Reduced complexity decoding of convolutional codes based on the M-algorithm and the minimal trellis

In this paper, we propose three new sub-optimum, reduced complexity decoding algorithms for convolutional codes. The algorithms are based on the minimal trellis representation for the convolutional code and on the M-algorithm. Since the minimal trellis has a periodically time-varying state profile, each algorithm has a different strategy to select the number of surviving states in each trellis depth. We analyse both the computational complexity, in terms of arithmetic operations, and the bit error rate performance of the proposed algorithms over the additive white Gaussian noise channel. Results demonstrate that considerable complexity reductions can be obtained at the cost of a small loss in the performance, as compared to the Viterbi decoder.     

A new high rate code scheme with highly parallel and low complexity decoding algorithm

A new high rate code scheme is proposed in this paper. It consists of serial concatenated recursive systematic ordinary (nonpunctured) convolutional codes with only 8 states in the trellis of the corresponding reciprocal dual codes. With a low complexity and highly parallel decoding algorithm, over additive white Gaussian noise channels, the proposed codes can achieve good bit error rate (BER) performance comparable to that of turbo codes and low density parity check (LDPC) codes. At code rate R=16/17, the overall decoding complexity of the proposed code scheme is almost half that of the LDPC codes.     

Performance of STTC-MC-CDMA Systems with Imperfect Channel Estimation

Since, in a practical system perfect channel state information (CSI) is not possible due to presence of noise. This paper deals with the performance of space-time trellis code (STTC) in multi-carrier code-division multiple-access systems in presence of channel estimation (CE) error and results are compared with perfect CSI at the receiver. The pilot symbol assisted (PSA) technique is used for CE employing minimum mean-square error method. The symbol error rate (SER) performance is observed by employing Viterbi decoding algorithm to decode STTC code at the receiver in multi-path fading channel. The simulated SER performances in presence of CE error and with perfect CSI are compared with the theoretical performances to validate the theoretical analysis.     

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     

Adaptive resource allocation for multicast orthogonal frequency division multiple access systems with guaranteed BER and rate

In this paper, we propose a resource allocation scheme to minimize transmit power for multicast orthogonal frequency division multiple access systems. The proposed scheme allows users to have different symbol error rate (SER) across subcarriers and guarantees an average bit error rate and transmission rate for all users. We first provide an algorithm to determine the optimal bits and target SER on subcarriers. Because the worst‐case complexity of the optimal algorithm is exponential, we further propose a suboptimal algorithm that separately assigns bit and adjusts SER with a lower complexity. Numerical results show that the proposed algorithm can effectively improve the performance of multicast orthogonal frequency division multiple access systems and that the performance of the suboptimal algorithm is close to that of the optimal one. Copyright © 2012 John Wiley & Sons, Ltd.     

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.     

Quasi-orthogonal STBC with minimum decoding complexity: performance analysis, optimal signal transformations, and antenna selection diversity

This letter presents a new method to directly analyze and optimize symbol error rate (SER) performance of minimum decoding complexity (MDC) ABBA space-time block codes based on a tight union bound on SER. Additionally, a new signal transformation for rectangular quadrature amplitude modulation is proposed to provide better performance than the existing ones with lower encoding/decoding complexities. It is also shown that MDC-ABBA codes achieve full-diversity with antenna selection and limited feedback.     

