Linear subcodes of turbo codes with improved distance spectra

In this correspondence, we present a technique for generation of linear subcodes of a given turbo code with better distance spectrum than the original mother turbo code, via an iterative process of trace-bit injection which minimally reduces code rate, followed by selective puncturing that allows recovery of the rate loss incurred during the trace-bit injection. The technique allows for asymptotic performance improvement of any linear turbo code. In effect, we trim the distance spectrum of a turbo code via elimination of the low distance and/or high multiplicity codewords from the output space of the code. To this end, we perform a greedy minimization of a cost function closely related to the asymptotic bit error probability (or frame error probability) of the code. This improves the performance of the code everywhere, but its main impact is a reduction in the error floor of the turbo code which is important for delay constrained applications employing short interleavers.     

On the theory and performance of trellis termination methods forturbo codes

The performance of a turbo code can be severely degraded if no trellis termination is employed. This paper investigates the implications of the choice of trellis termination method for turbo codes, and explains the origin of the performance degradation often experienced without trellis termination. An efficient method to derive the distance spectrum of turbo codes for different trellis termination methods is presented. Further, we present interleaver design rules that are tailored to each termination method. Using interleavers designed with these restrictions, we demonstrate that the performance difference between various termination methods is very small, including no trellis termination at all. For example, we demonstrate a turbo code with a 500-bit interleaver that exhibits no sign of an error floor for frame error rates as low as 10^-8, even though no trellis termination is employed     

Improving the distance properties of turbo codes using a third component code: 3D turbo codes - [transactions letters]

Thanks to the probabilistic message passing performed between its component decoders, a turbo decoder is able to provide strong error correction close to the theoretical limit. However, the minimum Hamming distance (d_min) of a turbo code may not be sufficiently large to ensure large asymptotic gains at very low error rates (the so-called flattening effect). Increasing the d_min of a turbo code may involve using component encoders with a large number of states, devising more sophisticated internal permutations, or increasing the number of component encoders. This paper addresses the latter option and proposes a modified turbo code in which a fraction of the parity bits are encoded by a rate-1, third encoder. The result is a noticeably increased d_min, which improves turbo decoder performance at low error rates. Performance comparisons with turbo codes and serially concatenated convolutional codes are given.     

A Bandwidth Efficient Turbo Coding Scheme for VDSL Systems

The current forward error correction (FEC) scheme for very high bit-rate digital subscriber line (VDSL) systems in the ANSI standard employs a 16-state four-dimensional (4D) Wei code as the inner code and the Reed-Solomon (RS) code as the outer code. The major drawback of this scheme is that further improvement cannot be achieved without a substantial increase in the complexity and power penalty. Also, a VDSL system employing the 4D Wei-RS scheme operates far below the channel capacity. In 1993, powerful turbo codes were introduced whose performance closely approaches the Shannon limit. In this paper, we propose a bandwidth and power efficient turbo coding scheme for VDSL modems in order to obtain high data rates, extended loop reach and increased transmission robustness. We also propose a pipelined decoding scheme to reduce the latency at the receiver end. The objective of the proposed scheme is to provide a higher coding gain than that given by the 4D Wei-RS scheme, resulting in an improved performance of the VDSL modems in terms of bit rate, loop length and transmitting power. The scheme is investigated for various values of transmitting power, signaling frequencies and numbers of crosstalkers for a targeted bit error rate of 10⁻⁵ and is implemented in a system with a quadrature amplitude modulation in which a mixed set partitioning mapping is employed to reduce the decoding complexity. The effects of code complexity, interleaver length, the number of decoding iterations and the level of modulation on the performance of VDSL modems are explored. Simulation results are presented and compared to those of the 4D Wei-RS scheme. The results show that the choice of turbo codes not only provides a significant coding gain over the standard FEC scheme but also efficiently maximizes the loop length and bit rate at a very low transmitting power in the presence of dominant far-end crosstalk and intersymbol interference. In order to compare the hardware complexity, we synthesize the proposed and 4D Wei-RS schemes using SYNOPSYS with the target technology of Xilinx 4020e-3. The Xilinx field programmable gate array statistics of the proposed scheme is compared with that of the 4D Wei-RS scheme.     

