Iterative maximum-likelihood trellis decoding for block codes

An iterative trellis search technique is described for the maximum-likelihood (ML) soft decision decoding of block codes. The proposed technique derives its motivation from the fact that a given block code may be a subcode for a parent code whose associated trellis has substantially fewer edges. Through the use of list-Viterbi (1967) decoding and an iterative algorithm, the proposed technique allows for the use of a trellis for the parent code in the ML decoding of the desired subcode. Complexity and performance analyses, as well as details of potential implementations, indicate a substantial reduction in decoding complexity for linear block codes of practical length while achieving ML or near-ML soft decision performance     

Efficient maximum likelihood decoding of linear block codes using a trellis

It is shown that soft decision maximum likelihood decoding of any(n,k)linear block code overGF(q)can be accomplished using the Viterbi algorithm applied to a trellis with no more thanq^{(n-k)}states. For cyclic codes, the trellis is periodic. When this technique is applied to the decoding of product codes, the number of states in the trellis can be much fewer thanq^{n-k}. For a binary(n,n - 1)single parity check code, the Viterbi algorithm is equivalent to the Wagner decoding algorithm.     

Sequence MAP decoding of trellis codes for Gaussian and Rayleighchannels

This paper considers the use of sequence maximum a posteriori (MAP) decoding of trellis codes. A MAP receiver can exploit any “residual redundancy” that may exist in the channel encoded signal in the form of memory and/or a nonuniform distribution, thereby providing enhanced performance over very noisy channels, relative to maximum likelihood (ML) decoding. The paper begins with a first-order two-state Markov model for the channel encoder input. A variety of different systems with different source parameters, different modulation schemes, and different encoder complexities are simulated. Sequence MAP decoding is shown to substantially improve performance under very noisy channel conditions for systems with low-to-moderate redundancy, with relative gain increasing as the rate increases. As a result, coding schemes with multidimensional constellations are shown to have higher MAP gains than comparable schemes with two-dimensional (2-D) constellations. The second part of the paper considers trellis encoding of the code-excited linear predictive (CELP) speech coder's line spectral parameters (LSPs) with four-dimensional (4-D) QPSK modulation. Two source LSP models are used. One assumes only intraframe correlation of LSPs while the second one models both intraframe and interframe correlation. MAP decoding gains (over ML decoding) as much as 4 dB are achieved. Also, a comparison between the conventionally designed codes and an I-Q QPSK scheme shows that the I-Q scheme achieves better performance even though the first (sampler) LSP model is used     

Non‐binary SOVA algorithms for decoding of block codes on a sectionalized trellis

In this paper, we explore computationally efficient implementations of the soft output viterbi algorithm (SOVA) applied to Soft‐Input Soft‐Output (SISO) decoding of linear block codes. In order to simplify the trellis‐based decoding of binary block codes with SOVA, we use the technique of sectionalization of the trellis, which has been successfully applied to the simplification of the MAP and Max‐Log‐MAP algorithms. Due to the branch complexity of the sectionalized trellis, we define a generalization of a non‐binary version of SOVA. However, the computational complexity of directly applying this approach remains too high for efficient implementation; we thus introduce the concept of non‐binary SOVA (NSOVA) with propagation of bit‐level reliabilities (BLR). This new algorithm is analyzed from a computational complexity viewpoint. Both serial and parallel implementations are explored. Finally, optimal sectionalizations are derived for selected codes; since the normal SOVA decoding is a particular case of NSOVA with BLR, we show that our approach is more efficient than a bit‐level trellis by showing that, for all the codes tested, the optimal trellis is a sectionalized one. Copyright © 2007 John Wiley & Sons, Ltd.     

On the BCJR trellis for linear block codes

In this semi-tutorial paper, we will investigate the computational complexity of an abstract version of the Viterbi algorithm on a trellis, and show that if the trellis has e edges, the complexity of the Viterbi algorithm is Θ(e). This result suggests that the “best” trellis representation for a given linear block code is the one with the fewest edges. We will then show that, among all trellises that represent a given code, the original trellis introduced by Bahl, Cocke, Jelinek, and Raviv in 1974, and later rediscovered by Wolf (1978), Massey (1978), and Forney (1988), uniquely minimizes the edge count, as well as several other figures of merit. Following Forney and Kschischang and Sorokine (1995), we will also discuss “trellis-oriented” or “minimal-span” generator matrices, which facilitate the calculation of the size of the BCJR trellis, as well as the actual construction of it     

Soft-decision decoding of binary linear block codes using reducedbreadth-first search algorithms

The article discusses soft-decision decoding of binary linear block codes using the t-algorithm and its variants. New variants of the basic algorithm are presented that reduce the decoding complexity using a threshold adaptive to the signal-to-noise ratio and address the variable decoding complexity by either limiting the memory or using a generalized M-algorithm with a nonconstant state profile     

Design of ring convolutional trellis codes for MAP decoding of MPEG-4 images

We propose a trellis-coded modulation system using continuous-phase frequency-shift keying (CPFSK) and ring convolutional codes for transmitting the bits generated by an embedded zerotree wavelet encoder. Improved performance is achieved by using maximum a posteriori decoding of the zerotree symbols, and ring convolutional trellis codes are determined for this decoding method. The CPFSK transmitter is decomposed into a memoryless modulator and a continuous phase encoder over the ring of integers modulo 4; the latter is combined with a polynomial convolutional encoder over the same ring. In the code design process, a search is made of the combined trellis, where the branch metrics are modified to include the source transition matrix. Simulation results of image transmission are provided using the optimized system, including mismatched channel cases.     

