Erasure-free sequential decoding of trellis codes

An erasure-free sequential decoding algorithm for trellis codes, called the buffer looking algorithm (BLA), is introduced. Several versions of the algorithm can be obtained by choosing certain parameters and selecting a resynchronization scheme. These can be categorized as block decoding or continuous decoding, depending on the resynchronization scheme. Block decoding is guaranteed to resynchronize at the beginning of each block, but suffers some rate loss when the block length is relatively short. The performance of a typical block decoding scheme is analyzed, and we show that significant coding gains over Viterbi decoding can be achieved with much less computational effort. A resynchronization scheme is proposed for continuous sequential decoding. It is shown by analysis and simulation that continuous sequential decoding using this scheme has a high probability of resynchronizing successfully. This new resynchronization scheme solves the rate loss problem resulting from block decoding. The channel cutoff rate, demodulator quantization, and the tail's influence on performance are also discussed. Although this paper considers only the decoding of trellis codes, the algorithm can also be applied to the decoding of convolutional codes     

Complexity Reduction in SISO Decoding of Block Codes

SISO decoding for block codes can be carried out based on a trellis representation of the code. However, the complexity entailed by such decoding is most often prohibitive and thus prevents practical implementation. This paper examines a new decoding scheme based on the soft-output Viterbi algorithm (SOVA) applied to a sectionalized trellis for linear block codes. The computational complexities of the new SOVA decoder and of the conventional SOVA decoder, based on a bit-level trellis, are theoretically analyzed and derived for different linear block codes. These results are used to obtain optimum sectionalizations of a trellis for SOVA. For comparisons, the optimum sectionalizations for Maximum A Posteriori (MAP) and Maximum Logarithm MAP (Max-Log-MAP) algorithms, and their corresponding computational complexities are included. The results confirm that the new SOVA decoder is the most computationally efficient SISO decoder, in comparisons to MAP and Max-Log-MAP algorithms. The simulation results of the bit error rate (BER) performance, assuming binary phase -- shift keying (BPSK) and additive white Gaussian noise (AWGN) channel, demonstrate that the performance of the new decoding scheme is not degraded. The BER performance of iterative SOVA decoding of serially concatenated block codes shows no difference in the quality of the soft outputs of the new decoding scheme and of the conventional SOVA.     

A constraint-length based modified Viterbi algorithm with adaptiveeffort

A modified Viterbi (1971) algorithm for convolutional codes is described that provides for signal-to-noise ratio (SNR) adaptive computational effort. The algorithm has three levels of prioritized effort. Movement from one level to the next is controlled by parameters that can be selected according to desired output bit error rate performance. For 3-bit soft decision detected signals, a coding gain within 0.06 dB of Viterbi at a 3-dB SNR is achieved for the same constraint-length code with modest parameter values and computational effort. At values of SNR above 6 dB, the algorithm decodes with very low computational effort. Effort levels are controlled by spanning the decoding trellis in steps that are one constraint-length long     

A Multiple Stack Algorithm for Erasurefree Decoding of Convolutional Codes

A new algorithm for erasurefree sequential decoding of convolutional codes is introduced which achieves low error probabilities at substantially higher decoding speeds than the Viterbi decoding algorithm. The algorithmic properties of the Multiple Stack Algorithm (MSA) are investigated and it is demonstrated that the MSA reaches a decision with an exponentially rather than Pareto distributed computational effort. The MSA's error probability on the binary symmetric channel is studied as a function of its parameters and its performance and complexity compared to that of the Viterbi algorithm. The MSA is seen to achieve equal and lower error probabilities with a significantly lower average decoding effort. The new algorithm can thus be considered an attractive alternative to the Viterbi algorithm where low error probabilities and high decoding speeds are required simultaneously.     

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.     

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     

Probabilistic construction of large constraint length trellis codesfor sequential decoding

Probabilistic algorithms are given for constructing good large constraint length trellis codes for use with sequential decoding that can achieve the channel cutoff rate bound at a bit error rate (BER) of 10^-5-10^-6. The algorithms are motivated by the random coding principle that an arbitrary selection of code symbols will produce a good code with high probability. One algorithm begins by choosing a relatively small set of codes randomly. The error performance of each of these codes is evaluated using sequential decoding and the code with the best performance among the chosen set is retained. Another algorithm treats the code construction as a combinatorial optimization problem and uses simulated annealing to direct the code search. Trellis codes for 8 PSK and 16 QAM constellations with constraint lengths v up to 20 are obtained. Simulation results with sequential decoding show that these codes reach the channel cutoff rate bound at a BER of 10^-5-10^-6 and achieve 5.0-6.35 dB real coding gains over uncoded systems with the same spectral efficiency and up to 2.0 dB real coding gains over 64 state trellis codes using Viterbi decoding     

