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.     

Multiple serial and parallel concatenated single parity-check codes

Single parity-check (SPC) codes are applied in both parallel and serial concatenated structures to produce high-performance coding schemes. The number of concatenations or stages, M, is increased to improve system performance at moderate-to-low bit-error rates without changing the overall code parameters (namely, code rate and code block length). Analytical bounds are presented to estimate the performance at high signal-to-noise ratios. The SPC concatenated codes are considered with binary phase-shift keying and with 16-quadrature amplitude modulation bit-interleaved coded modulation on the additive white Gaussian noise channel and the independent Rayleigh fading channel. Simulations show that the four-stage serial or parallel concatenated SPC codes can, respectively, outperform or perform as well as 16-state turbo codes. Furthermore, decoding complexity is approximately 9-10 times less complex than that of 16-state turbo codes. The convergence behavior of both serial and parallel concatenated SPC codes is also discussed.     

A simple remedy for the exaggerated extrinsic information produced by the SOVA algorithm

In this paper, we propose a novel and simple approach for dealing with the exaggerated extrinsic information produced by the soft-output Viterbi algorithm (SOVA). The proposed remedy is based on mathematical analysis and it involves using two attenuators, one applied to the immediate output of the SOVA and another applied to the extrinsic information before it is passed to the other decoder (assuming iterative decoding). The use of these attenuators aims at reducing the inherent strong correlation between the intrinsic information (input to the SOVA) and extrinsic information (output of the SOVA). We examine the modified SOVA (MSOVA) on additive white Gaussian noise (AWGN) and flat fading channels for parallel concatenated codes (PCCs) and serial concatenated codes (SCCs). We show that the MSOVA provides substantial performance improvements over both channels. For example, it provides improvements of about 0.8 to 1.0 dB at P/sub b/ = 10/sup -5/ in AWGN, and about 1.4 to 2.0 dB at P/sub b/ = 10/sup -5/ on fading channels. We also show that there are cases where the MSOVA is superior to the a posteriori probability (APP) algorithm. With this motivation, we extend the proposed modification to the APP algorithm with favorable results. We demonstrate that the modified APP (MAPP) provides performance improvements between 0.3 to 0.6 dB at P/sub b/ = 10/sup -5/ relative to the APP. We lastly mention that the proposed modifications, while they provide considerable performance improvements, keep the complexity of these decoders almost the same, which is remarkable.     

Improved upper bounds on the ML decoding error probability ofparallel and serial concatenated turbo codes via their ensemble distancespectrum

The ensemble performance of parallel and serial concatenated turbo codes is considered, where the ensemble is generated by a uniform choice of the interleaver and of the component codes taken from the set of time-varying recursive systematic convolutional codes. Following the derivation of the input-output weight enumeration functions of the ensembles of random parallel and serial concatenated turbo codes, the tangential sphere upper bound is employed to provide improved upper bounds on the block and bit error probabilities of these ensembles of codes for the binary-input additive white Gaussian noise (AWGN) channel, based on coherent detection of equi-energy antipodal signals and maximum-likelihood decoding. The influence of the interleaver length and the memory length of the component codes is investigated. The improved bounding technique proposed here is compared to the conventional union bound and to a alternative bounding technique by Duman and Salehi (1998) which incorporates modified Gallager bounds. The advantage of the derived bounds is demonstrated for a variety of parallel and serial concatenated coding schemes with either fixed or random recursive systematic convolutional component codes, and it is especially pronounced in the region exceeding the cutoff rate, where the performance of turbo codes is most appealing. These upper bounds are also compared to simulation results of the iterative decoding algorithm     

Performance evaluation of superorthogonal turbo codes in AWGN andflat Rayleigh fading channels

Turbo codes are parallel concatenated codes whose performance in the additive white Gaussian noise (AWGN) channel has been shown to be near the theoretical limit. In this paper, we describe a low-rate superorthogonal turbo code that combines the principles of low-rate convolutional coding and that of parallel concatenation. Due to the bandwidth expansion, this code outperforms the ordinary turbo code both in AWGN and especially in fading channels. Thus, superorthogonal turbo codes are suited mainly for spread-spectrum applications. For the purposes of iterative decoding, we concisely describe the connection between the optimal maximum a posteriori symbol estimation and suboptimal soft-output decoding based on sequence estimation. The suboptimal decoder produces outputs that can directly be used as additive metrics at successive decoding iterations, without the need for estimating channel noise variance. Simulation results in AWGN and flat Rayleigh fading channels are also presented, along with analytical upper bounds of bit- and frame-error probabilities     

