Joint source-channel decoding of variable-length codes for convolutional codes and turbo codes

Several recent publications have shown that joint source-channel decoding could be a powerful technique to take advantage of residual source redundancy for fixed- and variable-length source codes. This letter gives an in-depth analysis of a low-complexity method recently proposed by Guivarch et al., where the redundancy left by a Huffman encoder is used at a bit level in the channel decoder to improve its performance. Several simulation results are presented, showing for two first-order Markov sources of different sizes that using a priori knowledge of the source statistics yields a significant improvement, either with a Viterbi channel decoder or with a turbo decoder.     

Symbol-by-symbol reception of signals corresponding to high-rate convolutional codes and turbo codes based on these convolutional codes

Computational procedures for symbol-by-symbol reception of signal ensembles corresponding to binary high-rate convolutional codes and to turbo codes formed by these convolutional codes are described. It is shown that the developed procedures are based on an optimum symbol-by-symbol reception algorithm that uses the fast Walsh-Hadamard transform algorithm.     

An extensive search for good punctured rate-k/(k+1) recursive convolutional codes for serially concatenated convolutional codes

In many practical applications requiring variable-rate coding and/or high-rate coding for spectral efficiency, there is a need to employ high-rate convolutional codes (CC), either by themselves or in a parallel or serially concatenated scheme. For such applications, in order to keep the trellis complexity of the code constant and to permit the use of a simplified decoder that can accommodate multiple rates, a mother CC is punctured to obtain codes with a variety of rates. This correspondence presents the results of extensive search for optimal puncturing patterns for recursive convolutional codes leading to codes of rate k/(k+1) (k an integer) to be used in serially concatenated convolutional codes (SCCC). The code optimization is in the sense of minimizing the required signal-to-noise ratio (SNR) for two target bit-error rate (BER) and two target frame-error rate (FER) values. We provide extensive sample simulation results for rate-k/(k+1) SCCC codes employing our optimized punctured CC.     

A cost-function based technique for design of good prunable interleavers for turbo codes

This paper addresses the design of semi-random, prunable interleavers for parallel concatenated convolutional codes (PCCC). The proposed technique is iterative and is based on the growth of a smaller interleaver up to the desired length N. The optimization is achieved via a minimization using a cost-function strictly related to both the correlation properties of the extrinsic information and the concept of spread of an interleaver. Performance of the designed interleavers are given in terms of bit error rate (BER) and frame error rate (FER). Comparisons are given with respect to other prunable and ad-hoc interleaver design techniques already proposed in literature. The designed interleavers are prunable and have a behavior very similar to the interleavers designed with techniques which maximize the spread of the permutation.     

Analysis of recursive convolutional codes and turbo codes as sources with memory

The paper proposes a general method to analyze discrete sources with memory. Besides the classical entropy, we define new information measures for discrete sources with memory, similar to the information quantities specific to discrete channels. On the base of this method, we show for the first time that, as result of convolutional and turbo encoding, sources with memory are obtained. We apply this information analysis method for the general case of a recursive convolutional encoder of rate R_CC = 1/n₀ and memory of order m, and for a turbo encoder of rate R_TC = 1/3, with two systematic recursive convolutional component encoders. Each component encoder has memory of order m, and is built based on the same primitive feedback polynomial. For the convolutional and turbo codes, the information quantities H(Y/S), H(S,Y), H(S/Y), H(Y), H(S) and I(S,Y) have been computed, where S and Y denote the set of states and the set of messages of the encoder, respectively. The analysis considered two cases: n₀ ≤ m + 1 and n₀ > m + 1. When n₀ = m + 1, the mutual information I(S,Y) is maximum and equal to m, as is the entropy of the set of states. For turbo codes, the quantity I(S,Y) also depends on the input bit and on its probability.     

