2008 Congress on Image and Signal Processing
Modified Lock-in Detection for Extraction of Impressed EEG Signals in Lowfrequency Bounded-EIT Studies of the Human Head Pieter Poolman Electrical Geodesics, Inc. Eugene, OR 97403, USA
[email protected]
Robert M. Frank NeuroInformatics Center 5219 University of Oregon Eugene, OR 97403, USA
[email protected]
resolution finite difference models (FDM) of the human head based on the subject specific co-registered computed tomography (CT) and magnetic resonance imaging (MRI) scans, it was possible to extract three and four tissues conductivities. Because the same EEG spectral range and electrodes are used for bEIT as for measuring EEG, the bEIT procedure provides an efficient, low-cost specification of the electrical volume conduction through head tissues. With densearray bEIT measured as routinely as testing scalp electrode impedance, we can realize the promise of recent biophysics simulations suggesting that, with accurate correction for head tissue conductivity, EEG provides spatial resolution of brain activity that is equal to or better than magnetoencephalography (MEG). Low-frequency EIT studies of the human head (based on frequencies up to a few hundred Hz) differ in a significant way from high-frequency efforts (with frequencies in excess of kHz) in terms of the signal-tonoise ratio (SNR) of the impressed signal component as measured. In particular, the signal quality at low impressed frequencies are worse given multiple sources of other high amplitude signals, e.g. various electrophysiological sources (brain activity, eye blinks, etc.), movement artifacts, electrochemical activity at electrode/skin interfaces, and electromagnetic interference. The proposition of merely increasing the amplitude of the impressed current, in order to boost SNR, is limited by operational standards for ensuring patient safety and comfort. EIT applications usually comply with the International Electrotechnical Commission (IEC) standard [9], which specify a ‘‘patient auxiliary current’’ limit of 100 A from 0.1 Hz to 1 kHz; then 100f A from 1 to 100 kHz where f is the frequency in kHz; and 10 mA above 100 kHz. This standard is based on limitation of the impressed current to 10% of the average threshold of sensation.
Abstract In the paper we describe a method to extract the topography of an impressed current for our bounded electrical impedance tomography (bEIT) studies. The frequency of the impressed current is low (up to a few hundred Hz), and is buried in background EEG and other noise. For the development of the extraction method, special consideration is given to maximize the signal-to-noise ratio. The standard lock-in detection framework, with its remarkable sensitivity at the locked-in frequency, is modified to suit our acquisition and post-processing environment. Simulation results are provided to showcase the accuracy and robustness of the modification in extracting both the amplitude and phase offset of the impressed signal in the face of different types of noise.
1. Introduction Electroencephalography (EEG) is a powerful, inexpensive, and underutilized neurological diagnostic tool. If it were possible to accurately localize the neural sources of EEG activity to specific cortical networks, dense-array EEG could provide new insights in both clinical and research applications. These applications range from localizing seizure onset in neurosurgical planning for epilepsy [1] to identifying the neural foundations of language comprehension in infancy [2]. However, the spatial accuracy of EEG will remain limited because i) mostly simplistic models of the human head (like multi-shell spheres [3]) are commonly used in the inverse procedure of back-tocortex projection [4], and ii) the regional conductivities of the human head tissues are largely unknown. Recently we have shown in our group [5], [6], [7], [8] that using the non-invasive bounded EIT (bEIT) measurements procedure and realistically shaped high-
978-0-7695-3119-9/08 $25.00 © 2008 IEEE DOI 10.1109/CISP.2008.680
Sergei I. Turovets NeuroInformatics Center 5219 University of Oregon Eugene, OR 97403, USA
[email protected]
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topography was extracted by averaging the voltage measurements over the positive and negative pulses (after inverting the response to the negative pulses) at each recording site. For the Hoekema et al. study [11], a freshly removed skull part from a patient was placed in a specially constructed measurement device (to prevent the skull part from cooling and drying), consisting of two matrices of 4 4 electrodes, placed on either side of the skull part. A 1 A 10 Hz current was applied through two electrodes and the resulting potentials on all other electrodes was measured. Very little information is given in the paper about post processing methodology, except that the measured data was bandpass filtered around 10 Hz using FFT. In our case, we have based our EIT studies on the existing NetAmps 300 EEG amplifier (manufactured by