NEW ITERATIVE SUPER-TRELLIS DECODING WITH SOURCE A PRIORI INFORMATION FOR VLCS WITH TURBO CODES

A novel Joint Source and Channel Decoding （JSCD） scheme for Variable Length Codes （VLCs） concatenated with turbo codes utilizing a new super-trellis decoding algorithm is presented in this letter. The basic idea of our decoding algorithm is that source a priori information with the form of bit transition probabilities corresponding to the VLC tree can be derived directly from sub-state transitions in new composite-state represented super-trellis. A Maximum Likelihood （ML） decoding algorithm for VLC sequence estimations based on the proposed super-trellis is also described. Simulation results show that the new iterative decoding scheme can obtain obvious encoding gain especially for Reversible Variable Length Codes （RVLCs）, when compared with the classical separated turbo decoding and the previous joint decoding not considering source statistical characteristics.     

Codes defined on graphs

Low-density parity-check codes, turbo codes, and indeed most practically decodable capacity-approaching error correcting codes can all be understood as codes defined on graphs. Graphs not only describe the codes, but, more important, they structure the operation of the sum-product decoding algorithm (or one of many possible variations), which can be used for iterative decoding. Such coding schemes have the potential to approach channel capacity, while maintaining reasonable decoding complexity. In this tutorial article we review factor graphs, which can be used to describe codes and the joint probability distributions that must be dealt with in decoding. We also review the sum-product algorithm, and show how this algorithm leads to iterative decoding algorithms for codes defined on graphs.     

Bandwidth-efficient turbo trellis-coded modulation using puncturedcomponent codes

We present a bandwidth-efficient channel coding scheme that has an overall structure similar to binary turbo codes, but employs trellis-coded modulation (TCM) codes (including multidimensional codes) as component codes. The combination of turbo codes with powerful bandwidth-efficient component codes leads to a straightforward encoder structure, and allows iterative decoding in analogy to the binary turbo decoder. However, certain special conditions may need to be met at the encoder, and the iterative decoder needs to be adapted to the decoding of the component TCM codes. The scheme has been investigated for 8-PSK, 16-QAM, and 64-QAM modulation schemes with varying overall bandwidth efficiencies. A simple code choice based on the minimal distance of the punctured component code has also been performed. The interset distances of the partitioning tree can be used to fix the number of coded and uncoded bits. We derive the symbol-by-symbol MAP component decoder operating in the log domain, and apply methods of reducing decoder complexity. Simulation results are presented and compare the scheme with traditional TCM as well as turbo codes with Gray mapping. The results show that the novel scheme is very powerful, yet of modest complexity since simple component codes are used     

Product accumulate codes: a class of codes with near-capacity performance and low decoding complexity

We propose a novel class of provably good codes which are a serial concatenation of a single-parity-check (SPC)-based product code, an interleaver, and a rate-1 recursive convolutional code. The proposed codes, termed product accumulate (PA) codes, are linear time encodable and linear time decodable. We show that the product code by itself does not have a positive threshold, but a PA code can provide arbitrarily low bit-error rate (BER) under both maximum-likelihood (ML) decoding and iterative decoding. Two message-passing decoding algorithms are proposed and it is shown that a particular update schedule for these message-passing algorithms is equivalent to conventional turbo decoding of the serial concatenated code, but with significantly lower complexity. Tight upper bounds on the ML performance using Divsalar's (1999) simple bound and thresholds under density evolution (DE) show that these codes are capable of performance within a few tenths of a decibel away from the Shannon limit. Simulation results confirm these claims and show that these codes provide performance similar to turbo codes but with significantly less decoding complexity and with a lower error floor. Hence, we propose PA codes as a class of prospective codes with good performance, low decoding complexity, regular structure, and flexible rate adaptivity for all rates above 1/2.     

