise in frequency (and simultaneous drop in amplitude and hence SNR), the frequency errors are limited to within± 5 Hz. In Figure 1228 the frequency error exhibits a low frequency characteristic. [001000] Figures 123A and 124A show the amplitude results on different timescales, while

Figures 1238 and 1248 show the corresponding errors. The DST successfully tracks the reduction in amplitude, while the magnitude of the error stays roughly constant at around 50 mV throughout the simulation. Figure 1248 demonstrates that the amplitude error has a low frequency characteristic. [001001] Figures 125A and 126A show the phase results on different timescales, while

Figures 1258 and 1268 show the corresponding errors. There is a significant increase in the phase error magnitude range from ± 0.2 radians before the change in amplitude and frequency, to ± 0.5 radians after

t = 3.0 s, attributable to the lower SNR. Figures 126A

and1268 show the variations in true and tracked phase in detail. Again, the error has a low frequency characteristic. [001002] Finally, Figures 127A and 128A show the tracked signal results on different

timescales, while Figures 1278 and 1288 show the corresponding errors. The low frequency characteristic of the tracked signal error is clear in Figure 1288. [001003] Having illustrated the tracking performance of the DST, a set of simulation results

are presented in which the consequences of using different numbers of Romberg Integration (RI) stages for Prism signal processing are demonstrated. Specifically, the performance of the DST is examined where the number of RI stages is varied, for input signals both with and without white noise present. Figures 129 to 136 (graph 1290, graph 1300, graph 1310, graph 1320, graph 1330, graph 1340, graph 1350 and graph 1360) show the results from a simulation in which the sample rate fs is 48 kHz, m is 187.5 Hz, while the input sensor signal has a frequency of 100 Hz, an amplitude of 1 V, and is noisefree. In each figure data is shown over the same 0.1 s duration time period. Figures 129, 131 and 133 show the calculated values of the three parameters frequency, amplitude and

- 190 -



phase, respectively, while Figures 130, 132 and 134 show the residual errors between the true value of the corresponding parameter and the DST calculation. In each case, plots are provided for the results obtained using up to four stages of Romberg Integration. Stage 0 indicates Trapezoidal integration alone without the application of the RI technique, while Stages 1 to 4 indicate the application of up to four levels of the Romberg calculation. On the residual error plots, in each legend, the root sum square error (rss, as defined above) of the parameter estimate is given, based on 4s of simulation. In addition to frequency, phase, and amplitude, the tracked signal is plotted in Figure 135, while the corresponding error plot is shown in Figure 136. This is the reconstructed signal based on the parameter values of amplitude and phase (as described above), which is compared in the error plot against the original signal input to the DST. [001004] In the absence of noise, the behaviour of the DST estimates in this example may

be simply characterised. The rss errors are reduced by roughly three orders of magnitude for each RI stage added, up to the fourth stage. The results obtained from applying an additional, fifth stage of RI (not shown in the figures) usually demonstrate only marginal further improvements for each parameter, suggesting that with a noise-free input signal, the precision limit may be approached for this particular implementation with four RI stages. [001005] The limiting error for each parameter value may primarily be a function of the data

types used for the signal data and the implementations of the Prism, RI and DST algorithms. In the implementation used to generate these results, new_datum in Figure 21 is a double precision value; this new_datum value has been calculated based on the product of the original s(t) value (calculated to double precision) and a modulation function value (also calculated to double precision). The new_datum value is mapped onto a 63 bit signed integer, while the totalizers are implemented as 128 bit signed integers: this approach eliminates the accumulation of rounding errors in the totalizers as data are added and subtracted. Alternative implementations will have corresponding cost/benefits in terms of precision and computational overhead, but in the absence of noise it is likely that several RI stages may be required to deliver the minimum achievable error.

- 191 -



[001006] By contrast, Figures 137 to 144 (graph 1370, graph 1380, graph 1390, graph

1400, graph 1410, graph 1420, graph 1430 and graph 1440) show the corresponding results when white noise is added to the signal. Here the signal-to-noise ratio (SNR) is 100 dB, which, for the purposes of many real-time measurement applications, may be considered a moderate to low level of noise. Figures 137, 139 and 141 show the frequency, phase and amplitude estimates respectively, while Figures 138, 140 and 142 show the corresponding error time series. In the error plots, in addition to providing the rss errors statistics for each RI stage, each legend further indicates the value of the Cramer-Rao Lower Bound (CRLB) for the parameter concerned. The CRLB is further shown as a dashed line in each error plot. Figures 143 and 144 show the results and errors for the reconstructed signal, for which a CRLB value is not available, but which otherwise follows the broad pattern of behaviour observed in Figures 137 - 142. [001007] In summary, the results in Figures 137 to 144 show that, even with a relatively

low SNR of 100 dB, little improvement is gained using more than one RI calculation stage. However, the error reduction achieved from Trapezoidal (Stage 0) to Stage 1 RI is typically 2 to 3 orders of magnitude in this simulation, thus demonstrating the accuracy improvement achieved by applying Romberg Integration in this case. In Figure 138, for example, the Stage 0 rss error is 7.2e-3 Hz, while for Stage 1 the rss error is significantly improved to 9.89e-6 Hz. The next three stages yield scarcely any further improvement, while the CRLB is only roughly half the error performance achieved by Stage 1. The plots suggest why this might be the case. The instantaneous errors of the Stage 1 to 4 measurements are essentially tied together as they drift randomly above and below the CRLB during the time series: the most significant source of measurement error at any time is the particular pattern of white noise within the DST data window, which higher levels of Romberg Integration are unable to alter. Even for a low SNR of 100 dB, it is this pattern of noise that overcomes any further measurement performance improvements that might be achieved using more RI stages. A similar pattern of behaviour is shown in Figures 140, 142 and 144.

- 192 -



For example, in Figure 140, the amplitude error at RI Stage 1, 3.2e-7 V, is already close to the CRLB of 2.6e-7 V. [001008] In summary, the results shown in Figures 129 to 144 demonstrate that with zero

noise (and subject to implementation details), up to 4 RI stages may usefully provide improved measurement performance for the Deep Signal Tracker; by contrast with even relatively low levels of noise, a single RI stage may be sufficient to provide close to optimal performance for the DST. The Recursive Signal Tracker [001009] As discussed above, the FST and DST are examples of FIR trackers, where the

calculation of sinusoid parameter values are based on the instantaneous output values of the Prisms in the tracker's Prism network. A second class of trackers, llR or Recursive trackers, may be constructed using one or more Prisms, in which results generated from previous calculations may be carried over into the current estimation of sinusoid parameter values. Those familiar with the art will recognise that llR signal processing objects are liable to be more computationally efficient that their FIR equivalents, but care must be taken to ensure that the resulting calculations remain stable, for example if the sequence of input data contains sudden changes or anomalous data. [00101 O] One general approach for developing Recursive Prism-based trackers is as

follows. In general, when both the Gs h and Geh outputs of a Prism are generated (Equations 25 and 26), if the value of r is known (and it can be assumed that hand mare known), then the values of f, A and

~

may be calculated in a straightforward manner, using, for example

a root-sum-square calculation of A and an arctangent calculation of

~;

note also that f is

simply the product of rand m. See Equations 59, 61 and 63 below for examples of such calculations. An additional relationship that can be employed to calculate or correct the value of

r is the well-known dependency between frequency and phase: namely that

frequency is the rate of change of phase with time. More simply put, assuming a constant

- 193 -



frequency over a given time period, then the frequency may be calculated as the change in phase over the time period, divided by the duration of the time period, where appropriate steps are taken to manage any phase wrap-around (i.e.

TT~ -TT,

as discussed above) that

may occur during the time period. [001011] This leads to the following general approach:

1) Assume an initial estimate of

r (for example, a guess, or based on a previously

calculated value of r or f) 2) Based on this value of r, calculate the current value of the signal phase, and also a value of phase for an earlier time. Note that this stage requires storing previous values of Gs h and Geh . 3) Given these two values of phase, and the known time interval between them, calculate a revised estimate off, and hence r. 4) If the new value of

r is significantly different from the original estimate, repeat

stages 2 and 3, starting with the new estimate of r, until convergence is achieved for the new estimate of r. [001012] Figure 145 is an illustrative example of an llR tracker. This embodiment of a

Recursive Signal Tracker (RST) 1450 uses a single Prism 1451, with harmonic number h

=

1, to generate output values Gs and Ge that are passed to block 1452. Past values of Gs and Ge are stored in a vector, for example implemented as a circular buffer of fixed length, while at least one past value - typically the most recent - of the estimate of r, labelled r , is also stored in block 1452. Equations and other notations in block 1452 and below are for this embodiment only and do not limit the disclosure. [001013] For the purposes of explaining the operation of the RST 1450, the current or

latest values of the Prism output are labelled Gso and Geo. While a variety of storage lengths may equally be used in the RST, it may be mathematically convenient to use previous

- 194 -



values of Gs and Ge from 1/m seconds earlier: these values are labelled Gsm and Gem· Note this arrangement implies that the circular buffers for storing earlier values of Gs and Ge may have the same length as the integrals used within the Prism. Other buffer lengths and/or time intervals may be considered, requiring modifications to the equations given below, and resulting in tradeoffs between accuracy and dynamic response.

[001014] Given the current and previous values of Gs and Ge, and assuming an initial estimate of r as P , then current and past estimates of phase are given by the following: Equation (58) Equation (59) A new estimate of r can be calculated using: Equation (60) where, as previously discussed, care is taken to deal with phase wrap-around issues: here the desired range of the calculated phase difference is [O, 2n] in order to provide a [O, 1] range of values for P. More generally, if the time interval over which the calculation takes place is longer than the period of the input signal, care must be taken to ensure the calculated phase difference includes the correct number of whole cycles (i.e. multiples of 2n) to ensure the frequency calculation is performed correctly. Note also that, for the purposes of computation efficiency, the term

21rr

in equations (58) and (59) may be

excluded for the purposes of calculating Pin equation (60), as long as the term is restored for the final calculation of

¢0 ,

as discussed below.

[001015] Equations 58 to 60 may be repeated as often as deemed necessary in order to achieve convergence in the value of P . Corresponding estimates of f and A may then be calculated as follows:

- 195 -



]=rm

Equation (61)

Equation (62)

while the best estimate of phase is given by the latest value of

¢0 calculated in Equation 58.

[001016] An important consideration is whether the iterative estimation of r generated by

repeated application of Equations 58 to 60, is likely to be convergent. A related question is as follows: when an RST is initialized and there is no earlier estimate of r, how is an initial estimate to be selected, and what are the convergence properties in this case? [001017] Simulation studies suggest that, over the range

r

E

[0.1, 0.9], the iterative

calculation is, on average, likely to converge on the true value of r, because any initial error will be reduced in a subsequent iteration. This is illustrated in Figures 146A to 1488 (graph 1460, graph 1462, graph 1470, graph 1472, graph 1480 and graph 1482), which show, for different true values of r, the sequence of consecutive values of r arising from repeated applications of Equations 58 to 60. In each case the results are generated by an exemplary simulation, in which the sample rate fs is 51.2 kHz, m

= 200

Hz and the

simulation is noise free. [001018] Figure 146A shows the calculated values of r, starting from an initial estimate of

0.0, while Figure 1468 shows the error in each estimate. Both plots use the iteration number as the x-axis, i.e. the number of times equations (58)-(60) have been calculated. Note that in a practical instantiation, one or more iterations could be used for each sample update as required.

In Figures 146A and 1468, the true value of

iteration achieves a magnitude of error of less than 1o-

5

.

r is 0.505. The first

Subsequent iterations reduce the

error until a precision limit is reached, so that after six or more iterations the error magnitude remains close to 10-15 .

- 196 -



[001019] In Figures 147A and 1478, the true value of

r is lower, at 0.12. The pattern of

behavior is similar to that of Figures 146A and 1468, such that the error magnitude reduces with each iteration, but the rate of convergence is much slower: after 25 iterations, the error magnitude has reduced to approximately 1o-6 . [001020] Figures 148A and 1488 illustrate the same procedure with a high true value of r,

set to 0.93. As previously, the magnitude of the error reduces with each iteration until a precision limit is reached, so that after eight or more iterations the error magnitude remains close to 10-15. [001021] In each of the Figures 1468, 1478 and 1488, the error reduction against iteration

number appears to show an approximately 'linear' change on the logarithmic y-axis. This suggests a geometric progression of the form: Erron+1

= k * Erron

Equation (63)

i.e. the error at iteration i+1 is some constant k times the error at iteration i, where the error reduction factor, k, depends upon, among other factors, the true value of r. Assuming this equation is applicable, then one familiar with the art would recognize the following claim: if the magnitude of k is less than one, then the magnitude of the error is likely to decrease with each iteration. [001022] Figure 149 (graph 1490) shows the results obtained in a simulation study in which

the error reduction factor k is estimated against rand the error in the initial estimate of r. The simulation was carried out as follows. At each experimental point, a value of r (ranging from 0.1 to 0.9) and the initial error in r (ranging from 10-7 to 10-1) is selected. The sample rate fs is kept constant at 51.2 kHz, m second simulation, the value of initial estimate of

=200 Hz and the simulation is noise free. Over a five

r is calculated based on Equations 58 to 60, where the

r is reset every sample to match the selected initial error value. Only a

single iteration of the r estimation is performed each sample, so that the error reduction observed is the result of just a single iteration. The root sum square of the error in

- 197 -

r is



calculated over the last four seconds of the simulation. This root sum square error is divided by the original error in r to determine the error reduction factor. Note that calculating the root sum square error over a four second period provides a more comprehensive estimate of the noise reduction factor than would using the result from just a single sample, as it includes the influence of other factors, for example, the current value

of~-

[001023] The results shown in Figure 149 are consistent with the examples shown in

Figures 146A to 1488. The calculated error reduction factor is approximately constant with the value of the error, suggesting that the assumption of a geometric progression shown in Equation 65 may be reasonable. In other words, for any value of r, the error reduction

r varies from 10-1 down to 1o- 7 . The error reduction factor is shown to be highly dependent upon the value of r: With r close to 0.5, the error reduction per iteration appears to be most effective; with values of r greater than 0.5, the error reduction factor shown remains below 0.1, while for values of r less than

factor appears to be approximately constant as the error in

0.5 the error reduction factor is generally higher, but, for the simulated examples used, does not exceed one. Accordingly, these simulation studies suggest that the calculation of r using a recursive, and possibly iterative technique, may be numerically stable. [001024] While a range of different strategies may be applied for initialization and iteration

r calculation, the following simple strategy has been used in the examples which follow. On initialization, a first estimate of r = 0.5 is selected and two iterations are applied of the

to the first sample with complete data (i.e. as soon as the past history vectors for Gs and Ge are filled for the first time). On subsequent samples, the most recent estimate of

r is used

as the starting point, and a single iteration is applied. An important alternative approach, which would maintain a strictly FIR characteristic of the tracker, would be always to restart the estimate of

r at a fixed value, for example 0.5, and iterate to convergence every

sample. However, this strategy would be more computationally expensive, especially for low values of rfor which many iterations may be required.

- 198 -



[001025] The generally efficient error reduction properties of this method for values of

r

close to 0.5 provides one motivation for an approach to sinusoidal tracking as follows. When a sinusoid is being tracked and the value of suitable bounds, for example

r is not reasonably close to 0.5 (where

r < 0.4 or r > 0.6, may be selected as required for the

particular application), it may prove advantageous to instantiate new Prism-based signal processing blocks where the newly selected value(s) of m result in corresponding values of

r close to 0.5, in order to reduce tracking error. [001026] The RST performance as a sinusoidal tracker will now be demonstrated through a series of simulations. In broad terms, the RST may be characterized as having a similarly fast dynamic response as the FST, while having an accuracy and a robustness to noise similar to or better than the DST. These general properties will now be demonstrated in a series of simulation studies following those previously shown for the two FIR trackers.

[001027] The RST tracking performance has been evaluated for a range of noise-free sinusoids in simulation. m was fixed at 200 Hz, so that 1/m is 5ms. The sample rate fs was set at 51.2 kHz to provide 256 (+1) samples in each integral, and hence a total data window length of 512 samples. Four stages of Romberg Integration were applied to provide high precision. The amplitude of the sinusoid was kept constant for each simulation at a nominal 1 Volt, while the frequency was varied from 0.5 Hz to 199.5 Hz in steps of 0.5 Hz. In terms of the frequency ratio r, this range is equivalent to 0.0025 to 0.9975 in steps of 0.0025. For each value of r, a 5 second simulation was carried out, with the RST estimates of frequency, amplitude and phase recorded and compared with their true values. An overall root sum square error for each parameter was calculated, using every sample over the final four seconds of the simulation, as stated above in Equations 46 and 47.

[001028] Figures 150 to 152 (graph 1500, graph 1510 and graph 1520) show the results generated by the RST in noise-free simulations, for frequency, amplitude and phase respectively. Figure 153 (graph 1530) shows the tracked signal error, i.e. the difference between a reconstructed signal based on the calculated values Aand

- 199 -

¢ (i.e.

Asin(J) ),

and



the original sinusoid. All plots show broadly similar trends. Even for low frequencies (r < 0.1 ), tracking is good, with relative errors for all parameters below 1o-8 . For frequencies

r = 0.1, errors are consistently below 10-12 for all parameters. Each of the parameters shows a general lowering of the error as r increases towards 1.

above

[001029] A second set of simulations have been carried out in which white noise was

added to the signal s(t). The sample rate fs and detection frequency m were kept at 51.2 kHz and 200 Hz respectively. Simulations have been carried out over a range of noise levels and frequencies. The signal/noise ratio (SNR) in decibels is defined above in Equation 13, and in the simulation varies from 140 dB down to 20dB, with the frequency ratio

r varying from 0.2 to 0.8. The results are shown in Figures 154 to 157 (graph 1540,

graph 1550, graph 1560, graph 1570), which show the relative error in decibels for the given frequency and SNR, for the calculated values of frequency, amplitude, phase and signal. To a first approximation, all parameters show a roughly linear relationship between the error and the level of noise. [001030] Figures 158 to 161 compare the RST performance to the CRLB for each

parameter, defined above in Equations 10 - 12, in both absolute and relative terms. Given the approximately linear performance with noise exhibited in Figures 154 - 157, a single SNR ratio of 40 dB was selected, and a fixed nominal amplitude of 1 Volt. The previously applied values of sample rate fs

= 51.2

kHz and m

= 200

Hz were reused. Note that the

number of samples used, N, is 768. This is based on the two integral windows in the Prism, of 256 samples each, plus the window of old Gs and Ge values, which has a length of 256 samples in this case. [001031] Given these values, Figures 158 to 160 (graph 1580, graph 1590 and graph

1600) show the frequency, amplitude and phase errors generated by the RST in hertz, volts and radians respectively, alongside the corresponding CRLB values. Here each CRLB is expressed as a standard deviation, in the same linear units as the parameter being measured, by taking the square root of each value calculated in Equations 8 to 10. Figure

- 200 -



161 (graph 1610) shows the ratio of the RST errors to the CRLB for each parameter. The phase and frequency error ratios track each other closely, achieving roughly 1.4:1 or better for the approximate interval

r E[0.4, 0.6], and dropping to around 1.2: 1 close to r = 0.5. This

phase and frequency performance against the CRLB is better than either the FST (Figure 78) or the DST (Figure 113). The RST amplitude achieves a 1.4:1 ratio compared with the CRLB for the approximate interval

r E[0.4, 0.6], and is broadly similar to the performance of

the DST, and an improvement on the FST performance. [001032] This embodiment of the Recursive Signal Tracker combines the FIR behaviour of the Prism with the llR characteristic of recursively reusing the previously calculated estimate of r. Accordingly, when the value of r remains essentially constant but the amplitude changes, then the dynamics are similar to those of the FST. However, where the value of r changes, additional delays are introduced. [001033] Two simulation examples are used to demonstrate the dynamic response of the RST. In the first example, the sinusoid amplitude is changed linearly with time but the frequency is kept constant. This example demonstrates that the delay in the RST tracking of amplitude remains 1/m over a wide range of r, as long as r remains constant. [001034] The second example demonstrates how the RST responds to a sudden change in frequency, using the example previously used with the FST. Thus the frequency of a tracked (noise-free) sinusoid jumps instantly from 190 Hz down to 10 Hz. The RST is able to track the new frequency with high accuracy, but not as rapidly as the FST, due to the additional dynamics of tracking a step change in the value of r. [001035] Figures 162A to 166 (graph 1620, graph 1622, graph 1630, graph 1632, graph 1640, graph 1642, graph 1650, graph 1652 and graph 1660) show the results of a simulation in which the amplitude of the input sinusoid is varied with time. As previously, the sample rate fs

= 51.2 kHz, and m is 200 Hz.

Figure 162A shows the input sinusoid time

series while Figure 1628 shows the corresponding power spectrum. The sinusoidal input frequency is fixed at 133.33 Hz, so that

r = 0.666. The amplitude initially takes a value of - 201 -


1 V, but after


t = 1 s the amplitude is reduced by 0.25 V/s until t = 3 s; thereafter the

amplitude remains constant at 0.5 V. [001036] Figures 163A and 1638 shows how the RST tracks the signal over the entire time

series. Figure 163A shows the true and RST tracked values of the signal, while Figure 1638 shows the error in the RST tracked signal. The errors are small during the periods of steady amplitude, but between

t = 1 s and t = 3 s when the amplitude is dropping, the

tracked signal error increases to approximately± 1.25e-3 V. [001037] Figures 164A and 1648 show the tracked amplitude and its error. Figure 164A

shows the true and RST tracked amplitude, while Figure 1648 shows the error in the RST tracked amplitude. The error has an average value of approximately 1.25e-3 V during the period of amplitude reduction, while showing some variation around this mean value. For the amplitude rate of change of 0.25 V/s, the average amplitude error implies a delay of 5ms between input and output. [001038] Figures 165A and 1658 shows the amplitude and its error in more detail, around

t

= 2.0 s. The difference between the true and RST calculated value can be seen in Figure 165A, and the time delay estimated directly. The true amplitude takes the value 0. 75 V at

t = 2.0 s (left arrow), while the RST calculated amplitude reaches the value 0. 75 V at exactly t = 2.005 s (right arrow). This indicates a delay of 5 ms, corresponding to 1/m exactly

where m is 200 Hz. Figure 1658 shows the amplitude error, which averages 1.25e-3 V, but which shows a degree of modulation, which has a higher degree of variation than in the equivalent result for the FST shown in Figure 828. This RST modulation is at a minimum at

r = 0.666, and hence the selection off= 133.33 Hz in this example. [001039] To demonstrate that the delay is 1/m over a range of

r values, the simulation of

Figures 162A to 1658 has been repeated using different input frequencies. Figure 166 shows the results of this calculation for 1/m for frequency ratio

r between zero and 1. The delay is approximately

r E [0.2, 0.9].

- 202 -



[001040] The second simulation demonstrates the response of the RST to a sudden

change in frequency input. The same RST structure was used as previously, with a sample rate of 51.2 kHz and m

= 200

Hz. The input signal is noise free, and, as shown in Figure

167A (graph 1670), has the following behaviour: the amplitude is constant throughout the 2s simulation at 1 V, but at

t = 1s, the frequency steps from 190 Hz down to 10 Hz instantly.

Figure 1678 (graph 1672) shows the corresponding power spectrum of the entire time sequence, which manifests peaks at both 10 Hz and 190 Hz. [001041] Figures 168A to 1758 (graph 1680, graph 1682, graph 1690, graph 1692, graph

1700, graph 1702, graph 1710, graph 1712, graph 1720, graph 1722, graph 1730, graph 1732, graph 1740, graph 1742, graph 1750 and graph 1752) show the parameter outputs from the RST, over two timescales. Figures 168A to 1718 show frequency, amplitude, phase and signal values respectively over the full simulation period, while Figures 172A to 1758 show the same parameter values during the transition period around

t = 1.0 s. In

each case, the true value of each parameter is shown in the upper plot (for example Figure 172A) alongside the value calculated by the RST, while in the lower plot (for example Figure 1728) the logarithm of the (absolute value of) the RST parameter error is shown. Note that absolute error, rather than relative error in decibels, has been used here because of the very large step change in frequency value during the simulation, rendering the relative error difficult to interpret. [001042] Figures 168A to 1718 show a similar pattern: there is good agreement between

the true and calculated parameter values both before and after the frequency step change, but large deviations at the point of transition itself. Errors before and after the transition are roughly 10-12 for all parameters. [001043] Figures 172A to 1758 show the transition behaviour of each of the RST

parameter outputs between

t = 0.995 s and t = 1.012 s. While there is some variation in

behaviour, all parameters have error magnitudes consistently close to or below 10-12 after approximately

t = 1.016 s.

- 203 -



[001044] The calculated frequency (graph 1720 and graph 1722) swings in value between

t = 1.0s and 1 = 1.016s, but stays within the range of 10 Hz and 190 Hz. Similarly, the calculated phase (graph 1740 and graph 1742) stays within its nominal range of ±n radians, though of course it is constrained to do so: its behaviour is more erratic during the transition than that of frequency. Finally, the calculated amplitude (graph 1730 and graph 1732), and hence also the magnitude of the calculated signal value (graph 1750 and graph 1752), vary between 1.5 V and 0.3 V during the transition. [001045] Comparing this performance with the corresponding example for the FST (shown

in Figures 84A to 928), the RST requires approximately 16 ms to recover from the step change in frequency, compared to the FST's requirement for only 10 ms. The explanation

= 10 ms delay associated with their respective Prisms, but the RST has an additional delay of 1/m = 5 ms

for the additional delay is as follows. Both trackers share the common 2/m

associated with the record of Gs and Ge values which must be refilled with values corresponding to the new frequency in order for the recalculated values of r to be accurate.

r is 0.05, a low value. As discussed above and illustrated in Figures 147A, 1478 and 149, a low value of r may result in slow Finally, in this particular example, the final value of

convergence of the RST tracking algorithm, and this may be responsible for the final 1 ms

t = 1.015s, the first Gs and Ge values with the new frequency data become available for use in calculating r. From that time onwards the

of delay. In graph 1720 and graph 1722, at

frequency error reduces steadily until it settles close to 10-12 . This error reduction takes approximately 1 ms, but at a higher value of rthis might take less time. [001046] The RST, like the DST, is able to operate in the presence of moderate levels of

white noise, without necessarily requiring pre-filtering of the signal. Figures 176A to 1858 (graph 1760, graph 1762, graph 1770, graph 1772, graph 1780, graph 1782, graph 1790, graph 1792, graph 1800, graph 1802, graph 1810, graph 1812, graph 1820, graph 1822, graph 1830, graph 1832, graph 1840, graph 1842, graph 1850 and graph 1852) illustrate this behaviour in a simulation with characteristics similar to the DST example shown in Figures 119Ato 1288. As previously, the sample rate fs

- 204 -

= 51.2

kHz and m

= 200Hz.

The



input sinusoid has steady amplitude and frequency values for

t

E

[0.0, 1.0] s and

t

E

[2.0,

3.0] s. Between these steady state conditions there are simultaneous linear changes in amplitude and frequency. The amplitude is reduced from 1.0 V to 0.5 V, while the frequency increases from 80 Hz to 120 Hz. White noise of constant standard deviation 0.5/~2 Vis added to the signal. This results in a SNR of 6 dB at the start of simulation, and

of 0 dB after the reduction in amplitude. [001047] Figures 176A to 177B provide an overview of the simulation. Figure 176A shows

the whole time sequence. The noisy input signal is shown in grey, while the true and RST calculated signal values (i.e. Asin(¢)) are shown as the dashed and continuous lines respectively. Figure 176B shows the noise time series (i.e. the true signal subtracted from the noisy signal), and the RST calculated signal error. The noise amplitude remains constant through the simulation, while the true signal reduces. [001048] Figures 177A and 177B show a plot of the same parameters over a more limited

time period, i.e. during a 100 ms period after the parameter changes have been complete, and for which the SNR ratio is 0 dB. In Figure 177A the difference between the true and calculated signal can be perceived more clearly, along with the relative magnitudes of signal and noise. In Figure 177B, the RST signal error retains a low frequency characteristic previously observed in the DST signal error (Figures 120A and 120B). [001049] Figures 178A to 183B show the frequency, amplitude and phase estimates on

both full and detailed timescales. In addition, Figures 184A to 185B show the tracked signal of Figures 176A to 177B with the noise signal excluded. [001050] Figures 178A to 179B show the frequency results. Figures 178A and 179A show

true and RST calculated values for frequency, while Figures 178B and 179B show the error in the RST tracked frequency. The RST successfully tracks the rise in frequency. Before and during the rise in frequency, the errors are approximately within ± 3 Hz; after the rise in frequency (and simultaneous drop in amplitude), the frequency errors fall mostly within ± 5

- 205 -



Hz. In Figures 179A and 1798 the frequency error exhibits a low frequency characteristic, as a result of the filtering effect of the Prism. [001051] Figures 180A to 181 B show the amplitude results. Figures 180A and 181A show

true and RST calculated values for amplitude, while Figures 1808 and 181 B show the error in the RST tracked amplitude. The RST successfully tracks the reduction in amplitude, while the magnitude of the error stays roughly constant at around 50 mV throughout the simulation. Figures 180A and 1808 demonstrate that the amplitude error has a low frequency characteristic, similar to that observed in the corresponding DST example (shown in Figure 124A and 1248). However, the extended DST Prism network applies a greater filtering effect, at higher computational cost. [001052] Figures 182A to 1838 show the phase results. Figures 182A and 183A show true

and RST calculated values for phase, while Figures 1828 and 1838 show the error in the RST tracked phase. There is a significant increase in the phase error magnitude from approximately ± 0.2 radians before the change in amplitude and frequency, to approximately± 0.5 radians after

t = 3.0 s, attributable to the lower SNR. Figures 183A and

1838 show the variations in true and tracked phase on the reduced timescale. Finally, Figures 184A to 1858 show the true and tracked signals, excluding the noise signal shown in Figures 176A to 1778. Figures 184A and 185A show the true signal and the RST tracked signal, while Figures 1848 and 1858 show the error in the RST tracked signal. [001053] A graph by graph comparison between the DST and RST performance in this

example (starting at Figure 119A and Figure 176A respectively) will reveal that generally, the DST errors are of smaller magnitude than those of the RST, even though the performance of the RST against the CRLB is generally better. An explanation of this apparent paradox is given in Table 4, which compares the DST and FST performances for SNR 40 dB and r

= 0.5.

Table 4: Comparison of DST and RST Performance, with r = 0.5

- 206 -


Parameter


CLRB

RST, N = 768 RST

Ratio

CLRB

DST, N = 1536 DST

Ratio

Freq. (Hz)

1.3e-2

1.6e-2

1.2

4.7e-3

9.2e-3

2.0

Phase (rad.)

7.2e-4

8.7e-4

1.2

5.1e-4

9.2e-4

1.8

Amplitude (V)

3.6e-4

5.0e-4

1.4

2.6e-4

3.5e-4

1.4

[001054] As the DST includes three layers of Prisms, it uses 1536 sample points in this

case, which is twice the number used by the equivalent RST. Accordingly, the CRLB limits are smaller for the DST for each parameter, so that even though the RST performance as a ratio of the CRLB is better than that of the DST, the root sum square errors are generally lower for the DST. This improved absolute performance of the DST is obtained at the expense of higher computational burden (operating 5 Prisms instead of the RST's single Prism) and a slower dynamic response, with a typical time delay of 3/m against that of the RST which is typically in the range [1, 1.5] Im. RST effectiveness for low sample rates [001055] A further issue to be considered is the influence of integral sample length on

measurement performance. This is an important issue for evaluating the scalability of Prism-based solutions. For example, if Prism-based signal processing is effective for small window sizes (for example associated with low sampling rates and/or low computational power), then it can be applied in a range of applications from the simplest, cheapest, low power sensors up to high powered, high cost sophisticated sensors. Accordingly, the performance of Prism-based signal processing is presented over a range of sampling rates. [001056] Figures 186 to 1978 show results from a series of simulations where only the

length n in samples of each Prism integral is varied, from 32 samples up to 1024 in steps of 32 samples. Note that these simulations demonstrate it is not necessary for the length of the integral to be an exact power of 2. All that is required is for the integral length n to be a whole multiple of

1

2n_stages+ ,

where n_stages is the maximum number of Romberg

- 207 -



Integration stages applied in the integral calculation. As the highest number of RI stages applied here is 4, any multiple of 32 samples can be a valid integral length. [001057] Figures 186 to 188 (graph 1860, graph 1870, graph 1880) show the results of a

noise-free simulation, while each subsequent set of three figures show results for signal-tonoise ratios of 100 dB, 60 dB and 20 dB respectively. Other parameter values remain the same throughout the simulations: mis 200 Hz, the amplitude of the input signal is constant at 0.1 V, and the input sinusoid frequency is constant at 105 Hz. The adjustment of the number of samples in each integral is achieved by varying the sample rate fs using fs

=m x

n. In each case the results given are based on the mean root sum square error over 4

seconds of simulation. [001058] Figures 186 to 188 show the results for one and four RI stages for a noise-free

signal. For each of the three parameters frequency (Figure 186), amplitude (Figure 187) and phase (Figure 188), the one RI stage measurement errors exhibit a steady decline with increasing sample count. By contrast, the four RI Stage errors drop rapidly and then essentially stabilize (with some perturbation) around their respective minimum values for integral lengths of 128 samples or greater. [001059] In Figures 189A to 197B white noise is added to the input sinusoid in order to

evaluate RST performance with variable length integrals and different levels of noise. Given the findings discussed above with regard to the effectiveness of RI integration with white noise present, only one stage of RI is applied in these examples. In the graph for each parameter value the corresponding CRLB is calculated and compared with the performance of the Recursive Signal Tracker. [001060] In Figures 189A to 191 B (graph 1890, graph 1892, graph 1900, graph 1902,

graph 1910 and graph 1912), the SNR is 100 dB. For each parameter (frequency, amplitude, phase) the corresponding graph plots the CRLB and the RST performance in the upper graph (for example in Figure 189A for frequency), while the ratio of the RST performance to the CRLB is shown in the lower graph (for example Figure 189B for - 208 -



frequency). For all three parameters, the RST calculated value and the CRLB show a steady decline in error value as the integral sample size increases, while the ratio of RST error to CRLB is close to its optimal value (and less than 2) for all integral lengths exceeding 32 samples. [001061] In Figures 192A- 194B (graph 1920, graph 1922, graph 1930, graph 1932, graph

1940 and graph 1942), the SNR is 60 dB, while in Figures 195A to 197B (graph 1950, graph 1952, graph 1960, graph 1962, graph 1970 and graph 1972) the SNR is 20 dB. For all three parameters, the ratio of RST error to CRLB is, but for random variation, close to its optimal value for all integral lengths including the smallest, with only 32 samples. [001062] These results suggest that the RST offers an efficient algorithm for tracking

frequency, amplitude and phase even for relatively low sampling rates, and is therefore suitable for low cost, low power sensing applications, with, for example SNR of between 20 dB and 100 dB. Prism Pre-filtering for sinusoid tracking [001063] The sinusoid tracking capability of Prism-based trackers may be extended and

improved by using pre-filtering techniques, for example to reduce noise and/or the influence of undesired signal components. One familiar with the art will appreciate that Prism-based trackers can be used in combination with conventional filtering techniques in order to provide pre-filtering. In addition or alternatively, it is possible to use Prism-based filters to provide pre-filtering for a tracker. This approach has the advantages associated with other Prism-based techniques: the low computation burden associated with a recursive FIR calculation; simple, exact analytic expressions for the phase and gain at any frequency; a linear phase shift; and the simplicity and low computational burden of filter design. [001064] Three examples are given to illustrate Prim-based pre-filtering, as follows:

- 209 -



1) A single Prism used as a low-pass pre-filter for an FST; 2) Two Prisms used to provide notch filtering of a fixed frequency (for example in order to suppress mains electricity supply noise) as a pre-filter for a DST. 3) A bandpass filter chain used as a pre-filter to an RST. [001065] In each example, improvements may be demonstrated over the performance of

trackers without pre-filtering. Such improvements may be achieved at the expense of the additional computational requirement for the pre-filtering stage, as well as any additional dynamic response delays associated with the pre-filtering stage. Those skilled in the art will appreciate that many other combinations of Prism pre-filtering and trackers may be possible, and all such combinations are included in these disclosures. [001066] Figure 198 shows an exemplary illustration of a single Prism 1980 used as a low

pass pre-filter to an FST 1982. The input signal s(t) is passed through the Prism 1980 with characteristic frequency m. The Ge output from the Prism 1980 becomes the input to the FST 1982. The Ge output may typically be preferred to the Gs output for the purposes of low pass pre-filtering, as it generates generally higher attenuation of high frequencies (compare Figures 398 to 408). Similarly, the harmonic number h is typically 1 for a low pass prefiltering duty, while higher harmonic numbers may be used in band-pass pre-filters as discussed below. The value of m may be used to select the pass band for the pre-filter. Typically, m for the Prism 1980 may equal the value of m for the FST 1982, but other arrangements may be preferable according to the needs of the application. [001067] The FST 1982 may provide additional corrections to its frequency and phase

outputs to compensate for the effects of the Prism 1980, or this may be done at a later stage. As discussed above, such amplitude and phase compensation may be applied in a straightforward manner, by calculating the pre-filter gain and phase shift associated with the frequency detected by the FST 1982, and correcting the FST 1982 outputs accordingly.

