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Signal Processing

Volume 76, Issue 3, August 1999, Pages 323-326
Signal Processing

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On LLRT detection of deterministic signals in multiplicative noise

https://doi.org/10.1016/S0165-1684(99)00053-5Get rights and content

Abstract

The LLRT detector for a known deterministic signal in the presence of multiplicative noise is studied. An interesting property of this “multiplicative” detector is discovered. For the exponential class of noise covariances, it is shown that an unrestricted increase of output SNR (detection index) can be achieved when increasing the sampling rate. Such effect does not occur in the case of LLRT detector in the presence of additive noise, where the detection index is limited by the noise correlation time.

Introduction

Signal detection/estimation in multiplicative noise is an important problem in radar, sonar and image processing [1], [2], [5], [6], [7]. Below, we consider the log-likelihood ratio test (LLRT) detector for a known deterministic signal observed in multiplicative noise. An interesting property of this detector is discovered: for the exponential class of noise covariances, an unrestricted increase of output SNR can be achieved when increasing the sampling rate. In the case of additive noise, the LLRT detector is known to have quite different behavior, i.e., when increasing the sampling rate, the detection index is known to be severely restricted by the noise correlation time.

Section snippets

Problem formulation and LLRT detector

Let the samples of the known deterministic complex signal yi be observed in the presence of multiplicative noise, i.e.,xi=ξiyi,i=1,2,…,N,where ξi is assumed to be a complex zero-mean Gaussian random process. Eq. (1) in vector notation is expressed asx=Yξ,where x=(x1,x2,…,xN)T, ξ=(ξ1,ξ2,…,ξN)T, Y=diag{y1,y2,…,yN}, and (·)T stands for transpose. In the most general statement, the detection problem can be formulated as deciding between two hypothesesx=Y0ξ,H=H0,Y1ξ,H=H1,where the signal matrices Y0

Performance analysis

The detection performance is measured in terms of the receiver operating characteristic (ROC), or, alternatively, in terms of the detection index [4]η=(S1−S0)/D0,whereSi=E{Λ(x)}|Hi,Di=var{Λ(x)}|Hi,i=0,1.In fact, Eq. (11) represents the output SNR of the detector. From , , it follows thatE{Λ(x)}=tr{R(R0−1−R1−1)}.

Noting that for a zero-mean Gaussian processE{xixk∗xlxm∗}=[R]ik[R]lm+[R]im[R]lk,we obtain thatvar{Λ(x)}=trR(R0−1−R1−1)2.From , , , we haveS0−S1=2N−tr{R0R1−1+R1R0−1},D0=N−2tr{R0R1−1}+tr{(R

Sampling rate and performance

Let us study how the detection index (18) depends on the sampling rate. Assume that the input signal x is observed within the fixed interval [ta,tb] with the uniform samplingtn=ta+(n−1)Δ,Δ=tb−taN−1,n=1,2,…,N.Assume also that the noise covariance matrix (4) belongs to the exponential class, i.e., let[Q]nm=σξ2r|n−m|,r=exp(−Δ/τc),where σξ2 and τc are the noise variance and its correlation time, respectively. Exponential covariances are widely used for the modeling of multiplicative noise in sonar

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