Various log-likelihood-ratio-based belief-propagation (LLR-BP) decoding algorithms and their reduced-complexity derivatives for low-density parity-check (LDPC) codes are presented. Numerically accurate representations of the check-node update computation used in LLR-BP decoding are described. Furthermore, approximate representations of the decoding computations are shown to achieve a reduction in complexity by simplifying the check-node update, or symbol-node update, or both. In particular, two main approaches for simplified check-node updates are presented that are based on the so-called min-sum approximation coupled with either a normalization term or an additive offset term. Density evolution is used to analyze the performance of these decoding algorithms, to determine the optimum values of the key parameters, and to evaluate finite quantization effects. Simulation results show that these reduced-complexity decoding algorithms for LDPC codes achieve a performance very close to that of the BP algorithm. The unified treatment of decoding techniques for LDPC codes presented here provides flexibility in selecting the appropriate scheme from performance, latency, computational-complexity, and memory-requirement perspectives.

Reduced-complexity decoding of LDPC codes

In this letter, we address the problem of decoding nonbinary low-density parity-check (LDPC) codes over finite fields GF(q), with reasonable complexity and good performance. In the first part of the letter, we recall the original belief propagation (BP) decoding algorithm and its Fourier domain implementation. We show that the use of tensor notations for the messages is very convenient for the algorithm description and understanding. In the second part of the letter, we introduce a simplified decoder which is inspired by the min-sum decoder for binary LDPC codes. We called this decoder extended min-sum (EMS). We show that it is possible to greatly reduce the computational complexity of the check-node processing by computing approximate reliability measures with a limited number of values in a message. By choosing appropriate correction factors or offsets, we show that the EMS decoder performance is quite good, and in some cases better than the regular BP decoder. The optimal values of the factor and offset correction are obtained asymptotically with simulated density evolution. Our simulations on ultra-sparse codes over very-high-order fields show that nonbinary LDPC codes are promising for applications which require low frame-error rates for small or moderate codeword lengths. The EMS decoder is a good candidate for practical hardware implementations of such codes

Decoding Algorithms for Nonbinary LDPC Codes Over GF $(q)$

In this letter, we analyze the performance of two improved belief propagation (BP) based decoding algorithms for LDPC codes, namely the normalized BP-based and the offset BP-based algorithms, by means of density evolution. The numerical calculations show that with one properly chosen parameter for each of these two improved BP-based algorithms, performances very close to that of the BP algorithm can be achieved. Simulation results for LDPC codes with code length moderately long validate the proposed optimization.

Density evolution for two improved BP-Based decoding algorithms of LDPC codes

This paper discusses construction of protograph-based low-density parity-check (LDPC) codes. Emphasis is placed on protograph ensembles whose typical minimum distance grows linearly with block size. Asymptotic performance analysis for both weight enumeration and iterative decoding threshold determination is provided and applied to a series of code constructions. Construction techniques that yield both low thresholds and linear minimum distance growth are introduced by way of example throughout. The paper also examines implementation strategies for high throughput decoding derived from first principles of belief propagation on bipartite graphs.

Capacity-approaching protograph codes

The effects of clipping and quantization on the performance of the min-sum algorithm for the decoding of low-density parity-check (LDPC) codes at short and intermediate block lengths are studied. It is shown that in many cases, only four quantization bits suffice to obtain close to ideal performance over a wide range of signal-to-noise ratios. Moreover, we propose modifications to the min-sum algorithm that improve the performance by a few tenths of a decibel with just a small increase in decoding complexity. A quantized version of these modified algorithms is also studied. It is shown that, when optimized, modified quantized min-sum algorithms perform very close to, and in some cases even slightly outperform, the ideal belief-propagation algorithm at observed error rates.

On implementation of min-sum algorithm and its modifications for decoding low-density Parity-check (LDPC) codes

In this paper, we propose a belief-propagation (BP)-based decoding algorithm which utilizes normalization to improve the accuracy of the soft values delivered by a previously proposed simplified BP-based algorithm. The normalization factors can be obtained not only by simulation, but also, importantly, theoretically. This new BP-based algorithm is much simpler to implement than BP decoding as it requires only additions of the normalized received values and is universal, i.e., the decoding is independent of the channel characteristics. Some simulation results are given, which show this new decoding approach can achieve an error performance very close to that of BP on the additive white Gaussian noise channel, especially for low-density parity check (LDPC) codes whose check sums have large weights. The principle of normalization can also be used to improve the performance of the max-log-MAP algorithm in turbo decoding, and some coding gain can be achieved if the code length is long enough.

Near optimum universal belief propagation based decoding of low-density parity check codes

Iterative decoding techniques have been receiving more and more attentions with the invention of turbo codes and the rediscovery of low-density parity-check (LDPC) codes. An important aspect in the study of iterative decoding is the tradeoff between decoding performance and complexities. For both LDPC codes and turbo codes, optimum decoding algorithms can provide very good performance. However, complicated operations are involved in the optimum decoding, and prohibit the wide applications of LDPC codes and turbo codes in the next generation digital communication and storage systems. 
This research investigates the reduced complexity decoding algorithms of LDPC codes and turbo codes. For decoding LDPC codes, new algorithms, namely the normalized BP-based algorithm and the offset BP-based algorithm, are proposed. Density evolution algorithms are derived and are used to determine the best decoder parameters associated with each of the algorithms. Both numerical results and simulations show that the new algorithms can achieve near optimum decoding performances. In addition to the advantage of low computational complexities, the two new algorithms are less subject to quantization errors and correlation effects than the optimum BP algorithm, and are more suitable for hardware implementation. For a special kind of LDPC codes—the geometric LDPC codes, we propose the normalized APP-based algorithm, which is even more simplified yet still can achieve the near optimum performance. For decoding turbo codes, two new sub-optimum decoding algorithms are proposed. The first is the bi-directional soft-output Viterbi algorithm (bi-SOVA), which is based on utilizing a backward SOVA decoding in addition to the conventional forward one. The second is the normalized Max-Log-MAP algorithm, which improves the performance of the Max-Log-MAP decoding by scaling the soft outputs with some predetermined factors.

Reduced complexity decoding algorithms for low-density parity check codes and turbo codes

edu Ahstract- In this paper, we study Reed-Solomon (RS) coded beamforming in multiple-inp ut-multiple-outp ut (MIMO) Rayleigh fading channels. To combat the fading, we introduce a block interleaver into RS-coded beamforming systems and we present the performance analysis of such systems. We show that the interleaver can greatly improve the performance of RS coded beamforming systems, especially in the systems with a small number of antennas. We also observe that the interleaver gain in a RS coded beamforming system vanishes as the number of antennas increases. We illustrate that the proposed approach has potential to evaluate the tradeoffs between spatial diversity and time diversity via symbol interleaving.

http://ieeexplore.ieee.org/iel5/9623/30414/01404746.pdf

Jinghu Chen

Papers

Reduced-complexity decoding of LDPC codes

Near optimum universal belief propagation based decoding of low-density parity check codes

Reduced complexity decoding algorithms for low-density parity check codes and turbo codes

Performance of Reed-Solomon Coded Beamforming