- 210 -



[001068] Figures 199 to 201 (graph 1990, graph 2000 and graph 2010) show the results of a series of simulations using the FST and pre-filter arrangement illustrated in Figure 198. These results can be compared with those obtained using the FST without pre-filtering, which is shown in Figures 75 to 77. In both examples the sample rate fs is 51.2 kHz and FST m

= 200

Hz. The same value of m is used in the pre-filter applied in Figures 199 to

201, so that the total data window length used in the pre-filter and tracker combination is 1024 samples, with 512 each for the pre-filter and FST. This compares with 512 samples for the FST-only results shown in Figures 75 to 77. [001069] Figures 199 and 75 show the frequency results with and without pre-filtering respectively, in otherwise identical simulations. In Figure 75, the FST achieves a best rootsum-square error slightly above 0.1 Hz for values of

r in the approximate range r E [0.6,

0. 7]. By contrast, in Figure 199, the inclusion of the pre-filter generates rss errors below 0.1 Hz for the approximate range range

r E [0.3, 0.8], and errors below 0.03 Hz for the approximate

r E [0.5, 0.6]. Thus the inclusion of the pre-filter has generally reduced the frequency

error over a range of r values. [001070] Figures 200 and 76 show the amplitude results with and without pre-filtering respectively, in otherwise identical simulations. In Figure 76, the FST achieves a best rootsum-square error of approximately 6 x 10-4 V at a reasonably sharp minimum at approximately

r = 0.55; the rss amplitude error is below 1 x 10-3 V for values of r in the

approximate range

r E [0.6, 0. 7]. By contrast, in Figure 200, the inclusion of the pre-filter

generates rss errors below 6 x 104 V for the approximate range of approximately 4 x 10-4 V for the approximate range

r E [0.4, 0.65], and errors

r E [0.45, 0.55]. Thus the inclusion of

the pre-filter has generally reduced the amplitude error over a range of rvalues. [001071] Figures 201 and 77 show the phase results with and without pre-filtering respectively, in otherwise identical simulations. In Figure 77, the FST achieves a best rootsum-square error of approximately 4 x 10-3 radians for values of r in the approximate range

r E [0.6, 0. 75]. By contrast, in Figure 201, the inclusion of the pre-filter generates rss errors - 211 -



below 4 x 10-3 radians for the approximate range 10-3 radians for the approximate range

r E [0.35, 0. 75], and errors of below 2 x

r E [0.45, 0.65]. Thus the inclusion of the pre-filter

has generally reduced the phase error over a range of r values. [001072] Given the longer data window length in Figures 199 to 201 compared with

Figures 75 to 77, the Cramer-Rao Lower Bound (CRLB), as specified in Equations 8 to 10 above, are correspondingly smaller in Figures 199 to 201. Even so, the addition of the prefilter results in generally better error ratios against with the CRLB, as illustrated in Figure 202, which can be compared with Figure 78 where no pre-filtering occurs. The best error ratios achieved in Figure 202 (graph 2020) are generally lower than those of Figure 78. For example in the case of the amplitude parameter, the lowest error ratio achieved in Figure 78 is around 1.4, while the error ratio is below 2 for the approximate range of r values

rE

[0.5, 0.62]. By contrast, the best error ratio in Figure 202 is around 1.2, while the error ratio falls below 2 for the wider approximate range of rvalues

r E [0.4, 0.65].

[001073] Similarly, for the frequency and phase parameters, the best error ratios in Figure

202 are generally lower than those in Figure 78. For example, the error ratios for frequency and phase do not drop below 4 in Figure 78. However, in Figure 202 both the frequency and phase error ratios drop below 4 for the approximate range of r values and are close to or below 3 for the approximate range of rvalues

r E [0.42, 0.68],

r E [0.5, 0.6].

[001074] Figure 203 (graph 2030) illustrates a further benefit pre-filtering. The results

shown are for simulations which are identical to those of Figure 202, except that the signalto-noise ratio has been reduced by 20 dB (i.e. the amplitude of the white noise has been increased relative to the signal amplitude by a factor of 10). Apart from some minor variations due to the random nature of white noise, the results of Figures 202 and 203 are essentially identical. This illustrates the general finding that the use of pre-filtering may facilitate the successful use of Prism-based trackers on signals with higher levels of noise than would be possible without pre-filtering.

- 212 -



[001075] The increased robustness to high levels of white noise is further illustrated in a

second pre-filtering example, in which a bandpass filter is used as a pre-filter to an RST tracker. Figure 204 illustrates the arrangement: a three-stage bandpass filter 2040 provides pre-filtering for a Recursive Signal Tracker (RST) 2042. The bandpass filter 2040 design is similar to that shown in Figure 48 and discussed above. In this example the sample rate fs

= 192 kHz, the centre frequency of the bandpass filter 2040 c_f = 10 Hz, and for the RST 2042 m = 10 Hz. Thus the passband of the bandpass frequency is approximately 10 Hz ± 1 Hz. [001076] Figure 205A (graph 2050) shows the input signal, both with and without noise.

The signal-to-noise ratio is -20 dB, indicating that the amplitude of the white noise exceeds that of sinusoid to be tracked by an order of magnitude. The amplitude and frequency of the sinusoid are both constant throughout the simulation, with values of 1 V and 10.2 Hz, respectively, while the instantaneous value of the noisy signal may exceed 20 V. Figure 205B (graph 2052) indicates in general terms the noise reduction achieved by the RST 2042 and pre-filter 2040: while the pre-filter 2040 noise magnitude may exceed 20 V, the error in the RST 2042 estimated signal is significantly smaller. The performance is shown in more detail in the following figures. Note that the results shown excluded any required warmup period where the signal processing components are first filled with data. [001077] Figure 206A (graph 2060) shows the power spectrum of the raw (unfiltered)

signal, while Figure 206B (graph 2062) shows the power spectrum of the filtered signal i.e. after it has passed through the bandpass pre-filter 2040. In each case the relative amplitude in decibels is shown, scaled so that the peak amplitude at 10.2 Hz has an amplitude of 0 dB. In the raw signal, the noise floor is constant at approximately -40 dB, while in the filtered spectrum the noise floor drops rapidly both above and below the peak band, dropping below approximately -80 dB at 100 Hz. The bandpass filter has thus been effective in reducing the signal noise outside the desired passband.

- 213 -



[001078] Figure 207A (graph 2070) shows the frequency output of the RST 2042, which

remains close to the true signal frequency of 10.2 Hz. Figure 207B (graph 2072) shows the error in the RST 2042 frequency output, which in the time series shown remains within ± 0.015 Hz. The frequency error varies slowly with time, reflecting the dynamic characteristic of the pre-filter. [001079] Figure 208A (graph 2080) shows the amplitude output of the RST 2042, which

remains close to the true signal amplitude of 1 V. Figure 208B (graph 2082) shows the error in the RST amplitude output, which in the time series shown, remains within± 0.05 V. The amplitude error also varies slowly with time, reflecting the dynamic characteristic of the pre-filter 2040. [001080] Figure 209A (graph 2090) shows the phase output of the RST 2042, which

remains close to the true signal phase as it cycles around ± n radians. Figure 209B (graph 2092) shows the error in the RST 2042 phase output, which in the time series shown remains within ± 0.15 radians. The phase error also varies slowly with time, reflecting the dynamic characteristic of the pre-filter 2040. [001081] Figure 21 OA (graph 2100) shows the tracked signal output of the RST 2042 (i.e.

Asin(¢)), together with the noise-free original sinusoid to be tracked, which has a fixed amplitude of 1 V. Figure 21 OB (graph 2102) shows the error in the RST 2042 signal output, which has a maximum value of approximately 0.15 V. The signal errors have a broadly sinusoidal characteristic, reflecting the slow variation in amplitude and phase errors shown in Figures 208B and 209B. [001082] Figures 211A (graph 2110) and 211B (graph 2112) show the same data as

Figures 21 OA and 21 OB but on a reduced timescale. The true and tracked signals can be separately distinguished in Figure 211A, while the sinusoidal-like variation in the tracked signal error is shown in Figure 211 B.

- 214 -



[001083] Overall, this example demonstrates that a Prism-based bandpass filter may

enable an RST to track a signal with an unfiltered SNR ratio of - 20dB. [001084] Another type of pre-filtering that can be applied using a Prism chain is notch

filtering. This technique may be effective when it is known that an unwanted signal component occurs at a certain frequency, or within a narrow frequency range. A common example might be the presence of mains noise in a signal, where it is desirable to remove the mains noise (typically at 50 Hz or 60 Hz) to enable the tracking of another, possibly variable frequency, component. Notch filtering can be achieved simply by using one or more Prisms where the value of m is selected to be equal or close to the frequency to be notched out. The frequency responses of both the Gs and the Ge Prism outputs have notches at all multiples of m (see for example Figures 39 and 40), thus providing the desired pre-filtering characteristic. [001085] Figure 212 shows an embodiement of a Deep Signal Tracker 2126 and a

pre-filter 2120 designed to suppress mains noise, which for this example is taken to be at 50 Hz. The pre-filter 2120 comprises a chain of Prism 2122 and Prism 2124, each with m 50 Hz and the harmonic number h

= 1, where the

=

Gs output is used to propagate the signal

through the Prism chain. Alternative arrangements could be used, for example depending upon application requirements. The input signal 2121 s(t) is passed through both pre-filter 2120 Prisms before continuing into the DST 2126 where tracking occurs. As with the previous pre-filtering example, the DST 2126 outputs of amplitude and phase may be compensated for the influence of the pre-filter 2120 so that their original values in input signal 2121 may be calculated correctly. Numbers, equations and notations in Figure 212 are for this embodiment only and do not limit the disclosure. [001086] Figure 213 shows the frequency response of the pre-filter 2120 over the

frequency range 0 - 160Hz. It can be seen that a notch occurs at 50 Hz as well as its multiples 100 Hz and 150 Hz. These notches will act to suppress any signal components at or close to these frequencies.

- 215 -



[001087] Three examples are shown demonstrating the effectiveness of the notch filter in

blocking an undesired component at a fixed frequency. Certain parameters are common to all examples. The sample rate fs is 51.2 kHz and the DST 2126 uses m

= 160 Hz. Both the

mains frequency at 50 Hz, and the desired signal frequency (which varies with each example) have a common amplitude of 1 V, thus setting a significant challenge for the prefiltering and DST. Each of the examples differ in their desired component frequency (55Hz, 85 Hz, and 70Hz respectively), and the level of white noise (with SNR levels relative to the desired component amplitude only - thus excluding the mains noise in this calculation - of 80 dB, 40 dB and 0 dB respectively). These three examples demonstrate that the Prismbased signal processing system shown in Figure 212 may be suited to tracking a range of desired component frequencies, while successfully suppressing 50 Hz mains noise and dealing with varying levels of white noise. [001088] Figures 214A to 216B (graph 2140, graph 2142, graph 2150, graph 2152, graph

2160 and graph 2162) show results from the first simulation, where the desired signal component frequency is 55 Hz, and the SNR is 80 dB. Figure 214A shows a section of the raw signal time series, which manifests the interaction between the undesired 50 Hz mains component and the 55Hz desired signal component. Figure 214B shows the corresponding spectrum of the raw signal, where two approximately equal peaks occur at 50Hz and 55 Hz. Figure 215A shows the corresponding time series of the filtered signal, to which compensation for the pre-filter gain has been applied. The time series suggests the mains component has been largely removed, as the signal appears to be a pure sinusoid. This is further suggested by Figure 215B, which shows that the spectrum of the filtered signal has retained the 55 Hz desired signal component while the undesired 50 Hz mains signal component has been largely removed. Figure 216 shows the output of the DST 2126 in the form of the tracked signal (i.e. Asin(¢)) compared with the original, noise-free, 55 Hz signal component. Note that the individual estimates of frequency, amplitude and phase are also generated by the DST 2126 in the usual manner, but are not given in this example purely for reasons of brevity. Figure 216A shows the DST 2126 calculated signal output and the

- 216 -



original 55 Hz signal component, which largely coincide. Figure 216B shows the error in the tracked signal i.e. the difference between the tracked signal as estimated by the DST 2126 and the original true 55Hz signal component. The error remains within ± 3 x 104 V for the time series shown. [001089] Figures 217A to 219B (graph 2170, graph 2172, graph 2180, graph 2182, graph

2190 and graph 2192) show results from the second simulation, where the desired signal component frequency is 85 Hz, and the SNR is 40 dB. Figure 217 A shows part of the time series of the raw signal, which manifests the interaction between the unwanted 50 Hz mains component and the 85Hz desired signal component. Figure 217B shows the corresponding spectrum of the raw signal, where two approximately equal peaks occur at 50Hz and 85 Hz. Figure 218A shows the corresponding time series of the filtered signal, to which compensation for the pre-filter gain has been applied. The time series suggests the mains component has been largely removed, as the signal appears to be a pure sinusoid. This is further suggested by Figure 218B, which shows that the spectrum of the filtered signal has retained the 85 Hz desired signal component while the undesired 50 Hz mains signal component has been largely removed. Figure 219 shows the output of the DST in the form of the tracked signal (i.e., Asin(¢)) compared against the original 85 Hz signal component. Figure 219A shows the DST calculated signal output and the original 85Hz signal component, which largely coincide. Figure 207B shows the error in the tracked signal i.e. the difference between the tracked signal as estimated by the DST and the original true 85Hz signal component. The error remains within ± 5 x 10-2 V for the time series shown. [001090] Figures 220A to 222B (graph 2200, graph 2202, graph 2210, graph 2212, graph

2230 and graph 2232) show results from the third and final simulation of using a notch filter as a pre-filtering stage, where the desired signal component frequency is 70 Hz, and the SNR is 0 dB. Figure 220A shows the raw signal as a time series. The signal manifests the interaction between the 50 Hz mains component and the 70 Hz desired signal component; the high level of white noise is also apparent. Figure 220B shows the corresponding spectrum of the raw signal, where two approximately equal peaks occur at 50Hz and 70

- 217 -



Hz. The spectrum also shows an elevated noise floor at just below -50 dB amplitude. Figure 221A shows the corresponding time series of the filtered signal, to which compensation for the pre-filter gain has been applied. The time series suggests that the pre-filtering has been reasonably effective: much of the influence of both the mains component and the white noise has been removed. The signal appears to be reasonably sinusoidal, but note the variations in amplitude, as indicated in the maxima and minima values of the signal, which vary around the true 70 Hz component amplitude of exactly 1 V. Figure 221 B shows that the spectrum of the filtered signal has retained the 70 Hz desired signal component while the undesired 50 Hz mains signal component has been largely removed. The noise floor around 30 Hz remain close to -50 dB, but elsewhere has been reduced. Figures 222A and 222B show the output of DST 2126 in the form of the tracked signal (i.e. Asin(¢)) compared with the original 70 Hz signal component. Figure 222A shows the DST 2126 calculated signal output and the original 70 Hz signal component. Given the high level of noise in the original raw signal, there is greater discrepancy between the original and tracked signal. This is confirmed in Figure 222B which shows the error in the tracked signal i.e. the difference between the tracked signal as estimated by the DST 2126 and the original true 70 Hz signal component. The error remains within approximately ± 0.25 V for the time series shown. [001091] Three different types of Prism-based pre-filtering have been shown: low pass, band-pass and notch filtering. One familiar with the art will recognise that many other arrangements or combinations of these techniques might also usefully be deployed as required by a particular application. [001092] The notch filtering example is the first case in this disclosure where more than one signal component has been analysed using Prism signal processing. While the notch filter can be applied as a simple and effective pre-filtering technique, it suffers from a number of limitations. Firstly, the notch is applied to a fixed and narrow frequency band, so that it is not suited to blocking frequency components that may vary with time. Secondly, all the higher multiples of the notch frequency are also notched, so this may interfere with

- 218 -



tracking of a higher frequency component, especially if the notch frequency is low. Thirdly, it may be desirable to track some of the parameter values of the notched frequency component, for example its amplitude, for diagnostic purposes. These limitations can be overcome by using the Prism signal processing based technique called dynamic notch filtering, which is described below. Dynamic Notch Filtering [001093] Figure 5 illustrates the 'brick wall' filter, which is conventionally considered the

ideal filter, having the properties of a perfectly flat passband gain, an instant transition region, and a zero gain (which is equivalent to minus infinity on the decibel scale) in the stopband region. This filter is not realizable in practice, but remains an ideal in conventional signal processing, in that one or more aspects of the brick wall filter may be considered desirable. For example, one class of llR filters is the 'Maximally Flat' filter which has a maximally flat passband gain. Even where a particular filter design technique does not generate an approximately flat passband, minimization of 'passband ripple' may be seen as an inherently desirable goal. [001094] With

Prism-based filtering,

the

'brick wall'

ideal

is

exchanged for the

computational efficiency of a truly recursive FIR filter, as discussed above. This over-riding design requirement places constraints on other aspects of Prism-based filter performance. Specifically, there is large gain variation in what may be considered the passband of a Prism filter. For example, consider Figures 39A to 408, with h

= 1,

where the passband

may for example be considered as frequencies below m. In each figure the gain at 0 Hz is zero; this rises to a maximum magnitude (but negative value) at approximately m/2 Hz, and then the gain reduces in magnitude, returning to zero gain at m Hz. In the ideal filter, the issue of gain variation within the passband can be ignored. In practice, with real filters, it may be necessary to use polynomial fits or other techniques to approximate the relatively small changes in gain over the passband so that amplitude compensation may be applied.

- 219 -



[001095] With Prism-based filtering simple analytic expressions are available for the gain at

any frequency, so that compensation may be applied to obtain the original amplitude of any tracked frequency component in the unfiltered signal. In practice, only limited accuracy may be possible for frequencies close to the notches generated by Prism-based filters. [001096] However, the relatively wide variation in gain over the passband frequency

generated by Prism-based filters may facilitate the use of dynamic notch filtering (DNF), which the idealized flat passband of the brick wall filter would specifically exclude, as DNF relies on gain variations with frequency. Dynamic notch filtering is a technique for removing one or more frequency components from a signal, where the frequency to be removed can be selected in real time from a range of values. This provides far greater flexibility than the previously described (static) notch filtering, where a Prism chain is used to block a specific frequency (for example mains noise at 50 Hz). [001097] The principle of dynamic notch filtering is explained by considering the outputs of

two Prisms, having a common value of m, but with different harmonic numbers, say h and h

= 2.

= 1,

Considering the cosine outputs only from each Prism, for the conventional

sinusoidal input s(t) Equation 1 these are given by the following equations (derived from Equation 26): 2

Ge= Asinc (r)-i---cos(¢-2;rr) r -l

Equation (64)

Asinc 2 (r)~cos(¢-2;rr)

Equation (65)

Gk=

r -4

where, as previously, the subscript c is used to denote h denote h

= 2.

= 1, and the subscript k is used to

Note that while the phase offsets for these filter outputs are identical, taking

the value 2nr, the gains are generally different. The gain functions may be defined as: Equation (66)

- 220 -



Equation (67)

[001098] Figure 223 (graph 2230) shows the variation in gain for Ge and Gk as

r varies

between 0 and 1. The two gain functions are distinct from each other over the range, excluding the points

r = 0 and r = 1. Figure 224 (graph 2240) shows the ratio of the two

gains, given by:

Equation (68)

[001099] P(r) shows a monotonic increase from a minimum of 2, for

towards an unboundedly large positive value as

r very close to 0,

r approaches 1. In other words, for each

frequency ratio value r, there is a distinct and unique value of the ratio between the two gains. It is this property that enables the technique of dynamic notch filtering. [001100] Figure 225 illustrates an embodiment of dynamic notch filtering. Here the input

signal s(t) 2250 consists of two or more frequency components: s1(t) with possibly timevarying frequency f1(t) 2253, and a summation of one or more other frequency components si(t) 2250. It is assumed that f1(t) is known, for example by a separate calculation. Two

Prisms, Prism 2251 and Prism 2252 may be used to remove the frequency component s1(t) from s(t) 2250 by the following steps. Firstly, s(t) 2250 is passed through Prism 2251 and Prism 2252 in parallel, with the same value of m and with harmonic numbers of 1 and 2 respectively. The cosine outputs of the respective Prisms, sc(t) and sk(t), are passed into a calculation block 2254 along with the frequency to be blocked, f1(t) 2253. The value of r required to apply dynamic notch filtering is calculated from the ratio of f1(t) and m. The value of P(r) is calculated based on Equation 68. The output Snew(t) is calculated using s

new

(t)

=sc(t) -

Equation (69)

P(r) . sk(t)

[001101] This function eliminates the s1(t) component, for the following reasons. The

phases of the f1 components in sc(t) and sk(t) are identical, as shown in Equations 64 and

- 221 -



65. As P(r) is the unique ratio between the gains of f1 in sc(t) and sk(t), sk(t) scaled by P(r) has the same amplitude, as well as phase, as sc(t). Thus the scaled difference in Equation 69 will exactly eliminate the f1 component. All other frequency components in the Snew(t) have adjustments applied to their respective gains, but as discussed previously, these adjustments can be calculated for any frequency component and compensation applied in later calculations. While the example here uses the cosine outputs of each Prism, and harmonic numbers h

= 1 and 2, the same methodology can

be applied using, for example,

the sine outputs of each Prism and any combination of differing harmonic numbers, where the ratio function P(r) is changed to match the algebraic formulation of the Prism outputs selected. [001102] Note that the only factor which determines which frequency is to be notched is

P(r). Accordingly, multiple calculations can be carried out in parallel, using the same pair of Prisms outputs, in order to generate separate signal streams in which a different frequency has been notched. A simple example using two frequency components can be used to illustrate the principle. [001103] Figure 226 shows an embodiment for separating two frequency components

using dynamic notch filtering. This may also be described as signal splitting. The input signal s(t) 2260 consisting of two components, s1(t) and s2(t), with respective frequencies f1(t) and f2(t) 2263, which are assumed known. The combined input signal s(t) 2260 is fed

into Prism 2261 and Prism 2262 with common characteristic frequency m and harmonic numbers h

= 1 and 2 respectively. The cosine outputs of Prism 2261

and Prism 2262, sc(t)

and sk(t) respectively, are fed into a calculation block 2264, where the following calculations are performed. The r value for each frequency component, labelled r1 and r2, are determined, by calculating the ratios of the respective frequencies f1(t) and f2(t) to m. The desired weighting for the elimination of each component, P(r1) and P(r2), is calculated, using Equation 68. Finally new signals are calculated in which one or other frequency components are eliminated, using Equation 69 applying the value of P(r1) to eliminate frequency component 1 and the value of P(r2) to eliminate frequency component 2. By

- 222 -



repeating this calculation every sample, new separated component signals s1an1y(t) and s2an1y(t) may be generated as outputs to the calculation block that can used for further signal processing, for example to determine the amplitude and phase of each component. [001104] A simulation may be used to illustrate the technique. Suppose the frequency components are constant at 75 Hz and 91 Hz respectively. The dynamic notch filtering scheme of Figure 226 is applied to split each frequency component from s(t), using the sample rate fs

= 51.2 kHz and m = 200 Hz.

[001105] Given these parameter values, it is straightforward to calculate the following: r1 75 Hz I 200 Hz

= 0.375. r2 = 91

Hz I 200 Hz

approximate ratio values are obtained: P(r1)

= 0.455.

=

Using Equation 68, the following

=2.2455 and P(r2) =2.3916.

[001106] Figures 227A and 2278 (graph 2270 and graph 2272) show the frequency responses of the two filters obtained by applying Equation 69 with the calculated values of P(r1) and P(r2). Figure 227A shows the gains of the two filters on a linear scale, while Figure 2278 shows the magnitude of the same gains on a logarithmic scale, where the notches become apparent as the gains cross zero in Figure 227 A, at 75 Hz and 91 Hz respectively, as required. When applied to the two component signal s(t), these filters are effective in separating the two components. [001107] This separation or signal splitting is demonstrated in Figures 228A to 2328 (graph 2280, graph 2282, graph 2290, graph 2292, graph 2300, graph 2302, graph 2310, graph 2312, graph 2320 and graph 2322), which are the results obtained from a simulation of Figure 226 and Figures 227 A and 2278. For simplicity, both components have a fixed amplitude of 1 V and there is no noise. Figures 228A and 2288 show the input signal s(t), which evidently contains more than a single sinusoid. A time series is shown in Figure 228A, while its power spectrum is shown in Figure 2288, with peaks at both 75 Hz and 91 Hz.

- 223 -



[001108] Figures 229A and 2298 show the output signal S1an1y(t) which contains the 75 Hz

component and has had the 91 Hz component largely removed. Figure 229A shows the time series, which appears to be a sinusoid, while Figure 2298 shows the corresponding power spectrum, which retains the peak at 75 Hz while the peak at 91 Hz has been effectively eliminated. Note that in this calculation amplitude and phase compensation has been applied to the output signal, so that the apparent amplitude of approximately 1 V is close to that of the original 75 Hz signal component in s(t). [001109] Similarly, Figures 230A and 2308 show the output signal S2an1y(t) which contains

the 91 Hz component and has had the 75 Hz component largely removed. Figure 230A shows the time series, which appears to be a sinusoid, while Figure 2308 shows the corresponding power spectrum, which retains the peak at 91 Hz while the peak at 75 Hz has been effectively eliminated. [00111 O] Conventional Prism-based tracking can be applied to the pure sinusoidal outputs

S1an1y(t) and S2an1y(t), for example by using an FST, DST or RST. Use of such a tracker may enable the verification of the frequency of each pure sinusoidal output (which, for example, may vary with time), as well as the calculation of the corresponding amplitude and phase. Alternatively, a simplified tracking technique can be used, in which the supplied frequencies

f1(t) and f2(t) (and hence corresponding values of r1 and r2 ) are assumed correct. In order to calculate only the amplitude and phase for a pure sinusoidal output, a simplified version of the RST can be applied: a single Prism is used, but no history of past Gs and Ge values is required as the value of r is supplied. Equation 58 (using the current values of Gs and Ge) and Equation 62 are sufficient to calculate the phase and amplitude respectively of a sinusoidal signal of known frequency when passed through a single Prism with h

= 1. Other

equivalent equations may be used for other tracking arrangements. [001111] The results of tracking these outputs, assuming the supplied frequencies are

accurate, are shown in Figures 231A to 2328 respectively, where, for brevity, tracked values of the individual parameters are not shown, but only the value of the tracked signal

- 224 -



(i.e. Asin(¢)), compared with the true value of each component in the original signal s(t). Figure 231A shows the tracked 75 Hz component, together with the original signal component, while Figure 231 B shows the difference between the two. The error is kept within approximately ± 1e-12 V. Similarly, Figure 232A shows the tracked 91 Hz component, together with the original signal component, while Figure 2328 shows the difference between the two. The error in this case is kept within approximately ± 5e-13 V. [001112] A second example demonstrates how four frequency components in a signal can

be isolated and tracked individually using dynamic notch filtering. Figure 233 shows the signal processing scheme, which is similar to Figure 226, with the following differences. There are four frequency components 2333 in the input signal 2330, with known frequencies

f1 ... f4,

and the two Prisms (Prism 2331 and Prism 2332) have outputs, sc(t)

and sk(t), are combined with difference scaling values P(n), to generate new signals s_1(t) ... s_4(t), where in s-i(t) the i1h frequency component has been eliminated using dynamic notch

filtering to leave only three frequency components. Each of the signals s_1(t) ... s_4(t) may then be subject to further signal processing, including further stages of dynamic notch filtering, until individual frequency components are isolated and tracked to provide amplitude and phase information. In Figure 233, each filtered output s-i(t) is further processed to generate parameter values for the amplitude and phase of frequency component i+1 in the original input signal s(t), but many other arrangements are possible. [001113] Two simulation studies are described using the four component dynamic notch

filtering scheme shown in Figure 233. The first excludes noise and demonstrates that dynamic notch filtering may be used to separate frequency components even where the individual frequencies are close together. The second example demonstrates the technique in the presence of noise. The simulations share certain common parameters: the sampling rate fs is 51.2 kHz and the value of mused for the Prisms shown in Figure 233 is 200 Hz. [001114] The noise free example uses the following frequency and amplitude values: 101

Hz at 0.101 V, 102 Hz at 0.102 V, 103 Hz at 0.103 V, and finally 104 Hz at 0.104 V. These

- 225 -



values are selected to demonstrate the precision possible with dynamic notch filtering in the absence of noise. Figures 234A and 2348 (graph 2340, graph 2342, graph 2350 and graph 2352) show the frequency response of each of the dynamic notch filters generated by the application of Equation 69 with the appropriate value of P(ri) for each frequency component. Figure 234A shows the gain variation over the frequency range 0 Hz to 200 Hz (i.e. 0 to m), while Figure 2348 shows the same (absolute) gain variation on a logarithmic scale in the region of the notches at each of the required frequencies. [001115] Figures 235A and 2358 show the original input signal s(t), which contains all four

frequency components. Figure 235A shows a segment of the time series, while Figure 2358 shows the power spectrum of the signal, indicating the presence of all four components, with approximately similar amplitudes. [001116] Figures 236A to 2398 show the corresponding outputs s_1(t) ... s_4(t) to which

dynamic notch filtering has been applied in order to remove a single frequency component in each case. Figure 236A shows a time series of s_1(t) over the same period as that shown in Figure 235A, while Figure 2368 shows the corresponding power spectrum, which indicates that the 101 Hz component has been largely removed using dynamic notch filtering. Note that the power spectrum suggests the amplitudes of the three remaining components are different, as the gain of the notch filtering function is different for each of the remaining frequencies. However, these gains are readily calculated based on Equation 69 and the gains of the Prism outputs, so that compensation can be applied to calculate the amplitudes of the original components, as will be demonstrated in later figures. Figures 237A to 2398 have the same format as Figures 236A and 2368: in each case the upper graph (e.g. Figure 237A) shows the corresponding time series of the output, while the lower graph (e.g. Figure 2378) shows the power spectrum of the output time series. In each case the targeted frequency component has been effectively removed even when, in the cases of s_2(t) (Figures 237A and 2378) and s_3(t) (Figures 238A and 2388) there are adjacent components both above and below the target frequency, at a distance of only 1 Hz.

- 226 -



[001117] Figure 233 shows further processing of the outputs s_1(t) . . . s_4(t) so that eventually the amplitudes and phases of each of the individual components may be calculated. This additional processing was carried out in the simulation, applying further stages of dynamic notch filtering until individual components were isolated and tracked. In each case amplitude and phase compensation was applied for all intermediary steps so that the amplitude and phase values of the original signal components in s(t) were estimated. Figures 240A to 2558 show the results of these calculations.

