  Sparse superposition codes, also called 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.
