Finite dictionary techniques for MSER equalization in RKHS

Adaptive channel equalization is a signal processing technique to mitigate inter-symbol interference in a time dispersive channel. For adaptive equalization, minimum mean square error (MMSE) criterion-based reproducing kernel Hilbert spaces (RKHS) approaches such as the kernel least mean squares (KLMS) algorithm and its variants have been suggested in the literature for nonlinear channels. Another optimality criterion, based on minimum bit/symbol error rate (MBER/MSER), is a better choice for adapting an equalizer as compared to MMSE criterion. A kernel-based minimum symbol error rate (KMSER) equalization algorithm combines minimum symbol error rate (MSER)-based approaches with RKHS techniques. However, most algorithms in RKHS such as KMSER/KLMS require infinite storage requirement and hence cannot be practically implemented. To curtail the infinite memory requirement, and make adaptive algorithm suitable for implementation with finite memory and processing power, we propose quantized KMSER (QKMSER) and fixed-budget quantized KMSER (FBQKMSER)-based equalizers in this paper. In this paper, we derive the dynamical equation for MSE evolution of the QKMSER and FBQKMSER and find their performance to be asymptotically close to the MSE behavior of the KMSER. Also, it is found via simulations that the tracking performance of FBQKMSER is better than all the compared algorithms in this paper which is particularly useful for non-stationary 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.     

最低误码率非线性均衡器的快速自适应学习算法

针对最低误码率非线性均衡器的参数在线自适应学习问题,本文提出基于拟牛顿方法的快速自适应学习算法。采用Parzen窗函数方法估计误码率,通过设定切换条件,使参数学习在滑窗随机梯度法与滑窗拟牛顿法之间切换。这既增加了新算法的数值稳定性,又可提高收敛速度。通过对拟牛顿方法进行修改,还使新算法既可以在线自适应学习,也可用于高维参数的快速学习。仿真采用最低误码率非线性均衡器对通信系统进行干扰抑制和信道均衡,结果表明了新算法的高效性。     

Low complexity soft-input soft-output block decision feedback equalization

A low complexity soft-input soft-output (SISO) block decision feedback equalizer (BDFE) is presented for turbo equalization. The proposed method employs a sub-optimum sequence-based detection, where the soft-output of the equalizer is calculated by evaluating an approximation of the sequence-based a posteriori probability (APP) of the data symbol. The sequence-based APP approximation is enabled by the adoption of both soft a priori information and soft decision feedback, and it leads to better performance and faster convergence compared to symbol-based detection methods as used by most other low complexity equalizers. The performance and convergence property of the proposed algorithm is analyzed by using extrinsic information transfer (EXIT) chart. Both analytical and simulation results show that the new equalizer can achieve a performance similar to that of trellis-based equalization algorithms, with a complexity similar to linear SISO minimum mean square error equalizers.     

基于概率软切换的两级双模盲均衡器

文中提出一种基于概率软切换的两级双模盲均衡器。它实时统计两级盲均衡器输出硬判决值相同的概率,并利用它切换盲均衡算法。该均衡器结合了级联两级均衡结构和双模算法的优点。仿真表明,它能够纠正相位偏移,相对于波特间隔（BSE）的并发常模＋判决导引（CMA＋DD）盲均衡器,以非常小的计算复杂度代价,获得稳态均方误差（MSE）性能和误比特率（BER）性能的较大提高。     

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.      

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     

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     

Block channel equalization in the frequency domain

In this paper, channel equalization algorithms processing two samples of the received signal per channel symbol and operating in the frequency domain are described in a unifying framework. First, minimum mean-square error linear and decision-feedback equalizers are derived, and a synthesis technique based on the well-known Levinson-Durbin algorithm is proposed for the latter. Then, iterative linear and decision-feedback equalization algorithms for turbo processing are devised. Performance results for both uncoded and coded phase-shift keying transmissions show the efficacy of the proposed equalization techniques and their superiority over other existing frequency-domain equalization strategies.     

