Decoding Of Convolutional Codes Research Articles

On the decoding of convolutional codes on an unknown channel

An algorithm is proposed for universal decoding of convolutional/trellis codes employed over unknown channels. On discrete memoryless channels and at rates below the channel's computational cutoff rate (for a uniform input distribution), the algorithm achieves an asymptotic complexity-performance tradeoff similar to the tradeoff achieved by the Viterbi (1979) algorithm, but with the benefit that the algorithm's implementation does not require knowledge of the channel law. The algorithm is also applicable to channels with memory, and in particular to intersymbol interference (ISI) channels, to channels with nonlinear ISI, and even to general finite-state channels.

Single trellis decoding of Reed-Solomon and convolutional concatenated code

By using the trellis approaches suggested by Wolf, block codes can be decoded by trellis decoding methods. The authors apply these methods to RS-convolutional concatenated codes and propose a decoding method using a single combined trellis structure comprising a convolutional code and RS code to easily obtain the soft decision decoding gain. The proposed decoding algorithm eliminates the need for a symbol interleaver, thus removing the time delay due to the symbol interleaver. The performance of the soft decision single trellis decoding of RS-convolutional concatenated codes over a fading channel is evaluated by simulation.

Low-power Viterbi decoder for CDMA mobile terminals

An efficient state-sequential very large scale integration (VLSI) architecture and low-power design methodologies ranging from the system-level to the layout-level are presented for a large-constraint-length Viterbi decoder for code division multiple access (CDMA) digital cellular/personal communication services (PCS) applications. The low-power design approaches are also applicable to many other systems and algorithms. VLSI implementation issues and prototype fabrication results for a state-sequential Viterbi decoder for convolutional codes of rate 1/2 and constraint-length 9 are also described. The chip's core, consisting of approximately 65 k transistors, occupies 1.9 mm by 3.4 mm in a 0.8-/spl mu/m triple-layer-metal n-well CMOS technology. The chip's measured total power dissipation is 0.24 mW at a 14.4 kb/s data-rate with 0.9216 MHz clocking at a supply voltage of 1.65 V. The Viterbi decoder presented here is the lowest power and smallest area core in its class, to the best of our knowledge.

Concatenated decoding with a reduced-search BCJR algorithm

We apply two reduced-computation variants of the BCJR algorithm to the decoding of serial and parallel concatenated convolutional codes. One computes its recursions at only M states per trellis stage; one computes only at states with values above a threshold. The threshold scheme is much more efficient, and it greatly reduces the computation of the BCJR algorithm. By computing only when the channel demands it, the threshold scheme reduces the turbo decoder computation to one-four nodes per trellis stage after the second iteration.

Iterative decoding of compound codes by probability propagation in graphical models

We present a unified graphical model framework for describing compound codes and deriving iterative algorithms. After reviewing a variety of graphical models (Markov random fields, Tanner graphs, and Bayesian networks), we derive a general distributed marginalization algorithm for functions described by factor graphs. From this general algorithm, Pearl's (1986) belief propagation algorithm is easily derived as a special case. We point out that iterative algorithms for various codes, including turbo decoding of parallel-concatenated convolutional codes, may be viewed as probability propagation in a graphical model of the code. We focus on Bayesian network descriptions of codes, which give a natural input/state/output/channel description of a code and channel, and we indicate how iterative decoders can be developed for parallel-and serially concatenated coding systems, product codes, and low-density parity-check codes.

Iterative MAP decoding for serial concatenated convolutional codes

The paper provides detailed computational steps for implementing an iterative serial concatenated convolutional code (SCCC) decoder. These are based on maximum a posteriori probability (MAP) decoding of a single, rate 1/2, recursive, systematic convolutional code, which is reduced to easily implemented equations for forward and backward recursion. In particular, the crucial information exchange between MAP decoders is clarified. Simulation of a rate 1/3 SCCC with memory-2 codes and a coding delay of N = 1000 shows a bit error rate of 10-6 for Eb/N0 = 1.5 dB, and gives a typical interleaver gain of N-3.

An intuitive justification and a simplified implementation of the MAP decoder for convolutional codes

An intuitive shortcut to understanding the maximum a posteriori (MAP) decoder is presented based on an approximation. This is shown to correspond to a dual-maxima computation combined with forward and backward recursions of Viterbi algorithm computations. The logarithmic version of the MAP algorithm can similarly be reduced to the same form by applying the same approximation. Conversely, if a correction term is added to the approximation, the exact MAP algorithm is recovered. It is also shown how the MAP decoder memory can be drastically reduced at the cost of a modest increase in processing speed.