Electrical Geodesics, Inc.) and an isolated current generator. The NetAmps 300 platform synchronously digitizes 256 analog channels at 20 kHz and 24 bits, and uses a field-programmable gate array (FPGA) to collate and transfer data in IEEE 1394 (Firewire) format to a computer. The current source is batterypowered and isolated from the amplifier circuitry in order to eliminate leakage current through the amplifier. The frequency and amplitude of the impressed sinusoid are settable via a software interface. The waveform of the impressed current is sensed across a resistor with known value and in series with the impedance load containing the head and injector/sink electrodes. However, the extraction of the impressed current topography from the data, recorded during our EIT experiments, is being complicated by a few factors. As mentioned before, background EEG and other sources of random noise substantially reduce the signal-tonoise ratio of the impressed signal, and degrade the accuracy in the extraction of the impressed topography. Also, high electrode-scalp impedances magnify capacitive effects at the electrode-skin interface. Although the impact of high electrode-scalp impedance, acting as a voltage divider, is negated by the high input impedance of the Net Amps 300, it results in a phase delay between the sensing circuitry of the isolated current source and the EEG (on-scalp) measurements. The nature of differential EEG measurement, with respect to a shared on-scalp reference, could also combine with varying capacitive effects to generate substantially varying phase delays. Furthermore, an upgrade to the waveform generator onboard the current source, which will allow us in the near future to apply multiple frequencies simultaneously, contribute to the need of an extraction algorithm which minimizes crosstalk and interference
For frequencies below 1 kHz, the impressed signal is therefore buried in the background EEG. Recent experiments [10], [11], [12], and [13] on non-invasive conductivity extraction have been based on existing EEG recording systems, and several different algorithms have been published describing attempts to extract the potential distribution generated by impressed current in low-frequency EIT experiments. For example, in the case of Gonçalves et al. [10], current was injected via a pair of scalp electrodes while measuring the potential distribution on the remaining 60+ EEG sensors. The current generator was fed with a sinusoidal signal of 60 Hz and 10 V pp and produced an electric current with the same frequency and wave shape, having an intensity of 10 A root mean square. Data were acquired at a rate of 1250 Hz, using on-line high and low-pass filters at 0.16 and 300 Hz, respectively. Epochs of 105 s were recorded for each injection pair, each epoch consisting of 32 trials of 3.28 s, recorded in sequence. Post processing was based on the fact that the wave shape of the impressed current is known since it is coincident with the wave shape of the signal feeding the current generator. Assuming that a linear relationship exists between the impressed current and measured potential at each electrode, it is possible to correlate the latter with the signal feeding the current generator through a multiplication factor, i.e. through a regression analysis. Finally, multiplication factors, one for each electrode, constitute the impressed current topography to be used in the subsequent EIT analysis. The mean and standard deviation of the multiplication factor for each electrode over all trials are used to classify channels as good or bad. In the paradigm employed by Lai et al. [12], a sinusoidal electrical current was delivered at 50 Hz for 2 seconds and sampled simultaneously at multiple scalp recording sites at 400 Hz. To acquire the impressed topography, the recorded data were first band-pass filtered with cut-off frequencies at 10 and 70 Hz for noise suppression. The effect of the impressed current at each electrode was then calculated at each of the 8 measurement time points per cycle by averaging over the available 100 cycles in the 2 second epoch. Finally, to further improve the signal-to-noise ratio (SNR), these 8 estimates were averaged together. For the method described by Oostendorp et al. [13], the applied current consisted of a 1 A square pulse with a duration of 100 ms, followed by a -1 A pulse with the same duration. The potential at multiple measuring electrodes was sampled at 1250 Hz. Multiple traces were recorded over multiple sessions, from which 48–80 traces free of major artifacts were selected. Subsequently the impressed current
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from non-impressed signal components when demodulating the mixed signal. Our approach for extracting the impressed topography is based on a modified version of lock-in detection performed in software. The details of the method are discussed in the next section. Note that in traditional EIT studies, lock-in detection is usually implemented on the hardware level – see [14].
