The paper discusses parameter estimation and detection in Laplace distributed noise. The received signal is modeled as r(·)=As(·,&thetas;)+n(·), where A is an unknown amplitude, &thetas; is the parameter vector to be estimated and n(·) is independent Laplace distributed noise. The simultaneous maximum likelihood estimator of (A,&thetas;) is derived. The derived estimator is based on a combination of a weighted median filter [Astola and Nuevo, 1992] and a generalized form of the ordinary matched filter [Gustavsson and Borjesson, 1992]. Examples of performance for four different detectors are given for a case of binary detection, when the amplitude A or the signal shape s(·,&thetas;) are varied. Simulations indicate that the performance of detectors based on the generalized matched filter is not particularly dependent on either the estimate of the amplitude A or the signal shapeGodkänd; 1994; 20080222 (ysko

Börjesson, Per Ola

Gustavsson, Jan-Olof

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A simultaneous maximum likelihood estimator based on a generalized matched filter

Luleå University of Technology Publications

Digitala Vetenskapliga Arkivet - Academic Archive On-line

The paper discusses parameter estimation and detection in Laplace distributed noise. The received signal is modeled as r(·)=As(·,&thetas;)+n(·), where A is an unknown amplitude, &thetas; is the parameter vector to be estimated and n(·) is independent Laplace distributed noise. The simultaneous maximum likelihood estimator of (A,&thetas;) is derived. The derived estimator is based on a combination of a weighted median filter [Astola and Nuevo, 1992] and a generalized form of the ordinary matched filter [Gustavsson and Borjesson, 1992]. Examples of performance for four different detectors are given for a case of binary detection, when the amplitude A or the signal shape s(·,&thetas;) are varied. Simulations indicate that the performance of detectors based on the generalized matched filter is not particularly dependent on either the estimate of the amplitude A or the signal shap

