A Maximum Entropy Enhancement for a Family of High-Resolution Spectral Estimators

Abstract

Abstract-Structured covariances occurring in spectral analysis, filtering and identification need to be estimated from a finite observation record. The corresponding sample covariance usually fails to possess the required structure. This is the case, for instance, in the Byrnes-Georgiou-Lindquist THREE-like tunable, high-resolution spectral estimators. There, the output covariance 6 of a linear filter is needed to initialize the spectral estimation technique. The sample covariance estimate6, however, is usually not compatible with the filter. In this paper, we present a new, systematic way to overcome this difficulty. The new estimate 6 is obtained by solving an ancillary problem with an entropic-type criterion. Extensive scalar and multivariate simulation shows that this new approach consistently leads to a significant improvement of the spectral estimators performances.

A new metric for multivariate spectral estimation leading to lowest complexity spectra

“…4) In general the estimateΣ does not satisfy the conditions stated in Theorem 3.1. In order to guarantee feasibility, we compute a suitable matrixΣ by solving an ancillary optimization problem as in [6]. 5) Introduce a prior spectral density Ψ.…”

“…4 Indeed, following the lines detailed in [6] it is possible to obtain directly a basis of Range(Γ) by solving only N Lyapunov equations.…”

On the Geometry of Covariance Matrices

“…However, due to statistical noise, a "sample covariance" may only approximately satisfy those structural constraints. Thus, it is standard to consider covariance approximation problems of the form: (12) where is an appropriate distance/divergence measure [16]- [18]. In particular, using either or in (12), this becomes (13) which, when the set is defined by linear constraints as is the case for Toeplitz matrices, is a convex program.…”

A Maximum Entropy Approach to the Realizability of Spin Correlation Matrices

“…An example is provided by the covariance extension problem and its generalization; see [9][10][11][12][13][14]. These problems pose a number of theoretical and computational challenges, especially in the multivariable framework, for which we also refer the reader to [15][16][17][18][19][20][21][22]. Besides signal processing, significant applications of this theory are found in modeling and identification [23][24][25], H ∞ robust control [26,27], and biomedical engineering [28].…”