Entropy minimization for supervised digital communications channelequalization

This paper investigates the application of error-entropy minimization algorithms to digital communications channel equalization. The pdf of the error between the training sequence and the output of the equalizer is estimated using the Parzen windowing method with a Gaussian kernel, and then, the Renyi's quadratic entropy is minimized using a gradient descent algorithm. By estimating Renyi's entropy over a short sliding window, an online training algorithm is also introduced. Moreover, for a linear equalizer, an orthogonality condition for the minimum entropy solution that leads to an alternative fixed-point iterative minimization method is derived. The performance of linear and nonlinear equalizers trained with entropy and mean square error (MSE) is compared. As expected, the results of training a linear equalizer are very similar for both criteria since, even if the input noise is non-Gaussian, the output filtered noise tends to be Gaussian. On the other hand, for nonlinear channels and using a multilayer perceptron (MLP) as the equalizer, differences between both criteria appear. Specifically, it is shown that the additional information used by the entropy criterion yields a faster convergence in comparison with the MSE     

A robust adaptive DFE receiver for DS-CDMA systems under multipathfading channels

This paper presents a novel receiver design from signal processing viewpoint for direct-sequence code-division multiple access (DS-CDMA) systems under multipath fading channels. A robust adaptive decision-feedback equalizer (DFE) is developed by using optimal filtering technique via minimizing the mean-square error (MSE). The multipath fading channels are modeled as tapped-delay-line filters, and the tap coefficients are described as Rayleigh distributions in order to imitate the frequency-selective fading channel. Then, a robust Kalman filtering algorithm is used to estimate the channel responses for the adaptation of the proposed DFE receiver under the situation of partially known channel statistics. The feedforward and feedback filters are designed by using not only the estimated channel responses but the uncertainties and error covariance of channel estimation as well. As shown in the computer simulations, the proposed adaptive DFE receiver is robust against the estimation errors and modeling dynamics of the channels. Hence, it is very suitable for receiver design in data transmissions through multipath fading channels encountered in most wireless communication systems     

Robust Tomlinson-Harashima Precoders for Multiuser MISO Downlink with Imperfect CSI

In this paper, we consider robust non-linear precoding for the downlink of a multiuser multiple-input single-output (MISO) communication system in the presence of imperfect channel state information (CSI). The base station (BS) is equipped with multiple transmit antennas and each user terminal is equipped with a single receive antenna. We propose two robust Tomlinson-Harashima precoder (THP) designs. The first design is based on the minimization of the total BS transmit power under constraints on the mean square error (MSE) at the individual user receivers. We show that this problem can be solved by an iterative procedure, where each iteration involves the solution of a pair of convex optimization problems that can be solved efficiently. A robust linear precoder with MSE constraints can be obtained as a special case of this robust THP. The second design is based on the minimization of a stochastic function of the sum MSE under a constraint on the total BS transmit power. We formulate this design problem as an optimization problem that can be solved by the method of alternating optimization, the application of which results in a second-order cone program that can be numerically solved efficiently. Simulation results illustrate the improvement in performance of the proposed precoders compared to other robust linear and non-linear precoders in the literature.     

Robust Kalman filters for linear time-varying systems withstochastic parametric uncertainties

We present a robust recursive Kalman filtering algorithm that addresses estimation problems that arise in linear time-varying systems with stochastic parametric uncertainties. The filter has a one-step predictor-corrector structure and minimizes an upper bound of the mean square estimation error at each step, with the minimization reduced to a convex optimization problem based on linear matrix inequalities. The algorithm is shown to converge when the system is mean square stable and the state space matrices are time invariant. A numerical example consisting of equalizer design for a communication channel demonstrates that our algorithm offers considerable improvement in performance when compared with conventional Kalman filtering techniques     

Robust$H_infty$Power Control for CDMA Cellular Communication Systems

Power control is an important factor for direct-sequence code division multiple access (DS-CDMA) cellular radio systems to achieve higher communication link quality and better system capacity. In order to track the desired signal-to-interference-plus-noise ratio (SINR) under round-trip delay, multiple access interference (MAI), channel fading, and noise, a time delay-based state-space model is developed for representing the tracking error dynamics and a state feedback controller is introduced for SINR tracking control. Then the power tracking problem can be regarded as a control problem. In this paper, a robust$H_infty$power tracking control design is proposed to achieve a robust optimal SINR tracking from the minimization of the worst-case effect point of view. This robust optimal power tracking design problem can be transformed to solving the eigenvalue problem (EVP) under some linear matrix inequality (LMI) constraints. The LMI Matlab toolbox can be used to efficiently solve the EVP via convex optimization to achieve a robust optimal SINR tracking design. Under the proposed distributed framework, the information of channel gain is not needed.     

