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     

Study of energy and performance of space-time decoding systems in concatenation with turbo decoding

Recent studies have shown that using space-time code is an effective approach to increase the data rate over wireless channels. Space-time turbo (ST-Turbo) codes formed by concatenating space-time codes with turbo codes, take advantage of both the high diversity order of space-time systems and the randomness of the turbo codes. In this paper, we compare two ST-Turbo codes, i.e., simple space-time turbo codes (SiSTT) and turbo trellis-coded modulation space-time block codes (TTCM-STBCs), and their approximate versions with respect to performance and energy consumption for both general-purpose processor and synthesized implementations. The approximations are aimed at reducing the computational complexity and include reduction in the number of paths, number of iterations, and datapath computations. Analysis of the simulation results show that SiSTT-based versions should be used for higher SNR applications where low energy consumption is the primary design objective, and TTCM-STBC-based versions should be used where performance is the primary design objective. Finally, four ST-Turbo algorithms (i.e., baseline SiSTT and its energy-efficient approximate version and the baseline TTCM-STBC and its energy-efficient approximate version) have been synthesized in 0.18-/spl mu/m CMOS technology and the implementations compared with respect to area, power, and latency.     

RS码与LDPC码的联合迭代译码

针对RS码与LDPC码的串行级联结构,提出了一种基于自适应置信传播(ABP)的联合迭代译码方法.译码时,LDPC码置信传播译码器输出的软信息作为RS码ABP译码器的输入;经过一定迭代译码后,RS码译码器输出的软信息又作为LDPC译码器的输入.软输入软输出的RS译码器与LDPC译码器之间经过多次信息传递,译码性能有很大提高.码长中等的LDPC码采用这种级联方案,可以有效克服短环的影响,消除错误平层.仿真结果显示:AWGN信道下这种基于ABP的RS码与LDPC码的联合迭代译码方案可以获得约0.8 dB的增益.     

VLSI architectures for turbo codes

A great interest has been gained in recent years by a new error-correcting code technique, known as “turbo coding”, which has been proven to offer performance closer to the Shannon's limit than traditional concatenated codes. In this paper, several very large scale integration (VLSI) architectures suitable for turbo decoder implementation are proposed and compared in terms of complexity and performance; the impact on the VLSI complexity of system parameters like the state number, number of iterations, and code rate are evaluated for the different solutions. The results of this architectural study have then been exploited for the design of a specific decoder, implementing a serial concatenation scheme with 2/3 and 3/4 codes; the designed circuit occupies 35 mm², supports a 2 Mb/s data rate, and for a bit error probability of 10^-6, yields a coding gain larger than 7 dB, with ten iterations     

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     

An FPGA Implementation of High‐Speed Flexible 27‐Mbps 8‐StateTurbo Decoder

In this paper, we propose a flexible turbo decoding algorithm for a high order modulation scheme that uses a standard half‐rate turbo decoder designed for binary quadrature phase‐shift keying (B/QPSK) modulation. A transformation applied to the incoming I‐channel and Q‐channel symbols allows the use of an off‐the‐shelf B/QPSK turbo decoder without any modifications. Iterative codes such as turbo codes process the received symbols recursively to improve performance. As the number of iterations increases, the execution time and power consumption also increase. The proposed algorithm reduces the latency and power consumption by combination of the radix‐4, dual‐path processing, parallel decoding, and early‐stop algorithms. We implement the proposed scheme on a field‐programmable gate array and compare its decoding speed with that of a conventional decoder. The results show that the proposed flexible decoding algorithm is 6.4 times faster than the conventional scheme.     

A 690-mW 1-Gb/s 1024-b, rate-1/2 low-density parity-check codedecoder

A 1024-b, rate-1/2, soft decision low-density parity-check (LDPC) code decoder has been implemented that matches the coding gain of equivalent turbo codes. The decoder features a parallel architecture that supports a maximum throughput of 1 Gb/s while performing 64 decoder iterations. The parallel architecture enables rapid convergence in the decoding algorithm to be translated into low decoder switching activity resulting in a power dissipation of only 690 mW from a 1.5-V supply     

Comparative study of turbo decoding techniques: an overview

We provide an overview of the novel class of channel codes referred to as turbo codes, which have been shown to be capable of performing close to the Shannon limit. We commence with a discussion on turbo encoding, and then move on to describing the form of the iterative decoder most commonly used to decode turbo codes. We then elaborate on various decoding algorithms that can be used in an iterative decoder, and give an example of the operation of such a decoder using the so-called soft output Viterbi (1996) algorithm (SOVA). Lastly, the effect of a range of system parameters is investigated in a systematic fashion, in order to gauge their performance ramifications     

