Combined RLS-LMS initialization for per tone equalizers in DMT-receivers

In discrete multitone receivers, the classical equalizer structure consists of a (real) time domain equalizer (TEQ) combined with complex one-tap frequency domain equalizers. An alternative receiver is based on a per tone equalization (PTEQ), which optimizes the signal-to-noise ratio (SNR) on each tone separately and, hence, the total bitrate. In this paper, a new initialization scheme for the PTEQ is introduced, based on a combination of least mean squares (LMS) and recursive least squares (RLS) adaptive filtering. It is shown that the proposed method has only slightly slower convergence than full square-root RLS (SR-RLS) while complexity as well as memory cost are reduced considerably. Hence, in terms of complexity and convergence speed, the proposed algorithm is in between LMS and RLS.     

Bitrate-maximizing time-domain equalizer design for DMT-based systems

A time-domain equalizer (TEQ) is inserted in discrete multitone (DMT) receivers to impose channel shortening. Many algorithms have been developed to initialize this TEQ, but none of them really optimizes the bitrate. We present a truly bitrate-maximizing TEQ (BM-TEQ) cost function that is based on an exact formulation of the subchannel signal-to-noise ratio as a function of the TEQ taps. The performance of this BM-TEQ comes close to the performance of the per-tone equalizer.     

Linear and decision-feedback per tone equalization for DMT-based transmission over IIR channels

The per-tone equalizer (PTEQ) has been presented as an attractive alternative for the classical time-domain equalizer (TEQ) in discrete multitone (DMT) based systems, such as ADSL systems. The PTEQ is based on a linear minimum mean-square-error (L-MMSE) equalizer design for each separate tone. In this paper, we reconsider DMT modulation and equalization in the ADSL context under the realistic assumption of an infinite impulse response (IIR) model for the wireline channel. First, optimum linear zero-forcing (L-ZF) block equalizers for arbitrary IIR model orders and cyclic prefix (CP) lengths are developed. It is shown that these L-ZF block equalizers can be decoupled per tone, hence they lead to an L-ZF PTEQ. Then, based on the L-ZF PTEQ, low-complexity L-MMSE PTEQ extensions are developed: the linear PTEQ extension exploits frequency-domain transmit redundancy from pilot and unused tones; alternatively, a closely related decision-feedback PTEQ extension can be applied. The PTEQ extensions then add flexibility to a DMT-based system design: the CP overhead can be reduced by exploiting frequency-domain transmit redundancy instead, so that a similar bitrate as with the original PTEQ is achieved at a lower memory and computational cost or, alternatively, a higher bitrate is achieved without a considerable cost increase. Both PTEQ extensions are also shown to improve the receiver's robustness to narrow-band interference.     

Joint window and time domain equalizer design for bit rate maximization in DMT-receivers

In discrete multitone (DMT) receivers, as for instance in asymmetric digital subscriber lines (ADSLs), the classical equalizer structure consists of a (real) time domain equalizer (TEQ) combined with complex 1-tap frequency domain equalizers (FEQs). Additionally, receiver windowing can be applied to mitigate the bad spectral containment of the demodulating DFT sidelobes. We focus on a combined equalizer and windowing design procedure to maximize the achievable bit rate in DMT-based modems. Whereas the combination of a TEQ with a single window treats all the data carrying tones in a common way, the presented design method can also be used in a "per group" fashion, where smaller groups of tones receive each a different equalizer-window pair. When such groups contain only one single tone, the design procedure can be linked to the performance of an unbiased minimum mean square error (MMSE) per tone equalizer (PTEQ), which then also implicitly implements a per tone window. The general framework introduced Allows us to treat equalizer-only and window-only designs as well, which appear as special cases in a natural way. This set of bit rate maximizing techniques can serve either as practical design methods or as upper bounds for existing (suboptimal) methods. We will also show that for the same achievable bit rate, equalizer taps can be exchanged for windowing coefficients to reduce complexity during data transmission.     

