An implementation of the generalized stack algorithm using a partially sorting algorithm

This letter considers the average complexity of maximum-likelihood (ML) decoding of convolutional codes. ML decoding can be modeled as finding the most probable path taken through a Markov graph. Integrated with the Viterbi algorithm (VA), complexity reduction methods often use the sum log likelihood (SLL) of a Markov path as a bound to disprove the optimality of other Markov path sets and to consequently avoid exhaustive path search. In this letter, it is shown that SLL-based optimality tests are inefficient if one fixes the coding memory and takes the codeword length to infinity. Alternatively, optimality of a source symbol at a given time index can be testified using bounds derived from log likelihoods of the neighboring symbols. It is demonstrated that such neighboring log likelihood (NLL)-based optimality tests, whose efficiency does not depend on the codeword length, can bring significant complexity reduction. The results are generalized to ML sequence detection in a class of discrete-time hidden Markov systems.

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.

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.

This paper presents a parallel computing approach that is employed to reconstruct original information bits from a non-recursive convolutional codeword in noise, with the goal of reducing the decoding latency without compromising the performance. This goal is achieved by means of cutting a received codeword into a number of sub-codewords (SCWs) and feeding them into a two-stage decoder. At the first stage, SCWs are decoded in parallel using the Viterbi algorithm or equivalently the brute force algorithm. Major challenge arises when determining the initial state of the trellis diagram for each SCW, which is uncertain except for the first one; and such results in multiple decoding outcomes for every SCW. To eliminate or more precisely exploit the uncertainty, an Euclidean-distance minimization algorithm is employed to merge neighboring SCWs; and this is called the merging stage, which can also run in parallel. Our work reveals that the proposed two-stage decoder is optimal and has its latency growing logarithmically, instead of linearly as for the Viterbi algorithm, with respect to the codeword length. Moreover, it is shown that the decoding latency can be further reduced by employing artificial neural networks for the SCW decoding. Computer simulations are conducted for two typical convolutional codes, and the results confirm our theoretical analysis.

In this paper, use of recurrent neural network equalizer (RNNE) in place of linear equalizer (LE) to combat both linear and nonlinear distortions corrupting the Magneto-optical (MO) readback signal is discussed. It is shown that RNNE can outperform LE without introducing significant complexity. RNNE is used to equalize the MO recording readback signal corrupted by transition jitter, intersymbol interference (ISI) and additive white Gaussian Noise (AWGN) at a density of 50 kbpi. The MO signal is equalized to a partial response (PR) (1 + D) using either the RNNE or the LE and the equalizer's mean- squared-error (MSE) performance is compared. Then, the equalized signal is passed through a detector and it is shown that a signal equalized to a PR (1 + D) shape can be detected using either a bit-by-bit type of detector (BD) or a sequence detector implemented via Viterbi Algorithm (VA). The bit-error-rate (BER) performance of BD is compared to that of the Viterbi detector and it is shown that PR equalization of MO readback signals using RNNE improves MSE performance over linear equalizer, allowing use of BD rather than LE + Viterbi Algorithm with comparable BERs.© (1998) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

An efficient parallel processing mode is proposed for the implementation of fast forward-only iterative belief propagation decoding of orthogonal convolutional codes based on one-step decoding. In the proposed parallel processing mode, the iterative decoding of orthogonal convolutional codes makes use of the complete set of orthogonal parity-check equations on each information symbol, thus avoiding the error performance loss. The proposed parallel processing allows the speed of iterative decoding to approach that of one-step decoding. Therefore, an advantageous tradeoff can be available between data rate and latency.

The research on quantum convolutional codes is aimed at protecting a flow of information over long distance communications. The polynomial representation of a quantum state is defined. Based on the Calderbank-Shor-Steane(CSS)-type construction of quantum codes, a new method for encoding and decoding of CSS-type quantum convolutional codes is presented and corresponding networks are described. The basis state of the code is transformed into the product of an information polynomial by the generator polynomial. Then networks can be realized by operations of polynomial multiplication. Finally, inspired by classical convolutional decoding idea, a quantum Viterbi algorithm with linear complexity is put forward.

Sequence detection is studied for communication channels with intersymbol interference and non-Gaussian noise using a novel adaptive receiver structure. The receiver adapts itself to the noise environment using an algorithm which employs a Gaussian mixture distribution model and the expectation maximization algorithm. Two alternate procedures are studied for sequence detection. These are a procedure based on the Viterbi algorithm and a symbol-by-symbol detection procedure. The Viterbi algorithm minimizes the probability the sequence is in error and the symbol-by-symbol detector minimizes symbol error rate, which are different.

Convolutional codes are used extensively in numerous applications in order to achieve reliable data transfer. In this paper, the parameters of Convolutional codes are introduced and used to investigate its applications, characteristics and performance when employed as inner and outer code in decoding method. To evaluate the parameters, four schemes of Convolutional codes: non-systematic-recursive, systematic-recursive, non-systematic-non-recursive and systematic-non-recursive; are applied. In addition, the implementation of BCJR (Bahl, Cocke, Jelinek and Raviv) algorithm is discussed by applying APP-SISO (a posteriori probabilities — Soft Input Soft Output) decoding for Convolutional codes. The proposed scheme employing inner and outer decoding of Convolutional codes are analysed and compared by using EXIT characteristic of Convolutional decoders. The channel capacity measurements for different rates inner Convolutional codes are described by using BPSK modulation over AWGN channel. BER performances of outer decoding of Convolutional codes are observed for different schemes which are described the process of area beneath in the inverted EXIT function. This is essential to obtain better performance of BER plot indeed. The result shows that inner decoding of Convolutional codes is the best performance for the EXIT function, particularly for non-systematic-recursive. However, outer decoding of this code has not good performance due to having quite larger area beneath than inner code.

A reduced complexity algebraic type algorithm is described for decoding of convolutional codes over GF(q), q>2. It is founded on the same principles as algebraic-sequentia1 decoding. It is proved that for large q, the algorithm has better complexity-reliability tradeoff than the conventional Viterbi algorithm.

We compared the performance of trellis-coded partial response (TCPR) and fixed delay tree search with decision feedback (FDTS/DF) in a recording system with both electronic and medium noise characterized by Thurlings' model. To distinguish influences of the noise correlation on the Viterbi algorithm and the tree search algorithm, the bit error rate with and without noise correlation was calculated by using a noise correlation coefficient after equalization. We obtained the signal-to-noise ratio required for a bit-error-rate of 10/sup -9/, versus a normalized recording density for recording system with 50% electronic noise and 50% medium noise. The results indicate that TCPR gives the best performance for low linear recording density.

A differentially coherent receiver which uses the Viterbi algorithm is presented. This receiver compensates for intersymbol interference (ISI) and avoids noise enhancement. A phase estimate which takes into account the presence of inherent ISI in the signal is derived. This phase estimate is then used in the decision metric of a coherent receiver. The resulting decision rule can be implemented using the Viterbi algorithm. The complexity of the resulting receiver increases with the duration of observation used by the implicit phases estimator. A simple four-state Viterbi receiver with good performance is obtained when the phase estimate is based on one observable. Simulation results are given. >

Unified low-complexity decision feedback equalizer with adjustable double radius constraint

We introduce a very efficient (linear-time) method for finding the best metrics from among a number of contenders during breadth-first decoding of block or convolutional codes. This method can provide a sorted set of survivor metrics during M algorithm decoding of binary linear block codes using only /spl Lambda/1 comparisons in the worst case. This is a significant improvement over comparison-based sorting, which requires O(M log/sub 2/ M) comparisons. A similar improvement is attained over Hadian and Sohel's comparison-based selection method (where the best metrics are found without regard to order). The method is also readily applicable to the decoding of convolutional codes. >

A novel high-precision decomposition algorithm for EMG (electromyography) signals based on the Viterbi algorithm is described. Under certain assumptions this algorithm yields the least squares solution in the time domain. It is shown that the EMG signal source can be modeled as a (M+1)-ary signaling system with intersymbol interference. Direct application of the Viterbi algorithm would lead to a decoder with approximately=10/sup 20/ states, which cannot be implemented. It is shown that according to the special nature of the EMG signal essential simplifications can be done. The resulting algorithm requires only on the order of 10/sup 4/ states. The test of the implemented algorithm shows that, in the mean, over 90% correctly detected innervation time points and only about 5% additional time points can be expected.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>