On the Factorization of Rational Discrete-Time Spectral Densities

In this paper, we consider an arbitrary matrix-valued, rational spectral density $\Phi(z)$. We show with a constructive proof that $\Phi(z)$ admits a factorization of the form $\Phi(z)=W^\top (z^{-1})W(z)$, where $W(z)$ is stochastically minimal. Moreover, $W(z)$ and its right inverse are analytic in regions that may be selected with the only constraint that they satisfy some symplectic-type conditions. By suitably selecting the analyticity regions, this extremely general result particularizes into a corollary that may be viewed as the discrete-time counterpart of the matrix factorization method devised by Youla in his celebrated work (Youla, 1961).Comment: 34 pages, no figures. Revised version with partial rewriting of Section I and IV, added Section VI with a numerical example and other minor changes. To appear in IEEE Transactions of Automatic Contro

On Minimal Spectral Factors with Zeroes and Poles lying on Prescribed Region

In this paper, we consider a general discrete-time spectral factorization problem for rational matrix-valued functions. We build on a recent result establishing existence of a spectral factor whose zeroes and poles lie in any pair of prescribed regions of the complex plane featuring a geometry compatible with symplectic symmetry. In this general setting, uniqueness of the spectral factor is not guaranteed. It was, however, conjectured that if we further impose stochastic minimality, uniqueness can be recovered. The main result of his paper is a proof of this conjecture.Comment: 14 pages, no figures. Revised version with minor modifications. To appear in IEEE Transactions of Automatic Contro

On the existence of a solution to a spectral estimation problem \emph{\`a la} Byrnes-Georgiou-Lindquist

A parametric spectral estimation problem in the style of Byrnes, Georgiou, and Lindquist was posed in \cite{FPZ-10}, but the existence of a solution was only proved in a special case. Based on their results, we show that a solution indeed exists given an arbitrary matrix-valued prior density. The main tool in our proof is the topological degree theory.Comment: 6 pages of two-column draft, accepted for publication in IEEE-TA

On a Fejer-Riesz factorization of generalized trigonometric polynomials

Function theory on the unit disc proved key to a range of problems in statistics, probability theory, signal processing literature, and applications, and in this, a special place is occupied by trigonometric functions and the Fejer-Riesz theorem that non-negative trigonometric polynomials can be expressed as the modulus of a polynomial of the same degree evaluated on the unit circle. In the present note we consider a natural generalization of non-negative trigonometric polynomials that are matrix-valued with specified non-trivial poles (i.e., other than at the origin or at infinity). We are interested in the corresponding spectral factors and, specifically, we show that the factorization of trigonometric polynomials can be carried out in complete analogy with the Fejer-Riesz theorem. The affinity of the factorization with the Fejer-Riesz theorem and the contrast to classical spectral factorization lies in the fact that the spectral factors have degree smaller than what standard construction in factorization theory would suggest. We provide two juxtaposed proofs of this fundamental theorem, albeit for the case of strict positivity, one that relies on analytic interpolation theory and another that utilizes classical factorization theory based on the Yacubovich-Popov-Kalman (YPK) positive-real lemma.Comment: 11 page

Spectral Factorization of Rank-Deficient Rational Densities

Though there have been hundreds of methods on solving rational spectral factorization, most of them are based on a positive definite density matrix assumption. In this work, we propose a novel approach on the spectral factorization of a low-rank spectral density, to a minimum-phase full-rank factor. Compared with other several approaches on low-rank spectral factorizations, our approach uses the deterministic relation inside a factor, leading to a high computation efficiency. In addition, we shall show that this method is easily used in identification of low-rank processes and Wiener Filter.Comment: 25 page

The Spectral Approach to Linear Rational Expectations Models

This paper considers linear rational expectations models in the frequency domain under general conditions. The paper develops necessary and sufficient conditions for existence and uniqueness of particular and generic systems and characterizes the space of all solutions as an affine space in the frequency domain. It is demonstrated that solutions are not generally continuous with respect to the parameters of the models, invalidating mainstream frequentist and Bayesian methods. The ill-posedness of the problem motivates regularized solutions with theoretically guaranteed uniqueness, continuity, and even differentiability properties. Regularization is illustrated in an analysis of the limiting Gaussian likelihood functions of two analytically tractable models.Comment: JEL Classification: C10, C32, C62, E3

Choosing between identification schemes in noisy-news models

This paper is about identifying structural shocks in noisy-news models using structural vector autoregressive moving average (SVARMA) models. We develop a new identification scheme and efficient Bayesian methods for estimating the resulting SVARMA. We discuss how our identification scheme differs from the one which is used in existing theoretical and empirical models. Our main contributions lies in the development of methods for choosing between identification schemes. We estimate specifications with up to 20 variables using US macroeconomic data. We nd that our identification scheme is preferred by the data, particularly as the size of the system is increased and that noise shocks generally play a negligible role. However, small models may overstate the importance of noise shocks

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

Rational Covariance Extension, Multivariate Spectral Estimation, and Related Moment Problems: Further Results and Applications

This dissertation concerns the problem of spectral estimation subject to moment constraints. Its scalar counterpart is well-known under the name of rational covariance extension which has been extensively studied in past decades. The classical covariance extension problem can be reformulated as a truncated trigonometric moment problem, which in general admits infinitely many solutions. In order to achieve positivity and rationality, optimization with entropy-like functionals has been exploited in the literature to select one solution with a fixed zero structure. Thus spectral zeros serve as an additional degree of freedom and in this way a complete parametrization of rational solutions with bounded degree can be obtained. New theoretical and numerical results are provided in this problem area of systems and control and are summarized in the following. First, a new algorithm for the scalar covariance extension problem formulated in terms of periodic ARMA models is given and its local convergence is demonstrated. The algorithm is formally extended for vector processes and applied to finite-interval model approximation and smoothing problems. Secondly, a general existence result is established for a multivariate spectral estimation problem formulated in a parametric fashion. Efforts are also made to attack the difficult uniqueness question and some preliminary results are obtained. Moreover, well-posedness in a special case is studied throughly, based on which a numerical continuation solver is developed with a provable convergence property. In addition, it is shown that solution to the spectral estimation problem is generally not unique in another parametric family of rational spectra that is advocated in the literature. Thirdly, the problem of image deblurring is formulated and solved in the framework of the multidimensional moment theory with a quadratic penalty as regularization