Lund University Publications

LUND UNIVERSITYPO Box 117221 00 Lund+46 46-222 00 00A simultaneous maximum likelihood estimator based on a generalized matched filterGustavsson, Jan-Olof; Börjesson, Per OlaPublished in:[Host publication title missing]DOI:10.1109/ICASSP.1994.3897751994Link to publicationCitation for published version (APA):Gustavsson, J-O., & Börjesson, P. O. (1994). A simultaneous maximum likelihood estimator based on ageneralized matched filter. In [Host publication title missing] (pp. 481-484)https://doi.org/10.1109/ICASSP.1994.389775Total number of authors:2General rightsUnless other specific re-use rights are stated the following general rights apply:Copyright and moral rights for the publications made accessible in the public portal are retained by the authorsand/or other copyright owners and it is a condition of accessing publications that users recognise and abide by thelegal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private studyor research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portalRead more about Creative commons licenses: https://creativecommons.org/licenses/Take down policyIf you believe that this document breaches copyright please contact us providing details, and we will removeaccess to the work immediately and investigate your claim.Download date: 08. Oct. 2022A SIMULTANEOUS MAXIMUM LIKELIHOOD ESTIMATOR BASED O N  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 This paper discusses parameter estimation and detection in laplace distributed noise. The received signal is modeled as F(.) = A S ( . , @ )  + n(.), where A is an unknown amplitude, 6 is t.he parameter vector t o  be estimated and n(.) is indepen- dent laplace distributed noise. The simultaneous maximum likelihood estimator of (A ,$ )  is derived. The derived es- timator is based on a combmation of a weighted median filter I and a generaliset1 form of the ordinary matched filter121. Examples of performance for four different detectors are given for a case of binary detection, when the amplitude A or the signal shape . ( . , e )  are varied. Simulations indicate that the performance of detectors based on the generalized matched filter is not particularly dependent on either the estimate of the amplitude A or the signal shape. INTRODUCTION There are many problems where the task is to estimate a parameter in an observed signal(31. Examples of such problems are those of estimating the numerical value of a parameter (e.g. arrival time estimation) or choosing one hypothesis out of several hypotheses (e.g. detection or clas- sification of a signal). In many cases these problems can be dealt with by using a signal model of the form r( . )  = As(,, 8) + U(.), where .4 is a known amplitude and 8 is the parameter vector to be estimated from the observed sig- nal r(.). There are a number of studies of how to estimate 6 in an optimal way according to given optimization crite- ria assuming the signal model given above. Examples of such criteria are the maximum likelihood (ML), maximum a posteriori (MAP) and the minimum mean square error (MMSE) criterion[3]. It is usually assumed that the noise, IJ(,), is gaussian distributed[3, 4 ,  51. There are at least two reasons why this assumption is so comnion. One reason is that the noise in many applications is at least approxi- mately aussian distributed according to  the central limit theorem761, and another reason is that the gaussian assump tion is analytically tractable. However, situations also exist where the statistical distribution of the uoise in the above model is non-gaussian[7, 8, Y Johnson and Rao[B] state 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. When the noise is white gaussian, the optimal estimat,es of@ according to many difisrent crit,eria can be derived from the output from ordinary matched filters[3] when r ( . )  is the inp'iit l o  the filters'. This follows from the fact that when 'The output from filten marclied with s l . . S )  for all possible t.he noise is white gaussian, the output from the ordinary matched filters is a sufficient statistic for calculating the a posteriori distribution for 8, given the observed signal r(.)  and a known a priori distribution for 8. When the noise is independent non-gaussian it can be shown that  the output from a modified form of the ordinary matched filter is a suf- ficient statistic for calculating the a posteriori distribution for 8. Gustavsson and BBrjesson have derived and discussed this filter in 121. The modified form of the ordinary matched filter can be seen as an ordinary matched filter where the multipliers have been replaced by non-linearities. The non- linearity is known from several works dealing with the prob- lem of detection in non-gaussian noise[lO]. The estimation problem generally becomes much more difficult if the amplitude A in the model r( .) = As(. ,  @)+U( .) 