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Convex Optimization Problems Involving Finite autocorrelation sequences
, 2001
"... We discuss convex optimization problems where some of the variables are constrained to be finite autocorrelation sequences. Problems of this form arise in signal processing and communications, and we describe applications in filter design and system identification. Autocorrelation constraints in opt ..."
Abstract

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We discuss convex optimization problems where some of the variables are constrained to be finite autocorrelation sequences. Problems of this form arise in signal processing and communications, and we describe applications in filter design and system identification. Autocorrelation constraints in optimization problems are often approximated by sampling the corresponding power spectral density, which results in a set of linear inequalities. They can also be cast as linear matrix inequalities via the KalmanYakubovichPopov lemma. The linear matrix inequality formulation is exact, and results in convex optimization problems that can be solved using interiorpoint methods for semidefinite programming. However, it has an important drawback: to represent an autocorrelation sequence of length n, it requires the introduction of a large number (n(n + 1)/2) of auxiliary variables. This results in a high computational cost when generalpurpose semidefinite programming solvers are used. We present a more efficient implementation based on duality and on interiorpoint methods for convex problems with generalized linear inequalities.
Properties of the convex cone of vectors with autocorrelated components
"... This paper reviews some properties of the set of vectors with autocorrelated components. This set appears in some signal processing problems, in particular filter synthesis or statistical estimation. It turns out to be a closed convex cone enjoying several representations and various geometrical pr ..."
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This paper reviews some properties of the set of vectors with autocorrelated components. This set appears in some signal processing problems, in particular filter synthesis or statistical estimation. It turns out to be a closed convex cone enjoying several representations and various geometrical properties. The aim of this paper is to gather different aspects of the geometry of this cone. We adopt a convex analysis point of view to present known and new results.