smult ( t ) = 2sref ( t ) ssys ( t )
(
= 2Asys cos( ref t ) cos ref t + sys
)
+ 2 cos( ref t ) Anoise cos( noise t + noise ) noise
[ (
( )]
)
= Asys cos 2 ref t + sys + cos sys
+ Anoise cos[( ref + noise ) t + noise ]
2. Methodology
noise
+ Anoise cos[( ref noise ) t - noise ]
2.1. Standard lock-in detection
(4)
noise
Since the reference signal and the system signal have the same frequency, the difference frequency is at DC (zero frequency). Applying a low-pass filter to smult retains only the DC term and noise components with frequency near the reference signal:
A lock-in detector takes as input a periodic reference signal and a noisy system signal, and extracts only that part of the system signal that matches the reference signal in frequency [15], [16], [17]. The lockin detector yields remarkable sensitivity at the lockedin frequency, and is capable of discarding the impact of offset errors, 1/f noise, etc. efficiently. To demonstrate the functioning of the lock-in detector, consider a reference signal sref that is a pure sinusoid with frequency ref [16]:
sref ( t ) = cos( ref t )
( )
sfilt ( t ) = Asys cos sys +
A
noise@ref
cos[( ref noise ) t - noise ]
noise@ ref
( )
Asys cos sys
(1)
+
A
noise@ref
(5)
cos( noise@ref )
noise@ ref
and apply it as the input to the system. For a timeinvariant linear system, the response to the sinusoidal reference signal sref will be a sinusoid at the same frequency, ref, scaled by a factor Asys and phase shifted by an angle sys. With the addition of broadband noise, the system signal can then be written as [16]:
(
ssys ( t ) = Asys cos ref t + sys
)
+ Anoise cos( noise t + noise )
with
( ref noise ) 0 If the phase of the reference signal is adjusted to zero out sys, then:
sfilt ( t ) Asys + (2)
noise@ref
cos(˜ noise@ref )
(6)
with
noise
It is apparent that the goal of the lock-in detector is to extract the magnitude of Asys. To accomplish this, the functioning of the lock-in detector is based on the result of the following trigonometric identity:
cos(a) cos(b) = 12 [ cos(a +b) + cos(a - b)]
A noise@ ref
˜ noise@ref = ref noise@ref The amplitude of the filtered signal includes Asys, the amplitude of the system output signal, plus a (significantly reduced) contribution from noise. As the phase of the noise signal will vary randomly with respect to the reference signal, the contribution of noise will tend to disappear when averaging over time. In short, the building blocks of the lock-in detector consist of a phase-adjustable reference signal, multiplication, low-pass filtering, and time averaging (see Figure 1).
(3)
Applying the identity to twice the product of sref and ssys yields [16]:
176
Although the assumption of orthogonality is widely accepted in statistics, it is important to evaluate the validity of the orthogonality assumption in the current context and the impact of its violation on the accuracy of the lock-in detector. To shed more light on the meaningfulness of this assumption, consider the regression of sref and in the frequency domain. The Discrete Fourier transform (DFT) can express any realvalued signal x over N time points as a linear combination of complex harmonics for n = 0,…,N - 1: Figure 1. Flow diagram of a lock-in detector (adapted from [16]).