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 then becomes independent of the amplitude A. Another case where this can be done is discussed in this paper. The simultaneous ML estimator of A and 6 is derived assuming independent laplace distributed noise. The ML estimator is based on a combination of a matched median filter[l], which gives an estimate of the amplitude A, and a modified form of the ordinary matched filter[2]. The derivation can easily be modified by using Rayes theorem to give a simultaneous ML estimator of A and a MAP or MMSE estimator of 8. I n  a case of binary detection discussed in this paper, the performance of the estimator is calculated by means of sim- ulations, and the results are compared to the performance of three other estimators. THE MAXIMUM LIKELIHOOD ESTIMATOR Consider the problem of how to  estimate a parameter vector 6 from an observed signal r ( . ) ,  where r(.)  is mod- eled as In the model, s ( . , 8  is a signal dependent on the unknown parameter vector d, '  A is an unknown amplitude factor, A > 0, A E R, U(.) is independent laplace distributed noise with probability density function f"(.), k is a discrete 'time' variable and I the observation interval. The probability density function f v ( . )  is given by r ( k )  = A s ( k ,  e) + ~ ( k ) ,  k E I .  (1) where u2 is the variance of the noise. For all 8, the signal s (k ,8)  = 0 for all k outside the interval 10 : [m(8), m(8) + .(e)] and outside the observation interval I, i.e. the signal s(., 8) is of finite length, n(6 -t 1, and completley contained ill the observation interval 1. The parameter 8, which can 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- dom or non-random vector independent of the noise .I(.). g and a denote estimates of the unknown parameters 0 and A. A structure of a filter for estimating 8 according to the 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 distribution of 0 is known. The  second part, the estimator, uses this information about 6 to make an estimate of 6 according to a desired criterion. + Pre-processor 1 - 4  &timator + Figure 1. A geneml block-structure of a two-step process for estimating9. The output from the pw-processor i s  L(Olr( .)), for all possible values of 6 .  The ML estimate of the parameters ( A , B )  is achieved by maximizing the likelihood function for the parame- ters ( A , # ) .  The likelihood function is L i ( A , B )  = n f v ( d E )  - A ~ ( k . 0 ) )  * € I  = exp { &ln(f.(r(k) - A.q(k,@))} Maximizing L l ( A , 6 )  is equivalent to maximizin the exponent in (3). The exponent can be rewritten as[2e) laplace distributed, the maximum likelihood estimate A s  of A given 0 is the weighted median of the observations, P(.), normalized by the signal s(., O ) ,  where the weights are given by the absolute values of the signal s(.,6)[1]. That  is, A e  is given by Since the parameter vector 0 can assume only a finite num- ber of values, & can be calculated for all possible val- ues of 0.  Denote the value L,(Ae,O) with L ( 6 ) .  A block structure illustrating the calculation procedure IS shown in figure 2. In the figure the weighted median filter is based on equation (8) and the generalized matched filter is based on equation (7) wit,li A replaced by A e .  This structure has to be repeated for all possible values of 0 to give L(O),VO. The simultaneous maximum likelihood estimate of ( A ,  6) is then (Ai, i) ,  where 8 = arg { n1;x L ( 6 ) )  , c.f. the generalized likelihood ratio test in [3]. With the aid of Bayes theorem the estimation procedure can easily 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 a priori distribution. Figure 2 A block structure for calculating L ( 0 )  when the signal amplitude 18 unknown and the noise is indepen- dent laplace distributed. WMF: Weighted median filter, GMF: Generalized matched filter EXAMPLES OF PERFORMANCE [lenote the second sum on the hand side of by Lz(A 6 and note that tlle first sum is independellt, of botll A an6 Q, ~ ~ ~ ~ ~ ~ ~ ~ ~ l ~ ,  the likelihood functioll L ~ ( A , ~ )  is equivalent to , r 2 ( A , q ,  B~ using s ( k ,  6) = 0, E 6 [m(6), m(6) + n(8)] and by defining Consider the classical problem of deciding if a signal is Present Or not, to exemplify the perfornlance of the simulta- neous maximum likelihood estimator. The noise is indepen- dent laplace distributed with zero mean and unit variance. In this example. the prohlmi consists of cho'osing between the hypotheses bk(A,6) = As(m(B) + 4 6 )  - k , 6 )  and where = I n  { -} = 2(Iz1 - 1' - *I)* (')) s ( k , O  = 0 , V k  s ( k , 1 {  = 1 , k  E [1,5] och s(k,l) = 0,k [1 ,5]  For this case the performance of four different detectors, D1 - D4, has been calculated by means of simulations. In the different detectors, the pre-processor in figure 1 is based on: L2(A,6) can be written D1 the generalized matched filter assuming that the am- plitude A is kriown[?], that is, As = A. D2 the proposed simultaneous M L  estimator of A and 6,  which is the generalized matched filter with the amplitude A estimated using the weighted median[]], that is. A e  is given by (8 Henceforth, this detector will be referred to as e ide r  D2 or the simultaneous ( 7 )  = 2 1 ( b 4 A , B ) , r ( r n ( O )  + n(6) - k)). k=O maximum likelihood detector. D3 the generalized matched filter with A e  := 1 D4 t,he ordinary niatclied filter[3]. A filter structure for calculating L z ( A . 