New results on the rational covariance extension problem with degree constraint☆
Introduction
In recent years, a new theory for the rational covariance extension problem (with degree constraint) (RCEP) has emerged [16], [17], [18], [11], [8], [9] with applications in high-resolution spectral estimation, speech synthesis and other areas. The new theory gives a complete parametrization of all degree constrained solutions of the Carathéodory extension problem [16] which is different from the classical linear-fractional parametrization of the set of all solutions without the degree constraint. In the following, we provide a brief account of previous works on the RCEP.
The breakthrough concept leading to a constructive theory of rational interpolation with degree constraint is convex optimization, first proposed by Byrnes et al. [8]. In that paper, a constructive proof of complete parametrization of all strictly positive real solutions of the rational covariance extension problem was given, by convex optimization techniques, in terms of positive-definite pseudopolynomials of a bounded degree. However, the paper does not treat positive, but not necessarily strictly positive, real solutions corresponding to non-negative, but not positive definite, pseudopolynomials. The complete parametrization of all solutions of the rational interpolation problem with degree constraint (including the RCEP and its Nevanlinna–Pick counterpart) was finally settled by Georgiou [18] by non-constructive arguments. The convex optimization technique was then adapted to solve the related Nevanlinna–Pick interpolation problem with degree constraint in [6], but the positive, but not strictly positive, solutions were still left untreated. More recently, the ideas in [8], [18], [6] were unified and extensively generalized to develop a theory of generalized interpolation with a complexity constraint in [7]. The general theory includes the RCEP as a special case and covers the case where parametrizing pseudopolynomials are non-negative, but not positive definite.
Our present study is also based on an extension of the convex optimization method of [8] in a spirit similar to [7], for the special case of the RCEP (but readily extends to the Nevanlinna–Pick interpolation case). However, our analysis proceeds differently from [7] and continues a partial extension of [8] given in the papers [22], [21]. It is noted in [22] that there are important differences between the approach of that paper and [7]. For example, results of [22], such as a certain necessary and sufficient condition for boundedness of a solution, do not follow obviously from [7]. The analysis in [7] is carried out by reposing the problem in the setting of contractive functions on the unit disc via a certain bilinear transformation. This transformation effectively avoids complications or awkward details which may arise when dealing with positive real functions. Interestingly enough, in connection with the last point, the functional studied in [7] always has a stationary maximizer (see the penultimate part of the proof of [7, Theorem 1] on uniqueness of a solution, p. 13), whereas the minimizer of the functional , which was introduced in [8] (the real case) and [6] (the complex case) and investigated further in [22], [21] and this paper, need not be stationary. In this paper we tackle the problem directly in the original positive-real setting, without recourse to the space of contractive functions. A possible advantage of this, for the special case of the RCEP, is that the analysis is done purely on a complex Euclidean space instead of a function space as in [7]. Moreover, we conclude that solving the RCEP is essentially equivalent to finding the minimizers of a class of (strictly) convex functionals defined on a subset of the complex Euclidean space. This is done by establishing a new result on a bijective correspondence between denominator polynomials of non-strictly-positive solutions of the RCEP and the minimizers of the class of convex functionals associated with non-strictly-positive pseudopolynomials (Theorem 8). As a corollary to that result, we obtain an alternative constructive derivation of a theorem of Georgiou on complete parametrizationof all solutions of the RCEP [18], and a new proof of a homeomorphism which was established in [3] for the special case of real interpolators. Furthermore, the bijective correspondence result raises the interesting question of whether efficient numerical techniques can be developed to find minimizers which are not stationary, but lie on the boundary of the domain of their associated functionals. Previously, it has been proposed that all real solutions of the RCEP may be computed by solving a set of non-linear equations via a continuation method [4], but no convergence results are provided. We suggest that it may be possible to develop alternative numerical schemes for computing solutions of the RCEP.