On dualizing trellis-based APP decoding algorithms

The trellis of a finite Abelian group code is locally (i.e., trellis section by trellis section) related to the trellis of the corresponding dual group code which allows one to express the basic operations of the a posteriori probability (APP) decoding algorithm (defined on a single trellis section of the primal trellis) in terms of the corresponding dual trellis section. Using this local approach, any algorithm employing the same type of operations as the APP algorithm can, thus, be dualized, even if the global dual code does not exist (e.g., nongroup codes represented by a group trellis). Given this, the complexity advantage of the dual approach for high-rate codes can be generalized to a broader class of APP decoding algorithms, including suboptimum algorithms approximating the true APP, which may be more attractive in practical applications due to their reduced complexity. Moreover, the local approach opens the way for mixed approaches where the operations of the APP algorithm are not exclusively performed on the primal or dual trellis. This is inevitable if the code does not possess a trellis consisting solely of group trellis sections as, e.g., for certain terminated group or ring codes. The complexity reduction offered by applying dualization is evaluated. As examples, we give a dual implementation of a suboptimum APP decoding algorithm for tailbiting convolutional codes, as well as dual implementations of APP algorithms of the sliding-window type. Moreover, we evaluate their performance for decoding usual tailbiting codes or convolutional codes, respectively, as well as their performance as component decoders in iteratively decoded parallel concatenated schemes.     

Joint Source and Space Time Decoding over Fading Channels

For high data rate communications, variable length codes (VLC) and multiple input multiple output (MIMO) techniques are required in the application layer and the physical layer, respectively. In this paper, we study the joint decoding for systems that consist of reversible VLC (RVLC) as the source codes and recursive space time trellis codes (STTC) as the coded modulation. The maximum a posteriori probability (MAP) decoding algorithm is modified to accommodate issues including the inseparable systematic information, the unsymmetric trellis structure, and information in both bit and symbol domains. Further, by converting extrinsic information based on the modulation mode and the mapping method, iterative information exchange can be performed so as to fully utilize the redundancy in both RVLC and recursive STTC. The performance of the proposed joint decoding is evaluated over both rapid fading and quasi- static flat fading channels, which shows significant improvement in coding gain in comparison with the non-iterative receiver.     

Soft-decision decoding using punctured codes

Let a q-ary linear (n,k)-code be used over a memoryless channel. We design a soft-decision decoding algorithm that tries to locate a few most probable error patterns on a shorter length s ∈ [k,n]. First, we take s cyclically consecutive positions starting from any initial point. Then we cut the subinterval of length s into two parts and examine T most plausible error patterns on either part. To obtain codewords of a punctured (s,k)-code, we try to match the syndromes of both parts. Finally, the designed codewords of an (s,k)-code are re-encoded to find the most probable codeword on the full length n. For any long linear code, the decoding error probability of this algorithm can be made arbitrarily close to the probability of its maximum-likelihood (ML) decoding given sufficiently large T. By optimizing s, we prove that this near-ML decoding can be achieved by using only T≈q^(n-k)k(n+k)/ error patterns. For most long linear codes, this optimization also gives about T re-encoded codewords. As a result, we obtain the lowest complexity order of q^(n-k)k(n+k)/ known to date for near-ML decoding. For codes of rate 1/2, the new bound grows as a cubic root of the general trellis complexity q^min{n-k,k}. For short blocks of length 63, the algorithm reduces the complexity of the trellis design by a few decimal orders     

Super-orthogonal space-time trellis codes with eight dimensional phase-shift keying signal constellations

We propose a new method of designing super-orthogonal space-time trellis codes (SOSTTCS) by labeling each state transition of the trellis diagram with an ordered pair of 2×2 orthogonal matrices instead of only a single 2×2 orthogonal matrix. An 8D instead of a 4D signal constellation is thus used for modulation and the delay elements of the trellis encoder are 4T instead of 2T, where T is the 2D symbol duration. Therefore, at the receiver, the sequence decoding using Viterbi algorithm requires two times less frequent updating of the branch metrics. We investigate by computer simulation the performance of the new sosttcs compared to sosttcs designed by the known method. The frame error rate (fer) and the bit error rate (ber) simulation results show that the performance of the new sosttcs is practically as good as that of the sosttcs of similar complexity designed along classical lines, with the advantage that the number of arithmetic and logic operations performed by the decoder per time unit is smaller, which is crucial in high data-rate applications.     

