Frequency-Selective Vandermonde Decomposition of Toeplitz Matrices with Applications

TL;DR

This paper introduces a frequency-selective Vandermonde decomposition for Toeplitz matrices, establishing conditions for existence and uniqueness, and applies it to spectral estimation and moment problems with numerical validation.

Contribution

It extends classical Vandermonde decomposition to frequency-restricted cases, linking it to the positive real lemma and enabling new semidefinite programming approaches.

Findings

Provides explicit conditions for existence and uniqueness.

Connects the decomposition to the positive real lemma.

Offers a semidefinite program for frequency-selective atomic norm.

Abstract

The classical result of Vandermonde decomposition of positive semidefinite Toeplitz matrices, which dates back to the early twentieth century, forms the basis of modern subspace and recent atomic norm methods for frequency estimation. In this paper, we study the Vandermonde decomposition in which the frequencies are restricted to lie in a given interval, referred to as frequency-selective Vandermonde decomposition. The existence and uniqueness of the decomposition are studied under explicit conditions on the Toeplitz matrix. The new result is connected by duality to the positive real lemma for trigonometric polynomials nonnegative on the same frequency interval. Its applications in the theory of moments and line spectral estimation are illustrated. In particular, it provides a solution to the truncated trigonometric $K$-moment problem. It is used to derive a primal semidefinite program formulation of the frequency-selective atomic norm in which the frequencies are known {\em a priori} to lie in certain frequency bands. Numerical examples are also provided.