一种基于SOVA的改进的Turbo码译码方案

本文在研究Ｔｕｒｂｏ 码反向ＳＯＶＡ（Ｓｏｆｔ－Ｏｕｔｐｕｔ ＶｉｔｅｒｂｉＡｌｇｏｒｉｔｈｍ ）译码性能的基础上,提出了一种同时利用正向和反向ＳＯＶＡ译码软输出信息的基于ＳＯＶＡ 的改进译码结构及其相应的软输出修正公式。计算机模拟结果表明,所提出的改进方案与传统的ＳＯＶＡ算法相比,其译码性能有明显的改善,并略优于Ｍａｘ－Ｌｏｇ－ＭＡＰ的性能     

Coding for Parallel Channels: Gallager Bounds and Applications to Turbo-Like Codes

The transmission of coded communication systems is widely modeled to take place over a set of parallel channels. This model is used for transmission over block-fading channels, rate-compatible puncturing of turbo-like codes, multicarrier signaling, multilevel coding, etc. New upper bounds on the maximum-likelihood (ML) decoding error probability are derived in the parallel-channel setting. We focus on the generalization of the Gallager-type bounds and discuss the connections between some versions of these bounds. The tightness of these bounds for parallel channels is exemplified for structured ensembles of turbo codes, repeat-accumulate (RA) codes, and some of their recent variations (e.g., punctured accumulate-repeat-accumulate codes). The bounds on the decoding error probability of an ML decoder are compared to computer simulations of iterative decoding. The new bounds show a remarkable improvement over the union bound and some other previously reported bounds for independent parallel channels. This improvement is exemplified for relatively short block lengths, and it is pronounced when the block length is increased. In the asymptotic case, where we let the block length tend to infinity, inner bounds on the attainable channel regions of modern coding techniques under ML decoding are obtained, based solely on the asymptotic growth rates of the average distance spectra of these code ensembles.     

On improved bounds on the decoding error probability of block codesover interleaved fading channels, with applications to turbo-like codes

We derive here improved upper bounds on the decoding error probability of block codes which are transmitted over fully interleaved Rician fading channels, coherently detected and maximum-likelihood (ML) decoded. We assume that the fading coefficients during each symbol are statistically independent (due to a perfect channel interleaver), and that perfect estimates of these fading coefficients are provided to the receiver. The improved upper bounds on the block and bit error probabilities are derived for fully interleaved fading channels with various orders of space diversity, and are found by generalizing some previously introduced upper bounds for the binary-input additive white Gaussian nose (AWGN) channel. The advantage of these bounds over the ubiquitous union bound is demonstrated for some ensembles of turbo codes and low-density parity-check (LDPC) codes, and it is especially pronounced in a portion of the rate region exceeding the cutoff rate. Our generalization of the Duman and Salehi bound (Duman and Salehi 1998, Duman 1998) which is based on certain variations of Gallager's (1965) bounding technique, is demonstrated to be the tightest reported upper bound. We therefore apply it to calculate numerically upper bounds on the thresholds of some ensembles of turbo-like codes, referring to the optimal ML decoding. For certain ensembles of uniformly interleaved turbo codes, the upper bounds derived here also indicate good match with computer simulation results of efficient iterative decoding algorithms     

Tight exponential upper bounds on the ML decoding error probability of block codes over fully interleaved fading channels

We derive tight exponential upper bounds on the decoding error probability of block codes which are operating over fully interleaved Rician fading channels, coherently detected and maximum-likelihood decoded. It is assumed that the fading samples are statistically independent and that perfect estimates of these samples are provided to the decoder. These upper bounds on the bit and block error probabilities are based on certain variations of the Gallager bounds. These bounds do not require integration in their final version and they are reasonably tight in a certain portion of the rate region exceeding the cutoff rate of the channel. By inserting interconnections between these bounds, we show that they are generalized versions of some reported bounds for the binary-input additive white Gaussian noise channel.     

