Generalized Approximate Message-Passing Decoder for Universal Sparse Superposition Codes

Sparse superposition (SS) codes were originally proposed as a capacity-achieving communication scheme over the additive white Gaussian noise channel (AWGNC) [1]. Very recently, it was discovered that these codes are universal, in the sense that they achieve capacity over any memoryless channel under generalized approximate message-passing (GAMP) decoding [2], although this decoder has never been stated for SS codes. In this contribution we introduce the GAMP decoder for SS codes, we confirm empirically the universality of this communication scheme through its study on various channels and we provide the main analysis tools: state evolution and potential. We also compare the performance of GAMP with the Bayes-optimal MMSE decoder. We empirically illustrate that despite the presence of a phase transition preventing GAMP to reach the optimal performance, spatial coupling allows to boost the performance that eventually tends to capacity in a proper limit. We also prove that, in contrast with the AWGNC case, SS codes for binary input channels have a vanishing error floor in the limit of large codewords. Moreover, the performance of Hadamard-based encoders is assessed for practical implementations

Replica Analysis and Approximate Message Passing Decoder for Superposition Codes

Superposition codes are efficient for the Additive White Gaussian Noise channel. We provide here a replica analysis of the performances of these codes for large signals. We also consider a Bayesian Approximate Message Passing decoder based on a belief-propagation approach, and discuss its performance using the density evolution technic. Our main findings are 1) for the sizes we can access, the message-passing decoder outperforms other decoders studied in the literature 2) its performance is limited by a sharp phase transition and 3) while these codes reach capacity as $B$ (a crucial parameter in the code) increases, the performance of the message passing decoder worsen as the phase transition goes to lower rates.Comment: 5 pages, 5 figures, To be presented at the 2014 IEEE International Symposium on Information Theor

Spatially Coupled Sparse Regression Codes: Design and State Evolution Analysis.

We consider the design and analysis of spatially coupled sparse regression codes (SC-SPARCs), which were recently introduced by Barbier et al. for efficient communication over the additive white Gaussian noise channel. SC-SPARCs can be efficiently decoded using an Approximate Message Passing (AMP) decoder, whose performance in each iteration can be predicted via a set of equations called state evolution. In this paper, we give an asymptotic characterization of the state evolution equations for SC-SPARCs. For any given base matrix (that defines the coupling structure of the SC-SPARC) and rate, this characterization can be used to predict whether or not AMP decoding will succeed in the large system limit. We then consider a simple base matrix defined by two parameters $(\omega, \Lambda)$, and show that AMP decoding succeeds in the large system limit for all rates $R < \mathcal{C}$. The asymptotic result also indicates how the parameters of the base matrix affect the decoding progression. Simulation results are presented to evaluate the performance of SC-SPARCs defined with the proposed base matrix.Comment: 8 pages, 6 figures. A shorter version of this paper to appear in ISIT 201

Capacity-achieving Sparse Regression Codes via approximate message passing decoding

Sparse superposition codes were recently introduced by Barron and Joseph for reliable communication over the AWGN channel at rates approaching the channel capacity. In this code, the codewords are sparse linear combinations of columns of a design matrix. In this paper, we propose an approximate message passing decoder for sparse superposition codes. The complexity of the decoder scales linearly with the size of the design matrix. The performance of the decoder is rigorously analyzed and it is shown to asymptotically achieve the AWGN capacity. We also provide simulation results to demonstrate the performance of the decoder at finite block lengths, and introduce a power allocation that significantly improves the empirical performance.RV would like to acknowledge support from a Marie Curie Career Integration Grant (GA Number 631489). AG is supported by an EPSRC Doctoral Training Award.This is the author accepted manuscript. The final version is available from IEEE via http://dx.doi.org/10.1109/ISIT.2015.728280