[001118] Figures 240A and 2408 (graph 2400 and graph 2402) show the isolated 101 Hz component. This signal (like the corresponding signals for the other frequency components described below) is the output of several stages of dynamic notch filtering, after which essentially only the 101 Hz component remains. Amplitude compensation has been applied for the earlier processing stages. The time series is shown in Figure 240A: its form is suggestive of a pure sinusoid with an amplitude close to the original value of 0.101 V. Figure 2408 shows the corresponding power spectrum, which suggests that all other components have been largely removed, leaving only the 101 Hz component. Figures 241A to 2438 (graph 2410, graph 2412, graph 2420, graph 2422, graph 2430 and graph 2432) show the results obtained by applying a simplified tracker to the isolated 101 Hz signal of Figures 240A and 2408 to obtain amplitude (Figures 241A and 2418) and phase (Figures 242A and 2428) parameter values, and hence a tracked version of the original signal component (Figures 243A and 2438). In each case, the tracked parameter value or signal is compared with that of the original component in s(t). Figure 241A shows a section of the time series for the calculated value of the amplitude of the 101 Hz component, while Figure 2418 shows the corresponding error, which remains within approximately± 1e-10 V. Figure 242A shows the corresponding time series of the calculated value of the phase of the 101 Hz component, while Figure 2428 shows the corresponding error, which remains within approximately± 1e-9 radians. Finally, Figure 243A shows the corresponding time series of the calculated value of the tracked 101 Hz component signal, while Figure 2438 shows the corresponding error, which remains within approximately± 4e-11 V.

- 227 -



[001119] Figures 244A and 2448 (graph 2440 and graph 2442) show the isolated 102 Hz

component. The time series is shown in Figure 244A: its form is suggestive of a pure sinusoid with an amplitude close to the original value of 0.102 V. Figure 2448 shows the corresponding power spectrum, which suggests that all other components have been largely removed, leaving only the 102 Hz component. Figures 245A to 24 78 (graph 2450, graph 2452, graph 2460, graph 2462, graph 2470 and graph 2472) show the results obtained by applying a simplified tracker to the isolated 102 Hz signal of Figures 244A and 2448 to obtain amplitude (Figures 245A and 2458) and phase (Figures 246A and 2468) parameter values, and hence a tracked version of the original signal component (Figures 247A and 2478). In each case, the tracked parameter value or signal is compared with that of the original component in s(t). Figures 245A and 2458 show a section of the time series for the calculated value of the amplitude of the 102 Hz component, while Figure 2458 shows the corresponding error, which remains within approximately ± 3e-10 V. Figure 246A shows the corresponding time series of the calculated value of the phase of the 102 Hz component, while Figure 2468 shows the corresponding error, which remains within approximately± 3e-9 radians. Finally, Figure 247A shows the corresponding time series of the calculated value of the tracked 102 Hz component signal, while Figure 24 78 shows the corresponding error, which remains within approximately± 1.5e-10 V. [001120] Figures 248A and 2488 (graph 2480 and 2482) show the isolated 103 Hz

component. The time series is shown in Figure 248A: its form is suggestive of a pure sinusoid with an amplitude close to the original value of 0.103 V. Figure 2488 shows the corresponding power spectrum, which suggests that all other components have been largely removed, leaving only the 103 Hz component. Figures 249A to 2518 (graph 2490, graph 2492, graph 2500, graph 2502, graph 2510 and graph 2512) show the results obtained by applying a simplified tracker to the isolated 103 Hz signal of Figures 248A and 2488 to obtain amplitude (Figures 249A and 2498) and phase (Figures 250A and 2508) parameter values, and hence a tracked version of the original signal component (Figures 251A and 2518). In each case, the tracked parameter value or signal is compared with that

- 228 -



of the original component in s(t). Figure 249A shows a section of the time series for the calculated value of the amplitude of the 103 Hz component, while Figure 2498 shows the corresponding error, which remains within approximately± 3e-10 V. Figure 250A shows the corresponding time series of the calculated value of the phase of the 103 Hz component, while Figure 2508 shows the corresponding error, which remains within approximately ± 3e-9 radians. Finally, Figure 251A shows the corresponding time series of the calculated value of the tracked 103 Hz component signal, while Figure 2518 shows the corresponding error, which remains within approximately± 1.5e-10 V. [001121] Figures 252A and 2528 (graph 2520 and graph 2522) show the isolated 104 Hz

component. The time series is shown in Figure 252A: its form is suggestive of a pure sinusoid with an amplitude close to the original value of 0.104 V. Figure 2528 shows the corresponding power spectrum, which suggests that all other components have been essentially removed, leaving only the 104 Hz component. Figures 253A to 2558 (graph 2530, graph 2532, graph 2540, graph 2542, graph 2550 and graph 2552) show the results obtained by applying a simplified tracker to the isolated 104 Hz signal of Figures 252A and 2528 to obtain amplitude (Figures 253A and 2538) and phase (Figures 254A and 2548) parameter values, and hence a tracked version of the original signal component (Figures 255A and 2558). In each case, the tracked parameter value or signal is compared with that of the original component in s(t). Figure 253A shows a section of the time series for the calculated value of the amplitude of the 104 Hz component, while Figure 2538 shows the corresponding error, which remains within approximately± 1e-10 V. Figure 254A shows the corresponding time series of the calculated value of the phase of the 104 Hz component, while Figure 2548 shows the corresponding error, which remains within approximately ± 1e-9 radians. Finally, Figure 255A shows the corresponding time series of the calculated value of the tracked 104 Hz component signal, while Figure 2558 shows the corresponding error, which remains within approximately± 5e-11 V. [001122] Another simulation

based

on

the

embodiment depicted

in

Figure

233

demonstrates that dynamic notch filtering can be used to track for example four frequency

- 229 -



components when noise is also present in the signal. In this example, the frequencies of the four components are irregularly spaced at 79.9 Hz, 90.2 Hz, 99.7 Hz and 110.4 Hz. All frequency components have an amplitude of 0.2V, while the white noise standard deviation is 1.4e-4 V, corresponding to a SNR of 60 dB (when scaled to the amplitude of a single component).

[001123] Figures 256A and 2568 (graph 2560 and graph 2562) show the original signal s(t). Figure 256A shows the time series of the signal, while Figure 2568 shows the power spectrum, which indicates the presence of the four frequency components, with approximately equal amplitudes.

[001124] For brevity, the intermediary output signals s_1(t) ... s_4(t) are not shown, but the isolated components, along with their tracked parameters, are presented in Figures 257A to 2728.

[001125] Figures 257A and 2578 (graph 2570 and graph 2572) show the isolated 79.9 Hz component. The time series is shown in Figure 257A: its form is suggestive of a sinusoid with an amplitude close to the original value of 0.2 V, although some variation in amplitude is visible. Figure 2578 shows the corresponding power spectrum, which suggests that all other components have been largely removed, leaving only the 79.9 Hz component. Figures 258A to 2608 (graph 2580, graph 2582, graph 2590, graph 2592, graph 2600 and graph 2602) show the results obtained by applying a simplified tracker to the isolated 79.9 Hz signal of Figure 257A. Figure 258A shows a section of the time series for the calculated value of the amplitude of the 79.9 Hz component, while Figure 2588 shows the corresponding error, which remains within approximately ± 0.02 V. Figure 259A shows the corresponding time series of the calculated value of the phase of the 79.9 Hz component, while Figure 2598 shows the corresponding error, which remains within approximately ± 0.1 radians. Finally, Figure 260A shows the corresponding time series of the calculated value of the tracked 79.9 Hz component signal, while Figure 2608 shows the corresponding error, which remains within approximately± 0.015 V.

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[001126] Figures 261A and 2618 (graph 2610 and graph 2612) show the isolated 90.2 Hz

component. The time series is shown in Figure 261A: its form is suggestive of a sinusoid with an amplitude close to the original value of 0.2 V, although some variation in amplitude is visible. Figure 2618 shows the corresponding power spectrum, which suggests that all other components have been largely removed, leaving only the 90.2 Hz component. Figures 262A to 2648 (graph 2620, graph 2622, graph 2630, graph 2632, graph 2640 and graph 2642) show the results obtained by applying a simplified tracker to the isolated 90.2 Hz signal of Figure 261A. Figure 262A shows a section of the time series for the calculated value of the amplitude of the 90.2 Hz component, while Figure 2628 shows the corresponding error, which remains within approximately ± 0.05 V. Figure 263A shows the corresponding time series of the calculated value of the phase of the 90.2 Hz component, while Figure 2638 shows the corresponding error, which remains within approximately ± 0.2 radians. Finally, Figure 264A shows the corresponding time series of the calculated value of the tracked 90.2 Hz component signal, while Figure 2648 shows the corresponding error, which remains within approximately± 0.05 V. [001127] Figures 265A and 2658 (graph 2650 and graph 2652) show the isolated 99.7 Hz

component. The time series is shown in Figure 265A: its form is suggestive of a sinusoid with an amplitude close to the original value of 0.2 V, although some variation in amplitude is visible. Figure 2658 shows the corresponding power spectrum, which suggests that all other components have been largely removed, leaving only the 99. 7 Hz component. Figures 266A to 2688 (graph 2660, graph 2662, graph 2670, graph 2672, graph 2680 and graph 2682) show the results obtained by applying a simplified tracker to the isolated 99.7 Hz signal of Figure 265A. Figure 266A shows a section of the time series for the calculated value of the amplitude of the 99. 7 Hz component, while Figure 2668 shows the corresponding error, which remains within approximately ± 0.04 V. Figure 267A shows the corresponding time series of the calculated value of the phase of the 99. 7 Hz component, while Figure 2678 shows the corresponding error, which remains within approximately ± 0.2 radians. Finally, Figure 268A shows the corresponding time series of the calculated

- 231 -



value of the tracked 99.7 Hz component signal, while Figure 2688 shows the corresponding error, which remains within approximately± 0.015 V. [001128] Figures 269A and 2698 (graph 2690 and graph 2692) show the isolated 110.4 Hz

component. The time series is shown in Figure 269A: its form is suggestive of a sinusoid with an amplitude close to the original value of 0.2 V, although some variation in amplitude is visible. Figure 2698 shows the corresponding power spectrum, which suggests that all other components have been largely removed, leaving only the 110.4 Hz component. Figures 270A to 2728 (graph 2700, graph 2702, graph 2710, graph 2712, graph 2720 and graph 2722) show the results obtained by applying a simplified tracker to the isolated 110.4 Hz signal of Figure 269A. Figure 270A shows a section of the time series for the calculated value of the amplitude of the 110.4 Hz component, while Figure 2708 shows the corresponding error, which remains within approximately± 0.01 V. Figure 271A shows the corresponding time series of the calculated value of the phase of the 110.4 Hz component, while Figure 2718 shows the corresponding error, which remains within approximately ± 0.05 radians. Finally, Figure 272A shows the corresponding time series of the calculated value of the tracked 110.4 Hz component signal, while Figure 2728 shows the corresponding error, which remains within approximately± 0.015 V. [001129] The same Dynamic Notch filtering technique can be extended by using more than

two Prisms: for example, if n Prisms with a shared value of m and different values of h are used in parallel to filter a common input signal, then their outputs may be combined to create one or more signals each with up to n-1 notches, where the frequencies of the notches for each signal can be selected independently and in real time. Examples will be given of using three Prisms to create two notches, but one familiar with the art will appreciate that the techniques described are extensible to higher numbers of Prisms. [001130] Figure 273 (graph 2730) is similar to Figure 223 in showing the variation in gain

for Ge and Gk as r varies between 0 and 1, but adds a third line to plot the gain of Gv, which is here defined as the cosine output of a Prism with harmonic number h

- 232 -

= 3. In other words,



using the notation of Equation 26 Ge is equivalent to Ge l , Gk is equivalent to Ge 2 , and Gv is equivalent to Ge . The corresponding gain functions Equation 42 are 1c (r), 1k (r), and 1v (r) J

respectively. The outputs of the three Prisms can be combined to produce signals with frequency notches, as with the two Prism method, and gain ratio functions can be used to achieve this purpose. Figure 274 shows two gain ratio functions. P(r), formed from the ratio of 1c (r) I 1k (r), is equivalent to the function shown in Figure 224 for the two Prism technique. Q(r) is formed from the ratio of 1c (r) I 1v (r) and, like P(r), consists of a monotonically increasing function, so that each value of r has a unique corresponding value of Q(r). A third ratio, for example 1k (r) I 1v (r), could also be formed from the outputs of the three Prisms, but is not discussed further in these disclosures. [001131] Figure 275 shows an embodiment using Dynamic Notch Filtering to remove two components s1(t) and s2(t) from an input signal s(t) 2750 (which includes an arbitrary number of additional frequency components) using three Prisms 2752, where the frequencies of s1(t) and s2(t) are f1 and f2 respectively, with corresponding r values r1 and r2. The values of the gain ratio functions P(r) and Q(r) defined above, evaluated at r1, give

the ratios needed to create two different signals both with notches at frequency f1. These are defined as follows: s (t) ck

=sc(t) - P(r1)

. sk(t)

Equation (70)

s (t)

=s (t) - Q(r1)

. s (t)

Equation (71)

CV

C

V

where s (t) is a weighted combination of s (t) and sk(t), while s (t) is a weighted ck

combination of

C

CV

s c(t) and s v(t), and where in each case the weight is selected to provide a

notch at the desired frequency f1. A weighted combination of s (t) and s (t) will retain the ck

CV

notch at f1 and may be used to create a second notch at f2, as discussed below.

- 233 -



[001132] An example is used to illustrate the generation of the two weighted combination

signals s (t) and s (t) with a common notch frequency. Figures 276A and 2768 (graph ck

CV

2760 and graph 2762) show the gains of the two signals, where m

= 200

Hz, and where a

common notch has been created at 40 Hz. Figure 276A shows the value of the gain of each signal against frequency, while Figure 2768 shows the absolute value of the gain on a logarithmic plot, so that the notch frequency is apparent at 40Hz, corresponding to a zero crossing of both gain functions in Figure 276A. [001133] To combine the two signals

s ck(t) ands CV(t) to create a second notch at

f2

requires

calculation of the appropriate weighting, which may be done as follows. Defining: Equation (72) Equation (73) w

= gain_f2_in_ck1/gain_f2_in_cv1

Then the desired output function with notches at Snew(t)

= S ck(t) -

W*S CV

Equation (74) f1

and at

f2

is given by: Equation (75)

(t)

[001134] In the above calculations, the values gain_f2_in_ck1 and gain_f2_in_ck1 are the

values of the gain of the s (t) ands (t) signals at frequency ck

CV

f2.

These values can be used

to facilitate the calculation of a suitable weight w which may be applied in the combination of these two signals in order to create a second notch at

f2.

[001135] Figures 277A and 2778 illustrate how the two functions in Figures 276A and

2768 can be combined using the weighting calculations shown above to create a signal with notches at 40 Hz and 140Hz. Figure 277A shows the value of the gain of the new signal against frequency, while Figure 2778 shows the absolute value of the gain on a

- 234 -



logarithmic plot: notch frequencies are apparent at 40 Hz and 140 Hz, corresponding to zero crossings in Figure 277A. [001136] The values of the notch frequencies,

f1

and

f2,

are essentially arbitrary (although

they may be constrained to fall within the range 0 Hz to m Hz) and may vary with time. The dynamic notch filtering technique may also be used to isolate individual components for tracking, when several are to be tracked in parallel simultaneously. For example, suppose there are three components in the original signal, at 40 Hz, 100 Hz, and 140 Hz, all of which are to be isolated and tracked in parallel. Figures 276A to 2778 have already demonstrated how to construct a signal from the three Prism 2752 outputs that has notched out 40Hz and 140 Hz, so this signal could be used for tracking the 100 Hz component. Figures 278A to 2798 (graph 2780, graph 2782, graph 2790 and graph 2792) show alternative combinations of the three Prism 2752 outputs weighted to generate notches at 100 Hz and 140Hz respectively. Again, further weighted combinations can be generated to create second notches at desired frequencies. Finally, Figures 280A and 2808 (graph 2800 and graph 2802) show the construction of three weighted signals with the following properties. In each case, one of the three frequencies 40 Hz, 100 Hz, and 140 Hz has a gain of 1, while the other two have notches. Thus the solid line labelled 'Pass 40.0 Hz' has a gain of 1 at 40 Hz, and a gain of zero (a notch) at 100 Hz and at 140 Hz. The dashed line labelled 'Pass 100.0 Hz' has a gain of 1 at 100 Hz, and a gain of zero at 40 Hz and at 140 Hz. Finally, the dotted line labelled 'Pass 140.0 Hz' has a gain of 1 at 140 Hz, and a gain of zero at 40 Hz and at 100 Hz. [001137] These weighted combinations of the outputs of the three Prisms 2752 therefore

permit the isolation of individual frequency components for tracking. It has already been demonstrated that this can be achieved using dynamic notch filtering with only two Prisms, for more than two inputs. The advantage of using three or more Prisms is that more signals can be separated at a single dynamic notch filtering stage, and so, depending upon the number of signal components that need to be isolated, fewer such stages may be needed,

- 235 -



which may lead to a reduction in the computational requirement and delay in the dynamic response. [001138] A simulation example demonstrates the application of the three Prism Dynamic

Notch Filtering technique described above and illustrated in Figure 275. Figure 281A (graph 2810) shows an input signal consisting of three frequency components: 91 Hz with an amplitude of 0.21 V, 102 Hz with an amplitude of 0.22 V, and 115 Hz with an amplitude of 0.23 V, while Figure 2818 (graph 2812) shows the corresponding power spectrum. The sampling rate fs is 51.2 kHz, while the value of mused for the three Prisms 2752is 200 Hz. In this example no additional noise is added. Assuming the component frequencies are known, dynamic notch filtering is applied using the steps described above to separate and track each of the three components. [001139] Figures 282A and 2828 (graph 2820 and graph 2822) show the isolated 91 Hz

component. The time series is shown in Figure 282A: its form is suggestive of a sinusoid with an amplitude close to the original value of 0.21 V. Figure 2828 shows the corresponding power spectrum, which suggests that all other components have been largely removed, leaving only the 91 Hz component. Figures 283A to 2858 (graph 2830, graph 2832, graph 2840, graph 2842, graph 2850 and graph 2852) show the results obtained by applying a simplified tracker to the isolated 91 Hz signal of Figure 282A. Figure 283A shows a section of the time series for the calculated value of the amplitude of the 91 Hz component, while Figure 2838 shows the corresponding error, which remains within approximately ± 1e-10 V. Figure 284A shows the corresponding time series of the calculated value of the phase of the 91 Hz component, while Figure 2848 shows the corresponding error, which remains within approximately ± 4e-10 radians. Finally, Figure 285A shows the corresponding time series of the calculated value of the tracked 91 Hz component signal, while Figure 2858 shows the corresponding error, which remains within approximately± 1e-10 V.

- 236 -




component. The time series is shown in Figure 286A: its form is suggestive of a sinusoid with an amplitude close to the original value of 0.22 V. Figure 2868 shows the corresponding power spectrum, which suggests that all other components have been largely removed, leaving only the 102 Hz component. Figures 287A to 2898 (graph 2870, graph 2872, graph 2880, graph 2882, graph 2890 and graph 2892) show the results obtained by applying a simplified tracker to the isolated 102 Hz signal of Figure 286A. Figure 287 A shows a section of the time series for the calculated value of the amplitude of the 102 Hz component, while Figure 2878 shows the corresponding error, which remains within approximately ± 2e-10 V. Figure 288A shows the corresponding time series of the calculated value of the phase of the 102 Hz component, while Figure 2888 shows the corresponding error, which remains within approximately ± 5e-10 radians. Finally, Figure 289A shows the corresponding time series of the calculated value of the tracked 102 Hz component signal, while Figure 2898 shows the corresponding error, which remains within approximately± 2e-10 V. [001141] Figures 290A and 2908 (graph 2900 and graph 2920) show the isolated 115 Hz

component. The time series is shown in Figure 290A: its form is suggestive of a sinusoid with an amplitude close to the original value of 0.23 V. Figure 2908 shows the corresponding power spectrum, which suggests that all other components have been largely removed, leaving only the 115 Hz component. Figures 291A to 2938 (graph 2910, graph 2912, graph 2920, graph 2922, graph 2930 and graph 2932) show the results obtained by applying a simplified tracker to the isolated 115 Hz signal of Figure 290A. Figure 291A shows a section of the time series for the calculated value of the amplitude of the 115 Hz component, while Figure 2918 shows the corresponding error, which remains within approximately ± 1e-10 V. Figure 292A shows the corresponding time series of the calculated value of the phase of the 115 Hz component, while Figure 2928 shows the corresponding error, which remains within approximately ± 3e-10 radians. Finally, Figure 293A shows the corresponding time series of the calculated value of the tracked 115 Hz

- 237 -



component signal, while Figure 2938 shows the corresponding error, which remains within approximately± 1e-10 V. [001142] A second example of dynamic notch filtering using the three Prisms 2752

includes noise. In this example, the frequency components and their amplitudes are as follows: 52 Hz at 0.26 V, 71 Hz at 0.24 V, and 103 Hz at 0.1 V. The sampling rate fs is 51.2 kHz, while the value of m used for the three Prisms 2752 is 200 Hz.

White noise of

standard deviation 1.8e-4 V has been added to yield a SNR relative to the 52 Hz component of 60 dB. Figures 294A to 3068 show the results of applying the same three Prism dynamic notch filtering technique to this example, where the true frequencies of the three components are known. Figures 294A and 2948 (graph 2940 and graph 2942) show the time series (Figure 294A) and power spectrum (Figure 2948) of the original signal. The power spectrum clearly indicates the presence of the three distinct frequency components at 52 Hz, 71 Hz, and 103 Hz respectively. [001143] Figures 295A and 2958 (graph 2950 and graph 2952) show the isolated 52 Hz

component. The time series is shown in Figure 295A: its form is suggestive of a sinusoid with an amplitude close to the original value of 0.26 V. Figure 2958 shows the corresponding power spectrum, which suggests that all other components have been largely removed, leaving only the 52 Hz component. Figures 296A to 2988 (graph 2960, graph 2962, graph 2970, graph 2972, graph 2980 and graph 2982) show the results obtained by applying a simplified tracker to the isolated 52 Hz signal of Figure 295A. Figure 296A shows a section of the time series for the calculated value of the amplitude of the 52 Hz component, while Figure 2968 shows the corresponding error, which remains within approximately ± 0.015 V. Figure 297A shows the corresponding time series of the calculated value of the phase of the 52 Hz component, while Figure 2978 shows the corresponding error, which remains within approximately ± 0.06 radians. Finally, Figure 298A shows the corresponding time series of the calculated value of the tracked 52 Hz component signal, while Figure 2988 shows the corresponding error, which remains within approximately± 4e-3 V.

- 238 -




component. The time series is shown in Figure 299A: its form is suggestive of a sinusoid with an amplitude close to the original value of 0.24 V. Figure 2998 shows the corresponding power spectrum, which suggests that all other components have been largely removed, leaving only the 71 Hz component. Figures 300A to 3028 (graph 3000, graph 3002, graph 3010, graph 3012, graph 3020 and graph 3022) show the results obtained by applying a simplified tracker to the isolated 71 Hz signal of Figure 299A. Figure 300A shows a section of the time series for the calculated value of the amplitude of the 71 Hz component, while Figure 3008 shows the corresponding error, which remains within approximately ± 0.015 V. Figure 301A shows the corresponding time series of the calculated value of the phase of the 71 Hz component, while Figure 3018 shows the corresponding error, which remains within approximately ± 0.06 radians. Finally, Figure 302A shows the corresponding time series of the calculated value of the tracked 71 Hz component signal, while Figure 3028 shows the corresponding error, which remains within approximately± 0.015 V. [001145] Figures 303A and 3038 (graph 3030 and graph 3032) show the isolated 103 Hz

component. The time series is shown in Figure 303A: its form is suggestive of a sinusoid with an amplitude close to the original value of 0.1 V. Figure 3038 shows the corresponding power spectrum, which suggests that all other components have been largely removed, leaving only the 103 Hz component. Figures 304A to 3068 (graph 3040, graph 3042, graph 3050, graph 3052, graph 3060 and graph 3062) show the results obtained by applying a simplified tracker to the isolated 103 Hz signal of Figure 303A. Figure 304A shows a section of the time series for the calculated value of the amplitude of the 103 Hz component, while Figure 3048 shows the corresponding error, which remains within approximately± 3e-3 V. Figure 305A shows the corresponding time series of the calculated value of the phase of the 103 Hz component, while Figure 3058 shows the corresponding error, which remains within approximately± 0.02 radians. Finally, Figure 306A shows the corresponding time series of the calculated value of the tracked 103 Hz component signal,

- 239 -



while Figure 3068 shows the corresponding error, which remains within approximately ± 1e-3 V. [001146] The three Prism Dynamic Notching technique shown in Figure 275 is capable of

eliminating any two frequency components in a signal where several components are present. An example demonstrates this capability using a signal with four components. The frequencies components are at 60 Hz, 76 Hz, 93 Hz, and 114 Hz, all with a common amplitude of 0.24 V. As previously, the sample rate used fs Prisms used to perform dynamic notching used m

= 200

= 51.2

kHz, and the three

Hz. For the given input signals,

there are six alternative outcomes for notching out any two from a total of four frequency components, and all six are shown here, in which each possible set of two components are eliminated simultaneously. The same input signal is used each time, and the only difference in the input is the selection of which frequency values

(f1

and

f2

in Figure 275)

are to be eliminated. For brevity, no further processing is shown (such as isolation and tracking of individual frequency components using further dynamic notch filtering stages, or gain compensation), as these steps are an obvious continuation of the techniques already described and demonstrated in previous examples. [001147] Figure 307A (graph 3070) shows the original input signal (s(t) 2750) as a time

series, while Figure 3078 (graph 3072) clearly shows the four frequency components in the Power Spectrum with frequencies of 60 Hz, 76 Hz, 93 Hz, and 114 Hz. [001148] Figures 308A and 3088 (graph 3080 and graph 3082) shows the results of

notching out the frequency components at 60 Hz and 76 Hz. Figure 308A shows a section of the resulting time series, while Figure 3088 shows the resulting power spectrum. It is clear that the components at 60 Hz and 76 Hz have been largely removed, while components at 93 Hz and 114 Hz remain. [001149] Figures 309A and 3098 (graph 3090 and graph 3092) show the results of

notching out the frequency components at 60 Hz and 93 Hz. Figure 309A shows a section of the resulting time series, while Figure 3098 shows the resulting power spectrum. It is

- 240 -



clear that the components at 60 Hz and 93 Hz have been largely removed, while components at 76 Hz and 114 Hz remain. [001150] Figures 310A and 3108 (graph 3100 and graph 3102) show the results of

notching out the frequency components at 60 Hz and 114 Hz. Figure 310A shows a section of the resulting time series, while Figure 31 OB shows the resulting power spectrum. It is clear that the components at 60 Hz and 114 Hz have been largely removed, while components at 76 Hz and 93 Hz remain. [001151] Figures 311A and 3118 (graph 3110 and graph 3112) show the results of

notching out the frequency components at 76 Hz and 93 Hz. Figure 311A shows a section of the resulting time series, while Figure 311 B shows the resulting power spectrum. It is clear that the components at 76 Hz and 93 Hz have been largely removed, while components at 60 Hz and 114 Hz remain. [001152] Figures 312A and 3128 (graph 3120 and graph 3122) show the results of

notching out the frequency components at 76 Hz and 114 Hz. Figure 312A shows a section of the resulting time series, while Figure 3128 shows the resulting power spectrum. It is clear that the components at 76 Hz and 114 Hz have been largely removed, while components at 60 Hz and 93 Hz remain. [001153] Finally, Figures 313A and 3138 (graph 3130 and graph 3132) show the results of

notching out the frequency components at 93 Hz and 114 Hz. Figure 313A shows a section of the resulting time series, while Figure 3138 shows the resulting power spectrum. It is clear that the components at 93 Hz and 114 Hz have been largely removed, while components at 60 Hz and 76 Hz remain. [001154] Overall the Prism-based technique of dynamic notch filtering provides a powerful

method of removing

unwanted frequency components and/or isolating frequency

components for further signal processing, for example so that they can be tracked using Prism-based tracking techniques.

- 241 -



Signal diagnostics and modifying the signal processing scheme in real time [001155] When dealing with signals consisting of more than one frequency component, the

examples given in these disclosures so far have assumed knowledge of at least one of the frequencies. Thus in the case of static notch filtering example, the mains frequency was known, while in the dynamic notch filtering examples given, it has been assumed that the frequencies of each component is known. While there are many signal processing applications where signal component frequencies are known (and where, for example, only the amplitude and I or phase information is to be tracked), equally there are other applications where multiple frequency components may be expected within the signal to be analysed, but where detailed knowledge of the frequencies may not be available. For example, in the monitoring of resonant mechanical systems, it is commonly found that the frequencies generated may vary with, for example, temperature, or mass, or operating point (for example the number of revolutions per minute) of the system concerned. Furthermore, where more than one frequency component is present in the signal to be analyzed, if frequency variations do occur over time, then each component may change its frequency to a different extent, subject to the physical behavior of the underlying system. [001156] Generally, the designer of a signal processing scheme will have some knowledge

of the likely number of frequency components and the likely frequency range of each. Those familiar with the art will recognize that the use of low pass and bandpass filtering, such as the Prism-based techniques that have been described above, will in many cases be sufficient to isolate desired components and suppress unwanted components. However, there may be applications where static filtering techniques may not be adequate for isolating individual frequency components, and where for example dynamic notch filtering may be required to isolate and track more than one component. This technique, however, requires knowledge of the current frequencies of the various components. An additional consideration is that faults may occur, in which additional, unexpected frequency components may be introduced into the signal, for example arising from an internal anomaly or external interference.

- 242 -



[001157] An example is now given of how Prism-based techniques may be used to carry

out the following tasks: the tracking of a signal consisting of a single frequency component; the detection of the presence of an anomalous frequency component in the signal; and then the instantiation of additional Prism-based signal processing to isolate and track both the original frequency component (where the influence of the anomaly has been significantly attenuated) and the anomalous frequency component. [001158] Figure 314 shows an exemplary arrangement where an input signal 3140 is being

tracked by a Recursive Signal Tracker 3142, which generates sample-by-sample estimates of frequency, amplitude and phase under the assumption that the input signal has only a single frequency component to be tracked. [001159] In the exemplary simulation study that follows, the sample rate fs

the RST uses m

= 200

= 51.2 kHz, and

Hz. For the first 100 seconds of the simulation, only a single

frequency component, denoted the Primary component, is present. This has a frequency of 81 Hz and an amplitude of 0.5 V. The signal includes white noise with a standard deviation of 1e-5 V. After 100 seconds, an additional frequency component, denoted the Interference component, is added to the input signal, thus simulating the onset of an unexpected anomaly, as might be caused by an internal fault or some form of external interference. This additional frequency component has a constant frequency of 53 Hz and an amplitude of 1 mV (1 e-3 V). [001160] Figures 315A to 3278 demonstrate the impact such an unexpected anomaly has

on the RST signal tracking. Figure 315A (graph 3150) shows a time sequence of the input signal 3140 s(t), during the period before the Interference component begins. Figure 3158 (graph 3152) shows the power spectrum of the entire time sequence before the onset of the Interference component: only the Primary frequency component at 81 Hz is visible. [001161] Figure 316A (graph 3160) shows a time sequence of the input signal 3140 s(t),

after the onset of the Interference component. Given the low amplitude of the Interference component there is little apparent change in the time series. However, in Figure 3168

- 243 -



(graph 3162) the corresponding power spectrum (for the time t

= 100 s

... 200 s) includes

the Interference frequency component at 53 Hz as well as the Primary frequency components at 81 Hz. [001162] Figures 317A and 3178 (graph 3170 and 3172) show the estimate of frequency

generated by the RST 3142 supplied with input signal 3140 during the onset of the Interference component. Figure 317A (graph 3170) shows the time sequence: a clear step change in behavior takes place at t

= 100 s, when the Interference component is manifest.

Figure 3178 (graph (3172) shows the magnitude of the frequency error on a logarithmic scale. This shows that prior to the onset of the Interference component the frequency error was approximately within the range ± 1e-4 Hz, but after the onset of the Interference component the frequency error range increased to approximately ± 5e-2 Hz. [001163] Figures 318A and 3188 (graph 3180 and graph 3182) shows the estimate of

amplitude generated by the RST 3142 supplied with input signal 3140 over the same time interval as shown in Figures 317A and 3178. Figure 318A shows the time sequence: a clear step change in behavior takes place at t

= 100 s, when the Interference component is

manifest. Figure 3188 shows the magnitude of the amplitude error on a logarithmic scale. Prior to the onset of the Interference component the amplitude error was approximately within the range ± 5e-6 V, but after the onset of the Interference component the amplitude error range increased to approximately± 1e-3 V. [001164] Figures 319A and 3198 (graph 3190 and 3192) show the estimate of phase

generated by the RST 3142 supplied with input signal 3140 over the same time interval as shown in Figures 317A and 3178. Figure 319A shows the time sequence: no obvious change in behavior takes place at t

= 1OOs, when

the Interference component is manifest,

because of the continuous change in phase value. However, Figure 3198 shows the magnitude of the phase error on a logarithmic scale, where a step change in error at t

= 1OOs is clearly present.

Prior to the onset of the Interference component the phase error

- 244 -



was approximately within the range ± 1e-5 radians, but after the onset of the Interference component the phase error range increased to approximately ± 1e-2 radians. [001165] Figures 320A and 3208 (graph 3200 and graph 3202) show the tracked signal

generated by the RST 3140 supplied with input signal 3140 over the same time interval as shown in Figure 317. Figure 320A shows the time sequence: no obvious change in behavior takes place at t

= 1OOs, when the Interference component is manifest, because of

the continuously changing value of the signal. However, Figure 3208 shows the magnitude of the tracked signal error on a logarithmic scale, where a step change in error at t

= 1OOs

is clearly present. Prior to the onset of the Interference component the tracked signal error was approximately within the range ± 1e-5 V, but after the onset of the Interference component the tracked signal error range increased to ± 5e-3 V. [001166] In Figures 317 A to Figure 3188, the onset of the Interference component may be

associated with an apparent cycling in the calculated values of frequency and amplitude respectively. Figures 321A to 3248 (graph 3210, graph 3212, graph 3220, graph 3222, graph 3230, graph 3232, graph 3240 and graph 3242) demonstrate this more clearly. Figure 321A shows a short time sequence of the RST frequency estimate before the onset of the Interference component, while Figure 3218 shows the power spectrum of the entire RST frequency estimate for t

= 0.1

s to t

= 100

s, the period before the onset of the

interference signal. It is important to appreciate this is not the power spectrum of the original signal s(t). Rather, the frequency output of the RST is treated as if it were a separate signal, with sample by sample variation, and a power spectrum calculation is performed upon the sample-by-sample variation of the frequency estimate. Accordingly, the RST frequency estimate has a large DC (zero hertz) component, corresponding to the true parameter value of 81 Hz, but the rest of the spectrum is essentially smooth, suggesting an essentially random variation around the true value of 81 Hz. [001167] In Figures 322A and 3228 (graph 3220 and graph 3222), the same analysis is

performed of the RST frequency signal after the introduction of the Interference component.