Downlink transmission of broadband OFCDM systems-part IV: soft decision

In this paper, the performance of turbo-coded orthogonal frequency and code-division multiplexing (OFCDM) systems is investigated with soft multicode interference (MCI) cancellation and minimum mean-square error (MMSE) detection for downlink transmission in future high-speed wireless communications. To regenerate the soft interference signal, the conventional turbo decoding algorithm must be modified to provide log-likelihood ratio (LLR) values for all coded bits. Based on the LLR outputs of turbo decoder, two soft-decision functions are proposed, called LLR-soft-decision and Gaussian-soft-decision functions. The Gaussian assumptions used for deriving these two soft functions are verified by simulation results, and simple methods are proposed to estimate parameters used in the soft functions in practical systems. By means of computer simulations, the performance of soft MCI cancellation is studied extensively and compared with that of hard ones. It is shown that in a highly frequency-selective channel, soft MCI cancellation and MMSE detection can significantly improve the performance of turbo-coded OFCDM systems. Two iterations in turbo decoding are sufficient for both hard and soft-decision functions. The proposed soft-decision functions outperform the hard-decision function with various channel conditions and system parameters, such as the channel correlation, the quality of channel estimation, the number of iterations in turbo decoding and the frequency-domain spreading factor (N/sub F/). Furthermore, the Gaussian-soft-decision function provides better performance than the LLR-soft-decision function. Finally, although frequency diversity gain is saturated for large channel correlation when N/sub F/ is large as in , the gain increases further with increasing N/sub F/ for small channel correlation even when N/sub F/ is large.     

Improved Turbo Decoding Using Soft Combining Principle

One great challenge in wireless communication systems is to ensure reliable communications. Turbo codes are known by their interesting capabilities to deal with transmission errors. In this paper, we present a novel turbo decoding scheme based on soft combining principle. Our method improves decoding performance using soft combining technique inside the turbo decoder. Working on Max-Log-Maximum a Posteriori (Max-Log-MAP) turbo decoding algorithm and using an Additive White Gaussian Noise (AWGN) channel model and 16 Quadrature Amplitude Modulation (16QAM), simulation results show that the suggested solution is efficient and outperforms the conventional Max-Log-MAP algorithm in terms of Bit Error Rate (BER). The performance analysis is carried out in terms of BER by varying parameters such as the Energy per bit to Noise power spectral density ratio ( \(\text {E}_{\text {b}}/\text {N}_{\text {o}}\) ), and decoding iterations number. We call our proposed solution Soft Combined Turbo Codes.     

Turbo processing in transmit antenna diversity systems

We consider turbo-trellis-coded transmission over fading multiple-input-multiple-output (M1M0) channels with transmit diversity using space-time block codes. We give a new view on space-time block codes as a transformation of the fading MIMO channel towards a Gaussian single-input-single-output (siso) channel and provide analytical results on the BER of space-time block codes. Furthermore, we describe the concatenation of Turbo-TCM with a space-time block code and show that in addition to the transmit diversity substantial benefits can be obtained by turbo iterations as long as the channel is time-varying during transmission of a coded block or frequency hopping is applied. Finally, a double iterative scheme for turbo equalization and turbo decoding of the concatenation of Turbo-TCM and space-time block code in frequency-selective MIMO channels is described.     

FPGA implementation of low complexity LDPC iterative decoder

Low-density parity-check (LDPC) codes, proposed by Gallager, emerged as a class of codes which can yield very good performance on the additive white Gaussian noise channel as well as on the binary symmetric channel. LDPC codes have gained lots of importance due to their capacity achieving property and excellent performance in the noisy channel. Belief propagation (BP) algorithm and its approximations, most notably min-sum, are popular iterative decoding algorithms used for LDPC and turbo codes. The trade-off between the hardware complexity and the decoding throughput is a critical factor in the implementation of the practical decoder. This article presents introduction to LDPC codes and its various decoding algorithms followed by realisation of LDPC decoder by using simplified message passing algorithm and partially parallel decoder architecture. Simplified message passing algorithm has been proposed for trade-off between low decoding complexity and decoder performance. It greatly reduces the routing and check node complexity of the decoder. Partially parallel decoder architecture possesses high speed and reduced complexity. The improved design of the decoder possesses a maximum symbol throughput of 92.95 Mbps and a maximum of 18 decoding iterations. The article presents implementation of 9216 bits, rate-1/2, (3, 6) LDPC decoder on Xilinx XC3D3400A device from Spartan-3A DSP family.     

Distributed Arithmetic Coding

We propose a distributed binary arithmetic coder for Slepian-Wolf coding with decoder side information, along with a soft joint decoder. The proposed scheme provides several advantages over existing schemes, and its performance is equal to or better than that of an equivalent scheme based on turbo codes at short and medium block lengths.     

