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.     

Enhanced space-time block coded systems by concatenating turbo product codes

The BER performance of a turbo product code (TPC) based space-time block coding (STBC) wireless system has been investigated. With the proposed system, both the good error correcting capability of TPC and the concurrent large diversity gain characteristic of STBC can be achieved. The BER upper bound has been derived taking BPSK modulation as an example. The simulation results show that the proposed system with the concatenated codes outperforms the one with only TPC or STBC and other reported schemes that concatenate STBC with convolutional Turbo codes or trellis codes.     

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     

Convergence analysis of turbo decoding of product codes

Geometric interpretation of turbo decoding has founded an analytical basis, and provided tools for the analysis of this algorithm. We focus on turbo decoding of product codes, and based on the geometric framework, we extend the analytical results and show how analysis tools can be practically adapted for this case. Specifically, we investigate the algorithm's stability and its convergence rate. We present new results concerning the structure and properties of stability matrices of the algorithm, and develop upper bounds on the algorithm's convergence rate. We prove that for any 2×2 (information bits) product codes, there is a unique and stable fixed point. For the general case, we present sufficient conditions for stability. The interpretation of these conditions provides an insight to the behavior of the decoding algorithm. Simulation results, which support and extend the theoretical analysis, are presented for Hamming [(7,4,3)]² and Golay [(24,12,8)]² product codes     

Refinements and asymptotic performance of bandwidth-efficient turbo product codes

In this letter, a turbo product code (TPC) is combined with multilevel modulations (8-phase-shift keying and 16-quadrature amplitude modulation). The component codes are Bose-Chaudhuri-Hocquengem (BCH) or extended BCH. We derive soft-input/soft-output modules based on the dual code, with exact Euclidean metrics, and we show that the iterative TPC decoder gains no advantage in performance from this. Next, we evaluate asymptotic approximations for maximum-likelihood (ML) decoding from a combinatorial approach that can be applied to any bit-interleaved multilevel modulated code, once the first term (or terms) of the Hamming weight spectrum are known. For the TPCs and modulations studied in this letter, random bit interleaving before modulation leads to improved ML asymptotes. Simulations confirm that this advantage is maintained also under iterative decoding.     

An efficient Chase decoder for turbo product codes

In this letter, we propose an efficient decoding algorithm for turbo product codes as introduced by Pyndiah. The proposed decoder has no performance degradation and reduces the complexity of the original decoder by an order of magnitude. We concentrate on extended Bose-Chaudhuri-Hocquengem codes as the constituent row and column codes because of their already low implementation complexity. For these component codes, we observe that the weight and reliability factors can be fixed, and that there is no need for normalization. Furthermore, as opposed to previous efficient decoders, the newly proposed decoder naturally scales with a test-pattern parameter p that can change as a function of iteration number, i.e., the efficient Chase algorithm presented here uses conventionally ordered test patterns, and the syndromes, even parities, and extrinsic metrics are obtained with a minimum number of operations.     

Asymptotically Gaussian weight distribution and performance of multicomponent turbo block codes and product codes

It was suggested by Battail that a good long linear code should have a weight distribution close to that of random coding, rather than a large minimum distance, and a turbo code should be also designed using a random-like criterion. In this paper, we first show that the weight distribution of a high-rate linear block code is approximately Gaussian if the code rate is close enough to one, and then proceed to construct a low-rate linear block code with approximately Gaussian weight distribution by using the turbo-coding technique. We give a sufficient condition under which the weight distribution of multicomponent turbo block (MCTB) codes (multicomponent product (MCP) codes, respectively) can approach asymptotically that of random codes, and further develop two classes of MCTB codes (MCP codes) satisfying this condition. Simulation results show that MCTB codes (MCP codes) having asymptotically Gaussian weight distribution can asymptotically approach Shannon's capacity limit. MCTB codes based on single parity-check (SPC) codes have a far poorer minimum distance than MCP codes based on SPC codes, but we show by simulation that when the bit-error rate is in the important range of 10/sup -1/-10/sup -5/, these codes can still offer similar performance for the additive white Gaussian noise channel, as long as the code length of the SPC codes is not very short. These facts confirm in a more precise way Battail's inference about the "nonimportance" of the minimum distance for a long code.     

