A Simultaneous Maximum Likelihood Estimator Based on a Generalized Matched Filter

A simultaneous maximum likelihood estimator based on a generalized matched filter

Gustavsson, Jan-Olof; Börjesson, Per Ola

Published in: [Host publication title missing]

DOI: 10.1109/ICASSP.1994.389775

1994

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Citation for published version (APA): Gustavsson, J-O., & Börjesson, P. O. (1994). A simultaneous maximum likelihood estimator based on a generalized matched filter. In [Host publication title missing] (pp. 481-484) https://doi.org/10.1109/ICASSP.1994.389775

Total number of authors: 2

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Download date: 25. Sep. 2021 A SIMULTANEOUS MAXIMUM LIKELIHOOD ESTIMATOR BASED ON A GENERALIZED MATCHED FILTER

Jan-Olof Gustavsson' Per Ola Borjesson' 'University College of Karlskrona/Ronneby, Department of Signal Processing, S-372 25 Ronneby, Sweden 'Lulei University of Technology, Division of Signal Processing, S-971 87 Lulei, Sweden

ABSTRACT t.he noise is white gaussian, the output from the ordinary matched filters is a sufficient statistic for calculating the This paper discusses parameter estimation and detection in a posteriori distribution for given the observed signal r(.) laplace distributed noise. The received signal is modeled 8, as and a known a priori distribution for 8. When the noise is F(.) =AS(.,@)+ n(.), where A is an unknown amplitude, 6 is t.he parameter vector to be estimated and is indepen- independent non-gaussian it can be shown that the output n(.) from a modified form of the ordinary matched filter is a suf- dent laplace distributed noise. The simultaneous maximum ficient statistic for calculating the a posteriori distribution likelihood estimator of is derived. The derived es- (A,$) for 8. Gustavsson and BBrjesson have derived and discussed timator is based on a combmation of a weighted median this filter in 121. The modified form the ordinary matched filter I and a generaliset1 form of the ordinary matched of filter can be seen an ordinary matched filter where the filter121. as multipliers have been replaced by non-linearities. The non- Examples of performance for four different detectors are linearity is known from several works dealing with the prob- given for a case of binary detection, when the amplitude A lem of detection in non-gaussian noise[lO]. or the signal shape .(.,e) are varied. Simulations indicate that the performance of detectors based on the generalized The estimation problem generally becomes much more difficult if the amplitude A in the model r( = As(., @)+U( matched filter is not particularly dependent on either the .) .) estimate of the amplitude A or the signal shape. is both continuous valued and unknown. A special case when the ML estimate of 6 can still be calculated exactly is when the noise is gaussian distributed, since the estimator INTRODUCTION then becomes independent of the amplitude A. Another case where this can be done is discussed in this paper. The There are many problems where the task is to estimate simultaneous ML estimator of A and 6 is derived assuming a parameter in an observed signal(31. Examples of such independent laplace distributed noise. The ML estimator is problems are those of estimating the numerical value of based on a combination of a matched median filter[l], which a parameter (e.g. arrival time estimation) or choosing one gives an estimate of the amplitude A, and a modified form hypothesis out of several hypotheses (e.g. detection or clas- of the ordinary matched filter[2]. The derivation can easily sification of a signal). In many cases these problems can be modified by using Rayes theorem to give a simultaneous be dealt with by using a signal model of the form r(.) = ML estimator of A and a MAP or MMSE estimator of 8. As(,,8) + U(.), where .4 is a known amplitude and 8 is the In a case of binary detection discussed in this paper, the parameter vector to be estimated from the observed sig- performance of the estimator is calculated by means of sim- nal r(.). ulations, and the results are compared to the performance There are a number of studies of how to estimate 6 of three other estimators. in an optimal way according to given optimization criteria assuming the signal model given above. Examples of THE MAXIMUM LIKELIHOOD ESTIMATOR such criteria are the maximum likelihood (ML), maximum Consider the problem of how to estimate a parameter a posteriori (MAP) and the minimum mean square error vector 6 from an observed signal r(.),where r(.) is mod- (MMSE) criterion[3]. It is usually assumed that the noise, eled as IJ(,), is gaussian distributed[3, 4, 51. There are at least two reasons why this assumption is so comnion. One reason r(k) = As(k, e) + ~(k),k E I. (1) is that the noise in many applications is at least approxi- In the model, s(.,8 is a signal dependent on the unknown mately aussian distributed according to the central limit parameter vector d,' A is an unknown amplitude factor, theorem761, and another reason is that the gaussian assump A > 0, A E R, U(.) is independent laplace distributed noise tion is analytically tractable. However, situations also exist with probability density function f"(.), k is a discrete 'time' where the statistical distribution of the uoise in the above variable and I the observation interval. The probability model is non-gaussian[7, 8, Y Johnson and Rao[B] state density function fv(.) is given by that "Gaussian processes woul]d seem to be imprecise repre- sentations of physical measurements.". During the last few years the problem of parameter estimation in non-gaussian noise has been an active field of research. and many resnlts have been published, e.g. Kassam[lO] including references. where u2 is the variance of the noise. For all 8, the signal When the noise is white gaussian, the optimal estimat,es s(k,8) = 0 for all k outside the interval 10 : [m(8),m(8) + of@according to many difisrent crit,eria can be derived from .(e)] and outside the observation interval I, i.e. the signal the output from ordinary matched filters[3] when r(.)is the s(., 8) is of finite length, n(6 -t 1, and completley contained inp'iit lo the filters'. This follows from the fact that when ill the observation interval 1. The parameter 8, which can 'The output from filten marclied with sl..S) for all possible values of 0 are necessary

