Doubly-selective fading channel equalization: A comparison of the Kalman filter approach with the basis expansion model-based equalizers

In this paper, we exploit the Kalman filter as a time-varying linear minimum mean-square error equalizer for doubly-selective fading channels. We use a basis expansion model (BEM) to approximate the doubly-selective channel impulse response. Several time-varying linear equalizers have been proposed in the literature where both the channel and the equalizer impulse responses are approximated by complex exponential (CE) BEMs. Our proposed Kalman filter formulation does not rely on a specific BEM for the underlying channel, therefore, it can be applied to any BEM, including the CE-BEM and the discrete prolate spheroidal (DPS) BEM. Moreover, the Kalman filter relies solely on the channel model and therefore, does not incur any approximation error inherent in the CE-BEM representation of the equalizer. Through computer simulations, we show that compared to two of the existing algorithms, the proposed Kalman filter formulation yields the same or an improved bit error rate at a much lower computational cost, where the latter is measured in terms of the number of flops needed for the equalizer design and implementation.     

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.<>     

Second-order statistical approaches to channel shortening in multicarrier systems

In multicarrier systems with a cyclic prefix, interblock interference can be limited by shortening the channel impulse response using time-domain equalizers. In this paper, we present two blind channel-shortening algorithms that exploit the second-order statistics of the channel outputs. The equalizer parameter vector is chosen from a space that forces the length constraint on the effective channel impulse response. The first algorithm is less sensitive to channel-order estimation errors, whereas the second one is simpler to implement. Furthermore, we show that the two algorithms are equivalent for noiseless channels. Simulation examples are provided to demonstrate their superior performance over some existing algorithms.     

A channel subspace post-filtering approach to adaptive least-squares estimation

A major challenge while communicating in dynamic channels, such as the underwater acoustic channel, is the large amount of time-varying inter-symbol interference (ISI) due to multipath. In many realistic channels, the fluctuations between different taps of the sampled channel impulse response are correlated. Traditional least-squares algorithms used for adapting channel equalizers do not exploit this correlation structure. A channel subspace post-filtering algorithm is presented that treats the least-squares channel estimate as a noisy time series and exploits the channel correlation structure to reduce the channel estimation error. The improvement in performance of the post-filtered channel estimator is predicted theoretically and demonstrated using both simulation and experimental data. Experimental data is also used to demonstrate the improvement in performance of a channel estimate-based decision feedback equalizer that uses this post-filtered channel estimate to determine the equalizer coefficients.     

Modified Metaheuristic Algorithms for the Optimal Design of Multiplier-Less Non-uniform Channel Filters

Reconfigurable non-uniform channel filters are now being widely used in software define radio (SDR). The hardware implementation of these filters requires low complexity, low chip area and low power consumption. The frequency response masking (FRM) approach is proved to be a good candidate for the realization of a sharp digital finite impulse response (FIR) filter with low complexity. To reduce the complexity further, this paper gives an optimal design method which makes the channel filters totally multiplier-less. This is done in two steps. The channel filters are designed using the FRM approach with continuous filter coefficients. To obtain multiplier-less design, these filter coefficients are converted to finite-precision coefficients using signed power of two (SPT) space and the filter coefficients are synthesized in the canonic signed-digit (CSD) format. But this may lead to degradation of the filter performance. Hence the filter coefficients synthesis in the CSD format is formulated as an optimization problem. Several meta-heuristic algorithms like Differential Evolution (DE), Artificial Bee Colony (ABC), Harmony Search Algorithm (HSA) and Gravitational Search Algorithm (GSA) are modified and deployed and the best one is selected.     

Performance bounds for multistep prediction-based blindequalization

Blind equalization attempts to remove the interference caused by a communication channel without using any known training sequences. Blind equalizers may be implemented with linear prediction-error filters (PEFs). For many practical channel types, a suitable delay at the output of the equalizer allows for achieving a small estimation error. The delay cannot be controlled with one-step predictors. Consequently, multistep PEF-based algorithms have been suggested as a solution to the problem. The derivation of the existing algorithms is based on the assumption of a noiseless channel, which results in zero-forcing equalization. We consider the effects of additive noise at the output of the multistep PEF. Analytical error bounds for two PEF-based blind equalizers in the presence of noise are derived. The obtained results are verified with simulations. The effect of energy concentration in the channel impulse response on the error bound is also addressed     