Enhanced trellis extracted synchronisation for block codes usinginterleavers

Previously, trellis extracted synchronisation techniques (TEST) have been presented as a purely digital method of providing bit/symbol and word synchronisation for block codes using auxiliary data obtained through trellis decoding. One of the factors affecting the complexity of the algorithm has been that any cyclic properties within the code must be countered by building confidence in a particular synchronisation point. This problem is further exasperated by a long sequence of the all-zeros (or all-ones) codeword, which presents all cyclic shifts as valid codewords. The novel algorithm presented in this Letter uses an interleaver with an inversion to overcome these two problems and significantly reduce the complexity without reducing performance     

Energy-efficient node position identification through payoff matrix and variability analysis

This paper investigates the use of punctured recursive systematic convolutional codes for turbo coding in a 2-user binary adder channel (2-BAC) in the presence of additive white Gaussian noise, aiming to achieve a higher transmission sum rate with reduced decoding complexity. The encoders for the 2-BAC are assumed to be block synchronized and to employ identical puncturing patterns. Iterative decoding combining the Bahl Cocke Jelinek Raviv algorithm and a two-user punctured trellis is employed. For each user and for a fixed puncturing pattern, random interleavers of length 256 bits or 1024 bits, respectively, are simulated and corresponding curves relating bit error rate versus signal to noise ratio are presented for performance comparison purposes. Computer simulation indicates that the loss in performance of a punctured turbo code can be negligible when longer interleavers are used for both users, similarly to the single user case.     

Performance analysis of stack decoding on block coded modulation schemes using tree diagram

The channel encoder adds redundancy in a structured way to provide error control capability. Modulator converts the symbol sequences from the channel encoder into waveforms which are then transmitted over the channel. Usually channel coder and modulator are implemented independently one after the other. But in a band limited channel better coding gains without sacrificing signal power are achieved when coding is combined with modulation. Block Coded Modulation (BCM) is such a scheme that results from the combination of linear block codes and modulation. In this paper we are proposing a stack decoding of rate 2/3 and rate 1/2 BCM schemes using tree structure and performance is compared with the Viterbi decoding that uses trellis representation. Simulation result shows that at reasonable bit error rate stack decoder performance is just 0.2 to 0.5 dB inferior to that of Viterbi decoding. Since stack decoding is a near optimum decoding scheme and whose decoding procedure is adaptable to noise level, we can consider this method in place of Viterbi decoding which is optimum and its decoding complexity grows exponentially with large code lengths.     

Tight error bounds for space-time orthogonal block codes under slow Rayleigh flat fading

The performance of space-time orthogonal block (STOB) codes over slow Rayleigh fading channels and maximum-likelihood (ML) decoding is investigated. Two Bonferroni-type bounds (one upper bound and one lower bound) for the symbol error rate (SER) and bit error rate (BER) of the system are obtained. The bounds are expressed in terms of the pairwise error probabilities (PEPs) and the two-dimensional pairwise error probabilities (2-D PEPs) of the transmitted symbols. Furthermore, the bounds can be efficiently evaluated and they hold for arbitrary (nonstandard) signaling schemes and mappings. Numerical results demonstrate that the bounds are very accurate in estimating the performance of STOB codes. In particular, the upper and lower bounds often coincide even at low channel signal-to-noise ratios, large constellation sizes, and large diversity orders.     

一种基于PPM的LDPC编译码方案

该文基于LDPC码和PPM调制方式构造了一种适用于UWB无线通信系统的低码率的编译码方案LDPC-PPM。本方案在保证系统性能的前提下,通过改变编码比特到调制符号的映射方式,不但避免了一般编码调制系统中译码和解调之间的迭代运算,而且可以应用快速Hadamard变换(FHT)和基于FHT的后验概率译码(APP-FHT)来进一步降低接收端的译码复杂度。可以证明,该方案等价于BPSK调制下的低码率的LDPC-Hadamard码。仿真结果表明,在信息比特长度是65536,该方案可以在-1.18dB处达到误比特率为 的性能,仅比采用BPSK调制的Turbo-Hadamard码差0.02dB。     