Application of Fast Kalman Estimation to Adaptive Equalization

Very rapid initial convergence of the equalizer tap coefficients is a requirement of many data communication systems which employ adaptive equalizers to minimize intersymbol interference. As shown in recent papers by Godard, and by Gitlin and Magee, a recursive least squares estimation algorithm, which is a special case of the Kalman estimation algorithm, is applicable to the estimation of the optimal (minimum MSE) set of tap coefficients. It was furthermore shown to yield much faster equalizer convergence than that achieved by the simple estimated gradient algorithm, especially for severely distorted channels. We show how certain "fast recursive estimation" techniques, originally introduced by Morf and Ljung, can be adapted to the equalizer adjustment problem, resulting in the same fast convergence as the conventional Kalman implementation, but with far fewer operations per iteration (proportional to the number of equalizer taps, rather than the square of the number of equalizer taps). These fast algorithms, applicable to both linear and decision feedback equalizers, exploit a certain shift-invariance property of successive equalizer contents. The rapid convergence properties of the "fast Kalman" adaptation algorithm are confirmed by simulation.     

平均自适应滤波的信道均衡算法研究

近年来数字传输系统的信道均衡侧重于训练时间的缩短和跟踪速度的加快,需要研究快速收敛的自适应算法。从这点考虑递归最小二乘(RLS)均衡器是最佳的选择,但RLS算法的运算非常复杂而且存在稳定性问题,因而有必要研究一种能够代替传统RLS的算法。在本文中介绍一种基于平均自适应滤波(AFA)算法的均衡器,其主要优点是与RLS算法相当的快速收敛速度,同时运算复杂度较低。     

Low-complexity adaptive decision-feedback equalization of MIMO channels

A new adaptive MIMO channel equalizer is proposed based on adaptive generalized decision-feedback equalization and ordered-successive interference cancellation. The proposed equalizer comprises equal-length subequalizers, enabling any adaptive filtering algorithm to be employed for coefficient updates. A recently proposed computationally efficient recursive least squares algorithm based on dichotomous coordinate descents is utilized to solve the normal equations associated with the adaptation of the new equalizer. Convergence of the proposed algorithm is examined analytically and simulations show that the proposed equalizer is superior to the previously proposed adaptive MIMO channel equalizers by providing both enhanced bit error rate performance and reduced computational complexity. Furthermore, the proposed algorithm exhibits stable numerical behavior and can deliver a trade-off between performance and complexity.     

Importance sampling methodologies for simulation of communicationsystems with time-varying channels and adaptive equalizers

Two importance sampling (IS) methodologies for Monte Carlo simulation of communication links characterized by time-varying channels and adaptive equalizers are presented. One methodology is denoted as the twin system (TS) method. A key feature of the TS method is that biased noise samples are input to the adaptive equalizer, but the equalizer is only allowed to adapt to these samples for a time interval equal to the memory of the system. In addition to the TS technique, the IA method, a statistically biased, but simpler, technique for using IS with adaptive equalizers that is based on the independence assumption between equalizer input and equalizer taps is presented. Experimental results show run-time speedup factors of two to seven orders of magnitude for a static linear channel with memory, and of two to almost five orders of magnitude for a slowly-varying random linear channel with memory for both the IA and TS methods     

Channel Equalization Using Adaptive Lattice Algorithms

In this paper, a study of adaptive lattice algorithms as applied to channel equalization is presented. The orthogonalization properties of the lattice algorithms make them appear promising for equalizing channels which exhibit heavy amplitude distortion. Furthermore, unlike the majority of other orthogonalization algorithms, the number of operations per update for the adaptive lattice equalizers is linear with respect to the number of equalizer taps.     

Multipath Outage Performance of Digital Radio Receivers Using Finite-Tap Adaptive Equalizers