Performance of a turbo coded multicarrier DS/CDMA system withnonuniform repetition coding

This paper presents a turbo coded multicarrier direct sequence code division multiple access (DS/CDMA) system, where the outputs of a turbo encoder are repetition coded at multiple rates and transmitted in parallel over a number of subchannels. A performance bound useful in the so-called error floor region is obtained for Rayleigh fading channels when different diversity orders are given to each turbo code symbol. Simulation results are also provided for the low signal-to-noise ratio (SNR) region where the bound is not applicable. It is observed that the error floor can be lowered, with some performance loss in the low SNR region, by applying nonuniform repetition coding to the turbo code symbols     

Near‐Optimum Blind Decision Feedback Equalization for ATSC Digital Television Receivers

This paper presents a near‐optimum blind decision feedback equalizer (DFE) for the receivers of Advanced Television Systems Committee (ATSC) digital television. By adopting a modified trellis decoder (MTD) with a trace‐ back depth of 1 for the decision device in the DFE, we obtain a hardware‐efficient, blind DFE approaching the performance of an optimum DFE which has no error propagation. In the MTD, the absolute distance is used rather than the squared Euclidean distance for the computation of the branch metrics. This results in a reduction of the computational complexity over the original trellis decoding scheme. Compared to the conventional slicer, the MTD shows an outstanding performance improvement in decision error probability and is comparable to the original trellis decoder using the Euclidean distance. Reducing error propagation by use of the MTD in the DFE leads to the improvement of convergence performance in terms of convergence speed and residual error. Simulation results show that the proposed blind DFE performs much better than the blind DFE with the slicer, and the difference is prominent at the trellis decoder following the blind DFE.     

A turbo code interleaver design criterion based on the performanceof iterative decoding

The performance of a turbo code is dependent on two code properties: its distance spectrum and its suitability to be iteratively decoded. Both these properties are influenced by the choice of interleaver used in the turbo encoder. This paper presents an interleaver design criterion that focuses on the performance of iterative decoding, based on the correlation properties of the extrinsic inputs. Interleavers designed with the proposed criterion achieve very competitive performances, both in terms of convergence rates and error correcting capabilities     

Turbo codes-based image transmission for channels with multiple types of distortion

Product codes are generally used for progressive image transmission when random errors and packet loss (or burst errors) co-exist. However, the optimal rate allocation considering both component codes gives rise to high-optimization complexity. In addition, the decoding performance may be degraded quickly when the channel varies beyond the design point. In this paper, we propose a new unequal error protection (UEP) scheme for progressive image transmission by using rate-compatible punctured Turbo codes (RCPT) and cyclic redundancy check (CRC) codes only. By sophisticatedly interleaving each coded frame, the packet loss can be converted into randomly punctured bits in a Turbo code. Therefore, error control in noisy channels with different types of errors is equivalent to dealing with random bit errors only, with reduced turbo code rates. A genetic algorithm-based method is presented to further reduce the optimization complexity. This proposed method not only gives a better performance than product codes in given channel conditions but is also more robust to the channel variation. Finally, to break down the error floor of turbo decoding, we further extend the above RCPT/CRC protection to a product code scheme by adding a Reed-Solomon (RS) code across the frames. The associated rate allocation is discussed and further improvement is demonstrated.     

Maximal diversity algebraic space-time codes with low peak-to-mean power ratio

The design requirements for space-time coding typically involves achieving the goals of good performance, high rates, and low decoding complexity. In this paper, we introduce a further constraint on space-time code design in that the code should also lead to low values of the peak-to-mean envelope power ratio (PMEPR) for each antenna. Towards that end, we propose a new class of space-time codes called the "low PMEPR space-time" (LPST) codes. The LPST codes are obtained using the properties of certain cyclotomic number fields. The LPST codes achieve a performance identical to that of the threaded algebraic space-time (TAST) codes but at a much smaller PMEPR. With M antennas and a rate of one symbol per channel use, the LPST codes lead to a decrease in PMEPR by at least a factor of M relative to a Hadamard spread version of the TAST code. For rates beyond one symbol per channel use and up to a guaranteed amount, the LPST codes have provably smaller PMEPR than the corresponding TAST codes. Additionally, with the concept of punctured LPST codes proposed in this paper, significant performance improvement is obtained over the full diversity TAST schemes of comparable complexity. Numerical examples are provided to illustrate the advantage of the proposed codes in terms of PMEPR reduction and performance improvement for very high rate wireless communications.     