IV-481 0-1803-1175-0/94$3.00 0 1994 IEEE only assume a countable number of values, is either a ran- laplace distributed, the maximum likelihood estimate As of dom or non-random vector independent of the noise .I(.). g A given 0 is the weighted median of the observations, P(.), normalized by the signal s(., O), where the weights are given and a denote estimates of the unknown parameters 0 and A. by the absolute values of the signal s(.,6)[1]. That is, Ae is A structure of a filter for estimating 8 according to the given by ML, MAP or the MMSE criterion is proposed in [Z] and shown in figure 1. This filter is built up of two parts. The first part, the preprocessor, processes the observed signal r ) and gives as output L(6(r(.) , for all possible values of 0. deally, the preprocessor should be chosen so that these outputs are a (minimal) sufficient statistic for calculating the a posteriori distribution of 6, provided that the a priori Since the parameter vector 0 can assume only a finite num- distribution of 0 is known. The second part, the estimator, ber of values, & can be calculated for all possible val- uses this information about 6 to make an estimate of 6 ues of 0. Denote the value L,(Ae,O) with L(6). A block according to a desired criterion. structure illustrating the calculation procedure IS shown in figure 2. In the figure the weighted median filter is based +Pre-processor 1-4&timator + on equation (8) and the generalized matched filter is based on equation (7) wit,li A replaced by Ae. This structure has Figure 1. A geneml block-structure of a two-step process for to be repeated for all possible values of 0 to give L(O),VO. estimating9. The output from the pw-processor is L(Olr( .)), The simultaneous maximum likelihood estimate of (A,6) is for all possible values of 6. then (Ai,i),where The ML estimate of the parameters (A,B) is achieved by maximizing the likelihood function for the parameters (A,#).The likelihood function is 8 = arg { n1;x L(6)) ,

Li(A,B) = nfv(dE) A~(k.0)) c.f. the generalized likelihood ratio test in [3]. With the - aid of Bayes theorem the estimation procedure can easily *€I be modified to give a simultaneous ML estimate of A and a maximum a posteriori or minimum mean square error estimate of 6 given that 0 is a stocastic vector with known = exp { &ln(f.(r(k) - A.q(k,@))} a priori distribution.

