SIAM
Review

Volume 43,
Number 4

pp. 643-643

© 2001 Society for Industrial and Applied Mathematics
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SIGEST

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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.