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IEEE International Symposium on Information Theory (ISIT)

Importance sampling for loop-free decoding trees with application to low-density parity-check codes

B. Xia, W.E. Ryan

Published Jan 1, 2003• 2 citations
Publisher

Abstract

An importance sampling scheme for the decoding of loop-free decoding trees is proposed. It is shown that this scheme is asymptotically efficient and that, for an arbitrary tree and a given estimation precision, the number of simulation runs needed is inversely proportional to the standard deviation of the channel noise. This work has its application to the decoding of low-density parity-check codes.

DOI
10.1109/isit.2003.1228134

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IEEE International Symposium on Information Theory (ISIT)•Dec 23, 2004•11 citations
An importance sampling scheme for the decoding of loop-free decoding trees is proposed. It is shown that this scheme is asymptotically efficient and that, for an arbitrary tree and a given estimation precision, the number of simulation runs needed is inversely proportional to the standard deviation of the channel noise. This work has its application to the decoding of low-density parity-check codes.
Low-density parity-check (LDPC) codes have shown capacity-approaching performance with soft iterative decoding algorithms. Simulating LDPC codes at very low error rates normally takes an unacceptably long time. We consider importance sampling (IS) schemes for the error rate estimation of LDPC codes, with the goal of dramatically reducing the necessary simulation time. In IS simulations, the sample distribution is biased to emphasize the occurrence of error events and efficiency can be achieved with properly biased sample distributions. For LDPC codes, we propose an IS scheme that overcomes a difficulty in traditional IS designs that require codebook information. This scheme is capable of estimating both codeword and bit error rates. As an example, IS gains on the order of 105 are observed at a bit error rate (BER) of 10-15 for a (96, 48) code. We also present an importance sampling scheme for the decoding of loop-free multiple-layer trees. This scheme is asymptotically efficient in that, for an arbitrary tree and a given estimation precision, the required number of simulations is inversely proportional to the noise standard deviation. The motivation of this study is to shed light on an asymptotically efficient IS design for LDPC code simulations. For an example depth-3 regular tree, we show that only 2400 simulation runs are needed to achieve a 10% estimation precision at a BER of 10-75. Similar promising results are also shown for a length-9 rate-1/3 regular code after being converted to a decoding tree. Finally, we consider a convolutionally coded CDMA system with iterative multiuser detection and decoding. In contrast to previous work in this area, a differential encoder is inserted to effect an interleaver gain. We view the CDMA channel as a periodically time-varying ISI channel. The receiver jointly decodes the differential encoders and the CDMA channel with a combined trellis, and shares soft output information with the convolutional decoders in an iterative (turbo) fashion. Dramatic gains over conventional convolutionally coded systems are demonstrated via simulation. We also show that there exists an optimal code rate under a bandwidth constraint. The performance and optimal code rates are also demonstrated via density evolution analysis.
We introduce an importance sampling (IS) scheme for fast performance evaluation of the linear block codes with message-passing decoding. This novel scheme overcomes the existing difficulties in IS to some extent when code length is large and requires codebook information. Experiments show very high IS gains for iteratively decoded linear block codes such as single parity-check (SPC) codes and block product codes, as well as low density parity-check (LDPC) codes. We also provide a scheme based on our IS method to find the minimum Hamming distance and pick up those codewords which are very useful when calculating asymptotic performance.
Low-density parity-check codes are commonly decoded using iterative message-passing decoders, such as the min-sum and sum-product decoders. Computer simulations demonstrate that these suboptimal decoders are capable of achieving low probability of bit error at signal-to-noise ratios close to capacity. However, current methods for analyzing the behavior of the min-sum and sum-product decoders fails to produce usable bounds on the probability of bit error. Thus, the resulting probability of bit error when using these decoders remains largely unknown for signal-to-noise ratios beyond the reach of simulation. For this reason, it is worth considering alternative methods for decoding low-density parity-check codes. New methods for decoding low-density parity-check codes, known as finite tree-based decoders, are presented as alternative decoders for low-density parity-check codes. The goal of the finite tree-based decoders is to achieve probability of bit error comparable to that of the min-sum and sum-product decoders, while allowing for computationally tractable performance analysis. Finite tree-based decoding requires the construction of finite trees derived from the Tanner graph of the low-density parity-check code. The resulting size of the finite trees allows for current analytical techniques, such as deviation bounds and density evolution, to be used to predict the probability of bit error of finite tree-based decoding of short-to-moderate length low-density parity-check codes. Simulation results show that finite tree-based decoders are capable of outperforming current iterative decoders at high signal-to-noise ratios. Examples are also given where finite tree-based decoding provably approaches maximum-likelihood performance as the signal-to-noise ratio grows large. A new method is also presented for lower bounding the minimum distance of low-density parity-check codes. This new lower bound is used as a cost criteria for the construction of low-density parity-check codes with both large girth and minimum-distance properties. Codes generated with this new construction technique are shown in simulations to outperform codes generated with the progressive edge-growth algorithm, using both iterative decoding and finite tree-based decoding.