Achieving near-capacity on a multiple-antenna channel

Recent advancements in iterative processing of channel codes and the development of turbo codes have allowed the communications industry to achieve near-capacity on a single-antenna Gaussian or fading channel with low complexity. We show how these iterative techniques can also be used to achieve near-capacity on a multiple-antenna system where the receiver knows the channel. Combining iterative processing with multiple-antenna channels is particularly challenging because the channel capacities can be a factor of ten or more higher than their single-antenna counterparts. Using a "list" version of the sphere decoder, we provide a simple method to iteratively detect and decode any linear space-time mapping combined with any channel code that can be decoded using so-called "soft" inputs and outputs. We exemplify our technique by directly transmitting symbols that are coded with a channel code; we show that iterative processing with even this simple scheme can achieve near-capacity. We consider both simple convolutional and powerful turbo channel codes and show that excellent performance at very high data rates can be attained with either. We compare our simulation results with Shannon capacity limits for ergodic multiple-antenna channel.     

A low-complexity noncoherent iterative space-time demodulator

Turbo coded unitary space-time modulation (USTM) can provide large coding gain as compared to uncoded USTM. Because the noncoherent space-time maximum a posteriori demodulator is very complicated, in this letter, we propose a new low-complexity noncoherent iterative space-time demodulator for the USTM constructed from pilot symbol-assisted modulation. The proposed demodulator utilizes both hard and soft decisions from the turbo decoder to simplify the computational task as well as produce reliable soft outputs. Several examples demonstrate that this demodulator has both low complexity and good error performance.     

Interleaver design for turbo codes

The performance of a turbo code with short block length depends critically on the interleaver design. There are two major criteria in the design of an interleaver: the distance spectrum of the code and the correlation between the information input data and the soft output of each decoder corresponding to its parity bits. This paper describes a new interleaver design for turbo codes with short block length based on these two criteria. A deterministic interleaver suitable for turbo codes is also described. Simulation results compare the new interleaver design to different existing interleavers     

Iterative decoding and channel parameter estimation algorithms for repeat-accumulate codes

The sensitivity of the iterative decoder for repeat-accumulate (RA) codes to carrier phase and channel signal-to-noise ratio estimation errors is investigated, and efficient algorithms to estimate and correct these errors are developed. The behavior of RA codes with imperfect channel estimation is different from that of turbo codes, and correction algorithms specific to RA codes must be formulated. The proposed algorithms use the soft information generated within the iterative decoder, and thus, are not only hardware-efficient, but also offer excellent performance.     

Space-time codes and concatenated channel codes for wirelesscommunications

Following a brief historical perspective on channel coding, an introduction to space-time block codes is given. The various space-time codes considered are then concatenated with a range of channel codecs, such as convolutional and block-based turbo codes as well as conventional and turbo trellis codes. The associated estimated complexity issues and memory requirements are also considered. These discussions are followed by a performance study of various space-time and channel-coded transceivers. Our aim is first to identify a space-time code/channel code combination constituting a good engineering tradeoff in terms of its effective throughput, bit-error-rate performance, and estimated complexity. Specifically, the issue of bit-to-symbol mapping is addressed in the context of convolutional codes (CCs) and convolutional coding as well as Bose-Chaudhuri-Hocquenghem coding-based turbo codes in conjunction with an attractive unity-rate space-time code and multilevel modulation is detailed. It is concluded that over the nondispersive or narrow-band fading channels, the best performance versus complexity tradeoff is constituted by Alamouti's twin-antenna block space-time code concatenated with turbo convolutional codes. Further comparisons with space-time trellis codes result in similar conclusions     

Turbo space-time codes with time varying linear transformations

Turbo space-time codes with symbols precoded by randomly chosen unitary time variant linear transformations (TVLT) are investigated in this paper. It is shown that turbo codes with TVLT achieve full diversity gain and do not require exhaustive tests of the rank criterion. We prove that the coding gain performance of turbo space-time codes with TVLT improves with the Hamming distance between codewords (number of different columns). As an additional benefit of the application of TVLT, with the removal of the constant modulation condition, we prove that throughput rates achieved by these codes are significantly higher than those for conventional space-time codes. Numerical results are provided to demonstrate diversity gains, coding gains and rates of turbo space-time codes with TVLT     

Convergence behavior of iteratively decoded parallel concatenatedcodes

Mutual information transfer characteristics of soft in/soft out decoders are proposed as a tool to better understand the convergence behavior of iterative decoding schemes. The exchange of extrinsic information is visualized as a decoding trajectory in the extrinsic information transfer chart (EXIT chart). This allows the prediction of turbo cliff position and bit error rate after an arbitrary number of iterations. The influence of code memory, code polynomials as well as different constituent codes on the convergence behavior is studied for parallel concatenated codes. A code search based on the EXIT chart technique has been performed yielding new recursive systematic convolutional constituent codes exhibiting turbo cliffs at lower signal-to-noise ratios than attainable by previously known constituent codes