LDPC List Decoding for Low-Complexity Check Node Processing
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Solution Overview
Problem
Current decoding algorithms for Non-Binary Low Density Parity Check (LDPC) codes are computationally intensive and complex, particularly in the Check Node Processing (CNP) step, leading to instability and high computational requirements, especially when dealing with finite digit representations and large Galois fields.
Innovation Solution
A simplified algorithm for decoding Non-Binary LDPC codes that generates output tables containing corrected symbol values and their corresponding probability ratios, using a LIST decoder to process input tables and reduce complexity by sorting symbol candidates and normalizing probabilities, allowing for efficient computation of most probable values and their logarithmic ratios.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If the full complexity log-domain symbol based belief propagation algorithm is used, then decoding accuracy is improved, but computational complexity increases as the square of the size of the Galois field
Solution Approach 1:
The patent segments the Galois field elements into multiple bits (e.g., GF(2^m) is segmented into m binary bits). This allows the complex non-binary LDPC decoding to be broken down into multiple stages of binary processing, where each bit plane can be processed separately through binary-to-check-node processing, check-node processing, and check-to-variable processing stages, thereby reducing the computational complexity from O(q^2) to O(m*q) where q is the field size and m is the number of bits per symbol.
Solution Approach 2:
The patent transforms the problem from operating directly in the Galois field domain to operating in the binary bit-plane domain. By representing each Galois field element as multiple binary bits and processing these bits through separate stages (binary-to-CNP, CNP, CNP-to-variable), the algorithm changes the dimensional representation of the data, enabling more efficient computation while maintaining decoding accuracy.
2Quantity of substance
If probability values are represented in finite digits, then memory usage is reduced, but stability problems occur in the probability domain
Solution Approach 1:
The patent substitutes the probability domain operations with log-domain operations. Instead of directly manipulating probability values P which suffer from numerical underflow and stability issues when represented in finite precision, the algorithm uses logarithmic values L = log(P). This transformation converts multiplicative operations into additive operations, improving numerical stability while allowing for efficient memory representation using normalized logarithmic values.
3Measurement precision
If the check node processing is performed using the forward-backward approach over a fully connected trellis, then decoding accuracy is maintained, but the complexity increases significantly
Solution Approach 1:
The patent segments the fully connected trellis processing into multiple independent bit-plane processing stages. Instead of processing all q possible Galois field values simultaneously in a single complex trellis, the algorithm processes each bit plane separately through simplified binary trellises, reducing the processing complexity while maintaining the ability to capture all necessary probability information through the multi-stage approach.
Data Source
AI summary
A method is for generating, for each check node related to a parity check equation of a LDPC code, signals representing a first output table of corrected values of symbols of a word received through a communication channel and transmitted according to the LDPC code, and signals representing a second output table of the logarithm of the ratio between the respective probability of correctness of the values of same coordinates in the first output table and their corresponding maximum probability of correctness. The method is implemented by processing the components of a first input table of values of a Galois Field of symbols that may have been transmitted and of a second input table of corresponding probability of correctness of each value.