Recent analysis/simulation studies have quantified the multipath outage statistics of digital radio systems using ideal adaptive equalization. In this paper, we consider the use of finite-tap delay line equalizers, with the aim of determining how many taps are needed to approximate ideal performance. To this end, we assume anM-level QAM system using cosine rolloff spectral shaping and an adaptive equalizer with either fractionally spaced or synchronously spaced taps. We invoke a widely used statistical model for the fading channel and computer-simulate thousands of responses from its ensemble. For each trial, we compute a detection signal-to-distortion measure, suitably maximized with respect to the tap gains. We can thereby obtain probability distributions of this measure for specified combinations of system parameters. These distributions, in turn, can be interpreted as outage probabilities (or outage seconds) versus the number of modulation levels. A major finding of this study is that, for the assumed multipath fading model, very few taps (the order of five) are needed to approximate the performance of an ideal infinite-tap equalizer. We also find that a simple, suboptimal form of timing recovery is generally quite adequate, and that fractionally spaced equalizers are more advantageous than synchronously spaced equalizers with the same number of taps. This advantage is minor for rolloff factors of 0.5 and larger but increases dramatically as the rolloff factor approaches zero.     

IQ-Imbalance Compensation for OFDM in the Presence of IBI and Carrier-Frequency Offset

In this paper, we propose a frequency-domain equalization technique for orthogonal frequency division multiplexing (OFDM) transmission over frequency-selective channels. We consider the case where the receiver analog front-end suffers from IQ-imbalance and the local oscillator suffers from carrier-frequency offset (CFO). While the IQ-imbalance results in a mirroring effect, the CFO induces inter-carrier interference (ICI). In addition to ICI, we consider the channel delay spread is larger than the cyclic prefix (CP). This means that inter-block interference (IBI) is present. The frequency-domain equalizer is obtained by transferring a time-domain equalizer (TEQ) to the frequency-domain resulting in a per-tone equalizer (PTEQ). Due to the presence of IQ-imbalance the conventional TEQ (where only one TEQ is applied to the received sequence) is not sufficient to cope with the mirroring effect. A sufficient TEQ consists of two time-domain filters; one applied to the received sequence and another applied to a conjugated version of the received sequence. For the case of IQ-imbalance and CFO, the TEQs are designed according the basis expansion model (BEM) which showed to be able to cope with the ICI problem. Finally, in addition to the frequency-domain PTEQ design procedure, a training-based RLS type initialization scheme for direct per-tone equalization is also proposed     

Non-uniformly spaced tapped-delay-line equalizers

A systematic way to design nonuniformly spaced tapped-delay-line (TDL) equalizers is described, and the performance of such equalizers is compared to that of uniformly spaced TDL equalizers with the same number of tap coefficients. It is shown that the signal-to-mean-squared-error ratio at the output of a TDL equalizer can be improved by optimally choosing the positions of the tap weights. An algorithm to find both the tap positions and the corresponding tap weights for a given delay span and a given minimum tap spacing of the equalizer is presented. Typical results are illustrated by using, as an example, the magnetic recording channel. For two target waveforms at different densities of recording, it is shown that there is a potential for saving up to seven equalizer taps     

基于QR-RLS算法的OFDM系统盲自适应时域均衡器

在正交频分复用(OFDM)系统中,使用时域均衡器来消除由于循环前缀长度小于信道时延扩展长度而导致的符号间干扰。为了克服Merry算法收敛速度较慢的缺点,提出了一种适用于无线时变信道环境的改进的盲自适应时域均衡器。该算法利用QR-RLS算法实现均衡器抽头的迭代计算,改善了Merry算法的收敛速度和鲁棒性。理论分析和仿真结果表明,该算法收敛速度明显优于Merry算法,且性能接近MSSNR算法最优解。     

Equalization of digital radio channels with large multipath delayfor cellular land mobile applications

A new maximum a posteriori (MAP) equalizer is proposed for digital radio links affected by large multipath delays. The “sparse” nature of the channel, where a few nonzero powerful taps are spaced by many negligible taps, is exploited to achieve a complexity proportional to the number of nonzero taps. When the channel is time-varying, an efficient nonlinear Kalman like channel estimator is employed to track only the nonzero taps     

Functional link neural network cascaded with Chebyshev orthogonal polynomial for nonlinear channel equalization

