Realizable linear and decision feedback equalizers: properties and connections

Recently, there has been renewed interest in the use of infinite impulse response (IIR) linear equalizers (LEs) for digital communication channels as a means for both improving performance and blindly initializing decision feedback structures (DFEs). Theoretical justification for such an approach is usually given assuming unconstrained filters, which are not causal and therefore not implementable in practice. We present an analysis of realizable (i.e., causal, stable, and of finite degree) minimum mean square error (MMSE) equalizers for single-input multiple-output channels, both in the LE and DFE cases, focusing on their structures and filter orders, as well as the connections between them. The DFE resulting from rearranging the MMSE LE within a decision feedback loop is given special attention. It is shown that although this DFE does not in general coincide with the MMSE DFE, it still enjoys certain optimality conditions. The main tools employed are the Wiener theory of minimum variance estimation and Kalman filtering theory, which show interesting properties of the MMSE equalizers not revealed by previous polynomial approaches.     

Block channel equalization in the presence of a cochannelinterferent signal

Novel algorithms for block equalization of M-ary phase shift keying (PSK) signals transmitted over multipath fading channels in the presence of an interferent cochannel signal are introduced and analyzed. The algorithms exploit the intrinsic statistical properties of cochannel interference (CCI) in order to mitigate its effects. Both linear and decision feedback equalizers (DFEs) are derived under the assumption that the overall channel impulse responses of both the useful and the inteferent signal are known. Simulation results show that: (a) whereas zero-forcing block equalizers yield a large noise enhancement effect, a minimum mean-square block DFE (MMSE-BDFE) can efficiently compensate for the distortion in the useful channel and reduce the effect of CCI at the same time, and (b) the MMSE-BDFEs outperform conventional DFEs, at least in the idealized conditions of our analysis     

Error propagation and recovery in decision-feedback equalizers fornonlinear channels

Nonlinear intersymbol interference is often present in communication and digital storage channels. Decision-feedback equalizers (DFEs) can decrease this nonlinear effect by including appropriate nonlinear feedback filters. Although various applications of these types of equalizers have been published in the literature, the analysis of their stability and error recovery has not appeared. We consider a DFE with a nonlinear feedback filter based on a discrete Volterra series. We extend error propagation, error probability, stability, and error recovery time results for Nth order nonlinear channels     

Adaptive minimum symbol-error-rate decision feedback equalization for multilevel pulse-amplitude modulation

The design of decision feedback equalizers (DFEs) is typically based on the minimum mean square error (MMSE) principle as this leads to effective adaptive implementation in the form of the least mean square algorithm. It is well-known, however, that in certain situations, the MMSE solution can be distinctly inferior to the optimal minimum symbol error rate (MSER) solution. We consider the MSER design for multilevel pulse-amplitude modulation. Block-data adaptive implementation of the theoretical MSER DFE solution is developed based on the Parzen window estimate of a probability density function. Furthermore, a sample-by-sample adaptive MSER algorithm, called the least symbol error rate (LSER), is derived for adaptive equalization applications. The proposed LSER algorithm has a complexity that increases linearly with the equalizer length. Computer simulation is employed to evaluate the proposed alternative MSER design for equalization application with multilevel signaling schemes.     

High-Speed Architecture Design of Tomlinson–Harashima Precoders

Like decision feedback equalizers (DFEs), Tomlinson-Harashima precoders (TH precoders) contain nonlinear feedback loops, which limit their use for high-speed applications. Unlike in DFEs where the output levels of the nonlinear devices are finite, in TH precoders, theoretically, the output levels of the modulo devices are infinite. Thus, it is difficult to apply look-ahead and pre-computation techniques to pipeline TH precoders, which were successfully applied to pipeline infinite-impulse response (IIR) filters and DFEs in the past. In this paper, three approaches are proposed to design high-speed TH precoders. In the first approach, the traditional block processing technique for DFEs is generalized to the design of high-speed TH precoders. In the second approach, based on the equivalent form of a TH precoder where the precoder can be viewed as an IIR filter with an input equal to the sum of the original input to the TH precoder and a finite-level compensation signal, two high-speed pipelined designs are developed. In the third approach, parallel processing techniques for fast IIR filters are generalized to the design of parallel TH precoders.     

