Advanced digital and analog error correction codes

Practical communication channels are inevitably subject to noise uncertainty, interference, and/or other channel impairments. The essential technology to enable a reliable communication over an unreliable physical channel is termed as channel coding or error correction coding(ECC). The profound concept that underpins channel coding is distance expansion. That is, a set of elements in some space having small distances among them are mapped to another set of elements in possibly a different space with larger distances among the elements. Distance expansion in terms of digital error correction has been a common practice, but the principle is by no means limited to the discrete domain. In a broader context, a channel code may be mapping elements in an analog source space to elements in an analog code space. As long as a similar distance expansion condition is satisfied, the code space is expected to provide an improved level of distortion tolerance than the original source space. For example, one may treat the combination of quantization, digital coding and modulation as a single nonlinear analog code that maps real-valued sources to complex-valued coded symbols. Such a concept, thereafter referred to as analog error correction coding (AECC), analog channel coding, or, simply, analog coding, presents a generalization to digital error correction coding (DECC). This dissertation investigates several intriguing aspects of DECC and especially of AECC. The research of DECC focuses on turbo codes and low-density-parity-check (LDPC) codes, two of the best performing codes known to date. In the topic of 1 turbo codes, this dissertation studies on interleaver design, which plays an important role in the overall performance of turbo codes (at small to medium code lengths) but does not affect the decoding architecture. Before this work, the theoretical foundation of interleaver design and evaluation were rather incomplete, e.g. efficient approaches in measuring “randomness” (one of the most important characteristics for interleavers) were rigorously established. This work proposes two powerful metrics, cycle correlation sum (CCS) and variance of the second order spread spectrum (VSSS), to quantify spread and randomness, two fundamental properties of interleavers, while accounting for the iterative nature of turbo decoding and the weight spectrum of turbo encoding. We evaluate the ensemble of algebraic interleavers, propose design approaches specific to coprime interleavers, a subclass of algebraic interleavers, and provide theoretical insights on selecting parameters. Simulation results show superior performance of the newly designed coprime interleavers to the existing ones. The second topic analyzes the Gaussian assumption for the stochastic analysis in iterative decoding. Gaussian distribution is widely believed to match the real message density in analyzing iterative decoding, but the justification is largely pragmatic, except for the messages directly coming from Gaussian channels. This work investigates when and how well the Gaussian distribution approximates the real message density and why. We show that the Gaussian assumption is statistically sound when the LLRs extracted from the channel are reasonably reliable to start with, and when the check node degrees of the LDPC code are not very high; but the assumption is much less accurate when one or both conditions are violated. Extensive simulation results are provided to exemplify and verify this discussion.