Minimum-Rate Spectrum-Blind Sampling Based on Sparse-Graph Codes

Abstract

We study the problem of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">spectrum-blind</i> sampling, that is, sampling signals with sparse spectra whose frequency support is unknown. The minimum sampling rate for this class of signals has been established as twice the measure of its frequency support; however, constructive sampling schemes that achieve this minimum rate are not known to exist, to the best of our knowledge. We propose a novel <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">constructive</i> sampling framework by leveraging a mix of tools from <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">modern coding theory</i> , which has been largely untapped in the field of sampling. We make interesting connections between the fundamental problem of spectrum-blind sampling, and that of designing erasure-correcting codes based on sparse graphs, which have both theoretically and practically revolutionized the design of modern communication systems. Our key idea is to cleverly exploit, rather than avoid, the resulting aliasing artifacts induced by subsampling, which introduce linear mixing of spectral components in the form of parity constraints for <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">sparse-graph codes</i> . We achieve this by subsampling the input signal after filtering it using a carefully designed ‘sparse-graph coded filter-bank’ structure, where the pass-band on/off patterns of the filters are designed to match the parity-check constraints of a sparse-graph code. We show that the signal reconstruction under this sampling scheme is equivalent to the fast message-passing based “peeling” decoding of sparse-graph codes for reliable transmission over erasure channels. Most importantly, we further show that the achievable sampling rate is determined by the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">rate</i> of the sparse-graph codes used in the filter bank. As a result, based on insights derived from the design of capacity-achieving sparse-graph codes (such as Low Density Parity Check codes), we can <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">simultaneously</i> approach the minimum sampling rate for spectrum-blind sampling and low computational complexity based on fast peeling-based decoding with operations per unit of time scaling <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">linearly</i> with the sampling rate.