Fun and exciting textbook on the mathematics underpinning the most dynamic areas of modern science and engineering.

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Information Theory, Inference and Learning Algorithms

Mutual information transfer characteristics of soft in/soft out decoders are proposed as a tool to better understand the convergence behavior of iterative decoding schemes. The exchange of extrinsic information is visualized as a decoding trajectory in the extrinsic information transfer chart (EXIT chart). This allows the prediction of turbo cliff position and bit error rate after an arbitrary number of iterations. The influence of code memory, code polynomials as well as different constituent codes on the convergence behavior is studied for parallel concatenated codes. A code search based on the EXIT chart technique has been performed yielding new recursive systematic convolutional constituent codes exhibiting turbo cliffs at lower signal-to-noise ratios than attainable by previously known constituent codes.

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Convergence behavior of iteratively decoded parallel concatenated codes

Recent advancements in iterative processing of channel codes and the development of turbo codes have allowed the communications industry to achieve near-capacity on a single-antenna Gaussian or fading channel with low complexity. We show how these iterative techniques can also be used to achieve near-capacity on a multiple-antenna system where the receiver knows the channel. Combining iterative processing with multiple-antenna channels is particularly challenging because the channel capacities can be a factor of ten or more higher than their single-antenna counterparts. Using a "list" version of the sphere decoder, we provide a simple method to iteratively detect and decode any linear space-time mapping combined with any channel code that can be decoded using so-called "soft" inputs and outputs. We exemplify our technique by directly transmitting symbols that are coded with a channel code; we show that iterative processing with even this simple scheme can achieve near-capacity. We consider both simple convolutional and powerful turbo channel codes and show that excellent performance at very high data rates can be attained with either. We compare our simulation results with Shannon capacity limits for ergodic multiple-antenna channel.

http://www.ece.ualberta.ca/~hcdc/Library/StCommClass/HocTen01.pdf

Achieving near-capacity on a multiple-antenna channel

Zehavi (1992) showed that the performance of coded modulation over a Rayleigh fading channel can be improved by bit-wise interleaving the encoder output and by using an appropriate soft-decision metric as an input to a Viterbi decoder. The goal of this paper is to present in a comprehensive fashion the theory underlying bit-interleaved coded modulation, to provide tools for evaluating its performance, and to give guidelines for its design.

/pdf/bit-interleaved-coded-modulation-2ld6uk564f.pdf

Bit-Interleaved Coded Modulation

It has been recognized by Zehavi (1992) that the performance of coded modulation over a Rayleigh fading channel can be improved by bit-wise interleaving at the encoder output, and by using an appropriate soft-decision metric as an input to a Viterbi (1990) decoder. The paper presents in a comprehensive fashion the theory underlying bit-interleaved coded modulation, provides tools for evaluating its performance, and gives guidelines for its design.

/pdf/bit-interleaved-coded-modulation-fakv4whsdn.pdf

Bit-interleaved coded modulation

A novel method for visualising the convergence behaviour of iterative decoding schemes is proposed. Each constituent decoder is represented by a mutual information transfer characteristic which describes the flow of extrinsic information through the soft in/soft out decoder. The exchange of extrinsic information between constituent decoders is plotted in an extrinsic information transfer chart. The concepts are illustrated for an iterative demapping and decoding scheme.

Convergence of iterative decoding

Iterative decoding algorithms for spectrally efficient modulation have become a vital field of research in digital communications. We show how iterative demapping reduces the bit error rate in conventional communication systems applying multilevel modulation combined with simple channel coding. For this the soft demapping has to be modified in order to accept a priori information. The system can be regarded as a serially concatenated iterative decoding scheme whereby the inner decoder is replaced by the soft demapping device. A design rule based on mutual information to find the 'best' mapping is proposed. Numerical results are presented for 4-PAM and 16-QAM examples.

Iterative demapping and decoding for multilevel modulation

The authors present an iterative decoding scheme which is particularly suited for quadrature phase shift keying (QPSK) modulation with anti-Gray mapping. At the receiver, a soft demapper accepting a priori information and a convolutional decoder are applied in a serial concatenation. Numerical results are presented for QPSK in additive white Gaussian noise and Rayleigh-fading channels. The iterated system applying anti-Gray mapping outperforms the conventional system using Gray mapping after only one iteration.

Iterative demapping for QPSK modulation

A serially concatenated code is presented which exhibits a turbo cliff at 0.28 dB. The concatenation consists of an outer rate one-half repetition code and an inner rate one recursive convolutional code. The iterative decoding scheme was designed using the extrinsic information transfer chart (EXIT chart).

Rate one-half code for approaching the Shannon limit by 0.1 dB

The design of serially concatenated codes has yet been dominated by optimizing asymptotic slopes of error probability curves. We propose mutual information transfer characteristics for soft in/soft out decoders to design serially concatenated codes based on the convergence behavior of iterative decoding. The exchange of extrinsic information is visualized as a decoding trajectory in the Extrinsic Information Transfer Chart (EXIT chart).

S. ten Brink

Papers

Convergence of iterative decoding

Iterative demapping and decoding for multilevel modulation

Iterative demapping for QPSK modulation

Rate one-half code for approaching the Shannon limit by 0.1 dB

Design of Serially Concatenated Codes based on Iterative Decoding Convergence