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Figure 322A shows a short time sequence while Figure 3228 shows the power spectrum for the time t = 100 s to t = 200 s. The time sequence suggests a pattern of cycling in the frequency estimate, and this is confirmed in the Power Spectrum, which shows two peaks at approximately 28 Hz and 134 Hz. These are the so-called beat frequencies, found by forming the difference and sum respectively of the two original frequencies (81 Hz - 51 Hz = 28 Hz, and 81 Hz+ 53 Hz= 134 Hz). [001168] Similarly, Figures 323A to 3248 (graph 3230, graph 3232, graph 3240 and graph

3242) show the sample plots for the RST amplitude estimate. Figure 323A shows the time sequence of the amplitude estimate before the onset of the Interference component, which shows an apparently random variation around the true value of 0.5 V, while Figure 3238 shows a smooth power spectrum. Figure 324A shows the time sequence of the amplitude estimate after the onset of the Interference component, which manifests a strong cycling variation, while Figure 3248 shows peaks at the same peak frequencies of 28 Hz and 134 Hz. [001169] Figure 325 shows an embodiement of a signal processing scheme, as an

extension of the original RST tracker shown in Figure 314, to detect the presence of an interference frequency component in the input signal 3250 s(t). RST 3251 is comparable with the RST 3142 of Figure 314, with input signal 3250 and with estimates of the frequency, phase and amplitude parameters of s(t) as its outputs. However, additional signal processing is provided to detect an interference component. This comprises a second RST 3252 which accepts the frequency estimate of RST 3251 and estimates the frequency of variation of this parameter - in other words it is being used to track the (lower) beat frequency. The value of m selected for RST 3252 and the use of pre-filtering may be a means of rejecting the presence of the higher beat frequency; alternatively the higher beat frequency could be tracked by suppressing the lower beat frequency. The estimated raw beat frequency jbr (t) is fed into signal processing block 3253 . Signal processing block 3253 calculates a sliding window estimate of the mean and standard deviation of the beat frequency, using any of the techniques well known to those practiced in the art. A low - 246 -



standard deviation may indicate the presence of a steady, as opposed to a random, beat frequency, and hence the presence of an interference frequency. The mean beat frequency

jbr (t) may be more useful in later calculation stages than the raw value as it has reduced noise. A threshold test may be applied to the estimated standard deviation, where a low value may be used to flag the detection of an interference frequency. At the same time, the estimated mean beat frequency is passed to signal processing block 3254, together with the frequency output of the RST 3251, in order to provide an improved estimate of the prime component frequency

JP (t).

This might be done, for example, by calculating the

moving average of the prime component frequency, where the moving average is calculated over the period of the mean beat frequency (or nearest whole sample equivalent period), in order to minimise the variation induced by the beat frequency. [001170] Figures 326A, 3268 and 326C (graph 3260, graph 3262 and graph 3264) show

the simulated output resulting from the signal processing scheme shown in Figure 325, when input signal 3250 is used as an input. Figure 326A shows the calculated mean value of the raw beat frequency. Prior to the onset of the Interference component at t

= 100 s, the

mean frequency shows random variation. After the onset of the Interference component, the mean beat frequency rapidly settles on 28 Hz. Figure 3268 shows the calculated standard deviation of the raw beat frequency. Prior to the onset of the Interference component at t

= 100 s,

the standard deviation is typically around 10 Hz. Note that this is

the standard deviation of the raw beat frequency (not shown), not the standard deviation of the mean beat frequency (shown in Figure 326A); the latter has reduced variation due to averaging. After the onset of the Interference component, the calculated standard deviation of the raw beat frequency drops steadily, settling at around 0.1 Hz. A threshold level for detecting the onset of an interference component has been set at 0.8 Hz, although alternative thresholds and/or statistical tests might equally be applied. Figure 326C shows the flagged output of the diagnostic test, with two possible states: either the original signal is clean, or an Interference component has been detected. As long as the standard deviation of the beat frequency remains above the selected limit of 0.8 Hz, the diagnostic

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state value is set to 'Clean'. At approximately t

= 100.4 s,

when the standard deviation of

the beat frequency drops below the threshold, the flag is set to 'Interference'. In this case, the presence of the interference component has been detected within 0.4s of its onset. [001171] Once the presence of an interfering component has been detected, additional

Prism-based signal processing blocks may be created in response. For example, Figure 327 shows an extension of Figure 325 in which a Dynamic Notch Filtering block 3270 is introduced to isolate and track both the Primary component and the Interference component. The Dynamic Notch Filtering block 3270 requires as inputs the input signal 3250 and the estimated frequencies of the (in this case) two components. The Interference frequency is calculated using Equation 76. Equation (76) [001172] Because Prism design and instantiation is simple, a signal processing system can

create additional processing blocks as required in response to newly detected conditions, as this example illustrates. Furthermore, the ability to create Dynamic Notch Filtering blocks as a means of ameliorating the effects of unwanted components and/or tracking such components for the purpose of further diagnostic analysis, provides a powerful set of strategies that are suitable for implementation by automated systems. Unlike prior art static signal processing schemes, new processing blocks can be instantiated (and deleted) as on-line conditions require, so that the associated computational effort is only expended as needed. Furthermore, because the design effort required to design new Prism-based signal processing is low (for example compared with conventional FIR filter design as discussed above), it is possible to instantiate new Prism-based signal processing blocks in real time, selecting design parameters (e.g., m) to match the observed signal properties. [001173] Figures 328A to 3358 show the simulated response of the system illustrated in

Figure 325, which is extended into the system shown in Figure 327 by the creation in real

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time of the additional signal processing blocks which occurs once the presence of the interfering component is detected. [001174] Figures 328A and 3288 (graph 3280 and 3282) show the calculated frequency

estimate of the Primary component, and can be compared with Figures 317A and 3178 which shows the corresponding parameter output from the initial RST 3142. Figure 328A shows the time sequence: step changes in behavior occur at t Interference component is manifest, and at approximately t

=

100 s, when the

= 100.4 s when the correction is

enabled. Figure 3288 shows the magnitude of the frequency error on a logarithmic scale. This shows that prior to the onset of the Interference component the frequency error was approximately within the range ± 1e-4 Hz; after the onset of the Interference component the frequency error range increased to approximately ± 5e-2 Hz; after the correction for interference is applied the frequency error range reduced to approximately ± 5e-4 Hz. [001175] Figures 329A and 3298 (graph 3290 and 3292) show the calculated amplitude

estimate of the Primary component, and can be compared with Figures 318A and 3188 which show the corresponding parameter output from the initial RST 3142. Figure 329A shows the time sequence: step changes in behavior occur at t Interference component is manifest, and at approximately t

=

100 s, when the


enabled. Figure 3298 shows the magnitude of the amplitude error on a logarithmic scale. This shows that prior to the onset of the Interference component the amplitude error was approximately within the range± 5e-6 V; after the onset of the Interference component the amplitude error range increased to approximately ± 1e-3 V; after the correction for interference is applied the amplitude error range reduced to approximately ± 1e-4 V. [001176] Figures 330A and 3308 (graph 3300 and graph 3302) show the calculated phase

estimate of the Primary component, and can be compared with Figures 319A and 3198 which shows the corresponding parameter output from the initial RST 3140. Figure 330A shows the time sequence: step changes in behavior occur at t Interference component is manifest, and at approximately t

- 249 -

=

100 s, when the




enabled. Figure 330B shows the magnitude of the phase error on a logarithmic scale. This shows that prior to the onset of the Interference component the phase error was approximately within the range ± 1e-5 radians; after the onset of the Interference component the phase error range increased to approximately ± 1e-2 radians; after the correction for interference is applied the phase error range reduced to approximately ± 5e-5 radians. [001177] Figures 331A and 331B (graph 3310 and graph 3312) show the tracked signal

estimate of the Primary component, and can be compared with Figures 320A and 320B which show the corresponding tracked signal output from the initial RST 3142. Figure 331A shows the time sequence: step changes in behavior occur at t Interference component is manifest, and at approximately t

=

100 s, when the


enabled. Figure 331 B shows the magnitude of the tracked signal error on a logarithmic scale. This shows that prior to the onset of the Interference component the error was approximately within the range± 1e-5 V; after the onset of the Interference component the tracked signal error range increased to approximately ± 1e-3 V; after the correction for interference is applied the tracked signal error range reduced to approximately ± 1e-4 V. [001178] Figures 332A and 332B (graph 3320 and graph 3322) show the estimate value of

the frequency of the interference component. In Figure 332A the time sequences of the estimated and true values of the frequency are given. Both are zero before the onset of the interference component at t

= 1OOs.

Approximately 0.4 s elapses before the interference

component is detected, the new signal processing blocks are instantiated, and an estimate of the frequency of the interference component is provided. Figure 332B shows the error in the frequency estimate, which after t

= 100.4 s is within the approximate range ± 1e-3 Hz.

[001179] Figures 333A and 333B (graph 3330 and graph 3332) show the estimate value of

the amplitude of the interference component. In Figure 333A the time sequences of the estimated and true values of the amplitude are given. Both are zero before the onset of the interference component at t

= 1OOs.


- 250 -



component is detected, the new signal processing blocks are instantiated, and an estimate of the amplitude of the interference component is provided. Figure 333B shows the error in the amplitude estimate, which after t

= 100.4 s is within the approximate range ± 1e-5 V.

[001180] Figures 334A and 334B (graph 3340 and graph 3342) show the estimate value of

the phase of the interference component. In Figure 334A the time sequences of the estimated and true values of the phase are given. Both are zero before the onset of the interference component at t

= 1OOs.


component is detected, the new signal processing blocks are instantiated, and an estimate of the phase of the interference component is provided. Figure 334B shows the error in the phase estimate, which after t

= 100.4 s is within the approximate range ± 1e-2 radians.

[001181] Figures 335A and 335B (graph 3350 and graph 3352) show the estimated

tracked signal for the interference component. In Figure 335A the time sequences of the tracked and true interference signals are given. Both are zero before the onset of the interference component at t

= 1OOs.


component is detected, the new signal processing blocks are instantiated, and an estimate of the tracked interference component is provided. Figure 335B shows the error in the tracked signal, which after t

= 100.4 s is within the approximate range ± 1e-5 V.

[001182] Once the signal processing scheme of Figure 327 is instantiated, it may continue

to operate for as long as the interference component is deemed to persist. For example if the amplitude of the interference drops below a certain threshold, or the standard deviation of the beat frequency rises above another threshold, it may be determined by the system that the interference component need no longer be tracked and the associated signal processing may be discontinued. Note that the system of Figure 327 is capable of tracking simultaneous changes in frequency and amplitude for both the Primary and Interference frequency components, as will be shown shortly. Thus additionally, if it is deemed that the current signal processing scheme or associated parameters are no longer suited to tracking the current set of signal components, then new signal processing blocks may be

- 251 -



instantiated to better match the current signal properties. Note that such instantiation and filter warmup may occur in parallel with the continued operation of the current signal processing arrangement in order to prevent any interruption in tracking function before switching to the new signal processing scheme. [001183] Figures 336A to 3438 show the results of another simulation using the same

embodiment of Figure 327, but different signal properties. The simulation demonstrates the ability of the signal processing scheme to track simultaneous changes in frequency and amplitude of both the Primary and the Interference components. In the simulation, multiple linear changes occur between t

= 10 s

and t

= 20s,

as follows: the Primary Component

frequency increases from 105 Hz up to 107 Hz, while the Primary Component amplitude decreases from 0.55V down to 0.45 V; simultaneously the Interference Component frequency drops from 78 Hz down to 77 Hz, while the Interference Component amplitude increases from 2 mV up to 3 mV. Throughout the simulation, the standard deviation of added white noise remains constant at 1e-5 V. The sampling rate remains at 51.2 kHz. [001184] Figures 336A and 3368 (graph 3360 and graph 3362) show the Primary

Component frequency as it is tracked during its linear change from 105 Hz at t 107 Hz at t

= 20s.

= 10 s up to

Figure 336A shows the true and calculated (estimated) value of the

frequency, while Figure 3368 shows the absolute error in the frequency estimate, on a logarithmic scale. This shows that prior to the onset of the increase in frequency value the frequency error was approximately within the range ± 5e-4 Hz; during the linear increase the frequency error range increased to approximately ± 5e-3 Hz; after the frequency has settled at its new steady value the frequency error range reduced to approximately ± 5e-4 Hz. [001185] Figures 337A and 3378 (graph 3370 and 3372) show the Primary Component

amplitude as it is tracked during its linear change from 0.55 Vat t

=20s.

= 10 s down to 0.45 Vat t

Figure 337A shows the true and calculated (estimated) value of the amplitude, while

Figure 3378 shows the absolute error in the amplitude estimate, on a logarithmic scale.

- 252 -



This shows that prior to the onset of the decrease in amplitude value the amplitude error was approximately within the range ± 5e-5 V; during the linear increase the amplitude error range increased to approximately ± 5e-3 V; after the amplitude has settled at its new steady value the amplitude error range reduced to approximately± 5e-5 V. [001186] Figures 338A and 3388 (graph 3380 and 3382) show the Primary Component

phase as it is tracked during the simulation. Figure 338A shows the true and calculated (estimated) value of the phase, which on this timescale varies rapidly, while Figure 3388 shows the absolute error in the phase estimate, on a logarithmic scale. This shows that prior to the onset of the linear changes the phase error was approximately within the range ± 5e-5 radians; during the linear changes the phase error range increased to approximately ± 5e-4 radians; after the linear changes are complete the phase error range reduced to approximately ± 5e-5 radians. [001187] Figures 339A and 3398 (graph 3390 and graph 3392) show the Primary

Component tracked signal as it is tracked during the simulation. Figure 339A shows the true and calculated (tracked) value of the signal, which on this timescale varies rapidly, while Figure 3398 shows the absolute error in the tracked signal, on a logarithmic scale. This shows that prior to the onset of the linear changes the signal error was approximately within the range ± 5e-5 V; during the linear changes the signal error range increased to approximately ± 1e-3 V; after the linear changes are complete the signal error range reduced to approximately ± 5e-5 V. [001188] Figures 340A and 3408 (graph 3400 and graph 3402) show the Interference

Component frequency as it is tracked during its linear change from 78 Hz at t to 77 Hz at t

= 20s.

= 10 s down

Figure 340A shows the true and calculated (estimated) value of the

frequency, while Figure 3408 shows the absolute error in the frequency estimate, on a logarithmic scale. This shows that prior to the onset of the increase in frequency value the frequency error was approximately within the range ± 5e-3 Hz; during the linear decrease

- 253 -



the frequency error range increased to approximately ± 5e-2 Hz; after the frequency has settled at its new value the frequency error range reduced to approximately± 5e-3 Hz. [001189] Figures 341A and 3418 (graph 3410 and graph 3412) show the Interference

Component amplitude as it is tracked during its linear change from 2 mV at t mV at t

= 20s.

= 10 s up to 3

Figure 341 A shows the true and calculated (estimated) value of the

amplitude, while Figure 3418 shows the absolute error in the amplitude estimate, on a logarithmic scale. This shows that prior to the onset of the increase in amplitude value the amplitude error was approximately within the range ± 1e-5 V; during the linear decrease the amplitude error range increased to approximately ± 2e-4 V; after the amplitude has settled at its new steady value the amplitude error range reduced to approximately ± 1e-5 V. [001190] Figures 342A and 3428 (graph 3420 and 3422) show the Interference

Component phase as it is tracked during the simulation. Figure 342A shows the true and calculated (estimated) value of the phase, which on this timescale varies rapidly, while Figure 3428 shows the absolute error in the phase estimate, on a logarithmic scale. This shows that prior to the onset of the linear changes the phase error was approximately within the range ± 1e-2 radians; during the linear changes the phase error range increased to approximately ± 5e-2 radians; after the linear changes are complete the phase error range reduced to approximately ± 1e-2 radians. [001191] Figures 343A and 3438 (graph 3430 and 3432) show the Interference

Component tracked signal as it is tracked during the simulation. Figure 343A shows the true and calculated (tracked) value of the signal, which on this timescale varies rapidly, while Figure 3438 shows the absolute error in the tracked signal, on a logarithmic scale. This shows that prior to the onset of the linear changes the signal error was approximately within the range ± 1e-5 V; during the linear changes the signal error range increased to approximately ± 2e-4 V; after the linear changes are complete the signal error range reduced to approximately ± 1e-5 V.

- 254 -



[001192] In these simulation results, higher errors are observed during the linear changes

in parameter values than when the parameter values are steady. This is due in part to the previously described dynamic response effect (see for example Figures 80 - 83 and associated discussion) i.e. the time delay between a change in parameter value and it being reported by a Prism-based tracker. These delays may in turn lead to larger errors in other signal processing blocks, such as dynamic notch filtering. Nevertheless, the system is capable of tracking the changes of the two signal components in real time, and the error magnitudes generally reduce once the signals settle at their new parameter values. A Prism-based signal processing design for Coriolis Mass Flow metering [001193] Coriolis metering is one of the most widely used industrial techniques for

measuring mass flow. The last two decades have seen a number of significant technological advances in Coriolis metering, as detailed in the survey paper of Wang and Baker (Wang, T, and Baker, R. "Coriolis flowmeters: a review of developments over the past 20 years, and an assessment of the state of the art and likely future directions", Flow Measurement

and

Instrumentation

40

(2014),

pp99-123.

DOI:

http://dx.doi.org/10.1016/j.flowmeasinst.2014.08.015), which is incorporated by reference. [001194] A Coriolis meter consists of two parts. The flowtube is essentially a mechanical

conduit inserted into the process pipework. As conventionally described, the flowtube is induced to resonate at the natural frequency (typically 100 - 1000 Hz) of a selected mode of mechanical vibration: the instantaneous frequency of the mode is a function of several parameters, most significantly the density of the fluid passing through the flowtube. Sensors, often simple velocity coils, are used to generate sinusoidal signals from which the resonant frequency may be extracted. In addition, Coriolis forces induce a phase difference between the sensor signals, roughly proportional to mass flow. The transmitter is essentially an electronic unit which simultaneously maintains flowtube oscillation and performs measurement calculations. Oscillation is maintained by generating drive signals

- 255 -



with suitable amplitude, frequency and phase characteristics, which are transmitted to drivers (typically electromagnetic coils) in the flowtube. [001195] In practice, a Coriolis flowtube is a mechanical system with many modes of

vibration. Of these, the 'drive' mode (typically the lowest mode of vibration) is normally selected for high amplitude oscillation, while Coriolis forces apply themselves in an adjacent, 'Coriolis' mode. Meter calibration (i.e. the mapping from observed phase difference to mass flow) requires a known relationship between the frequencies of the Coriolis and drive modes across all fluid densities. Commonly, the ratio of frequencies is assumed to be constant (the attribute of a 'balanced' flowtube); this imposes constraints on the mechanical design, while never being perfectly achieved in practice. A significant shift in the frequency ratio can indicate flowtube damage, such as erosion or cracking, and manufacturers have developed off-line diagnostic procedures to detect such shifts. Currently, however, such tests may be performed only occasionally (say six monthly or yearly) as the tests may require manual intervention and may disable measurement function for their duration. [001196] In practice, several mechanical modes of a Coriolis flowtube are liable to

excitation by external vibration present in the environment. Often, vibration in undesired modes causes only low level measurement noise, which can be suppressed using mechanical and/or signal processing means. However, with gas/liquid mixtures, significant mechanical energy is expended inside the flowtube, and the amplitude of the Coriolis (and other) modes, while highly erratic, can approach or even temporarily exceed that of the drive mode. When conventional, typically single component, signal processing techniques are applied to such sensor signals, very high levels of measurement noise may be generated, and gas/liquid mixtures may further cause the transmitter to lose track of the flowtube vibration, and (temporarily) cease to operate. [001197] It is therefore desirable to actively measure and control more than one mode of

flowtube vibration. An initial goal might be the active monitoring and control of two modes of

- 256 -



vibration, the conventional 'drive' mode and the adjacent 'Coriolis' mode. The advantages of providing accurate Coriolis mode measurement and control are several.

The meter

calibration may be adjusted in real time as the modal frequency ratio varies so that more accurate (as opposed to more precise) measurement may be achieved. The resulting diagnostic monitoring (e.g. detecting mechanical damage to the flowtube) may be provided on a continuous basis without interrupting measurement function. In addition, removing the requirement for balanced flowtubes may broaden design possibilities, which may be of significant benefit to manufacturers and users. For example, short-path flowtubes with reduced pressure drop might be developed for metering viscous fluids such as heavy oils. However, the greatest benefits may accrue when metering multiphase flow. Here, active tracking and control of the Coriolis mode may result in substantial reductions in noise, with corresponding

improvements in

uncorrected measurement precision and

dynamic

response; this in turn could lead to significantly improved correction strategies. [001198] Previous workers have proposed signal processing techniques for tracking the

Coriolis and drive modes simultaneously. Examples include the technique of model based phasor control by Helmut Rock and colleagues (see Rock, H., Koschmieder, F. "ModelBased Phasor Control of a Coriolis Mass Flow Meter (CMFM) for the Detection of Drift in Sensitivity and Zero Point", in Recent Advances in Sensing Technology, Lecture Notes in Electrical Engineering Volume 49, 2009, pp 221-240, and Rock, H., Koschmieder, F. "Operating a Coriolis Mass Flow Meter at two different Frequencies simultaneously using Phasor Control", 10th International Symposium of Measurement Technology and Intelligent Instruments, Daejeon, Korea, June 2011 ), and the Coriolis mode processing techniques developed by the current author (described in US Patent 7,313,488). However, such techniques do not incorporate Prism-based signal processing, with its associated advantageous properties as previously stated herein. [001199] Two exemplary Prism-based signal processing schemes are presented for

tracking two modes of Coriolis mass flow meter vibration. Both schemes use a two-stage process: firstly the current frequency of each mode is identified, using a different technique

- 257 -



in each scheme, and in the second stage dynamic notch filtering is used to track the amplitude and phase of each component in each sensor signal, for both schemes. The first scheme uses band-pass filtering to separate the two modes of vibration in order to track the frequency of each, while the second uses an initial dynamic notch filtering stage to separate and track the frequencies. [001200] The Coriolis meter signal processing problem may be described as follows: s1(t)

=

A1 sin(2eft +¢i) + B1 sin(2;zgt +t'i) + o-1(t)

Equation (77) Equation (78)

Where: s1(t) is the signal from sensor signal 1 A1 is the (time-varying) amplitude of the drive mode component in sensor signal 1 f is the (time-varying) frequency of the drive mode component (common to both sensor signal 1 and sensor signal 2) ~1

is the (time-varying) phase of the drive mode component in sensor signal 1

B1 is the (time-varying) amplitude of the Coriolis mode component in sensor signal 1 g is the (time-varying) frequency of the Coriolis mode component (common to both sensor signal 1 and sensor signal 2) 81 is the (time-varying) phase of the Coriolis mode component in sensor signal 1 01 is the (time-varying) noise in sensor signal 1 s2(t) is the signal from sensor signal 2 A2 is the (time-varying) amplitude of the drive mode component in sensor signal 2

- 258 -

Inventor: Manus Henry University of Oxford ~2


2 is the (time-varying) phase of the drive mode component in sensor signal 2

82 is the (time-varying) amplitude of the Coriolis mode component in sensor signal 2 82 is the (time-varying) phase of the Coriolis mode component in sensor signal 2 02 is the (time-varying) noise in sensor signal 2 [001201] The two sensor signals thus share common frequencies f and g for the driven and Coriolis modes respectively, but are otherwise nominally independent from one another. In practice it would be expected that the amplitudes of each mode observed on the two signals, A1 and A2, as well as 81 and 82, are reasonably close in value. Similarly, the phases on the two modes are reasonably aligned. Thus the phase differences for each mode, defined as: Equation (79) Equation (80) are both likely to be small, for example with a magnitude of less than 10 degrees, or even, in the case of some flowtube geometries, less than one degree. [001202] The form of Equations 77 and 78 show that s1(t) is defined as a sum of the drive and Coriolis mode components (plus noise), while s2(t) is the difference of the two mode components (plus noise).This arrangement is understood by those familiar with the art as arising from the different 'modal shapes' of vibration for each mode, as viewed from the fixed location of each sensor. Effectively, this arrangement ensures that Equations 79 and 80 have small phase differences, which are approximately linear with mass flow rate. Under alternative forms of Equations 77 and 78, the phase differences of Equations 79 and/or 80 might have an offset of 180 degrees. Equivalently the issue may be analysed in terms of the 'polarity' of each mode for each sensor signal. Other, but essentially similar, equations may be used to represent the signal contents for different designs of Coriolis flowtube.

- 259 -



[001203] While for mass flow and density measurement purposes, the requirements are to

calculate the phase differences and frequencies for each mode, it is also valuable to calculate the amplitude and instantaneous phase of each mode on each sensor signal to facilitate flowtube control and drive generation, as discussed for example in GB Patent 2,412,437 (which considers drive signal generation for the drive mode only, but uses amplitude and phase measurement information when it is available). [001204] Given the similarity in value for the amplitude and phase for each mode on the

sensor 1 and sensor 2 signals, and the polarity of each mode on each signal, as discussed above, a commonly used technique, recognised by those familiar with the art, is to use the sum and/or difference of the sensor signals to create new signals in which one mode is amplified and the other is reduced by partial cancellation. For example, using Equations 77 and 78, the sum of the two sensor signals will reinforce the drive mode and partially cancel the Coriolis mode (due to sign difference for the Coriolis mode), while the difference between the two sensor signals will suppress the drive mode and reinforce the Coriolis mode. The signal separation via this technique can be further improved by using a weighted sum to reduce the effect of any amplitude difference between the modes in each signal, but the mode cancellation is unlikely to be exact as long as there may be phase difference between the two sensor signals. [001205] Nevertheless, the sum and difference technique is a useful starting point for both

signal processing approaches described here, as the results provide pooled signals, using data from both sensors simultaneously, from which to extract the frequencies of the drive and Coriolis modes, which are common to both sensor signals. In the example used, the drive mode frequency is in the range 110 - 120 Hz, while the Coriolis mode frequency is in the range 157 - 167 Hz, but other frequency ranges may be used for other Coriolis flowtube designs. [001206] Figure 344 shows an embodiment of the disclosures to track the frequency of the

drive mode, fd(t), denoted the drive frequency 3448 for brevity, using the sum of the sensor

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signal one 3440, s1(t), and sensor signal two 3442, s2(t), passed through a Prism-based bandpass filter 3444 with, for example, a central frequency of 115 Hz. The filtered signal may then be passed to a Deep Signal Tracker 3446, which generates an estimate of the drive frequency. Figure 345 shows a corresponding exemplary signal processing scheme to track the frequency of the Coriolis mode, fc(t), denoted the Coriolis frequency 3454 for brevity. Here the difference of the sensor signals is calculated, to reinforce the Coriolis mode as discussed above, and the result is passed through a Prism-based bandpass filter 3450 with a suitably selected central frequency, such as 162 Hz. A separate Deep Signal Tracker 3452 is used to track the Coriolis frequency. [001207] With both frequencies established, Prism based signal splitters, using the

technique of dynamic notching filtering as illustrated above, can be used to track the phase and frequency of each mode, where separate splitters are used for each sensor signal. For example, Figure 346 shows a signal splitter 3462 for inputs 3460 including sensor signal 1, in which the drive frequency 3448 and Coriolis frequency 3454 may be used to apply dynamic notch filtering in order to separately track outputs 3464 including the amplitude and phase of each mode. The value of m for signal splitter 3462 is selected as 250 Hz, but other values may be used according to the values of the input frequencies. In parallel, a second splitter (not shown) may be used to track the corresponding parameters for sensor signal 2. [001208] A simulation study demonstrates the effectiveness of the signal processing

scheme illustrated in Figures 344 to 346, where it is assumed that all of these networks are applied once per sample (with a sampling rate of 48 kHz), using as inputs supplied values of s1(t) 3440 and s2(t) 3442. [001209] Firstly, Figures 347A to 3488 show the frequency responses of the two bandpass

filters, centred on 115 Hz and 162 Hz respectively. Figure 34 7A (graph 34 70) shows the gain of the bandpass filter for the drive mode, centred on 115Hz, while Figure 34 78 (graph 3472) shows the absolute gain on a logarithmic scale. Lines above each graph show the

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range of frequencies for each mode. The drive mode frequency range has a consistently high gain, while the Coriolis mode frequency range has a significantly reduced gain. Further Prism-based bandpass filter design adjustment might provide a notch in the centre of the Coriolis frequency range to further attenuate this component for the purposes of tracking the drive mode. [00121 O] Similarly, Figure 348A (graph 3480) shows the gain of the bandpass filter for the

Coriolis mode, centred on 162Hz, while Figure 3488 (graph 3482) shows the absolute gain on a logarithmic scale. Lines above each graph show the range of frequencies for each mode. The Coriolis mode frequency range has a consistently high gain, while the drive mode frequency range has a significantly reduced gain. [001211] In the simulation example provided, the frequency, amplitudes and phase

= 115 Hz, g = 162 Hz, A1 = 0.05 V, A2 ~d = 1.0 degrees, and ed = 0.1 degrees. Note that as

differences for each mode are constant, as follows: f

= 0.051

V, 81

= 0.01

V, 82

= 0.011

V,

the phase difference is typically small, it is commonly reported using degrees rather than radians. Independent white noise is added to each sensor signal with a standard deviation of 1e-6 V. [001212] Figures 349A and 3498 (graph 3490 and graph 3492) show the sensor 1 signal,

which contains the two frequency components for the drive and Coriolis modes respectively. The drive mode, having the high amplitude of 0.05 V, is the most significant component in the time series, shown in Figure 349A. Figure 3498 shows the power spectrum, which indicates both frequency components. [001213] Figures 350A and 3508 (graph 3500 and graph 3502) show the sensor 2 signal,

which appears similar to Figure 349, except for the following detail. As the Coriolis mode has negative polarity in the sensor 2 signal (compare Equation 79 and 80), the beat pattern in the time series of Figure 350A differs from that of Figure 349A. For example the first peak in Figure 349A, immediately after t

= 1.0

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s, is higher than the subsequent peak,



whereas in Figure 350A, the first peak is lower than the subsequent peak. The spectra of Figure 3498 and 3508 are essentially similar. [001214] Figures 351A and 3518 (graph 3510 and graph 3512) show the sum of the

sensor 1 and sensor 2 signals, and demonstrates the effectiveness of summing the signals in order to accentuate one component and attenuate another. The time series of Figure 351A shows much less amplitude variation than FIGS 349A or 350A, and the power spectrum of Figure 3518 shows why: the power of the Coriolis mode has been significantly reduced by the cancellation effect of adding the two signals together. Naturally, the effectiveness of this cancellation is dependent upon the phase difference and amplitude matching of the Coriolis modes in the two sensor signals. [001215] Figures 352A and 3528 (graph 3520 and graph 3522) show the difference of the

sensor 1 and sensor 2 signals, which reduces the drive mode and accentuates the Coriolis mode. In Figure 352A, the apparent frequency of the signal is higher than that of Figure 351A; this is explained by the power spectrum in Figure 3528, which shows that the Coriolis mode component has higher power than the drive mode component, so that its higher frequency is visible in the time series. [001216] Figure 353A (graph 3530) shows the time series for the tracked drive mode

frequency, arising from passing the sensor 1 and sensor 2 signals through the signal processing blocks of Figure 344. The tracked frequency remains close to the true value of 115 Hz. Figure 3538 (graph 3532) shows the error in the tracked drive mode frequency, which remains approximately within ± 6e-6 Hz. [001217] Figure 354A (graph 3540) shows the time series for the tracked Coriolis mode

frequency, arising from passing the sensor 1 and sensor 2 signals through the signal processing blocks of Figure 345. The tracked frequency remains close to the true value of 162 Hz. Figure 3548 (graph 3542) shows the error in the tracked Coriolis mode frequency, which remains approximately within ± 5e-4 Hz.

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[001218] Figures 355A and 3558 (graph 3550 and graph 3552) shows the results of

tracking the drive mode amplitude for sensor 1, which is achieved by using the drive and Coriolis frequency values shown in Figure 353 and 354, together with the sensor 1 signal, as inputs to the signal splitter shown in Figure 346. Figure 355A shows the time series of the amplitude estimate, which remains close to the true value of 0.05 V. Figure 3558 shows the error in the amplitude estimate, which remains approximately within ± 1e-6 V. [001219] Figures 356A and 3568 (graph 3560 and graph 3562) show the results of tracking

the drive mode amplitude for sensor 2, which is achieved by using the drive and Coriolis frequency values shown in Figure 353A and 354A, together with the sensor 2 signal, as inputs to a signal splitter equivalent to that shown in Figure 346, working in parallel to split and track sensor signal 2. Figure 356A shows the time series of the amplitude estimate, which remains close to the true value of 0.051 V. Figure 3568 shows the error in the amplitude estimate, which remains approximately within ± 1e-6 V. [001220] Figures 357A and 3578 (graph 3570 and graph 3572) shows the results of

tracking the Coriolis mode amplitude for sensor 1, which is achieved by using the drive and Coriolis frequency values shown in Figure 353 and 354, together with the sensor 1 signal, as inputs to the signal splitter shown in Figure 346. Figure 357A shows the time series of the amplitude estimate, which remains close to the true value of 0.01 V. Figure 3578 shows the error in the amplitude estimate, which remains approximately within ± 1e-6 V. [001221] Figures 358A and 3588 (graph 3580 and graph 3582) show the results of tracking

the Coriolis mode amplitude for sensor 2, which is achieved by using the drive and Coriolis frequency values shown in Figure 353A and 354A, together with the sensor 2 signal, as inputs to the signal splitter equivalent to that shown in Figure 346, working in parallel to split and track sensor signal 2. Figure 358A shows the time series of the amplitude estimate, which remains close to the true value of 0.011 V. Figure 3588 shows the error in the amplitude estimate, which remains approximately within ± 1e-6 V.

