589 research outputs found
Capacity-Achieving Sparse Superposition 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. The codebook is defined in terms of a Gaussian design matrix, and
codewords are sparse linear combinations of columns of the matrix. In this
paper, we propose an approximate message passing decoder for sparse
superposition codes, whose decoding complexity 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 with an appropriate power
allocation. Simulation results are provided to demonstrate the performance of
the decoder at finite blocklengths. We introduce a power allocation scheme to
improve the empirical performance, and demonstrate how the decoding complexity
can be significantly reduced by using Hadamard design matrices.Comment: 25 pages, 4 figures. IEEE Transactions on Information Theor
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 (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
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
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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
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