Application of turbo codes in satellite mobile systems

Turbo codes are considered in a mobile channel with FSK modulation and LDI detection. Numerical results are presented for narrowband Gaussian, land mobile and satellite mobile channels separately. Numerical results show that turbo codes can significantly improve the performance over uncoded signalling     

Nonsystematic turbo codes

In this paper, we introduce the concept of nonsystematic turbo codes and compare them with classical systematic turbo codes. Nonsystematic turbo codes can achieve lower error floors than systematic turbo codes because of their superior effective free distance properties. Moreover, they can achieve comparable performance in the waterfall region if the nonsystematic constituent encoder has a low-weight feedforward inverse. A uniform interleaver analysis is used to show that rate R=1/3 turbo codes using nonsystematic constituent encoders have larger effective free distances than when systematic constituent encoders are used. Also, mutual information-based transfer characteristics and extrinsic information transfer charts are used to show that rate R=1/3 turbo codes with nonsystematic constituent encoders having low-weight feedforward inverses achieve convergence thresholds comparable to those achieved with systematic constituent encoders. Catastrophic encoders, which do not possess a feedforward inverse, are shown to be capable of achieving low convergence thresholds by doping the code with a small fraction of systematic bits. Finally, we give tables of good nonsystematic turbo codes and present simulation results comparing the performance of systematic and nonsystematic turbo codes.     

Stream-oriented turbo codes

This work considers the design and performance of a stream-oriented approach to turbo codes which avoids the need for data framing. The stream paradigm applies to both serial and parallel turbo codes using continuous, free-running constituent encoders along with continuous, periodic interleavers. A stream-oriented turbo code based on parallel concatenated convolutional codes (PCCC) is considered and interleaver design criteria are developed for both block and nonblock periodic interleavers. Specifically, several nonblock interleavers, including convolutional interleavers, are considered. Interleaver design rules are verified using simulations where it is shown that nonblock interleavers with small-to-moderate delay and small synchronization ambiguity can outperform block interleavers of comparable delay. For large-delay designs, nonblock interleavers are found which perform within 0.8 dB of the capacity limit with a synchronization ambiguity of N=11     

Symbol-based turbo codes

We present a new turbo-coding method which parses the input block into n-bit symbols and interleaves on a symbol-by-symbol basis. This is used in conjunction with different modulation techniques to take advantage of tradeoffs between bit error rate performance, code-rate, spectral efficiency, and decoder complexity. The structure of the encoder and decoder of these codes, which We call symbol-based turbo codes, are outlined. The bit error rate performance of a few specific codes are examined. A discussion on decoder complexity is also included. Symbol-based turbo codes are good candidates for low delay transmission of speech and data in spread spectrum communication systems     

Time-varying turbo codes

Turbo codes typically utilize time-invariant component codes whose total encoder memory is no greater than 4. In this letter, turbo codes with time-varying component codes are studied via EXIT analysis, distance spectrum analysis and simulation. Turbo codes with time-varying component codes and memory 6 are shown to have similar performance in the waterfall region and superior performance in the error floor region when compared to both the original memory 4 Berrou turbo code and the memory 8 big numerator-little denominator turbo codes.     

A parallel decoder for low latency decoding of turbo product codes

There has been intensive focus on turbo product codes (TPCs) which have low decoding complexity and achieve near-optimum performances at low signal-to-noise ratios. Different than the original TPC decoder, which performs row and column decoding in a serial fashion, we propose a parallel decoder structure. Simulation results show that with this approach, decoding latency of TPCs can be halved while maintaining virtually the same performance level     

Multilayer turbo space-time codes

This letter describes a multilayer turbo space-time coding scheme. Based on a carefully designed power allocation strategy, performance reasonably close to the theoretical limits can be achieved at a rate of two bits per channel use with very low complexity.     

Differential space-time turbo codes

Serial concatenation of simple error control codes and differential space-time modulation is considered. Decoding is performed iteratively by passing symbol-wise a posteriori probability values between the decoders of the inner space-time code and the outer code. An extrinsic information transfer analysis is used to predict thresholds for outer convolutional codes of various memory orders and a simple outer parity-check code. This parity-check code is well matched to the inner differential space-time code and achieves a bit-error rate (BER) of 10/sup -6/ less than 2 dB from the Shannon capacity of the fast fading multiple antenna channel. The differential space-time code can also be used to generate a priori information in the absence of channel knowledge. This information can be exploited by a channel estimator inserted into the decoding iteration. It is demonstrated that the inner space-time code provides soft training symbols from periodically inserted training symbols. The reliability of these soft training symbols does not depend on the speed of the channel variations, but on the structure of the inner code and the signal-to-noise ratio (SNR). Simulation studies confirm these findings and show that the proposed system with no initial channel knowledge achieves a performance very close to that of the system with perfect channel knowledge.     

Concatenated parity and turbo codes

Turbo codes have an error floor that is caused by low-weight error events. Here, it is shown that a concatenated code with a simple rectangular parity-check outer code and a turbo inner code can significantly reduce the error floor. It is also shown that in several situations, the concatenated parity-check and turbo code performs significantly better than a turbo code alone     

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     

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.     

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     

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.