Iterative turbo decoder analysis based on density evolution

We track the density of extrinsic information in iterative turbo decoders by actual density evolution, and also approximate it by symmetric Gaussian density functions. The approximate model is verified by experimental measurements. We view the evolution of these density functions through an iterative decoder as a nonlinear dynamical system with feedback. Iterative decoding of turbo codes and of serially concatenated codes is analyzed by examining whether a signal-to-noise ratio (SNR) for the extrinsic information keeps growing with iterations. We define a “noise figure” for the iterative decoder, such that the turbo decoder will converge to the correct codeword if the noise figure is bounded by a number below zero dB. By decomposing the code's noise figure into individual curves of output SNR versus input SNR corresponding to the individual constituent codes, we gain many new insights into the performance of the iterative decoder for different constituents. Many mysteries of turbo codes are explained based on this analysis. For example, we show why certain codes converge better with iterative decoding than more powerful codes which are only suitable for maximum likelihood decoding. The roles of systematic bits and of recursive convolutional codes as constituents of turbo codes are crystallized. The analysis is generalized to serial concatenations of mixtures of complementary outer and inner constituent codes. Design examples are given to optimize mixture codes to achieve low iterative decoding thresholds on the signal-to-noise ratio of the channel     

Approaching Shannon performance by iterative decoding of linear codes with low-density generator matrix

We propose the use of linear codes with low density generator matrix to achieve a performance similar to that of turbo and standard low-density parity check codes. The use of iterative decoding techniques - message passing -over the corresponding graph achieves a performance close to the Shannon theoretical limit. As an advantage with respect to turbo and standard low-density parity check codes, the complexity of the decoding and encoding procedures is very low.     

Multilevel turbo coding with short interleavers

The impact of the interleaver, embedded in the encoder for a parallel concatenated code, called the turbo code, is studied. The known turbo codes consist of long random interleavers, whose purpose is to reduce the value of the error coefficients. It is shown that an increased minimum Hamming distance can be obtained by using a structured interleaver. For low bit-error rates (BERs), we show that the performance of turbo codes with a structured interleaver is better than that obtained with a random interleaver. Another important advantage of the structured interleaver is the short length required, which yields a short decoding delay and reduced decoding complexity (in terms of memory). We also consider the use of turbo codes as component codes in multilevel codes. Powerful coding structures that consist of two component codes are suggested. Computer simulations are performed in order to evaluate the reduction in coding gain due to suboptimal iterative decoding. From the results of these simulations we deduce that the degradation in the performance (due to suboptimal decoding) is very small     

Design and performance of turbo Gallager codes

The most powerful channel-coding schemes, namely, those based on turbo codes and low-density parity-check (LDPC) Gallager codes, have in common the principle of iterative decoding. However, the relative coding structures and decoding algorithms are substantially different. This paper shows that recently proposed novel coding structures bridge the gap between these two schemes. In fact, with properly chosen component convolutional codes, a turbo code can be successfully decoded by means of the decoding algorithm used for LDPC codes, i.e., the belief-propagation algorithm working on the code Tanner graph. These new turbo codes are here nicknamed "turbo Gallager codes." Besides being interesting from a conceptual viewpoint, these schemes are important on the practical side because they can be decoded in a fully parallel manner. In addition to the encoding complexity advantage of turbo codes, the low decoding complexity allows the design of very efficient channel-coding schemes.     

Turbo greedy multiuser detection

Previously, a novel scheme for iterative multiuser detection and turbo decoding was proposed by Damnjanovic and Vojcic (2000, 2001). In this scheme, multiuser detection and single-user turbo decoding are tightly coupled to maximize the overall gain. The extrinsic probabilities for the coded bits of the interfering users, obtained after each turbo decoding iteration, are used as a priori probabilities in the following multiuser iteration and the extrinsic information for the systematic bits of the desired user is used as a priori information in the next single-user turbo decoding iteration. Turbo decoding of parallel concatenated convolutional codes is carried out in parallel fashion. It has been shown that the proposed detector approaches the multiuser capacity limit within 1 dB in the low signal-to-noise ratio region. However, the main drawback of the scheme is its exponential complexity in the number of users, which is due to the complexity of the maximum a posteriori probability (MAP) multiuser detector. In this paper, we show that the complexity of the scheme can be significantly reduced by replacing the (MAP) multiuser detector with an iterative detector derived from the greedy multiuser detector proposed by AlRustamani and Vojcic (2000). In this paper, we demonstrate that, for both the additive white Gaussian noise and the frequency-nonselective Rayleigh fading, the substantial reduction in complexity of the iterative scheme proposed by Damnjanovic and Vojcic when the greedy detector is utilized introduces a slight degradation in performance     