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.<>     

一种不受信道阶数估计影响的SIMO直接盲均衡算法

针对单输入多输出（Single-Input-Multiple-Output, SIMO）模型提出一种完全不需要信道阶数估计的直接盲均衡算法。文章利用接收数据的截短协方差矩阵和信号子空间的关系设计一种零延迟均衡器,并通过信道矩阵和均衡器系数的合响应特性克服了算法相位偏转的问题,最后得到一种对信道阶数估计鲁棒并且没有相位偏转的盲均衡算法。该算法不同于一般子空间类算法,不需要直接对接收信号的协方差矩阵进行信号子空间和噪声子空间的分解,因此对信道阶数估计具有很强的鲁棒性。文章给出了算法的Batch实现过程,同时为更好适应一般时变信道环境和实现实时处理的要求,通过递归迭代得到算法的自适应实现过程。仿真实验表明该算法几乎不受信道阶数过估计或欠估计的影响,同时该算法具有良好的均方误差（Mean Square Error, MSE）和误符号率SER（Symbol Error Rate, SER）性能,并且具有很快的收敛速度。      

Statistical detection for communication channels with intersymbol interference

The optimum detector under a fixed delay constraint D is derived for channels having memory and additive noise. The resulting receiver is recursive and does not grow with the message length. Its structure is presented for linear channels (AM and PSK) with known (or estimated) impulse response and noise statistics. The intersymbol interference is assumed to extend for L sampling periods and the receiver is allowed a "look-ahead" at D future received samples. Simulation results using actual channel characteristics show the detector to outperform a transversal equalizer even for relatively small values of D.     

Robust joint optimization for MIMO AF multiple-relay systems with correlated channel uncertainties

In the paper, robust joint optimization of the source/relays precoders and destination equalizer is proposed for non-regenerative dual-hop multiple-input multiple-output (MIMO) amplify-and-forward (AF) multiple-relay systems with correlated channel uncertainties. By taking the imperfect channel state information (CSI) into consideration, the robust transceiver/relays joint optimization is developed based on the minimum mean-squared error (MMSE) criterion under individual power constraints at the source and relays. The optimization problem of precoding and amplifying matrices under power constraints belongs to neither concave nor convex so that a nonlinear matrix-form conjugate gradient (MCG) algorithm is applied to explore local optimal solutions. Simulation results illustrate that the robust transceiver/relays joint architecture for an AF-MIMO multiple-relay system outperforms the non-robust transceiver/relays design.     

An ATSC DTV receiver with improved robustness to multipath and distributed transmission environments

This paper presents the design and implementation of an 8-VSB DTV receiver for indoor and distributed transmission environments. The receiver is designed to handle severe multipath distortion from indoor and Single Frequency Network (SFN) transmission conditions. The architecture of the receiver is first introduced. The adaptive equalizer structure and design are then discussed in detail. A channel-matched filter is employed as a pre-filter such that the signal energies from different echoes are combined optimally and the signal to noise ratio of the equalizer input is maximized. Feedforward and feedback equalizers are used to handle the pre-echo (pre-cursor) and post-echo (post-cursor), respectively. The feedforward filter is designed to minimize the pre-cursors or convert them into post-cursors, while the feedback equalizer is used to eliminate the post-cursors. Initial tap coefficients are computed to speed up the convergence of these two filters based on the channel estimation. Laboratory tests show that the new prototype DTV receiver has very robust performance in multipath environments. 0 dB echoes can be handled with this receiver due to the enhanced design of the equalizer. It can withstand a -10 dB single echo within a -29.5 to +38.5 microsecond range and a 0 dB echo within a 12 microsecond range.     

