Distributed detection of a signal in generalized Gaussian noise

The problem of distributed detection of a signal in incompletely specified noise is considered. The noise assumed belongs to the generalized Gaussian family and the sensors in the distributed network employ the Wilcoxon test. The sensors pass the test statistics to a fusion center, where a hypothesis testing results in a decision regarding the presence or the absence of a signal. Three monotone and admissible fusion center tests are formulated. Restricted numerical evaluation over a certain parameter range of the noise distribution and the range of signal level indicates that these tests yield performances at comparable levels. I . INTRODUCTION The problem of detection of a signal using a distributed network of sensors has been analyzed in the literature. In order to save transmission bandwidths, the sensors process the information they receive and pass condensed information, such as the test statistics or the decisions with regard to the presence or the absence of a signal, to the fusion center. For the best performance, it is essential that the processing at the sensors and at the fusion be optimized So far, the problem analyzed in the literature assumes a complete statistical knowledge of the received signal. However, in sonar and other underwater detection problems, the signal is embedded in a noise whose characteristics are not completely known and are changing with time. In such situations, the sensors’ statistics must be based on some general characteristics of the noise density function rather than on some specific form of noise density function. In this correspondence, we consider the distributed detection of a constant signal in generalized Gaussian noise. Such a noise density function approximates physical noise encountered in different situations [ lo] , [ l l ] . In Section I1 we discuss test statistics at the sensors and at the fusion. In Section 111 we present the performance analysis of three [11-[91. Manuscript received April 29, 1988; revised October 5, 1988. This work was supported by the SDIOIIST and managed by the Office of Naval Research under Contract N00014-86-K-0515. The authors are with the Department of Electrical Engineering, Southem Illinois University, Carbondale, IL 62901-6603. IEEE Log Number 8926688. different tests at the fusion center. Numerical results are shown for a three sensors network with three samples per sensor. We conclude our discussion in Section IV. 11. THE GENERALIZED GAUSSIAN OISE A N D DISTRIBUTED TESTS The problem of detection of a constant signal in additive noise is described by the following hypotheses testing: Ho: XI = nl ( 1 ) Hi: X , = n, + 8 , j an integer. We assume that the noise nl has a symmetric density function described by the following equation [ 1 I]: The noise has unit variance and hence a satisfies the relation a 2 / c = r ( 3 / c ) / r ( i / c ) . (3) By varying the parameter c , we can control the tail of the noise density. When c equals 2 the noise reduces to the Gaussian, and for c equals 1 it becomes Laplace. In general, smaller values of c represent heavy tails. For detecting a signal in symmetric noise at a sensor, a variety of nonparametric tests such as the sign test and the Wilcoxon test exist [12]. Our choice of the Wilcoxon test is motivated by the fact that i) the Wilcoxon test is nonparametric, i i ) its performance is comparable to other nonparametric tests, i i i ) it performs better than the sign test in most cases, and iv) the Wilcoxon statistic takes on a finite number of discrete values. Fig. 1 shows the distributed network of sensors and the fusion center. The statistics T, , T I , . . . , TN are the Wilcoxon statistics, and the test at the fusion is given as follows: