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.