Date of Award
Ph.D. in Engineering Science
Codes that are capable of generating any number of encoded symbols from a given number of source symbols are called rateless codes. Luby transform (LT) codes are the first practical realization of rateless codes while Raptor codes are constructed by serially concatenating LT codes with high-rate outer low-density parity-check (LDPC) codes. Although these codes were originally developed for binary erasure channel (BEC), due to their rateless feature, they are being investigated and designed for their use in noisy channels. It is known that LT codes are the irregular non-systematic rateless counterpart of low-density generator-matrix (LDGM) codes. Therefore, the first part of our work is focused on LDGM codes and their serially concatenated scheme called serially concatenated LDGM (SCLDGM) codes. Though single LDGM codes are asymptotically bad codes, the SCLDGM codes are known to perform close to the Shannon limit. We first study the asymptotic behaviour of LDGM codes using a discretized density evolution method. We then show that the DDE method can be used in two-steps to provide the detailed asymptotic performance analysis of SCLDGM codes. We also provide the detailed error-floor analysis of both the LDGM and SCLDGM codes. We also prove a necessary condition for the successful decoding of such concatenated codes under sum-product (SP) decoding in binary input additive white Gaussian noise (BIAWGN) channels. Based on this necessary condition, we then develop a DDE-based optimization approach which can be used to optimize such concatenated codes in general. We present both the asymptotic performance and simulation results of our optimized SCLDGM codes that perform within 0.26 dB to the Shannon limit in BIAWGN channels. Secondly, we focus on the asymptotic analysis and optimization design of LT and Raptor codes over BIAWGN channels. We provide the exact asymptotic performance of LT codes using the DDE method. We apply the concept of the two-step DDE method to the Raptor codes and obtain their exact asymptotic performance in BIAWGN channels. We show that the existing Raptor codes using solely the same output degree distribution can perform within 0.4 dB to the Shannon limit for various realized code-rates. We then develop a DDE-based optimization technique to optimally design such physical layer Raptor codes. Our optimized Raptor codes are shown to perform within 0.2 dB to the Shannon limit for most of the realized code-rates. We also provide the asymptotic curves, decoding thresholds, and simulation results showing that our optimized Raptor codes outperform the existing Raptor codes in BIAWGN channels. Finally, we present the asymptotic analysis and optimization design of systematic version of these codes namely systematic LT and systematic Raptor codes as well.
Kharel, Amrit, "On The Design Of Physical Layer Rateless Codes" (2018). Electronic Theses and Dissertations. 530.
Emphasis: Electrical Engineering