A convex optimization approach to arma(n,m) model design from covariance and cepstral data

• Methods for determining ARMA(n, m) filters from covariance and cepstral estimates are proposed. In [C. I. Byrnes, P. Enqvist, and A. Lindquist, SIAM J. Control Optim., 41 ( 2002), pp. 23-59], we have shown that an ARMA( n, n) model determines and is uniquely determined by a window r(0), r(1),..., r(n) of covariance lags and c(1), c(2),..., c(n) of cepstral lags. This unique model can be determined from a convex optimization problem which was shown to be the dual of a maximum entropy problem. In this paper, generalizations of this problem are analyzed. Problems with covariance lags r(0), r(1),..., r(n) and cepstral lags c(1), c(2),..., c(m) of different lengths are considered, and by considering different combinations of covariances, cepstral parameters, poles, and zeros, it is shown that only zeros and covariances give a parameterization that is consistent with generic data. However, the main contribution of this paper is a regularization of the optimization problems that is proposed in order to handle generic data. For the covariance and cepstral problem, if the data does not correspond to a system of desired order, solutions with zeros on the boundary occur and the cepstral coefficients are not interpolated exactly. In order to achieve strictly minimum phase filters for estimated covariance and cepstral data, a barrier-like term is introduced to the optimization problem. This term is chosen so that convexity is maintained and so that the unique solution will still interpolate the covariances but only approximate the cepstral lags. Furthermore, the solution will depend analytically on the covariance and cepstral data, which provides robustness, and the barrier term increases the entropy of the solution.

Subject headings

NATURVETENSKAP  -- Matematik (hsv//swe)

NATURAL SCIENCES  -- Mathematics (hsv//eng)

Keyword

cepstrum

covariance

ARMA

entropy

convex optimization

MATHEMATICS

MATEMATIK

Publication and Content Type

ref (subject category)

art (subject category)