Convolutional Compressed Sensing Using Deterministic Sequences

TL;DR

This paper introduces a new class of deterministic circulant matrices for convolutional compressed sensing, enabling efficient recovery of sparse signals in multiple domains, with promising theoretical and practical implications.

Contribution

It proposes a novel deterministic sequence-based circulant matrix construction for convolutional CS, expanding beyond random convolution methods.

Findings

Effective recovery of sparse signals in time, frequency, and DCT domains.

Analysis of coherence parameters for deterministic sensing matrices.

Demonstrated advantages over random convolution in certain scenarios.

Abstract

In this paper, a new class of circulant matrices built from deterministic sequences is proposed for convolution-based compressed sensing (CS). In contrast to random convolution, the coefficients of the underlying filter are given by the discrete Fourier transform of a deterministic sequence with good autocorrelation. Both uniform recovery and non-uniform recovery of sparse signals are investigated, based on the coherence parameter of the proposed sensing matrices. Many examples of the sequences are investigated, particularly the Frank-Zadoff-Chu (FZC) sequence, the \textit{m}-sequence and the Golay sequence. A salient feature of the proposed sensing matrices is that they can not only handle sparse signals in the time domain, but also those in the frequency and/or or discrete-cosine transform (DCT) domain.