The application of adaptive noise cancellation to pyroelectric detectors

Infrared Phys. Vol. 27, No. 4, pp. 267-273, 1987

0020-0891/87 $3.00 + 0.00 Copyright © 1987 Pergamon Journals Ltd

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THE

APPLICATION OF ADAPTIVE NOISE CANCELLATION TO PYROELECTRIC DETECTORS

P. R. WHITE, I P. M. CLARKSONI and I. C. CARMICHAEL 2 qnstitute of Sound and Vibration Research, Southampton University, Southampton, U.K. and 2Royal Signals and Radar Establishment, Malvern, U.K. (Received 11 February 1987) A~tract--This paper discusses the application of a signal processing technique known as Adaptive Noise Cancellation to the problem of reducing noise levels at the output of a pyroelectric detector. The detection system is considered in relation to self-modulating sources so that the input signals can be considered as pseudo-periodic. Two forms of adaptive processing, namely the Adaptive Line Enhancer and spike sequence input, are compared and contrasted, both methods are shown to improve signal-to-noise and thus increase detector performance.

INTRODUCTION

The performance of detector systems is limited by the noise levels present. Considerable work has been directed at the reduction of these noises directly, for example the design of, antimicrophony mountings and improvements in pre-amplifiers and detector materials/1,:) Noise reduction can also be achieved by the reduction of the thickness of the pyroelectric material, this reduces dielectric, resistor and current noises, although this approach is now reaching the limits of manufacturing technology. This paper discusses another approach, namely that of signal processing, specifically the application of a technique known as Adaptive Noise Cancellation (ANC). The signals under consideration in this paper will be those from a pyroelectric detector which will be modulated, typically at a frequency of less than 50 Hz, and contain a set of harmonics. There are many forms of signal processing which are applicable to this type of problem, such as exploiting the periodicity of the signal by averaging over several periods, this is known as comb filtering. The comb filter has two major disadvantages, firstly it requires an accurate estimate of the signal period, secondly if the signal has a time dependent form the comb filter is unable to vary with it. Another commonly used signal processing technique is that of matched filtering in which the signal is passed through a time reversed version of itself, this requires a priori knowledge of the signal and causes signal distortion. Adaptive Noise Cancellation (ANC) has the advantage of requiring little a priori knowledge and has the ability to adapt to variations in the signal. Using ANC we can increase the signal-to-noise ratio and hence improve the performance statistics of the detector. The predominant noise sources which will be of interest in the frequency range specified are the dielectric loss noise in the pyroelectric and current noise in the amplifiers. Microphony could be a problem but can be greatly reduced by the use of anti-microphony mountings for the detector, these can be expensive and comparatively large, ANC may offer a more cost effective alternative. 1If noise is often of little practical importance since it is predominant only in the low frequency end of the spectrum; thus its effects can often be eliminated by high-pass filtering. However in this case we are interested in low frequency signals which makes the use of a high-pass filter impractical. ADAPTIVE NOISE C A N C E L L A T I O N Given an arbitrary signal dn, as an input to the detector system and assuming that it consists of a correlated component sn, together with an additive noise v,, so that: d. = s~ + v., the objective is to enhance s. at the expense of v.. 267

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On

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Fig. 1. General adaptive noise canceller.

The principle behind Adaptive Noise Cancellation (ANC), shown in Fig. 1, is to take a second input which is in some way related to the noise in the primary input, v.. This secondary input is filtered in such a way as to produce a replica of v., which can then be subtracted from the primary input, leaving an enhanced version of s.. ~3) Alternatively the second input can be another measurement of the signal, in this case it is the filter output which is an enhanced version of s., so that after the subtraction it is v. which is left. In most detector systems, however it is not possible to obtain a second noise (or signal) measurement, although there are a number of ways to construct an appropriate reference without actually making a second reading. We shall consider two such techniques, the first, known as the Adaptive Line Enhancer (ALE) involves forming a reference from a delayed version of the input and the second uses a sequence of unit pulses each pulse being separated by the period of the input signal. Before discussing these approaches further we must consider the basic properties of the adaptation process.

THE LEAST MEAN SQUARES A L G O R I T H M For a general ANC system, where dn is the primary input, x. is the secondary input and y. is the filter output defined by: !

l

y,,= x . _ f . = f . x . ,

(1)

with x'. = [ x . x . _ ~ - " x._L+ i] and f ' . = [ f . ( 1 ) f . ( 2 ) . . - f . ( L ) ] in w h i c h f ( j ) is the j th filter element at time step i and where L is the number of coefficients in the filter. The error, e., is defined as: e. = d. - x'. L = d. - f ' . x . .

