Beyond Nyquist: Efficient Sampling of Sparse Bandlimited Signals

TL;DR

This paper introduces a random demodulator system that efficiently samples sparse bandlimited signals at rates much lower than Nyquist, enabling stable reconstruction using nonlinear methods.

Contribution

It presents a novel sampling system for sparse signals that reduces sampling rates exponentially compared to Nyquist, with theoretical analysis supporting empirical results.

Findings

Requires O(K log(W/K)) samples per second for stable reconstruction

Sampling rate is exponentially lower than Nyquist rate of W Hz

Supports stable signal recovery using convex programming

Abstract

Wideband analog signals push contemporary analog-to-digital conversion systems to their performance limits. In many applications, however, sampling at the Nyquist rate is inefficient because the signals of interest contain only a small number of significant frequencies relative to the bandlimit, although the locations of the frequencies may not be known a priori. For this type of sparse signal, other sampling strategies are possible. This paper describes a new type of data acquisition system, called a random demodulator, that is constructed from robust, readily available components. Let K denote the total number of frequencies in the signal, and let W denote its bandlimit in Hz. Simulations suggest that the random demodulator requires just O(K log(W/K)) samples per second to stably reconstruct the signal. This sampling rate is exponentially lower than the Nyquist rate of W Hz. In contrast with Nyquist sampling, one must use nonlinear methods, such as convex programming, to recover the signal from the samples taken by the random demodulator. This paper provides a detailed theoretical analysis of the system's performance that supports the empirical observations.