|M.Sc Thesis||Department of Electrical Engineering|
|Supervisor:||Assoc. Prof. Sason Igal|
|Full Thesis text|
Traditionally, low-density parity-check (LDPC) code ensembles are designed using numerical methods suited to the channel on which they are to be used. In practice, the actual channel statistics are hardly ever known in advance. Moreover, even if the channel statistics were known, it is desirable to design universal codes that will be robust enough to perform well on a range of channels, rather than a specific channel only. Therefore, a universal design of LDPC code ensembles that enables to operate reliably over various channels is of great theoretical and practical interest.
In this thesis we consider the universality of LDPC code ensembles over families of memoryless binary-input output-symmetric (MBIOS) channels, both under belief propagation (BP) decoding and maximum-likelihood (ML) decoding.
For the BP decoding case, we rely on the density evolution approach, to derive an analytical method for universal LDPC code design over various families of MBIOS channels. We analyze this regime for several families of MBIOS channels. The density evolution approach also enables us to derive a necessary condition for universality of LDPC code ensembles under BP decoding. This necessary condition sits at the heart of an LP bound on the universal achievable fraction of capacity. It also enables us to provide analytical and easy-to-calculate bounds on the threshold of LDPC code ensembles under BP decoding that are based on the Bhattacharyya parameter of the channel. The results for LDPC code ensembles are also extended to irregular repeat-accumulate (IRA) code ensembles under BP decoding.
For the ML decoding case, we prove that properly selected regular LDPC code ensembles are universally capacity-achieving for the set of equi-capacity MBIOS channels. We extend this result also to prove that punctured regular LDPC code ensembles are also universally capacity-achieving for the set of equi-capacity MBIOS channels.