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[001222] Figures 359A and 3598 (graph 3590 and graph 3592) show the results of tracking

the drive mode phase for sensor 1, which is achieved by using the drive and Coriolis frequency values shown in Figure 353A and 354A, together with the sensor 1 signal, as inputs to the signal splitter shown in Figure 346. Figure 359A shows the time series of the phase estimate tracking the true value. Figure 3598 shows the error in the phase estimate, which remains approximately within ± 1.5e-5 radians. [001223] Figures 360A and 3608 (graph 3600 and graph 3602) show the results of tracking

the drive mode phase for sensor 2, which is achieved by using the drive and Coriolis frequency values shown in Figure 353A and 354A, together with the sensor 2 signal, as inputs to a signal splitter equivalent to that shown in Figure 346, working in parallel to split and track sensor signal 2. Figure 360A shows the time series of the phase estimate tracking the true value. Figure 3608 shows the error in the phase estimate, which remains approximately within ± 2e-5 radians. [001224] Figures 361A and 3618 (graph 3610 and graph 3612) show the results of tracking

the Coriolis mode phase for sensor 1, which is achieved by using the drive and Coriolis frequency values shown in Figure 353A and 354A, together with the sensor 1 signal, as inputs to the signal splitter shown in Figure 346. Figure 361A shows the time series of the phase estimate tracking the true value. Figure 3618 shows the error in the amplitude estimate, which remains approximately within ± 1e-4 radians. [001225] Figures 362A and 3628 (graph 3620 and graph 3622) show the results of tracking

the Coriolis mode phase for sensor 2, which is achieved by using the drive and Coriolis frequency values shown in Figure 353A and 354A, together with the sensor 2 signal, as inputs to a signal splitter equivalent to that shown in Figure 346, working in parallel to split and track sensor signal 2. Figure 362A shows the time series of the phase estimate tracking the true value. Figure 3628 shows the error in the phase estimate, which remains approximately within ± 1e-4 radians.

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[001226] Figures 363A and 3638 (graph 3630 and graph 3632) show the calculated drive

mode phase difference, which is given by the difference (modulo ±TT) in the drive mode phase for sensor 1 and sensor 2 (shown in Figures 359A and 360A, respectively) converted to degrees from radians. Figure 363A shows the time series, which remains close to the true value of 1.0 degrees, while Figure 3638 shows the error in the drive mode phase difference, which remains approximately within ± 1e-3 degrees. [001227] Figures 364A and 3648 (graph 3640 and graph 3642) show the calculated

Coriolis mode phase difference, which is given by the difference (modulo ±TT) in the Coriolis mode phase for sensor 1 and sensor 2 (shown in Figures 361A and 362A, respectively) converted to degrees from radians. Figure 364A shows the time series, which remains close to the true value of 0.1 degrees, while Figure 3648 shows the error in the Coriolis mode phase difference, which remains approximately within ± 5e-3 degrees. [001228] An alternative exemplary design of Prism-based signal processing blocks for a

Coriolis mass flow meter is shown in part in Figures 365 to 367. The principle applied is to replace the bandpass filtering of Figures 344 and 345 with dynamic notch filtering. [001229] In Figure 365 the drive frequency 3656, fd(t), is estimated by applying dynamic

notch filtering 3652 to the sum of sensor signal one 3440 and sensor signal two 3442 (as previously discussed) in order to remove the Coriolis frequency, based upon the value of a mean Coriolis Frequency 3650, explained below. The output of dynamic notch filtering 3652, with the Coriolis mode frequency component removed, is then passed to a Recursive Signal Tracker 3654 in order to track the drive mode frequency. [001230] Similarly, in Figure 366 the Coriolis mode frequency 3666, f c(t), is estimated by

applying dynamic notch filtering 3662 to the difference of the sensor signal one 3440 and sensor signal two 3442 (as previously discussed) in order to remove the drive frequency component, based upon the value of a mean drive Frequency 3660, explained below. The output of the dynamic notch filtering 3662, with the drive mode frequency component

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removed, is then passed to a Recursive Signal Tracker 3664 in order to track the Coriolis frequency 3666. [001231] Figure 367 illustrates the method by which the mean drive frequency 3676 (and,

using a similar calculation block, the mean Coriolis frequency) may be produced. As previously discussed in the context of Figures 325 and 326, a frequency estimate which is possibly contaminated by the influence of another frequency component, may be improved by averaging over the period of the beat frequency (i.e. the difference between the frequency being estimated and the contaminant frequency). Accordingly, in Figure 367, an improved mean drive frequency 3676 may be produced as follows. Calculate the beat frequency (i.e. difference) between the drive frequency 3670 and Coriolis frequency 3672, and calculate the mean drive frequency 3676 over a window of data with a length corresponding to the period of the beat frequency (or nearest whole sample equivalent). The same technique can be applied to calculate the mean Coriolis frequency. [001232] Once the drive frequency 3656 and Coriolis frequency 3666 estimates, and their

mean estimates have been calculated, then the amplitude and phase of each frequency component for each sensor signal may be calculated as previously, as illustrated for example in Figure 346 and discussed above. Either the mean frequencies or the instantaneous frequencies may be used to drive the signal splitter 3462 when used in the current design. The potential advantages of using this design is that it may reduce the computational requirement and delay in responding to changes in the signal parameters, compared with the bandpass filtering design described above. [001233] A simulation is used to demonstrate the results obtained using this alternative

design for Coriolis meter signal processing. For convenience and brevity, and to simplify performance comparison, the same parameter values are used in this example as shown in Figures 349A to 3648 and listed above. Accordingly, Figures 349A to 3528 showing the input sensor signals, their sum and their difference, apply equally to the new simulation.

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[001234] Figure 368A (graph 3680) shows the time series for the tracked drive mode

frequency, arising from passing the sensor 1 and sensor 2 signals through the signal processing blocks of Figure 365. The tracked frequency remains close to the true value of 115 Hz. Figure 3688 (graph 3682) shows the error in the tracked drive mode frequency, which remains approximately within ± 4e-4 Hz. [001235] Figure 369A (graph 3690) shows the time series for the tracked Coriolis mode

frequency, arising from passing the sensor 1 and sensor 2 signals through the signal processing blocks of Figure 366. The tracked frequency remains close to the true value of 162 Hz. Figure 3698 (graph 3692) shows the error in the tracked Coriolis mode frequency, which remains approximately within ± 4e-3 Hz. [001236] Figures 370A and 3708 (graph 3700 and 3702) show the results of tracking the

drive mode amplitude for sensor 1, which is achieved by using the drive frequency 3656 and Coriolis frequency 3666, together with the sensor one signal 3440, as inputs to the signal splitter 3462. Figure 370A shows the time series of the amplitude estimate, which remains close to the true value of 0.05 V. Figure 3708 shows the error in the amplitude estimate, which remains approximately within ± 1e-6 V. [001237] Figures 371A and 3718 (graph 3710 and graph 3712) show the results of tracking

the drive mode amplitude for sensor 2, which is achieved by using the drive frequency 3656 and Coriolis frequency 3666, together with the sensor two signal 3442, as inputs to a signal splitter 3462, working in parallel to split and track sensor signal 3442. Figure 371A shows the time series of the amplitude estimate, which remains close to the true value of 0.051 V. Figure 3718 shows the error in the amplitude estimate, which remains approximately within

± 1e-6 V. [001238] Figures 372A and 3728 (graph 3720 and graph 3722) show the results of tracking

the Coriolis mode amplitude for sensor 1, which is achieved by using the drive frequency 3656 and Coriolis frequency 3666, together with the sensor one signal 3440, as inputs to the signal splitter 3462. Figure 372A shows the time series of the amplitude estimate, which

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remains close to the true value of 0.01 V. Figure 3728 shows the error in the amplitude estimate, which remains approximately within ± 1e-6 V. [001239] Figures 373A and 3738 (graph 3730 and graph 3732) show the results of tracking

the Coriolis mode amplitude for sensor 2, which is achieved by using the drive frequency 3656 and Coriolis frequency 3666, together with the sensor two signal 3442, as inputs to the signal splitter 3462, working in parallel to split and track sensor signal 3442. Figure 373A shows the time series of the amplitude estimate, which remains close to the true value of 0.011 V. Figure 3738 shows the error in the amplitude estimate, which remains approximately within ± 1e-6 V. [001240] Figures 374A and 3748 (graph 3740 and graph 3742) show the results of tracking

the drive mode phase for sensor 1, which is achieved by using the drive frequency 3656 and Coriolis frequency 3666, together with the sensor one signal 3440, as inputs to the signal splitter 3462. Figure 37 4A shows the time series of the phase estimate tracking the true value. Figure 3748 shows the error in the phase estimate, which remains approximately within ± 4e-5 radians. [001241] Figures 375A and 3758 (graph 3750 and graph 3752) show the results of tracking

the drive mode phase for sensor 2, which is achieved by using the drive frequency 3656 and Coriolis frequency 3666, together with the sensor two signal 3442, as inputs to a signal splitter 3462, working in parallel to split and track sensor signal two 3442. Figure 375A shows the time series of the phase estimate tracking the true value. Figure 3758 shows the error in the phase estimate, which remains approximately within ± 3e-5 radians. [001242] Figures 376A and 3768 (graph 3760 and graph 3762) show the results of tracking

the Coriolis mode phase for sensor 1, which is achieved by using the drive frequency 3656 and Coriolis frequency 3666, together with the sensor one signal 3440, as inputs to the signal splitter 3462. Figure 376A shows the time series of the phase estimate tracking the true value. Figure 3768 shows the error in the amplitude estimate, which remains approximately within ± 2e-4 radians.

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the Coriolis mode phase for sensor 2, which is achieved by using the drive frequency 3656 and Coriolis frequency 3666, together with the sensortwo signal 3442, as inputs to a signal splitter 3462, working in parallel to split and track sensor signal two 3442. Figure 377A shows the time series of the phase estimate tracking the true value. Figure 3778 shows the error in the phase estimate, which remains approximately within ± 2e-4 radians. [001244] Figures 378A and 3788 (graph 3780 and graph 3782) show the calculated drive

mode phase difference, which is given by the difference (modulo ±TT) in the drive mode phase for sensor 1 and sensor 2 (shown in Figures 374 and 375, respectively) converted to degrees from radians. Figure 378A shows the time series, which remains close to the true value of 1.0 degrees, while Figure 3788 shows the error in the drive mode phase difference, which remains approximately within± 3e-3 degrees. [001245] Figures 379A and 3798 (graph 3790 and graph 3792) show the calculated

Coriolis mode phase difference, which is given by the difference (modulo ±TT) in the Coriolis mode phase for sensor 1 and sensor 2 (shown in Figures 376 and 377, respectively) converted to degrees from radians. Figure 379A shows the time series, which remains close to the true value of 0.1 degrees, while Figure 3798 shows the error in the Coriolis mode phase difference, which remains approximately within ± 2e-2 degrees. [001246] The Prism-based signal processing scheme exemplified in Figures 365 to 367

and Figure 346 has been implemented to operate on a prototype Coriolis mass flow meter transmitter. Figure 380 provides a simplified schematic of the prototype transmitter design. The Coriolis flowtube 3800 includes a number of sensors, including at least two sensors for monitoring oscillation or vibration (for example magnetic coils), as well as other sensors for example indicating temperature. As an alternative to using the flowtube itself, for example, in order to test measurement performance using accurately known signal properties, one or more signal generators may be used to simulate the analog sensor signals output by a flowtube. Irrespective of their source, the analog sensor signals 3801 are passed through

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an analog signal conditioning block 3802 in which each signal is treated in a suitable manner as understood by those familiar with the art (for example removing high frequency noise through analog filtering, and/or the application of a gain factor). Each conditioned analog sensor may then be sampled via an analog to digital converter (ADC) 3803, in order to generate a sampled time series suitable for digital signal processing. With the data in the digital domain, measurement and control algorithms can be applied, for example by a processor 3804. Measurement values 3805 may be generated by the processor 3804, including values for frequency, amplitude, phase, phase difference, mass flow, density, temperature and other parameters, where several of these may be calculated for each of several sensors and/or for each of one or more modes of vibration. In addition, the processor may determine appropriate values for one or more drive signals, based upon the measurement analysis, and selected to maintain the oscillation of the flowtube based on desired operational criteria, such as maintaining a steady oscillation amplitude for one or more modes of vibration. These drive signals may be converted into analog signals via one or more digital to analog converter (DAG) channels 3805. Additional signal conditioning 3807 (for example power amplification) may be applied before the drive signals 3808 are transmitted to the flowtube 3800 drivers in order to maintain flowtube operation. [001247] Those familiar with the art will recognise that this exemplary description may be

modified in a variety of ways while still retaining the essential functionality. In particular, these disclosures relate primarily to measurement calculations, whether within a Coriolis meter transmitter or another measurement application. While the calculation of frequency, phase and amplitude of one or more modes of vibration on one or more sensor signals may facilitate more sophisticated Coriolis flowtube control, it is also possible to combine a simpler technique (for example purely analogue), for drive generation and flowtube control along with a Prism-based analysis of the sensor signals for measurement purposes. [001248] In the prototype system used in the following examples, the sampling rates of the

ADC and DAG channels are 48 kHz. Two flowtube vibration monitoring sensor channels and two drive output channels are employed.

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[001249] Two examples are given to demonstrate the use of Prism-based signal

processing on this prototype Coriolis meter system. In the first example, two signal generators are used to provide sensor input data. As is well understood by those familiar with the art, the use of a signal generator for testing purposes is advantageous in the development of instrumentation in that the true values of at least some of the parameters of the input signals are known to within the certified accuracy of the signal generator. For example, the frequency, phase and phase difference of multiple frequency components may be controlled to tight specifications using state of the art signal generators. However, the instantaneous phase of each frequency component may not generally by available, due to its constantly changing value. [001250] In the second example, the prototype is interfaced to a commercial flowtube in

order to demonstrate that the algorithms disclosed here may be used for the real-time operation of a Coriolis mass flow meter. [001251] Results from using the signal generator to simulate a Coriolis flowtube are shown

in Figures 381A to 3968. The sinusoidal signal parameter values generated by the signal generator are as follows. The drive mode frequency is 115 Hz and the Coriolis mode frequency is 159 Hz. The amplitude of the drive mode on sensor 1 is 40 mV, while that on sensor 2 is 41 mV. The amplitude of the Coriolis mode on sensor 1 is 10 mV, while that on sensor 2 is 11 mV. The phase difference between the two sensor signals for the drive mode is 6 degrees, while the phase difference for the Coriolis mode is 0.6 degrees. [001252] Figures 381A and 3818 (graph 3810 and graph 3812) shows the sensor 1 signal

as received by the processor 3804 as a 48 kHz time series. Figure 381A shows a typical section of the time series, while Figure 3818 shows the corresponding power spectrum, which indicates the drive and Coriolis mode peaks. Figures 382A and 3828 (graph 3820 and graph 3812) show the corresponding sensor 2 signal, with the time series in Figure 382A and the power spectrum in Figure 3828. Figures 383A to 3848 (graph 3830, graph 3832, graph 3840 and graph 3842) show the sums and difference of the signal generated

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sensor 1 and 2 signals of Figures 381A to 3828, and as discussed above, these combined signals have reduced amplitude for one or other of the modes. Figure 383A shows the time series of the sum, while Figure 3838 shows the corresponding power spectrum in which the Coriolis mode component has reduced power compared with Figures 3818 and 3828. Figure 384A shows the time series of the difference, while Figure 3848 shows the corresponding power spectrum in which the drive mode component has reduced power compared with Figures 3818 and 3828. [001253] Figure 385A (graph 3850) shows the time series for the tracked drive mode

frequency, arising from the previously described processing of the signal generator data. The tracked frequency remains close to the true value of 115 Hz. Figure 3858 (graph 3852) shows the error in the tracked drive mode frequency, which remains approximately within ± 1e-2 Hz. [001254] Figure 386A (graph 3860) shows the time series for the tracked Coriolis mode

frequency, arising from the previously described processing of the signal generator data. The tracked frequency remains close to the true value of 159 Hz. Figure 3868 (graph 3862) shows the error in the tracked Coriolis mode frequency, which remains approximately within± 1e-1 Hz. [001255] Figure 387A (graph 3870) shows the results of tracking the drive mode amplitude

for sensor 1, arising from the previously described processing of the signal generator data. Figure 387A shows the time series of the amplitude estimate, which remains close to the true value of 0.04 V. Figure 3878 (graph 3872) shows the error in the amplitude estimate, which remains approximately within ± 2e-5 V. [001256] Figures 388A and 3888 (graph 3880 and graph 3882) show the results of tracking

the drive mode amplitude for sensor 2, arising from the previously described processing of the signal generator data. Figure 388A shows the time series of the amplitude estimate, which remains close to the true value of 0.041 V. Figure 3888 shows the error in the amplitude estimate, which remains approximately within ± 4e-5 V.

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the Coriolis mode amplitude for sensor 1, arising from the previously described processing of the signal generator data. Figure 389A shows the time series of the amplitude estimate, which remains close to the true value of 0.01 V. Figure 3898 shows the error in the amplitude estimate, which remains approximately within ± 1e-5 V. [001258] Figures 390A and 3908 (graph 3900 and graph 3902) show the results of tracking

the Coriolis mode amplitude for sensor 2, arising from the previously described processing of the signal generator data. Figure 390A shows the time series of the amplitude estimate, which remains close to the true value of 0.011 V. Figure 3908 shows the error in the amplitude estimate, which remains approximately within ± 3e-5 V. [001259] Figure 391 (graph 3910) shows the results of tracking the drive mode phase for

sensor 1, arising from the previously described processing of the signal generator data. Only the time series is shown, as it is not possible to obtain its continuously changing true value from the signal generator in real time. However, the calculate phase behaves as expected, with a smooth, linear change in value, and the related phase difference calculation shown in Figure 395 can be compared with the known, static phase difference value assigned by the signal generator. The same comments apply to Figure 392 (graph 3920) , which shows the tracked drive mode phase for sensor 2. Similarly, Figures 393 (graph 3930) and 394 (graph 3940) show the tracked phase for the Coriolis mode for sensors 1 and 2 respectively. These cannot be verified directly, but their difference is verified in Figures 396A and 3968 (graph 3960 and graph 3962), whether the Coriolis mode phase difference can be compared with the value used by the signal generator. [001260] Figures 395A and 3958 (graph 3950 and graph 3952) show the calculated drive

mode phase difference, which is given by the difference (modulo ±n) in the drive mode phase for sensor 1 and sensor 2 (shown in Figures 391 and 392, respectively) converted to degrees from radians. Figure 395A shows the time series, which remains close to the true

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value of 6.0 degrees, while Figure 3958 shows the error in the drive mode phase difference, which remains approximately within ± 6e-2 degrees. [001261] Figures 396A and 3968 (graph 3960 and graph 3962) show the calculated

Coriolis mode phase difference, which is given by the difference (modulo ±n) in the Coriolis mode phase for sensor 1 and sensor 2 (shown in Figures 394 and 395A/B, respectively) converted to degrees from radians. Figure 396A shows the time series, which remains close to the true value of 0.6 degrees, while Figure 3968 shows the error in the Coriolis mode phase difference, which remains approximately within± 1e-1 degrees. [001262] Thus, the signal generator example demonstrates that Prism-based signal

processing can be applied in real-time to the analysis and tracking of multi-component signals as for example may be generated by a Coriolis mass flow meter flowtube oscillating in two modes simultaneously. [001263] The final example demonstrates the use of Prism-based algorithms on sensor

data generated by a Coriolis meter flowtube, operated using a prototype transmitter matching the schema of Figure 380. Additional technical difficulties may be introduced when using Coriolis flowtube sensor signals, because many frequency components may be present, which complicates the task of tracking the desired frequency components. Figures 397 (graph 3970) and 398 (graph 3980) show, for a particular commercial Coriolis flowtube design, typical power spectra of the sensor signals when the flowtube is being operated by a transmitter such as Figure 380 to vibrate in two primary modes, the drive and Coriolis modes. Both figures show several peaks in each spectra, as discussed below. Note that, because of the role of feedback control in the operation of the flowtube, different control algorithms may result in characteristically different spectra. Most obviously, if a second mode of vibration is tracked and controlled to match a desired amplitude, then the power observed at that frequency is likely to be higher than if the mode of vibration is uncontrolled and is excited only by, for example, random environmental noise.

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[001264] For clarity, the same frequency peaks in the spectra of Figures 397 and 398 are

given the same labels, and are explained in Table 5 below.

[001265] TABLE 5: Frequency peaks in Figures 397 and 398

Label

Frequency (Hz)

Description

1

49.4

Coriolis Mode - Drive Mode

2

115.9

Drive Mode

3

165.2

Coriolis Mode

4

231.7

2 x Drive Mode

5

281.1

Drive Mode + Coriolis Mode

6

330.2

2 x Coriolis Mode

7

347.4

3 x Drive Mode

[001266] Note that peaks 1 and 6 are scarcely visible in Figure 397, while they are clearly

visible in Figure 398. It is not uncommon for Coriolis sensor signals to have differing

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frequency spectra characteristics even when observing a common flowtube vibration. It can be be seen from Table 5 that all of the labelled frequency peaks are essentially derived from the two forced (i.e. transmitter driven) modes of oscillation, the drive frequency and the Coriolis frequency, respectively. It is important to appreciate that these additional modes do not necessarily arise from actual flowtube vibration, but may arise from nonlinear behaviour in the flowtube sensors themselves, for example as a result of hysteresis in electromagnetic sensing coils, or from the control action of the transmitter itself. Thus different sensing technologies operating on identical flowtubes may result in different sensor frequency spectra, which in turn may require different signal processing strategies for the extraction of the desired parameter values. [001267] However, these additional, undesired, frequency components are present in the

sensor signal as received by the transmitter from the flowtube in use, and so signal processing techniques as outlined above may be required to minimise their impact on the tracking of the drive and Coriolis frequency components. An additional technique that may be well-suited to the particular characteristics of this flowtube, is the use of a pre-filter, consisting of a chain of (for example) four Prisms, each with a distinct value of m selected to create a notch at an undesired frequency. As discussed above, compensating the phase and amplitude of any tracked component for the effect of such pre-filtering is straightforward. The dashed line shown in both Figures 397 and 398 is the frequency response of an exemplary pre-filter, designed as follows. [001268] A four Prism chain, using the cosine output only, is constructed with values of m

corresponding to the frequencies of components 4, 5, 6, and 7 in the sensor signals, resulting in notches at these frequencies. Note that the generally low pass filtering characteristic of each Prism will have an attenuating effect on higher frequencies, as shown in the generally declining power of the filter characteristic with frequency. [001269] Note also that in this case the frequency of each notch can be calculated from the

tracked frequencies of the drive and Coriolis modes. When a flowtube starts up, default

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values may be used for the initial values of the notch frequencies. This may result in incomplete notching and higher noise levels in the tracked parameter values. However, when averaged values of the drive and Coriolis frequencies are established over a suitably selected time period then, as previously discussed, a new pre-filtering Prism chain may be instantiated with more suitable notch frequencies. Such notch frequency adjustment may take place on a regular basis or in response to a detected change in one or more resonant frequencies of the flowtube. Note further that even when additional genuine modes of flowtube vibration do occur, which may not be simple sums and/or differences of tracked frequencies) a similar technique can be applied where, for example, the approximate ratio between a tracked frequency and an undesired frequency component is known. The results of applying the pre-filter to the sensor 1 and 2 data shown in Figures 397 and 398 can be seen in Figures 399 and 400. [001270] Figures 399A and 3998 (graph 3990 and graph 3992) show the sensor 1 signal of

Figure 397 after pre-filtering has been applied. Figure 399A shows a typical section of the time series, while Figure 3998 shows the corresponding frequency spectrum, where the drive and Coriolis mode peaks of Figure 397 remain, while the other peaks have been attenuated. Figures 400A and 4008 (graph 4000 and graph 4002) show the corresponding sensor 2 signal, with the time series in Figure 400A and the power spectrum in Figure 4008. Again, the drive and Coriolis mode peaks remain in place while the other modes have been attenuated compared to the raw sensor 2 signal spectrum of Figure 398. [001271] Further tracking calculations of the drive and Coriolis modes may proceed based

upon the pre-filtered sensor 1 and sensor 2 signals, as discussed below. However, in the case of data from a Coriolis flowtube, there are no 'true' values available for the sinusoidal parameters. Furthermore, parameter variations may occur as a result of, for example flowtube control behaviour. The data of Figures 397 and 398 were generated by a flowtube with the following characteristics: the flowtube was empty of liquid with no flow passing through, so the expected phase difference for both modes is approximately zero (as well understood by those familiar in the art of Coriolis mass flow metering). The frequencies of

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both modes are expected to be steady. The drive control algorithms act to maintain the mean value of the sensor 1 and 2 drive mode amplitudes at 0.04 V, while maintaining the mean Coriolis mode amplitude at 0.01V. Given these characteristics, it is possible to evaluate in general terms the performance of the measurement algorithms. [001272] Figure 401 (graph 4010) shows the time series for the tracked drive mode

frequency, arising from the previously described signal processing technique for the signal generator data, applied to the Coriolis flowtube data. The tracked frequency remains close to 115.842 Hz, with a variation over the time sequence shown of approximately± 5e-3 Hz. [001273] Figure 402 (graph 4020) shows the time series for the tracked Coriolis mode

frequency, arising from the previously described signal processing technique for the signal generator data, applied to the Coriolis flowtube data. The tracked frequency remains close to 165.1 Hz, with a variation over the time sequence shown of approximately ± 5e-2 Hz. [001274] Figure 403 (graph 4030) shows the time series results of tracking the drive mode

amplitude for sensor 1, arising from the previously described processing of the signal generator data, applied to the Coriolis flowtube data. Figure 404 (graph 4040) shows the corresponding time series results of tracking the drive mode amplitude for sensor 2. The sensor 1 amplitude in the sequence shown has a mean value of approximately 0.04178 V, while the corresponding mean value of the sensor 2 amplitude is approximately 0.03822 V, so that the average of both is approximately 0.04 V, as expected. Note that both amplitude measurement decline at the end of the time period shown. It is likely that the low frequency variations in the observed amplitudes are due to actual adjustments to the flowtube oscillation, for example as a result of a change in the supplied drive signal power, whereas high frequency variations in the observed amplitudes are likely attributable to measurement noise. [001275] Figure 405 (graph 4050) shows the time series results of tracking the Coriolis

mode amplitude for sensor 1, arising from the previously described processing of the signal generator data, applied to the Coriolis flowtube data. Figure 406 (graph 4060) shows the

- 279 -



corresponding time series results of tracking the Coriolis mode amplitude for sensor 2. The sensor 1 amplitude in the sequence shown has a mean value of approximately 0.00926 V, while the corresponding mean value of the sensor 2 amplitude is approximately 0.0107 V, so that the average of both is approximately 0.01 V, as expected. Note that both amplitude measurements show reasonably strong correlation in their low frequency variations over the time period shown, most obviously with a common rise in value between

t = 1.3 s and t

= 1.8 s, with a subsequent more rapid decline. It is likely that the low frequency variations in the observed amplitudes are due to actual adjustments to the flowtube oscillation, for example as a result of a change in the supplied drive signal power, whereas high frequency variations in the observed amplitudes are likely attributable to measurement noise. [001276] Figure 407 (graph 4070) shows the results of tracking the drive mode phase for

sensor 1, arising from the previously described processing of the signal generator data, applied to the Coriolis flowtube data. Only the time series is shown, as it is not possible to obtain its continuously changing true value from the flow tube in real time. However, the calculate phase behaves as expected, with a smooth, linear change in value. The same comments apply to Figure 408 (graph 4080), which shows the tracked drive mode phase for sensor 2. Similarly, Figures 409 (graph 4090) and 410 (graph 4100) show the tracked phase for the Coriolis mode for sensors 1 and 2 respectively. [001277] Figure 411 (graph 4110) shows the calculated drive mode phase difference,

which is given by the difference (modulo ±TT) in the drive mode phase for sensor 1 and sensor 2 (shown in Figures 407 and 408, respectively) converted to degrees from radians. The time series remains close to zero, as expected, with a mean value of around -0.02 degrees and a variation of approximately ± 4e-2 degrees for the time sequence shown. The observed phase zero offset is well understood by those familiar with the art, and may fall within the expected range for an empty flowtube. [001278] Figure 412 (graph 4120) shows the calculated Coriolis mode phase difference,

which is given by the difference (modulo ±TT) in the Coriolis mode phase for sensor 1 and

- 280 -



sensor 2 (shown in Figures 409 and 410, respectively) converted to degrees from radians. The mean value of the time series shown is close to zero degrees, as expected, while instantaneous values vary by as much as± 2.5e-1 degrees. General Examples [001279] Finally, a set of scenarios for sensor signal processing requirements are described, starting from the simplest and adding increased sophistication at each step, in which exemplary Prism-based instantiations are proposed, in order to demonstrate the suitability of Prism-based solutions for a wide range of signal processing requirements e.g. for the new sensor types required for the Internet of Things. [001280] A brief summary of the advantageous properties of the Prism and its associated signal processing blocks is provided below: 1) The Prism is a recursive FIR calculation: the computational burden of calculating one or two Prism outputs is generally low and generally constant, irrespective of the length of the Prism windows. 2) Prism design is essentially trivial: A) There are only a few design parameters for a Prism (principally m, where the window length fs/m is constrained to be an integer, and the harmonic number h, also an integer) and so design choice is limited, and therefore simple;

B) Once the design parameters are determined, the required coefficients are readily calculated as equally-spaced sine and cosine values. Accordingly, even a low power sensor system can design and instantiate new Prisms as required, for example in response to the properties of signal being tracked which may vary over time.

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3) The Prism-based Trackers such as the Recursive Signal Tracker (RST) generate accurate estimates of sinusoid parameter close to the Cramer-Rao Lower Bounds, even in high noise conditions (down to 0 dB SNR) and with small window sizes (e.g. 32 samples). [001281] Prism-based pre-filtering, for example low-pass, bandpass and notch filtering,

may be used to reduce the influence of noise and undesired frequency components. Such pre-filtering may be instantiated in real-time in response to observed and/or changing signal conditions. [001282] Dynamic notch filtering may be used to remove time-varying frequency

components, and may be deployed repeatedly to enable the isolation and tracking of multiple components in a signal, even when their respective frequencies are in close proximity to one another. [001283] The signal processing inside a sensor can respond adaptively to changing

conditions by the instantiation of new Prism objects that better suit the current operating environment and performance requirements. [001284] Based on these advantageous Prism properties, it is possible to envisage a wide

range of applications, from the simplest low power sensor through to high powered, sophisticated devices, where additional functionality is added at each step. Exemplary (but not exclusive) scenarios include the following. Note that although different sinusoidal trackers might be deployed, in the examples given the RST is used throughout, for brevity.

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1) A low power sensor harvests energy from its local environment, with sufficient energy storage capacity to perform a single measurement cycle, assuming infrequent ad-hoc polling. On being polled for data (for example via radio frequency identification - RFID), the sensor collects sufficient data samples to fill the integrals of a short RST, performs a single measurement calculation, and reports the results. 2) The low power sensor records, when powering down, a recommended value of

m for its next measurement, based on the current value of the frequency, for example, in order to achieve a value of

r close to 0.5 on the next measurement

calculation. 3) The sensor has sufficient power to collects data samples regularly and updates the RST's Prism each sample. On being polled, it calculates frequency, amplitude and phase values for reporting. 4) The sensor has a secure power supply and reasonable computational resources. It performs RST measurements every sample, and reports the latest value upon request. 5) A Prism or Prism chain is used as a pre-filter to the RST to reduce noise and improve measurement accuracy. 6) Self-optimization is introduced. In addition to a static, 'background' RST, a 'tracking' RST is deployed. It is instantiated with the value of m selected to provide optimal measurement performance. If, say,

r falls outside the range [0.4,

0.6] a new RST is instantiated with a new value of m and the previous tracker RST is deleted. Similar adaptation may be applied to any pre-filtering in use. 7) Three tracking RSTs are deployed. These have values of m near to, above and below the currently optimal value of m (typically 2*f where f is the currently observed sinusoid frequency of the input signal), thus broadening the band of

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optimal measurement performance. Old RSTs (with their associated pre-filtering) are replaced by better positioned RSTs as the input signal frequency changes over time. 8) Multiple RSTs are used to track the sensor signal, including some with high m values, to enable faster tracking of any change in signal parameter values (as the dynamic response associated with parameter change is a function of 1/m). Simple rules (for example based on the observed rate of change of sinusoidal parameters) are used to determine a 'best' estimate of each sinusoid parameter from the RST outputs, which may be based on some weighted combination of the estimates from more than one RST. 9) Generic sensor diagnostics are introduced. A residual signal is calculated by subtracting the best estimate of the signal (i.e.

Asin(~1))

from the raw data.

Alternatively, dynamic notch filtering could be applied to remove the tracked signal component and leave the residual signal. The SNR is estimated from the residual signal, for example by calculating a moving window estimate of its standard deviation, from which a simple estimate of the measurement uncertainty can be derived. 1O)The residual signal and or the sinusoidal parameter estimates are subject to further analysis for example via further RST tracking to detect any harmonics or beat frequencies which may be characteristic of sensor faults (e.g. additional modes of vibration, mains noise). When additional frequency components are detected, suitable Prism/RST signal processing networks may be instantiated to track the new components and/or to remove their influence on the previously tracked signal components for example using dynamic notch filtering. Diagnostic algorithms are introduced to examine the residual analysis for evidence of known fault modes.

