Shannon's determination of the capacity of the linear Gaussian channel has posed a magnificent challenge to succeeding generations of researchers. This paper surveys how this challenge has been met during the past half century. Orthogonal minimum-bandwidth modulation techniques and channel capacity are discussed. Binary coding techniques for low-signal-to-noise ratio (SNR) channels and nonbinary coding techniques for high-SNR channels are reviewed. Recent developments, which now allow capacity to be approached on any linear Gaussian channel, are surveyed. These new capacity-approaching techniques include turbo coding and decoding, multilevel coding, and combined coding/precoding for intersymbol-interference channels.

Modulation and coding for linear Gaussian channels

An overview of the many practical applications of channel coding theory in the past 50 years is presented. The following application areas are included: deep space communication, satellite communication, data transmission, data storage, mobile communication, file transfer, and digital audio/video transmission. Examples, both historical and current, are given that typify the different approaches used in each application area. Although no attempt is made to be comprehensive in the coverage, the examples chosen clearly illustrate the richness, variety, and importance of error-control coding methods in modern digital applications.

Applications of error-control coding

Starting from Shannon's celebrated 1948 channel coding theorem, we trace the evolution of channel coding from Hamming codes to capacity-approaching codes. We focus on the contributions that have led to the most significant improvements in performance versus complexity for practical applications, particularly on the additive white Gaussian noise channel. We discuss algebraic block codes, and why they did not prove to be the way to get to the Shannon limit. We trace the antecedents of today's capacity-approaching codes: convolutional codes, concatenated codes, and other probabilistic coding schemes. Finally, we sketch some of the practical applications of these codes.

Channel Coding: The Road to Channel Capacity

Wireless Optical Communication Systems addresses the problem of designing efficient signaling and provides a link between the areas of communication theory and modem design for amplitude constrained linear optical intensity channel. Topics include historical perspective, channel impairments, amplitude constraints and the characteristics of popular optoelectronic components. A variety of wireless optical channel topologies are presented along with a survey and analysis of present day signalling techniques employed for these channels. The author provides a unifying framework for signalling design which allows the channel constraints to be represented geometrically and permits the use of modem design principles from electrical channels. Modulation schemes are designed using the formalism of lattice codes and a design process for signalling sets is specified. The use of multiple-input/multiple-output (MIMO) wireless optical channels to improve the spectral efficiency of links is explored. The basic spatio-temporal modem design problem is specified and a spatial multiplexing gain is quantified. New spatial discrete multitone modulation is proposed and the unique features are discussed. Based on measurements on an experimental prototype, a channel model is formulated and a realizable spatio-temporal coding scheme is simulated to quantify performance gains. This volume is organized for professional and academic readers engaged in modem design for wireless optical intensity channels. Significant background material is presented on both the properties as well as on fundamental communications principles. Wireless Optical Communication Systems can be used by physicists and experimentalists as an introduction to signalling design as well as communication systems designers.

Wireless Optical Communication Systems

We review probabilistic constellation shaping (PCS), which has been a key enabler for several recent record-setting optical fiber communications experiments. PCS provides both fine-grained rate adaptability and energy efficiency (sensitivity) gains. We discuss the reasons for the fundamentally better performance of PCS over other constellation shaping techniques that also achieve rate adaptability, such as time-division hybrid modulation, and examine in detail the impact of sub-optimum shaping and forward error correction (FEC) on PCS systems. As performance metrics for systems with PCS, we compare information-theoretic measures such as mutual information (MI), generalized MI (GMI), and normalized GMI, which enable optimization and quantification of the information rate (IR) that can be achieved by PCS and FEC. We derive the optimal parameters of PCS and FEC that maximize the IR for both ideal and non-ideal PCS and FEC. To avoid plausible pitfalls in practice, we carefully revisit key assumptions that are typically made for ideal PCS and FEC systems.

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Probabilistic Constellation Shaping for Optical Fiber Communications

A 24-dimensional block code based on the Leech lattice is described for use in a 19200-b/s modem. Encoding (the process of converting binary data into message point coordinates) and detection (conversion of the received coordinates into the coordinates of the nearest code point of the lattice) are described. Decoding (the further conversion of these into binary data) is not described, since it consists of an easily achieved reversal of the encoding process. Block (1000-bit-length) error rates in the presence of white noise for baud rates of 2400 and 2743, the two baud rates at which the modem can operate, are presented. Encoding plus decoding was successfully programmed for an ADSP2100 signal processor requiring altogether slightly less than one half the processing capability. >

G.R. Lang

Papers

A Leech lattice modem