X0 j n +X e j n + K+ X 1e ( ) 1 1 x(n) = N +X cos n j n +X +1e ( ) + K+ X N1e j N 1 2
1
2.2. Modified lock-in detection
N
It is not always practical, or even efficient, to attempt to phase shift the reference signal in order to zero out the unknown sys. Furthermore, one may be more interested in the ratio of the amplitudes of the system and reference signals, specifically when the amplitude of the reference signal is not unity. To cater to these requirements, a different implementation of the lock-in detector is needed. To start, consider the system signal with noise (t) included:
+ (t )
= hs
(
2
N
2
)
2
N +1 2
N
N-1 2
1
N-1 2
N1 2
N1 2
e jn + e j(-n) = 2 cos(n), (7)
2 1 sref ( n ) = S0 + Sk cos k n + ks N k=1
sys
(
ref Asys
=
Aref
(
(11)
)
with
s ref , s sys s ref ,
k = 2 Nk Sk , ks = amplitude and phase of the kth harmonic of the 1-sided DFT of the reference signal Qk , k = amplitude and phase of the kth harmonic of the 1-sided DFT of the noise signal
s ref , sreft s ref , s sys
)
,
for n = 0,K, N - 1 N-1 2 1 (n) = Q0 + Qk cos k n + k , N
k=1 for n = 0,K, N - 1
Assuming that the reference and noise signals are orthogonal and h is constant over time, the magnitude of h can be estimated from a time-series regression of sref and ssys over interval T (e.g. [10]):
hˆ =
(10)
the complex 2-sided DFTs for sref and can be expressed as real 1-sided DFTs (N odd): N-1
h=
-1
Based on the identity:
with
t =
-1
2
X0 j n 1 x(n) = +X1e j n + K+ X e N j n +X e + K+ X N1e j +1
s sys (t ) = hs ref (t +t ) + (t ) t ref
N
, N even n (9) , N odd n
(8)
s ref , sreft
For this case, the orthogonality constraint, given by the inner product of sref and over N time points for each k, can now be expanded to:
with
sref , = 0
177
mean correct sref to ensure that the DC term S0 (and therefore Q0) does not bias the estimation of h in (8). As in the case of the NetAmps 300, EEG measurements collected from DC amplifiers often contain substantial DC offsets (tens of mV). Finally, if SKQK cos( K ) could be assumed to vanish, then
N1 NS Q + S Q k k 0 0 1 k=1 n= 0 2 N s cos k n + k cos k n + k N1 NS Q + S Q k k 1 0 0 k=1 n= 0 2 N s s 1 2 cos 2 k n + k + k + cos k - k NS0Q0 1 2 s 1 N + Sk Qk 2 0 + N cos k - k
k=1 2S Q 1 0 0 2N +S1Q1 cos( 1 ) + K+ S Q cos (12) N-1 2
sref , =
(
) (
)
N-1 2
=
[ (
=
=
)
N-1 2
[
)]
(
N1 2
hˆ =
k = where, for
=
( )
=
ref K
S S
ref K
(
sys
SKref hSKref ref K
S S
ref K
sys
sys
However, if sys is unknown, by sref:
N1
)
Q0 Sk cos k n + ks = 0
= for i j
sreft has to be replaced
s ref , s ref S
ref K
(hS ) cos ( ) (S ) ref K
sys
ref 2 K
N1
SiQ j cos( i n + is ) cos( j n + j ) = 0,
= hcos ( sys )
n= 0
(13)
s ref , s sys s ref ,
hˆ =
n= 0
n= 0
( ) cos( ) ) cos( ) cos( )
SKref SKsys cos sys 0
=h k
k,i, j = 1,K, N21 :
(
with
sref ,sreft
N1 2
S0Qk cos( k n + k ) = 0 N1
(11)–(12)
sref ,ssys sref ,
with s k
from
= ssys , and sys known, yields:
)]
(
N1 2
( )
sref , ssys = SKref SKsys cos sys
(14)
For the case of sys = 0 , the time-series regression will work. Before we solve the case for sys 0 , it is important to point out that the choice of N only partially solves the problem of spectral leakage, as the broadband nature of the noise will tend to inflate QK, and therefore SKQK cos( K ) too. To trace the effect of spectral leakage, i.e. the capture of signal/noise energy at non-DFT frequencies by multiple DFT components, it is useful to formulate the complete spectral decomposition of a pure harmonic at a given frequency f (or equivalent ) onto the set of DFT frequencies (i.e. k’s). Define:
Recalling that the noise is assumed to be broadband (i.e. all Qk’s nonzero), chances are that the orthogonality constraint will be violated (i.e. sref , 0 ), compromising the sensitivity and robustness of the lock-in detector. Fortunately, the reference signal is assumed to be a pure sinusoid (with frequency ref), offering the possibility of zeroing out most Sk’s. However, spectral leakage will result in nonzero Sk’s, unless N is chosen to ensure that ref coincides with one of the k harmonics (or DFT frequencies), say K, in which case only SK will remain nonzero. To accomplish this for a pure sinusoid, the choice of N must be based on selecting N consecutive time points to span an interval that includes an integer number of cycles; or, in practical terms, and as accurately as possible, to regress over a time interval containing an even number of zero crossings subsequent to the first zero crossing. Please note that, regardless of the choice of N, it is advisable to at least
x ( n) = Ae j(n+ ) , for n = 0,K,N - 1 w k ( n ) = e j n , for n = 0,K,N - 1 k
with
178
(15)
= 2
f fs
k = 2
k N
ensuring the continuity of the projection Pk ( x ) at = k .
f s = sampling frequency [ Hz] then the orthogonal projection of x onto the kth DFT frequency is given by: x ,w k
Pk ( x ) =
w k ,w k 1 N
=
N-1
x (n) wk (n) n= 0
N-1 1 j n Ae j e ( k ) N n= 0
=
N-1 1 = Ae j e j( k ) N n= 0
[
Figure 2. Spectral leakage of A cos(t ) for f =
]
40 to 80 Hz (A = 1000 units) onto DFT sinusoid at 60 Hz (DFT f = 1 Hz).
n
(16)
Recalling the geometric series: N-1
zn = n= 0
1 z N 1 z
From the above result it is clear that the broadband nature of the noise has to be reduced to lower the contribution of non-DFT frequencies to QK. An easy solution is to apply a band-pass filter at ref to the system signal (and to the reference signal, if corrupted by measurement noise) ahead of the computation of the time-series regression. To illustrate spectral leakage, Figure 2 shows how a pure sinusoid (A = 1000 units) at different frequencies (both DFT and non-DFT) contributes to the spectral power of a specific DFT frequency (chosen as 60 Hz for this example). As shown in Figure 3, the leakage from very low frequency components, e.g. due to ramping, spreads across all DFT frequencies, and could easily mask or distort low-amplitude higher frequency components.
(17)
for z 1,
the case k reduces to:
1 1 e jN ( ) Ae j j N 1 e ( ) k
Pk ( x ) =
k
j N2 k
=
-j N2 ( k )
1 e ( )e Ae j j ( ) -j N e e 1 2
k
j N2 k
[ [
1 2
( k )
k
e
-j N2 ( k )
e ( ) e 1 j = Ae j -j N e ( ) e 1 2
e
1 2
( k )
j N2 ( k ) j 21 ( k )
e e
e
j N2 ( k )
e
j 21 ( k )
j N2 ( k ) j 21 ( k )
] ]
(18)
Finally, applying the identity:
e j e-j = j2sin( ),
(19)
the complete spectral decomposition simplifies to:
Ae j , Pk ( x ) = 1 sin[ N2 ( k )] j [ e A 1 N sin[ 2 ( k )]
= k ( k )+ ]
N1 2
, k
(20) Figure 3. Spectral leakage of A cos(t ) for f =
Note that, from L’Hospital’s rule, for close to k:
lim
k
sin[ N2 ( k )] sin[ 12 ( k )]
=N
{0.1, 0.01, 0.001} Hz (A = 1000 units) onto DFT sinusoids (0 to 60 Hz, f = 1 Hz).
