| The fundamental goal of communication is to transmit information reliably over a noisy communication channel with the most efficient use of the resources available, namely bandwidth, power, and computational complexity. Shannon showed in his seminal 1948 papers, that reliable transmission is possible over a noisy channel if and only if the transmission rate, R, is less than some quantity C, which he called the channel capacity. Unfortunately his proof of this result provided no guidance as to the construction of good coding schemes, nor did it address the complexity of implementing the required encoders and decoders. Although significant progress had been made in the field of coding theory during the first 45 years following Shannon's work, the promise of closely approaching capacity using practical coding schemes was left unfulfilled. The development of turbo codes and the re-birth of low-density parity-check (LDPC) codes in the early to mid-1990's finally offered the opportunity to achieve this long sought after goal. In this thesis, LDPC codes are designed, analyzed and evaluated for their use on both the bandwidth (W) constrained additive white Gaussian noise (AWGN) channel with R/W > 1 and on the optical, direct detection M-ary pulse position modulated (PPM) channel. On the bandwidth constrained AWGN channel, LDPC codes are combined with multilevel coding (MLC) and multistage decoding (MSD) in order to achieve power and bandwidth efficient communications using two-dimensional signal constellations. Trellis shaping of the two-dimensional signal constellation is also employed to improve performance. We demonstrate theoretically and empirically that excellent performance, with reasonable complexity, can be achieved using well-designed irregular LDPC component codes with trellis shaping. In fact, a practical scheme is demonstrated that beats the capacity of equally-likely signaling using 64-ary QAM on the AWGN channel operating at a data rate of two bits per signaling dimension. We also study the use of LDPC codes with MLC/MSD on the direct-detection, M-ary PPM, optical channel, where the received statistics are Poisson distributed. We demonstrate both theoretically and empirically that performance close to the channel capacity is realizable. Finally, we derive the exact channel capacity of the direct detection optical channel with continuous-PPM modulation. Two variants of the continuous-PPM optical channel are introduced and their capacities are derived. |
