4 research outputs found
A globally convergent matricial algorithm for multivariate spectral estimation
In this paper, we first describe a matricial Newton-type algorithm designed
to solve the multivariable spectrum approximation problem. We then prove its
global convergence. Finally, we apply this approximation procedure to
multivariate spectral estimation, and test its effectiveness through
simulation. Simulation shows that, in the case of short observation records,
this method may provide a valid alternative to standard multivariable
identification techniques such as MATLAB's PEM and MATLAB's N4SID
Uncertainty Bounds for Spectral Estimation
The purpose of this paper is to study metrics suitable for assessing
uncertainty of power spectra when these are based on finite second-order
statistics. The family of power spectra which is consistent with a given range
of values for the estimated statistics represents the uncertainty set about the
"true" power spectrum. Our aim is to quantify the size of this uncertainty set
using suitable notions of distance, and in particular, to compute the diameter
of the set since this represents an upper bound on the distance between any
choice of a nominal element in the set and the "true" power spectrum. Since the
uncertainty set may contain power spectra with lines and discontinuities, it is
natural to quantify distances in the weak topology---the topology defined by
continuity of moments. We provide examples of such weakly-continuous metrics
and focus on particular metrics for which we can explicitly quantify spectral
uncertainty. We then consider certain high resolution techniques which utilize
filter-banks for pre-processing, and compute worst-case a priori uncertainty
bounds solely on the basis of the filter dynamics. This allows the a priori
tuning of the filter-banks for improved resolution over selected frequency
bands.Comment: 8 figure
Novel Results on the Factorization and Estimation of Spectral Densities
This dissertation is divided into two main parts. The first part is concerned with one of the most classical and central problems in Systems and Control Theory, namely the factorization of rational matrix-valued spectral densities, commonly known as the spectral factorization problem. Spectral factorization is a fundamental tool for the solution of a variety of problems involving second-order statistics and quadratic cost functions in control, estimation, signal processing and communications. It can be thought of as the frequency-domain counterpart of the ubiquitous Algebraic Riccati Equation and it is intimately connected with the celebrated Kálmán-Yakubovich-Popov Lemma and, therefore, to passivity theory. Here, we provide a rather in-depth and comprehensive analysis of this problem in the discrete-time setting, a scenario which is becoming increasingly pervasive in control applications. The starting point in our analysis is a general spectral factorization result in the same vein of Dante C. Youla. Building on this fundamental result, we then investigate some key issues related to minimality and parametrization of minimal spectral factors of a given spectral density. To conclude, we show how to extend some of the ideas and results to the more general indefinite or J-spectral factorization problem, a technique of paramount importance in robust control and estimation theory.
In the second part of the dissertation, we consider the problem of estimating a spectral density from a finite set of measurements. Following the Byrnes-Georgiou-Lindquist THREE (Tunable High REsolution Estimation) paradigm, we look at spectral estimation as an optimization problem subjected to a generalized moment constraint. In this framework, we examine the global convergence of an efficient algorithm for the estimation of scalar spectral densities that hinges on the Kullback-Leibler criterion. We then move to the multivariate setting by addressing the delicate issue of existence of solutions to a parametric spectral estimation problem. Eventually, we study the geometry of the space of spectral densities by revisiting two natural distances defined in cones for the case of rational spectra. These new distances are used to formulate a "robust" version of THREE-like spectral estimation