Polynomial Fourier Domain as a Domain of Signal Sparsity

TL;DR

This paper introduces a novel compressive sensing method for polynomial phase signals using the Polynomial Fourier transform, enabling sparse representation where traditional domains fail, and extends it with local polynomial Fourier transforms for enhanced analysis.

Contribution

It proposes a new CS reconstruction approach leveraging Polynomial Fourier transform for polynomial phase signals, including a generalized method using Local Polynomial Fourier transform.

Findings

Polynomial Fourier transform ensures sparsity of polynomial phase signals.

The approach improves reconstruction quality over traditional methods.

Local Polynomial Fourier transform captures local signal behavior effectively.

Abstract

A compressive sensing (CS) reconstruction method for polynomial phase signals is proposed in this paper. It relies on the Polynomial Fourier transform, which is used to establish a relationship between the observation and sparsity domain. Polynomial phase signals are not sparse in commonly used domains such as Fourier or wavelet domain. Therefore, for polynomial phase signals standard CS algorithms applied in these transformation domains cannot provide satisfactory results. In that sense, the Polynomial Fourier transform is used to ensure sparsity. The proposed approach is generalized using time-frequency representations obtained by the Local Polynomial Fourier transform (LPFT). In particular, the first-order LPFT can produce linear time-frequency representation for chirps. It provides revealing signal local behavior, which leads to sparse representation. The theory is illustrated on examples.