sampling and digitisation case study

Good Practice in Signal Processing: sampling and digitisation case study D Georgakopoulos and T J Esward

Introduction When one samples and digitises an analogue signal one produces in the digital domain a signal that is an approximation to the original analogue signal. If sampling rates are too slow or the quantisation is too coarse, the digital signal will have the appearance of a “staircase” and the frequency content of the digitised signal will be different from that of the original analogue signal, as the sharp transitions of the edges of the “staircase” will introduce spurious frequency components into the spectrum. The reverse problem - generating an analogue signal from a digitised source - presents similar problems. Analogue signals obtained from digital arbitrary waveform generators, where a signal is generated digitally and then filtered to produce an analogue output, may also not possess the required frequency content if the original digital signal is undersampled or inadequately quantised. The case study summarised here is concerned with the generation of an analogue signal with a specific harmonic content that can be used as a calibration signal for a dynamic fatigue testing machine. Inadequate sampling rates are shown to introduce higher harmonic distortion in the required analogue waveform.

The problem The example application which is considered is the calibration of dynamic force measurement machines or fatigue testing machines. Fatigue testing machines are used to apply time varying forces to a specimen under test. Many industrial force measurements are carried out in the field of materials testing. There are established standards for the static calibration of tensile and compressive testing machines but not for the calibration of fatigue testing machines that apply dynamic forces. To fill this need, BSI has produced a draft British Standard (BS 7935) “Method for Constant Amplitude Dynamic Force Calibration”. Part 1 of BS 7935 covers the “Calibration of non-resonant uniaxial dynamic testing systems” and Part 2 the “Calibration of proving devices to be used for the dynamic calibration of non-resonant uniaxial dynamic testing systems” [2]. The first part of the standard is complete but at the time of writing, the second part of this standard was undergoing consultation and had not yet been finalised. This second part is concerned with the method of “calibration of the calibration device instrumentation to be used for the dynamic calibration of non-resonant uniaxial dynamic testing systems”. As the draft standard points out, in a dynamic test the true force experienced by the test-piece can differ significantly from the force indicated by the test system. This difference arises from inertia 1

force acting on the load cell and from errors in the electronics of the dynamic force indicating system. We are concerned in this case study with this second source of difference, the electronics of the device instrumentation. The methodology for verifying the performance of the machine’s electronics is to provide what is described in the BSI document as a dynamic reference standard, that is, instrumentation providing a traceable ±2mV/V excitation voltage for the calibration device instrumentation. The dynamic reference standard is to be used in two ways. Firstly, it will be used to generate a set of DC voltages and the difference will be determined between the values displayed on the calibration device instrumentation and the nominal values generated by the dynamic reference standard. Secondly, the dynamic reference standard is used to generate a set of AC waveforms in the range from DC to the maximum test frequency with varying amplitudes and offsets. The peak and trough values displayed on the calibration device instrumentation are then compared with the nominal values generated by the dynamic reference standard. To simulate laboratory conditions, the AC calibration is to be repeated with a known amount of harmonic distortion to ensure the instrumentation is capable of measuring such peak and trough values correctly. The task of developing the dynamic reference standard was carried out at NPL. The specification for this device was as follows: The dynamic reference standard enables the amplitude and frequency of the output waveform, and the DC offset value, to be independently set, for the nominal specified impedances. In addition, it enables a specified amount of harmonic distortion to be added to the waveform to allow the performance of the calibration device instrumentation under non-ideal conditions to be determined. The uncertainty in the peak and trough voltages generated by the dynamic reference standard shall not exceed 0.2 % of the range (i.e. peak voltage - trough voltage). In the DC case, the uncertainty of the generated voltage shall not exceed 0.000002 VE (e.g. for an excitation voltage of 10 V, the reference standard shall be capable of generating differential DC voltages in the range from −20 mV to +20 mV , with an uncertainty of 20 muV at specified impedances). For AC calibration of the instrument seven waveforms are to be generated using the dynamic reference standard. The frequency is to be varied over the range of interest and, at a minimum of three discrete frequencies, the peak and trough voltage ratios that are displayed on the instrument under test are to be recorded. The seven calibration waveforms each have a different DC offset in the range from −1.5 mV /V to +1.5 mV /V . A sinusoidal variation is then superimposed on the DC level, The amplitudes of the sinusoidal variation lie in the range from 0.5 mV /V to 2.0 mV /V . In addition, the dynamic reference standard is to allow the AC calibration process to be repeated but with a fixed amount of total harmonic distortion (0.125% ± 0.01%) to mimic realistic laboratory measurement conditions.

The solution NPL’s DC and LF group were asked to build a device that could be employed as the dynamic reference standard that could meet the specification in the draft standard. The approach that was adopted was to generate the required waveforms digitally using a arbitrary waveform generator and then to filter the digitally generated waveform to produce an analogue AC waveform with the 2

