Decoding algebraic-geometric codes up to the designed minimumdistance

A simple decoding procedure for algebraic-geometric codes C _Ω(D,G) is presented. This decoding procedure is a generalization of Peterson's decoding procedure for the BCH codes. It can be used to correct any [(d*-1)/2] or fewer errors with complexity O(n³), where d * is the designed minimum distance of the algebraic-geometric code and n is the codelength     

The iterative turbo decoding algorithm has fixed points

It is shown that the iterative decoding algorithm proposed to decode “turbo codes” has fixed points regardless of the specific constituent codes and regardless of the noise variance of the additive white Gaussian noise (AWGN) channel     

Modeling fading channel-estimation errors in pilot-symbol-assisted systems, with application to turbo codes

In this paper, we address the issue of imperfect channel estimation in coded systems on fading channels. Since performance of channel codes is influenced in different ways by different components of channel-estimation errors, we develop a simplified model which separates the estimation errors of a Wiener-filtered received signal into the amplitude error and the phase error. Based on the model, we derive tight bounds on component error variances. Moreover, we prove that the classical Wiener filter results in a biased estimate of the channel amplitude. We also show that the probability of having a phase-estimation error large enough to cause decision errors in the receiver is significant. Using our model, we derive an approximate upper limit on the optimum pilot-symbol spacing and approximate lower limit on bit-error rate performance of coded systems with a given pilot-symbol separation. The proposed model and derivations are confirmed by extensive simulations.     

Joint synchronization and SNR estimation for turbo codes in AWGN channels

Turbo codes are sensitive to both (timing) synchronization errors and signal-to-noise ratio (SNR) mismatch. Since turbo codes are intended to work in environments with very low SNR, conventional synchronization methods often fail. This paper investigates blind symbol-timing synchronization and SNR estimation based on oversampled data frames. The technique is particularly suitable for low-rate turbo codes operating in additive white Gaussian noise at low SNR and modest data-transfer rates, as in deep space, satellite, fixed wireless, or wireline communications. In accordance with the turbo principle, intermediate decoding results are fed back to the estimator, thereby facilitating decision-directed estimation. The analytical and simulated results show that with three or more samples per symbol and raised cosine-rolloff pulse shaping, performance approaches that of systems with perfect timing and SNR knowledge at the receiver.     

Method to analyse convergence of turbo codes

The use of cross-entropy to analyse the convergence behaviour of a turbo decoder is proposed. Based on the new method, E/sub b//N/sub 0/ thresholds are predicted and compared with those predicted by existing techniques. Simulation results show that the new technique is effective and advantageous in practical applications.     

Analysis of turbo codes with asymmetric modulation

If different energies are assigned to two outputs of a turbo encoder, the information bit and parity bit, then the performance will be changed according to the ratio of the information bit energy to the parity bit energy. The optimum point of the ratio may not be 1. As the rate of the turbo code is changed, the optimum point will also be changed. Rate 1/2, 1/3, 1/4 and 1/5 turbo codes with asymmetric modulation are considered     

Joint estimation and compression of correlated nonbinary sources using punctured turbo codes

We propose a system to perform data compression of correlated nonbinary sources when the correlation between sources is not known, either at the encoder or the decoder. The sequences of nonbinary symbols are transformed into sequences of bits, and then source coded using punctured turbo codes, with the puncturing adjusted to achieve the desired compression rate. Each source is compressed without knowledge about the other source, and the correlation model is not assumed to be known at the encoder. The source decoder uses iterative schemes over the compressed binary sequences, and recovers the nonbinary symbol sequences from both sources. The correlation model between sources does not need to be known at the decoder, since it can be estimated jointly with the iterative decoding process. Compared with the case in which the correlation is known at the decoder, no significant performance loss is observed. The performance of the proposed scheme is close to the Slepian-Wolf theoretical limit.     

Test-pattern-reduced decoding for turbo product codes with multi-error-correcting eBCH codes

We present a method to reduce the number of test patterns (TPs) decoded in the Chase-II algorithm for turbo product codes (TPCs) constructed with multi-error-correcting extended Bose-Chaudhuri-Hocquengem (eBCH) codes. We classify TPs into different conditions based on the relationship between syndromes and the number of errors so that TPs with the same codeword are not decoded except the one with the least number of errors. For eBCH with code length of 64, simulation results show that over 50% of TPs need not to be decoded without any performance degradation.     

Space-time turbo codes with full antenna diversity

In attempting to find a spectrally and power efficient channel code which is able to exploit maximum diversity from a wireless channel whenever available, we investigate the possibility of constructing a full antenna diversity space-time turbo code. As a result, both three-antenna and two-antenna (punctured) constructions are shown to be possible and very easy to find. To check the decodability and performance of the proposed codes, we derive non-binary soft-decoding algorithms. The performance of these codes are then simulated and compared with two existing space-time convolutional codes (one has minimum worst-case symbol-error probability; the other has maximal minimum free distance) having similar decoding complexity. As the simulation results show, the proposed space-time turbo codes give similar or slightly better performance than the convolutional codes under extremely slow fading. When fading is fast, the better distance spectra of the turbo codes help seize the temporal diversity. Thus, the performance advantage of the turbo codes becomes evident. In particular, 10^-5 bit-error rate and 10^-3 frame-error rate can be achieved at less than 6-dB E_b/N₀ with 1 b/s/Hz and binary phase-shift keying modulation. The practical issue of obtaining the critical channel state information (CSI) is also considered by applying an iteratively filtered pilot symbol-assisted modulation technique. The penalty when the CSI is not given a priori is about 2-3 dB     

Distance-based-decoding of block turbo codes

List-based algorithms for. decoding block turbo Codes (BTC) have gained popularity due to their low computational complexity. The normal way to calculate the soft outputs involves searching for a decision code word D and a competing codeword B. In addition, a scaling factor /spl alpha/ and an estimated reliability value /spl beta/ are used. In this letter, we present a new approach that does not require /spl alpha/ and /spl beta/. Soft outputs are generated based on the Euclidean distance property of decision code words. By using the new algorithm, we achieve better error performance with even less complexity-for certain BTCs.     

List decoding of turbo codes

List decoding of turbo codes is analyzed under the assumption of a maximum-likelihood (ML) list decoder. It is shown that large asymptotic gains can be achieved on both the additive white Gaussian noise (AWGN) and fully interleaved flat Rayleigh-fading channels. It is also shown that the relative asymptotic gains for turbo codes are larger than those for convolutional codes. Finally, a practical list decoding algorithm based on the list output Viterbi algorithm (LOVA) is proposed as an approximation to the ML list decoder. Simulation results show that the proposed algorithm provides significant gains corroborating the analytical results. The asymptotic gain manifests itself as a reduction in the bit-error rate (BER) and frame-error rate (FER) floor of turbo codes     

SNR estimation in Nakagami-m fading with diversity combining and its application to turbo decoding

We propose an online signal-to-noise ratio (SNR) estimation scheme for Nakagami-m (1960) fading channels with L branch equal gain combining (EGC) diversity. We derive the SNR estimate based on the statistical ratio of certain observables over a block of data, and use the SNR estimates in the iterative decoding of turbo codes on Nakagami-m fading channels with L branch EGC diversity. We evaluate the turbo decoder performance using the SNR estimate under various fading and diversity scenarios (m = 0.5, 1, 5 and L = 1, 2, 3) and compare it with the performance using perfect knowledge of the SNR and the fade amplitudes.     

Analysis of three-dimensional turbo codes

Our paper presents a detailed study of the three-dimensional turbo code (3D TC). This code which combines both parallel and serial concatenation is derived from the classical TC by concatenating a rate-1 post-encoder at its output. The 3D TC provides very low error rates for a wide range of block lengths and coding rates, at the expense of an increase in complexity and a loss in convergence. This paper deals with the performance improvement of the 3D TC. First, we optimize the distance spectrum of the 3D TC by means of the adoption of a non regular post-encoding pattern. This allows us to increase the minimum hamming distance (MHD) and thereby to improve the performance at very low error rates. Then, we propose a time varying construction of the post-encoded parity in order to reduce the observable loss of convergence at high error rates. Performance comparisons are made between the 3GPP2 standardized TC and the corresponding 3D code. The different improvement stages are illustrated with simulation results, asymptotical bounds, and EXIT charts.     