Maximizing Ll(A,6) is equivalent to maximizin the exponent in (3). The exponent can be rewritten as[2e)

Figure 2 A block structure for calculating L(0) when the signal amplitude 18 unknown and the noise is independent laplace distributed. WMF: Weighted median filter, GMF: Generalized matched filter

EXAMPLES OF PERFORMANCE [lenotethe second sum on the hand side of by Consider the classical problem of deciding if a signal is Lz(A 6 and note that tlle first sum is independellt, of botll Present Or not, to exemplify the perfornlance of the simulta- A an6 Q, ~~~~~~~~~l~,the likelihood functioll neous maximum likelihood estimator. The noise is indepen- L~(A,~)is equivalent to ,r2(A,q, B~ using dent laplace distributed with zero mean and unit variance. s(k, 0, [m(6), m(6) and by defining In this example. the prohlmi consists of cho'osing between 6) = E 6 + n(8)] the hypotheses bk(A,6) = As(m(B)+ 46)- k,6) and where s(k,O = 0,Vk s(k,1{ = 1,k E [1,5]och s(k,l) = 0,k [1,5] =In { -} = 2(Iz1- 1' - *I)* (')) For this case the performance of four different detectors, D1 - D4, has been calculated by means of simulations. In L2(A,6) can be written the different detectors, the pre-processor in figure 1 is based on: D1 the generalized matched filter assuming that the amplitude A is kriown[?], that is, As = A. D2 the proposed simultaneous ML estimator of A and (7) 6, which is the generalized matched filter with the amplitude A estimated using the weighted median[]], = 21(b4A,B),r(rn(O) + n(6) - k)). that is. Ae is given by (8 Henceforth, this detector k=O will be referred to as eider D2 or the simultaneous maximum likelihood detector. A filter structure for calculating Lz(A.0) for a given am- D3 the generalized matched filter with Ae := 1 plitude A is ,~~~~~~~~din [2], N~~ for each @, maxilnize L2(A,0)with respect to A. Since the noise is independent D4 t,he ordinary niatclied filter[3].

IV-482 Note that the detectors D1 - D3 have in common that they all are based on the generalized matched filter. D1 is assumed to know the amplitude A, while D2 tries to estimate the amplitude A from the received signal r(.) using (8). Since A > 0 in (I), an estimate As < 0 of A is changed to Ae = 0. It follows from (7) that under the hypothesis Ho : B = 0, the output from the pre-processor L el.(.)) = 0, VF(.). Con- sequently, the estimator must base tie decision between Ha and HIsolely on the output from the pre-processor in figure 2, L(O), with 0 = 1, that is L(1). In the four different detectors, the estimator used is based on a Neyman-Pearson test[3]. The estimator compares L(1) to a threshold A, where X is chosen so that a given probability of false alarm, p, (that is, to decide HI when HO is true), is achieved. The decision rule is:

