Exploring connections between Sparse Fourier Transform computation and decoding of product codes

We show that the recently proposed Fast Fourier Aliasing-based Sparse Transform (FFAST) algorithm for computing the Discrete Fourier Transform (DFT) [1] of signals with a sparse DFT is equivalent to iterative hard decision decoding of product codes. This connection is used to derive the thresholds for sparse recovery based on a recent analysis by Justensen [2] for computing thresholds for product codes. We first extend Justesen´s analysis to d-dimensional product codes and compute thresholds for the FFAST algorithm based on this. Additionally, this connection also allows us to analyze the performance of the FFAST algorithm under a burst sparsity model in addition to the uniformly random sparsity model which was assumed in prior work [1].