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11 )Process diagnostics are introduced. The sensor accepts requests from other system components to apply one or more RSTs (with Prism pre-filtering and/or dynamic notch filtering as required) to detect arbitrary frequency components for process diagnostics. 12)An FFT procedure is included. A complete analysis of the sensor signal is performed

by

a

dynamic

and

recursive

process

of

building

Prism

chain/RST/dynamic notch filtering blocks and networks to reflect the current structure of the signal. At each stage, the raw sensor signal, or the latest residual, is submitted to the FFT to identify the next largest component. A prefilter Prism/RST chain is constructed to track the new component. If significant change occurs in the signal structure this is detected and the Prism networks are rebuilt as required. The current set of signal components and values are made available to sensors and other devices in the system. [001285] This list of scenarios is not intended to be prescriptive, but rather simply to

illustrate the ways in which the Prism might be deployed in increasingly complex ways to solve sensor signal processing requirements by autonomic sensors. [001286] Note also that while these techniques have been developed primarily with real-

time signal processing, many of them are also applicable to off-line analysis of signals. [001287] Further applications of Prism signal processing may be found in recent

conference and journal papers. The paper 'Fast Coriolis mass flow metering for monitoring diesel fuel injection' (by Felix Leach, Salah Karout, Feibiao Zhou, Michael Tombs, Martin Davy, and Manus Henry, in Flow Measurement and Instrumentation 58 (2017) pp 1-5, December

2017,

available

online

20

September

2017,

incorporated here by reference, describes the application of Prism signal processing to provide fast (48 kHz) measurement updates of the mass flow rate for monitoring short (1 ms) pulses of fuel in a laboratory diesel engine experiment using Coriolis metering. The flowtube is driven in only a single mode of

- 285 -



vibration, but the high noise environment of the experimental facility results in the excitation of other modes of vibration. Prism-based notch filtering is used to remove these unwanted modes of vibration from the sensor signals in order to facilitate the tracking of individual fuel pulses, which previously has not been possible in this application. [001288] The conference paper "Prism Signal Processing for Sensor Condition Monitoring"

(by Manus Henry, Oleg Bushuev and Olga lbryaeva, IEEE International Conference on Industrial Electronics, Edinburgh, UK, June 2017. DOI: 10.1109/ISIE.2017.8001451 ), incorporated by reference, describes a sensor validation application for pressure sensing using Prism signal processing. Here the mechanical integrity of an industrial pressure sensor is checked via a series of ultrasonic pulses transmitted from within the device, whereby the returning signal contains a number of exponentially decaying sinusoids and where the frequency, amplitude and rate of decay of each frequency component may provide diagnostic information on the mechanical integrity of the pressure sensor. Prism signal processing, specifically a mixture of notch filtering and dynamic notch filtering, is applied to isolate each of six frequency components, after which RST tracking is used to calculate frequency and amplitude information. This application demonstrates the utility of Prism signal processing in analyzing complex multicomponent signals. [001289] The journal paper "The Prism - Efficient Signal Processing for the Internet of

Things" (by Manus Henry, Felix Leach, Martin Davy, Oleg Bushuev, Michael Tombs, Feibiao Zhou, and Salah Karout, IEEE Industrial Electronics Magazine, December 2017, pp 2 - 11, DOI: 10.1109/MIE.2017.2760108), incorporated by reference, provides additional demonstrations of the computational efficiency provided by Prism signal processing compared with previous FIR filtering techniques, gives further examples of bandpass filtering, and describes how Prism signal processing may be particularly beneficial in applications for the Internet of Things. [001290] Heterodyning Tracker

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[001291] Heterodyning is a well-known technique within signal processing. It may be used

in combination with Prism signal processing to create new types of tracker. The key advantage of including heterodyning is that it may result in a faster dynamic response then would be possible otherwise i.e. the delay of a heterodyning tracker may be reduced. This performance improvement may be achieved at the expense of some additional calculation, and possibly reduced accuracy in the calculated frequency, amplitude and phase parameters of the sinusoid to be tracked. [001292] Given the pure sinewave input signal with frequency f described by Equations (2)

and (3), Prism-based trackers such as the FST, DST or RST may provide good accuracy performance if the expected range of values for f is known. As has been shown above, the value of m providing the lowest errors in the calculation of frequency, amplitude and/or phase is 2f, and any corresponding tracker may have a known and fixed time delay of at least 1/m

=

1/2 f Heterodyning may offer a means of perform such tracking with

significantly reduced delay. [001293] One exemplary technique entails the generation of one or more heterodyning

signals h(t) with a known, fixed, higher frequency h, and an amplitude of unity. The input signal s(t) is multiplied by the one or more heterodyning signals to create new signals, each with two frequency components at frequencies f - h and f + h. Let a first heterodyning signal be defined as [001294]

hs(t)

=

Equation (81)

sin(w(t))

[001295] where [001296]

w(t)

=

Equation (82)

2nht +\if 0

[001297] Here l/l(t) is the instantaneous phase of the heterodyning function, and I/lo is its

initial value at

t = 0. Multiplying the input signal by the heterodyning function, the following

signal is obtained:

- 287 -


[001298]


Hs (t) = hs(t) .s(t) =

--f cos( 1h (t)) + -f cos( ¢ (t)) 1

Equation (83)

[001299] where the higher and lower frequency components and their phases are defined

as follows: [001300]

fh = h + f

Equation (84)

[001301]

ft =h-f

Equation (85)

[001302] and

Equation (86) [001304]

¢1 (t)

=

2eftt +If/ 0

-

Equation (87)

¢0

[001305] Equations 83 - 87 thus describe the results of heterodyning the input signal to

create a new signal containing two frequency components, with their corresponding frequency and phase relations. [001306] Given the heterodyned signal, one exemplary means of extracting the frequency,

amplitude and phase of each component is to use dynamic notch filtering as described above, where the value of m used to perform DNF may typically be 2h; this is likely to provide a more rapid dynamic response, as illustrated below. Given the estimates of the parameters of each component, those skilled in the art will recognise that elementary additiona calculations will yield estimates of the amplitude, frequency and phase of the original signal. [001307] As an alternative exemplary approach, a second heterodyne function may further

be defined, with the same frequency but orthogonal to the first: [001308]

Equation (88)

he (t) =cos( fj/(t))

[001309] Independently applying this heterodyne function to the same input:

- 288 -



Equation (89) [001311] Prism signal processing may now be applied to each of these heterodyned

signals in order to separate and track the components. Each of the heterodyned signals Hs(t) and Hc(t) may be passed through its own Prism with characteristic frequency m

= 2h.

The Gs and Ge outputs of each Prism are given as follows:

Equation (90)

Equation (91) [001314] and

Equation (92)

Equation (93) [001317] where rh

= fh/m and Ii= fJ/m.

[001318] Each Prism output consists of two components, where the high and low

frequencies have the same (if unknown) phase shifts. Accordingly, simple weighted combinations of these Prism outputs are sufficient to eliminate one or other component for conventional tracking, in a similar technique to dynamic notch filtering as discussed above. To eliminate the lower frequency component, it is noted that its phase in Gs(Hc) and Gc(Hs) is identical, and the gain varies only by the factor -fi. Accordingly,

- 289 -


[001320]


=

A( . 2( ) 2 smc rh

rh (rh -r1 ) 2

rh -I

· (d-. sm 'f'h

-

21rrh ))

Equation (95)

[001321] A similar combination of Gs(Hs) and Gc(Hc) yields a cosine term with the lower frequency component eliminated.

Equation (96) [001323] Equations (95) and (96) provide a pair of orthogonal functions with the same scaling, so that frequency, phase and amplitude calculations may be performed in a similar manner to those previously described for the FST, DST and RST. Note that the requirement for values of rh and

n in

Equations (94) and (96) may be satisfied using initial

estimates and iteration in a similar technique for that used for the RST. Other combinations of Prism outputs may be used to isolate the lower frequency component:

Equation (97)

Equation (98) [001326] With estimates of the amplitude, frequency and phase of both the upper and lower frequency components, corresponding estimates of the original low frequency signal may also be determined. [001327] The potential benefits of using heterodyning in combination with Prism signal processing are illustrated with the example given in Figures 413A - 4168 (graph 4130, graph 4132, graph 4140, graph 4142, graph 4150, graph 4152, graph 4160 and graph 4162) which is comparable to an earlier example which showed the dynamic response of the RST in Figures 162A - 1658, where m

=200 Hz.

- 290 -



[001328] Figure 413A (compare with Figure 162A) shows a sinusoidal input signal with a

fixed frequency of 133.33 Hz, and an initial amplitude of 1 V which drops linearly to 0.5 V between

t = 1s and t= 3s. Figure 4138 shows (compare with Figure 1628) that the signal

contains a single peak frequency at approximately 133 Hz. [001329] Using the heterodyning technique outlined above, combined with dynamic notch

filtering, an improved dynamic performance is achieved compared with the RST example. A heterodyning frequency of 1000 Hz is selected, and a higher sampling rate of 128 kHz is used (compared with the 51.2 kHz sampling frequency used in the earlier RST example of Figure 162A). [001330] Figure 414A shows the resulting tracked signal, while Figure 4148 shows the

corresponding error. These may be compared with Figures 163A and 1638 from the RST example. The tracking errors in Figure 1638 fall within approximately ± 1.5 e-3 V, while the tracking errors in Figure 4148 fall within approximately ± 2.0 e-4 V, a significant reduction. Figures 415A and 4158 show the amplitude tracking and its error; the average error during the period of amplitude reduction is approximately 2.5e-4 V; given that the rate of amplitude reduction is 0.25 V/s, this average error corresponds to an average delay of 1 ms. This contrasts with the earlier RST example (Figures 164A and 1648) where the average error during the period of amplitude decline is 1.25e-3V, corresponding to an average delay of 5 ms. Figures 416A and 4168 show the amplitude tracking and its error in more detail and demonstrate that the tracking delay is approximately 1ms,

contrasting with the

corresponding figures 165A and 1658 for the RST technique where the tracking delay is shown to be approximately 5ms. In Figure 415A, the true amplitude is exactly 0.75V at

t=

2.0 seconds, while the tracked amplitude reaches 0. 75V at approximately 2.001 s, 1 ms later, corresponding to the expected delay. The equivalent delay shown in Figure 165A is 5ms. Note also that the errors in Figure 4168 do not exceed approximately 3.5e-4 V, while the maximum error in Figure 1658 is approximately 1.35e-3 V, a considerably larger value.

- 291 -



[001331] Thus heterodyning is a useful technique, when combined with Prism signal

processing, to reduce tracking delay. An alternative use of heterodyning in combination with Prism signal processing may be to reduce the frequency of an input signal. A simple example illustrates the concept. Suppose the sampling rate is 48 kHz, and the input signal to be processed has a main frequency component with a frequency f of approximately 12 kHz. With only around four samples per period of the component to be tracked, it would be difficult to apply Prism signal processing in this case. However, if a heterodyning signal is applied with a frequency of say, 11.5 Kz, then the resulting output with have components of 500 Hz (h - f) and 23.5 kHz (h + f). The lower frequency component may be tracked using a conventional RST, for example, where the higher component will be naturally attenuated due to its relatively high frequency. This provides a means of tracking frequency and amplitude of the 12 kHz signal, but further calculations will be necessary in order to obtain the phase of the original signal, if required. Tracking Rapidly Changing Amplitude [001332] A common problem in signal processing is the tracking of rapid amplitude

changes, often in cases where the frequency of the signal is known and relatively constant. One such example is the monitoring of an AC power supply, where the signal frequency (for example 50 Hz or 60 Hz) may remain relatively constant, whereas changes in the amplitude of the signal may be rapid and may need to be tracked accurately with minimum time delay. A second example is the tracking of ultrasonic pulses, for example in ultrasonic flowmeters, where once more the frequency of the signal to be tracked may be known to reasonable accuracy, while the amplitude of the signal may vary rapidly with time. [001333] Figure 417 shows an example of an exponentially decaying signal 4170. Here the

= 50 Hz, while the amplitude of the sinusoid = 0.25V at t = 0.25 s. The sample rate used is

frequency of the sinusoid remains constant at f drops from A

= 1 V at t = 0.1

s to around A

48 kHz. Equation (2), which describes a sinusoidal signal with constant amplitude, may be replaced with the following equation in order to incorporate an exponential amplitude decay: - 292 -



[001334] s(t) = Aoe-at sin¢(t)

Equation (99)

[001335] where Ao is the amplitude at time

t = 0, a is the exponential term, and r/f.J) is the

time varying phase with constant frequency fas defined in Equation (3) above. [001336] To provide an analysis independent of the specific frequency of the sinusoid, it is

useful to define an Amplitude Reduction Factor (ARF): [001337] ARF

= a/f

Equation (100)

[001338] For example, the ARF in Figure 417 is 0.2. Note that this definition remains valid

for exponentially increasing sinusoids, in which case the ARF value is negative. [001339] Figure 418 shows a Prism-based signal processing system 4180 which may be

used to track a sinusoid s(t) 4181 with rapidly varying amplitude. It consists of two RSTs 4182, 4184 and a further processing block 4186. The upper RST (denoted "RST1 ") uses the conventional value of h

=1

for its tracking Prism, in order to generate one set of

parameter estimates 4183 as described above. The lower RST (denoted "RST2") uses h

=

2 for its tracking Prism in order to generate a separate set of parameter estimates 4185. Equivalent results to Equations (58) - (61) may be derived for an RST calculation where h

= 2.

The purpose of having the second RST is that, when tracking an exponentially

decaying sinusoid, the parameter estimates of the RSTs differ. Accordingly, an additional processing block may be used to analyse the estimates of frequency, amplitude and phase from the two RSTs and perform detection and correction, thereby yielding corrected estimates 4187 of the instantaneous frequency, amplitude and phase of s(t), denoted as f·(t), A·(t) and

~·(t)

respectively.

[001340] Figure 419A (graph 4190) shows the true amplitude of the input signal from

Figure 417, together with the amplitude calculated by RST1 when the signal 4170 is passed through the signal processing scheme 4180 of Figure 418. Note that in this case, given that the frequency is known to be approximately 50 Hz, the value of m used in both

- 293 -



RSTs 4182 and 4184 is 100 Hz. Figure 4198 (graph 4192) shows the corresponding error in the amplitude estimate generated by RST1. The amplitude calculation appears to lag behind the true value, resulting in relatively large errors. Figure 420A and 4208 (graph 4200 and graph 4202) show the equivalent results for RST2, with essentially similar results. [001341] Figure 421A and 4218 (graph 4210 and graph 4212) show the RST1 phase

tracking performance. Figure 421A shows the RST1 calculated value together with the true phase value, while Figure 421 B shows the corresponding error. The RST1 phase error has a non-zero, positive mean value, and exhibits an oscillation at twice the frequency of the input signal s(t). [001342] Figure 422A and 4228 (graph 4220 and graph 4222) show the corresponding

results for the RST2 phase tracking performance. Figure 422A shows the RST2 calculated value together with the true phase value, while Figure 4228 shows the corresponding error. By contrast with the RST1 phase error of Figure 421 B, the RST2 phase error in Figure 4228 has a negative mean value, while also exhibiting an oscillation at twice the frequency of the input signal s(t). [001343] Figure 423A and 4238 (graph 4230 and graph 4232) show the signal tracking

performance of RST1 for this example. Figure 423A shows the true and tracked values of the signal, while Figure 4238 shows the error in the tracked signal, which exceeds 0.1 V, or 10% of the signal value, at the start of the simulation. [001344] Tracking of the exponentially decaying signal may be improved through

calculations based upon the difference between the RST1 and RST2 phase values. Figure 424 (graph 4240) shows the phase difference between RST1 and RST2, which has a mean value of approximately 0.017 radians, and oscillates with an amplitude of approximately 0.005 radians at twice the frequency of the input signal s(t). Tthe phase difference between RST1 and RST2 provides a means of estimating the ARF, and hence of correcting for the effects of exponential amplitude decay.

- 294 -



[001345] Figure 425 (graph 4250) shows how the average phase errors of RST1 and RST2 vary with ARF. The mean RST1 error shows an approximately linear increase with ARF, while the mean RST2 error shows an approximately linear decrease with ARF. As the negative slope of the RST2 error is approximately twice the positive slope of the RST1 error, these results indicate a means of providing a corrected phase, as follows:

[001346] ~c

= (2~1

[001347] where

+ ~2

~c,

)

Equation (101)

/3

the corrected phase, is calculated from a simple linear combination of

the observed phases

~1

~2

and

from RST1 and RST2 respectively. The results of this

correction will be shown below.

[001348] Figure 426 (graph 4260) shows how the mean phase difference between RST1 and RST2 (calculated using

~1 - ~2)

varies with the ARF; the plot shows a near linear

relationship. Accordingly, for an observed phase difference between RST1 and RST2, the ARF may be estimated, for example using a simple polynomial model. Additional calculation steps, understood by those familiar with the art, may be used to compensate for the modulation observed in the instantaneous phase difference (for example as shown in Figure 424 ), noting that the phase of the modulation is readily computed from the (corrected) phase of the input signal, while the amplitude of the modulation is also readily modelled from the observed phase difference.

[001349] Thus, given the observed phase difference between RST1 and RST2, the ARF factor may be calculated, and further, for example polynomial, corrections may be applied to calculate a corrected amplitude for the input signal.

[001350] Figures 427A - 4298 (graph 4270, graph 4272, graph 4280, graph 4282, graph 4290 and graph 4292) illustrate how such corrections may improve signal tracking in the case of the input signal given in Figure 417. Figure 427A shows the true and corrected phase, while Figure 4278 shows the corresponding error in the corrected phase. This stays within± 1.5e-3 radians, a significant reduction in error compared with the RST1 phase error

- 295 -



(Figure 4218) which varies within ± 2e-2 radians. Figure 428A shows the true and corrected amplitude, while Figure 4288 shows the corresponding error in the corrected amplitude. This stays within ± 1e-3 V, a significant reduction in error compared with the RST1 amplitude error (Figure 4198) which initially exceeds 0.1V. Finally, Figure 429A shows the true signal and the corrected tracked signal, while Figure 4298 shows the corresponding error in the corrected tracked signal. The error remains within ± 1.5e-3 V, while the RST1 tracked signal error (Figure 4238) exceeds± 0.1 V. [001351] A further example illustrates that this technique may perform well when tracking

other types of amplitude variation. Figure 430 (graph 4300) shows a 50 Hz signal with an average amplitude of 1V, but where the amplitude modulates± 0.25 V with a frequency of 3 Hz. Using the same signal processing system shown in Figure 418, and applying the same technique used to generate the results of Figures 427A - 4298, reductions in tracking error are achieved compared with a conventional RST. Note that this example demonstrates the correction technique operating over a continuously changing amplitude reduction factor (ARF), including both positive and negative values. [001352] Figures 431A (graph 4310) shows the true amplitude and the amplitude estimate

generated by the RST1 tracker. Figure 4318 (graph 4312) shows the corresponding RST1 error, which varies between ± 0.05 V. There is a clear delay seen in the tracker's response to the constantly varying amplitude. Figure 432A (graph 4320) shows the true phase and the phase estimate generated by the RST1 tracker. Figure 4328 (graph 4322) shows the corresponding RST1 phase error, which varies between ± 0.01 radians.

Figure 433A

(graph 4330) shows the true signal and the corresponding RST1 tracked signal, while Figure 4338 (graph 4332) shows the error in the RST1 tracked signal, which falls within ± 0.05 V. [001353] Applying the correction technique described above, the phase difference between

RST1 and RST2 is shown in Figure 434 (graph 4340). This manifests an approximately sinusoidal characteristic, reflecting the modulation of the amplitude of the input signal, and

- 296 -



hence facilitating a suitable correction calculation. Figure 435A (graph 4350) shows the true phase of the input signal along with the corrected phase, while Figure 4358 (graph 4352) shows the error in the corrected phase, which falls within ± 0.001 radians, and order of magnitude reduction compared with the error of the RST1 calculation shown in Figure 4328. Figure 436A (graph 4360) contrasts the true and corrected amplitude, with the error in the corrected amplitude shown in Figure 4368 (graph 4360). The error remains approximately within ± 0.005 V, an order of magnitude improvement over the error shown by the RST1 amplitude given in Figure 4318. Finally, Figure 437A (graph 4370) shows the true signal value and the corrected tracked signal, while Figure 4378 (graph 4372) shows the corresponding error. This error remains approximately within ± 0.005 V, an order of magnitude less than the error shown by the RST1 tracked signal given in Figure 4338. The heterodyning technique and the correction for rapidly varying amplitude further demonstrate, but by no means exhaust, the variety of signal processing techniques that may be developed using the properties of the Prism as disclosed herewith.

- 297 -



CLAIMS 1.

A method for filtering an original signal, the method comprising:

passing the original signal through two or more integration paths, each path including a sequence of two or more integration stage blocks; and calculating at least one final output signal, where the at least one final output signal is based on an arithmetical combination of the outputs of at least two integration paths; and whereby: the original signal, the final output signal and at least one intermediary signal consist of a sequence of samples, where a sample is a numerical value, and where consecutive samples represent the corresponding signal over consecutive fixed intervals of time; each integration stage block accepts an input signal and generates an output signal; and in a sequence of integration stage blocks, the output signal of a first block in the sequence forms the input signal to the next block in the sequence. 2.

The method of claim 1, wherein:

each integration stage block has a data window, with an associated length in samples, which contains a sequence of the most recent input samples received by the integration stage block, whereby only those input samples which are contained within the data window may influence the value of the corresponding integration stage block output sample. 3.

The method of claim 2, wherein:

- 298 -



each integration stage block assigns to every newly received input sample a coefficient whereby the sample value is multiplied by the coefficient to generate a corresponding product value; and the product value associated with each input sample remains unchanged for as long as the input sample remains within the data window; and for each newly received input sample, each integration stage block calculates a corresponding output sample by carrying out a numerical integration over the product values corresponding to the input samples in the data window. 4. The method of claim 3, wherein: the total number of coefficients used in each integration stage block corresponds to the length of its data window, irrespective of the length of the input sample sequence; and the coefficients have generally different values. 5.

The method of claim 4, wherein the coefficient assigned to the newest input

sample is equal to that of the oldest sample in the data window. 6.

The method of claim 2, wherein all integration stage blocks have the same

data window length. 7.

The method of claim 3, where the coefficients consist of linearly spaced sine

values. 8.

The method of claim 7, where the set of coefficients for each integration stage

block corresponds to a whole number of sinusoid periods. 9.

The method of claim 8, where all integration blocks use either of two sets of

coefficients: a first set of linearly spaced sinusoidal values ("sine coefficients"), or a second

- 299 -



set ("cosine coefficients") where each coefficient is shifted by n/2 radians compared with the corresponding coefficient in the first set. 10.

The method of claim 9, where there are two integration paths: a first

integration path consisting of a sequence of two integration stage blocks, each with sine coefficients, and a second integration path consisting of a sequence of two integration stage blocks, each with cosine coefficients; whereby the final output signal is based on the sum of the output signals from the two integration paths. 11.

The method of claim 9, where there are two integration paths: a first

integration path consisting of a sequence of two integration stage blocks, the first with sine coefficients and the second with cosine coefficients, and a second integration path consisting of a sequence of two integration stage blocks, the first with cosine coefficients and the second with sine coefficients; whereby the final output signal is based on the difference of the output signals from the two integration paths. 12.

The method of claim 10 where there are an additional two integration paths

and a second final output: the third integration path consisting of a sequence of two integration stage blocks, the first with sine coefficients and the second with cosine coefficients, and the fourth integration path consisting of a sequence of two integration stage blocks, the first with cosine coefficients and the second with sine coefficients; whereby the second final output signal is based on the difference of the output signals from the third and fourth integration paths. 13.

The method of claim 3, where numerical integration is performed using

Romberg Integration. 14.

The method of claim 3, where the numerical integration in each integration

stage block is performed recursively, whereby the result for the current set of input samples in the data window is derived from values calculated for the previous set of input samples in the data window, wherein the calculation includes removing the influence of the oldest data

- 300 -



sample in the previous data window and including the influence of the newest data sample in the current data window. 15.

The method of claim 14, whereby, for a particular instantiation, the

computational burden of performing the filtering calculation is substantially similar irrespective of the length of the data windows. 16.

The method of claim 12, further comprising additional calculations to estimate

one or more parameter values of a sinusoid component within the original signal. 17.

The method of claim 16, whereby the one or more parameter values include

the frequency, amplitude and/or phase of the sinusoidal component. 18.

The method of claim 16, where the parameter estimates are calculated for

each new sample in the original signal. 19.

The method of claim 17, further comprising estimating the amplitude with an

error less than 1.5 times the Cramer-Rao Lower Bound ('CRLB') for selected values of the frequency and a signal to noise ratio of 40 dB via said calculating. 20.

The method of claim 17, further comprising estimating the frequency with an

error less than 1.3 times the CRLB for selected values of the frequency and a signal to noise ratio of 40 dB via said calculating. 21.

The method of claim 17, further comprising estimating the phase with an error

less than 1.3 times the CRLB for selected values of the frequency and a signal to noise ratio of 40 dB via said calculating. 22.

The method of claim 9, wherein said filtering of the original signal is a

constituent operation of a bandpass filter. 23.


constituent operation of a notch filter.

- 301 -


24.



constituent operation of a dynamic notch filter. 25.


constituent operation of a measurement system. 26.


constituent operation of a Coriolis meter. 27.


constituent operation of an ultrasonic flow meter. 28.


constituent operation of a pressure sensor. 29.


constituent operation of an electrical power monitoring system. 30.


constituent operation of a processing mode operable in the Internet of Things. 31.

A system for filtering an original signal, the system comprising:

a filter being capable of receiving the said original signal; and said filter being further capable of passing the said original signal through two or more integration paths, each path including a sequence of two or more integration stage blocks; and said filter being further capable of calculating at least one final output signal, where the at least one final output signal is based on an arithmetical combination of the outputs of at least two integration paths; and whereby:

- 302 -



the said original signal, said final output signal and at least one intermediary signal consist of a sequence of samples, where a sample is a numerical value, and where consecutive samples represent the corresponding signal over consecutive fixed intervals of time; each said integration stage block accepts an input signal and generates an output signal; and in a sequence of said integration stage blocks, the output signal of a first block in the sequence forms the input signal to the next block in the sequence.

32.

The system of claim 31 wherein:

each integration stage block has a data window, with an associated length in samples, which contains a sequence of the most recent input samples received by the integration stage block, whereby only those input samples which are contained within the data window may influence the value of the corresponding integration stage block output sample. 33.


each integration stage block assigns to every newly received input sample a coefficient whereby the sample value is multiplied by the coefficient to generate a corresponding product value; and the product value associated with each input sample remains unchanged for as long as the input sample remains within the data window; and for each newly received input sample, each integration stage block calculates a corresponding output sample by carrying out a numerical integration over the product values corresponding to the input samples in the data window.

- 303 -


34.



the total number of coefficients used in each integration stage block corresponds to the length of its data window, irrespective of the length of the input sample sequence; and the coefficients have generally different values. 35.

The system of claim 34, wherein all said integration stage blocks have the

same fixed window length. 36.

The system of claim 32, wherein all integration stage blocks have the same

data window length. 37.

The system of claim 33, where the coefficients consist of linearly spaced sine

values. 38.

The system of claim 37, where the set of coefficients for each integration

stage block correspond to a whole number of sinusoid periods. 39.

The system of claim 38, where all integration blocks use either of two sets of

coefficients: a first set of linearly spaced sinusoidal values ("sine coefficients"), or a second set ("cosine coefficients") where each coefficient is shifted by n/2 radians compared with the corresponding coefficient in the first set. 40.

The system of claim 39, where there are two integration paths: a first

integration path consisting of a sequence of two integration stage blocks, each with sine coefficients, and a second integration path consisting of a sequence of two integration stage blocks, each with cosine coefficients; whereby the final output signal is based on the sum of the output signals from the two integration paths. 41.

The system of claim 39, where there are two integration paths: a first

integration path consisting of a sequence of two integration stage blocks, the first with sine coefficients and the second with cosine coefficients, and a second integration path

- 304 -



consisting of a sequence of two integration stage blocks, the first with cosine coefficients and the second with sine coefficients; whereby the final output signal is based on the difference of the output signals from the two integration paths. 42.

The system of claim 40 where there are an additional two integration paths

and a second final output: the third integration path consisting of a sequence of two integration stage blocks, the first with sine coefficients and the second with cosine coefficients, and the fourth integration path consisting of a sequence of two integration stage blocks, the first with cosine coefficients and the second with sine coefficients; whereby the second final output signal is based on the difference of the output signals from the third and fourth integration paths. 43.

The system of claim 33, where numerical integration is performed using

Romberg Integration. 44.

The system of claim 33, where the numerical integration in each integration

stage block is performed recursively, whereby the result for the current set of input samples in the data window is derived from values calculated for the previous set of input samples in the data window, wherein the calculation includes removing the influence of the oldest data sample in the previous data window and including the influence of the newest data sample in the current data window. 45.

The system of claim 44, whereby, for a particular instantiation, the

computational burden of performing the filtering calculation is substantially similar irrespective of the length of the data windows. 46.

The system of claim 42, being further capable of estimating one or more

parameter values of a sinusoid component within the original signal. 47.

The system of claim 46, whereby the one or more parameter values include

the frequency, amplitude and/or phase of the sinusoidal component.

- 305 -


48.


The system of claim 46, where the parameter estimates are calculated for

each new sample in the original signal. 49.

The system of claim 47, being further capable of estimating the amplitude

with an error less than 1.5 times the Cramer-Rao Lower Bound ('CRLB') for selected values of the frequency and a signal to noise ratio of 40 dB via said calculating. 50.

The system of claim 47, being further capable of estimating the frequency

with an error less than 1.3 times the CRLB for selected values of the frequency and a signal to noise ratio of 40 dB via said calculating. 51.

The system of claim 47, being further capable of estimating the phase with an

error less than 1.3 times the CRLB for selected values of the frequency and a signal to noise ratio of 40 dB via said calculating. 52.

The system of claim 39, wherein said filter is a constituent part of a bandpass

53.

The system of claim 39, wherein said filter is a constituent part of a notch

54.

The system of claim 39, wherein said filter is a constituent part of a dynamic

filter.

filter.

notch filter. 55.

The system of claim 39, wherein said filter is a constituent part of a

measurement system. 56.

The system of claim 55, wherein said filter is a constituent part of a Coriolis

57.

The system of claim 55, wherein said filter is a constituent part of an

meter.

ultrasonic meter.

- 306 -


58.


The system of claim 55, wherein said filter is a constituent part of a pressure

sensor. 59.

The system of claim 55, wherein said filter is a constituent part of an electrical

power monitoring system. 60.

The system of claim 39, wherein said filter is a constituent part of a

processing mode operable in the Internet of Things.

- 307 -



METHOD AND SYSTEM FOR TRACKING SINUSOIDAL WAVE PARAMETERS FROM A RECEIVED SIGNAL THAT INCLUDES NOISE

ABSTRACT A system for tracking selected wave parameters from a received sinusiodal wave with noise and methods for making and using the same. The method includes performing a multi-track double integral analysis of the sinusoidal wave with noise and creating time dependent outputs.

These time dependent outputs may be analyzed mathematically to

determine the amplitude, frequency and/or phase of the wave with reduced noise. In one embodiment, the method may employ multiple passes through double integral analysis. The method advantageously can measure output sinusoidal wave parameters with reduced noise, measurements that are close to theoretical noise reduction limits.

- 308 -

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=

=

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FIG. 172A

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RST Tracked Freauency Error (Detail

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FIG. 1728

~

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0 1--'

N

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00

RST Tracked Amplitude (Detail

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- - - - True - - - - - - - - - - - - - - I- - - - - - - - - ..___

__..1

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FIG. 173A

1730

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IL

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FIG. 1738

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0

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00

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c

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1

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1740

1

FIG. 174A

..,. .....i

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RST Tracked Phase Error

(,,.)

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c

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IV

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FIG. 1748

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____ __.............-----

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1750

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FIG. 175A

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FIG. 1758

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Noise Free, Noisy and RST Tracked Signal

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c

OJ

3

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1760

FIG. 176A

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FIG. 1768

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and RST Trac,ked

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(Detail

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FIG. 177A

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rn

~

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c

r

rn

1

N

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FIG. 1908

~

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1

1

1

9 9 9 9 9 Q 9 9 9 9 9 Q 9 9 9 9 Q 9 9 Q 9 9 Q 9 9 Q 9 9 9 9

(/)

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1910

FIG. 191A

_..

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rn

w

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r

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FIG. 1918

~

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1

9 9 Q Q Q Q Q Q Q Q Q Q Q Q Q Q 9 Q 9 9 Q 9 Q Q Q 9 9 9 Q

1

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1920

FIG. 192A

_..

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00

FIG. 192B

~

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1

0 x

0 x

1

0

x 0x 0 0 0 x x 0

x

(/)

0

x

0 x

0 x

0

x

0

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x

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0

x

0 x

0

x

0 x

0 x

0 x

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0

x x

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0

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0

x

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1930

FIG. 193A

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w

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9,

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rn

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c

r

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rn

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1932

0

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00

FIG. 1938

~

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N

0 1--'

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Phase CRLB and RST error. SNR

1

= 60 dB 0

9

Q

1

RST

9 9 Q

9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 Q 9 9 9 9 Q 9

(/)

c

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1940

FIG. 194A

_..

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rn

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c

r

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1942

FIG. 1948

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1

Q 1

9

Q

9 9 Q9 9 9

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Q Q Q

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9 Q R9 9 9 9 9 9 9 R9 RR

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1950

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FIG. 195A

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01

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rn

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c

r

rn

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1

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1~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

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1952

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FIG. 195B

~

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1

0 x

f5- 0

1

x

0

x 0x 0 0 x x

(/)

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0

x

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x

0

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1

1960

FIG. 196A

c.o

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9, w

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rn

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c

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FIG. 2188

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FIG. 222A

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....

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I

'

L

\

I

I

\

0

,......_ ._

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0

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N N

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""'O

n

-I rn ""'O N

0

1--'

'-I

2253

0

2254

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'-I

0

FIG. 225

1--' ~

0 1--'

N

0 1--'

00

2261

\

2260 (/)

c OJ

(/)

-I -I

c -I rn

\

s(t)

Dynamic Notch Filtering - Two components 5 1only(t)

sc(t) Calculate

ri = fi(t)/m

r2 = h(t)/m

I(= s (t) + s (t) ) 1

2

Then

N N

I

s 10n 1/t) = sc(t)- P(r2) . sk(t)

~

s20n 1/t) = sc(t)- P(r1) . sk(t)

(/)

rn

,......_ ;:;o

N CJ)

9, .j:>.. w -...J

5 2only(t)

sk(t)

c r rn

CJ)

s 10n1/t) has removed the f 2 component irrespective of its

r -

2262

f l(t)l

'-'

I /

f,(t)

L -

phase and amplitude.

""O

n

S20n1/t) has removed the f 1 component irrespective of its J

phase and amplitude.

-I rn ""O

N

0

1--'

'-I

2263

0

'-I 00

2264 FIG. 226

~

'-I

0

1--' ~

0 1--'

N

0 1--'

00

Weighted combinations of Gc and Gk to create notches at 75.0 Hz and 91.0 Hz

---------(/)

c

OJ (/)

-I -I

c

-I

rn

0

2270

FIG. 227A

""

(/)

Q,

I

rn

-----

~

,......_

--- --------

;:;o

c

r

rn

N CJ) '-'

I\.) I\.)

