Analysis of Signal Reconstruction Algorithms Based on Consistency Constraints

Abstract

A fundamental problem in signal processing called signal reconstruction, or signal recovery, is the determination of a signal from a sequence of samples obtained from the signal. The sampling process can be viewed as obtaining measurements from a set of measurement vectors in an N-dimensional space. Studies on the reconstruction problem have resulted in major breakthroughs in technology in the past century, and practical solutions to the problem are still essential in the advancement of fields such as image processing and speech recognition. Consistent reconstruction and Rangan-Goyal algorithm are two algorithms that produce estimates of a signal from consistency constraints when the measurements are corrupted with i.i.d uniformly distributed noises. Under the assumption that the measurements are taken with i.i.d. unit-norm random vectors, the second error moments of both algorithms are known to converge with a rate of O(N^2). In this work, we showed that the general p-th error moments of both algorithms converge with a rate of O(N^p) under general admissibility conditions on the sampling distribution that no longer require the measurement vectors to be unit-norm.