Reconstruction of Bandlimited Functions from Unsigned Samples

TL;DR

This paper demonstrates that real-valued bandlimited functions can be reconstructed from absolute samples if sampled above twice the Nyquist rate, introducing an FFT-based algorithm with exponential convergence for uniform samples.

Contribution

It establishes the conditions for reconstructing bandlimited functions from unsigned samples and presents a fast, convergent FFT-based reconstruction algorithm for uniform sampling.

Findings

Reconstruction possible above twice the Nyquist rate

FFT-based algorithm converges exponentially

Numerical tests confirm effectiveness

Abstract

We consider the recovery of real-valued bandlimited functions from the absolute values of their samples, possibly spaced nonuniformly. We show that such a reconstruction is always possible if the function is sampled at more than twice its Nyquist rate, and may not necessarily be possible if the samples are taken at less than twice the Nyquist rate. In the case of uniform samples, we also describe an FFT-based algorithm to perform the reconstruction. We prove that it converges exponentially rapidly in the number of samples used and examine its numerical behavior on some test cases.