Multiple Signal Counting

• Published in: Journal of the Society for Industrial and Applied Mathematics Vol. 11; no. 2; pp. 360 - 397
• Main Authors: Gersch, Will, Ash, Robert
• Format: Journal Article
• Language: English
• Published: Philadelphia Society for Industrial and Applied Mathematics 01.06.1963
• Subjects: A priori knowledge; Asymptotic theory; Estimators; Legends; Mathematical independent variables; Mathematical moments; Maximum likelihood method; Noise; Parameter estimation; Radar; Random variables; Signal noise; Supernova remnants
• ISSN: 0368-4245; 0036-1399; 2168-3484; 1095-712X
• DOI: 10.1137/0111026

This research is concerned with estimation of the number of signals that are present in a sample containing signals plus noise. The sample is divided into M cells, of which N contain signals and noise, and M - N contain noise alone. The signals are assumed distributed in amplitude, with the distribution known to within p parameters. The noise statistics are assumed known. The observable quantities are Ri, i = 1, 2, ⋯, M the magnitude of the envelope of the cell contents. The problem is to estimate N. Multiple Signal Counting is an idealized model for a radar operating in a multiple signal environment. The problem is characterized by p + 1 parameters, namely N, the number of signals, and p unknown signal parameters. The minimum data case is considered in which the p + 1 functions of the Ri; v1, v2, ⋯, vp + 1 are formed. The random variables vj, j = 1, 2, ⋯, p + 1 are functions of N and thus form a model from which it is possible to estimate N. Subject to certain mild restrictions on the mean and the central moments of the vj, the estimator N̂ of N is defined as the solution of the set of p + 1 nonlinear equations, implicit in N, in which the observation vj is equated to its expected value. A solution of the equations for N̂ is described. The estimator N̂ is asymptotically unbiased and the asymptotic relative variance, Var $\hat N/N$, goes to zero. Var N̂ can be evaluated without requiring that the equations for N̂ be solved. In addition if the vj are asymptotically normal, N̂ is asymptotically normal. Two means of physically realizing the data v1, v2, ⋯, vp + 1 are available. These correspond to taking p + 1 integral powers of the Ri and comparing the Ri with p + 1 different thresholds. The functions v1, v2, ⋯, vp + 1 are formed by summing the results of each over all the cells. The two methods are assessed and compared numerically for several signal amplitude distributions and several unknown signal distribution parameters. The comparison is based on the computation of the variance of the fractional error in the estimate of N. In both methods the quantity N appears linearly and the estimator N̂ is asymptotically normal. The results indicate that under some circumstances it is possible to reliably estimate the number of signals that are present when in fact all the signals are imbedded in the noise.