Performance of Reduced-Rank Equalization

We evaluate the performance of reduced-rank equalizers for both single-input single-output (SISO) and multiple-input multiple-output (MIMO) frequency-selective channels. Each equalizer filter is constrained to lie in a Krylov subspace, and can be implemented as a reduced-rank multistage Wiener filter (MSWF). Both reduced-rank linear and decision-feedback equalizers (DFEs) are considered. Our results are asymptotic as the filter length goes to infinity. For SISO channels, the output mean-squared error (MSE) is expressed in terms of the moments of the channel spectrum. For MIMO channels, both successive and parallel interference cancellation are considered. The asymptotic performance in that case requires the computation of moments, which depend on shifted versions of the channel impulse response for different users. Those are also expressed in terms of the MIMO channel frequency response. Numerical results are presented, which show that near full-rank performance can be achieved with relatively low-rank equalizers     

A new adaptive equalizer with channel estimator for mobile radiocommunication

For unknown mobile radio channels with severe intersymbol interference (ISI), a maximum likelihood sequence estimator, such as a decision feedback equalizer (DFE) having both feedforward and feedback filters, needs to handle both precursors and postcursors. Consequently, such an equalizer is too complex to be practical. This paper presents a new reduced-state, soft decision feedback Viterbi equalizer (RSSDFVE) with a channel estimator and predictor. The RSSDFVE uses maximum likelihood sequence estimation (MLSE) to handle the precursors and truncates the overall postcursors with the soft decision of the MLSE to reduce the implementation complexity. A multiray fading channel model with a Doppler frequency shift is used in the simulation. For fast convergence, a channel estimator with fast start-up is proposed. The channel estimator obtains the sampled channel impulse response (CIR) from the training sequence and updates the RSSDFVE during the bursts in order to track changes of the fading channel. Simulation results show the RSSDFVE has nearly the same performance as the MLSE for time-invariant multipath fading channels and better performance than the DFE for time-variant multipath fading channels with less implementation complexity than the MLSE. The fast start-up (FS) channel estimator gives faster convergence than a Kalman channel estimator. The proposed RSSDFVE retains the MLSE structure to obtain good performance and only uses soft decisions to subtract the postcursor interference. It provides the best tradeoff between complexity and performance of any Viterbi equalizers     

Applications of Linear Programming to the Time-Domain Design of Digital-Filter Equalizers

A time-domain technique is presented for the design of frequency-sampling and transversal digital filters for use in equalizing channels with known impulse responses. The technique uses a linearprogramming algorithm to specify filter multiplier coefficients that satisfy constraint equations based upon specified values and derivatives of the desired output signal. It is shown that the transversal equalizer design requires fewer multipliers and performs in a superior fashion in the presence of timing jitter. The effects of changing the number and type of constraints upon such parameters as equalizer complexity, transmitted energy, and residual distortion are presented.     

Determination of Tap Positions for Sparse Equalizers

Sparse equalizers, in which only a small subset of the filter taps is selected to be nonzero, were recently proposed as a low-complexity solution for receivers operating in wireless frequency-selective channels with sparse power profiles. The performance of the sparse equalizer heavily depends on its tap-positioning algorithm. This paper presents efficient low-complexity algorithms for determination of sparse equalizer tap positions based on a forward sequential search. We develop low-complexity metrics for the evaluation of the candidate tap positions in the search space as well as methods to effectively reduce the search space size. The proposed algorithms are shown to be superior over previously proposed algorithms in a wide range of channel conditions. Actually, the proposed algorithms yield, in most of the tested cases, performance identical to the optimal, prohibitively complex, tap-positioning algorithm. The main emphasis is on linear equalization suitable for wideband code-division multiple-access systems but the algorithm can be extended to a variety of equalization schemes and channels.     

The Choice of the Desired Impulse Response in Combined Linear-Viterbi Algorithm Equalizers

Equalizer structures using the Viterbi Algorithm achieve at least order of magnitude performance improvement over linear equalizers on some intersymbol interference channels. Using a linear equalizer to shape the original channel impulse response to some shorter desired impulse response (DIR) is a technique which reduces the complexity of the Viterbi Algorithm equalizer. This paper looks at three techniques for choosing a DIR. These are choosing the DIR by truncation, minimum mean square error and matching the power spectrum to that of the original channel. Using effective signal to noise ratio as the figure of merit for comparison, results are given for one particular channel.     