Power Optimization of Three Dimensional Turbo Code Using a Novel Modified Symbiotic Organism Search (MSOS) Algorithm

In modern communication system error-control coding scheme is used to elevate the immunity of noisy communication channel. Turbo code (TC) is considered as one of the significant channel coding schemes which approaches to the Shannon limit. An upgraded version of TC named as 3 dimensional turbo code (3D-TC) has been emerged as a challenging research area in recent past. Meanwhile, considerable improvement in bit error rate (BER) performance of the TC has been achieved by incorporating suitable optimization algorithms. Motivated by above research trends, a modified symbiotic organisms search (MSOS) algorithm has been proposed by changing the organism structure and selection criteria of a newly developed symbiotic organisms search (SOS) algorithm. Subsequently the proposed MSOS has been used to design an improved 3D-TC. Here an optimal power allocation scheme of a new class of 3 dimensional turbo encoder has been investigated to improve its BER characteristics mainly in high SNR regions. Furthermore, the BER performance of the proposed 3D-TC code has been compared with conventional 2D serially concatenated and parallel concatenated turbo code as well as conventional 3D-TC. Finally, the BER performance of the proposed MSOS optimized 3D-TC has been compared with the SOS optimized 3D-TC and harmony search optimized 3D-TC.     

Performance analysis of coded space‐time adaptive detection in DS/CDMA systems over Rayleigh fading channels

In this paper, we study the use of channel coding in a direct‐sequence code‐division multiple‐access (DS‐CDMA) system that employs space‐time adaptive minimum‐mean square‐error (MMSE) interference suppression over Rayleigh fading channels. It is shown that the employment of adaptive antenna arrays at the receiver can assist in attenuating multiuser interference and at the same time speeds‐up the convergence rate of the adaptive receiver. In this work, we assess the accuracy of the theoretical results developed for the uncoded and convolutionally coded space‐time multiuser detector when applied to the adaptive case. It is found that the use of antenna arrays brings the receiver performance very close to its multiuser counterpart. Using performance error bounds, we show that a user‐capacity gain of approximately 200% can easily be achieved for the space‐time adaptive detector when used with a rate 1/2 convolutional code (CC) and a practical channel interleaver. This capacity gain is only 10% less than the gain achieved for the more complicated multiuser‐based receiver. Finally, we perform a comparison between convolutional and turbo coding where we find that the latter outperforms the former at all practical bit‐error rates (BER). Copyright © 2006 John Wiley & Sons, Ltd.     

Combined turbo codes and interleaver design

The impact of the distance spectrum and interleaver structure on the bit error probability of turbo codes is considered. A new turbo code design method for Gaussian channels is presented. The proposed method combines a search for good component codes with interleaver design. The optimal distance spectrum is used as the design criterion to construct good turbo component codes at low signal-to-noise ratios (SNRs). In addition, an interleaver design method is proposed. This design improves the code performance at high SNR. Search for good component codes at low SNR is combined with a code matched interleaver design. This results in new turbo codes with a superior error performance relative to the best known codes at both low and high SNR. The performance is verified by both analysis and simulation     

Error Correcting Encoders for Parallel-Type ADCs

Full-parallel and folding and interpolation (F&;I) ADCs have a bank of latched comparators, the output of which is a “thermometer code” in full-parallel ADCs and a “circular code” in F&;I ADCs. These special codes have to be transformed into a binary code. Nonideal effects can originate more than one “0” to “1” transition in the thermometer or circular code, causing large errors in the output binary code. A comparison of existing error correction techniques is presented in this paper, which shows that 1st order error correction provides considerable performance improvement, without a significant increase of circuit complexity. Higher order correction in full-parallel ADCs using Wallace tree encoding is very effective, but has a high cost in terms of circuit area and power consumption. In this paper it is shown that Wallace tree encoding can be extended to F&;I ADCs, and that this leads to a very compact circuit for higher order error correction in low and medium resolution ADCs. This is demonstrated by designing and simulating a 6-bit, 200 Msample/s F&;I ADC which has low power consumption and a small area.     

Design and analysis of simple irregular LDPC codes with finite‐lengths and their application in CDMA systems

In this paper, we design and optimize simple irregular low‐density parity‐check (LDPC) codes for finite‐length applications where the asymptotic noise threshold of the channel cannot play as a dominant optimization factor. Our design procedure is based on some observations resulted from analytical study of these codes. Although we present our design procedure for some specified rates but it can generally be used for any rate. Specifically, we design a simple irregular LDPC code for IS‐95 and compare its performance with the other reported codes 1 - 3 for this application. Our results show a 3.7‐fold increase in the capacity at bit error rate (BER) equal to 10⁻⁵ compared to the low‐rate orthogonal convolutional codes and 1.2 times increase compared to a high performance LDPC code of Reference 3 . Copyright © 2004 John Wiley & Sons, Ltd.     

The performance of trellis-coded MDPSK with multiple symboldetection

The idea of using a multiple (more than two) symbol observation interval to improve error probability performance is applied to differential detection of trellis-coded multiple phase-shift keying (MPSK) over an additive white Gaussian noise (AWGN) channels. An equivalent Euclidean distance measure per trellis branch is determined for this detection scheme. This is used to define an augmented (larger multiplicity) trellis code whose distance measure is the conventional squared Euclidean distance typical of conventional trellis-coded modulation on the AWGN. Such an augmented multiple trellis code is a convenient mathematical tool for simplifying the analysis. Results are obtained by a combination of analysis and computer simulation. It is shown that only a slight increase (e.g. one symbol) in the length of the observation interval will provide a significant improvement in bit error probability performance     

Construction and generation of OCDMA code families using a complete row-wise orthogonal pairs algorithm

A new code construction algorithm for incoherent Multi-Dimensional Optical Code Division Multiple Access (MD-OCDMA) for asynchronous fiber optic communication is proposed. We refer multi-dimensionality to two-dimensional (2D) wavelength–time or space–time domains and three-dimensional (3D) space–wavelength–time domains. The application of the algorithm in constructing 2D multiple pulses per row codes and 3D multiple pulses per plane codes is given. The performance of the codes is discussed. In the applications discussed, this construction ensures a maximum crosscorrelation of 1 between any two codes. The proposed codes have complete 1D code allocation, which increases the cardinality. The performance of some codes in literature is compared with the proposed codes. The analyzed performance measure is bit error rate due to multiple access interference for different numbers of active users. The performance analysis shows that the proposed 2D construction offers very low bit error rate at lower spectral efficiency when compared with other 2D constructions. A comparison of the proposed 3D construction with existing 3D constructions shows lower bit error rate for equivalent code dimension. New integrated optic designs for the generation of OCDMA codes using titanium indiffused lithium niobate technology are explored, which can enable compact encoders and decoders for computer communications.     

Some block- and trellis-coded modulations for the Rayleigh fadingchannel

The error performance of a modulation code over a channel depends on several distance parameters and the path multiplicity of the code. For the AWGN channel, the error performance of a modulation code depends mainly on its minimum squared Euclidean distance and path multiplicity. For the Rayleigh fading channel, however, the error performance of a modulation code depends strongly on its minimum symbol distance, minimum product distance, and path multiplicity. It depends on the minimum squared Euclidean distance in a lesser degree. This paper is concerned with the construction of block and trellis MPSK modulation codes for the Rayleigh fading channel. In each construction, the distance parameters are chosen to achieve good error performance with reduced decoding complexity     

Unit-rate complex orthogonal space-time block code concatenated with turbo coding

Space-Time Block （STB） code has been an effective transmit diversity technique for combating fading due to its orthogonal design, simple decoding and high diversity gains. In this paper, a unit-rate complex orthogonal STB code for multiple antennas in Time Division Duplex （TDD） mode is proposed. Meanwhile, Turbo Coding （TC） is employed to improve the performance of proposed STB code further by utilizing its good ability to combat the burst error of fading channel. Compared with full-diversity multiple antennas STB codes, the proposed code can implement unit rate and partial diversity; and it has much smaller computational complexity under the same system throughput. Moreover, the application of TC can effectively make up for the performance loss due to partial diversity. Simulation results show that on the condition of same system throughput and concatenation of TC, the proposed code has lower Bit Error Rate （BER） than those full-diversity codes.     

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.     

Trellis code‐aided high‐rate M‐QAM space‐time labeling diversity using a unitary expansion

In this paper, we present a high‐rate M‐ary quadrature amplitude modulation (M‐QAM) space‐time labeling diversity (STLD) system that retains the robust error performance of the conventional STLD system. The high‐rate STLD is realised by expanding the conventional STLD via a unitary matrix transformation. Robust error performance of the high‐rate STLD is achieved by incorporating trellis coding into the mapping of additional bits to high‐rate codes. The comparison of spectral efficiency between the proposed trellis code‐aided high‐rate STLD (TC‐STLD) and the conventional STLD shows that TC‐STLD with 16‐QAM and 64‐QAM respectively achieves a 12.5% and 8.3% increase in spectral efficiency for each additional bit sent with the transmitted high‐rate codeword. Moreover, we derive an analytical bound to predict the average bit error probability performance of TC‐STLD over Rayleigh frequency‐flat fading channels. The analytical results are verified by Monte Carlo simulation results, which show that the derived analytical bounds closely predict the average bit error probability performance at high signal‐to‐noise ratios (SNR). Simulation results also show that TC‐STLD with 1 additional bit achieves an insignificant SNR gain of approximately 0.05 dB over the conventional STLD, while TC‐STLD with 2 additional bits achieves an SNR gain of approximately 0.12 dB.