In Section 5, we generalize the homeomorphism result to also allow variation in the covariance data. For the special case of strictly positive pseudopolynomials, Byrnes and Lindquist [12, Theorem 6.6] have shown a stronger version of this generalization, i.e., where a diffeomorphism is established, in the context of the Nevanlinna–Pick interpolation problem with degree constraint (but holds analogously for the RCEP). Our contribution is allowing for non-strictly-positive pseudopolynomials, but we only establish a weaker homeomorphism property. Another work related to variation of covariance data is [13]. However, there are two features of this paper which contrast it to [13]. The first contrasting feature is that [13] derives the unique pair of (normalized) partial covariance sequence and positive-definite bounded spectral density which minimizes a certain Kullback–Leibler divergence criterion under some moment constraints, whereas here we are not interested in such an optimal pair, but we show that pairs of partial covariance sequence and pseudopolynomial data are in homeomorphic correspondence with the graph symbols of solutions of the RCEP. In particular, we may perform a continuous coordinate transformation from the first pair to the latter pair and vice versa. Secondly, the case where the associated pseudopolynomial is non-negative, but not positive definite, is not considered in [13]. Indeed, in this case, the solution of the RCEP may be unbounded and not integrable, while [13] restricts the solution to be integrable (see Eq. 6 therein). On the other hand, we allow for non-negative, but not positive definite, pseudopolynomials and do not impose integrability of the solutions. The importance of considering simultaneous variation of the covariance and pseudopolynomial data lies in the fact that in practice, for example in spectral estimation, both data are typically unknown and have to be estimated. Continuity implies that the resulting spectral density estimate will be robust to small errors in the estimates of the pair of data.
We also mention the forthcoming paper [14] which was brought to our attention by a referee for this paper. It solves a generalized moment problem with complexity constraint; however, the problem treated there is rather different since the non-negative functions , which are monotone non-decreasing and of bounded variation on a compact interval of the real line, sought in [14] must satisfy a finite set of moment conditions and can be expressed as for some functions and which are non-negative for almost all and for which the ratio is integrable on (the latter conditions on are also referred to collectively as “complexity constraint”). This is not the case in general for the RCEP; (unbounded) solutions f of the RCEP that have one or more poles on the unit circle do not correspond, via the moment constraints and for , to absolutely continuous functions on (see, e.g., [16, Eqs. (3.10)–(3.12), p. 36]).
Our interest in the RCEP is motivated by the problem of approximation of stochastic systems with non-coercive (i.e., can have zeros on the unit circle), possibly non-rational, spectral densities arising in practice. These processes appear in applications such as aircraft control under the influence of turbulence [1] and control of adaptive optics [23]. Specifically, we are interested in a new algorithm for computing canonical spectral factors of spectral densities of the type mentioned above. Many spectral factorization algorithms, such as the Bauer and Schur algorithms, which are based on Cholesky decomposition of a semi-infinite Toeplitz matrix [24], [26] are known to converge slowly when the spectral density has zeros close to or on the unit circle. A new approach to spectral factorization based on the RCEP (with solutions which can be specifically chosen to correspond to pseudopolynomials with roots on the unit circle) has recently been proposed in [19], [20]. In particular, in [19], we have successfully applied a new algorithm to compute approximate spectral factors of the non-coercive and non-rational Kolmogorov and von Karman power spectra that arise in the study of atmospheric turbulence.
Section snippets
Notation and definitions
In this section we introduce the main notation and definitions which are used throughout the paper.
and denote the completion and boundary of a set A, respectively,
, , and denote the set of real numbers, complex numbers, the open unit and the unit circle, respectively,
denotes the real part of .
and denote the transpose and conjugate transpose of a complex matrix C, respectively,
,
denotes the parahermitian conjugate of a complex
The rational covariance extension problem (with degree constraint) (RCEP)
Definition 1 A sequence of complex numbers (with ) is a partial covariance sequence (PCS) (of order n) if the Toeplitz matrix , with , is positive definite. Problem 2 RCEP Given a PCS , find all rational functions of McMillan degree at most such that the first coefficients of the Taylor series expansion of about 0 is .
