High-rate punctured convolutional codes for Viterbi and sequentialdecoding

An investigation is conducted of the high-rate punctured convolutional codes suitable for Viterbi and sequential decoding. Results on known short-memory codes M⩽8 discovered by others are extended. Weight spectra and upper bounds on the bit error probability of the best known punctured codes having memory 2⩽M ⩽8, and coding rates 2/3⩽R⩽7/8 are provided. Newly discovered rate-2/3 and -3/4 long-memory punctured convolutional codes with 9⩽M⩽23 are provided together with the leading terms of their weight spectra and their bit error performance bounds. Some results of simulation with sequential decoding are given     

BER analysis of convolution coded QDPSK, in digital mobile radio

The bit error rate (BER) performance of convolutional coded quaternary differential phase-shift keying (QDPSK) with Viterbi decoding is theoretically investigated in Rayleigh fading environments. The probability density functions of the path and branch metric values of Viterbi decoding are derived. The BERs after decoding due to additive white Gaussian noise and cochannel interference are theoretically analyzed. Rate 1/2 codes and their symbol punctured high-rate codes are considered, and the symbol positions for deletion to minimize the BER after decoding are presented for the codes with a constraint length K=3-7. It is shown that Viterbi decoding considerably reduces the desired signal-to-interference power ratio as well as the signal energy per information bit-to-noise power spectrum density ratio necessary to achieve a certain BER. The spectrum efficiency of the cellular mobile radio system, achievable by the use of the symbol punctured codes, is also evaluated     

Free distance bounds for convolutional codes

The best asymptotic bounds presently known on free distance for convolutional codes are presented from a unified point of view. Upper and lower bounds for both time-varying and fixed codes are obtained. A comparison is made between bounds for nonsystematic and systematic codes which shows that more free distance is available with nonsystematic codes. This result is important when selecting codes for use with sequential or maximum-likelihood (Viterbi) decoding since the probability of decoding error is closely related to the free distance of the code. An ancillary result, used in proving the lower bound on free distance for time-varying nonsystematic codes, furnishes a generalization of two earlier bounds on the definite decoding minimum distance of convolutional codes.     

High-rate punctured convolutional self-doubly orthogonal codes for iterative threshold decoding

The puncturing technique allows obtaining high-rate convolutional codes from low-rate convolutional codes used as mother codes. This technique has been successfully applied to generate good high-rate convolutional codes which are suitable for Viterbi and sequential decoding. In this paper, we investigate the puncturing technique for convolutional self-doubly orthogonal codes (CSO/sup 2/C) which are decoded using an iterative threshold-decoding algorithm. Based on an analysis of iterative threshold decoding of the rate-R=b/(b+1) punctured systematic CSO/sup 2/C, the required properties of the rate-R=1/2 systematic convolutional codes (SCCs) used as mother codes are derived. From this analysis, it is shown that there is no need for the punctured mother codes to respect all the required conditions, in order to maintain the double orthogonality at the second iteration step of the iterative threshold-decoding algorithm. The results of the search for the appropriate rate-R=1/2 SCCs used as mother codes to yield a large number of punctured codes of rates 2/3/spl les/R/spl les/6/7 are presented, and some of their error performances evaluated.     

On the performance of sequential and Viterbi decoders for high-ratepunctured convolutional codes

This paper presents new computationally efficient and accurate techniques for estimating the performance of specific high-rate punctured convolutional codes and uses these techniques to evaluate the performance of sequential and Viterbi decoders for the best known codes. In particular, it demonstrates that the disparity between sequential and Viterbi decoding increases dramatically for long memory codes with high rates and for such codes, the union bound cannot be used as a criterion for selecting good codes for sequential decoders. In contrast, it shows that the proposed methods can be used as efficient tools for performance evaluation and/or identification of good high-rate punctured convolutional codes for use with sequential decoders     

VLSI Structures for Viterbi Receivers: Part I--General Theory and Applications

A taxonomy of VLSI grid model layouts is presented for the implementation of certain types of digital communication receivers based on the Viterbi algorithm. We deal principally with networks of many simple processors connected to perform the Viterbi algorithm in a highly parallel way. Two interconnection patterns of interest are the "shuffleexchange" and the "cube-connected cycles." The results are generally applicable to the development of area-efficient VLSI circuits for decoding: convolutional codes, coded modulation with multilevel/phase signals, punctured convolutional codes, correlatively encoded MSK signals and for maximum likelihood sequence estimation ofM-ary signals on intersymbol interference channels. In a companion paper, we elaborate on how the concepts presented here can be applied to the problem of building encoded MSK Viterbi receivers. Lower bounds are established on the product (chip area) * (baud rate)^-2and on the energy consumption that any VLSI implementation of the Viterbi algorithm must obey, regardless of the architecture employed or the intended application.     