Punctured recursive convolutional encoders and their applicationsin turbo codes

Puncturing is the predominant strategy to construct high code rate convolutional encoders, and infinite impulse response (IIR) convolutional encoders are an essential building block in turbo codes. In this paper, various properties of convolutional encoders with these characteristics are developed. In particular the closed-form representation of a punctured convolutional encoder and its generator matrix is constructed, necessary and sufficient conditions are given such that the punctured encoders retain the IIR property, and various lower bounds on distance properties, such as effective free distance, are developed. Finally, necessary and sufficient conditions are given on the inverse puncturing problem: representing a known convolutional encoder as a punctured encoder     

Comparative study of turbo equalization schemes usingconvolutional, convolutional turbo, and block-turbo codes

Turbo equalizers have been shown to be successful in mitigating the effects of inter-symbol interference introduced by partial response modems and by dispersive channels for code rates of R⩽ 1/2. We comparatively studied the performance of a range of binary phase-shift keying turbo equalizers employing block-turbo codes, namely Bose-Chaudhuri-Hocquenghen (1960, 1959) turbo codes, convolutional codes, and convolutional turbo codes having high code rates, such as R=3/4 and R=5/6, over a dispersive five-path Gaussian channel and an equally weighted symbol-spaced five-path Rayleigh fading channel. These turbo equalization schemes were combined with an iterative channel estimation scheme in order to characterize a realistic scenario. The simulation results demonstrated that the turbo-equalized system using convolutional turbo codes was the most robust system for all code rates investigated     

Limited search trellis decoding of convolutional codes

The least storage and node computation required by a breadth-first tree or trellis decoder that corrects t errors over the binary symmetric channels is calculated. Breadth-first decoders work with code paths of the same length, without backtracking. The Viterbi algorithm is an exhaustive trellis decoder of this type; other schemes look at a subset of the tree or trellis paths. For random tree codes, theorems about the asymptotic number of paths required and their depth are proved. For concrete convolutional codes, the worst case storage for t error sequences is measured. In both cases the optimal decoder storage has the same simple dependence on t. The M algorithm and algorithms proposed by G.J. Foschini (ibid., vol.IT-23, p.605-9, Sept. 1977) and by S.J. Simmons (PhD. diss., Queens Univ., Kingston, Ont., Canada) are optimal, or nearly so; they are all far more efficient than the Viterbi algorithm     

Comparison of convolutional and turbo codes for OFDM with antennadiversity in high-bit-rate wireless applications

This letter presents simulation results for the application of turbo coding to an OFDM system with diversity. First, results are presented for convolutional and Reed-Solomon codes. It is shown that even with short constraint lengths, convolutional codes have the potential to outperform Reed-Solomon codes, provided that sufficient precision is used in the soft decoder. We then evaluate the performance of turbo codes under slow fading conditions and study the effects of varying codeword size. Increasing codeword size theoretically provides better interleaving between the two component codes. However, this advantage is less clear when the fading rate is significantly lower than the symbol rate, which is typical of the high-data-rate systems considered here. Under such conditions, the advantage of using two component convolutional codes in turbo codes is limited. A single convolutional code with a long constraint length may be a better choice     

There are many good periodically time-varying convolutional codes

It is shown empirically that the number of good periodically time-varying convolutional codes increases exponentially with the period for any set of parameters. Hence, they can be used to enhance the security of cryptosystems without sacrificing error correction capability. It is shown that some periodically time-varying convolutional codes improve the free distance compared with fixed codes     

Low rate convolutional and turbo codes based on non‐linear cyclic codes

Low rate convolutional and turbo codes that output non‐linear cyclic (NLC) codewords of length n = 2^m, m being a positive integer, are described. These codes have a very low coding rate, which makes them especially suitable for spread spectrum systems where they can be used for simultaneously achieving error correction and bandwidth expansion. Due to the cyclic properties and codeword length of the component codes, branch metrics can be efficiently computed using the fast Fourier transform (FFT), enabling simple implementation of the encoder and decoder. Among the possible NLC base codes, special attention is given to the Tomlinson, Cercas, Hughes (TCH) codes family due to their good autocorrelation properties. It is shown by simulation that the turbo codes schemes studied usually perform better than traditional turbo codes (in this paper the universal mobile telecommunications system (UMTS), rate 1/3 turbo code was used as a reference). This improvement is accomplished at the cost of bandwidth expansion. One of the advantages of the presented solutions over other low rate codes is their ability to improve the synchronization process at the receiver due to the good autocorrelation properties of the available NLC codes (especially TCH codes). A comparison of performance between the UMTS uplink connection and an equivalent system using the proposed codes for a multiuser scenario in a multipath fading channel is presented showing the possibility of capacity increase when using these codes. Copyright © 2006 John Wiley & Sons, Ltd.     

Hybrid ARQ scheme based on recursive convolutional codes and turbo decoding

We propose a hybrid ARQ scheme using recursive convolutional codes, the turbo principle and combining diversity. While spending the same energy for transmission, the proposed scheme presents a better throughput for most of the SNR range and a smaller decoding complexity than another well-known method.     

