Turbo Source Extraction Algorithm and Noncancellation Source Separation Algorithms by Kurtosis Maximization

The kurtosis maximization criterion has been effectively used for blind spatial extraction of one source from an instantaneous mixture of multiple non-Gaussian sources, such as the kurtosis maximization algorithm by Ding and Nguyen, and the fast kurtosis maximization algorithm (FKMA) by Chi and Chen. By empirical studies, we found that the smaller the normalized kurtosis magnitude of the extracted source signal, the worse the performance of these algorithms. In this paper, with the assumption that each source is a non-Gaussian linear process, a novel blind source extraction algorithm, called turbo source extraction algorithm (TSEA), is proposed. The ideas of the TSEA are to exploit signal temporal properties for increasing the normalized kurtosis magnitude, and to apply spatial and temporal processing in a cyclic fashion to improve the signal extraction performance. The proposed TSEA not only outperforms the FKMA, but also shares the convergence and computation advantages enjoyed by the latter. This paper also considers the extraction of multiple sources, also known as source separation, by incorporating the proposed TSEA into the widely used multistage successive cancellation (MSC) procedure. A problem with the MSC procedure is its susceptibility to error propagation accumulated at each stage. We propose two noncancellation multistage (NCMS) algorithms, referred to as NCMS-FKMA and NCMS-TSEA, that are free from the error propagation effects. Simulation results are presented to show that the NCMS-TSEA yields substantial performance gain compared with some existing blind separation algorithms, together with a computational complexity comparison. Finally, we draw some conclusions.     

Low-Complexity Block Turbo Equalization for OFDM Systems in Time-Varying Channels

We propose low-complexity block turbo equalizers for orthogonal frequency-division multiplexing (OFDM) systems in time-varying channels. The presented work is based on a soft minimum mean-squared error (MMSE) block linear equalizer (BLE) that exploits the banded structure of the frequency-domain channel matrix, as well as a receiver window that enforces this banded structure. This equalization approach allows us to implement the proposed designs with a complexity that is only linear in the number of subcarriers. Three block turbo equalizers are discussed: two are based on a biased MMSE criterion, while the third is based on the unbiased MMSE criterion. Simulation results show that the proposed iterative MMSE BLE achieves a better bit error rate (BER) performance than a previously proposed iterative MMSE serial linear equalizer (SLE). The proposed equalization algorithms are also tested in the presence of channel estimation errors.      

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.     

Partial turbo detection with reduced complexity

In this article, we propose a novel method for reducing the complexity of the turbo detector MAP (maximum a posteriori). The basic idea consists in turbo detecting a part of intersymbol interference (ISI) after decomposing the channel in two parts. We show that we can reduce the trellis complexity in the turbo process at a certain cost, i.e. performance loss.     

Extended tail-biting schemes for turbo codes

In this letter, we investigate the problem of using tailbiting recursive systematic convolutional (RSC) codes in turbo codes. Tailbiting is not always possible for a given RSC code with fixed length. We propose an extended tailbiting method for RSC codes and compare it with another extension method proposed by Van Stralen, et al. (IEE Electron. Lett., vol. 35, pp. 1461-1462, 1999). Several schemes using these extended tailbiting RSC codes in turbo code systems are developed and compared.     

Relation between encoder and syndrome former variables and symbol reliability estimation using a syndrome trellis

We derive a linear correspondence between the variables of an encoder and those of a corresponding syndrome former. Using the derived correspondence, we show that the log-likelihood ratio of an information bit conditioned on a received sequence can be equally calculated using the syndrome trellis. It is shown that the proposed method also applies to recursive systematic convolutional codes which are typical constituent codes for turbo codes. Moreover, we show that soft-in syndrome decoding considering a priori probabilities of information bits is possible in the same way as for Viterbi decoding based on the code trellis. Hence, the proposed method can be applied to iterative decoding such as turbo decoding. We also show that the proposed method is effective for high-rate codes by making use of trellis modification.     