The RCEP gives an added twist of degree constraint to the classical Carathéodory extension problem which can be solved with
An analysis of all solutions of the RCEP
Define the mapping byClearly Q is a bijective map.
For any we consider the functional defined byFor , the properties of this functional are given in [8], [6], and it was argued in [21], [22] that they continue to hold for . The properties are as follows: Theorem 5 has the following properties for any : is finite and continuous at any
Generalization of results to variations in both the covariance and pseudopolynomial data
Thus far we have only looked at the continuous relationship between and when is varied and the PCS is fixed. However, the ideas used in deriving Lemma 6, Theorem 8 and Corollary 11 can be adapted easily to analyze the case where the PCS is allowed to vary. In this section we shall state generalizations of Lemma 6 and Corollary 11. Since the main ideas here are the same as in the last section, we only sketch the proofs.
Let and define the
Conclusions and further research
The contributions of this paper is an alternative analysis of the rational covariance extension problem (with degree constraint) (RCEP) which yields new results in Theorem 8 for solutions parametrized by pseudopolynomials in and in the part of Theorem 13 which extends the domain of the homeomorphism to .
Previously, it has been shown that any real solution of the RCEP can be found by solving non-linear equations for [3]. Corollary 9 of this paper shows that elements of are
Acknowledgements
The author is grateful to the referees for their helpful suggestions for improving the quality of this paper and to Prof. A.C. Antoulas for coordinating the review process.
References (26)
Introduction to Random Processes in Engineering
(1995)Continuity of the spectral factorization mapping
J. London Math. Soc. (2)
(2004)- A. Blomqvist, G. Fanizza, R. Nagamune, Computation of bounded degree Nevanlinna–Pick interpolants by solving nonlinear...
- et al.
Computation of bounded degree Nevanlinna–Pick interpolants by solving nonlinear equations
- et al.
Matrix-valued Nevanlinna–Pick interpolation with complexity constraint: an optimization approach
IEEE Trans. Automat. Control
(2003) - et al.
A generalized entropy criterion for Nevanlinna–Pick interpolation with degree constraint
IEEE Trans. Automat. Control
(2001) - C.I. Byrnes, T.T. Georgiou, A. Lindquist, A. Megretski, Generalized interpolation in H∞ with a complexity constraint,...
- et al.
A convex optimization approach to the rational covariance extension problem
SIAM J. Control Optim.
(1998) - et al.
From finite covariance windows to modeling filters: a convex optimization approach
SIAM Rev.
(2001) - et al.
On the well-posedness of the rational covariance extension problem
A complete parametrization of all positive rational extensions of a covariance sequence
IEEE Trans. Automat. Control
On the duality between filtering and Nevanlinna–Pick interpolation
SIAM J. Control Optim.
The uncertain generalized moment problem with complexity constraint
Cited by (10)
Multidimensional rational covariance extension with approximate covariance matching
2018, SIAM Journal on Control and OptimizationLikelihood Analysis of Power Spectra and Generalized Moment Problems
2017, IEEE Transactions on Automatic ControlFurther results on multidimensional rational covariance extension with application to texture generation
2017, 2017 IEEE 56th Annual Conference on Decision and Control, CDC 2017The Multidimensional Moment Problem with Complexity Constraint
2016, Integral Equations and Operator TheoryMultidimensional rational covariance extension with applications to spectral estimation and image compression
2016, SIAM Journal on Control and Optimization
- ☆
This work is supported by National ICT Australia, Ltd. (NICTA). National ICT Australia is funded by the Australian Government's Department of Communications, Information Technology and the Arts and the Australian Research Council through Backing Australia's Ability and the ICT Centre of Excellence Program.