A Parallel Concatenated Convolutional-Based Distributed Coded Cooperation Scheme for Relay Channels

In this paper, we investigate a coded cooperation diversity scheme suitable for L-relay channels operating in the soft-decode-and-forward (soft-DF) mode. The proposed scheme is based on parallel concatenated convolutional codes (PCCC). To improve the overall performance through diversity, the coded cooperation operates by sending the systematic and the first parity outputs via L?+?1 independent fading paths. Instead of using only a centralized turbo code system at the source node, we have proposed a DCC scheme, where the first recursive systematic coding is done at both source and relay nodes. At the destination, the received replicas are combined using the maximal ratio combining (MRC). The entire codeword, comprising the MRC sequence and the second parity part, is decoded via the maximum a-posteriori (MAP) algorithm and turbo decoding principle. We analyze the proposed scheme in terms of bit error rate (BER). In fact, we define the explicit upper bounds for error rate assuming Binary phase shift keying (BPSK) transmission for fully interleaved channels with channel state information (CSI). We use the Rayleigh fading channels with independent fading. Our study shows that the full diversity order is achieved when the source-relay link is more reliable than the other links. Otherwise, the diversity decreases. However, in both cases, it is shown that significant performance improvements are possible to achieve over non-cooperative coded systems. Theorical and simulation results are presented to demonstrate the efficacy of the proposed scheme.     

Forward Error Correction Coding for Fading Compensation in Mobile Satellite Channels

Fading in mobile satellite communications severely degrades the performance of data transmission. The channel is modeled with nonfrequency selective Rice and Rayleigh fading. Also, stored channel simulation is used for hardware data transmission. FEC coding with Viterbi decoding of convolutional codes, and Berlekamp-Massey decoding of Reed-Solomon codes, are used to compensate for the fading. In addition to interleaving, channel state and erasure information improve the performance of the decoder. The BER after decoding is calculated for specific codes on several channels and for different transmission schemes. Using very simple channel state and erasure information gives 2-7 dB additional coding gain. These gains have been verified by hardware data transmission on synthetic fading channels and stored mobile satellite channels.     

On turbo decoding of nonbinary codes

Symbol-by-symbol maximum a posteriori (MAP) decoding algorithms for nonbinary block and convolutional codes over an extension field GF(p ^a) are presented. Equivalent MAP decoding rules employing the dual code are given which are computationally more efficient for high-rate codes. It is shown that these algorithms meet all requirements needed for iterative decoding as the output of the decoder can be split into three independent estimates: soft channel value, a priori term and extrinsic value. The discussed algorithms are then applied to a parallel concatenated coding scheme with nonbinary component codes in conjunction with orthogonal signaling     

Joint iterative channel estimation and decoding in flat correlatedRayleigh fading

This paper addresses the design and performance evaluation with respect to capacity of M-PSK turbo-coded systems operating in frequency-flat time-selective Rayleigh fading. The receiver jointly performs channel estimation and turbo decoding, allowing the two processes to benefit from each other. To this end, we introduce a suitable Markov model with a finite number of states, designed to approximate both the values and the statistical properties of the correlated flat fading channel phase, which poses a more severe challenge to PSK transmission than amplitude hiding. Then, the forward-backward algorithm determines both the maximum a posteriori probability (MAP) value for each symbol in the data sequence and the MAP channel phase in each iteration. Simulations show good performance in standard correlated Rayleigh fading channels. A sequence of progressively tighter upper bounds to the capacity of a simplified Markov-phase channel is derived, and performance of a turbo code with joint iterative channel estimation and decoding is demonstrated to approach these capacity bounds     

Finite-Dimensional Bounds on Zm and Binary LDPC Codes With Belief Propagation Decoders

This paper focuses on finite-dimensional upper and lower bounds on decodable thresholds of Zopf_m and binary low-density parity-check (LDPC) codes, assuming belief propagation decoding on memoryless channels. A concrete framework is presented, admitting systematic searches for new bounds. Two noise measures are considered: the Bhattacharyya noise parameter and the soft bit value for a maximum a posteriori probability (MAP) decoder on the uncoded channel. For Zopf _m LDPC codes, an iterative m-dimensional bound is derived for m-ary-input/symmetric-output channels, which gives a sufficient stability condition for Zopf_m LDPC codes and is complemented by a matched necessary stability condition introduced herein. Applications to coded modulation and to codes with nonequiprobably distributed codewords are also discussed. For binary codes, two new lower bounds are provided for symmetric channels, including a two-dimensional iterative bound and a one-dimensional noniterative bound, the latter of which is the best known bound that is tight for binary-symmetric channels (BSCs), and is a strict improvement over the existing bound derived by the channel degradation argument. By adopting the reverse channel perspective, upper and lower bounds on the decodable Bhattacharyya noise parameter are derived for nonsymmetric channels, which coincides with the existing bound for symmetric channels     

