Bandwidth efficient coded modulation using low density parity check codes

The power efficiency of low density parity check (LDPC) codes is combined with the bandwidth efficiency of various coded modulation techniques to improve system performance. Two different approaches to coded-modulation are studied—one, using binary LDPC codes, and the other, using non-binary LDPC codes. In the first approach, the block nature of (binary) LDPC codes is shown to be suited for bit-interleaved and multi-level coded modulation (i.e., BICM and MLC, respectively). Particular LDPC codes are designed specifically for use in these systems. In the process, an algebraic construction of LDPC codes, originally introduced by Tanner, is examined, first separately, and then, in the framework of BICM and MLC modulation. The convergence behaviors of different iterative decoding algorithms used to decode LDPC codes in these frameworks are analyzed and (random) LDPC codes are specifically optimized for the particular decoding algorithm(s).; In the second approach, the design of LDPC codes over groups and matching signal sets is considered. The resulting codes obtained in these designs are geometrically uniform codes, as introduced by Forney and Loeliger. The codewords of a LDPC code constructed over certain groups are mapped onto a matching signal set to give a geometrically uniform signal space code. The convergence behavior of the belief propagation (BP) algorithm on these non-binary LDPC codes is analyzed, and based on this analysis, the group LDPC codes are optimized to yield good performance. The relationship between the minimum Hamming distance of the non-binary LDPC code and the minimum Hamming distance of a “corresponding” binary LDPC code is established. This result in turn relates the minimum distance of the non-binary LDPC code to the choice of non-zero entries of the nonbinary LDPC matrix. Further, the relationship between the convergence of the BP decoder on the non-binary LDPC constraint graph and the choice of non-zero entries of the non-binary LDPC matrix is partially quantified. Most of the analyses on convergence of the BP algorithm are based on Monte Carlo simulations. A numerical technique, an extension of the method described in [1], to compute the limiting performance of the BP algorithm on LDPC codes over the ring $Z4$, is also described.