Linear prediction and maximum entropy spectral analysis of finite bandwidth signals in noise

The optimal minimum—mean—square—error discrete filters are obtained for finite bandwidth signals in noise represented by a rational pole— zero model in the spectral domain. An analytical technique called the method of undetermined coefficients is used to solve the Wiener equation for the optimal prediction coefficients. These coefficients are then used to examine (a) the frequency response of the Wiener prediction filter, and (b) the maximum entropy spectral estimator. Special cases considered include the limiting case of narrow bandwidth signals and large prediction filter lengths. N r (k) = o26(k) + I 2 exp(—ct kj)cosu k n 0 n n n n1 (1) In (1), o represents the white noise power; o2,e, and a represent the power, bandwidth, and frequency , rLpectively, of the n—tb signal; and 6(k) is the Kronecker delta function. The model represented by (1) is a mixed autoregressive—moving average (ANNA) model with 2N AR terms and 2N NA terms, Equivalently, this model can be represented by the transfer function 2N —p +jO n n AU (z—e I, Introduction Recently the use of linear prediction filters (LPF) for enhancing the detectability of mar— rowband signals in broadband noise has been considered [1—6], For the case of sinusoidal signals in uncorrelated noise it has been shown in [1—6] that am LPF forms narrow bandpass filters around the sinusoids. The widths of these filters is inversely porportional to the length L of the LPF and approach zero as L becomes very large. It is the purpose of this paper to extend the results in references [1—6] to the case of signals with non—zero bandwidth. An example could be that of a stationary bamdpass signal process corrupted by additive noise. Also, we will consider the closely related problem of obtaining spectral estimates by the. maximum entropy method (MEN) for the case of finite bandwidth signals in additive noise. The approach used will be to first develop a pole—zero model for the signal plus noise spectrum. Them the normal equations for the LPF coefficients will be solved by the method of undetermined coefficients. In this paper it will be assumed that the moise background is unoorrelated, although the extension of the pole—zero models presented here to the correlated noise case is straightforward0 A reasonably simple model for an input process consisting of N finite bandwidth signals in unoorrel— ated noise may be represented by the autooorrela— tion function r (k) where x(t) …