基于Turbo-TCM与时空分组码的级联系统的次优译码算法

针对瑞利信道中存在的严重的多径衰落,本文实现了Turbo-TCM方案与时空分组码的级联系统,以期利用空间分集改善系统的误码率性能。针对级联系统的译码,本文给出了一种具有低译码时延的次优译码算法,该算法的特点是各模块独立译码,先算比特对数似然比再进行二进制Turbo码译码。最后通过计算机仿真给出了使用该次优译码算法的Turbo－TCM方案与时空分组码的级联系统的译码性能。仿真结果说明,当发射天线数目一定时,随着接收天线数目的增加,译码性能的增益随之增加而帧长对译码性能的影响则随之减小。     

DSD (double soft decision) concatenated FEC scheme in mobilesatellite communication systems

In order to realize a higher-code-gain forward error correction scheme in mobile satellite communication systems, a novel concatenated coding scheme employing soft decision decoding for not only inner codes but also outer codes (double soft decision, or DSD, concatenated forward error correction scheme) is proposed. Soft-decision outer decoding can improve the bit error probability of inner decoded data. In this scheme, likelihood information from an inner Viterbi decoder is used in the decoding of outer codes. A technique using the path memory circuit status 1.0 ratio for likelihood information is proposed, and is shown to be the most reliable even though it requires the simplest hardware among the alternative methods. A computer simulation clarifies that the DSD scheme improves Pe performance to one-third of that of the conventional hard-decision outer decoding. Moreover, to reduce the interleaving delay time in fading channels or inner decoded data of concatenated codes, a parallel forward error correction scheme is proposed     

Performance analysis on LDPC-Coded systems over quasi-static (MIMO) fading channels

In this paper, we derive closed form upper bounds on the error probability of low-density parity-check (LDPC) coded modulation schemes operating on quasi-static fading channels. The bounds are obtained from the so-called Fano- Gallager?s tight bounding techniques, and can be readily calculated when the distance spectrum of the code is available. In deriving the bounds for multiple-input multiple-output (MIMO) systems, we assume the LDPC code is concatenated with the orthogonal space-time block code as an inner code. We obtain an equivalent single-input single-output (SISO) channel model for this concatenated coded-modulation system. The upper bounds derived here indicate good matches with simulation results of a complete transceiver system over Rayleigh and Rician MIMO fading channels in which the iterative detection and decoding algorithm is employed at the receiver.     

Achievable Performance of Orthogonal STBC Over Spatially Correlated Rician Channels

The effect of spatial correlation on the performance of orthogonal space-time block codes (OSTBCs) over multiple-input-multiple-output (MIMO) Rician fading channels is studied. Asymptotic error-rate formulas for OSTBC with high average signal-to-noise ratios (ASNRs) over arbitrarily correlated Rician MIMO channels are derived in terms of the diversity and coding gains. Our results show that, in correlated fading, the phase vector phi of the channel line-of-sight (LOS) components affects the effective Rice K-factor at the OSTBC receiver output and, hence, may result in a coding gain that is significantly higher than that for independent Rician MIMO channels. Furthermore, when the channel covariance matrix is rank deficient and under some additional mild conditions, the error and outage probabilities of OSTBC achieve those in a nonfading additive-white-Gaussian-noise channel. For both cases of full-rank and rank-deficient channel covariance matrices, analytical expressions of optimal and worst case phase vectors phi, and exact upper and lower bounds of OSTBC performance are derived. These results provide new insights into the achievable performance of OSTBC over correlated Rician MIMO channels and, if incorporated into future multiple antenna systems design, will bring about significant performance enhancement     

Performance analysis of convolutionally coded systems over quasi-static fading channels

This paper presents an improved upper bound on the performance of convolutionally coded systems over quasi-static fading channels (QSFC). The bound uses a combination of a classical union bound when the fading channel is in a high signal-to-noise ratio (SNR) state together with a new upper bound for the low SNR state. This new bounding approach is applied to both BPSK convolutional and turbo codes, as well as serially concatenated BPSK convolutional/turbo and space-time block codes. The new analytical technique produces bounds which are usually about 1 dB tighter than existing bounds. Finally, based on the proposed bound, we introduce an improved design criterion for convolutionally coded systems in slow flat fading channels. Simulation results are included to confirm the improved ability of the proposed criterion to search for convolutional codes with good performance over a QSFC.     

低莱斯因子衰落信道下的Turbo码性能研究

基于Turbo码编译码技术,在Rician衰落信道模型下,详细研究了Turbo码Log-MAP译码算法和SOVA译码算法在低Rician因子衰落信道下的性能,并结合调制技术对整个编译码系统进行Matlab仿真。仿真结果表明,在低Rician因子信道下,Log-MAP译码算法的性能优于SOVA译码算法。