SIAM Review
Volume 43, Number 4
pp. 643-643
© 2001 Society for Industrial and Applied Mathematics


The Editors

Abstract. Since its inception in March 1999, the SIGEST section of SIAM Review has highlighted excellent papers of broad interest from SIAM's specialized journals, making SIAM readers aware of outstanding work whose content and roots span multiple areas. This issue's SIGEST selection takes that (short) tradition one step further: Christopher I. Byrnes, Sergei V. Gusev, and Anders Lindquist present "From Finite Covariance Windows to Modeling Filters: A Convex Optimization Approach," an expanded, generalized version of their paper "A Convex Optimization Approach to the Rational Covariance Extension Problem," which originally appeared in SIAM Journal on Control and Optimization, volume 37 (1998).

The SIGEST paper begins with a careful survey of the rational covariance extension problem, tracing its historical developments from the early twentieth-century work of Carathéodory, Toeplitz, and Schur to the present. Connections and applications abound: potential theory, stochastic realization, spectral estimation, trigonometric moments, circuit and systems theory, signal and speech processing, robust control, and mobile communication. The authors have gone out of their way to relate mathematical developments to practical consequences---and to bring theory to life through well-chosen examples. In one instance among many, the authors clearly explain why the easily computable maximum entropy solution cannot capture valleys in the speech spectrum and hence produces "flat" synthesized speech.

The scientific contributions of the paper are substantial, including new theoretical results as well as an efficient Newton-based interior method. Of particular interest is the formulation of appropriate spaces leading to an optimization problem that solves the rational covariance extension problem with degree constraints. Appearance of a barrier-like integral analogous to barrier terms in interior methods leads to existence of interior minimizers, a well-posed optimization problem, and a systems-theoretic consequence.

We are grateful indeed to the authors for their efforts in giving us an impressive SIGEST paper of wide scope and fascinating content.