(2)

Now consider the problem of minimising the expected mean square value of this error, it is a quadratic function of the filter coefficients and as such has a unique solution, this solution has been shown to be: f~ = R ; ~ P . ,

(3)

where R. is the autocorrelation matrix defined by E[x.x'.] and P. is the cross-correlation vector given by E[d.x.], where E[ • ] indicates expectation. It is important to note that P. and R. are in general time varying and if this is the case it maybe virtually impossible to estimate them using only one time history. Due to this and the amount of calculation required to estimate the invert R. at each step, direct application of equation (3) is in general impractical. We shall thus use a gradient search method known as the Least Mean Squares (LMS) Algorithm, originally proposed by Widrow and Hoff. (4) The LMS algorithm attempts to find the optimal solution, given by equation (3), by seeking the bottom of the error surface, this is achieved by estimating the gradient at each step and progressing along this estimate by a pre-determined amount. The exact gradient is given by: q) = V(E[e2.]),

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where V is the vector differential with respect to _f. We can make a point estimate, by replacing E[e2.] with e.z, to give: = - 2e.x., this leads to the well known filter update equation:

f.+~ = _jr. + ~e.x.,

(4)

where ~ is a user selected constant, which controls the rate of convergence. Widrow has also shown (3~that each filter element will converge to its optimal value, under certain commonly made assumptions, at an average rate given by: exp(-1/zp),

where

"rp= l/o~J,p,

(5)

where 2p is the pth eigenvalue of R, and Zp is generally referred to as the time constant associated with the pth mode of convergence. Hence the final rate of convergence for the filter as a whole, can be considered as being dictated by the slowest mode of convergence, i.e. the smallest eigenvalue of R,. Another factor to take into consideration with the LMS algorithm is known as misadjustment, which is caused by the fact that even if the filter obtains its optimal value, given by equation (3), the error, e,, will not be identically zero. It will in general be zero mean but with some finite variance and it is these fluctuations about the desired value which cause the misadjustment. It has been shown (6) that under certain conditions the misadjustment is proportional to ~, this leads to a conflict of interests since for small misadjustment a small ~ is required, but we can see from equation (5), this will lead to slower rate of convergence. THE ADAPTIVE LINE ENHANCER The rationale behind the Adaptive Line Enhancer is to use a delayed form of the primary input, d., as the secondary input, so that xn = d~_A. The principle is to make A sufficiently large so that the noises will decorrelate, i.e. E[v.v._a] = 0, but the signal components being periodic, will only undergo a phase shift. The filter in attempting to minimise E[e2.] will reduce the noise level in its output as much as possible, as well as correcting the phase shift, this ensures that upon subtraction s. will tend to be cancelled out, leaving the noise, v.. We shall firstly consider the relatively simple problem of a single sine wave in white noise. It has been shown (4) that the R. matrix with a pure sine wave as input for the ALE has at least two non-zero eigenvalues given by: 2p = A 2[L + sin(~oLT)/sin(~T)]/4,

(6)

where A is the amplitude of the sine wave, T is the sampling interval and o~ is the frequency of the sine wave. Consequently R is at best rank 2, hence in general R -~ will not exist, this is indicative of the fact that this problem is under constrained. The effect of adding white noise to the input is to inflate the diagonal elements of the R, matrix by az, where a2 is the power of the white noise, hence it can be shown (6) that the time constants for a sine wave in white noise are given by:

~p = l/~( G2 + )-A.

(7)

Noting 2p is non-negative and that the rate of decay is finally limited by the largest time constant, which is given by the smallest value of a 2 + 2p which in turn is limited by a 2, the largest time constant possible is thus: Tp = 1/~r 2.

(8)

T H E S P I K E T R A I N AS S E C O N D A R Y I N P U T This form of secondary input, originally proposed by Elliott and Darlington (8), consists of a series of unit pulses each being separated by the period of the desired part of the input signal, s.. The

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reason for using this form of signal is that a spike sequence arranged in such a fashion, contains all the harmonics of the input signal aligned such that they are all in phase at the site of any spike. The filter now only has to re-align the harmonics and adjust their amplitudes in order to reconstruct the coherent part of the signal, s,, so to minimise E[eZn]. It is relatively simple to show that for this case the eigenvalues of R, are given by: 2e= 1/L. There are L such eigenvalues and since they are all of the same size there will be only one time constant for the filter as a whole. This is given by: rp = L /~.