On Blind Equalization of Biorthogonal Signaling

Motivated by increasing interest in energy efficient modulations, we provide the first look at adaptive equalization of biorthogonal signaling. While this modulation has historically been considered only for use in narrowband systems without intersymbol interference (ISI), recent attention has been given to its use in ISI channels. Due to the fact that biorthogonal modulation (BOM) results in a source that is not i.i.d., however, classical blind adaptive equalization techniques cannot be directly applied to equalization of BOM signals. We first examine minimum mean-squared error (MMSE) and least mean squares (LMS)-based equalizers, and identify some peculiarities that arise in equalization of BOM signals when compared to more traditional modulations like binary phase shift keying (BPSK). Next, we present two novel blind algorithms for the adaptive equalization of BOM signals: LTBOMB and TROMBONE. We discuss the convergence properties of these algorithms, and demonstrate their performance with numerical simulations     

Multichannel Algorithms for Simultaneous Equalization and Interference Suppression

This paper proposes techniques for simultaneous cancellation of intersymbol and interchannel or multi-access interference (ISI and ICI) that shows up in several multi-input, multi-output (MIMO) communication channels. Correlation and kurtosis based optimization criteria are derived for multi-channel decision feedback equalizers (MC-DFE) and compared with the popular Godard algorithm (CMA) and the minimum mean-square error in a decision directed mode (MMSE-DD). The proposed adaptive algorithms are easily extended to a scenario with more than two users with the computational complexity increasing linearly with the number of inputs. Simulation results show that the algorithms converge to the global minimum in a blind environment with channels that introduce moderate distortion.     

Space–Time Turbo Equalization With Successive Interference Cancellation for Frequency-Selective MIMO Channels

This paper addresses the issue of iterative space–time equalization for multiple-input–multiple-output (MIMO) frequency-selective fading channels. A new soft equalization concept based on successive interference cancellation (SIC) is introduced for a space–time bit-interleaved coded modulation (STBICM) transmission. The proposed equalizer allows us to separate intersymbol interference (ISI) and multiantenna interference (MAI) functions. Soft ISI is successively suppressed using a low-complexity suboptimum minimum mean square error (MMSE) criterion. The decoupling of ISI and MAI offers more flexibility in the design of the whole space–time equalizer. Different multiantenna detection criteria can be considered, ranging from simple detectors to the optimal maximum a posteriori (MAP) criterion. In particular, we introduce two soft equalizers, which are called SIC/SIC and SIC/MAP, and we show that they can provide a good performance-to-complexity tradeoff for many system configurations, as compared with other turbo equalization schemes. This paper also introduces an MMSE-based iterative channel state information (CSI) estimation algorithm and shows that attractive performance can be achieved when the proposed soft SIC space–time equalizer iterates with the MMSE-based CSI estimator.      

Space diversity reception and parallel blind equalization inshort-burst TDMA systems

We consider a q-antenna space diversity system combined with two parallel equalizer branches for the reception of short-burst time-division multiple-access signals. By applying a combination of several previously proposed blind equalization algorithms, we make significant improvements over the burst error probability performance reported to date. This is achieved by starting the two blind equalizers from different initial tap settings and applying a specific weighting of the equalizers' outputs in order to derive the symbol decision. A burst error probability of less than 10^-3 is achieved with the new strategy for a root mean square (rms) delay spread of less than 0.4 times the symbol duration     

复值的MBER非线性均衡器用于QAM信号

本文提出一种复值的最低误码率非线性滤波器用于非线性信道中QAM信号的均衡.推导了针对QAM信号的最低误码率准则训练算法的目标函数,并用Voherra序列来实现复值的非线性均衡器.为使非线性均衡器能在线自适应训练并增加训练算法的数值稳定性,提出～种滑窗随机梯度算法.大量仿真表明,对于非线性信道中QAM信号的均衡,最低误码率非线性均衡器的性能优于最小均方误差准则.     

