On sampling theorem with sparse decimated samples: exploring branching spectrum degeneracy

TL;DR

This paper explores a new spectrum degeneracy concept enabling the recovery of sequences from sparse decimated samples, allowing approximation of band-limited functions from fewer samples than the Nyquist rate.

Contribution

It introduces a novel spectrum degeneracy framework and demonstrates its application to sparse sampling, surpassing traditional Nyquist rate limitations.

Findings

Sequences with spectrum degeneracy can be recovered from sparse samples.

Band-limited functions can be approximated by functions recoverable from fewer samples.

The method bypasses the classical Nyquist sampling restriction.

Abstract

The paper investigates possibility of recovery of sequences from their decimated subsequences. It is shown that this recoverability is associated with certain spectrum degeneracy of a new kind, and that a sequences of a general kind can be approximated by sequences featuring this degeneracy. This is applied to sparse sampling of continuous time band-limited functions. The paper shows that these functions allow an arbitrarily close approximation by functions that can be recovered from sparse equidistant samples with sampling distance larger than the distance defined by the critical Nyquist rate for the underlying function. This allows to bypass, in a certain sense, the restriction on the sampling rate defined by the Nyquist rate.