RESULTS In this section we present the results of a comparison of these two forms of adaptive noise cancellers, using the results of trials on synthetic and laboratory data. The synthetic data consisted of four sine waves embedded in Gaussian white noise, the sine waves had amplitudes of 1, 0.35, 0.2, 0.15 and frequencies of 40, 80, 120 and 160 Hz, respectively. Each method was applied to three different Signal-to-Noise Ratios (SNRs), which we define as SNR = 10 loglo(as/a,), 2 2 where a ,2 is the noise power and a~ is the signal power, both defined over the whole bandwidth. For each trial ~ was selected so that the rates of convergence were identical, in the case of the ALE there are several modes, and hence several rates of convergence. For the ALE it was assumed that the dominant rate of convergence was the fastest, i.e. the one due to the largest sine wave, if this assumption is fallacious then we will under estimate the performance of the ALE. The filter length, L, was selected such that it spanned one whole period of the input signal. Noise levels were measured after the filter had converged, i.e. in steady state, to give the results shown in Table 1. These results show a significant improvement in SNR, further it is clear that the spike sequence produces a significantly larger improvement than the ALE especially at higher SNRs. Examining the power spectra of the output signals shown in Fig. 2 for a typical case, reveals that the spike sequence produces a set of spurious harmonics which are of the same power as the original noise floor. These spurious harmonics mean that this method would not be particularly applicable for studying harmonic structures of signals or for using thresholding detection methods. SNR is not a complete measure of performance, it is important to consider some other form of criteria on which to judge the two methods. We shall consider in more detail the distribution of the noise power, specifically in two regions. Firstly that region of the spectrum in the vicinity of the signal, the precise area of which is dictated by the finite response length of the filter, this will be referred as the "in-band" region. The remaining parts of the spectrum are dubbed the "out-of-band" region. Table 2 shows the power of the output noise normalised with respect to the original noise power per unit band width, in the two regions and for the same set of data as in Table 1, in this case the in-band region is approximately 0-200 Hz. It is clear from this that the ALE performs significantly better in the out-of-band region whereas it is inferior within the signal band, this may have been expected from Fig. 2. It should be noted however that this distinction becomes less marked as SNR decreases. Another problem with using the spike sequence is its dependence upon an estimate of the period of the input signal, it is not only the extra calculation involved, which can at least in part be traded Table 1. SNR for ALE and spike sequence with equal convergence rates Original SNR 10 0 - 10

SNR after ALE 12.0 7.2 - 1.9

SNR after spike sequence 17.7 10.3 0.06

Table 2. Normalised noise power/Hz Original SNR 10 0 - 10

in-band 2.5 0.84 0.52

ALE out-of-band 0.031 0.037 0A7

Spike sequence in-band out-of-band 0.19 0.19 0.18

0.2 0.18 0.18

Application

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F i g . 2. P o w e r spectral densities o f (a) s y n t h e t i c data, (b) A L E w i t h (a) as input, (c) o u t p u t o f spike s e q u e n c e

w i t h (a) as input.

off against the simplification of the update equation (4). Also any practical scheme for estimating the period will be only approximate and it is important to try to assess how robust the algorithm is in relation to errors of this type. Simulations were carried out with period estimates which had errors of 1 and 2 samples and the resulting SNRs are shown in Table 3. It can be seen that for small errors in the period estimate there is a large decrease in the corresponding SNR obtained. It should be noted that at high SNR the effects can be such that with even a single sample error the data is actually degraded. The ALE of course requires no such estimate and so suffers from none of these problems. Finally both methods were tested on data obtained from a pyroelectric detector, with a modulated source, with a fundamental frequency of approximately 15 Hz. The resulting spectra

Table 3. S N R for spike sequence with inexact period estimates S N R with exact

Original S N R 10 0 -- 10 INF 274~E

period est. 17.7 10.3 0.06

S N R with error of 1 sample 5.4 4.4 -0.04

SNR with error of 2 samples 2.6 2.0 - 1.7

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Fig. 3. Power spectral densities

of (a) data from a pyroelectric detector, (b) output of ALE with (a) as input, (c) output of spike sequence with (a) as input.

of two such tests are shown in Fig. 3. Again it can be seen from these spectra that the spike sequence produces spurious harmonics. By considering the mains pick-up, which is also present as a signal we can see the ALE enhances it where as the spike sequence tends to cancel it out. This points out another difference in the behaviour of the techniques, specifically that the ALE will enhance any set of periodic signals whereas the spike sequence will only enhance one fundamental and its associated harmonics. CONCLUSIONS In this paper it has been demonstrated that the use of adaptive signal processing methods can produce significant improvements in SNR and therefore in detector performance. Two methods were considered, ALE and spike sequence input, both were found to produce increases in SNR. Given that exact period estimates are available the spike sequence has a better in-band performance, whilst the ALE in general performs better out-of-band, since the spike sequence produces a set of spurious harmonics. These unwanted components in low SNR will degrade detection performance.

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For the more practical case of inaccurate period estimates the spike sequence's performance degrades rapidly with decrease in accuracy of period estimates for high SNR, whilst the ALE is almost immune from such errors. Overall the ALE performs better unless there is access to good estimates of the period of the input signal as well as a low SNR. Acknowledgements--The financial support of the Science and Engineering Research Council of Great Britain and of the Royal Signals and Radar Establishment is gratefully acknowledged. REFERENCES N. M. Shorrocks et al., 2nd Symp. Opt. Electron. Opt. Appl. Sei. Engng (1985). S. G. Porter, Ferroelectrics 33, 193 (1980). B. Widrow et al.. Proc. IEEE, 63, 1692 (1975). B. Widrow and M. E. Hoff, WESCON Conv. Rec., pp. 96-140 (1960). B. Widrow, Aspects of Network and System Theory (Edited by R. E. Kalman and N. DeClaris), pp. 563-587. Holt, Rhinehart and Winston, New York (1971). 6. J. R. Treichler, IEEE Trans. Acoustics, Speech Signal Processing 27, 53 (1979). 7. S. J. Elliott and P. Darlington, IEEE Trans. Acoustics Speech Signal Processing 33, 715 (1985). 8. E. L. Dereniak and D. G. Crowe, Optical Radiation Detectors. Wiley, London.

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