A modified Gradient Descent Bit Flipping (GDBF) algorithm is proposed for decoding Low Density Parity Check (LDPC) codes on the binary-input additive white Gaussian noise channel. The new algorithm, called Noisy GDBF (NGDBF), introduces a random perturbation into each symbol metric at each iteration. The noise perturbation allows the algorithm to escape from undesirable local maxima, resulting in improved performance. A combination of heuristic improvements to the algorithm are proposed and evaluated. When the proposed heuristics are applied, NGDBF performs better than any previously reported GDBF variant, and comes within 0.5 dB of the belief propagation algorithm for several tested codes. Unlike other previous GDBF algorithms that provide an escape from local maxima, the proposed algorithm uses only local, fully parallelizable operations and does not require computing a global objective function or a sort over symbol metrics, making it highly efficient in comparison. The proposed NGDBF algorithm requires channel state information which must be obtained from a signal to noise ratio (SNR) estimator. Architectural details are presented for implementing the NGDBF algorithm. Complexity analysis and optimizations are also discussed.
Because of its reliance on pseudo-random noise and single-bit messages, the proposed NGDBF algorithms bear some resemblance to the family of stochastic iterative decoders that were first introduced by Gaudet and Rapley @cite_16 . One of the authors (Winstead) introduced stochastic decoding for codes with loopy factor graphs @cite_24 , and Sharifi- later demonstrated stochastic decoding for LDPC codes @cite_19 @cite_4 . High throughput stochastic decoders have been more recently demonstrated by Sharifi @cite_3 @cite_21 @cite_18 and by @cite_17 . Stochastic decoders are known to have performance very close to BP, allow for fully-parallel implementations, and use very simple arithmetic while exchanging single-bit messages. They may therefore serve as an appropriate benchmark for comparing complexity against the proposed SM-NGDBF algorithm (an analysis of comparative complexity is presented in Section ).
@cite_18 @cite_4 @cite_21 @cite_3 @cite_24 @cite_19 @cite_16 @cite_17
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This letter presents the first successful method for iterative stochastic decoding of state-of-the-art low-density parity-check (LDPC) codes. The proposed method shows the viability of the stochastic approach for decoding LDPC codes on factor graphs. In addition, simulation results for a 200 and a 1024 length LDPC code demonstrate the near-optimal performance of this method with respect to sum-product decoding. The proposed method has a significant potential for high-throughput and or low complexity iterative decoding. A novel class of bit-flipping (BF) algorithm for decoding low-density parity-check (LDPC) codes is presented. The proposed algorithms, which are referred to as gradient descent bit flipping (GDBF) algorithms, can be regarded as simplified gradient descent algorithms. The proposed algorithms exhibit better decoding performance than known BF algorithms, such as the weighted BF algorithm or the modified weighted BF algorithm for several LDPC codes. We introduce a simple technique for analyzing the iterative decoder that is broadly applicable to different classes of codes defined over graphs in certain fading as well as additive white Gaussian noise (AWGN) channels. The technique is based on the observation that the extrinsic information from constituent maximum a posteriori (MAP) decoders is well approximated by Gaussian random variables when the inputs to the decoders are Gaussian. The independent Gaussian model implies the existence of an iterative decoder threshold that statistically characterizes the convergence of the iterative decoder. Specifically, the iterative decoder converges to zero probability of error as the number of iterations increases if and only if the channel E sub b N sub 0 exceeds the threshold. Despite the idealization of the model and the simplicity of the analysis technique, the predicted threshold values are in excellent agreement with the waterfall regions observed experimentally in the literature when the codeword lengths are large. Examples are given for parallel concatenated convolutional codes, serially concatenated convolutional codes, and the generalized low-density parity-check (LDPC) codes of Gallager and Cheng-McEliece (1996). Convergence-based design of asymmetric parallel concatenated convolutional codes (PCCC) is also discussed. This paper proposes majority-based tracking forecast memories (MTFMs) for area efficient high throughput ASIC implementation of stochastic Low-Density Parity-Check (LDPC) decoders. The proposed method is applied for ASIC implementation of a fully parallel stochastic decoder that decodes the (2048, 1723) LDPC code from the IEEE 802.3an (10GBASE-T) standard. The decoder occupies a silicon core area of 6.38 mm2 in CMOS 90 nm technology, achieves a maximum clock frequency of 500 MHz, and provides a maximum core throughput of 61.3 Gb s. The decoder also has good decoding performance and error-floor behavior and provides a bit error rate (BER) of about 4 × 10-13 at Eb N0=5.15 dB. To the best of our knowledge, the implemented decoder is the most area efficient fully parallel soft -decision LDPC decoder reported in the literature.