Book•
Error-Correction Coding for Digital Communications
30 Jun 1981-
TL;DR: This paper presents a meta-modelling architecture for Convolutional Code Structure and Viterbi Decoding, and some of the techniques used in this architecture can be applied to Group Codes and Block Codes.
Abstract: 1. Fundamental Concepts of Coding.- 2. Group Codes.- 3. Simple Nonalgebraic Decoding Techniques for Group Codes.- 4. Soft Decision Decoding of Block Codes.- 5. Algebraic Techniques for Multiple Error Correction.- 6. Convolutional Code Structure and Viterbi Decoding.- 7. Other Convolutional Decoding Techniques.- 8. System Applications.- Appendix A. Code Generators for BCH Codes.- Appendix B. Code Generators for Convolutional Codes.- B.1. Viterbi Decoding.- B.2. Table Look-up Decoding.- B.3. Threshold Decoding.- B.4. Sequential Decoding.- References.
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TL;DR: Using log-likelihood algebra, it is shown that any decoder can be used which accepts soft inputs-including a priori values-and delivers soft outputs that can be split into three terms: the soft channel and aPriori inputs, and the extrinsic value.
Abstract: Iterative decoding of two-dimensional systematic convolutional codes has been termed "turbo" (de)coding. Using log-likelihood algebra, we show that any decoder can be used which accepts soft inputs-including a priori values-and delivers soft outputs that can be split into three terms: the soft channel and a priori inputs, and the extrinsic value. The extrinsic value is used as an a priori value for the next iteration. Decoding algorithms in the log-likelihood domain are given not only for convolutional codes but also for any linear binary systematic block code. The iteration is controlled by a stop criterion derived from cross entropy, which results in a minimal number of iterations. Optimal and suboptimal decoders with reduced complexity are presented. Simulation results show that very simple component codes are sufficient, block codes are appropriate for high rates and convolutional codes for lower rates less than 2/3. Any combination of block and convolutional component codes is possible. Several interleaving techniques are described. At a bit error rate (BER) of 10/sup -4/ the performance is slightly above or around the bounds given by the cutoff rate for reasonably simple block/convolutional component codes, interleaver sizes less than 1000 and for three to six iterations.
2,632 citations
TL;DR: A generalization of the convex hull of a finite set of points in the plane leads to a family of straight-line graphs, "alpha -shapes," which seem to capture the intuitive notions of "fine shape" and "crude shape" of point sets.
Abstract: A generalization of the convex hull of a finite set of points in the plane is introduced and analyzed. This generalization leads to a family of straight-line graphs, " \alpha -shapes," which seem to capture the intuitive notions of "fine shape" and "crude shape" of point sets. It is shown that a-shapes are subgraphs of the closest point or furthest point Delaunay triangulation. Relying on this result an optimal O(n \log n) algorithm that constructs \alpha -shapes is developed.
1,648 citations
Book•
27 Sep 2001
1,221 citations
TL;DR: A class of nonlinear estimation algorithms is described to estimate the unknown phase of a carrier which is fully modulated by m -ary PSK modulation, and the effect of quantization and finite read-only-memory implementation of the nonlinearity are determined by computer simulation.
Abstract: Burst transmission of digital data and voice has become commonplace, particularly in satellite communication systems employing time-division multiple-access (TI)MA) and packet demand-assignment multiple-access (DAMA) techniques. In TDMA systems particularly, phase estimation on each successive burst is a requirement, while bit timing and carrier frequency can be accurately tracked between bursts. A class of nonlinear estimation algorithms is described to estimate the unknown phase of a carrier which is fully modulated by m -ary PSK modulation. Performance of the method is determined in closed form and compared to the Cramer-Rao lower bound for the variance of the estimation error in the phase of an unmodulated carrier. Results are also obtained when the carrier frequency is imprecisely known. Finally, the effect of quantization and finite read-only-memory (ROM) implementation of the nonlinearity are determined by computer simulation.
1,116 citations
IBM1
TL;DR: An introduction into TCM is given, reasons for the development of TCM are reviewed, and examples of simple TCM schemes are discussed.
Abstract: rellis-Coded Modulation (TCM) has evolved over the past decade as a combined coding and modulation technique for digital transmission over band-limited channels. Its main attraction comes from the fact that it allows the achievement of significant coding gains over conventional uncoded multilevel modulation without compromising bandwidth efficiency. T h e first TCM schemes were proposed in 1976 [I]. Following a more detailed publication [2] in 1982, an explosion of research and actual implementations of TCM took place, to the point where today there is a good understanding of the theory and capabilities of TCM methods. In Part 1 of this two-part article, an introduction into TCM is given. T h e reasons for the development of TCM are reviewed, and examples of simple TCM schemes are discussed. Part I1 [I51 provides further insight into code design and performance, and addresses. recent advances in TCM. TCM schemes employ redundant nonbinary modulation in combination with a finite-state encoder which governs the selection of modulation signals to generate coded signal sequences. In the receiver, the noisy signals are decoded by a soft-decision maximum-likelihood sequence decoder. Simple four-state TCM schemes can improve. the robustness of digital transmission against additive noise by 3 dB, compared to conventional , uncoded modulation. With more complex TCM schemes, the coding gain can reach 6 dB or more. These gains are obtained without bandwidth expansion or reduction of the effective information rate as required by traditional error-correction schemes. Shannon's information theory predicted the existence of coded modulation schemes with these characteristics more than three decades ago. T h e development of effective TCM techniques and today's signal-processing technology now allow these ,gains to be obtained in practice. Signal waveforms representing information sequences ~ are most impervious to noise-induced detection errors if they are very different from each other. Mathematically, this translates into therequirement that signal sequences should have large distance in Euclidean signal space. ~ T h e essential new concept of TCM that led to the afore-1 mentioned gains was to use signal-set expansion to I provide redundancy for coding, and to design coding and ' signal-mapping functions jointly so as to maximize ~ directly the \" free distance \" (minimum Euclidean distance) between coded signal sequences. This allowed the construction of modulation codes whose free distance significantly exceeded the minimum distance between uncoded modulation signals, at the same information rate, bandwidth, and signal power. The term \" …
874 citations