Iterative decoding for partial response (PR), equalized,magneto-optical (MO) data storage channels

Turbo codes and low-density parity check (LDPC) codes with iterative decoding have received significant research attention because of their remarkable near-capacity performance for additive white Gaussian noise (AWGN) channels. Previously, turbo code and LDPC code variants are being investigated as potential candidates for high-density magnetic recording channels suffering from low signal-to-noise ratios (SNR). We address the application of turbo codes and LDPC codes to magneto-optical (MO) recording channels. Our results focus on a variety of practical MO storage channel aspects, including storage density, partial response targets, the type of precoder used, and mark edge jitter. Instead of focusing just on bit error rates (BER), we also study the block error statistics. Our results for MO storage channels indicate that turbo codes of rate 16/17 can achieve coding gains of 3-5 dB over partial response maximum likelihood (PRML) methods for a 10^-4 target BER. Simulations also show that the performance of LDPC codes for MO channels is comparable to that of turbo codes, while requiring less computational complexity. Both LDPC codes and turbo codes with iterative decoding are seen to be robust to mark edge jitter     

Turbo decoding as an instance of Pearl's “beliefpropagation” algorithm

We describe the close connection between the now celebrated iterative turbo decoding algorithm of Berrou et al. (1993) and an algorithm that has been well known in the artificial intelligence community for a decade, but which is relatively unknown to information theorists: Pearl's (1982) belief propagation algorithm. We see that if Pearl's algorithm is applied to the “belief network” of a parallel concatenation of two or more codes, the turbo decoding algorithm immediately results. Unfortunately, however, this belief diagram has loops, and Pearl only proved that his algorithm works when there are no loops, so an explanation of the experimental performance of turbo decoding is still lacking. However, we also show that Pearl's algorithm can be used to routinely derive previously known iterative, but suboptimal, decoding algorithms for a number of other error-control systems, including Gallager's (1962) low-density parity-check codes, serially concatenated codes, and product codes. Thus, belief propagation provides a very attractive general methodology for devising low-complexity iterative decoding algorithms for hybrid coded systems     

Efficient encoding of low-density parity-check codes

Low-density parity-check (LDPC) codes can be considered serious competitors to turbo codes in terms of performance and complexity and they are based on a similar philosophy: constrained random code ensembles and iterative decoding algorithms. We consider the encoding problem for LDPC codes. More generally we consider the encoding problem for codes specified by sparse parity-check matrices. We show how to exploit the sparseness of the parity-check matrix to obtain efficient encoders. For the (3,6)-regular LDPC code, for example, the complexity of encoding is essentially quadratic in the block length. However, we show that the associated coefficient can be made quite small, so that encoding codes even of length n≃100000 is still quite practical. More importantly, we show that “optimized” codes actually admit linear time encoding     

新的降低复杂度的Turbo均衡

Turbo均衡是一种通过反复均衡和信道译码来提高接收性能的迭代接收机算法。通常的Turbo均衡算法采用均衡与软输出译码的迭代运算，由于均衡和译码的重复计算，使得复杂度大大提高。文中提出了2种降低复杂度的Turbo均衡器：第一种采用软判决维特比译码，第二种采用软输入硬输出的维特比译码。通过仿真表明，这2种算法在几乎没有损失接收性能的情况下，大大降低了计算复杂度，并且第二种的性能要好于第一种。     

