Discrete Signal Reconstruction by Sum of Absolute Values

TL;DR

This paper introduces a linear programming approach for reconstructing unknown discrete signals from incomplete measurements by minimizing a weighted sum of absolute values, extending compressed sensing techniques to finite alphabets.

Contribution

It proposes a novel linear programming method for discrete signal reconstruction that leverages known probability distributions, improving upon existing exponential-complexity methods.

Findings

Effective reconstruction demonstrated through examples

Linear programming formulation enables practical computation

Method extends compressed sensing to finite alphabet signals

Abstract

In this letter, we consider a problem of reconstructing an unknown discrete signal taking values in a finite alphabet from incomplete linear measurements. The difficulty of this problem is that the computational complexity of the reconstruction is exponential as it is. To overcome this difficulty, we extend the idea of compressed sensing, and propose to solve the problem by minimizing the sum of weighted absolute values. We assume that the probability distribution defined on an alphabet is known, and formulate the reconstruction problem as linear programming. Examples are shown to illustrate that the proposed method is effective.