Stochastic Unbiased Minimum Mean Error Rate Algorithm for Decision Feedback Equalizers

Capitalizing on a well-known minimum mean-square error (MMSE) property for decision feedback equalization (DFE) along with the use of stochastic gradient approach, we formulate an adaptive minimum error rate (MER) algorithm for DFE over M-ary PAM channels to be named as stochastic unbiased minimum mean-error rate (SUMMER). Comparisons are made between our algorithm and existing MER algorithms in the literature. Also, by invoking the central limit theorem, we present an analytical proof that an unbiased MMSE equalizer will approach an MER equalizer when the equalizer length approaches infinity; thereby, we obtain a lower bound expression for MER.     

Adaptive H∞ channel equalization for wirelesspersonal communications

In this paper, a new adaptive H_∞ filtering algorithm is developed to recursively update the tap-coefficient vector of a decision feedback equalizer (DFE) in order to adaptively equalize the time-variant dispersive fading channel of a high-rate indoor wireless personal communication system. Different from conventional L ₂ (such as the recursive least squares (RLS)) filtering algorithms which minimize the squared equalization error, the adaptive H _∞ filtering algorithm is a worst case optimization. It minimizes the effect of the worst disturbances (including input noise and modeling error) on the equalization error. Hence, the DFE with the adaptive H_∞ filtering algorithm is more robust to the disturbances than that with the RLS algorithm. Computer simulation demonstrates that better transmission performance can be achieved using the adaptive H_∞ algorithm when the received signal-to-noise ratio (SNR) is larger than 20 dB     

Adaptive Equalization of the Slow Fading Channel

This paper considers equalization of the slow fading channel for a serial data transmission application. Linear and decision-feedback adaptive equalization techniques are contrasted. The error propagation effect in decision-feedback equalizers is analyzed by a Markov process model. The error probability magnification is computed for both fixed and fading channels and for both binary and quaternary phase-shift-keying (PSK) transmission. The results show that the error propagation effect is small and in regions of practical error probabilities the decision-feedback equalizer is superior to its linear counterpart. Parameters of a practical decisionfeedback equalizer are estimated and a performance evaluation is performed. The implicit diversity gain is shown to be significant and the intersymbol interference penalty is found to be less than 1 dB. Because the intersymbol interference penalty is small, more complex nonlinear processors such as the Viterbi algorithm cannot be recommended for this application. Time jitter effects for the equalizer are included in a calculation of average error probability.     

Multichannel Fast QR-Decomposition Algorithms: Weight Extraction Method and Its Applications

Multichannel fast QR decomposition RLS (MC-FQRD-RLS) algorithms are well known for their good numerical properties and low computational complexity. The main limitation is that they lack an explicit weight vector term, limiting themselves to problems seeking an estimate of the output error signal. This paper presents techniques which allow us to use MC-FQRD-RLS algorithms with applications that previously have required explicit knowledge of the adaptive filter weights. We first consider a multichannel system identification setup and present how to obtain, at any time, the filter weights associated with the MC-FQRD-RLS algorithm. Thereafter, we turn to problems where the filter weights are periodically updated using training data, and then used for fixed filtering of a useful data sequence, e.g., burst-trained equalizers. Finally, we consider a particular control structure, indirect learning, where a copy of the coefficient vector is filtering a different input sequence than that of the adaptive filter. Simulations are carried out for Volterra system identification, decision feedback equalization, and adaptive predistortion of high-power amplifiers. The results verify our claims that the proposed techniques achieve the same performance as the inverse QRD-RLS algorithm at a much lower computational cost.     

MIMO DFE equalization for multitone DS/SS systems over multipathchannels

The combination of multitone modulation with direct sequence spectrum spreading (DS/SS) has been introduced in the past. The performance of a correlation receiver has been evaluated for a multipath channel and in the presence of an additional multiple access interference. We analyze the problem of decision feedback equalization (DFE) for such a system. In order to understand the potential of the system with equalization, we first study the steady-state behavior of the equalizer for a minimum mean square error (MMSE) criterion. The investigation is carried out for a receiver made of a bank of filters matched to both the symbol shape and the channel, and for a two path channel. Assuming transmission of binary phase shift keying (BPSK) symbols, an exact expression of the bit error probability is obtained in the form of an integral. Then adaptive least mean square (LMS) and recursive least square (RLS) structures are derived. The performance of the adaptive RLS algorithm is demonstrated by means of computer simulations