Advanced Theory of Signal Detection

In this and following two chapters, we will be concerned with detection of weak known, random, and composite signals in observations governed by the generalized noisy signal model proposed in Chapter 1, which accommodates multiplicative and signal-dependent noise as well as purely-additive noise. In particular, we will consider the locally optimum detection of known signals, based on the generalized version of the Neyman-Pearson fundamental lemma of statistical hypothesis testing in this chapter. Again, we will use the terms ’known signal’ and ’deterministic signal’ interchangeably in this book.

Although the problem of finding good detection schemes for signals buried in noise processes has been considered in various situations, much of such work has concentrated mainly on detection of known signals with or without a finite number of random parameters. Less attention has been paid to the problem of detecting random (stochastic) signals, although there does exist a reasonable literature on this subject. (Again, we will use the terms ’random signal’ and ’stochastic signal’ interchangeably in this book.) Random signals are of practical interest in a number of situations. For example, in underwater applications it is often impossible to represent desired signals by known or parametric models because of random dispersion resulting from turbulence and inhomogeneities in the propagation medium, and also often because of the very nature of the signal source.

Ever since the threshold detection theory has first been studied early in the last century, numerous authors have extended the theory in various directions. Therefore, we can certainly find many interesting studies on various detection schemes for either random or known signals from some early studies to more recent ones. For instance, locally optimum detection of known and random signals, locally optimum detection with a space diversity concept, and locally optimum detection in a generalized observation model are some of the typical examples.

The problem of signal detection can be considered as a parameter testing problem of a null hypothesis against an alternative hypothesis, as we have seen in Chapters 2–4, for example. As a consequence, the knowledge of a priori information on the parameter is essential for establishing the hypothesis testing problem. Unfortunately, it is very difficult, if not impossible, to exactly estimate the value of a parameter in practice. It is well-known that the design of a suitable discrete-time detector for signals in corrupting noise is sometimes complicated because of inexact knowledge of the statistics of the noise process: if we are not able to get a priori information on the distribution of the parameter, we cannot design an optimum parametric detector. Although we may estimate the parameters in some cases, small deviations of the parameters from the theoretic model in the real environment may lead sometimes to a significant performance degradation of the optimum parametric detector. Specifically, the performance of a detector is quite sensitive to the statistics of the noise process, and an optimum detector based on Neyman-Pearson lemma often performs poorly in the case where a priori knowledge of the noise statistics is not exact. In such a case, we shall need nonparametric detectors, which ensure a constant falsealarm probability. The best known nonparametric detectors are those based on signs and ranks of the received data samples.

In this chapter, we consider the locally optimum rank detection of random and composite signals in additive, multiplicative, and signal-dependent noise models. Applications of the basic ideas and concepts of locally optimum rank tests and their extensions to the particular problem of random and composite signal detection in various noise models are addressed in this chapter.

By viewing the weak signal detection problem as a problem of binary hypothesis testing in statistical inference, we get a convenient mathematical framework. When we consider the signal detection problem in this way, we often assume that the observations are independent. The assumption of statistically independent sampling is frequently violated, however, in discrete-time signal detection applications, and the optimum detectors designed under this assumption are no longer optimum in practice. Such a situation becomes more crucial as the sampling rate gets higher. It is also known that dependent noise models are quite useful in modeling the inter-user interference which arises frequently in the multiple access communication systems. Thus, investigations on signal detections in dependent noise should be considered. In several recent studies, the solutions under some dependent observation models have been considered for designing signal detection systems, which are then implemented with or without memory unit.

During the past years, it has been pointed out by several authors that imprecise information is common in signal processing areas. For example, information such as ’the noise characteristic of a certain channel is impulsive’ is imprecise in nature since we cannot clearly (quantitatively) describe the impulsive noise. It is needless to say that there are many other examples containing vagueness in information. Based on this observation, applications of the fuzzy set theory have been devoted to several signal processing problems including digital signal restoration, image segmentation, biomedical signal processing, and digital filter design.