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Capacity-achieving Spatially Coupled Sparse Superposition Codes with AMP Decoding
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