On Detection and Estimation of Wave Fields for Surveillance

This paper considers the detection and estimation of a signal field in the presence of a noise field. The wave field, which is a continuous space-time function, is converted into a discrete set of time functions by an array of transducer elements which convert the physical field quantities into other quantities appropriate for processing. The resulting set of time functions makes up a vector random process. A generalization of the one-dimensional Karhunen-Loeve expansion applied to the vector random process yields a series representation with uncorrelated coefficients. The effects of complex element weighting and of internal noise are considered in describing the noise and signal vector processes. If the noise field is Gaussian, the conditional probability density functions of the vector processes, under the hypotheses of noise alone and of signal pulse noise, are straightforwardly written, leading directly to the likelihood ratio for a completely known signal. The operation to obtain a test statistic based upon the likelihood ratio is interpreted as a set of filtering operations, time-varying in the general case where the noise field is not wide-sense stationary. When the noise field is wide-sense stationary, the field may be described by a spectral density matrix whose elements are the cross-spectral densities of the total noise at the transducers taken in pairs. The operation to obtain the test statistic is now interpreted as a set of filtering operations described by a filtering matrix.