Performance bounds of rate-? convolutional codes with QPSK on Rayleigh fading channel

An analytic expression for the bit error probability upper bounds of rate-? convolutional codes in conjunction with QPSK modulation and maximum-likelihood Viterbi decoding on the fully interleaved Rayleigh fading channel is presented. The given expression is evaluated numerically for selected rate-? optimum convolutional codes together with QPSK.     

BEAST decoding of block codes obtained via convolutional codes

BEAST is a bidirectional efficient algorithm for searching trees. In this correspondence, BEAST is extended to maximum-likelihood (ML) decoding of block codes obtained via convolutional codes. First it is shown by simulations that the decoding complexity of BEAST is significantly less than that of the Viterbi algorithm. Then asymptotic upper bounds on the BEAST decoding complexity for three important ensembles of codes are derived. They verify BEAST's high efficiency compared to other algorithms. For high rates, the new asymptotic bound for the best ensemble is in fact better than previously known bounds.     

Performance of Unequally Punctured Convolutional Codes

In practical applications, some coded symbols are unequally punctured for rate matching of terminated convolutional codes. In this letter, performance upper bounds are proposed for a terminated convolutional code punctured with an arbitrary pattern. To obtain the upper bounds, a weight enumerator is defined to represent the relation between the Hamming weight of the coded outputs and that of the selected input information bits. A method is presented to compute the weight enumerator based on a modified trellis diagram. Simulation and numerical analysis show that the proposed bounds are tight to the performance of terminated convolutional codes.     

A Tightened Upper Bound on the Error Probability of Binary Convolutional Codes with Viterbi Decoding

Tighter upper bounds on the error event and the bit error probabilities, respectively, for maximum-likelihood decoding of binary convolutional codes on the binary symmetric channel are derived from upper bounds previously published by Viterbi [1]. The measured bit error rateP_bfor a constraint length 3 decoder has been plotted versus the channel transition probabilitypand shows close agreement with the improved bound on the bit error probability.     

A convolutional single-parity-check concatenated coding scheme forhigh-data-rate applications

The coding scheme uses a set of n convolutional codes multiplexed into an inner code and a (n,n-1) single-parity-check code serving as the outer code. Each of the inner convolutional codes is decoded independently, with maximum-likelihood decoding being achieved using n parallel implementations of the Viterbi algorithm. The Viterbi decoding is followed by additional outer soft-decision single-parity-check decoding. Considering n=12 and the set of short constraint length K=3, rate 1/2 convolutional codes, it is shown that the performance of the concatenated scheme is comparable to the performance of the constraint length K=7, rate 1/2 convolutional code with standard soft-decision Viterbi decoding. Simulation results are presented for the K=3, rate 1/2 as well as for the punctured K=3, rate 2/3 and rate 3/4 inner convolutional codes. The performance of the proposed concatenated scheme using a set of K=7, rate 1/2 inner convolutional codes is given     

High-Rate Punctured Convolutional Codes for Soft Decision Viterbi Decoding

The high-rate punctured codes of rates 2/3 through 13/14 are derived from rate 1/2 specific convolutional codes with maximal free distance. Coding gains of derived codes are compared based on their bit error rate performances under soft decision Viterbi decoding.     

Optimal allocation of bandwidth for source coding, channel coding, and spreading in CDMA systems

This paper investigates the tradeoffs between source coding, channel coding, and spreading in code-division multiple-access systems, operating under a fixed total bandwidth constraint. We consider two systems, each consisting of a uniform source with a uniform quantizer, a channel coder, an interleaver, and a direct-sequence spreading module. System A is quadrature phase-shift keyed modulated and has a linear block channel coder. A minimum mean-squared error receiver is also employed in this system. System B is binary phase-shift keyed modulated. Rate-compatible punctured convolutional codes and soft-decision Viterbi decoding are used for channel coding in system B. The two systems are analyzed for both an additive white Gaussian noise channel and a flat Rayleigh fading channel. The performances of the systems are evaluated using the end-to-end mean squared error. A tight upper bound for frame-error rate is derived for nonterminated convolutional codes for ease of analysis of system B. We show that, for a given bandwidth, an optimal allocation of that bandwidth can be found using the proposed method.     

