Turbo codes and the iterative algorithm for decoding them sparked a new era in the theory and practice of error control codes. Turbo equalization followed as a natural extension to this development, as an iterative technique for detection and decoding of data that has been both protected with forward error correction and transmitted over a channel with intersymbol interference (ISI). In this paper, we review the turbo equalization approach to coded data transmission over ISI channels, with an emphasis on the basic ideas, some of the practical details, and many of the research directions that have arisen from this offshoot, introduced by Douillard, of the original turbo decoding algorithm. The subsequent relaxation of the maximum a posteriori (MAP) equalization algorithm to include linear and other simpler receivers sparked a decade and a half of research into iterative algorithms, spanning research problems ranging from trellis coded modulation to underwater acoustic communications.

We consider the problem of jointly decoding the correlated data picked up and transmitted by the nodes of a large-scale sensor network. Assuming that each sensor node uses a very simple encoder (a scalar quantizer and a modulator), we focus on decoding algorithms that exploit the correlation structure of the sensor data to produce the best possible estimates under the minimum mean-square error (MMSE) criterion. Our analysis shows that a standard implementation of the optimal MMSE decoder is unfeasible for large-scale sensor networks, because its complexity grows exponentially with the number of nodes in the network. Seeking a scalable alternative, we use factor graphs to obtain a simplified model for the correlation structure of the sensor data. This model allows us to use the sum-product decoding algorithm, whose complexity can be made to grow linearly with the size of the network. Considering large sensor networks with arbitrary topologies, we focus on factor trees and give an exact characterization of the decoding complexity, as well as mathematical tools for factorizing Gaussian sources and optimization algorithms for finding optimal factor trees under the Kullback-Leibler criterion.

We consider the sensor reachback problem, in which a large number of sensor nodes are deployed on a field, and the goal is to reconstruct at a remote location the correlated data collected and transmitted by all the nodes. In this paper, we assume that each sensor node uses a very simple encoder (a scalar quantizer and a modulator) and focus on decoding algorithms that exploit the correlation structure of the sensor data to produce the best possible estimates under the minimum mean square error (MMSE) criterion. Our analysis shows that the optimal MMSE decoder is unfeasible for large scale sensor networks, because its complexity grows exponentially with the number of nodes in the network. Seeking a scalable alternative, we use factor graphs to obtain a simplified model for the correlation structure of the sensor data. This model allows us to use an iterative decoding algorithm whose complexity can be made to grow linearly with the size of the network.

Based on extrinsic information transfer (EXIT) charts, the convergence behavior of iterative decoding is studied for a number of serially concatenated systems, such as a serially concatenated code, coded data transmission over an intersymbol interference channel, bit-interleaved coded modulation, or trellis-coded modulation. Efficient optimization algorithms based on simplified EXIT chart construction are devised to find irregular codes improving the convergence of iterative decoding. One optimization criterion is to find concatenated systems exhibiting thresholds of successful decoding convergence, which are close to information-theoretic limits. However, these thresholds are approached only for very long block lengths. To overcome this problem, the decoding convergence after a fixed, finite number of iterations is optimized, which yields systems performing very well for short block lengths, too. As an example, optimal system configurations for communication over an additive white Gaussian noise channel are presented.

We investigate turbo equalization, or iterative equalization and decoding, as a receiver technology for systems where data is protected by an error-correcting code, shuffled by an interleaver, and mapped onto a signal constellation for transmission over a frequency-selective channel with unknown time-varying channel impulse response. The focus is the concept of soft iterative channel estimation, which is to improve the channel estimate over the iterations by using soft information fed back from the decoder from the previous iteration to generate "extended training sequences" between the actual transmitted training sequences.

We consider the problem of coded data transmission over an inter-symbol interference (ISI) channel with unknown and possibly time-varying parameters. We propose a low-complexity algorithm for joint equalization, estimation, and decoding using an estimator, which is separate from the equalizer. Based on existing techniques for analyzing the convergence of iterative decoding algorithms, we show how to find powerful system configurations. This includes the use of recursive precoders in the transmitter. We derive a novel a-posteriori probability equalization algorithm for imprecise knowledge of the channel parameters. We show that the performance loss implied by not knowing the parameters pf the ISI channel is entirely a loss in signal-to-noise ratio for which a suitably designed iterative receiver algorithm converges.

