Random Detection Research Articles

Intensity Discrimination of Fluctuating Signals

In this paper, the theoretical aspects of the fluctuations inherent in random noise are discussed, insofar as these influence detection of random signals. Brief elucidations of the fundamentals of statistical detection theory and of sampling theory are given. The model developed for the detection process closely follows a proposal by Green, but with the major exception that a kind of internal noise is supposed to be involved in the process. The auditory mechanism is assumed to measure the average intensity of the signal presented over a certain time T. In doing so, the evaluation is hampered by random activity so that the measurement is carried out less accurately than possible. The detection finally involves a likelihood-ratio decision procedure. The theory explains experimental thresholds of random-noise signals very well. Agreement with data on thresholds of very short noise bursts is less convincing. Finally, the theory explains very elegantly why experimental psychometric functions for wide-band signals have the same slope as those for narrow-band signals. Instances in which higher slopes are reported for experimental psychometric functions for wide-band signals can most likely be traced down to having involved experimental procedures that leave too much uncertainty in the listener's observations.

Comments on “Detection of Random Acoustic Signals by Receivers with Distributed Elements: Optimum Receiver Structures for Normal Signal and Noise Fields” [David Middleton and Herbert L. Groginsky, J. Acoust. Soc. Am. 38, 727–737 (1965)

A previous paper has shown an interesting structure for the optimum array-processing system using long observation time to detect weak stationary Gaussian signal in stationary Gaussian noise fields. For dominant amplifier and system noise (Case I discussed by Middleton and Groginsky), this structure is “factored.” The previous result is generalized to linear nonidentical hydrophones. A valid procedure for finding causal optimum filters is presented. The results are extended to the case or known signal; the optimum filters are not matched to the signal, but to a waveform dependent on signal, noise, and array structure.

Detection of Random Acoustic Signals by Receivers with Distributed Elements: Optimum Receiver Structures for Normal Signal and Noise Fields

This paper deals with the passive detection of noiselike signals in the presence of both external (environmental) noise and self-generated (receiver) noise, using an array of transducers. Starting with the Bayesian formulation of the general detection problem, a set of matrix integral equations is derived whose solution yields the optimum detector function. By regarding the resultant time-varying filters as operators and the defining matrix integral equations as a set of operational equations, it is possible to examine the underlying structure of the optimum detector most easily. It is shown, thereby, that factorization of the space-time operations (i.e., separation of the required filter into two successive operations—the first depending only on the geometry of the array, the second depending only on the statistics of the noise processes) is not, in general, possible in optimum systems. Only in the strong-signal case has it been possible to show that factorization analogous to conventional beam forming can be utilized in optimum array detection. Another interesting conclusion is that an optimum detector is not superdirective in the limiting case of strong external directive noise.

Optimum detection of random signals in noise, with application to scatter-multipath communication--I

Solutions are obtained in open form for the optimum, probability-computing detector of either Gaussian signals, or known signals transmitted via scatter-paths, where the signals have been further perturbed by additive white Gaussian noise. The optimum receiver operates on the received waveforms with filter-functions and biasing constants determined by pairs of inhomogeneous and homogeneous integral equations, respectively. General solution in closed form has not been obtained, but it is possible to draw a few broad conclusions, among them that the filter-functions can be physically realizable. Approximate solution (for the optimum scatter-path receiver) at small signal-to-noise ratios yields a block diagram having interesting implications. For a single-scatter-path, the optimum receiver may be interpreted as the combination of a correlator with an optimum estimator of the Wiener type. Certain special cases in which complete solution is possible have been investigated in detail, and appropriate curves are presented. The role and performance of the probability-computing detector in an optimum decision-making receiver, for the types of channels considered, is deferred to a companion paper.

Optimum sequential detection of signals in noise

A device which performs a sequential test on a mixture of signal and noise is called a Sequential Detector. With such a device, two thresholds are introduced, each of which is associated with a terminal decision. The length of the detection process (integration time) is not fixed in advance of the experiment but is a random variable, depending on the progress of the test. An optimum form of such a test exists and is characterized by the fact that detection is performed on the average faster than with conventional; i.e., fixed sample size (optimum or non-optimum), devices. The sequential analysis developed by A. Wald is fully applied in this paper, but an important new feature is the treatment of correlated samples and its application to continuous sampling processes. In the introduction, the problem is presented within the framework of Wald's Statistical Decision Theory, and the optimum properties of sequential detectors are discussed accordingly. It is pointed out that a sequential detector is defined in terms of conditional probabilities and hence its operation is essentially independent of a priori information, although the average risk or cost of detection necessarily depends on the a priori signal data. The general theory is illustrated with some cases of special interest. The simplest example of detection involves independent, discrete observations; e.g., the case of a pulsed carrier in normal noise. Here the optimum detector still has the well-known <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\log I_0</tex> structure, but it is shown that the square law approximation for weak signals requires a bias correction due to the fourth order term. Coherent sequential detection of causal signals in normal noise provides another illustration of the theory. An interesting result is that the probabilities of error do not depend on the shape of the filter, provided the proper computer is used. The use of RC-filtered noise illustrates the treatment of continuous detection processes. Finally, the reduction in minimum detectable signal level resulting from the use of a sequential detector is computed. A third example is the sequential detection of random signals in normal noise. It is shown that, although the optimum computer involves the knowledge of the inverted correlation matrix, the average length of the test does not. Hence a curious result is obtained that in this instance detection can be performed in an arbitrarily short time. The paper concludes with a discussion of the practical necessity of truncating the detection process and exact expressions for the error probabilities of such truncated tests are derived and compared with Wald's original approximations.