(21)
The derivation of the complete spectral decomposition for the case = ref also solves the
179
problem of the estimation of h and sys for sys 0 . In terms of a real-world implementation, the estimation can be accomplished by computing the quotient of the complex DFT coefficients of the band-pass filtered ssys and sref at K = ref (after carefully choosing N, as explained earlier). Note that the computation of the DFT coefficients essentially represents a regression of ssys and sref onto the DFT sinusoids. It can also be shown that the regression of ssys onto the Hilbert transform of sref generates the same result (again, based on a carefully chosen N). To summarize, in the absence of a phase-adjustable reference signal, a high-fidelity lock-in detector can be assembled in software by integrating a band-pass filter, zero-crossing counter, and time-series regression, e.g. via a Fourier transform. The working principle of this modified lock-in detection is based on minimizing the impact of non-DFT frequency components in ssys and sref, and ensuring that the reference frequency ref coincides with a DFT frequency. The data flow through the modified lock-in detector is depicted in Figure 4. Its subsequent implementation in Matlab and application to simulated data is described in the following section.
frequencies onto the reference DFT frequency is negligible. We have designed the lock-in detector to satisfy these requirements by inclusion of the bandpass filter and zero-crossing counter processing steps. To measure the detector’s accuracy in the presence of noise and to validate the contribution of the aforementioned processing steps, we have designed a simulation protocol in Matlab that varies noise type, intensity and lock-in detector implementation. Specifically, the simulation quantifies the lock-in detector’s ability to ascertain the amplitude and phase offset of activity, embedded within a noisy system signal, at a specified reference frequency, while controlling aspects of the detector’s implementation.
3.2. Description of Noise We chose a 25 Hz unity amplitude sine wave, sampled at 1 kHz for 2000 milliseconds, as the reference signal, sref,. We then generated a system signal, ssys, by adding noise, , to a phase-shifted version of the reference signal:
sref = sin( ref nT ), n = 0,...,1999 ssys = sin( ref nT + ) + , n = 0,...,1999.
(22)
To simulate wide-band, narrow-band, 1 f or single-frequency interference, we used the following noise profiles: Gaussian, 1 f , linear ramp and sinusoid. Gaussian noise was generated via the Matlab randn function, which uses the Ziggurat algorithm for generating random variables [18]. The linear ramp was implemented as a continuous linear increase in signal intensity, with the rate of increase determined by the specified SNR. To model 1 f noise, we computed the complex sequence
0 , Xn = 1 e j n , n
Figure 4. Flow diagram of the modified lock-in detector.
3. Simulations
n 0, N 2
{
}
(
)
n 1, N 2 .
(23)
The X n specify the magnitude and phase of the positive frequency DFT harmonics from DC to Nyquist: the 1 factors give the desired frequency
3.1. Introduction We have introduced a software implementation of a lock-in detector designed to measure the amplitude and phase of a signal at frequency ref in the presence of both wide- and narrow-band noise. However, as detailed above, the lock-in detector’s precision is based upon two key requirements: that the reference frequency ref coincides with a DFT frequency, and that the spectral leakage of activity at non-DFT
n
response and the n are random phase offsets between and . Hermitian symmetry ensures the corresponding negative frequency harmonics are given by X n . To reduce frequency bin width and improve computational efficiency, we set N = 216. Applying an inverse FFT to X n and X n yielded a length N time domain signal with the desired 1 f frequency
180
response, from which we selected the first 2000 samples. The sinusoidal noise was modeled by a 20.83 Hz sine wave, with a randomly generated phase offset. We specified a 20.83 Hz frequency for two reasons: it was outside the passband of the lock-in detector’s bandpass filter, attenuated by -40 dB, and it ensured that the 2000 sample noise sinusoid contained a fractional number of cycles. This guaranteed that the sinusoidal noise was not captured by a single DFT harmonic, but rather leaked onto neighboring harmonics to maximize its effect. Note that this noise sinusoid also contains a fractional number of cycles in signal lengths of 1320 and 1400 samples, the relevance of which will be discussed below. Prior to its addition to the phase-shifted reference signal, the noise was scaled to the requested SNR with respect to reference, typically 0 dB or -3 dB in our analysis.