"' ...

' '\

-...!

I

\

1

.j::>.

w

""'O

I \ I

n

-I

1,

rn

II

""'O N

0

1--'

'-I

0

'-I

0

2272

00

FIG. 2278

~

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

sltl with comoonents at 91 Hz and 75 Hz

(/)

c

OJ (/)

-I -I

c

-I

rn

2280

FIG. 228A

I\.) I\.)

co

(/)

I

g,

Power Soectrum, sttl with cornoonents at 91 Hz and 75 Hz

rn rn

.f::>.

w

--.J

-I

,......_

;:;o

c

r

rn

N

""'O

n

-

CJ) '-'

-I

rn

""'O N

1

0

1--'

'-I

0

'-I

00

2282

FIG. 2288

~

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

- 91 Hz component removed ~~~·--.....~~~~~~~~7!'1;--~~~~~-,~~....~~~-,~~~~7"1;----,

I

(/)

c

OJ (/)

-I -I

c

-I

rn

1.1 2290

FIG. 229A

I\.)

"'9,

CD

(/)

Power Spectrum, s

I

rn rn

1 1ony

•

91 Hz component removed

.j:>..

w

--.!

-I

,......_

;:;o

c

r

rn

N

""'O

n

-

CJ) '-'

-I

rn

""'O N

0

1--'

'-I

0

'-I

00

2292

FIG. 2298

~

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

• 75 Hz component removed

(/)

c

OJ (/)

-I -I

c

-I

rn

1

1

2300

FIG. 230A

I\.)

w

0

(/)

Power Spectrum, s 2ony 1

I

rn rn

-

g,

Hz component removed

-1>-

w

~

-I

,......_

;:;o

c

r

rn

N

""'O

n

-

CJ) '-'

-I

rn

""'O N

1

0

1--'

'-I

0

'-I

00

2302

FIG. 2308

~

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

(/)

c

OJ (/)

1

-I -I

c

-I

rn

2310

FIG. 231A

I\.)

w

(/)

I

3

rn

9, w

Tracked 75 Hz component error

~

.(:>.

-...J

,......_

;:;o

c

r

5

rn

N

""'O

n

CJ) '-'

-I

rn

""'O N

0

1--'

'-I

0

0

1

2312

1

1.1 .. · · - \ - 1

FIG. 2318

'-I

00 ~

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

Tracked 91 Hz component

(/)

c

OJ (/)

1.1

-I -I

c

-I

rn

2320

FIG. 232A

N

w N

(/)

I

9, c.u

Tracked 91 Hz component error

rn rn

.i::. -..J

-I

,......_

;:;o

c

r

rn

N

""'O

n

-

CJ) '-'

-I

rn

""'O N

0

1--'

'-I

0

'-I

00

2322

FIG. 2328

~

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

2331

Dynamic Notch Filtering for four components

sc(t)

2330 (/)

c OJ

(/)

-I -I

\

\

For each frequency components.: I Calculate

s(t) I( =~si(t) )

sit)= s/t) - P(n) . sk(t)

c -I rn

where S-i is a signal with the ith frequency component removed.

(/)

I

rn

~

sk(t)

,......_

;:;o

I

c r rn

s_z(t)

s_ 3 (t)

All other frequency components in s-i have an

Further Processing

Further Processing

Further Processing

f( ) adjusted gain and phase, but compensation it , can be applied in later calculations.

N CJ) '-'

s_ 1 (t)

2332

s_4 (t)

I

Further Processing

A2(t)

-2(t)

A3(t)

-3(t)

A4(t) 4(t)

A1(t)

N

c..v c..v 9, +:>.

c..v )

..

-....i

""'O

n

-I

rn

4(t)

""'O

N

0

1--'

2333

'-I

0

'-I 00

2334

~

'-I

0

I

FIG. 233

1--' ~

0 1--'

N

0 1--'

00

Dynamic Notch Filters for 101.0 Hz, 102.0 Hz, 103.0 Hz and 104.0 Hz

(/)

c

1~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

OJ (/)

0

-I -I

c

-I

rn

2340

FIG. 234A

(/)

~

I

rn

~

I\.)

w .j:::. ~

1

w

--..!

,......_

;:;o

c

r

rn

N CJ) '-'

1

""'O

'''

--101.0Hz -------- 102.0 Hz --103.0Hz

n

"" ""

-I

l

rn

""'O N

I I

0

1--'

1

'-I

110

0

'-I

00

2342

FIG. 2348

~

'-I

0 1--' ~

0 1--'

N

0 1--'

00

Original Signal s(t). Components at 101, 102, 103, 104 Hz.

(/)

c

OJ (/)

-I -I

c

-I

rn

1 ~

2350

...

-

f

FIG. 235A

!\.)

w

(jl

(/)

0 ....,

I

rn rn

w

,......_

"""'

.j:>.

-I

;:;o

c

r

rn

N CJ) '-'

~ 1v 0

II

11

11

II

""'O

n

-I

rn

- -

~

~-

...........,

""'O N

0

1--'

'-I

0

'-I

00

2352

FIG. 2358

~

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

0

c (I) c

c..

0

E N

0

:c

....

~

-.. .

236 of 437

PCT/EP 2017/078 970 - 19.01.2018


1

" ~

s_2 (t): 102 Hz Component Removed

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

(/)

c

OJ (/)

-I -I

c

-I

rn

1

1

1

1.7

1

' ,

2370

2

1

FIG. 237A

I\.)

w

--..!

(/)

0 ....,

I

rn rn

.j>.

w

-...J

-I

,......_

;:;o

c

r

rn

N CJ) '-'

~

Cl

I

II

II

II

""'O

n

-I

rn

.....MnTl T~~r 1

- 1 ..

""'O N

1 ' l l 1n1a..........-

0

1--'

'-I

0

'-I

2372

-

00

'

FIG. 2378

~

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

238 of 437

0

co M N

PCT/EP 2017/078 970 - 19.01.2018


s.)t): 104 Hz Component Removed

(/)

c

OJ (/)

-I -I

c

-I

rn

1

1.

1.2

1

1

1.6

1

2390

1.7

FIG. 239A

(/)

I\.)

w co 0-+,

I

rn rn

.j:>. (;.)

-...!

-I

,......_

;:;o

c

r

rn

N CJ) '-'

..... 1 (I)

II

~

II

II

-0

n

-I

rn

1

-~~·•·••lliflff r ' '

....

-0 N

0

~_

1--'

'-I

0

'-I

00

2392

-·-,·-·-· ·-.;

,.

~-1

FIG. 2398

~

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

"C CD

s0

-

.!!l c: CD c:

-

0

a.

e0

N

u 0

::c

--

0

N

,,...

M

""""

240 of 437

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III

0

,,... N

~

0

.

.i::.

(/)

0 ....,

I

rn rn -I

.j:>. (.;.)

--.J

,......_ ;:;o

c rn

r

N

~

""'O

Il

CJ) '-'

n

-I rn ""'O N

1

rar1w10011M111W11Ml~tl'ln'lr111r ' ·1 ' ' ir n • 1 1 1 1 1

' I

11

I 11 I lf I

0

r ' I 1r11mnn1H1'11'NJMliwlllrilh4llwll1i1111w111..1111..

111u11••m. ........ ... .. ..

1--'

'-I

0

'-I 00

2442

FIG. 2448

~

'-I

0

I 1--' ~

0 1--'

N

0 1--'

00

102 Hz

(/)

c

OJ (/)

1.1

-I -I

c

-I

rn

2

2450

FIG. 245A

(/)

I

9,

102 Hz Component Amplitude Error . ..

rn rn

I\.)

.i::. 01 .i::.

w

'

-...J

-I

,......_

;:;o

c

r

rn

N

""'O

n

CJ) '-'

-I

rn

""'O N

0

1--'

'-I

0

2 2452

FIG. 2458

'-I

00 ~

'-I

0 1--' ~

0 1--'

N

0 1--'

00

102 Hz Component Phase TI

l

l

l

t

l

j;

it

l

(/)

c

OJ (/)

-I -I

2

c

-I

rn

FIG. 246A

2460

(/)

I

0

Phase Error

1

rn rn

I\.) .j:>.

CJ)

.j:>.

w

-....!

-I

,......_

;:;o

c

r

rn ""'O

N

n

-

CJ) '-'

-I

rn

""'O N

0

1--'

'-I

0

'-I

1 2462

1

1

1

1 I

I~

1

1

2

00

FIG. 2468

'-I

~

0 I 1--' ~

0 1--'

N

0 1--'

00

102 Hz Component Signal True

1 0 1 (/)

c

OJ (/)

-I -I

c

-I

rn

1 2470

1

1

2 FIG. 247A

I\.) .j:>. -..,j

Q,

(/)

I

.i:::.

w -..,j

rn rn

-I

,......_

;:;o

c

r

rn

""'O

n

-

N CJ) '-'

-I

rn

""'O N

0

1--'

'-I

0

'-I

00

2472

FIG. 2478

~

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

103 Hz Component Isolated

0

(/)

c

OJ (/)

-I -I

c

-I

rn

1

1

1.4

1.2

' '

2480

1.7

2

1.8

FIG. 248A

I\.) .j:>.

co

(/)

0

I

rn rn

.j:>.

w

-...!

-I

,......_

;:;o

c

r

rn

N CJ) '-'

~ 1

""'O

a

n

11

-I

rn

""'O N

1

'"' .111..1llullL1ill1Jl!IJ!lbllll!.lliWlllPlllWllMIHHmnnf1r '

'I , .., ~J I

~·

r 11

I

l

I

II

'

I

'

I' 1 r ~ ' 1 ·r 11· '

0

11nnnmH1~1H11M1w1W111M1111.J11w111iJ11i.d11.•11 ••11.

1--'

'-I

0

'-I

00

2482

FIG. 2488

~

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

103 Hz Component A111plitude

1 1

.

CD

(/)

I

0_,,

Error

103Hz

rn rn

-!>-

w

3

-I

"""

,......_

;:;o

c

r

rn

N

.J

CJ) '-'

""'O

n

-I

rn

""'O N

0

1--'

'-I

0

'-I

00

2492

i lllll:j \;:::,}

FIG. 2498

~

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00


0

It! 11111111111/ I I II !Ill! HI 1111Iii1111Hlift111111111t11llii111111 H II

(/)

c

OJ (/)

-I -I

c

-I

rn

1

1

1

2500

FIG. 250A

I\.)

01

(/)

0

I

rn rn

0 ...,

-I

w

,......_

"""'

.j:;.

;:;o

c

r

rn

N CJ) '-'

v

ro

""'O

n

f'I

-I

rn

""'O N

0

1--'

'-I

0

1

2502

1

1

1

1

2 FIG. 2508

'-I

00 ~

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

103 Hz Component Signal

1 0 1 (/)

c

OJ (/)

'"'

-I -I

c

-I

rn

... 1

1

1

1

1 fi

1

2 FIG. 251A

2510

(/)

0_,,

103 Hz Component Si nal Error

I

rn rn

.1111

I\.)

01 _.

.j:>.

w

ll!th

~

-I

,......_

;:;o

c

r

rn

""'O

n

N CJ) '-'

-I

rn

""'O N

0

1--'

'-I

0

1

2512

1

'Lb

1.4 I

Iv \'3/

.0

I f

.o

1

2 FIG. 2518

'-I

00 ~

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

"'C

0

m

(I)

0

J!l c(I) c

0

a. E

N

0 ~

::c: 0 "F"

0

N

"": .....

252 of 437

PCT/EP 2017/078 970 - 19.01.2018


5

104 Hz

(/)

c OJ

(/)

-I -I

c

-I rn

1

1.1

1

1

1

1

2530

2 FIG. 253A

I\.)

01

w

(/)

104 Hz

I

rn rn -I

0

Error

.j:>.

w

10

"'-I

,......_ ;:;o

c rn

r

N

""'O

CJ)

11111111,

'-'

n

.111111111.

.1111111111

-I rn ""'O N

0

1--'

'-I

1

2532

1.1

1

1

2 ... ·-

,-

FIG. 2538

0

'-I 00 ~

'-I

0

I 1--' ~

0 1--'

N

0 1--'

00


0

(/)

c

OJ (/)

-I -I

1

1

c

-I

rn

1

1

1

1

2540

2 FIG. 254A

(/)

I

rn rn

9, ..,..

10

-I

I\.)

01 .i::.

(,,)

-...J

,......_

;:;o

c

r

rn

N

u m

CJ) '-'

-

""'O

n

-I

rn

""'O N

0

1--'

'-I

0

1 2542

.1

1

1

1 Tima le:\

1

1

2 FIG. 2548

'-I

00 ~

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

104 Hz I

I

1 0 1 (/)

c

OJ (/)

-I -I

1

1

1

c

-I

rn

2550

FIG. 255A

(/)

I

rn rn

2

-I

l!11i.

()1

9,

104 Hz Component Si nal Error

,......_

I\.)

01

.j:>.

.>!

illllih.

w

--J

;:;o

c

r

rn

N CJ) '-'

""'O

n

-I

rn

""'O N

0

1--'

'-I

0

'-I

00

2552

~

FIG. 2558

'-I

0 1--' ~

0 1--'

N

0 1--'

00

ca

·-·;::cm 0

N

256 of 437

I.()

w

m

PCT/EP 2017/078 970 - 19.01.2018

,,... N

I.()

-

w

......i

-I

,......_

;:;o

c

r

rn

N CJ) '-'

""'O

n

-I

u..

rn

1

""'O N

0

1--'

'-I

110

115

0

'-I

00

2572

FIG. 2578

~

'-I

0 1--' ~

0 1--'

N

0 1--'

00

79.9 Hz Component Amplitude

(/)

c

OJ

16

(/)

-I -I

c

-I

rn

1

1

1

1

1

1

2580

2 FIG. 258A

I\.)

01

co

(/)

79.9 Hz

I

rn rn

0

Error

-!>-

w

-...!

-I

,......_

;:;o

c

r

rn

N

lllWl•IHllWJIAIJ-lllWJll•MlllbJlllJllll'lljlllllilll-111•

CJ) '-'

""'O

n

-I

rn

""'O N

0

1--'

'-I

0

'-I

1 2582

1

1

1

1

2 FIG. 2588

00 ~

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

79.9 Hz Component Phase

(/)

c

OJ (/)

-I -I

c

-I

rn

2590

FIG. 259A

(/)

79.9 Hz Component Phase Error

I

rn rn

Q,

1

-I

I\.)

01

(0

.j:>. (;.)

-..,J

,......_

;:;o

c

r

rn

N CJ) '-'

""'O

n

-I

rn

""'O N

0

1--'

'-I

1 2592

2 FIG. 2598

0

'-I

00 ~

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

79.9 Hz Component Tracked Signal

(/)

c

OJ (/)

-I -I

c

-I

rn

1

1

2600

2 FIG. 260A

79.9 Hz Component Tracked Sianal Error

(/)

I

I\.)

O'l 0

9,

rn rn

.j:>.

w

-...J

-I

,......_

;:;o

c

r

rn

N

""'O

n

CJ) '-'

-I

rn

""'O N

0

1--'

'-I

1

2602

2 FIG. 2608

0

'-I

00 ~

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

a>

"C

.! 0 .! c: a> c:

-

0

0

E 0 N

:c 0

N

m

N

~ ..-

......

!'-

~

......

N

261 of 437

o

w

-I

--.j

,......_

;:;o

c

r

rn

N

""'O

CJ) '-'

n

-I

rn

""'O N

0

1--'

'-I

2

0

'-I

00

2672

FIG. 2678

~

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

99.7 Hz Component Tracked Signal

(/)

c

OJ (/)

-I -I

c

-I

rn

2

1

2680

FIG. 268A

(/)

99. 7 Hz Component Tracked Signal Error

I

rn rn

I\.)

O'l

co 9, .i::.

w

-....!

-I

,......_

;:;o

c

r

rn

N

""'O

n

CJ) '-'

-I

rn

""'O N

0

1--'

'-I

0

1

2682

2 FIG. 2688

'-I

00 ~

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

'D

sm

0

....

.!!}.

c: CD c:

0

a. E 0

(.) N

::c: 0

~

-

N

.,-

N

269 of 437

PCT/EP 2017/078 970 - 19.01.2018

III

en

.,-

N

.2>

3:

I

I

I

\

I

I

279 of 437

I I

I I

I

I

0

PCT/EP 2017/078 970 - 19.01.2018


N

::c 0 0

N' ::c

0

0

0

c:: cu N ::c

"'O

~

~

0 O')

~

fl)

Cl.>

....c:: f! cu a. cu c::

.!!1. rn

.2>

..

Cl.>

"'O

.c::

Cl.>

.2>

s:

N

I

I I

\

I

0

' ' ''

'\ \

\

I I

0

280 of 437

0 0

co N

\

\

'

\ \

0

PCT/EP 2017/078 970 - 19.01.2018


I

"""'

N

.....

281 of 437

. w


rn rn

--..I

-I

,......_

;:;o

c

r

rn

N

""'O

n

CJ) '-'

-I

rn

""'O N

0

1--'

'-I

1

3012

1

2 FIG. 3018

0

'-I

00 ~

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

71.0 Hz Comp()nent 'frC:lcked §ignal

(/)

c

OJ (/)

-I -I

c

-I

rn

1

3020

FIG. 302A

(/)

I

g,

71.0 Hz Comoonent Tracked Sianal Error

rn rn

.j::>.

w

5

-I

,......_

w

0 N

~

;:;o

c

r

rn

N

""'O

n

CJ) '-'

-I

rn

""'O N

0

1--'

'-I

0

1

3022

1

1

1 :11

j

J'lrJ

1 \.....,./

1

2

FIG. 3028

'-I

00 ~

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

103.0 Hz Component Isolated

(/)

c

OJ (/)

-I -I

c

-I

rn

1

1

1

1.4

1.5

1

1.7

1.8

2

FIG. 303A

3030

w w

0

-

(/)

0

I

rn rn

.j:>.

w

--.J

-I

,......_

;:;o

c

r

rn

I-

N

3:

CJ) '-'

1

(1)

0

1

-0

'

n

-I

rn

-0 N

0

1--'

'-I

0

'-I

00

3032

FIG. 3038

~

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

103.0 Hz Component Amplitude

(/)

c

OJ (/)

-I -I

c

-I

rn

1

2

FIG. 304A

3040

w

0

.j:>.

(/)

9, .j:>. w --..j

103.0 Hz Component Amplitude Error

I

rn rn

-I

,......_

;:;o

c

r

rn

N

""'O

n

-

CJ) '-'

-I

rn

""'O N

0

1--'

'-I

0

'-I

00

3042

FIG. 3048

~

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

(/)

c

OJ (/)

-I -I

c

-I

rn

3050

FIG. 305A

w

0

01

(/)

I

9, -1:>w


rn rn

-I

--..!

,......_

;:;o

c

r

rn

N

""'O

CJ) '-'

n

-I

rn

""'O N

0

1--'

'-I

0

'-I

00

3052

FIG. 3058

~

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

103.0 Hz Comconent Tracked Sianal

(/)

c

OJ (/)

-I -I

c

-I

rn

1

2

.1

3060

FIG. 306A

w

0

Q)

(/)

g,

103.0 Hz Comoonent Tracked Sianal Error

I

rn

.j:>.

w

~

---.I

,......_

;:;o

c

r

rn

N

""'O

n

-

CJ) '-'

-I

rn

""'O N

0

1--'

'-I

0

'-I

00

3062

FIG. 3068

~

'-I

0 1--' ~

0 1--'

N

0 1--'

00

(/)

c

OJ (/)

1

-I -I

c

-I

rn

.1

1

1

1.4

1

1.5

3070

FIG. 307A

w

0

~

(/)

0 ...,

I

rn rn

.!>-

w

--.J

-I

,......_

;:;o

c

r

rn

N CJ) '-'

§ a 1

~

~

~

~

~

""O

n

-I

rn

""O N

0

1--'

'-I

0

'-I

00

3072

FIG. 3078

~

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

Components at 60 Hz and 76 Hz notched

(/)

c

OJ (/)

-I -I

c

-I

rn

1

1

1.3

1.4

1.5

1.6

1.7

1.8

3080

2

FIG. 308A

w 0 co

-

(/)

0

I

rn rn

.j:>.

w

--..!

-I

,....._

;:;o

c

1

r

rn

N CJ) '-'

I.II

3:

n

1

l

l..

""'O

n

-I

rn

""'O N

0

1--'

'-I

0

'-I

00

3082

FIG. 3088

~

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

4

at 60 Hz and 93 Hz notched

,, .

(/)

c

OJ (/)

1

-I -I

c

-I

rn

11

1

1

.4

1

1

1

1.7

3090

2

FIG. 309A

w

0

.

w

--.J

-I

1

,...._

;:;o

c

r

rn

N CJ) '-'

w

~

I

!

""O

n

-I

rn

""O N

0

1--'

'-I

0

'-I

00

3092

FIG. 3098

~

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

4~~~~~~~~~~--'-~~~~~~~~~~~~~~~~~~~~~~

(/)

c

OJ (/)

1

-I -I

c

-I

1.1

1

3100

rn

FIG. 310A

w _.. 0

(/)

0 ....,

I

rn rn

w """' -...j

-I

,......_

;:;o

c

r

rn

N CJ) '-'

§ n

l

!

""'O

n

-I

rn

""'O N

0

1--'

'-I

1

1

0

'-I

00

3102

FIG. 3108

~

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

4.---~~-.--~~-,.-~~~---L~~~~~r---~~,.--~~-,-~~-,-~~-.-~~--,

(/)

c

OJ (/)

-I -I

c

-I

rn

3110

FIG. 311A

w

-

(/)

0

I

rn rn

.J:>.

w

1~-!'i

-I

,......_

.....i

;:;o

c

r

rn

N CJ) '-'

IJ)

~

I

l

""'O

n

-I

rn

""'O N

0

1--'

'-I

0

'-I

00

3112

FIG. 3118

~

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

Comoonents at 76 Hz and 114 Hz notched

(/)

c

OJ (/)

1

-I -I

c

-I

rn

1

1.1

1.4

1.5

1

1.7

1.9

3120

2

FIG. 312A

w _..

N

(/)

0

I

rn rn

.!>-

w -...j

-I

,......_

;:;o

c

r

rn

N CJ) '-'

s:......n

j

""'O

'

n

-I

rn

""'O N

0

1--'

1

a

'-I

1

1

0

'-I

00

3122

FIG. 3128

~

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

c

4

I

I

I

93 H

.

hed

d 114 H

I

I

I

2 ~

A

~

I

I

n

I

I

~

n

0 J

\

u

~

~

u

'

\

(/)

c

I

OJ

1

(/)

-I -I

c

-I

rn

1.1

'I

I

1

I

I

1

I

I

1

1

3130

FIG. 313A

(/)

I

rn

~

,......_

;:;o

w _.. w 9, .J:>. w --..J

1

c

r

rn

N

""'O

n

CJ) '-'

-I

rn

""'O N

0

1--'

'-I

1 0

1

1

0

'-I

00

3132

FIG. 3138

~

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

j ~ ~

,...,..

.. ~ ~.

'..._/

(~

I-

ro ,......

u

~

Q)

,......

314 of 437

.. ~

-ro c

V>

........ ...._....

.....-...

.. ~

(./")

·-bO

(~

,...,..

,...,..

~

'..._/

Q)

> ·ci:::

Q)

u

:J

(/') ,......

'..._/

(~

~

o/ """' M

PCT/EP 2017/078 970 - 19.01.2018


Input Signal before onset of Interference Component

(/)

c

OJ (/)

-I -I

1

c

-I

rn

(/)

1

1.1

.12

1.14

1. 6

3150

1.18

1

FIG. 315A

..,.9,

I

rn

Power Spectrum, Input Signal before onset of Interference Component

~

w _.. 01

VJ -....!

,......_

;:;o

c

r

rn

N CJ) '-'

-0

n

-

1

-I

rn

-0 N

0

1--'

'-I

0

'-I

00

0

~

3152

FIG. 3158

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

1

Input

,

~ignal

after onset of lnterfere,nce Coml?onent

(/)

c

OJ (/)

-1~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

-I -I

110

c

-I

rn

(/)

1

1

3160

FIG. 316A

I

rn

-!:>-

Power 5Dectrum, lmmt Sianal after onset of Interference ComDonent

~

w _.. en 9, (.;.)

-..,J

,......_

;:;o

c

r

rn

N CJ) '-'

""'O

n

-

1

-I

rn

""'O N

0

1--'

'-I

0

'-I

00

0

3162

~

FIG. 3168

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

Primarv Component: Frequencv (Uncorrected

(/)

c

OJ (/)

-I -I

1

c

-I

rn

(/)

1

1

3170

1

FIG. 317A

w _.. ---1

0 ....,

I

rn rn

-!'>

w

--.J

-I

,......_

;:;o

c

r

rn

1

N

I

CJ) '-'

II Illlll III IIllll IIIIllllll IIIIlllll III IIIHrl 1111111111111111111111 lllHll IIIlllll 111111111111111111111111111111 1

••u••ue1 • • ·1-

·rn 1TH 1 1111r I' 1111 11rn 1 11r1 • 111· r 1rn

• • •1

I

n "I rnii1 'ml' l'INN

""'O

n

-I

rn

""'O N

0

1--'

'-I

P!FI I D l P I

. . . . . .. . . ,. . . . . . . . . . . . . . . .!II. . . . . . . . . .

0

u

'-I

00

3172

.... ·-

,-,

~

FIG. 3178

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

Primary C:e>111Ponent: J.\111plitu(fe

(ljncorrecb~(f)

(/)

c

OJ (/)

-I -I

1

1

c

-I

rn

(/)

3180

FIG. 318A

I

rn rn

.p.

Primary C:o111ponent: J\111plitude (U11«;orrected) Error

-I

w _.. co 9, w

-...J

,......_

;:;o

c

r

rn

N

""'O

1

CJ) '-'

n

-I

rn

""'O N

0

1--'

'-I

0

1

'-I

00 ~

3182

FIG. 3188

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

Component: Phase (Uncorrected)

(/)

c

OJ (/)

-I -I

c

-I

rn

(/)

3190

FIG. 319A

I

9, .j::>. w -.,j

rn

~

Primary Component: Phase (ljrH::()rrected) Error

,......_

w _.. (0

;:;o

c

r

rn

N ""'O

CJ) '-'

n

-

1

-I

rn

""'O N

0

1--'

'-I

0

1

'-I

1

3192

FIG. 3198

00 ~

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

1~~~~~~~~~~~-,--'~~~~~,--0~

Uncorrected

(/)

c

OJ (/)

-I -I

c

-I

rn

(/)

3200

FIG. 320A

w

N 0

~

I

rn

.i::..

Primarv Component: Signal (Uncorrected) Error

~

w

-...!

,......_

;:;o

c

r

rn

N CJ) '-'

""'O

n

1

-I

rn

""'O N

0

1--'

'-I

0

1

'-I

00 ~

3202

FIG. 3208

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

RST Frequency estimate before onset of Interference Component

(/)

c OJ

(/)

-I -I

c -I rn (/)

1

1

1.1

1.12

1.14

1.16

3210

1 18

1.2

FIG. 321A Power Soectrum, RST Freauencv estimate before onset of Interference Comoonent

~

N

9, -1'>w

I

rn

w

-..J

,......_ ;:;o

c rn

r

N

""'O

n

CJ) '-'

-I

rn

""'O N

0

r-

-....."All'ftllmmttil'i""".....--

1--'

'.!

0

0

1

1

1

1

'.!

00 ~

3212

FIG. 3218

'.!

0 I 1--' ~

0 1--'

N

0 1--'

00

RST frequency estimate after onset of Interference Component

(/)

c

OJ (/)

-I -I

110

1

18

11

c

-I

rn

(/)

3220

FIG. 322A

I

Power Spectrum, RST frequency estimate after onset of Interference Component

rn

~

w N N

9, .j:>. w --..!

,......_

;:;o

c

r

rn

N

""'O

n

-

CJ) '-'

-I

rn

""'O N

1

0

1--'

'-I

0

'-I

0

00 ~

3222

FIG. 3228

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

RST Amclitude estimate before onset of Interference Comconent

(/)

c

OJ (/)

-I -I

1

c

-I

rn

(/)

1

1.12

1 14

1.16

3230

1.18

1 w N w

FIG. 323A

~

I

rn

Power

~

estimate before onset of Interference

.j:>.

w

--..J

,......_

;:;o

c

r

rn

N CJ) '-'

""'O

n

- 1

-I

rn

""'O N

0

1--'

'-I

0

0

1

1

'-I

00 ~

3232

FIG. 3238

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

RST Amplitude estimate after onset of Interference Component

(/)

c

OJ (/)

-I -I

c

-I

rn

(/)

3240

FIG. 324A

w N

~

9,

I

rn rn

Power Spectrum, RST Amplitude estimate after onset of Interference Component

-I

+>-

(;.)

-....!

,......_

;:;o

c

r

rn

N CJ) '-'

""'O

!llJ

n

-

1

-I

rn

""'O N

0

1--'

'-I

0

'-I

00

0

~

3242

FIG. 3248

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

3254

I -""'

Estimate

A

f

Prime

-...

(/)

c

p

(t)

...-

Frequency

OJ (/)

-I -I

A

fb (t)

c

-I

rn

(/)

I

rn

~

,......_

;:;o

c

r

rn

RST

3 250

\

-.

f(t) Recursive s(t)

calculate A

A

tracking beat

fbr (t)

.

mean & std;

-.

Tracker

A(t) _

.

CJ) '-'

test

.j::>.

w

detection flag

-.

-....J

A

¢(t)

N

frequency

w I\.) 01

Q,

threshold

A

Signal

-...

~

....

I

I

3252

3253

""'O

n

-I

rn

""'O N

3251

0

1--'

'-I

0

'-I

00 ~

'-I

0 1--'

FIG. 325

~

0 1--'

N

0 1--'

00

Mean of Beat Frequency

1

1

1

1

(/)

c

OJ

3260

FIG. 326A

(/)

-I -I

c

-I

w

rn

N Ol

(/)

9, w

I

rn

.p..

~

--.J

1

,......_

;:;o

1

c

1

1

r

rn

N

3262

FIG. 3268

""'O

n

CJ) '-'

-I

~

I

I

Diagnostic State -

rn

""'O N

0

1--'

'-I

0

I

1

I

1

I

1

-

1

'-I

00 ~

'-I

0

3264

FIG. 326C

1--' ~

0 1--'

N

0 1--'

00

3254

I Estimate Prime Frequency (/)

c

OJ (/)

-I -I

c

-I

rn

(/)

I

rn rn

-I

,......_

3250

\ s(t)

I

RST tracking Recursive Signal

beat

H

c

Dynamic

!;,

Notch

mean & std;

Filtering

threshold

frequency

test

I

and detection

Tracking

flag

Tracker

;:;o

r

"'''"'"e ,

I

3252

3253

rn

---.!

0

.j:>..

w

---.!

¢, s(t)

I

_.j

N

~

w I\.)

""'O

n

CJ) '-'

-I

rn

3270

""'O N

0

1--'

'-I

0

'-I

00 ~

'-I

0 I 1--' ~

FIG. 327

0 1--'

N

0 1--'

00

Primary Component: Frequency (Corrected)

(/)

c

OJ (/)

-I -I

1

1

1

.5

1

c

-I

rn

(/)

3280

FIG. 328A

I

rn

Primary Component: Fntquency (Corrected) Error

~

w N co 9, .j:>.

w

-....!

,......_

;:;o

c

r

rn

N

""'O

n

-

CJ) '-'

-I

rn

""'O N

0

1--'

'-I

0

'-I

00 ~

3282

FIG. 3288

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

Prima - - - - True

---Cale

(/)

c

OJ (/)

-I -I

1

1

1

c

-I

rn

(/)

FIG. 329A

3290

w

N CD

0

I

rn rn

.i:::.

w

--.J

-I

,......_

;:;o

c

r

rn

N

1

CJ) '-'

""'O

11111111111111111111111111.

111

i

1 Ill J .• I••• L

I

I. • 1

~I

I

I

I

I

~ ....

I

,.1

Ll. " 1l ... IL II

u

Ii

u

n

-I

rn

""'O N

0

1

u••••u••PpPR

, . . . , , .. . . . . .. . , ,• • m 1 • n , w • •

111'1

I

1111

I

I

llltB••.,..w

-

-

..

·---- -

--

1--'

'-I

0

'-I

00 ~

3292

FIG. 3298

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

(/)

c

OJ (/)

-I -I

c

-I

rn

(/)

3300

FIG. 330A

w w

0

I

9,

~

.f:>.

rn

Primary C()111ponent: Phase ((:orrected) Error

,......_

(,.) -._J

;:;o

c

r

rn

N ""'O

CJ) '-'

n

-

1

-I

rn

""'O N

0

1--'

'-I

0

1

'-I

00 ~

3302

FIG. 3308

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

1,---~~~~--r-~~~~~-,---"--~--"-~----,-~~--''--~-,---.

Interference Component: Amplitude Error

~

w

-.J

,......_

;:;o

c

r

rn

N CJ) '-'

""'O

1

n

-I

rn

""'O N

0

1--'

'-I

0

'-I

1

00 ~

3332

FIG. 3338

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

Interference Component: Phase

(/)

c

OJ (/)

-I -I

1

c

-I

rn

(/)

w w

FIG. 334A

3340

""'0_,, w ""'

I

rn rn

Phase Error

Interference

-I

-...J

,......_

;:;o

c

1

r

rn

N

""'O

n

CJ) '-'

-I

rn

""'O N

0

I

1

1- I

II

'I '''IH 11 I -

1 1r ,.. 11 r

·

1 11 '

1--'

11

'Ill' 'I 111

'-I

0

'-I

1

1

3342

' '

1

1

00 ~

FIG. 3348

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

2

·-

Interference

1 oi--~~~~~~~~~~

(/)

c

OJ (/)

-I -I

c

-I

rn

(/)

3350

FIG. 335A

9, .j:>. w

I

rn

~

w w

01

Interference Comp()nent: Signal Error

~

,......_

;:;o

c

r

rn

N CJ) '-'

""'O

n

-

1

-I

rn

""'O N

0

1--'

'-I

0

'-I

00 ~

3352

FIG. 3358

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

1 II

-------1

071-

(/)

c

Cale

"

OJ (/)

-I -I

1

8

c

10

12

16

18

-I

rn

(/)

FIG. 336A

3360

-I

0

I

rn rn

w w

0)

.i;:.