A new decoding algorithm for geometrically uniform trellis codes

A new decoding algorithm for geometrically uniform trellis codes is presented. The group structure of the codes is exploited in order to improve the decoding process. Analytical bounds to the algorithm performance and to its computational complexity are derived. The algorithm complexity does not depend on the number of states of the trellis describing the code. Extensive simulations yield results on the algorithm performance and complexity, and permit a comparison with the Viterbi algorithm and the sequential Fano algorithm     

A new sequential decoding algorithm based on branch metric

A new sequential decoding algorithm with an adjustable threshold and a new method of moving through the decoding tree is proposed. Instead of the path metric of the conventional sequential decoding algorithms, the proposed algorithm uses a branch metric based on maximum-likelihood criterion. Two new parameters, the jumping-back distance and going-back distance, are also introduced. The performance of the algorithm for long constraint length convolutional codes is compared to those of the other sequential decoding algorithms and the Viterbi algorithm. The results show that the proposed algorithm is a good candidate for decoding of convolutional codes due to its fast decoding capability and good bit error rate (BER) performance. This work was supported in part by the Research Foundation at Karadeniz Technical University under Grant 2004.112.004.01 and 2005.112.009.2.     

An analysis of sequential decoding for specific time-invariant convolutional codes

A new analysis of the computational effort and the error probability of sequential decoding is presented, which is based entirely on the distance properties of a particular convolutional code and employs no random-coding arguments. An upper bound on the computational distributionP(C_{t}>N_{t})for a specific time-invariant code is derived, which decreases exponentially with the column distance of the code. It is proved that rapid column-distance growth minimizes the decoding effort and therefore also the probability of decoding failure or erasure. In an analogous way, the undetected error probability of sequential decoding with a particular fixed code is proved to decrease exponentially with the free distance and to increase linearly with the number of minimum free-weight codewords. This analysis proves that code construction for sequential decoding should maximize column-distance growth and free distance in order to guarantee fast decoding, a minimum erasure probability, and a low undetected error probability.     

Computation of Posterior Marginals on Aggregated State Models for Soft Source Decoding

Optimum soft decoding of sources compressed with variable length codes and quasi-arithmetic codes, transmitted over noisy channels, can be performed on a bit/symbol trellis. However, the number of states of the trellis is a quadratic function of the sequence length leading to a decoding complexity which is not tractable for practical applications. The decoding complexity can be significantly reduced by using an aggregated state model, while still achieving close to optimum performance in terms of bit error rate and frame error rate. However, symbol a posteriori probabilities can not be directly derived on these models and the symbol error rate (SER) may not be minimized. This paper describes a two-step decoding algorithm that achieves close to optimal decoding performance in terms of SER on aggregated state models. A performance and complexity analysis of the proposed algorithm is given.     

Adaptive Viterbi decoding of convolutional codes over memorylesschannels

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     

Distance and Computation in Sequential Decoding

The relationship between the column distance function and the computational effort of sequential decoding is studied and the results of computer simulations are reported. A table ofR = 1/2codes having good free distance and optimum average column distance function (CDF) is presented.     

Two decoding algorithms for tailbiting codes

The paper presents two efficient Viterbi decoding-based suboptimal algorithms for tailbiting codes. The first algorithm, the wrap-around Viterbi algorithm (WAVA), falls into the circular decoding category. It processes the tailbiting trellis iteratively, explores the initial state of the transmitted sequence through continuous Viterbi decoding, and improves the decoding decision with iterations. A sufficient condition for the decision to be optimal is derived. For long tailbiting codes, the WAVA gives essentially optimal performance with about one round of Viterbi trial. For short- and medium-length tailbiting codes, simulations show that the WAVA achieves closer-to-optimum performance with fewer decoding stages compared with the other suboptimal circular decoding algorithms. The second algorithm, the bidirectional Viterbi algorithm (BVA), employs two wrap-around Viterbi decoders to process the tailbiting trellis from both ends in opposite directions. The surviving paths from the two decoders are combined to form composite paths once the decoders meet in the middle of the trellis. The composite paths at each stage thereafter serve as candidates for decision update. The bidirectional process improves the error performance and shortens the decoding latency of unidirectional decoding with additional storage and computation requirements. Simulation results show that both proposed algorithms effectively achieve practically optimum performance for tailbiting codes of any length.     