for each pair the DFT coefficients corresponding to the harmonic whose frequency bin was closest to ref.
3.4. Description of the Simulation Throughout the simulation, we fixed the target system gain to 1 and the phase offset to 30o, as illustrated by:
ssys = 1.0 sin( ref nT 30 o ) +
(24)
We tested our modified lock-in detector against 4 noise profiles, : 0 dB Gaussian, 0 dB 1 f , 0 dB sinusoid and 0 dB linear ramp. Note that all the noise profiles have a random component, except the linear ramp, and that his randomness is either explicit, as in the case of Gaussian noise, or implicit, as in the random phase offsets of the 1 f and sinusoidal noise. For each noise profile, we measured the performance of 4 different implementations of the lock-in detector algorithm, formed by activating and / or deactivating the bandpass-filter and zero-crossing counter lock-in detector components (see Table 1): 1. Bandpass filter and count zero crossings, 2. Do not bandpass filter but count zero crossings, 3. Bandpass filter but do not count zero crossings, 4. Neither bandpass filter nor count zero crossings.
3.3. Implementation of fractional cycles To FFT the reference and system signals over an integer number of cycles, and thereby assign activity at ref to a single DFT harmonic, requires that the number of samples in the FFT window correspond to an even integer of signal zero crossings. The 25 Hz reference sine wave, sampled at 1 kHz for 2000 milliseconds, has 100 zero crossings in its 50 cycles. Note that the 50th cycle is actually 1 sample short of completion, and so its last zero crossing is not part of the 2000 sample segment. This final zero crossing is not included to account for the periodicity of the DFT. In our simulations, the first 600 sample points of the time window may be contaminated by the startup transient of a 600 point FIR bandpass filter, and so are not used in the analysis, leaving 1400 sample points, beginning at the 601st sample, with which to DFT the reference and system signals. These 1400 sample points translate into 70 zero crossings and 35 cycles, with the 35th cycle one sample short of completion. To apply the DFT to a fractional number of signal cycles, we first selected the 1320 signal values contained in samples 601 to 1920 of the 2000-sample reference and system signals. This generated two new signals, containing 66 zero crossings and 33 full cycles each. We then increased their signal lengths by adding from 0 to 39 additional reference and system signal values, respectively, with the number added chosen at random from a uniform distribution. By iterating on this procedure, we could generate a family of reference and system signal pairs with fractional 34th cycles for analysis. As before, we applied DFTs to the fractionalcycle reference and system signal pairs. However, since ref no longer mapped directly to a DFT sinusoid, which all have an integer number of cycles, we chose
To account for randomness induced by the Gaussian noise, phase offsets and fractional cycles, we performed 20,000 iterations for each noise profile and detector implementation. Each iteration recomputed ssys and then re-estimated the system gain and phase offset via the following steps: 1. Generate sref, 2. Generate , 3. Generate ssys, 4. Bandpass filter (if applicable), 5. Zero crossings (if applicable), 6. FFT integer / fractional sref and ssys cycles, 7. Estimate gain and phase offset from ratio of sref and ssys DFT harmonics. We computed the mean and standard deviations of the estimated gain and phase offset across all 20,000 iterations for each noise profile and detector implementation, and present the results in Table 1. As seen in Table 1 and discussed further in Section 4, our modified lock-in detector accurately estimated the true system gain of 1.0 and phase offset of -30o, under a variety of noise conditions. It even performed reasonably well when the assumptions of bandpass filtering the reference and system signals and acquiring an integer number of signal cycles were violated.