Design of Parallel Tomlinson–Harashima Precoders

Like decision feedback equalizers (DFEs), Tomlinson-Harashima precoders (TH precoders) contain nonlinear feedback loops, which limit their use for high-speed applications. Unlike in DFEs, where the output levels of the nonlinear devices are finite, in TH precoders the output levels of the modulo devices are either infinite or finite but very large. Thus, it is difficult to apply look-ahead and pre-computation techniques to speed up TH precoders, which were successfully applied to design parallel and pipelined infinite impulse response (IIR) filters and DFEs in the past. However, a TH precoder can be viewed as an IIR filter with an input equal to the sum of the original input to the TH precoder and a finite-level compensation signal. Based on this point of view, a novel parallel architecture is proposed to speed up TH precoders. This architecture can be used in many high-speed applications, such as 10-Gb Ethernet over copper.     

Reduced complexity decision feedback equalization for digitalsubscriber loops

Decision feedback equalizers (DFEs) are widely used in modern local network digital transmission systems to remove the intersymbol interference caused by slowly decaying pulse tails. A gradient descent algorithm for adapting a coefficient to model the slowly decaying portion of the tail is described. An equalization strategy is described that exploits prior knowledge of the nature of the subscriber loop channel, together with the new adaptation algorithm, to give reduced complexity DFE structures. The use of this algorithm in FIR and IIR equalizer structures is described. The use of this algorithm in FIR and IIR equalizers is quantitatively compared to a conventional DFE in terms of performance and implementation complexity. An analysis is presented describing the operation of the adaptation algorithm in the presence of noise. Simulation results illustrate the training of the algorithm and its stability in the presence of near-end crosstalk noise     

The structure and design of realizable decision feedback equalizers for IIR channels with colored noise

A simple algorithm for optimizing decision feedback equalizers (DFEs) by minimizing the mean-square error (MSE) is presented. A complex baseband channel and correct past decisions are assumed. The dispersive channel may have infinite impulse response, and the noise may be colored. Consideration is given to optimal realizable (stable and finite-lag smoothing) forward and feedback filters in discrete time. They are parameterized as recursive filters. In the special case of transmission channels with finite impulse response and autoregressive noise, the minimum MSE can be attained with transversal feedback and forward filters. In general, the forward part should include a noise-whitening filter (the inverse noise model). The finite realizations of the filters are calculated using a polynomial equation approach to the linear quadratic optimization problem. The equalizer is optimized essentially by solving a system of linear equations Ax=B, where A contains transfer function coefficients from the channel and noise model. No calculation of correlations is required with this method. A simple expression for the minimal MSE is presented. The DFE is compared to MSE-optimal linear recursive equalizers. Expressions for the equalizer in the limiting case of infinite smoothing lags are also discussed.<>     

MIMO decision feedback equalization from an H/sup /spl infin// perspective

We approach the multiple input multiple output (MIMO) decision feedback equalization (DFE) problem in digital communications from an H/sup /spl infin// estimation point of view. Using the standard (and simplifying) assumption that all previous decisions are correct, we obtain an explicit parameterization of all H/sup /spl infin// optimal DFEs. In particular, we show that, under the above assumption, minimum mean square error (MMSE) DFEs are H/sup /spl infin// optimal. The H/sup /spl infin// approach also suggests a method for dealing with errors in previous decisions.     

Error-rate evaluation of linear equalization and decision feedbackequalization with error propagation

We propose applying an approximate Fourier series to evaluate efficiently the bit-error-rate (BER) performance of finite-length linear equalization (LE) and decision feedback equalization (DFE). By extending the Fourier series, we enable BER calculations for quadrature phase-shift keying (QPSK) transmission on complex channels with in-phase and crosstalk intersymbol interference (ISI). The BER calculation is based on determining the residual ISI samples and background Gaussian noise variance at the equalizer output for static channels or for realizations of quasi-static fading channels. A simple bound on the series error magnitude in terms of the Fourier series parameters ensures the required accuracy and precision. Improved state transition probability estimates are derived and verified by simulation for an approximate Markov model of the DFE error propagation for the case in which residual ISI exists even when the previous decisions stored in the feedback filter (FBF) are correct. We demonstrate the ease and widespread applicability of our approach by producing results which elucidate a variety of equalization tradeoffs. Our analysis includes symbol-spaced and fractionally spaced minimum mean-square error (MMSE)-LE, zero-forcing (ZF)-LE, and MMSE-DFE (with and without error propagation) on static ISI channels and multipath channels with quasi-static Rayleigh fading; a comparison between suboptimum and optimum receiver filtering in conjunction with equalization; and an assessment of the accuracy of some widely used equalization BER approximations and bounds     

