Optimality in detecting targets with unknown location

Abstract

An optimal test does not exist for the problem of detecting a known target with unknown location in additive Gaussian noise. A common solution uses a generalised likelihood ratio testing (GLRT) formalism, where a maximum likelihood estimate of the unknown location parameter is used in a likelihood ratio test. The performance of this test is commonly assessed by comparing it to the ideal matched filter, which assumes the target location known in advance. This comparison is of limited utility, however, since the fact that the location is unknown has a significant effect on the detectability of the target. We demonstrate that a uniformly most powerful invariant (UMPI) optimal test exists for a specific class of unknown target location problems, where observations are discrete and shifts are defined circularly. Since this approach explicitly models the location as unknown, an assessment of the suboptimality of competing tests becomes meaningful. It is shown that for certain examples in this class the GLRT performance is negligibly different from that of the optimal test.

Introduction

Many problems in signal and image processing involve detecting a known target with unknown location in a sequence of data. This constitutes a composite hypothesis testing problem, with the actual target location playing the role of an unknown parameter under the target present hypothesis.

The most common structured approach to this problem is to use the generalised likelihood ratio test (GLRT) [1], which has been studied at length in the engineering and applied statistics literature. It involves obtaining a maximum likelihood estimate of the location parameter, and performing a simple hypothesis test on the data under the assumption that the estimate is accurate. The procedure constitutes a special case of a plug-in classifier formulation [2], but is generally favoured on account of asymptotic optimality properties [3].

The popularity of the GLRT for the unknown target location problem is perhaps more accidental than planned: the formulation for transient targets very often leads naturally to an implementation in a sliding window framework. Here a detection statistic is calculated for sequential overlapping intervals of received data, and a decision of target presence is made if any of the test values exceeds some threshold. This is a natural procedure to adopt in detection systems which operate continuously through time.

The use of the GLRT in signal processing is usually justified in terms of its asymptotic optimality properties: as estimates of the unknown parameters become more accurate, the GLRT performance approaches that of the ideal matched filter which assumes the values known. It is for this reason that one commonly sees the GLRT performance assessed by comparing it to that of the ideal matched filter. Since the matched filter has perfect knowledge of all quantities involved in the detection, it cannot be improved upon, and its performance represents an upper bound on the achievable performance of any test.

The bound is not tight, however, and a test may perform considerably worse than the ideal matched filter while still not being significantly suboptimal. That is, the presence of the unknown parameter can significantly change the detectability properties of the target. A more meaningful comparison requires that the unknown parameters be explicitly modelled as such.

An optimal test does not exist for the general problem of detecting a target with unknown location in additive noise. Such a test does, however, exist for one specific case, in particular where samples form a finite discrete sequence and shifts are defined circularly over the observation interval. In this paper an optimal test is developed under these conditions: an invariant hypothesis testing formulation is followed, where the tests considered are restricted to those which are invariant to cyclic permutations of the observations, and the best test is found within this class. This test is uniformly most powerful invariant (UMPI) under the invariance condition described.

Although the UMPI test has certain advantages over the GLRT for the unknown cyclic permutation target problem, it is argued that its main benefit lies in providing a baseline against which the GLRT can be compared. It is demonstrated for some simple targets that the differences between the GLRT and the optimal UMPI test performance are very small. This provides definitive justification for the traditional GLRT solution to the problem.

The need to define shifts circularly is quite a restriction, although in some signal analysis scenarios the condition is at least approximately appropriate [4]. Many real problems, on the other hand, involve the sampling of continuous-time signals over a finite time interval, which will instead result in targets being truncated at the edges of the observation. Unfortunately, this situation does not exhibit sufficient symmetry to afford a sensible invariance condition, and therefore does not aid in the selection of optimal tests. Nonetheless, if the observation interval is long and the targets short and transient then these edge effects may be expected to play a minimal role, and results obtained for the one case should extend to the other. Therefore results obtained for the GLRT under cyclic permutation invariance may be considered to be at least approximately appropriate for related cases with shifts defined differently.

Even without the significance to unknown location target detection, the test presented is interesting purely from an academic point of view, as an exercise in invariant hypothesis testing. As such, the formulation complements other ideal invariant detector formulations that have appeared in the literature, for example invariance to signal scaling, invariance to subspace interference [5], [6], and invariance to unknown covariance elements [7], [8].

The structure of this paper is as follows. In Section 2 the cyclic permutation invariant detection problem formulation is presented, and the standard GLRT solution is presented. In Section 3 the optimal (UMPI) test for this problem is derived. Section 4 compares the performance of these two tests, using Monte Carlo simulations for a simple set of signals. Section 5 briefly discusses extensions of the testing principle to other problems.

Throughout this paper, data sequences are represented as column vectors denoted by bold roman characters. The notation $\mathbf{x}:N[\mathbf{m},\mathbf{C}]$ is used to signify that the random vector $\mathbf{x}$ has a multivariate normal (MVN) distribution, with mean $\mathbf{m}$ and covariance matrix $\mathbf{C}$.