Selective trellis pruning for block codes: theory and application to iterative decoding

In this paper we present a new method to simplify trellis‐based decoding of linear block codes. Our method is based on removing (pruning) some edges and states from the trellis representation of the code; the trellis is pruned through some hard‐decisions that are made on received bits whose likelihood exceeds a predefined threshold. We show in this paper how this simplification can be accounted for by a new generator matrix (and sometimes a coset leader) that totally parametrize the new trellis, or, equivalently, the set of allowed codewords after simplification. Extensive simulations show that significant computational savings can be achieved, at a very small loss in coding performance, as long as the operating point and threshold are carefully chosen. Moreover, the application of this technique to iterative decoding of product codes is outlined, and our results show that the simplifications do not hinder the convergence, again, as long as proper parameters are chosen. Copyright © 2007 John Wiley & Sons, Ltd.     

Efficient heuristic search algorithms for soft-decision decoding oflinear block codes

Efficient new algorithms are presented for maximum-likelihood and suboptimal soft-decision decoding algorithms for linear block codes. The first algorithm, MA*, improves the efficiency of the A* decoding algorithm, conducting the heuristic search through a code tree while exploiting code-specific properties. The second algorithm, H*, reduces search space by successively estimating the cost of the minimum-cost codeword with a fixed value at each of the most reliable and linearly independent components of the received message. The third algorithm, directed search, finds the codeword closest to the received vector by exploring a continuous search space. The strengths of these three algorithms are combined in a hybrid algorithm, applied to the (128,64), the (256,131), and the (256,139) binary-extended Bose-Chaudhuri-Hocquenghem (BCH) codes. Simulation results show that this hybrid algorithm can efficiently decode the (128,64) code for any signal-to-noise ratio, with near-optimal performance. Previously, no practical decoder could have decoded this code with such a performance for all ranges of signal-to-noise ratio     

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     

Differential trellis decoding of convolutional codes

This paper investigates the principle of metric differences for trellis decoding of convolutional codes. Based on this differential method, a new algorithm, referred to as differential trellis decoding (DTD), is proposed. DTD offers an alternative to the conventional “add-compare-select” (ACS) method for implementing the Viterbi algorithm     

Critical point for maximum likelihood decoding of linear block codes

In this letter, the SNR value at which the error performance curve of a soft decision maximum likelihood decoder reaches the slope corresponding to the code minimum distance is determined for a random code. Based on this value, referred to as the critical point, new insight about soft bounded distance decoding of random-like codes (and particularly Reed-Solomon codes) is provided.     

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 decoding algorithms of binary linear block codes inRayleigh fading channels with diversity reception

The word error probability of binary linear block codes is evaluated in Rayleigh fading channels with diversity reception for three decoding algorithms: error correction (EC), error/erasure correction (EEC), and maximum likelihood (ML) soft decoding algorithms. The performance advantage of EEC over EC in the required average SNR decreases as the number of diversity channels increases. The performance advantage of EEC over EC does not depend on the specific value of word error probability although the advantage of ML soft decoding over EC increases for lower word error probability     

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     

Soft decoding of linear block codes with differential phasedetection

Soft decoding of linear block codes is considered over a Gaussian channel. A noncoherent soft decision differential phase detector that operates on many preselected regions is considered for detection. Numerical results demonstrate the improvement of soft decoding over hard decoding and the effect of the number of regions in the phase detector     

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 adaptive two-stage maximum-likelihood decoding algorithm for linear block codes

In this paper, we propose a new two-stage (TS) structure for computationally efficient maximum-likelihood decoding (MLD) of linear block codes. With this structure, near optimal MLD performance can be achieved at low complexity through TS processing. The first stage of processing estimates a minimum sufficient set (MSS) of candidate codewords that contains the optimal codeword, while the second stage performs optimal or suboptimal decoding search within the estimated MSS of small size. Based on the new structure, we propose a decoding algorithm that systematically trades off between the decoding complexity and the bounded block error rate performance. A low-complexity complementary decoding algorithm is developed to estimate the MSS, followed by an ordered algebraic decoding (OAD) algorithm to achieve flexible system design. Since the size of the MSS changes with the signal-to-noise ratio, the overall decoding complexity adaptively scales with the quality of the communication link. Theoretical analysis is provided to evaluate the potential complexity reduction enabled by the proposed decoding structure.     

Reduced-search soft-decision trellis coding of linear block codes (Corresp.)

Algorithms for the soft-decision decoding of linear block codes are presented. These algorithms perform a reduced complexity search through a trellis derived from the parity check matrix of an(n, k)linear block code. The computational complexity of the algorithms is considerably reduced from that of a full maximum-likelihood algorithm. We demonstrate the trade-off between complexity and efficiency of the algorithms through computer simulation.