Stability of the Recovery of Missing Samples in Derivative Oversampling

TL;DR

This paper investigates the stability of reconstructing band-limited signals with missing samples and derivatives, analyzing eigenvalues of associated matrices and proposing regularization methods for ill-conditioned cases.

Contribution

It provides eigenvalue estimates for matrices involved in missing sample recovery and introduces a numerical regularization approach for ill-conditioned scenarios.

Findings

Eigenvalue estimates depend on oversampling and missing samples.

Recovery becomes ill-conditioned with consecutive missing samples.

Tikhonov regularization improves stability in noisy conditions.

Abstract

This paper deals with the problem of reconstructing a band-limited signal when a finite subset of its samples and of its derivative are missing. The technique used, due to P.J.S.G. Ferreira, is based on the use of a particular frame for band-limited functions and the relative oversampling formulas. We study the eigenvalues of the matrices arising in the procedure of recovering the lost samples, finding estimates of their eigenvalues and their dependence on the oversampling parameter and on the number of missing samples. When the missing samples are consecutive, the problem may become very ill-conditioned. We present a numerical procedure to overcome this difficulty, also in presence of noisy data, by using Tikhonov regularization techniques.