Singular non-Gaussian measures in detection and estimation theory

If a mathematical model of a signal detection problem is such that there exists a detector which achieves zero error, the model is called singular. Such models are usually not acceptable. In this paper, various sufficient conditions for singular detection and estimation are presented. For the case of a known signal, second-moment conditions are given which imply singularity of detection in the most general kind of noise. For the case of random signals, no such general result exists. Let the signal be a known function of some random parameter and let the detection problem corresponding to each value of be singular. It is shown that if has a discrete distribution or if the noise is Gaussian, then detection is singular. Finally, if is wide-sense stationary, if the signal is the sum of randomly spaced Fourier transformable signals, and if certain moment conditions are satisfied, then one can not only singularly detect the signal, but can also singularly estimate the unknown parameters of the signal--at least when is Gaussian.