Optimum linear joint transmit-receive processing for MIMO channels with QoS constraints

This paper considers vector communications through multiple-input multiple-output (MIMO) channels with a set of quality of service (QoS) requirements for the simultaneously established substreams. Linear transmit-receive processing (also termed linear precoder at the transmitter and linear equalizer at the receiver) is designed to satisfy the QoS constraints with minimum transmitted power (the exact conditions under which the problem becomes unfeasible are given). Although the original problem is a complicated nonconvex problem with matrix-valued variables, with the aid of majorization theory, we reformulate it as a simple convex optimization problem with scalar variables. We then propose a practical and efficient multilevel water-filling algorithm to optimally solve the problem for the general case of different QoS requirements. The optimal transmit-receive processing is shown to diagonalize the channel matrix only after a very specific prerotation of the data symbols. For situations in which the resulting transmit power is too large, we give the precise way to relax the QoS constraints in order to reduce the required power based on a perturbation analysis. We also propose a robust design under channel estimation errors that has an important interest for practical systems. Numerical results from simulations are given to support the mathematical development of the problem.     

Relationships between the constant modulus and Wiener receivers

The Godard (1980) or the constant modulus algorithm (CMA) is an effective technique for blind receiver design in communications. However, due to the complexity of the constant modulus (CM) cost function, the performance of the CM receivers has primarily been evaluated using simulations. Theoretical analysis is typically based on either the noiseless case or approximations of the cost function. The following question, while resolvable numerically for a specific example, remains unanswered in a generic manner. In the presence of channel noise, where are the CM local minima and what are their mean-squared errors (MSE)? In this paper, a geometrical approach is presented that relates the CM to Wiener (or minimum MSE) receivers. Given the MSE and the intersymbol/user interference of a Wiener receiver, a sufficient condition is given for the existence of a CM local minimum in the neighborhood of the Wiener receiver. The MSE bounds on CM receiver performance are derived and shown to be tight in simulations. The analysis shows that, while in some cases the CM receiver performs almost as well as the (nonblind) Wiener receiver, it is also possible that, due to its blind nature, the CM receiver may perform considerably worse than a (nonblind) Wiener receiver     

Adaptive DFE Multiuser Receiver for CDMA Systems using Two-Step LMS-Type Algorithm: An Equalization Approach

This paper presents an adaptive decision feedback equalizer (DFE) based multiuser receiver for code division multiple access (CDMA) systems over smoothly time-varying multipath fading channels using the two-step LMS-type algorithm. The frequency-selective fading channel is modeled as a tapped-delay-line filter with smoothly time-varying Rayleigh-distributed tap coefficients. The receiver uses an adaptive minimum mean square error (MMSE) multiuser channel estimator based on the reduced Kalman least mean square (RK-LMS) algorithm to predict these tap coefficients (Kohli and Mehra, Wireless Personal Communication 46:507–521, 2008). We propose the design of adaptive MMSE feedforward and feedback filters by using the estimated channel response. Unlike the previously available Kalman filtering algorithm based approach (Chen and Chen, IEEE Transactions on Signal Processing 49:1523–1532, 2001), the incorporation of RK-LMS algorithm reduces the computational complexity of multiuser receiver. The computer simulation results are presented to show the substantial improvement in its bit error rate performance over the conventional LMS algorithm based receiver. It can be inferred that the proposed multiuser receiver proves to be robust against the nonstationarity introduced due to channel variations, and it is also beneficial for the multiuser interference cancellation and data detection in CDMA systems.     

Robust hybrid beamforming design for millimeter-wave massive MIMO systems

Hybrid analog-digital beamforming is recognized as a promising solution for a practical implementation of massive multiple-input multiple-output(MIMO) systems based on millimeter-wave(mmWave) technology. In view of the overwhelming hardware cost and excessive power consumption and the imperfection of the channel state information(CSI), a robust hybrid beamforming design is proposed for the mmWave massive MIMO systems, where the robustness is defined with respect to imperfect knowledge or error of the CSI at the transmitter due to limited feedback and/or imperfect channel estimation. Assuming the errors of the CSI are bounded, the optimal hybrid beamforming design with robustness is formulated to a mean squared error(MSE) minimization problem. An iterative semidefinite programming(SDP) based algorithm is proposed to obtain the beamforming matrices. Simulation results show that the proposed robust design can provide more than 4 dB performance gain compared to that of non-robust design.     

