Power spectrum estimation by means of relative-entropy minimization with uncertain constraints

This paper presents a new multisignal spectrum-analysis method based on the relative-entropy principle ("principle of minimum cross entropy"). The method simultaneously estimates a number of power spectra given (1) an initial ("prior") estimate of each, (2) imprecise values of the autocorrelation function of their sum, and (3) estimates of the error in measurement of the autocorrelation values. One application is to the separate estimation of the spectra of a signal and independent additive noise, based on imprecise measurements of the autocorrelations of the signal plus noise. A derivation is sketched, beginning with a discussion of relative-entropy minimization subject to quadratic constraints that require certain linear constraints to hold approximately, to within a specified error bound. The new method is an extension of multisignal relative-entropy spectrum analysis (with exact autocorrelations). The two methods are compared, and connections with previous related work are indicated. Mathematical properties of the new method are discussed, and an illustrative numerical example is presented.