Turbo码概率-代数联合解码算法

Turbo码采用修正的BAHL et al.算法实现解码.这是一种基于软值的概率迭代解码算法.本文在保持Turbo码迭代软解码算法优点的基础上,充分利用Turbo码编码器结构这一确知条件,结合代数解码原理,提出了一种Turbo码概率-代数联合解码算法.该算法结合了概率解码和代数解码的优点,又有效避免了误差传播的发生,使Turbo码的纠错性能在原经典算法的基础上得到进一步的提高.该算法不仅为降低Turbo码的比特误码率和误差地板值提供了一种新的研究途径,而且因其更好的纠错性能而具有十分明显的实用价值.仿真实验结果显示,在比特误码率(BER)为10^-3~10^-4时,与经典Turbo码解码算法相比,采用该算法能获得0.1dB左右的编码增益.     

Information geometry of turbo and low-density parity-check codes

Since the proposal of turbo codes in 1993, many studies have appeared on this simple and new type of codes which give a powerful and practical performance of error correction. Although experimental results strongly support the efficacy of turbo codes, further theoretical analysis is necessary, which is not straightforward. It is pointed out that the iterative decoding algorithm of turbo codes shares essentially similar ideas with low-density parity-check (LDPC) codes, with Pearl's belief propagation algorithm applied to a cyclic belief diagram, and with the Bethe approximation in statistical physics. Therefore, the analysis of the turbo decoding algorithm will reveal the mystery of those similar iterative methods. In this paper, we recapture and extend the geometrical framework initiated by Richardson to the information geometrical framework of dual affine connections, focusing on both of the turbo and LDPC decoding algorithms. The framework helps our intuitive understanding of the algorithms and opens a new prospect of further analysis. We reveal some properties of these codes in the proposed framework, including the stability and error analysis. Based on the error analysis, we finally propose a correction term for improving the approximation.     

Non-sequential decoding algorithm for hard iterative turbo product codes - [transactions letters]

In this letter, we propose a new decoding algorithm to improve the bit error rate performance of the hard-input hard-output (HIHO) turbo product codes (TPC) with hard iterative decoding. The proposed algorithm iteratively, but not sequentially, decodes the received TPC blocks based on the reliability of the constituent codes. Simulation results confirm a noticeable coding gain improvement using the proposed decoding process with respect to standard HIHO TPC decoding. An efficient implementation of the new technique offers a negligible additional complexity when the channel-bit error probability is less than 10^?2.     

APPLICATION OF TURBO CODES IN HIGH-SPEED REAL-TIME CHANNEL

I. Introduction Turbo code has obtained comprehensive atten-tion and research due to its near-Shannon perform-ance since it was proposed in 1993[1], and has be-come a research hotspot in information and coding theory area. Application and realization methods of turbo codes in various communication systems have also attracted great interest of researchers. The good BER performance of turbo codes provides it a wide application prospect in deep space and mobile com- munication systems. The IT…     

Lower bound on BER of finite-length turbo codes based on EXIT characteristics

We present a generalized extrinsic information transfer characteristics of the iterative turbo decoding algorithm. By using this tool, we obtain a lower bound on the bit error rate performance of the finite-length turbo codes showing a gentle waterfall over a wide waterfall region.     

Iterative decoding with a hamming threshold for block turbo codes

This work introduces a novel approach to increase the performance of block turbo codes (BTCs). The idea is based on using a Hamming threshold to limit the search for the maximum-likelihood (ML) codeword within only those codewords that lie within this threshold. The proposed iterative decoding approach is shown to offer both significant coding gain and complexity reduction over the standard iterative decoding methods.     

Iterative Decoding Using Optimum Soft Input - Hard Output Module

Maximum likelihood sequence estimation (MLSE) in the shape of the Viterbi algorithm, although a strictly hard-output approach, is utilized for iterative decoding of turbo codes. It is a practical alternative when decoding complexity and speed may be traded for performance.