Cepstral coefficients, covariance lags, and pole-zero models for finite data strings

One of the most widely used methods of spectral estimation in signal and sp eech processing is linear predictive coding (LPC). LPC has some attractive features, which account for its popularity, including the properties that t he resulting modeling filter i) matches a finite window of n + 1 covariance lags, ii) is rational of degree at most n, and iii) has stable zeros and p oles. The only limiting factor of this methodology is that the modeling fil ter is "all-pole," i.e., an autoregressive (AR) model. In this paper, we present a systematic description of all autoregressive mo ving-average (ARMA) models of processes that have properties i)-iii) in the context of cepstral analysis and homomorphic filtering. Indeed, we show th at each such ARMA model determines and is completely determined by its fini te windows of cepstral coefficients and covariance lags. This characterizat ion has an intuitively appealing interpretation of a characterization by us ing measures of the transient and the steady-state behaviors of the signal, respectively. More precisely, we show that these nth-order windows form lo cal coordinates for all ARMA models of degree n and that the pole-zero mode l can be determined from the windows as the unique minimum of a convex obje ctive function. We refine this optimization method by first noting that the maximum entropy design of an LPC filter is obtained by maximizing the zero th cepstral coefficient, subject to the constraint i). More generally, we m odify this scheme to a more well-posed optimization problem where the covar iance data enters as a constraint and the linear weights of the cepstral co efficients are "positive"-in a sense that a certain pseudo-polynomial is po sitive-rather succinctly generalizing the maximum entropy method. This new problem is a homomorphic filter generalization of the maximum entropy metho d, providing a procedure for the design of any stable, minimum-phase modeli ng filter of degree less or equal to n that interpolates the given covarian ce window We conclude the paper by presenting an algorithm for realizing th ese biters in a lattice-ladder form, given the covariance window and the mo ving average part of the model. While we also show how to determine the mov ing average part using cepstral smoothing, one can make use of any good a p riori estimate for the system zeros to initialize the algorithm. Indeed, we conclude the paper with an example of this method, incorporating an exampl e from the literature on ARMA modeling.