Channel‐estimate‐based frequency‐domain equalization (CE‐FDE) for broadband single‐carrier transmission

A channel‐estimate‐based frequency‐domain equalization (CE‐FDE) scheme for wireless broadband single‐carrier communications over time‐varying frequency‐selective fading channels is proposed. Adaptive updating of the FDE coefficients are based on the timely estimate of channel impulse response (CIR) to avoid error propagation that is a major source of performance degradation in adaptive equalizers using least mean square (LMS) or recursive least square (RLS) algorithms. Various time‐domain and frequency‐domain techniques for initial channel estimation and adaptive updating are discussed and evaluated in terms of performance and complexity. Performance of uncoded and coded systems using the proposed CE‐FDE with diversity combining in different time‐varying, multi‐path fading channels is evaluated. Analytical and simulation results show the good performance of the proposed scheme suitable for broadband wireless communications. For channels with high‐Doppler frequency, diversity combining substantially improves the system performance. For channels with sparse multi‐path propagation, a tap‐selection strategy used with the CE‐FDE systems can significantly reduce the complexity without sacrificing the performance. Copyright © 2004 John Wiley & Sons, Ltd.     

Turbo space-time equalization of TCM for broadband wireless channels

This paper presents a space-time turbo (iterative) equalization method for trellis-coded modulation (TCM) signals over broadband wireless channels. For fixed wireless systems operating at high data rates, the multipath delay spread becomes large, making it impossible to apply trellis-based equalization methods. The equalizer proposed here consists of a broadband beamformer which processes antenna array measurements to shorten the observed channel impulse response, followed by a conventional scalar turbo equalizer. Since the applicability of trellis-based equalizers is limited to additive white noise channels, the beamformer is required to preserve the whiteness of the noise at its output. This constraint is equivalent to requiring that the finite-impulse response (FIR) beamforming filters must have a power complementarity property. The power complementarity property imposes nonnegative definite quadratic constraints on the beamforming filters, so the beamformer design is expressed as a constrained quadratic optimization problem. The composite channel impulse response at the beamformer output is shortened significantly, making it possible to use a turbo equalizer for the joint equalization and decoding of trellis modulated signals. The proposed receiver structure is simulated for two-dimensional TCM signals such as 8-PSK and 16-QAM and the results indicate that the use of antenna arrays with only two or three elements allows a large decrease in the channel signal-to-noise ratio needed to achieve a 10/sup -4/ bit-error rate.     

Set-membership adaptive equalization and an updator-sharedimplementation for multiple channel communications systems

This paper considers the problems of channel estimation and adaptive equalization in the novel framework of set-membership parameter estimation. Channel estimation using a class of set-membership identification algorithms known as optimal bounding ellipsoid (OBE) algorithms and their extension to tracking time-varying channels are described. Simulation results show that the OBE channel estimators outperform the least-mean-square (LMS) algorithm and perform comparably with the RLS and the Kalman filter. The concept of set-membership equalization is introduced along with the notion of a feasible equalizer. Necessary and sufficient conditions are derived for the existence of feasible equalizers in the case of linear equalization for a linear FIR additive noise channel. An adaptive OBE algorithm is shown to provide a set of estimated feasible equalizers. The selective update feature of the OBE algorithms is exploited to devise an updator-shared scheme in a multiple channel environment, referred to as updator-shared parallel adaptive equalization (USHAPE). U-SHAPE is shown to reduce the hardware complexity significantly. Procedures to compute the minimum number of updating processors required for a specified quality of service are presented     

一种基于信道缩短的深衰落稀疏均衡改进算法

针对较低信噪比下的深衰落稀疏多径信道,提出了一种基于信道缩短的自适应稀疏均衡改进算法。该算法采用前置分数间隔信道缩短均衡器与后置自适应稀疏均衡器级联的均衡器结构,其中,首先利用短训练序列设计基于最小均方误差准则的前置均衡器,前置均衡器与稀疏多径信道级联后得到能量集中于较短时间区域且分布稀疏的等效信道,使得原始信道的深衰落畸变得到部分有效补偿;然后采用能实现稀疏信号重构的随机梯度追踪算法调整后置自适应均衡器的抽头系数,后置均衡器用于消除等效信道的剩余符号间干扰。仿真结果表明,与传统的单级分数间隔自适应均衡器相比,该算法具有收敛速度快和运算复杂度低的优点。     

Adaptive MLSE equalizers with parametric tracking for multipathfast-fading channels