Performance of hybrid turbo codes

The authors present the latest results on turbo codes which are built from a serial concatenation between a block code (BCH) and a recursive systematic convolutional code. The performance on a Gaussian channel with QPSK modulation is evaluated     

Permutation decoding of abelian codes

A permutation decoding procedure for abelian codes is introduced by using the Groebner bases theory. This method is valid for decoding all the binary abelian codes. Some examples are given to show how powerful this method can be     

Average-entropy variation in iterative decoding of turbo codes and its application

A new stopping criterion for turbo codes is proposed. Based on the entropy concept, a metric called average-entropy to measure the average uncertainty of the estimated bits of each iteration is derived. This metric has a close relation to the bit error rate (BER). The average-entropy decreases as BER reduces and vice versa. The proposed stopping criterion stops the iterative algorithm when there is not a, or little, reduction of the average-entropy. Compared with other well-known criteria, this new criterion reduces the average number of iterations while maintaining error correction performance.     

Analysis of recursive convolutional codes and turbo codes as sources with memory

The paper proposes a general method to analyze discrete sources with memory. Besides the classical entropy, we define new information measures for discrete sources with memory, similar to the information quantities specific to discrete channels. On the base of this method, we show for the first time that, as result of convolutional and turbo encoding, sources with memory are obtained. We apply this information analysis method for the general case of a recursive convolutional encoder of rate R_CC = 1/n₀ and memory of order m, and for a turbo encoder of rate R_TC = 1/3, with two systematic recursive convolutional component encoders. Each component encoder has memory of order m, and is built based on the same primitive feedback polynomial. For the convolutional and turbo codes, the information quantities H(Y/S), H(S,Y), H(S/Y), H(Y), H(S) and I(S,Y) have been computed, where S and Y denote the set of states and the set of messages of the encoder, respectively. The analysis considered two cases: n₀ ≤ m + 1 and n₀ > m + 1. When n₀ = m + 1, the mutual information I(S,Y) is maximum and equal to m, as is the entropy of the set of states. For turbo codes, the quantity I(S,Y) also depends on the input bit and on its probability.     

Near-optimum decoding of product codes: block turbo codes

This paper describes an iterative decoding algorithm for any product code built using linear block codes. It is based on soft-input/soft-output decoders for decoding the component codes so that near-optimum performance is obtained at each iteration. This soft-input/soft-output decoder is a Chase decoder which delivers soft outputs instead of binary decisions. The soft output of the decoder is an estimation of the log-likelihood ratio (LLR) of the binary decisions given by the Chase decoder. The theoretical justifications of this algorithm are developed and the method used for computing the soft output is fully described. The iterative decoding of product codes is also known as the block turbo code (BTC) because the concept is quite similar to turbo codes based on iterative decoding of concatenated recursive convolutional codes. The performance of different Bose-Chaudhuri-Hocquenghem (BCH)-BTCs are given for the Gaussian and the Rayleigh channel. Performance on the Gaussian channel indicates that data transmission at 0.8 dB of Shannon's limit or more than 98% (R/C>0.98) of channel capacity can be achieved with high-code-rate BTC using only four iterations. For the Rayleigh channel, the slope of the bit-error rate (BER) curve is as steep as for the Gaussian channel without using channel state information     

Using partial unit memory codes to improve distance properties of woven turbo codes

Partial unit memory (PUM) codes are reputed for their excellent distance properties and lower decoding complexity compared to equivalent convolutional codes. Woven turbo codes (WTCs) are constructed, which outperform turbo codes, using component PUM codes. Simulation results confirm that WTCs based on PUM codes outperform those based on equivalent convolutional codes, as reflected by their respective minimum distances, which is also calculated.