Concatenated codes for fading channels based on recursive space-time trellis codes

We propose a class of codes which combine the principles of turbo coding and space-time trellis codes. It is first shown that several classes of space-time codes have an equivalent recursive realization. This fact is then exploited to design serial concatenated coding schemes with an outer code, interleaver, and an inner recursive space-time encoder. Two solutions are proposed in this paper - the use of convolutional outer codes aimed mainly to improve the power efficiency and the use of very high-rate outer codes to obtain significant improvement in power efficiency with a marginal decrease in spectral efficiency. We show that single parity check based turbo product codes are a good candidate for very high-rate outer codes. Finally, we propose an automatic repeat request scheme based on recursive realizations of space-time codes and show that the proposed scheme provides significant reduction in frame error rate.     

Turbo Decoding and Detection for Wireless Applications

A historical perspective of turbo coding and turbo transceivers inspired by the generic turbo principles is provided, as it evolved from Shannon's visionary predictions. More specifically, we commence by discussing the turbo principles, which have been shown to be capable of performing close to Shannon's capacity limit. We continue by reviewing the classic maximum a posteriori probability decoder. These discussions are followed by studying the effect of a range of system parameters in a systematic fashion, in order to gauge their performance ramifications. In the second part of this treatise, we focus our attention on the family of iterative receivers designed for wireless communication systems, which were partly inspired by the invention of turbo codes. More specifically, the family of iteratively detected joint coding and modulation schemes, turbo equalization, concatenated space-time and channel coding arrangements, as well as multi-user detection and three-stage multimedia systems are highlighted.     

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     

LDPC-based space-time coded OFDM systems over correlated fadingchannels: Performance analysis and receiver design

We consider a space-time coded (STC) orthogonal frequency-division multiplexing (OFDM) system with multiple transmitter and receiver antennas over correlated frequency- and time-selective fading channels. It is shown that the product of the time-selectivity order and the frequency-selectivity order is a key parameter to characterize the outage capacity of the correlated fading channel. It is also observed that STCs with large effective lengths and ideal built-in interleavers are more effective in exploiting the natural diversity in multiple-antenna correlated fading channels. We then propose a low-density parity-check (LDPC)-code-based STC-OFDM system. Compared with the conventional space-time trellis code (STTC), the LDPC-based STC can significantly improve the system performance by exploiting both the spatial diversity and the selective-fading diversity in wireless channels. Compared with the previously proposed turbo-code-based STC scheme, LDPC-based STC exhibits lower receiver complexity and more flexible scalability. We also consider receiver design for LDPC-based STC-OFDM systems in unknown fast fading channels and propose a novel turbo receiver employing a maximum a posteriori expectation-maximization (MAP-EM) demodulator and a soft LDPC decoder, which can significantly reduce the error floor in fast fading channels with a modest computational complexity. With such a turbo receiver, the proposed LDPC-based STC-OFDM system is a promising solution to highly efficient data transmission over selective-fading mobile wireless channels     

Improving the performance of turbo codes with a simple protection scheme for error-prone bit positions

It is well known that turbo codes provide highly unequal error protection within a transmitted frame. Previous attempts to exploit this fact focused mainly on adding additional redundancy to provide extra protection for the error-prone bit positions. Here, instead, we use the error-detection capability of the cyclic redundancy check (CRC), which is almost always employed in practical systems. Once a frame is declared uncorrectable by the CRC, a process termed the error-prone bit processing procedure is activated in an attempt to correct the probable error patterns which are a priori identified as being error-prone.     

