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IEEE Communications Letters

A Fast Approximate Check Polytope Projection Algorithm for ADMM Decoding of LDPC Codes

Qiaoqiao Xia, Yan Lin, Shidi Tang, Qinglin Zhang

Published Jul 1, 2019• 29 citations
Publisher

Abstract

Simplifying the Euclidean projection onto check polytope is an efficient way to reduce the computational complexity of alternating direction method of multipliers (ADMM) decoding algorithm for low-density parity-check (LDPC) codes. Existing algorithms for check polytope projection require sorting operation or iterative operation, which happens to be the most complex part of the projection. In this letter, a novel and fast projection algorithm is proposed without sorting and iterative operations. In the proposed algorithm, line segment projection replaces check polytope projection to approach approximate Euclidean projection at low computational complexity. Simulation results show that the proposed algorithm can substantially reduce the projection time while maintaining the frame error rate (FER) performance. In particular, the proposed algorithm can save the average projection time by 43% compared with cut search algorithm (CSA) when the dimension of the input vector is 20.

DOI
10.1109/lcomm.2019.2926085

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IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences•Sep 30, 2019•10 citations
Simplifying the Euclidean projection onto check polytope is an efficient way to reduce the computational complexity of alternating direction method of multipliers (ADMM) decoding algorithm for low-density parity-check (LDPC) codes. Existing algorithms for check polytope projection require sorting operation or iterative operation, which happens to be the most complex part of the projection. In this letter, a novel and fast projection algorithm is proposed without sorting and iterative operations. In the proposed algorithm, line segment projection replaces check polytope projection to approach approximate Euclidean projection at low computational complexity. Simulation results show that the proposed algorithm can substantially reduce the projection time while maintaining the frame error rate (FER) performance. In particular, the proposed algorithm can save the average projection time by 43% compared with cut search algorithm (CSA) when the dimension of the input vector is 20.
Alternating direction method of multipliers (ADMM) is a popular technique for linear-programming decoding of low-density parity-check codes. The computational complexity of ADMM is dominated by the Euclidean projection of a real-valued vector onto a parity-check polytope. Existing algorithms for such a projection all require sorting operations, which happen to be the most complex part of the projection. In this letter, we propose an iterative algorithm, without sorting operation, for projection onto the parity-check polytope. The proposed algorithm has a worst case complexity linear in the input dimension compared with the super-linear complexity of existing algorithms.
The Euclidean projection onto check polytope is the most complicated and time-consuming operation in the alternating direction method of multipliers (ADMM) decoding algorithm for low-density parity-check (LDPC) codes. Existing approximate even-vertex projection algorithm (EVA) can simplify the projection operation, but yields poor decoding performance. By alternately adopting EVA and other accurate projection algorithms, this letter proposes an innovative and simple hybrid projection algorithm (HPA) which can increase the percentage of unuseful projections. Simulation results show that the proposed algorithm can substantially reduce the projection time while achieving better frame error rate (FER) performance when compared with the standard ADMM decoding and the ADMM penalized decoding in the case of avoiding the tedious work of penalty parameter optimization. More importantly, compared with cut search algorithm (CSA), the proposed algorithm can significantly save the average projection time by nearly 80%, and the average decoding time by about 85%.
The Euclidean projection onto check polytope is the most complicated and time-consuming operation in the alternating direction method of multipliers (ADMM) decoding for Low-Density Parity-Check (LDPC) codes. The check polytope projection penalized algorithm can avoid the tedious work of penalty parameter optimization and simplified the calculation process of check polytope projection. However, the over-punishment of check polytope projection penalized algorithm may lead to poor decoding performance, when combined with the over-relaxation technique. In this paper, an improved check polytope projection penalized algorithm is proposed. In order to reduce the ratio of even-vertex projection, we restrict the points far away from the even-vertex of the check polytope by introducing the penalty constraint condition. Simulation results show that the frame error rate (FER) performance of the proposed algorithm is almost the same as that of the cut search projection algorithm (CSA), and it can save the average decoding time by 22.9%–52%.
Alternating Direction Method of Multipliers (ADM-M) is an effective method for Linear Programming (LP) decoding of Low-Density Parity-Check (LDPC) codes. In the ADMM-based LDPC decoding algorithm, Euclidean projection on the check polytope is the key step. Current Euclidean projection algorithms in general require either sorting or iterative operations. However, when decoding LDPC codes with high coding rate, the sorting operation brings a high level of computation complexity. Moreover, although the current iterative projection algorithms do not involve the complex operation of sorting, its convergence speed is slow. In this paper, based on trend extrapolation, a reduced-complexity iterative check polytope projection algorithm is proposed. In the proposed algorithm, the target projection point can be found based on the variation trend of the geometric position of the projection points obtained from previous iterations. Therefore, it ensures fast convergence and does not sacrifice on bit error rate (BER) performance. The simulation results confirm that the proposed algorithm can effectively reduce the decoding complexity, especially for LDPC codes with high coding rate.
ADMM penalized decoding method for Low-Density Parity-Check (LDPC) Codes can improve the frame error rate (FER) performance than the standard ADMM decoder by adding penalty terms to the objective function. However, penalty parameter optimization of the penalty terms is very difficult and extremely time-consuming. Additionally, the Euclidean projection onto the check polytope is the most complex part of ADMM-based decoding algorithms. In this paper, by dynamically choosing even-vertex in the check polytope closest to the input vector as the approximate projection, a simple and novel check polytope projection penalized algorithm for ADMM decoding is proposed which can avoid the tedious work of penalty parameter optimization and simplify the check polytope projection. The simulation results show that the proposed algorithm can substantially reduce the projection time while achieving better FER performance when compared with the existing penalized decoding. In particular, the proposed algorithm can save the decoding time by 23% to 41% compared with ADMM-PD-LSA algorithm.
Using linear programming (LP) decoding based on alternating direction method of multipliers (ADMM) for low-density parity-check (LDPC) codes shows lower complexity than the original LP decoding. However, the development of the ADMM-LP decoding algorithm could still be limited by the computational complexity of Euclidean projections onto parity check polytope. In this paper, we proposed a bisection method iterative algorithm (BMIA) for projection onto parity check polytope avoiding sorting operation and the complexity is linear. In addition, the convergence of the proposed algorithm is more than three times as fast as the existing algorithm, which can even be 10 times in the case of high input dimension.