Decoding performance of linear block codes using a trellis indigital mobile radio

Trellis decoding of linear block codes in a Rayleigh fading channel is discussed. Two methods for calculating metric values for each bit in a received block are considered: the values are calculated from the received signal envelope sample and from the demodulator output. Bit error rate (BER) performances of hard decision and trellis decoding are compared using Hamming (7, 4) and Golay (24, 12) codes in computer simulations and laboratory experiments. A simplified trellis decoding algorithm, in which the hard decision output of a bit with an envelope sample greater than the threshold value is accepted as correct, is presented. Laboratory experimental results for trellis decoding in combination with Gaussian minimum-shift-keying (GMSK) modulation and frequency detection are shown. The effect of n-bit A/D-conversion in signal envelope sampling is investigated experimentally. The results show that the trellis decoding algorithm improves BER performance     

一种改进的无线光通信LDPC码译码算法

针对无线光通信中低密度奇偶校验码(LDPC)置信传播(BP)译码算法复杂度高及置信度振荡造成译码错误等缺点,基于对数BP算法提出了一种改进的译码算法。改进的译码算法在校验节点运算时,判断输入到校验节点消息的最小值与某个门限的大小,根据比较结果,分别用消息最小值或若干个最小值进行运算,在损失很少性能的情况下降低了运算复杂度;同时在比特节点采用振荡抵消处理运算,提高了算法的性能增益。最后在对数正态分布湍流信道模型下,分别对比特充分交织和交织深度为16的情况进行了仿真实验。仿真结果表明,改进的译码算法与BP算法相比,大幅度降低了计算复杂度,而且译码性能有一定的优势,收敛速度损失很少;而相对于最小和算法,改进的算法虽然译码复杂度有所增加,但误码率性能有明显的优势,并且收敛速度也优于最小和算法。因此,改进的译码算法是无线光通信中LDPC码译码算法复杂度和性能之间一个较好的折中处理方案。     

Low-Density Parity-Check Codes with 2-State Trellis Decoding

A class of low-density parity-check (LDPC) codes with a simple 2-state trellis structure is presented. For LDPC decoding, the conventional belief propagation (BP) algorithm consists of numerous sub-decoders of single-parity check codes and exchanges information between sub-decoders in an iterative manner. If the single-parity check codes can be constructed and grouped in a proper way, the decoder can be decomposed into few identical 2-state trellis decoders. Therefore, instead of numerous sub-decoders of single-parity check codes, an iterative decoding algorithm based on few sub-decoders over 2-state trellis is proposed. The proposed decoding algorithm improves the efficiency of message passing between sub-decoders and hence provides a fast convergent rate as compared to the standard BP algorithm. Simulation results show that the proposed scheme provides a better performance and a fast convergent rate as compared to those of standard BP algorithm. The result also shows that the proposed algorithm has a similar performance as that of asynchronous replica shuffled BP algorithm and has a slightly inferior performance than that of synchronous replica shuffled BP algorithm. However, complexity analysis shows that our proposed algorithm has complexity that is lower than that of the replica shuffled BP algorithm.     

Multilevel block codes for Rayleigh-fading channels

New multilevel block codes for Rayleigh-fading channels are presented. At high signal-to-noise ratios (SNRs), the proposed block codes can achieve better bit error performance over TCM codes, optimum for fading channels, with comparable decoder complexity and bandwidth efficiency. The code construction is based on variant length binary component block codes. As component codes for the 8-PSK multilevel block construction, the authors propose two modified forms of Reed-Muller codes giving a good trade-off between the decoder complexity and the effective code rates. Code design criteria are derived from the error performance analysis. Multistage decoding shows very slight degradation of bit error performance relative to the maximum likelihood algorithm