Design And Applications Of Low Density Parity Check Codes

In 1948, Shannon proved the existence of the capacity approaching error correcting codes based on the theory of typical sequence, but didn’t give any details on construction method. In recent years, the researches on coding theory transfers from designing channel codes with low complexity and excellent performance to studying how to design capacity achieving codes. Low density parity check (LDPC) codes are a class of channel codes which have capacity approaching performance and relatively low decoding complexity. So it is a good candidate for many different communication scenarios. However, how to prove LDPC codes can approach capacity is still an open problem.Spatially coupled LDPC (SC-LDPC) codes, a particularly exciting new class of LDPC codes, attract much attention due to their "threshold saturation" properties, which can improve the belief propagation (BP) threshold up to the maximum a posterior (MAP) threshold and as the variable node degree increases, the MAP threshold can achieve Shannon capacity. Therefore, spatially coupled theory opens up a new way to achieve capacity. Since the researches on SC-LDPC codes are still at the start stage, many problems need further studied. This paper made a thorough research on the construction and design of LDPC and SC-LDPC codes.Firstly, a particular class of SC-LDPC codes constructed by parallelly connecting multiple different coupled chains is proposed. By varying the number of the connected chains and the degree of each chain, a family of SC-LDPC codes with wide rate range can be obtained. Different from the existed connection way, the proposed structure can keep the degrees of each coupled chain unchanged and no extra added edges are introduced. Density evolution analysis shows that the decoding thresholds of the proposed code ensembles are very close to Shannon limits and are slightly better than that constructed by a single chain over the binary erasure channel (BEC).Secondly, rate-compatible SC-LDPC code with low implementation complexity is proposed. Partial repetition expansion method is used to construct lower rate codes while randomly puncture scheme is for constructing higher rate codes. Different from conventional rate-compatible SC-LDPC codes, the proposed method requires no design extending matrix and puncture matrix for each rate and the rate compatibility is realized by adjusting three parameters:selection fraction, repetition times and puncture fraction. Density evolution analysis shows that the iterative decoding thresholds of all the member codes in the proposed rate-compatible SC-LDPC code family are very close to Shannon limits over the BEC and additive white Gaussian noise (AWGN) channel. Especially for very low rate codes, the performances are much better than those in existing schemes.Thirdly, due to the excellent threshold performance of SC-LDPC codes, a new structure of masking matrix with generalized spatially coupling structure is proposed. From the viewpoint of the protograph, the proposed masking matrix is obtained by coupling two or more identical subgraphs together. Asymptotic performance shows that although the proposed masking matrix has almost the same decoding threshold as the random one, the former converges faster than the latter. Simulation results show that the QC-LDPC code constructed by the proposed masking matrix has about 0.2 dB coding gains compared with the random one.Finally, for decode-and-forward relay network, an optimization algorithm for bilayer lengthened LDPC (BL-LDPC) codes based on Gaussian approximation is proposed, which can search the lower and upper variable node degree distributions simultaneously. Simulation results show that for the BL-LDPC codes obtained by the proposed optimization, the gaps between the thresholds and the theoretical limits are smaller and the BER performances are better. On this basis, combined with the extra added check nodes generated by the relay node, a joint degree distribution optimization for BL-LDPC code is proposed. Using this low complexity algorithm, the optimized BL-LDPC codes have better thresholds and BER performances.