181
components in ssys and sref, and ensuring that the reference frequency ref coincides with a DFT frequency. We proposed that this the lock-in detector can be assembled in software and the aforementioned working principle met by integrating a band-pass filter, zero-crossing counter, and time-series regression, via a Fast Fourier transform. An important aspect of our lock-in detector is that it incorporates two features that have appeared individually in separate designs. Oh et al. [19] incorporate a bandpass filter in the design of their phase sensitive detector to increase the signal to noise ratio. However, they do not require that the amplitude and phase offset be estimated from a portion of the signal containing an integer number of cycles at ref. Conversely, Smith et al. [20] are careful to choose a sampling frequency and time window that capture both an integer number of cycles and an integer number of samples per cycle. They do not, however, bandpass the reference signal. Our modified lock-in detector incorporates both a bandpass filter and a zero crossing counter to guarantee the acquisition of an integer number of cycles. This attention to detail is critical, as any artifactual activity captured by the ref DFT harmonic would degrade the performance of the lockin detector’s estimation of the system gain. We have implemented our detector in software and rigorously evaluated its performance in a series of tests conducted within the Matlab programming environment. We simulated applying a sinusoidal reference signal to a linear system containing robust narrow and wideband noise profiles, and employed our lock-in detection algorithm to extract its amplitude gain and phase offset at ref. Moreover, we tested multiple implementations of the detector to determine the effect of minimizing the impact of non-DFT frequency components in ssys and sref, and ensuring that the reference frequency ref coincides with a DFT frequency. As summarized in Table 1, our lock-in detector gave excellent results in its full implementation (Bandpass filter + Integer cycles). In the presence of Gaussian noise of equal intensity, our lock in detector estimated the amplitude and phase with a mean relative error of 0.04% and 0.20%, respectively. Furthermore, the standard deviations in our estimation of the amplitude gain and phase offset represented less than, respectively, 3% and 5% of their estimated values. And this was a worst-case scenario, as the standard deviations of the gain and phase offset estimations dropped appreciably in the remaining noise profiles, while the mean relative error improved. Our simulations also revealed that while our lock-in detector continued to perform reasonably well as we
Table 1: Simulation Results
4. Discussion In this paper, we briefly reviewed several techniques for low-frequency electrical impedance tomography. In particular, we emphasized the important role that lock-in detection, however implemented, must play in extracting the potential distribution generated by impressed currents in lowfrequency EIT experiments, due to the rich noise environment. We then summarized a standard lock-in detection technique and pointed out its chief weakness, which is the need to phase shift the reference signal in order to zero out the unknown sys. We introduced in detail a new high-fidelity lock-in detector design that does not rely upon a phaseadjustable reference signal. We demonstrated that the working principle of this modified lock-in detection is based on minimizing the impact of non-DFT frequency
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modified its implementation, loss of the zero crossing counter increased the standard deviations of the gain and phase offset estimations across all noise profiles. This is somewhat expected as our detector would apply an FFT to a fractional number of cycles, precluding ref from mapping to a unique DFT harmonic. Loss of the bandpass filter, on the other hand, was most acute in the sinusoidal noise profile, as was also to be expected. Since this noise was outside the filters bandwidth, the bandpass filter had eliminated most of it, except for some minor leakage. In a future update to our current generator, the synthesized waveform of the impressed current will be generated synchronously with the digitizing process of the NetAmps 300. Any selected frequency for the impressed current will be adjusted with respect to the sampling rate to ensure an integer number of sampling points per cycle. This feature will remove the need of the zero-crossing counter.
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