< X : decide Ho. { -> X : decide HI Amplitude, A

Let pd denote the probability of detection, that is, to decide Figure 3. The probability of detection, pd, for four diflerent HI when HI is true. detectors when the prolubility of false alarm is 0.1 and 0.01 as a function of the signal amplitude. The noise is laplace In figures 3 and 4, the simulated results of the prob- distributed. ability of detection, pd, respectively the probability of ,,, - 1 - pd, are shown as a function of the amplitude A in 1) for- the detectors D1 - D4, when the probability of false alarm, pf,is 0.1 and 0.01. To facilitate comparison of the different detectors, figures 5 and 6 show the quotient between pd for D2 - D4 and pd for the detector D1, when p, = 0.01 and pf = 0.1 respectively. In figures 3 - 8 t.hr following can be observed: the probability of detection is not part.icularly drpen- dent on the estimate A, of A. for small values of A, the detector based on the ordinary matched filter, D4, has a lower probability of detection than the other detectors used. for large values of A, the detector based on the ordinary matched filter, D4,has a higher probability of detection than the other detectors, i.e. D2 and D3, when the true value of the amplitude A is unknown. 0.5 I 1.5 2 25 3 Amplitude. A In figure 7 the pulse shape of s(k,I), k E [1.5] varies instead of t.he amplitude. The pulse shapes are given h) Figure 4. The probability of miss, pm = 1 - pd, for four s(.,1)= z,--,l,-,x different detectors when the probability of false alarm is 0.1 I 2 and 0.01 as a function of the signal amplitude. The noise I is laplace distributed. where x varies from 0.1 to 3. c.f. [I]. All signals are scaled so t.hat the signal energy becomes 5. This signal energy different detectors has been compared. The simulations in- corresponds to A = 1 in figures 3 -- 6. In figure 7 the dicate that, in general, the performance of the simultane- following can be observed: ous maximum likelihood detector is better than the per- the probability detection is not. particularly depen- formance of other detectors used, when the received signal of amplitude is unknown. dent on the signal shape. The simulations also indicate that the performance of the detector based on the ordinary matched filter, 114, detectors based on the generalized matched filter are not has a lower probabi1it.y of detection than the other de- particularly dependent on either the estimate of the ampli- tectors simulated for all signal shapes used. tude ,4 or the signal shape. CONCLUSIONS REFERENCES The simultaneous maximum likelihood estimator has been [I] J. Astola and Y. Neuvo. Matched median filtering. derived for estimating a parameter vector B and an unknown IEEE transactions on communications, 40(4):722-729, signal amplitude A in independent laplace distributed noise. April 1992. This estimator consists of a combination of a weighted median filter and a generalized form of the ordinary matched [?I J-0. Gnstavsson and P.O. Borjesson. A generalized filter. The estimator can easily be modified to become an matched filter. In Proceedings of the third international ML estimator of the amplitude and a MAP or MMSE esti- symposium on stgnal processing and its applications, mator of the parameter vector. pages 16--22, Gold Coast, Australia, August 1992. Simulations have been made for a cast of binary detec- [3] H.L. van Trees. Detection, estimation and modulation tion discussed in this paper, where the performance of fonr theory, volume 1. U'iley. IJSA, 1968.

IV-483 I I ' PI = 0.1 ' 0.9 - , ...... : "'...... __ __ . , . .~ ...... - ...... -...... 0.8 -

0.7. pj = 0.01

. n? 0.11 '...... _. .' I - . n4-. 0.5 01 a5 I 1.5 2 25 3 0 0.5 I IS 15 Amplitude, A Signal shape factor, x

Figure 5. The quotient between the probability of detection Figure 7. The probability of detection, pd, for three different for three different detectors, D2 - D4,and the probability of detectors when the probability of false alarm, pj, is 0.01 and detection for the detector, Df, when the probability of Jalse 0.1 as a Junction of the signal shope. The noise is laplace alarm is 0.01 as a junction of the signal amplitude. The distributed. noise is laplace distributed. Wegman and J.G. Smith, editors, Statistical signal processing. Marcel Dekker, New-York, USA, 1984. ___~ ~ ----_ [lo] S.A. Kassam. Signal detection in non-gaussian noise. Springer Verlag, USA, 1988.

0.9. '3,~ ,/' ,,. ~,,.. .. . : 0.8- ...... : : D2

0.7 - ~~ : D3 ...... : D4

0.5 1 0.) I 1.5 Amplitude, A

Figure 6. The quotient between the probability of detection for three different detectors, D.2 - 04,and the probability OJ detection for the detector, D1, when the probability of false alarm is 0.1 as a function of the srgnal amplitude. The noise is laplace distributed. [4] P.O. BGrjesson, 0. Pahlm, L. SGrnmo, and M-E. NygBrds. Adaptive QRS detection based on maximum a posteriori estimation. IEEE Transactrons on biomedical engineering, BMG?9(5), May 1982. [5] J.M. Mendel. Lessons in digital estimation theory. Prentice-Hall, USA, 1987. [SI A. Papoulis. Probabilrty, random r,ariables and stochastic processes. McCraw-Hill, USA, secood edi- tion, 1984. [7] V.R. Algazi and R.M. Lerner. Binary detectiou in white non-gaussian noise. Report DS-2138. Lincoln Laboratory, M.I.T., USA, 1964. [a] D.H. Johusori and P.S. Rao. On the existence 01 gaussian noise. In Proceedings ol ICASSP 91, Toronto, Canada, May 1991. [9] F.W. Machell and C.S. Penrod. Probability density functions of ocean acoustic noise processes. In E.J.

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