802.16a协议中多进制卷积Turbo码及其改进

本文介绍了802.16a协议中的多进制卷积Turbo码编码原理以及基于比特判决的MAP算法,并提出了一种改进的交织方案以及仿真结果.     

A code-matched interleaver design for turbo codes

A code-matched interleaver design for turbo codes in which a particular interleaver is constructed to match the code weight distribution is proposed. The design method is based on the code distance spectrum. The low weight paths in the code trellis which give large contributions to the error probability in the signal-to-noise ratio region of interest for practical communication systems are eliminated so that they do not appear in the overall code trellis after interleaving. The proposed interleaver improves the code error performance at moderate to high signal-to-noise ratio and considerably increases the asymptotic slope of the error probability curves     

A conceptual framework for understanding turbo codes

For understanding turbo codes, we propose to locate them at the intersection of three main topics: the random-like criterion for designing codes, the idea of extending the decoding role to reassess probabilities, and that of combining several codes by product or concatenation. Concerning the idea of designing random-like (RL) codes, we distinguish strongly and weakly random-like codes depending on how the closeness of their weight distribution to that obtained in the average by random coding is measured. Using, e.g., the cross entropy as a closeness measure results in weakly RL codes. Although their word-error rate is bad, their bit-error rate (BER) remains low up to the vicinity of the channel capacity. We show that pseudorandom recursive convolutional codes belong to this family. Obtaining reasonably good performance with a single code of this type involves high complexity, and its specific decoding is difficult. However, using these codes as components in the turbo-code scheme is a simple means for improving the low-weight tail of the distribution and to adjust the BER to any specification. In order to increase the encoder memory without inordinate complexity, it is suggested to use iterated nonexhaustive replication decoding     

Bolt interleavers for turbo codes

A procedure is described to produce efficient turbo-code interleavers. It is well suited to the selection of interleavers that produce a large minimum distance for the resulting turbo codes. A feature of these interleavers is that the final state of the two elementary encoders is simultaneously zero. Examples are given and the corresponding codes are simulated. They seem to be among the best ones for rate 1/3 turbo codes.     

Interleaver design for turbo codes

The performance of a turbo code with short block length depends critically on the interleaver design. There are two major criteria in the design of an interleaver: the distance spectrum of the code and the correlation between the information input data and the soft output of each decoder corresponding to its parity bits. This paper describes a new interleaver design for turbo codes with short block length based on these two criteria. A deterministic interleaver suitable for turbo codes is also described. Simulation results compare the new interleaver design to different existing interleavers     

VLSI architectures for turbo codes

A great interest has been gained in recent years by a new error-correcting code technique, known as “turbo coding”, which has been proven to offer performance closer to the Shannon's limit than traditional concatenated codes. In this paper, several very large scale integration (VLSI) architectures suitable for turbo decoder implementation are proposed and compared in terms of complexity and performance; the impact on the VLSI complexity of system parameters like the state number, number of iterations, and code rate are evaluated for the different solutions. The results of this architectural study have then been exploited for the design of a specific decoder, implementing a serial concatenation scheme with 2/3 and 3/4 codes; the designed circuit occupies 35 mm², supports a 2 Mb/s data rate, and for a bit error probability of 10^-6, yields a coding gain larger than 7 dB, with ten iterations     

A robust decoding for long convolutional codes

Sequential decoding is an attractive technique to achieve the reliability of communication promised by the channel coding theory. But, because it utilizes the Fano metric, its performance is sensitive to channel parameter variations and it cannot simultaneously minimize both decoding effort and probability of decoding error. Based on the distance properties of the codes, we have derived a new set of metric which not only can overcome the two drawbacks caused by the Fano metric but also can significantly reduce the decoding effort required by sequential decoding.     

A construction technique for random-error-correcting convolutional codes

A simple algorithm is presented for finding rate1/nrandom-error-correcting convolutional codes. Good codes considerably longer than any now known are obtained. A discussion of a new distance measure for convolutional codes, called the free distance, is included. Free distance is particularly useful when considering decoding schemes, such as sequential decoding, which are not restricted to a fixed constraint length. It is shown how the above algorithm can be modified slightly to produce codes with known free distance. A comparison of probability of error with sequential decoding is made among the best known constructive codes of constraint length36.