Iterative design and detection of a DFE in the frequency domain

Error-propagation phenomena and computational complexity of the filters' design are important drawbacks of existing decision-feedback equalizers (DFE) for dispersive channels. In this paper, we propose a new iterative block DFE (IBDFE) which operates iteratively on blocks of the received signal. Indeed, a suitable data-transmission format must be used to allow an efficient implementation of the equalizer in the frequency domain, by means of the discrete Fourier transform. Two design methods are considered. In the first method, hard detected data are used as input of the feedback, and filters are designed according to the correlation between detected and transmitted data. In the second method, the feedback signal is directly designed from soft detection of the equalized signal at the previous iteration. Estimators of the parameters involved in the IBDFE design are also derived. From performance simulations on a wireless dispersive fading channel, we observed that the IBDFE outperforms existing DFEs. Moreover, the IBDFE exhibits a reduction of the computational complexity when compared against existing schemes, both in signal processing and in filter design.     

Decision-feedback equalization via separating hyperplanes

The design of finite-length decision-feedback equalization (DFE) forward and feedback filters under the assumption of genie-aided feedback and independent and equally likely transmitted symbols is considered. It is shown that the problem of determining DFE filters that minimize the probability of symbol error at high signal-to-noise ratio (SNR) is equivalent to finding the hyperplane that maximally separates two given finite groups of points in a finite-dimensional Euclidean space. The latter task can be formulated as a quadratic program which is readily solved numerically. It is also shown that the problem of finding finite-length DFE filters that minimize the probability of symbol error at any SNR subject to a certain separation condition is a convex optimization problem. The case where the transmitted data is coded using a runlength-limited code is also investigated. Examples show that this criterion yields a performance that is better than zero-forcing DFE on severely distorted channels at high SNR     

Parallel adaptive decision feedback equalizers

Several algorithms for parallel implementation of adaptive decision feedback equalizers (DFEs) are proposed. The first is a double-row DFE algorithm that outperforms previous approaches. Under the no-error-propagation assumption, the algorithm will perform exactly like a serially adapting DFE. The multiplication complexity of the double-row DFE algorithm is of the same order as that of the parallel DFE algorithm and the extended least-mean-square (LMS) method. The previous algorithms and the double-row DFE algorithm may become impractical to implement due to their large computational complexity, so three additional parallel implementations of the DFE, which lead to considerable hardware savings and avoid the coding loss of the former approaches, are presented. The different algorithms are compared on the basis of convergence analysis and simulation results     

Zero forcing and minimum mean-square-error equalization formultiuser detection in code-division multiple-access channels

In code-division multiple-access (CDMA) systems transmitting over time-varying multipath channels, both intersymbol interference (ISI) and multiple-access interference (MAI) arise. The conventional suboptimum receiver consisting of a bank of matched filters is often inefficient because interference is treated as noise. The optimum multiuser detector is too complex to be implemented at present. Four suboptimum detection techniques based on zero forcing (ZF) and minimum mean-square-error (MMSE) equalization with and without decision feedback (DF) are presented and compared. They combat both ISI and MAI. The computational complexity of all four equalizers is essentially the same. All four equalizers are independent of the size of the data symbol alphabet. It is shown that the performance of the MMSE equalizers is better than that of the corresponding ZF equalizers. Furthermore, the performance of the equalizers with DF is better than that of the corresponding equalizers without DF. The impairing effect of error propagation on the equalizers with DF is reduced by channel sorting     

Equalizers for multiple input/multiple output channels and PAMsystems with cyclostationary input sequences

The author studies minimum mean square error (MMSE) linear and decision feedback (DF) equalisers for multiple input/multiple output (MIMO) communication systems with intersymbol interference (ISI) and wide-sense stationary (WSS) inputs. To derive these equalizers, one works in the D-transform domain and uses prediction theory results. Partial-response MMSE equalizers are also found. As an application, the author considers a pulse amplitude modulation (PAM) communication system with ISI and cyclostationary inputs. The MMSE linear and DF equalizers are determined by studying an equivalent MIMO system. The resulting filters are expressed in compact matrix notation and are time-invariant, whereas the corresponding single input/single output filters are periodically time-invariant. The author also considers MMSE equalizers for a wide-sense stationary process by introducing a `random phase'. To aid in the performance evaluation of various equalizers, the author derives their mean square errors     

