On a Parametric Spectral Estimation Problem

Abstract

We consider an open question posed in Zhu and Baggio (2017) on the uniqueness of the solution to a parametric spectral estimation problem.

Multivariable analytic interpolation with complexity constraints: A modified Riccati approach

“…CONCLUDING REMARKS We have extended our previous results [31] for the scalar case to the matrix case. However, multivariable versions of analytic interpolation with rationality constraints have been marred by difficulties to establish existence and, in particular, uniqueness in the various parameterizations [2], [22], [23], [24], [25], [26], [27], [29], [30], and we have encountered similar difficulties here. Our approach attacks these problems from a different angle and might put new light on these challenges.…”

“…On the other hand, in recent years there have been a number of results [25], [26], [27], [29], [30] on the question of existence and uniqueness of the multivariate analytic interpolation problem, mostly for the covariance extension problem (m = 0, n 0 = n + 1), but there are so far only partial results and for special structures of the prior (in our case Σ(z)). Especially the question of uniqueness has proven elusive.…”

“…The multivariable case (ℓ > 1) is much harder, and the nice spectral-zero assignability present in the scalar case appears to be lost or at lease elusive. Restrictive classes of such problems have been considered in large number of papers [2], [22], [23], [24], [25], [26], [28], [27], [29], [30], but the theory remains incomplete, and many problems have been left open.…”

On the Uniqueness Result of Theorem 6 in "Relative Entropy and the Multivariable Multidimensional Moment Problem"

“…The claim that ∇h(Λ) vanishes nowhere in L Γ + is true in the special case when the prior Ψ = ψI, namely, a scalar spectral density function times the identity matrix. Details can be found in [1] itself; see also [23], [31]. An important observation is that the Jacobian in that case is a self-adjoint operator, and in fact, it is equal to the negative Hessian of a certain cost function.…”

“…Since then it has been significantly developed and extended to the multivariate case. We mention an incomplete list of contributions [13]- [18] in the scalar case, and [1], [19]- [31] for the multivariate counterpart. In that framework, the steady-state covariance matrix of the output process of a rational filter is used as data for the reconstruction of the input spectrum, which naturally admits a formulation as a generalized moment problem.…”