Modern Coding Theory

Abstract

Having trouble deciding which coding scheme to employ, how to design a new scheme, or how to improve an existing system? This summary of the state-of-the-art in iterative coding makes this decision more straightforward. With emphasis on the underlying theory, techniques to analyse and design practical iterative coding systems are presented. Using Gallager's original ensemble of LDPC codes, the basic concepts are extended for several general codes, including the practically important class of turbo codes. The simplicity of the binary erasure channel is exploited to develop analytical techniques and intuition, which are then applied to general channel models. A chapter on factor graphs helps to unify the important topics of information theory, coding and communication theory. Covering the most recent advances, this text is ideal for graduate students in electrical engineering and computer science, and practitioners. Additional resources, including instructor's solutions and figures, available online: www.cambridge.org/9780521852296.

Extremal Problems of Information Combining

“…Many properties for LLRs, such as symmetry, are well established in [3], [5], [25], and similar properties can be obtained for soft bits as follows.…”

“…It is well known that the MMSE is equal to Further (25) where the second equality is due to (6). Hence (26) Theorem 2: Among binary-input symmetric-output channels with a fixed mutual information, the MMSE for estimating the channel input in a BSC is strictly minimal.…”

“…First, we can show that the maximum and minimum exist. It follows from sequential compactness of the space of symmetric distributions [25] (under point-wise convergence at points of continuity of the cumulative distributions) and continuity of mutual information and the moments. Assume the upper bound is (50) where needs to be determined.…”

A Performance Comparison of Systematic Polar Codes and Non-systematic Polar Codes

“…While the reliability can also be computed by density evolution or Gaussian approximation method for other channels [3] [4]. For a given N, polar codes are encoded in the same manner, namely GN is the generator matrix of order N, defined as…”

Statistical estimation: from denoising to sparse regression and hidden cliques

“…We next derive a message passing version of this iteration 3 (we refer for instance to [47,48] for background). The motivation for this modification is that message passing algorithms have appealing statistical properties.…”