ESPRIT versus ESPIRA for reconstruction of short cosine sums and its application

Abstract

In this paper we introduce two algorithms for stable approximation with and recovery of short cosine sums. The used signal model contains cosine terms with arbitrary real positive frequency parameters and therefore strongly generalizes usual Fourier sums. The proposed methods both employ a set of equidistant signal values as input data. The ESPRIT method for cosine sums is a Prony-like method and applies matrix pencils of Toeplitz + Hankel matrices while the ESPIRA method is based on rational approximation of DCT data and can be understood as a matrix pencil method for special Loewner matrices. Compared to known numerical methods for recovery of exponential sums, the design of the considered new algorithms directly exploits the special real structure of the signal model and therefore usually provides real parameter estimates for noisy input data, while the known general recovery algorithms for complex exponential sums tend to yield complex parameters in this case.

On the conditioning of some structured generalized eigenvalue problems

“…Of course, in the case of an overdetermined generalized eigenvalue problem, where the matrices become rectangular, the AAA algorithm may play a more important part in the conditioning of the Loewner generalized eigenvalue problem than in the square case. We learned from [6] that the latter can deliver more accurate results in the case of noisy data.…”