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Line Spectrum Representation for Vector Processes With Application to Frequency Estimation

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Document pages: 10 pages

Abstract: A positive semidefinite Toeplitz matrix, which often arises as the finitecovariance matrix of a stationary random process, can be decomposed as the sumof a nonnegative multiple of the identity corresponding to a white noise, and asingular term corresponding to a purely deterministic process. Moreover, thesingular nonnegative Toeplitz matrix admits a unique characterization in termsof spectral lines which are associated to an oscillatory signal. This is thecontent of the famous Carathéodory-Fejér theorem. Its importance liesin the practice of extracting the signal component from noise, providinginsights in modeling, filtering, and estimation. The multivariate counterpartof the theorem concerning block-Toeplitz matrices is less well understood, andin this paper, we aim to partially address this issue. To this end, we firstestablish an existence result of the line spectrum representation for a finitecovariance multisequence of some underlying random vector field. Then, we givea sufficient condition for the uniqueness of the representation, which indeedholds true in the special case of bivariate time series. Equivalently, weobtain the Vandermonde decomposition for positive semidefinite block-Toeplitzmatrices with $2 times 2$ blocks. The theory is applied to the problem offrequency estimation with two measurement channels within the recentlydeveloped framework of atomic norm minimization. It is shown that exactfrequency recovery can be guaranteed in the noiseless case under suitableconditions.

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