Analysis and performance of bidirectional decoding of convolutional codes over fading channels

The performance of suboptimal convolutional decoding over fading channels is explored. The suboptimal decoding algorithm used is the bidirectional algorithm. By estimating a "decoder weight spectrum" for the decoder, an "equivalent free distance" may be observed. Furthermore, by using this "decoder weight spectrum", useful estimations of the error probabilities are obtained and compared to computer-simulation results in the case of very slow and very fast fading. The resultant curves are shown to be very tightly related. Computer-simulation results are also shown for various signal-to-noise ratios, normalized Doppler spreads, and frame length on three typical fading channels: the Rayleigh fading channel with exponential and Bessel autocorrelation functions and the Rician fading channel with exponential autocorrelation function. We show that considerable gains (up to 4 dB) can be obtained with respect to a similar-complexity Viterbi decoder at a frame error probability P/sub e/=10/sup -3/ and a slightly smaller gain (up to 1.8 dB) at a bit error probability P/sub b/=10/sup -5/.

Optimum soft decision decoding with channel state information in the presence of fading

The article first presents the formulation for computing the optimum soft decision metrics for decoding of convolutional codes on fading channels and then demonstrate the performance gains achieved in using the optimum soft decision metrics. The optimum soft decision metrics use channel state information to account for signal strength variations due to fading. The formulation for computing the optimum soft decision metrics is given for the general case in which the statistics of the additive noise on the channel is nonstationary. However, the formulation is applied to a more simple case in which the noise is assumed to be stationary. The results indicate that at moderate signal-to-noise ratios, the 2 dB advantage usually claimed for soft decision decoding over hard decision decoding on Gaussian channels is achievable on fading channel, with the rate 3/4 code used, if the channel state information is incorporated in the soft decision metrics.

High-performance VLSI architecture for the Viterbi algorithm

The Viterbi (1967) algorithm (VA) is known to be an efficient method for the realization of maximum-likelihood (ML) decoding of convolutional codes. The VA is characterized by a graph, called a trellis, which defines the transitions between states. To define an area efficient architecture for the VA is equivalent to obtaining an efficient mapping of the trellis. We present a methodology that permits the efficient hardware mapping of the VA onto a processor network of arbitrary size. This formal model is employed for the partitioning of the computations among an arbitrary number of processors in such a way that the data are recirculated, optimizing the use of the PEs and the communications. Therefore, the algorithm is mapped onto a column of processing elements and an optimal design solution is obtained for a particular set of area and/or speed constraints. Furthermore, the management of the surviving path memory for its mapping and distribution among the processors was studied. As a result, we obtain a regular and modular design appropriate for its VLSI implementation in which the only necessary communications between processors are the data recirculations between stages.

Three types of robust maximum-likelihood Viterbi decoder

We propose three new types of maximum-likelihood Viterbi (1967) decoders (for convolutional or trellis codes) with each under the hypothesis of logistics, Laplace, of Cauchy noise distribution respectively. Unlike a Gaussian ML Viterbi decoder, the three newly-proposed Viterbi decoders exhibit a valuable robustness property, i.e. insensitivity to deviations from the assumed noise distribution.

Two-step trellis decoding of partial unit memory convolutional codes

We present a new soft-decision decoding method for high-rate convolutional codes. The decoding method is especially well-suited for PUM convolutional codes. The method exploits the linearity of the parallel transitions in the trellis associated with PUM codes. We provide bounds on the number of operations per decoded bit, and show that this number is dependent on the weight hierarchy of the linear block code associated with the parallel transitions. The complexity of the new decoding method for PUM codes is compared to the complexity of Viterbi decoding of comparable punctured convolutional codes. Examples from a special class of PUM codes show that the new decoding method compares favorably to Viterbi decoding of punctured codes.

Modified multiple stack algorithm for decoding convolutional codes

The multiple stack algorithm (MSA), devised by Chevillat and Costello, is an efficient algorithm for erasurefree decoding of long constraint length convolutional codes. In the MSA, substack size and the number of transferred survivors (or successors) are assumed to be small. Lower error probabilities can be achieved by increasing the first stack size and/or increasing the computational limit. A large storage capacity for survivors is required to prevent memory overflow and achieve a low error probability. The authors present a modified MSA, in which the storage capacity for survivors is kept as a constant, while the substacks are arranged in a ring-like structure to handle the overflow problem of storage for survivors. In addition, the substack size and the number of transferred survivors are made large to improve performance. The performance of the modified MSA in decoding a convolutional code with constraint length m = 23 is investigated and compared with the performance of the unmodified MSA.

