期刊
IEEE TRANSACTIONS ON INFORMATION THEORY
卷 67, 期 7, 页码 4446-4484出版社
IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC
DOI: 10.1109/TIT.2021.3083733
关键词
Decoding; Standards; Compressed sensing; AWGN channels; Modulation; Iterative decoding; Sparse matrices; Sparse superposition codes; spatial coupling; approximate message passing decoding; state evolution; AWGN channel; coded modulation
资金
- Engineering and Physical Sciences Research Council (EPSRC) Doctoral Training Award
- Turing Fellowship through the Alan Turing Institute
- National Science Foundation (NSF) through Computing and Communications Foundation (CCF) [1849883]
- Simons Institute for the Theory of Computing
- NTT Research
- Direct For Computer & Info Scie & Enginr
- Division of Computing and Communication Foundations [1849883] Funding Source: National Science Foundation
Sparse superposition codes, also known as sparse regression codes, are efficient for communication over the AWGN channel at rates approaching channel capacity. By utilizing the spatially coupled structure and AMP decoding, these codes can achieve channel capacity. Numerical simulations show that spatially coupled SPARCs perform well in finite lengths and outperform LDPC codes in terms of error performance.
Sparse superposition codes, also referred to as sparse regression codes (SPARCs), are a class of codes for efficient communication over the AWGN channel at rates approaching the channel capacity. In a standard SPARC, codewords are sparse linear combinations of columns of an i.i.d. Gaussian design matrix, while in a spatially coupled SPARC the design matrix has a block-wise structure, where the variance of the Gaussian entries can be varied across blocks. A well-designed spatial coupling structure can significantly enhance the error performance of iterative decoding algorithms such as Approximate Message Passing (AMP). In this paper, we obtain a non-asymptotic bound on the probability of error of spatially coupled SPARCs with AMP decoding. Applying this bound to a simple band-diagonal design matrix, we prove that spatially coupled SPARCs with AMP decoding achieve the capacity of the AWGN channel. The bound also highlights how the decay of error probability depends on each design parameter of the spatially coupled SPARC. An attractive feature of AMP decoding is that its asymptotic mean squared error (MSE) can be predicted via a deterministic recursion called state evolution. Our result provides the first proof that the MSE concentrates on the state evolution prediction for spatially coupled designs. Combined with the state evolution prediction, this result implies that spatially coupled SPARCs with the proposed band-diagonal design are capacity-achieving. Using the proof technique used to establish the main result, we also obtain a concentration inequality for the MSE of AMP applied to compressed sensing with spatially coupled design matrices. Finally, we provide numerical simulation results that demonstrate the finite length error performance of spatially coupled SPARCs. The performance is compared with coded modulation schemes that use LDPC codes from the DVB-S2 standard.
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