We use the parametric channel identification algorithm proposed by Chen and Paulraj (see Proc. IEEE Vehicular Technology Conf., p.710-14, 1997) and by Chen, Kim and Liang (see IEEE Trans. Veh. Technol., p.1923-35, 1999) to adaptively track the fast-fading channels for the multichannel maximum likelihood sequence estimation (MLSE) equalizer using multiple antennas. Several commonly-used channel tracking schemes, decision-directed recursive least square (DD/RLS), per-survivor processing recursive least square (PSP/RLS) and other reduced-complexity MLSE algorithms are considered. An analytic lower bound for the multichannel MLSE equalizer with no channel mismatch in the time-varying specular multipath Rayleigh-fading channels is derived. Simulation results that illustrate the performance of the proposed algorithms working with various channel tracking schemes are presented, and then these results are compared with the analytic bit error rate (BER) lower bound and with the conventional MLSE equalizers directly tracking the finite impulse response (FIR) channel tap coefficients. We found that the proposed algorithm always performs better than the conventional adaptive MLSE algorithm, no matter what channel tracking scheme is used. However, which is the best tracking scheme to use depends on the scenario of the system     

Fast coupled adaptation for sparse impulse responses using a partial haar transform

This work presents a novel scheme for identifying the impulse response of a sparse channel. The scheme consists of two adaptive filters operating sequentially. The first adaptive filter adapts using a partial Haar transform of the input and yields an estimate of the location of the peak of the sparse impulse response. The second adaptive filter is then centered about this estimate. Both filters are short in comparison to the delay uncertainty of the unknown channel. The principle advantage of this scheme is that two short adaptive filters can be used instead of one long adaptive filter, resulting in faster overall convergence and reduced computational complexity and storage. The scheme is analyzed in detail for a least mean squares (LMS) LMS-LMS type of structure, although it can be implemented using any combination of adaptive algorithms. Monte Carlo simulations are shown to be in good agreement with the theoretical model for the behavior of the peak estimating filter as well as for the mean square error (MSE) behavior of the second filter.     

An adaptive maximally decimated channelized UWB receiver with cyclic prefix

The frequency channelized receiver based on hybrid filter bank is a promising receiver structure for ultra-wideband (UWB) radio because of its relaxed circuit requirements and robustness to interference. The uncertainties in the analog analysis filters and the time varying nature of the propagation channels necessitate adaptive methods in practical frequency channelized receivers. Adaptive synthesis filters, however, suffer from slow convergence speed especially when maximally decimated to reduce the analog-digital converter sampling frequency. To improve the convergence speed, the cyclic prefix is applied to the transmitted data. The propagation channel and the channelizer can then be modeled as a circulant matrix and block CM, respectively. Such matrix representation enables the transmitted data to be recovered by two cascaded one-tap equalizers, one of which corresponds to the channelizer and the other to the propagation channel. The cascaded structure is attractive as it allows the estimation of the propagation channel and the channelizer, which vary at vastly different rates, to be updated separately. Adaptive algorithms for both the fractionally spaced equalizer and the symbol spaced equalizer are derived. After initial convergence during startup, the adaptive performance of the channelized receiver to different propagation channels is similar to that of an ideal full band receiver.     

An IIR adaptive electronic equalizer for polarization multiplexed fiber optic communication systems

An electronic digital equalizer for polarization multiplex coherent fiber optic communication systems is designed to compensate polarization mode dispersion (PMD) and residual chromatic dispersion (CD) of transmission channel. The proposed equalizer is realized with fraction spaced infinite impulse response (IIR) butterfly structure with 21 feedforward taps and 2 feedback taps. Compared with finite impulse response (FIR) structure, this structure can reduce implementation complexity of hardware under the same condition. To keep track of the random variation of channel characteristics, the filter weights are updated by least mean square (LMS) algorithm. The simulation results show that the proposed equalizer can compensate residual chromatic dispersion (CD) of 1600 ps/nm and differential group delay (DGD) of 90 ps simultaneously, and also can increase the PMD and residual CD tolerance of the whole communication system.     

基于Hammerstein模型的非线性信道广义线性盲均衡算法

为了提高非线性信道盲均衡的性能、降低运算复杂度,本文以Hammerstein模型代替传统的Volterra级数模型来模拟非线性信道,利用非线性信道接收信号呈现非圆性的特点,构造了一种新的基于Wiener非线性模型的广义线性盲均衡器,并在常模准则的基础上提出了NCWL-CMA和NCWL-CMA Newton-like两种非线性信道广义线性盲均衡器抽头系数更新算法.理论分析和仿真实验结果表明,与传统盲均衡算法相比,新算法显著地降低了剩余码间干扰,提高了收敛速度.