0 )  for a given am- plitude A is , ~ ~ ~ ~ ~ ~ ~ ~ d  in [2], N~~ for each @, maxilnize L2(A,0)  with respect to A. Since the noise is independent 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 esti- mate the amplitude A from the received signal r ( . )  using ( 8 ) .  Since A > 0 in ( I ) ,  an estimate A s  < 0 of A is changed I t  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 t i e  decision between Ha and HI solely on the output from the pre-processor in fig- ure 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 proba- bility of false alarm, p, (that is, to decide HI when HO is true), is achieved. The decision rule is: to Ae = 0. < X : decide Ho. { - > X : decide H I  Let p d  denote the probability of detection, that is, to decide H I  when H I  is true. In figures 3 and 4, the simulated results of the prob- 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, p f ,  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 p d  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 or- dinary matched filter, D4, h a s  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. 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) s ( . , 1 ) =  z , - - , l , - ,x  I 2 I 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 corresponds to A = 1 in figures 3 -- 6. In figure 7 the following can be observed: the probability of detection is not. particularly depen- dent on the signal shape. the detector based on the ordinary matched filter, 114, has a lower probabi1it.y of detection than the other de- tectors simulated for all signal shapes used. CONCLUSIONS The simultaneous maximum likelihood estimator has been derived for estimating a parameter vector B and an unknown signal amplitude A in independent laplace distributed noise. This estimator consists of a combination of a weighted me- dian filter and a generalized form of the ordinary matched filter. The estimator can easily be modified to become an ML estimator of the amplitude and a MAP or MMSE esti- mator of the parameter vector. Simulations have been made for a cast of binary detec- tion discussed in this paper, where the performance of fonr Amplitude, A Figure 3. The probability of detection, pd, for four diflerent detectors when the prolubility of false alarm is 0.1 and 0.01 as a function of the signal amplitude. The noise is laplace distributed. 0.5 I 1.5 2 2 5  3 Amplitude. A Figure 4.  The probability of  miss, pm = 1 - p d ,  for four different detectors when the probability of false alarm i s  0.1 and 0.01 as a function of the signal amplitude. The noise is laplace distributed. different detectors has been compared. The simulations in- dicate that, in general, the performance of the simultane- ous maximum likelihood detector is better than the per- formance of other detectors used, when the received signal amplitude is unknown. The simulations also indicate that the performance of detectors based on the generalized matched filter are not particularly dependent on either the estimate of the ampli- tude ,4 or the signal shape. REFERENCES [I] J .  Astola and Y.  Neuvo. Matched median filtering. IEEE transactions on communications, 40(4):722-729, April 1992. A generalized matched filter. In Proceedings of the third international symposium on stgnal processing and its applications, pages 16--22, Gold Coast, Australia, August 1992. [3] H.L. van Trees. Detection, estimation and modulation theory, volume 1. U'iley. IJSA, 1968. [?I J -0 .  Gnstavsson and P.O. Borjesson. IV-483 I I 0.9 0.8 0.7. '. .' . .._. . .  a5 I 1.5 2 2 5  0.5 3 Amplitude, A ' PI = 0.1 ' , ... .. ... . . . .. .......... . ........ . ...... .... ........... :"'. . ......... . __  __  . , . . ~ . . . . . . . . - . . . .... -.. ... . . . . - - p j  = 0.01 Figure 5.  The quotient between the probability of detection for three different detectors, D2 - D4, and the probability of detection for the detector, Df, when the probability of Jalse alarm is 0.01 as a junction of the signal amplitude. The noise is laplace distributed. ___~ ~ ----_ 0.9. 0.8- 0.7 '3,~ ,/' ,,  .. . ~,,. : .. : .... : D2 - ~~ : D3 : D4 ...... 0.) I 1.5 0.5 1 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 i s  laplace distributed. [4] P.O. BGrjesson, 0. Pahlm, L. SGrnmo, and M-E. NygBrds. Adaptive QRS detection based on maxi- mum 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 i n  white non-gaussian noise. Report DS-2138. Lincoln Laboratory, M.I.T., USA, 1964. On the existence 01 gaussian noise. In Proceedings o l  ICASSP 91, Toronto, Canada, May 1991. Probability density functions of ocean acoustic noise processes. I n  E.J .  [a] D.H. Johusori and P.S. Rao. [9] F.W. Machell and C.S. Penrod. . 0.11 . - .  01 0 0.5 I n? I - n4 IS 1 5  Signal shape factor, x Figure 7. The probability of detection, p d ,  for three different detectors when the probability of false alarm, p j ,  is 0.01 and 0.1 as a Junction of the signal shope. The 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. IV-484 

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A simultaneous maximum likelihood estimator based on a generalized matched filter

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