Andrew Thangaraj

Density evolution for protograph low-density paritycheck (LDPC) codes is considered, and it is shown that the message-error rate falls double-exponentially with iterations whenever the degree-2 subgraph of the protograph is cycle-free and noise level is below threshold. Conditions for stability of protograph density evolution are established and related to the structure of the protograph. Using large-girth graphs, sequences of protograph LDPC codes with block-error threshold equal to bit-error threshold and block-error rate falling near-exponentially with blocklength are constructed deterministically. Small-sized protographs are optimized to obtain thresholds near capacity for binary erasure and binary-input Gaussian channels.

The capacity-achieving input distribution of the discrete-time, additive white Gaussian noise (AWGN) channel with an amplitude constraint is discrete and seems difficult to characterize explicitly. A dual capacity expression is used to derive analytic capacity upper bounds for scalar and vector AWGN channels. The scalar bound improves on McKellips' bound and is within 0.1 bit of capacity for all signal-to-noise ratios (SNRs). The 2-D bound is within 0.15 bits of capacity provably up to 4.5 dB; numerical evidence suggests a similar gap for all SNRs. As the SNR tends to infinity, these bounds are accurate and match with a volume-based lower bound. For the 2-D complex case, an analytic lower bound is derived by using a concentric constellation and is shown to be within 1 bit of capacity.