Analysis and Design of Protograph Based LDPC Codes and Ensembles
Jeremy Thorpe
Channel coding is a key component of artificial communication systems, allowing reliable communication using unreliable channels. In the last decade, iteratively decoded channel codes have become or clearly will become standards in a wide range of applications where large amounts of information must be communicated using unreliable media. Of the class of iteratively decoded codes, Low-density parity check (LDPC) codes are arguably the simplest class to describe, and indeed were described more than four decades ago in 1963 by Robert Gallager. The current understanding of LDPC codes has progressed in several significant ways beyond what had been expressed by 1963 by Gallager. Importantly, irregular LDPC codes, whose parity check matrices do not have constant row and column sums, have been shown to significantly outperform their regular counterparts explicitly considered by Gallager. By 1999, researchers had defined a class of irregular ensembles, each characterized by a pair of polynomials. Along with this new class of ensemble, they defined an analytical technique, density evolution, that accurately predicted the channel coding performance of a typical code under iterative message-passing decoding. The pair of polynomials could be effectively be designed by optimizing the coefficients of the polynomial for density evolution threshold threshold. This thesis concerns a different class of ensembles, namely protograph ensembles. Protograph ensembles are characterized by a template graph called, intuitively, a protograph. The Tanner-graph representation of a code in the ensemble is a random lift of the protograph. Protograph-based codes have significant advantages over unstructured irregular codes in regard to implementation of their encoders and decoders. In the decoder, this structure can be used in at least two distinct ways to organize the computations defined by any message-passing algorithm. If, in addition to the protograph structure, circulant structure is imposed on each "section" of the matrix then a quasi-cyclic code results, bestowing even mores advantages, especially in the possible implementation of the encoder. A central difficulty in using protograph ensembles is finding a suitable protograph. Since graphs are discrete objects, there is no obvious correspondence to any optimization model using vectors of real numbers. Instead, the technique of simulated annealing has been applied with a remarkable degree of success. For example, on the AWGN channel, given a constraint on the node degrees, protograph ensembles can be found that achieve a threshold only half as far (measured in dB) from the Shannon limit as unstructured irregular ensembles. This simultaneously illustrates an inherent performance advantage of protograph codes over unstructured codes as well as the efficacy of simulated annealing as an optimization technique. A persistent problem which appears to be common in all codes optimized for density evolution threshold is that of error floors. On a superficial level, this is explained by the maxim that "There's no such thing as a free lunch." In some contexts, such as in codes designed for the erasure channel, the phenomenon can be explained on a much deeper level, though it is not clear why the phenomenon should persist so universally. Still, even without a detailed understanding the cause of this problem, there are techniques that can mitigate error floors. An important tool toward this end is weight enumerators, which are discussed in chapter 3. Codeword and stopping set enumerators can be efficiently computed if a certain (non-concave) function can be efficiently maximized. Protographs that are selected on the basis of their enumerators have shown some success in reducing error floors.
•Feb 13, 2006•7 citations