Error

, --?

w

-...I

,......_

;:;o

c

r

rn

N

""'O

n

-

CJ) '-'

-I

rn

""'O N

0

1--'

'-I

0

'-I

8

3362

10

12

16 .

~

- ,-,

00

18

~

FIG. 3368

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

Primary Component: Amplitude (Corrected)

(/)

c

OJ (/)

-I -I

8

0

12

16

18

c

-I

rn

(/)

3370

FIG. 337A

Prima

~

-...J

9, .j::>.. w

I

rn

w w

Component: Amplitude (Corrected) Error

---.I

,......_

;:;o

c

r

rn

N CJ) '-'

""'O

1

n

-I

rn

""'O N

0

1--'

'-I

0

1

'-I

00 ~

3372

FIG. 3378

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

(/)

c

OJ (/)

-I -I

c

-I

rn

(/)

3380

FIG. 338A

w w co

g,

I

rn

.j:>..

w

Primarv Component: Phase (Corrected) Error

~

-..,J

,......_

;:;o

c

r

rn

N

1

""'O

n

-

CJ) '-'

-I

rn

""'O N

1

0

1--'

'-I

0

1

'-I

00 ~

3382

FIG. 3388

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

1

~~~~~~~~~~~~~~~T--~~~'-,-~~~~_,,__~~~~~~~~~~

(/)

c

OJ (/)

-I -I

8

c

10

12

16

18

-I

rn

(/)

FIG. 339A

3390

9, -!>o w

I

rn

~

w w

.

g,

I

rn

.j:>.

Interference Component: Amplitude Error

~

w

-.,J

,......_

;:;o

c

r

rn

1

N

""'O

n

-

CJ) '-'

-I

rn

1

""'O N

0

1--'

'-I

0

1

'-I

00 ~

3412

FIG. 3418

'-I

0 1--' ~

0 1--'

N

0 1--'

00

Interference Component: Phase

(/)

c

OJ (/)

-I -I

c

-I

rn

(/)

FIG. 342A

3420

I

Interference Component: Phase Error

rn

~

w

.J:>.

""9, +:>.

w

""""

,......_

;:;o

c

r

1

rn

N

""'O

n

-

CJ) '-'

-I

rn

1

""'O N

0

1--'

'-I

1

0

'-I

00 ~

3422

FIG. 3428

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

Interference Component: Sianal

(/)

c

OJ (/)

-I -I

8

c

10

12

4

16

18

-I

rn

(/)

FIG. 343A

3430

w .J:>. w

0

I

rn rn

Interference

-I

.J:>.

Error

w

--.J

,......_

;:;o

c

1

r

rn

N CJ) '-'

""'O

n

-I

~ !...

rn

- 1

""'O N

0

1--'

'-I

0

1

'-I

00 ~

3432

FIG. 3438

'-I

0 1--' ~

0 1--'

N

0 1--'

00

344 of 437

u.. 0

u..

!....

a. ro

c

"'O

N

+I

LL

(.!)

~ ~ ("')

PCT/EP 2017/078 970 - 19.01.2018


3440

(/)

c OJ

\

3454

S1(t)

(/)

-I -I

c

-I rn

Bandpass

Deep

Filter

Signal Tracker

(/)

I

rn rn -I

,......_ ;:;o

c r rn N

CJ) '-'

162 ± 16 Hz

S2(t)

\ 3442

/ fc(t) Coriolis Frequency

w .j::>. 01 0

..... .i;.

w

m

\ 3450

=328 Hz

\

3452

....i

""'O

n

-I rn ""'O N

0

1--'

'-I

0

'-I 00 ~

'-I

0

1--'

FIG. 345

~

0 1--'

N

0 1--'

00

L

r

L

346 of 437

- QJ -··V1

u..

M

"d"

(£)

PCT/EP 2017/078 970 - 19.01.2018

:c 0 N II

..,


Bandpass Gain vs. Frequency. Centre Frequency = 115.0

1

" ""'

(/)

c

OJ

0

(/)

-I -I

1

0

c

1

-I

rn

(/)

I

I V''-!UCI !VJ

\I IL.J

FIG. 347A

3470

w

.J:>.

~

0

I

rn rn

.i:::.

w

--.J

-I

,......_

;:;o

c

r

rn

1

N CJ) '-'

~ I\

(I)

1

r-1

1 1

'"'

(\(\( ~ \ !\ I

\I

II

I

\

\A ( I

""O

"

n

-I

~

rn

""O N

\

I

\

I

\

/4

0

1--'

'-I

0

1

'-I

1

00 ~

3472

FIG. 3478

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

Bandpass Gain vs. Frequency. Centre Frequency= 162.0

(/)

c

OJ

0

(/)

-I -I

1

0

c

-I

rn

(/)

FIG. 348A

3480

w .J:>. co 0 .....,

I

rn rn

w """' -.._J

-I

,......_

;:;o

c

r

rn

N CJ) '-'

-

I

/

'\

/

\

/\ I

\I

\

I

\

""'O

n

-I

rn

""'O N

0

1--'

'-I

1

IL

0

'-I

00 ~

3482

FIG. 3488

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

Sensor 1 Signal (incl noise)

A

0

f

AA AA AAA AA AAriAA AAA AA AA A AA AA

fl

Ah AA

A

~I. 1} 1j V~ V~ ~ V~ V~ ~ V~ V~ ~ V~ V~ ~ ~ ~ V·v ·v· ·~ ·v ·v· ·1; ·v· ·u·~

(/)

c

OJ (/)

-I -I

1

1.1

1

.15

.2

1

1.3

c

-I

rn

(/)

(f)

------r-----~-----r~----~---------

9, -i:>. w

I

rn

- - - Power Spectrum, Sensor 1 Signal (incl noise)

~

w

FIG. 349A

3490

.J:>.

--..!

,......_

;:;o

c

r

rn

N

1

""'O

n

CJ) '-'

-I

rn

""'O N

0

1--'

'-I

0

50

150

200

250

0

'-I

00 ~

3492

FIG. 3498

'-I

0 1--' ~

0 1--'

N

0 1--'

00

Sensor 1 Signal+ Sensor 2 Signal

1

0 1 (/)

c

nnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnr

vvvvv v v v v v v v v v v v v v v v v v v v v v v v v v v v v

V-

OJ (/)

-I -I

1

c

-I

rn

(/)

I

I

1.1

1 5

I

1

3500

1.3

FIG. 351A

w

01

Q,

I

rn

Power Spectrum, Sensor 1

~

.j::>.

+ Sensor 2 Signal

w

-.,J

,......_

;:;o

c

r

rn

N

""'O

CJ) '-'

n

-

1

-I

rn

""'O N

0

1--'

'-I

0

'-I

00 ~

3512

FIG. 3518

'-I

0 1--' ~

0 1--'

N

0 1--'

00

Sensor1 I

I

0 (/)

c

OJ

I

(/)

-I -I

I

I

I

1.1

.2

I

1

1

c

-I

rn

(/)

3520

FIG. 352A

w

01 N

Q,

I

rn

Power Spectrum, Sensor 1 Signal - Sensor 2 Signal

~

.j:::.

w

--.J

,......_

;:;o

c

r

rn

1

N

I

II

II

I ""'O

n

-

CJ) '-'

-I

rn

""'O N

0

1--'

'-I

0

'-I

00 ~

3522

FIG. 3528

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

Tracked Drive Mode Frequency

(/)

c

OJ (/)

-I -I

1

.1

1.2

1.4

1

1

.6

1

.8

1.9

c

-I

rn

(/)

3530

w

FIG. 353A

01

w

0 ....,

I

rn rn

Error

Tracked Drive Mode

.j:>.

w

--.J

-I

,......_

;:;o

c

r

rn

N CJ) '-'

-

2

lfl' l'll.1l ~ ~ 1'i\I\ ~ ~ 111 A.

I. ~

Ii\' I T ,.ll.111

1.

1.11u ~, w·, r \t • , 1111"~.1111 jJl '~ I

""'O

n

-I

rn

""'O N

0

1--'

'-I

0

1

.1

1.2

1.3

1.4

1.5

.6

1.7

.8

1.9

2

'-I

00 ~

3532

FIG. 3538

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

Tracked Coriolis Mode Frequency

1

(/)

-

1

c

OJ (/)

-I -I

1

1.2

1

.6

1.4

.7

1.9

1

2

c

-I

rn

(/)

w

FIG. 354A

3540

01

+:>. 0

I

rn rn

Tracked Coriolis Mode

-I

.!>-

Error

w

---i

,......_

;:;o

5

c

r

rn

N

~ fHl JUl i1 nII JI" rl nJ ~Hll I, uII n~~ ru ~J lll f, HII JI rt rl HJ lfl uI~ uII nnn1i lfl ,, I~" 'u

CJ) '-'

J

""O

n

-I

rn

""O N

0

-5 HI II ii I

II 1 I I I I II I Ill I I I II I Ir I I I I I 'I I I I II II I I I II ,. I I Ir II II Ill I I 11 Ill II

11'1

1--'

'-I

0

1

1.4

1.5

.6

.7

1.8

1.9

2

'-I

00 ~

3542

FIG. 3548

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

.U;JUUUL

Tracked Drive Mode Amplitude Sensor 1 ~--~---~---1-·----i-------,-------1----·

~--~---~---~

(/)

c

OJ (/)

-I -I

1

c

-I

rn

(/)

1.2

1.3

1.4

1

.6

1.7

.8

3550

1.9

2 w

FIG. 355A

01 01 0_,,

I

rn rn

.j:>.

Tracked Drive Mode

-I

,...._

w

Sensor 1 Error

--.j

;:;o

c

r

rn

N CJ) '-'

u.o

rl I l ll I l I AI l ~ ' I af\ 11~ m~ I~ l I l Ii~ IJ1l.1~ .111 m~ ~ I n' I ~ Ail .I J\) l1 I rl1

""O

n

-I

rn

""O N

0

1--'

'-I

0

'-I

1

3552

1

1.3

1.4

.6

1 \

1.7

.8

1

FIG. 3558

2

00 ~

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

Tracked Drive Mode Amplitude Sensor 2

(/)

c

OJ (/)

-I -I

1

1.4

1.9

2

c

-I

rn

(/)

w

FIG. 356A

3560

01

O'l

..,.9,

I

rn rn

Tracked Drive Mode Amplitude Sensor 2 Error

w

-...I

-I

,......_

;:;o

c

r

rn

N

""'O

n

CJ) '-'

-I

rn

""'O N

0

1--'

'-I

0

1

1

1.3

1.4

.7

.8

1.9

2

'-I

00 ~

3562

FIG. 3568

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

Tracked Coriolis Mode Amplitude Sensor 1

(/)

c

OJ (/)

-I -I

1

1.2

1

1.4

1.7

.8

1.9

2

c

-I

rn

(/)

3570

FIG. 357A

w

01 -...!

~

I

rn

.j::.

Tracked Coriolis Mode Amplitude Sensor 1 Error

~

w

-...!

,......_

;:;o

c

r

rn

N

""'O

n

-

CJ) '-'

-I

rn

""'O N

0

1--'

'-I

0

'-I

00 ~

3572

FIG. 3578

'-I

0 1--' ~

0 1--'

N

0 1--'

00


Sensor 2

(/)

c

OJ (/)

-I -I

1

c

1.2

1.3

.8

1

-I

rn

(/)

1

2 w 01 co

FIG. 358A

3580

0

I

rn rn


-I

-!>-

Sensor 2 Error

w

-.J

,....._

;:;o

c

r

rn

N

111111111~111.J.JI l.dllM.1.l111iJ1l.l1 •.• L..HJ..11l111ll lltul~lll1.1•1 ..• 1•1 ILnll J lu~ 1llJtlll1llM1JI .l.1.~ ulad. 11l~UU1.ll. 1U

CJ) '-'

""'O

n

-I

rn

""'O N

0

1--'

'-I

0

1

1.2

1.3

1.4

.5

.6

.7

.8

1.9

2

'-I

00 ~

3582

FIG. 3588

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

Tracked Drive Mode Phase Sensor 1 ~~~-,-----~~-----,-~~~-----.--~~-------,

(/)

c

OJ (/)

-I -I

c

-I

rn

(/)

3590

w 01 co

FIG. 359A

~

I

rn rn

-1>-

Tracked Drive Mode Phase Sensor 1 Error

-I

w

-.._J

,......_

;:;o

c

r

rn

N

""'O

n

CJ) '-'

-I

rn

""'O N

0

1--'

'-I

0

1

1.2

1.3

1.4

.5

1.6

1.7

1.8

1.9

2

'-I

00 ~

3592

FIG. 3598

'-I

0 1--' ~

0 1--'

N

0 1--'

00

Tracked Drive Mode Phase Sensor 2

(/)

c

OJ (/)

-I -I

c

-I

rn

(/)

3600

w en

FIG. 360A

I

rn rn

0

g,


.i;:..

w

-I

-...J

,......_

;:;o

c

r

rn

N

-0

n

CJ) '-'

-I

rn

-0 N

0

1--'

'-I

1

1.2

.3

1.4

.5

1

.B

1.9

2

0

'-I

00 ~

3602

FIG. 3608

'-I

0 1--' ~

0 1--'

N

0 1--'

00

Tracked Coriolis Mode Phase Sensor 1

(/)

c

OJ (/)

-I -I

c

-I

rn

(/)

3610

w en

FIG. 361A

I

9, w

Tracked Coriolis Mode Phase Sensor 1 Error

rn

~

.i:::. ---i

,......_

;:;o

c

r

rn

-- 5

N CJ) '-'

""'O

p:!

n

0

-I

rn

""'O N

-5

0

1--'

'-I

1

.1

1.2

1

1.4

.5

1.6

1.8

1.9

2

0

'-I

00 ~

3612

FIG. 3618

'-I

0 1--' ~

0 1--'

N

0 1--'

00


(/)

c

OJ (/)

-I -I

c

-I

rn

(/)

3620

w en

FIG. 362A

N

9,

I

rn

.j:>.


~

w

~

,......_

;:;o

c

r

rn

N

""'O

n

CJ) '-'

-I

rn

""'O N

0

1--'

'-I

0

1

1.2

1.3

1.4

1.5

1.6

1.8

1.9

2

'-I

00 ~

3622

FIG. 3628

'-I

0 1--' ~

0 1--'

N

0 1--'

00

Tracked Drive Mode Phase Difference

(/)

c

OJ (/)

-I -I

1

1.1

1

.4

.5

1

1

.8

1.9

2

c

-I

rn

(/)

3630

w en w

FIG. 363A

0

I

rn rn

Tracked Drive Mode Phase Difference Error

.J:>.

w

.....i

-I

,......_

;:;o

c

r

rn

N

0

CJ) '-'

lllllU1llll'llU,lll1'rft1/111R\1t1,ml1llrrn. llllll1IVl'ml-.llll'IJ'IQ' "lm',1'1111'1

-0

n

-I

rn

-0 N

0

1--'

'-I

1

1.1

1.2

1.3

.4

.5

1.6

1.7

.8

1.9

2

0

'-I

00 ~

3632

FIG. 3638

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

Tracked Coriolis Mode Phase Difference

(/)

c

OJ (/)

-I -I

1

c

1.1

1

1

1.4

1.5

1

1

-I

rn

(/)

1.9

2 w en .j::>.

FIG. 364A

3640

..,.0_,,

I

rn rn

Tracked Coriolis Mode Phase Difference Error

-I

(;j

-..J

,......_

;:;o

c

r

rn

N

" IJUilJUIHllWll ~1mH~MUl1 UHIUlldlHIMlll L.1M .1Jllli.lllH.lh. ~IL~ IJl1H 1.JWllL. 111.ltilUdJ llUllM8IUlU1~llJJI

CJ) '-'

""'O

n

-I

rn

""'O N

0

1--'

'-I

0

1

1.1

12

.3

.4

1.5

1

1.7

1.9

2

'-I

00 ~

3642

FIG. 3648

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

365 of 437

u

E

rn

c

PCT/EP 2017/078 970 - 19.01.2018


366 of 437

u

0

0

c

--

PCT/EP 2017/078 970 - 19.01.2018

c Q)

:J

l..L


-LL

367 of 437

LL

--

PCT/EP 2017/078 970 - 19.01.2018



1

1 I

(/)

c

OJ (/)

-I -I -I

rn

(/)

1.2

1

c

1.4

1

.6

1

.8

3680

1.9

2 w en co 0 ......

FIG. 368A

I

rn rn

-I

,....._

+:>.

Error

Tracked Drive Mode

(;.)

-...J

4

;:;o

c

r

rn

N CJ) '-'

0

nrum1m11m1111111vmn~HIWIURIJ11fti1Ut,l11t~lifnm~ll-

""'O

n

-I

rn

""'O N

0

1--'

'-I

0

1

.1

1.2

1.3

1.4

.5

.6

.7

.8

1.9

2

'-I

00 ~

3682

FIG. 3688

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

Tracked Coriolis Mode Frequency

(/)

c

OJ (/)

-I -I -I

rn

(/)

1.9

1.3

c

3690

FIG. 369A

2 w en

CD

Q,

I

rn rn


-I

10 "..,."U

_.,.

Error

(;.)

--.J

4

,......_

;:;o

c

r

rn

N

""'O

n

-

CJ) '-'

-I

rn

""'O N

0

1--'

'-I

0

'-I

00 ~

3692

FIG. 3698

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00


(/)

c

OJ (/)

-I -I

1

1.2

1.3

1.4

1

1

1

2

c

-I

rn

(/)

3700

w

FIG. 370A

--..I 0

0

I

rn rn

Tracked Drive Mode

-l'>

Sensor 1 Error

VJ -..,J

-I

,......_

;:;o

c

r

rn

N

~l11l8llh1ilj111Jl1l111.lla1Ud ll,

CJ) '-'

mm 111111.-•.1. 6.l.iil. i..uf w.u 11 RIL Ii lllli..~1.11 llil nu. ILi

""O

n

-I

rn

""O N

0

1--'

'-I

0

1

1.2

1.3

1.4

.6

1.7

.8

1.9

2

'-I

00 ~

3702

FIG. 3708

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00


(/)

c

OJ (/)

-I -I

1

c

-I

rn

(/)

.1

1.2

1.3

1.4

.5

1.6

3710

1.9

FIG. 371A

2 w

-....!

g,

I

rn rn

.i::.

Tracked Drive Mode Amplitude Sensor 2 Error

-I

w

-....!

,......_

;:;o

c

r

rn

N

-0

n

-

CJ) '-'

-I

rn

-0 N

0

1--'

'-I

0

'-I

00 ~

3712

FIG. 3718

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00


(/)

c

OJ (/)

-I -I -I

rn

(/)

1.2

.1

1

c

1.3

1.4

.6

1

.8

3720

2

1

w

FIG. 372A

--..I N

0

I

rn rn

w """

Tracked Coriolis Mode A.mplitude Sensor 1 Error

-I

-...J

,......_

;:;o

c

r

rn

5

N

0~1j11'1~l.ll1~M,11~warihila11~~Mi

CJ) '-'

I

-5

11' I '.

I'

I

I

I

In

I

I

•

.

. I

II'

. I'

~

""'O

n

-I

rn

""'O N

0

1--'

'-I

0

1

1.2

1.3

1.4

.5

.6

.7

.8

1.9

2

'-I

00 ~

3722

FIG. 3728

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00


(/)

c

OJ (/)

-I -I

1

c

-I

rn

(/)

'1

1.2

1.3

1.4

2

1

3730

FIG. 373A

I

rn rn

Tracked Coriolis Mode Amplitude Sensor 2 Error

-I

w --..I w 9, ~ w -....!

,......_

;:;o

c

r

rn

N

""O

n

-

CJ) '-'

-I

rn

""O N

0

1--'

'-I

0

'-I

00 ~

3732

FIG. 3738

'-I

0 1--' ~

0 1--'

N

0 1--'

00


(/)

c

OJ (/)

-I -I

c

-I

rn

(/)

3740

w

FIG. 374A

--..I

.p.

I

S2,

rn rn

.j:>.


-I

w

--..I

,......_

;:;o

c

r

rn

N

""'O

n

CJ) '-'

-I

rn

""'O N

0

1--'

'-I

0

'-I

1

3742

1

1

1

1

FIG. 3748

2

00 ~

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00


(/)

c

OJ (/)

-I -I

c

-I

rn

(/)

3750

FIG. 375A

I

w

--..I

()1

9,

rn rn

.... w


-I

--..I

,......_

;:;o

c

r

rn

N

""'O

n

CJ) '-'

-I

rn

""'O N

0

1--'

'-I

0

1

.1

1.2

1

1.4

.5

1.6

1.7

1

1.9

2

'-I

00 ~

3752

FIG. 3758

'-I

0 1--' ~

0 1--'

N

0 1--'

00


(/)

c

OJ (/)

-I -I

c

-I

rn

(/)

3760

FIG. 376A

w

--..I

Q)

Q,

I

rn


~

.j:>.

VJ -....j

,......_

;:;o

c

r

rn

N

""'O

n

CJ) '-'

-I

rn

""'O N

0

1--'

'-I

0

1

1.2

.3

1.4

1.9

2

'-I

00 ~

3762

FIG. 3768

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00


(/)

c

OJ (/)

-I -I

c

-I

rn

(/)

3770

FIG. 377A

w

--..I -..-.i

~

I

rn

~

w


~

--.J

,......_

;:;o

c

r

rn

N

-0

n

CJ) '-'

-I

rn

-0 N

0

1--'

'-I

0

1

.1

1.2

1.3

1.4

.5

1.6

1.7

1.8

1.9

2

'-I

00 ~

3772

FIG. 3778

'-I

0 1--' ~

0 1--'

N

0 1--'

00


(/)

c

OJ (/)

-I -I -I

rn

(/)

1.1

1

c

1

1

1

3780

.8

1

2 w --..I co

FIG. 378A

0_,.,

I

rn rn

-I

-1:>-

w


1

--..I

,......_

;:;o

c

r

rn

_lllr\!.N~ 111~ .MIMl.illlllllMllllMl~l.ll a, ~Ulullll1ll~ijA~l Ill llll .M~Uli l JllllJl.Mll~,'-lull lltilllll bll

N CJ) '-'

""'O

n

-I

rn

""'O N

0

1--'

'-I

0

1

1.1

1

.3

1.5

1

1.7

1

2

'-I

00 ~

3782

FIG. 3788

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00


(/)

c

OJ (/)

-I -I

1

1.2

1

A

1.7

1

1

.9

2

c

-I

rn

(/)

3790

w

FIG. 379A

--..I

(!)

0

I

rn rn

.:::. w

Tracked Coriolis Mode Phase Difference Error

-I

--..I

,......_

;:;o

c

r

rn

N

" ~IUllllljllllll~lllilMlftllJI lllllrtlllllh~llll~llLlllBM~~~llHIM~ll dll1Udd.lllh\1lalJlllUwJJt.lllml~IL'IJlllmMHIJ

CJ) '-'

""'O

n

-I

rn

""'O N

0

1--'

'-I

0

1

1

A

1.5

1.6

1

1

.9

2

'-I

00 ~

3792

FIG. 3798

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

Processor For

• • (/)

c

OJ (/)

to

to

-I -I

c

-I

converters

rn

(/)

I

rn

~

•

Phase

•

Phase difference

• • • •

Mass flow f)pnc;itv

I

I Pmm:ir;:it11rP

I

etc

I

w co

0

0

.i:::.

(.;.)

---.I

,......_

;:;o

c

r

rn

N

""'O

n

CJ) '-'

-I

rn

""'O N

Coriolis Flowtube

0

1--'

'-I

Generator to simulate

0

'-I

00 ~

'-I

0 1--' ~

FIG. 380

0 1--'

N

0 1--'

00

Sensor 1 Signal

(/)

c

OJ (/)

-I -I

11

.1

.2

1

.3

c

-I

rn

(/)

3810

FIG. 381A

w co 9,

I

rn

-1>-

Power Spectrum, Sensor 1 Signal

~

w

--.!

,......_

;:;o

c

r

rn

N CJ) '-'

1

""'O

n

-I

rn

""'O N

0

1--'

'-I

0

'-I

00 ~

3812

FIG. 3818

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

Sensor 2 Signal I ~

~

I

~

A

~

-

I

n

A

n

A

-

0

u

(/)

c

\i

\

u

~

\

~

\

OJ (/)

I

-I -I

c

-I

rn

(/)

I

I

1.1

.1

I

.2

3820

I

1

FIG. 382A

9, .J:>. w

I

rn rn

Power Spectrum, Sensor 2 Sianal

-I

w co !\.)

---.!

,......_

;:;o

c

r

rn

N CJ) '-'

'- 1

""'O

n

-I

rn

""'O N

0

1--'

'-I

0

0

00

'-I

150

00 ~

3822

FIG. 3828

'-I

0 1--' ~

0 1--'

N

0 1--'

00

Sensor 1

+ Sensor 2 Signal

0

(/)

c

OJ (/)

-I -I

1

1.1

.2

1

c

-I

rn

(/)

3830

FIG. 383A

I

rn

~

Power Spectrum, Sensor 1

w co w 9, .i:::.

w --.j

+ Sensor 2 f;ignal

,......_

;:;o

c

r

rn

N CJ) '-'

""O

n

-

1

-I

rn

""O N

0

1--'

'-I

0

'-I

00 ~

3832

FIG. 3838

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

Sensor 1 Signal - Sensor 2

0

(/)

c

OJ (/)

-I -I

1

c

1

1.1

1 5

.2

-I

rn

(/)

1

FIG. 384A

3840

w co .j::>.

g,

I

_.,.

rn

~

w

Power Spectrum, Sensor 1 Signal - Sensor 2 §ignal

--.I

,......_

;:;o

c

r

rn

N

1

""'O

n

-

CJ) '-'

-I

rn

""'O N

0

1--'

'-I

0

'-I

00 ~

3842

FIG. 3848

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00


(/)

c

OJ (/)

-I -I

1 1.1

c

-I

rn

(/)

.2

1

1.4

2

1

3850

w co

FIG. 385A

01

~

I

rn rn

.j::.

w

Tracked Drive Mode Frequency Error

-I

,....._

-..!

;:;o

c

r

rn

N

""'O

n

-

CJ) '-'

-I

rn

""'O N

0

1--'

'-I

0

'-I

.2

3852

1.9

FIG. 3858

2

00 ~

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

1

Tracked Cor.iolis Mode Frequency

1

(/)

c

OJ (/)

-I -I

1.1

1.2

1.4

1

.8

1.9

2

c

-I

rn

(/)

3860

FIG. 386A

w co

0)

g,

I

rn

.j:>.

Tracked Coriolis Mode Frequency Error

~

w

--..!

,....._

;:;o

c

r

rn

N

-0

n

-

CJ) '-'

-I

rn

-0 N

0

1--'

'-I

0

'-I

00 ~

3862

FIG. 3868

'-I

0 1--' ~

0 1--'

N

0 1--'

00

Tracked Drive Mode Amplitude on Sensor 1

(/)

c

OJ (/)

-I -I

1

c

-I

rn

(/)

1.2

1

1A

.6

3870

1

2

FIG. 387A

Tracked Drive Mode Amplitude on Sensor 1 Error

~

-..J

9, .j:>. w

I

rn

w co

-..J

,......_

;:;o

c

r

rn

N

""'O

n

CJ) '-'

-I

rn

""'O N

0

1--'

'-I

0

1

.1

1.2

1.3

1A

1.9

2

'-I

00 ~

3872

FIG. 3878

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

Tracked Drive Mode Amplitude on Sensor 2

(/)

c

OJ (/)

-I -I

1

c

-I

rn

(/)

.1

1.2

1.3

1.4

.6

1

1.9

3880

2 w co co

FIG. 388A

-..,. 0

I

rn rn

Tracked Drive Mode

-I

on Sensor 2 Error

w

--..!

,......_

;:;o

2

c

r

rn

N

0

CJ) '-'

WI '8\ilW".. MA .Lift 1Jlll M..1 ll'~M ,N'ntt\»WJM .J'.tHl\t W ~~ d.ltll. •IU.11.till~

""'O

n

-I

rn

""'O N

0

1--'

'-I

0

1

.1

1.2

1.3

1.4

.6

1.7

.8

1.9

2

'-I

00 ~

3882

FIG. 3888

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

Tracked Coriolis Mode Amplitude on Sensor 1

(/)

c

OJ (/)

-I -I

1

c

-I

rn (/)

.1

1.2

1

1.9

.6

1.4

2 w co co 9,

FIG. 389A

3890

.J::>.

I

rn

w

Tracked Coriolis Mode Amplitude on Sensor 1 Error

~

-....I

,......_

;:;o

c

r

rn

-0

n

N

-I

CJ) '-'

rn

-0 N

0

1--'

'-I

0

1

.1

1.2

1

1.4

.5

.6

.7

1.8

1.9

2

'-I

00

~

'-I

3892

FIG. 3898

0 1--' ~

0 1--'

N

0 1--'

00

Tracked Coriolis Mode Amplitude on Sensor 2

(/)

c

OJ (/)

-I -I

1

c

-I

rn

(/)

1.2

1.3

1.4

1.8

3900

1.9

2

FIG. 390A

w

.


-I

w

on Sensor 2 Error

--.J

,......_

;:;o

3

c

r

rn

N

""O

n

-

CJ) '-'

-I

rn

""O N

0

1--'

'-I

0

'-I

00 ~

3902

FIG. 3908

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00

I

I

I

391 of 437

I

I

I

I

PCT/EP 2017/078 970 - 19.01.2018

I


c: 0

392 of 437

I

I

I

PCT/EP 2017/078 970 - 19.01.2018


393 of 437

PCT/EP 2017/078 970 - 19.01.2018


394 of 437

PCT/EP 2017/078 970 - 19.01.2018



(/)

c

OJ (/)

-I -I

.1

c

1.2

1

1.4

-I

rn

(/)

2

1.5

FIG. 395A

3950

w


,......_

-..!

;:;o

c

r

rn

N

-0

n

-

CJ) '-'

-I

rn

-0 N

0

1--'

'-I

0

'-I

00 ~

3952

FIG. 3958

'-I

0 I 1--' ~

0 1--'

N

0 1--'

00


(/)

c

OJ (/)

-I -I

1

c

-I

rn

(/)

1.1

.2

1.3

1.4

1.5

1

2

1

3960

FIG. 396A

w

ut

1

0.5 0 (/)

-0.5

c

OJ (/)

-I -I

c

4130

-1

-I

0.5

1.5

2

2.5

3

3.5

4 .j::>.

rn

FIGURE 413A

(/)

I

Power Spectrum, Sinusoid lnout

rn

~

_.

w 9, .J::>. w -...I

,......_

;:;o

c

r

rn

1

N

""'O

n

-

CJ) '-'

-I

rn

""'O N

0

1

1--'

'-I

0

4132

'-I

00

FIGURE 4138

~

'-I

0 1--' ~

0 1--'

N

0 1--'

00

Tracked Sianal

(/)

c OJ

(/)

-I -I

c -I rn

.5

4140

0.5

1

1.5

2

2.5

3

3.5

4 .j::>.

...... ~

(/)

0_,,

I

rn rn -I

,......_ ;:;o

c rn

.j:::.

w

--.J

1

r

N CJ)

-0

0

n

'-'

-I rn -0

-1

N

0

1--'

'-I

-2

0.5

1

1.5

2

2.5

3

3.5

4

4142

0

'-I 00 ~

'-I

FIGURE 4148

0

1--' ~

0 1--'

N

0 1--'

00

1.2

,---i-----i-----i-~~~~~~---i------.-----,-----

(/)

c

OJ (/)

-I -I

0.4

-I

4150

c

rn

0.5

1

2

.5

2.5

3

3.5

4 .j::>.

...... 01 0

(/)

I

rn rn

-I

,....._

Tracked

....,

FIGURE 415A

Error

.i:::. v:i

""

4

;:;o

c

r

rn

3

N CJ) '-'

~

2

""'O

n

-I

rn

""'O N

0

0

1--'

'-I

0

-1

4152

'-I

1

.5

2

4

3

00 ~

'-I

FIGURE 4158

0 I 1--' ~

0 1--'

N

0 1--'

00

Tracked Amplitude (Detail

0.751

--------True

0.7505 ....

-

-

·~

0.75

.,

-----

(/)

c OJ

(/)

0.7495

1.u ms

-I -I

c

-I rn

4160 0.749

1.999

1.9995

2

2.0005

2.001

2.0015

2.002

(/)

I

9,

FIGURE 416A

rn rn -I

.j::>.

VJ

Tracked Amplitude Error (Detail

1

,....._ ;:;o

.j::>.

...... (J)

--.j

3.5

c rn

r

3

N

""'O

n

-

CJ) '-'

-I rn

2.5

""'O N

2

0

1--'

'-I

1.5

0

'-I 00

4162 1.999

1.9995

2

2.0005

2.001

2.0015

FIGURE 4168

2.002

~

'-I

0

1--' ~

0 1--'

N

0 1--'

00

N

II

Q

LL i:::r::

o(

ci

:::s

'ti ~

a. E m c m

u

·~ Cl>

'ti

)-\

-0

m :;:::; c Cl> c

a.

)(

....

Cl> .t::.

ca

-'ic ti)

417 of 437

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4183

4182

""

"" .... I

OJ

-I -I

c -I rn (/)

I

rn rn

I;

1

""

I

I

I I . I

___ /

Tracker

(/)

c

- - - -I

Recursive Signal

(/)

4186

4181

(h

""

= 1)

I "

•I

r; I

Detection and

I

correction of

-----

S\-1

-- - - -

-I

,......_ ;:;o

c r rn N

I

CJ) '-'

I •I

1:12

)•

....I

Recursive

I

:"'* ) : "' .j::>.

......

exponential

a::i

change in

w

amp I ude

9, ..,.. ""-I

·----""'O

n

Signal

-I

Tracker (h

4180

I

4187

I

4184

= 2)

rn

""'O N

I~;--

0

1--'

'-I

0

'-I 00

4185

~

FIGURE 418

'-I

0

1--' ~

0 1--'

N

0 1--'

00

RST 1 Amolitude

(/)

OJ

-I -I

------- -

- - - - --- - -

I

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