Nonlinear intersymbol interference (ISI) leads to significant error rate in nonlinear communication and digital storage channel. In this paper, therefore, a novel computationally efficient functional link neural network cascaded with Chebyshev orthogonal polynomial is proposed to combat nonlinear ISI. The equalizer has a simple structure in which the nonlinearity is introduced by functional expansion of the input pattern by trigonometric polynomial and Chebyshev orthogonal polynomial. Due to the input pattern and nonlinear approximation enhancement, the proposed structure can approximate arbitrarily nonlinear decision boundaries. It has been utilized for nonlinear channel equalization. The performance of the proposed adaptive nonlinear equalizer is compared with functional link neural network (FLNN) equalizer, multilayer perceptron (MLP) network and radial basis function (RBF) along with conventional normalized least-mean-square algorithms (NLMS) for different linear and nonlinear channel models. The comparison of convergence rate, bit error rate (BER) and steady state error performance, and computational complexity involved for neural network equalizers is provided.     

The Estimate Feedback Equalizer: A Suboptimum Nonlinear Receiver

A recursive nonlinear equalizer has been developed. The Bayes estimation theory has been used to obtain an unrealizable nonlinear minimum mean-square error (MMSE) receiver for the reception of binary pulse-amplitude-modulation signals in the presence of intersymbol interference and noise. A realizable approximation to the Bayes structure has been derived as the combination of a matched filter and a nonlinear recursive equalizer structure. The equalizer has been made adaptive using a new algorithm that defines and maintains the time frame of reference and is constrained so that the equalizer's parameters always move toward their optimum values. Computer simulations have been used to demonstrate the properties of the estimate feedback equalizer (EFE) and to compare its performance to that of presently known equalizers.     

Design of Sparse Filters for Channel Shortening

Channel shortening equalizers are used in acoustics to reduce reverberation, in error control decoding to reduce complexity, and in communication receivers to reduce inter-symbol interference. The cascade of a channel and channel shortening equalizer ideally produces an overall impulse response that has most of its energy compacted into fewer adjacent samples. Once designed, channel shortening equalizers filter the received signal on a per-sample basis and need to be adapted or re-designed if the channel impulse response changes significantly. In this paper, we evaluate sparse filters as channel shortening equalizers. Unlike conventional dense filters, sparse filters have a small number of non-contiguous non-zero coefficients. Our contributions include (1) proposing optimal and sub-optimal low complexity algorithms for sparse shortening filter design, and (2) evaluating impulse response energy compaction vs. design and implementation stage computational complexity tradeoffs for the proposed algorithms. We apply the proposed equalizer design procedures to (1) asymmetric digital subscriber line channels and (2) underwater acoustic communication channels. Our simulation results utilize measured channel impulse responses and show that sparse filters are able to achieve the same channel energy compaction with half as many coefficients as dense filters.     

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.      

A NEW ADAPTIVE EQUALIZER STRUCTURE FOR NONLINEAR CHANNELS

Based on the analysis of nonlinear channel models,a new connectionist model ofadaptive equalizer is constructed.Comparing with the connectionist model using the Volterraseries to extend the input vector space,the number of weights with the new structure is reducedsignificantly.It is shown by simulations that the weight values of the new scheme converge to theoptimal values closely for non-minimum phase channels as well minimum phase channels,if thechannel noise is small enough.Testing results of the BER(Bit Error Rate)tell us that the newadaptive equalizer for nonlinear channels is superior to the conventional linear equalizers in theequalization performances.     

An adaptive algorithm for differentially coherent detection in the presence of intersymbol interference

An adaptive equalization method is proposed for use with differentially coherent detection of M-ary differential phase-shift keying (DPSK) signals in the presence of unknown carrier frequency offset. A decision-feedback or a linear equalizer is employed, followed by the differentially coherent detector. The equalizer coefficients are adjusted to minimize the post-detection mean squared error. The error, which is a quadratic function of the equalizer vector, is used to design an adaptive algorithm of stochastic gradient type. The approach differs from those proposed previously, which linearize the post-detection error to enable the use of least mean squares (LMS) or recursive least squares (RLS) adaptive equalizers. The proposed quadratic-error (Q) algorithm has complexity comparable to that of LMS, and equal convergence speed. Simulation results demonstrate performance improvement over methods based on linearized-error (L) algorithm. The main advantages of the technique proposed are its simplicity of implementation and robustness to carrier frequency offset, which is maintained for varying modulation level.