New ternary line codes based on trellis structure

The trellis coding technique is applied to line-coded baseband digital transmission systems. For R=n/n+1(n=1,2,3) coding rates, a new codeword assignment model is proposed to accomplish basic requirements for line coding in which each length n binary data sequence is encoded into a length n+1 ternary (+,0,-) line codeword chosen among the code alphabet with 2ⁿ⁺² elements. Assuming Viterbi decoding, the system error performance is improved by increasing the free Euclidean distance between coded sequences. A new algorithm is given for the calculation of the free distance between line-coded sequences so obtained. For R=1/2 and R=3/4 rates, the analytical error performance upper bounds are derived. The power spectral densities of the new line codes are also calculated and compared with those of known line codes     

A comparison of reduced complexity decoding algorithms for trelliscodes

Comparisons are made of a genie-aided sequential algorithm due to D. Haccoun and M.J. Ferguson (1975), the Viterbi algorithm, the M -algorithm, and the Fano algorithm for rate-1/2 and rate-2/3 trellis modulation codes on rectangular signal sets. The effects of signal-to-noise ratio and decoding-delay constraints on the choice of decoding algorithms for framed data are examined by computer simulation. Additionally, the genie-aided algorithm is used as a tool in estimating the asymptotic behavior of the M-algorithm. In general, the results conform closely to experience with convolutional codes due to the similar distance structure of the codes. The Fano algorithm produces good error performance with a low average number of computations when long decoding delay is permissible. The M-algorithm provides a savings in computation compared to the Viterbi algorithm if a small decoding delay is required     

Scarce-state-transition error-trellis decoding of block codes withBPSK signals

A novel decoding technique for linear block codes with coherent BPSK signals is proposed. The new system has the same error performance as and similar complexity to the conventional trellis decoding of block codes. Like the scarce-state-transition Viterbi decoding of convolutional codes, the proposed system is also well suited for CMOS VLSI implementation and has a lower power consumption     

A bidirectional multiple stack algorithm

Bidirectional sequential decoding (BSD) substantially reduces the computational variability of conventional sequential decoding without compromising the error performance. However, BSD does not completely eliminate the erasure problem. We propose an erasure-free decoding algorithm which combines the idea of BSD in conjunction with that of the multiple stack algorithm (MSA). It is found that the new bidirectional multiple stack algorithm (BMSA) offers substantial advantages over the MSA in terms of computational effort, memory requirements, and error performance. The BMSA appears as an attractive alternative to the Viterbi algorithm (VA) where low error probabilities and high decoding speeds are required     

Low-complexity iterative joint source-channel decoding for variable-length encoded Markov sources

In this paper, we present a novel packetized bit-level decoding algorithm for variable-length encoded Markov sources, which calculates reliability information for the decoded bits in the form of a posteriori probabilities (APPs). An interesting feature of the proposed approach is that symbol-based source statistics in the form of the transition probabilities of the Markov source are exploited as a priori information on a bit-level trellis. This method is especially well-suited for long input blocks, since in contrast to other symbol-based APP decoding approaches, the number of trellis states does not depend on the packet length. When additionally the variable-length encoded source data is protected by channel codes, an iterative source-channel decoding scheme can be obtained in the same way as for serially concatenated codes. Furthermore, based on an analysis of the iterative decoder via extrinsic information transfer charts, it can be shown that by using reversible variable-length codes with a free distance of two, in combination with rate-1 channel codes and residual source redundancy, a reliable transmission is possible even for highly corrupted channels. This justifies a new source-channel encoding technique where explicit redundancy for error protection is only added in the source encoder.     

Channel coding with multilevel/phase signals

A coding technique is described which improves error performance of synchronous data links without sacrificing data rate or requiring more bandwidth. This is achieved by channel coding with expanded sets of multilevel/phase signals in a manner which increases free Euclidean distance. Soft maximum--likelihood (ML) decoding using the Viterbi algorithm is assumed. Following a discussion of channel capacity, simple hand-designed trellis codes are presented for 8 phase-shift keying (PSK) and 16 quadrature amplitude-shift keying (QASK) modulation. These simple codes achieve coding gains in the order of 3-4 dB. It is then shown that the codes can be interpreted as binary convolutional codes with a mapping of coded bits into channel signals, which we call "mapping by set partitioning." Based on a new distance measure between binary code sequences which efficiently lower-bounds the Euclidean distance between the corresponding channel signal sequences, a search procedure for more powerful codes is developed. Codes with coding gains up to 6 dB are obtained for a variety of multilevel/phase modulation schemes. Simulation results are presented and an example of carrier-phase tracking is discussed.