基于总体最小二乘准则的OFDM系统时变信道估计

信道估计的准确程度直接影响正交频分复用系统的性能。为了提高时变信道估计算法的精度,基于总体最小二乘准则( TLS)提出了一种时变信道的估计方法。该方法用线性模型对时变信道进行建模,不仅考虑了信道噪声,同时也兼顾了模型误差。该方法能较好地跟踪信道的变化,显著消除模型误差。仿真结果表明所提算法的均方误差介于最小二乘算法与最小均方误差算法之间,在不同归一化多普勒频移下,该算法具有较好的稳健性。     

A novel expression for the achievable MSE performance obtained by blind adaptive equalizers

In this paper, I propose for the noisy, real, and two independent quadrature carrier case, an approximated closed-form expression for the achievable minimum mean square error (MSE) performance obtained by blind equalizers where the error that is fed into the adaptive mechanism which updates the equalizer’s taps can be expressed as a polynomial function of the equalized output of order three like in Godard’s algorithm. The proposed closed-form expression for the achievable MSE is based on the step-size parameter, on the equalizer’s tap length, on the channel power, on the signal to noise ratio (SNR), on the nature of the chosen equalizer, and on the input signal statistics. Since the channel power is measurable or can be calculated if the channel coefficients are given, there is no need anymore to carry out any simulation with various step-size parameters, different values for the signal to noise ratio (SNR) and equalizer’s tap length for a given equalization method, and input signal statistics in order to find the MSE performance in the convergence state.     

Joint MAP equalization and channel estimation forfrequency-selective and frequency-flat fast-fading channels

This paper presents a new fractionally-spaced maximum a posteriori (MAP) equalizer for data transmission over frequency-selective fading channels. The technique is applicable to any standard modulation technique. The MAP equalizer uses an expanded hypothesis trellis for the purpose of joint channel estimation and equalization. The fading channel is estimated by coupling minimum mean square error techniques with the (fixed size) expanded trellis. The new MAP equalizer is also presented in an iterative (turbo) receiver structure. Both uncoded and conventionally coded systems (including iterative processing) are studied. Even on frequency-flat fading channels, the proposed receiver outperforms conventional techniques. Simulations demonstrate the performance of the proposed equalizer     

Adaptive Interference Suppression for the Downlink of a Direct Sequence CDMA System with Long Spreading Sequences

A simple approach for adaptive interference suppression for the downlink (base-to-mobile link) of a direct sequence (DS) based cellular communication system is presented. The base station transmits the sum of the signals destined for the different mobiles, typically attempting to avoid intra-cell interference by employing orthogonal spreading sequences for different mobiles. However, the signal reaching any given mobile passes through a dispersive channel, thus destroying the orthogonality. In this paper, we propose an adaptive linear equalizer at the mobile that reduces interference by approximately restoring orthogonality. The adaptive equalizer uses the pilot's spreading sequence (which observes the same channel as the spreading sequence for the desired mobile) as training. Simulation results for the linear Minimum Mean Squared Error (MMSE) equalizer are presented, demonstrating substantial performance gains over the RAKE receiver. Long spreading sequences (which vary from symbol to symbol) are employed, so that the equalizer adapts not to the time-varying spreading sequences, but to the slowly varying downlink channel. Since the inter-cell interference from any other base station also has the structure of many superposed signals passing through a single channel, the adaptive equalizer can also suppress inter-cell interference, with the tradeoff between suppression of intra- and inter-cell interference and noise enhancement depending on their impact on the Mean Squared Error (MSE).     

Convergence analysis of the constant modulus algorithm

We study the global convergence of the stochastic gradient constant modulus algorithm (CMA) in the absence of channel noise as well as in the presence of channel noise. The case of fractionally spaced equalizer and/or multiple antenna at the receiver is considered. For the noiseless case, we show that with proper initialization, and with small step size, the algorithm converges to a zero-forcing filter with probability close to one. In the presence of channel noise such as additive Gaussian noise, we prove that the algorithm diverges almost surely on the infinite-time horizon. However, under suitable conditions, the algorithm visits a small neighborhood of the Wiener filters a large number of times before ultimately diverging.     

A Minimum Mean-Square Error Equalizer for Nonlinear Satellite Channels

The problem of designing and evaluating the performance of a minimum mean-square error equalizer (MMSEE) for binary PSK transmission over band-limited nonlinear satellite channels is considered in this correspondence. The effect of intersymbol interference followed by AM/AM and AM/PM conversions are taken into account while optimizing the performance in the presence of the downlink white Gaussian noise. In analyzing the problem, the decision is made on a typical signal in a received sequence taking into account past and future interfering signals, i.e., ISI. As an illustrative example of the receiver, a typical channel model is considered in details. Based on the analysis, an alternative receiver structure which is more suitable for implementation is introduced. The taps gain coefficients for minimum mean-square error, between the received sample and the actual transmitted bit, are obtained using numerical methods. The performance of the equalizer is evaluated using computer simulation techniques and it is shown that significant performance improvement over the single-sample sign detector can be obtained.