We study the convergence behavior of iterative decoding of various serially concatenated systems such as a concatenated code, coded transmission over a channel introducing inter-symbol interference, bit-interleaved coded modulation, trellis coded modulation, a.s.o. We use EXIT charts to construct simple irregular codes, which can significantly improve the convergence behavior of iterative decoding. An efficient optimization algorithm is presented yielding system which approach information theoretic limits very closely. However, these systems exhibit a satisfactory performance only for very long block lengths. To overcome this problem, we also show how to optimize a concatenated system such that the decoding performance is optimal after a certain fixed number of iterations. It turns out that these systems perform very well for short block lengths, too. As an example, optimal system configurations for data transmission over an AWGN channel are presented.

We show how to use EXIT charts for convergence prediction of a threefold serially concatenated system. The corresponding chart has three dimensions and allows us to appropriately select system parameters and to find an optimal schedule of decoding iterations between the three decoders of such a system. Convergence thresholds are obtained to determine the minimal signal-to-noise ratios for which convergence is possible. It turns out that threefold concatenated systems do not achieve any additional performance gain compared to suitably designed twofold systems. We conclude that a threefold concatenation should be considered only when the decoders cannot be chosen freely.

To combat the effect of intersymbol interference (ISI) while transmitting data over an ISI channel in a coded data transmission system, the impulse response of the channel is required. As part of the turbo equalization approach, which facilitates iterative equalization and decoding, we introduce a method to iteratively improve the quality of the estimate of the channel characteristics. This is done by incorporating soft information fed back by the decoder to improve the initial estimate, obtained for example using a training sequence. Decision criteria based on the analytical calculation of the variance of the channel estimation error are derived to decide whether the soft information improves the quality of the estimate. The considered estimation algorithm is the well-known recursive-least-squares algorithm. It turns out that incorporating soft information for iterative channel estimation does not always improve the quality of the estimate. If it does, the bit-error-rate performance improves significantly over a system not using soft iterative channel estimation.

Low-complexity soft-in soft-out (SISO) equalizers based on time-varying linear filters and soft inter-symbol interference cancellation are known to be a viable alternative to the optimal trellis-based SISO equalizers when used in receivers based on iterative equalization and decoding. In particular, when the signal constellation is large and/or the channel impulse response is long, the computational complexity of trellis-based equalizers becomes prohibitive while linear equalizers can still be used. In this paper, a SISO linear equalization algorithm is derived for the case of a time-varying channel impulse response, which is either known or estimated. Simulation results are presented showing that the error rate performance of the SISO linear equalizer is close to that using an optimal trellis-based equalizer. The performance of iterative channel estimation is also investigated.

We study the turbo equalization approach to coded data transmission over channels with intersymbol interference. In the original system invented by Douillard et al. (1995), the data are protected by a convolutional code and the receiver consists of two trellis-based detectors, one for the channel (the equalizer) and one for the code (the decoder). It has been shown that iterating equalization and decoding tasks can yield tremendous improvements in bit error rate. We introduce new approaches to combining equalization based on linear filtering, with decoding.. Through simulation and analytical results, we show that the performance of the new approaches is similar to the trellis-based receiver, while providing large savings in computational complexity. Moreover, this paper provides an overview of the design alternatives for turbo equalization with given system parameters, such as the channel response or the signal-to-noise ratio.

The performance of high-rate tail-biting convolutional codes serially concatenated with generalized partial response channels is studied. The effect of precoders on the overall performance is investigated. Extrinsic information transfer charts are used to guide the selection of appropriate tail-biting codes and precoders. Simulation results for a magnetic recording system modeled as a serial concatenation of tail-biting codes with a generalized partial response channel are presented. In particular, rate-8/9 and -16/17 short- and long-block-length tail-biting codes are studied. In the former case, hard-decision decoded interleaved Reed-Solomon (RS) codes are used as the outer-most code, whereas in the latter case the sector-size tail-biting codes replace the RS codes traditionally used in storage systems. The results indicate that high-rate tail-biting codes deliver significant performance gains when used in conjunction with a rate-1 precoder and iterative detection/decoding. The results also show that long tail-biting codes can outperform hard-decision decoding of RS codes by 2 dB at a sector error rate of approx. 10/sup -4/.

For coded data transmission over channels introducing inter-symbol interference, one approach for joint equalization and decoding in the receiver is turbo equalization. We rederive existing linear equalization algorithms applicable to turbo equalization for 2/sup m/-ary signal alphabets and compare their computational complexity. Moreover by evaluating the algorithm performance properly, we select for each iteration the most suitable of the two algorithms with lowest computational complexity and achieve at low bit error rates a performance close to that of optimal approaches for equalization, ie, maximum a-posteriori probability symbol detection.