On Compressed Sensing of Binary Signals for the Unsourced Random Access Channel

• Main Authors: Elad Romanov, Or Ordentlich
• Format: Article
• Language: English
• Published: MDPI AG 2021-05-01
• Series: Entropy
• Subjects: unsourced random access; compressed sensing; low-density parity-check codes; glauber dynamics
• Online Access: https://www.mdpi.com/1099-4300/23/5/605

Motivated by applications in unsourced random access, this paper develops a novel scheme for the problem of compressed sensing of binary signals. In this problem, the goal is to design a sensing matrix <i>A</i> and a recovery algorithm, such that the sparse binary vector <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="bold">x</mi></semantics></math></inline-formula> can be recovered reliably from the measurements <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="bold">y</mi><mo>=</mo><mi>A</mi><mi mathvariant="bold">x</mi><mo>+</mo><mi>σ</mi><mi mathvariant="bold">z</mi></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="bold">z</mi></semantics></math></inline-formula> is additive white Gaussian noise. We propose to design <i>A</i> as a parity check matrix of a low-density parity-check code (LDPC) and to recover <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="bold">x</mi></semantics></math></inline-formula> from the measurements <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="bold">y</mi></semantics></math></inline-formula> using a Markov chain Monte Carlo algorithm, which runs relatively fast due to the sparse structure of <i>A</i>. The performance of our scheme is comparable to state-of-the-art schemes, which use dense sensing matrices, while enjoying the advantages of using a sparse sensing matrix.