Adaptive Viterbi decoding of convolutional codes over memoryless channels

In this paper, an adaptive decoding algorithm for convolutional codes, which is a modification of the Viterbi algorithm (VA) is presented. For a given code, the proposed algorithm yields nearly the same error performance as the VA while requiring a substantially smaller average number of computations. Unlike most of the other suboptimum algorithms, this algorithm is self-synchronizing. If the transmitted path is discarded, the adaptive Viterbi algorithm (AVA) can recover the state corresponding to the transmitted path after a few trellis depths. Using computer simulations over hard and soft 3-bit quantized additive white Gaussian noise channels, it is shown that codes with a constraint length K up to 11 can be used to improve the bit-error performance over the VA with K=7 while maintaining a similar average number of computations. Although a small variability of the computational effort is present with our algorithm, this variability is exponentially distributed, leading to a modest size of the input buffer and, hence, a small probability of overflow.

Metrics for syndrome decoding of convolutional codes

AbstractThis paper derives soft decision metrics for syndrome decoding of convolutional codes. It is shown that, given a received sequence Z, syndrome decoding of convolutional codes can be seen as the problem of finding error sequence E. This sequence may be a possible cause of the syndrome sequence computed from Z and for which the joint probability μ(Z,E) is maximum. Then the soft decision metrics are derived by actually evaluating the probability μ(Z, E). In particular, a concrete representation of the symbol metrics for an AWGN channel is given. After expressing the derived metric in terms of the inner product, the basic properties of the metric for syndrome decoding are clarified and the similarity between the metrics for syndrome and Viterbi decoding are shown.

An algorithm for identifying rate (n-1)/n catastrophic punctured convolutional encoders

It is known that both Viterbi and sequential decoding of convolutional codes can be greatly simplified by employing punctured convolutional codes, which are obtained by periodically deleting a part of the bits of a low-rate convolutional code. Even if the original low-rate convolutional code is noncatastrophic, some deleting maps may result in rate (n-1)/n catastrophic punctured encoders. An algorithm is presented to identify such encoders when the original rate 1/b encoder is antipodal. The major part of the algorithm solves a linear equation of /spl nu/+1 variables, where /spl nu/ is the constraint length of the original rate 1/b code.

Iterative decoding of binary block and convolutional codes

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.

Variable-complexity trellis decoding of binary convolutional codes

Considers trellis decoding of convolutional codes with selectable effort, as measured by decoder complexity. Decoding is described for single parent codes with a variety of complexities, with performance near that of the optimal fixed receiver complexity coding system. Effective free distance is examined. Criteria are proposed for ranking parent codes, and some codes found to be best according to the criteria are tabulated, Several codes with effective free distance better than the best code of comparable complexity were found. Asymptotic (high SNR) performance analysis and error propagation are discussed. Simulation results are also provided.

The TAR decoder-a bandpass Viterbi/FFT decoder for convolutional encoded spread-spectrum signals

A new form of bandpass convolutional decoder termed the TAR decoder is presented. The decoder has as its basis the classic Viterbi algorithm, but uses a fast Fourier transform (FFT) in order to cope with a frequency offset, phase rotating, received signal. The TAR decoder also uses a two-dimensional (2-D) despread signal history array in addition to the usual path history registers. The advantage of the TAR decoder is that the decoder does not need to be preceded by a modem and is not subject to the problems of cycle slips. It can operate at negative input signal-to-noise ratio (SNR) values and is well suited to the decoding of low rate convolutional codes as may be used in code-division multiple-access (CDMA) systems. Symbol timing recovery is still required but this is straightforward in practical cases down to /spl sime/-15 dB SNR.

Hidden Markov models for the burst error statistics of Viterbi decoding

The method of the hidden Markov model (HMM) is used to develop a faithful model for the burst error statistics of Viterbi decoding of convolutional codes. One of the advantages of building such a model is that it can be used to generate the output sequence with little cost and can provide a basis for studying other system parameters. The HMM developed generally performs better than the geometric model and, in most cases, better than the previously proposed Markov model, and it requires much fewer parameters than those of the Markov model for convolutional codes of large constraint length.