Capacity-achieving Spatially Coupled Sparse Superposition Codes with AMP Decoding
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Sparse superposition codes (SPARCs) are a class of codes for efficient communication over the AWGN channel at rates approaching the channel capacity. In a SPARC, codewords are sparse linear combinations of columns of an i.i.d. Gaussian design matrix. 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 enhance theerror 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. The asymptotic mean squared error (MSE) of the AMP decoder 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 SPARCs. Combined with the state evolution prediction, this result implies that spatially coupled SPARCs are capacity-achieving. The proof technique used to establish the main result is also used to 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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