^a Institute of Applied Physics RAS, Nizhny Novgorod, Russia ^b Department of Electrical and Computer Engineering, McMaster University, Ontario, Canada

Received 11 May 1998.

Available online 29 July 1999.

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.

Author Keywords: LLRT detector; Multiplicative noise; Sampling rate

Article Outline

1. Introduction

2. Problem formulation and LLRT detector

3. Performance analysis

4. Sampling rate and performance

References

1. Introduction

Signal detection/estimation in multiplicative noise is an important problem in radar, sonar and image processing [1, 2, 5, 6 and 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.

2. Problem formulation and LLRT detector

Let the samples of the known deterministic complex signal y_i be observed in the presence of multiplicative noise, i.e.,

where ξ_i is assumed to be a complex zero-mean Gaussian random process. Eq. (1) in vector notation is expressed as

where x=(x₁, x₂, …, x_N)^T, ξ=(ξ₁, ξ₂, …, ξ_N)^T, Y=diag{y₁, y₂, …, y_N}, and (·)^T stands for transpose. In the most general statement, the detection problem can be formulated as deciding between two hypotheses

where the signal matrices Y₀ and Y₁ are known a priori. Let the covariance matrix of multiplicative noise

be of arbitrary structure but be known a priori (for example, measured prior to processing [5]). Here, (·)^H stands for Hermitian transpose.

The LLRT is given by

where p(x|H_i), i=0,1, is the probability density function of the received data under the hypothesis H_i, and ν is a threshold. The covariance matrix of the observed signal (2) is given by

From (5) and (6), we obtain

where the constant terms (not depending on x) are omitted. In what follows, let us assume that

where I is the identity matrix, and S=diag{s₁, s₂, …, s_N} is a discrete-time version of known deterministic signal s(t) satisfying the weakness assumption |s(t)|1. This model is quite typical for the long-path underwater transmission [3]. The weak (known) perturbations s_i occur when an inhomogeneity crosses the transmission path.

Let us use the following matrix expansion:

Neglecting the second- and higher-order terms, we obtain from (7), (8) and (9) the approximate LLRT

3. Performance analysis

The detection performance is measured in terms of the receiver operating characteristic (ROC), or, alternatively, in terms of the detection index [4]

where

In fact, Eq. (11) represents the output SNR of the detector. From (6) and (7), it follows that

Noting that for a zero-mean Gaussian process

we obtain that

From (12), (13) and (15), we have



Finally, the detection performance is given by (11), (16) and (17). For the approximate log-likelihood ratio (10), the detection index can be simplified to

4. 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 [t_a, t_b] with the uniform sampling

Assume also that the noise covariance matrix (4) belongs to the exponential class, i.e., let

where σ_ξ² and τ_c are the noise variance and its correlation time, respectively. Exponential covariances are widely used for the modeling of multiplicative noise in sonar applications [5].

Let us find the limit of η (18) for N→∞ (or, equivalently, Δ→0). From Eq. (18) it follows that, first of all, we should evaluate the term

with arbitrary diagonal matrices

Exploiting a 3-diagonal representation for the inverse of Eq. (20) [2]

Eq. (21) becomes

Using a Taylor series expansion,

and the following representation of the Riemann sum,

we find that

Hence, we obtain from (18) and (27) that

Therefore, the detection performance can increase unlimitedly (proportionally to ) if the sampling rate increases (i.e., Δ→0), provided the observing interval is fixed. Obviously, this property does not hold in the presence of additive noise. To demonstrate this fact, consider the following signal model with additive zero-mean Gaussian noise:

The detection index after the LLRT detector for additive noise is given by [4] , where s=(s₁, s₂, …, s_N)^T. Proceeding in the same manner as for the multiplicative noise case, we obtain that in the additive noise case

Hence, unlike the multiplicative noise case, the detection index in the additive noise case is limited by the constant terms in Eq. (30). In particular, if Δ<τ_c, then the further increase of the sampling rate cannot improve the performance more.

Corresponding author; email: gershman@ieee.org