Figure 1: Dynamic force standard schematic required DC offset and harmonic distortion. NPL’s engineers chose a fifth-order low-pass filter with linear phase for this purpose. For fuller details of the solution, including experimental results, see [1]. A dynamic force standard was built that can be used to simulate the dynamic characteristics of strain gauge bridges. The principles of operation of the standard were that: 1. Two digital-to-analogue converters (DAC) produce one DC and one AC signal according to the requirements of the BS 7935. 2. The output of each DAC is passed through a programmable gain amplifier (PGA) and a 5th order anti-imaging low-pass filter (LPF)1 with Bessel frequency response2 . 3. The output of the LPF of each channel is applied to a bridge amplifier and the output of each bridge amplifier is then connected to the output nodes of the strain gauge bridge simulator through a resistive attenuator. 4. The attenuator resistors are used to sum the DC and AC signals produced by the DAC and scale the output signals to the appropriate levels. 5. The strain gauge bridge is simulated by four precision, low-drift resistors. A schematic diagram of the system is shown in figure 1 BS 7935 [2] defines one static and two dynamic tests for fatigue testing machines. In the static test DC signals are used and in the dynamic test a single-frequency AC signal, and a multi-frequency AC signal, are superimposed on a DC signal. The standard defines AC signals with fundamental frequency in the range 0-100 Hz. In the single-frequency case, the standard defines exactly what is required. For the multi-frequency case, it is necessary to be able to reproduce the waveform shape shown in Annex A of the standard. However, BS 7935 does not define the exact magnitude and phase of the required test waveform. It requires harmonic distortion to be added to a waveform to 1 An anti-imaging filter is a low-pass filter used at the output of DACs to attenuate frequencies above the Nyquist frequency to eliminate image spectra present at multiples of the sampling frequency. 2 This is a variety of linear filter with a maximally flat group delay (i.e., linear phase response). Analog Bessel filters have almost constant group delay across the entire passband, so that the shape of filtered signals is preserved.

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Harmonic 1 2 3 4 5

Amplitude (% of fundamental) 100 3.1 0.09 0.03 1

Phase (degrees) 0 193.8 54.9 52.5 196.0

Table 1: Harmonic content of multi-frequency calibration waveform as described in BS 7935 Harmonic 1 2 5

Amplitude (% of fundamental) 100 3 1.26

Phase (degrees) 0 180 196.0

Table 2: Harmonic content of multi-frequency DFS calibration waveform used by NPL allow the performance of the calibration device instrumentation under non-ideal conditions to be determined [3]. Table 1 shows the amplitude and phase components of the multi-frequency AC signal that derived from the raw data (provided to us by the BSI) describing the time domain representation of the required distorted signal. To generate and maintain these values in a way that is traceable to the national standards is not an easy task. Table 2 shows the magnitude and phase components of the multi-frequency AC signal actually used in the DFS. BS 7935 requires the measurement of the peak and trough values of the output voltage of the strain gauge bridge. The peak and trough values of the time domain representation of the signals described in tables 1 and 2 agree within 0.1 %, i.e. the waveform generated from the data in table 2 can be used to calibrate fatigue testing machines according to BS7935. The DFS generates the analogue signals required by BS 7935 by using digital techniques, i.e. the DAC generates an AC staircase signal that is passed through an LPF. The accuracy of the generated DFS signal, as against the nominal signal, depends on the sampling frequency of the DAC and the cut-off frequency of the LPF. To obtain confidence in the results of the DFS it is important to model the DAC-LPF combination and to compare the theoretical to the real spectrum. The output of the DFS can be simulated simply by scaling the output of the simulated LPF by the gain of the amplifier and the attenuator. A model based on the theoretical LPF transfer function will deviate from the constructed LPF, especially at high frequencies, owing to the relatively large tolerance of the capacitors (e.g. 5%) used in the filter. Therefore, to overcome this limitation, the model of the LPF was based on the least-square approximation of the measured magnitude frequency response by using a first order transfer function in the frequency range 0-500 Hz. To demonstrate the effects of the sampling frequency and the LPF cut-off frequency on the DFS output signal, consider the graphs shown in figures 2 and 3. For LPF cut-off frequency 400 Hz and fundamental signal frequency 50 Hz, figures 2 and 3 show the simulated DAC output signal

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Figure 2: Dynamic force standard: 1000 samples per second

Figure 3: Dynamic force standard: 10000 samples per second 5

(continuous line) and the simulated filter output signal (dashed line) for sampling rates of 1000 samples/second and 10000 samples/second, respectively. As can be seen from the figure, the lower the sampling rate the coarser the “staircasing”. The amplitude of the third harmonic of the filter output signal for the sampling rate of 1000 samples/second is 3.96 % of the fundamental, while for the sampling rate of 10000 samples/second, it is 0.396 %. For the DFS the peak and trough values of the excitation signal are required. The percentage difference between the simulated filter’s output peak-to-trough voltages for sampling rates 1000 and 10000 samples/second is 0.048 %, which is sufficient to meet the BS 7935 requirements.

Conclusion This brief case study had demonstrated the importance the correct choices of sampling interval and quantisation when converting digital to analogue signals. Similar problems arise when analogue signals are sampled and digiised, as undersampling and quantisation that is too coarse can fail to represent adequately the underlying signal. This study had concentrated on repetitive sinusoidal signals. Broadband transient signals, however, can present an even more challenging task, but this is outside the scope of the work reported here.

References [1] D. Georgakopoulos, J.M. Williams, A. Knott, T.J. Esward, and P.S. Wright. Dynamic characterisation of the electronic instrumentation used in the calibration of fatigue testing machines. IEE Proceedings - Science, Measurement and Technology, 153(6):256–259, 2006. [2] British Standards Institution. Constant amplitude dynamic force calibration - part 2: calibration of device instrumentation to be used for the dynamic calibration of nonresonant uniaxial dynamic testing systems - method, 2004. [3] P. Martin and R.B.D Knight. Components and systems for AC/DC transfer at the ppm level. IEEE Transactions on Instrumentation and Measurement, IM-32:63–72, 1983.

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