Minimum Mean-Squared Error Iterative Successive Parallel Arbitrated Decision Feedback Detectors for DS-CDMA Systems

In this paper we propose minimum mean squared error (MMSE) iterative successive parallel arbitrated decision feedback (DF) receivers for direct sequence code division multiple access (DS-CDMA) systems. We describe the MMSE design criterion for DF multiuser detectors along with successive, parallel and iterative interference cancellation structures. A novel efficient DF structure that employs successive cancellation with parallel arbitrated branches and a near-optimal low complexity user ordering algorithm are presented. The proposed DF receiver structure and the ordering algorithm are then combined with iterative cascaded DF stages for mitigating the deleterious effects of error propagation for convolutionally encoded systems with both Viterbi and turbo decoding as well as for uncoded schemes. We mathematically study the relations between the MMSE achieved by the analyzed DF structures, including the novel scheme, with imperfect and perfect feedback. Simulation results for an uplink scenario assess the new iterative DF detectors against linear receivers and evaluate the effects of error propagation of the new cancellation methods against existing ones.     

Turbo-Coded OFDM Transmission Over a Nonlinear Channel

In this paper, deliberate level clipping and turbo coding are combined to achieve an orthogonal frequency division multiplexing (OFDM) transmission system with a low peak-to-average power ratio (PAR) and a good performance. Using the linear approximation technique based on the minimum mean square error (MMSE) criterion, we first modify the metric computation for the turbo decoding in order to consider the distortion effects of the nonlinearity, caused by the Cartesian clipper. Also, this paper introduces a modified turbo decoder which simultaneously performs the data estimation and signal reconstruction. In other words, the turbo decoder iteratively recovers the clipped signal by using the estimated data, and then improves the data estimation by using the newly recovered signal. Numerical results are presented showing an improvement in the performance of the OFDM transmission system over the nonlinear channel, an increase in the efficiency of the high power amplifier (HPA), and/or an expansion of the transmitter coverage area.     

Minimum mean squared error equalization using a priori information

A number of important advances have been made in the area of joint equalization and decoding of data transmitted over intersymbol interference (ISI) channels. Turbo equalization is an iterative approach to this problem, in which a maximum a posteriori probability (MAP) equalizer and a MAP decoder exchange soft information in the form of prior probabilities over the transmitted symbols. A number of reduced-complexity methods for turbo equalization have been introduced in which MAP equalization is replaced with suboptimal, low-complexity approaches. We explore a number of low-complexity soft-input/soft-output (SISO) equalization algorithms based on the minimum mean square error (MMSE) criterion. This includes the extension of existing approaches to general signal constellations and the derivation of a novel approach requiring less complexity than the MMSE-optimal solution. All approaches are qualitatively analyzed by observing the mean-square error averaged over a sequence of equalized data. We show that for the turbo equalization application, the MMSE-based SISO equalizers perform well compared with a MAP equalizer while providing a tremendous complexity reduction     

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.     

Some notes on rate-compatible punctured turbo codes (RCPTC) design

In this letter, we propose and compare some design criteria for the search of good rate-compatible systematic turbo codes (RCPTC) families. The considerations presented by Benedetto et al. to find "best" component encoders for turbo-code construction are extended to find good rate-compatible puncturing patterns leading to codes with promising performances.     

Design of Prunable Interleavers for Parallel Turbo Decoder Architectures

In this paper we propose a technique to implement in a parallel fashion a turbo decoder based on an arbitrary permutation, and to expand its interleaver in order to produce a family of prunable S-random interleavers suitable for parallel implementations. We show that the spread properties of the obtained interleavers are almost optimal and we prove by simulation that they are very competitive in terms of error floor performance. A few details on the decoder architecture are also provided     

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.     

Bit Error Rate Estimation for Turbo Decoding

In this paper we propose a method for on-line estimation of Bit Error Rate during turbo decoding. We model the log-likelihood ratios as a mixture of two Gaussian random variables and derive estimators for the mean and variance of these distributions based on a Maximum-Likelihood approach. The parameter estimates are then employed to calculate the cross-over area of the Gaussian tails to estimate BER at each decoder iteration. The performance of the BER estimator is analysed and compared.