Mitigating error propagation effects in a decision feedbackequalizer

We present an approximate analysis approach to the computation of the probability of error and mean burst error length for a decision feedback equalizer (DFE) that takes into account feedback of decision errors. The method uses a reduced-state Markov model of the feedback process and is applicable to linear modulation formats. We use this technique to analyze a DFE design that mitigates the effects of feedback error by incorporating a soft decision device into the feedback path and a norm constraint on the feedback filter weights. We apply the DFE design and analysis approach to a dispersive multipath propagation environment     

Decision-feedback equalization of pulse-position modulation onmeasured nondirected indoor infrared channels

We examine the performance of two decision-feedback equalizers (DFEs) for pulse-position modulation (PPM) on measured nondirected indoor infrared channels with intersymbol interference. PPM offers high average-power efficiency, but on ISI channels, unequalized PPM suffers severe performance penalties. We have previously examined the performance of the maximum-likelihood sequence detector (MLSD), and found that it yields significant improvements. However, the MLSD often requires such large complexity and delay that it may be impractical. We investigate suboptimal, reduced-complexity equalization techniques for PPM, providing a performance analysis of zero-forcing chip-rate and symbol-rate DFEs. Our results show that a symbol-rate DFE provides performance that closely approaches that of the optimal MLSD     

Spatial and temporal equalization for broadband wireless indoornetworks at millimeter waves

The combined use of adaptive antennas and decision feedback equalization (DFE) is analyzed in a realistic propagation scenario at millimeter waves, taking the direction of arrivals (DOA's) of the received paths into account. The joint antennas and DFE scheme, with one forward filter for each antenna and a single feedback filter (FBF), can be viewed as a spatial and temporal DFE (ST-DFE). The performance of this solution is compared with the cascade of adaptive antenna used for beamforming and DFE. It is found that ST-DFE achieves better performance since it combines the beamforming capability of the antenna array with the equalization properties of the DFE, with great advantages especially when rays arrive from similar angles. The mean square error (MSE) is analytically derived for infinitely long filters in a quasi-static environment with multiple rays having different DOAs, and compared (for the two-path model) with simulation results assuming filters with a small number of taps. Finally, service availability through coverage evaluation is developed and compared with that of a coded-orthogonal frequency division multiplexing (C-OFDM) system     

Novel Techniques to Minimize the Error Propagation of Decision Feedback Equalizer in 8VSB DTV System

In this paper, novel and yet simple techniques are presented to minimize the error propagation caused by the large precursors and postcursors of the decision feedback equalizer (DFE) in 8VSB DTV system. A technique that selects a reference tap (symbol timing of DFE) from an estimated channel impulse response (CIR) is presented to minimize the effect of the large precursors. Another technique that selects the reference tap position, i.e., decision delay in a feedforward filter (FFF), from the estimated CIR and the amplitude of the selected reference tap is proposed to minimize the effect of large postcursors. The combined structure of a feedback filter (FBF) and Viterbi decoder for use in 8VSB DTV system is also proposed to replace the past unreliable decision symbols in FBF as well as to reduce the decision error probability. Simulation results show that our proposed DFE can prevent effectively the error propagation, in particular, by changing the reference tap and its position in FFF according to the channel condition. It is also shown that an echo removing capability of the proposed DFE, where 400 and 620taps are used for the FFF and FBF, respectively, is greater than that of conventional DFEs by about -20 mus in the single pre-echo of -10 dB channel and by about 10 mus in the single post-echo of -1 dB channel     

Importance sampling simulation for evaluating lower-bound symbolerror rate of the Bayesian DFE with multilevel signaling schemes

For the class of equalizers that employs a symbol-decision finite-memory structure with decision feedback, the optimal solution is known to be the Bayesian decision feedback equalizer (DFE). The complexity of the Bayesian DFE, however, increases exponentially with the length of the channel impulse response (CIR) and the size of the symbol constellation. Conventional Monte Carlo simulation for evaluating the symbol error rate (SER) of the Bayesian DFE becomes impossible for high channel signal-to-noise ratio (SNR) conditions. It has been noted that the optimal Bayesian decision boundary separating any two neighboring signal classes is asymptotically piecewise linear and consists of several hyperplanes when the SNR tends to infinity. This asymptotic property can be exploited for efficient simulation of the Bayesian DFE. An importance sampling (IS) simulation technique is presented based on this asymptotic property for evaluating the lower bound SER of the Bayesian DFE with a multilevel pulse amplitude modulation (M-PAM) scheme under the assumption of correct decisions being fed back. A design procedure is developed, which chooses appropriate bias vectors for the simulation density to ensure asymptotic efficiency (AE) of the IS simulation