New very high rate punctured convolutional codes

New very high rate punctured convolutional codes suitable for Viterbi decoding are presented. These codes extend recent results on high rate convolutional codes by employing an efficient computer search technique and are optimum in that they minimise the residual bit error rate over all puncturing patterns given a particular basis code. Tables of new codes of rate (n-1)/n are given for n=9, 10, 11 and 12     

Joint Source and Channel Coding using Punctured Ring Convolutional Coded CPM

In this paper, a novel trellis source encoding scheme based on punctured ring convolutional codes is presented. Joint source and channel coding (JSCC) using trellis coded continuous phase modulation (CPM) with punctured convolutional codes over rings is investigated. The channels considered are the additive white gaussian noise (AWGN) channel and the Rayleigh fading channel. Optimal soft decoding for the proposed JSCC scheme is studied. The soft decoder is based on the a posteriori probability (APP) algorithm for trellis coded CPM with punctured ring convolutional codes. It is shown that these systems with soft decoding outperform the same systems with hard decoding especially when the systems operate at low to medium signal-to-noise ratio (SNR). Furthermore, adaptive JSCC approaches based on the proposed source coding scheme are investigated. Compared with JSCC schemes with fixed source coding rates, the proposed adaptive approaches can achieve much better performance in the high SNR region. The novelties of this work are the development of a trellis source encoding method based on punctured ring convolutional codes, the use of a soft decoder, the APP algorithm for the combined systems and the adaptive approaches to the JSCC problem.     

基于删余等效的高码率卷积码的低复杂度译码方法

根据删余卷积码具有较低的译码复杂度这一特征,提出了一种适用于普通高码率卷积码的低复杂度译码方法。通过多项式生成矩阵表示法,推导了删余卷积码的等效多项式生成矩阵,给出了等效多项式生成矩阵的计算准则。在分析删余卷积码与相同码率普通卷积码的等效关系和区别的基础上,提出了高码率卷积码的删余等效并给出了计算高码率卷积码删余等效后原始码和删余矩阵的方法。以原始码和删余矩阵构成的删余等效结构为译码基础,实现了高码率卷积码的低复杂度译码,其译码复杂度与原始码相当。仿真结果表明,删余等效译码方法相对于正常译码方法,其性能损失很小。     

一种应用于LTE系统的Viterbi译码算法

LTE(long term evolution,长期演进)系统中采用了咬尾卷积码和Turbo码来实现前向纠错,Viterbi译码是卷积码的一种杰出的译码算法,它是一种最大似然译码方法。本文基于LTE系统中的咬尾卷积码,详细分析了几种较成熟的Viterbi译码算法,并综合现有算法,提出了一种改进算法,减小了译码计算的复杂度。仿真结果表明,改进算法在降低译码计算复杂度的同时还降低了译码误比特率,因此非常适合LTE系统的译码要求。     

Rate k/(k+1) punctured convolutional encoders

G.D. Forney (1970, 1975) defined a minimal encoder as a polynomial matrix G such that G generates the code and G has the least constraint length among all generators for the code. Any convolutional code can be generated by a minimal encoder. High-rate k(k+1) punctured convolutional codes were introduced to simplify Viterbi decoding. An ordinary convolutional encoder G can be obtained from any punctured encoder. A punctured encoder is minimal if the corresponding ordinary encoder G is minimal and the punctured and ordinary encoders have the same constraint length. It is shown that any rate k/(k+1), noncatastrophic, antipodal punctured encoder is a minimal encoder.<>     

State diagram connectivity and its effects on the decoding ofshift-register-based codes

Coding performance is limited not only by Shannon's (1950) bounds but also by the complexity of decoders. Decoder complexity is in turn governed by the need for the different pieces of the machine to communicate with one another. This paper calculates lower limits on the intra-system information flow for the Viterbi decoding of shift-register-based codes, e.g., convolutional codes. These limits provide practical guidance for the construction of decoders for the current generation of convolutional and trellis codes. In particular, these bounds prove that a very specialized decoder family, called graph partition decoders, have an asymptotically optimum communications growth rate. The techniques used in this paper can, moreover, be applied to the design of new (non-shift-register-based) codes which may possibly circumvent the limits derived in the paper and to the design of parallel processors     

A new upper bound on the first-event error probability for maximum-likelihood decoding of fixed binary convolutional codes (Corresp.)

An upper bound on the first-event error probability for maximum-likelihood decoding of fixed binary convolutional codes on the binary symmetric channel is derived. The bound is evaluated for rate1/2codes, and comparisons are made with simulations and with the bounds of Viterbi, Van de Meeberg, and Post. In particular, the new bound is significantly better than Van de Meeberg's bound for rates aboveR_{comp}.