Recovering Signals from Lowpass Data

TL;DR

This paper investigates conditions for recovering signals from lowpass filtered data, especially in shift-invariant spaces, and proposes pre-processing methods like modulation to enhance recoverability.

Contribution

It provides necessary and sufficient conditions for signal recovery from low-frequency data and introduces modulation techniques to improve reconstruction when data is insufficient.

Findings

Derived conditions on cutoff frequency for recovery

Identified generators enabling perfect reconstruction

Proposed modulation methods to reduce bandwidth requirements

Abstract

The problem of recovering a signal from its low frequency components occurs often in practical applications due to the lowpass behavior of many physical systems. Here we study in detail conditions under which a signal can be determined from its low-frequency content. We focus on signals in shift-invariant spaces generated by multiple generators. For these signals, we derive necessary conditions on the cutoff frequency of the lowpass filter as well as necessary and sufficient conditions on the generators such that signal recovery is possible. When the lowpass content is not sufficient to determine the signal, we propose appropriate pre-processing that can improve the reconstruction ability. In particular, we show that modulating the signal with one or more mixing functions prior to lowpass filtering, can ensure the recovery of the signal in many cases, and reduces the necessary bandwidth of the filter.