Estimation of linear weighting functions in Gaussian noise

The measurement of linear time-varying systems and the mapping of densely distributed radar targets provide the motivation for the problem of estimation of linear multidimensional weighting functions in Gaussian noise. The assumption that the weighting function to be estimated is a sample of a Gaussian process of known autocorrelation function, and the adoption of either a maximum a posteriori probability or a minimum mean-square error criterion for the excellence of the estimate reduces the estimation problem to that of the solution of a Fredholm equation, the kernel of which is a generalization of Woodward\´s ambiguity function. A unique solution is assured if the statistics involved have sufficiently short correlation intervals. Closed-form solutions to the Fredholm equation, in particular two-dimensional cases of time- and frequency-shifted signals, are obtained over infinite domains of the weighting function by using Fourier and Wiener-Hopf techniques, and over finite domains by solving the eigenvalue problem of the ambiguity function. Generally speaking, the estimation error decreases with the decrease in the domain area; within a domain of unit area, perfect reproduction can be approached by increasing the SNR, in agreement with previous resuits on strictly noiseless measurement of linear systems. The appropriate "optimum" signal processor performs on the received signal a linear operation that is, essentially, an "inversion" of the eigenvalue spectrum of the ambiguity function; the extent of inversion is,limited:by its accentuation of the additive noise of the interference from neighboring areas of the domain. Only under extreme circumstances of constant eigenvalue spectra or of vanishing signal-to-noise ratios is the optimum processor equivalent to a "matched filter." The performances of the optimum processor and of the matched filter are compared in several examples.