What are the analytical conditions for which a blind equalizer will loose the convergence state?

In this paper, I propose for the noiseless, real and two independent quadrature carrier case some approximated conditions on the step-size parameter, on the equalizer’s tap length and on the channel power, related to the nature of the chosen equalizer and input signal statistics, for which a blind equalizer will not converge anymore. These conditions are valid for type of blind equalizers where the error that is fed into the adaptive mechanism that updates the equalizer’s taps can be expressed as a polynomial function of the equalized output of order three like in Godard’s algorithm. 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 and equalizer’s tap length for a given equalization method and input signal statistics in order to find the maximum step-size parameter for which the equalizer still converges.     

Mutual information and minimum mean-square error in Gaussian channels

This paper deals with arbitrarily distributed finite-power input signals observed through an additive Gaussian noise channel. It shows a new formula that connects the input-output mutual information and the minimum mean-square error (MMSE) achievable by optimal estimation of the input given the output. That is, the derivative of the mutual information (nats) with respect to the signal-to-noise ratio (SNR) is equal to half the MMSE, regardless of the input statistics. This relationship holds for both scalar and vector signals, as well as for discrete-time and continuous-time noncausal MMSE estimation. This fundamental information-theoretic result has an unexpected consequence in continuous-time nonlinear estimation: For any input signal with finite power, the causal filtering MMSE achieved at SNR is equal to the average value of the noncausal smoothing MMSE achieved with a channel whose SNR is chosen uniformly distributed between 0 and SNR.     

Nonlinear effects in LMS adaptive equalizers

An adaptive transversal equalizer based on the least-mean-square (LMS) algorithm, operating in an environment with a temporally correlated interference, can exhibit better steady-state mean-square-error (MSE) performance than the corresponding Wiener filter. This phenomenon is a result of the nonlinear nature of the LMS algorithm and is obscured by traditional analysis approaches that utilize the independence assumption (current filter weight vector assumed to be statistically independent of the current data vector). To analyze this equalizer problem, we use a transfer function approach to develop approximate analytical expressions of the LMS MSE for sinusoidal and autoregressive interference processes. We demonstrate that the degree to which LMS may outperform the corresponding Wiener filter is dependent on system parameters such as signal-to-noise ratio (SNR), signal-to-interference ratio (SIR), equalizer length, and the step-size parameter     

地空衰落信道下JTIDS信号信噪比盲估计

针对地空衰落信道下联合战术信息分发系统（JTIDS）信号信噪比的估计问题,提出了一种适用于单脉冲和双脉冲时隙结构JTIDS信号的接收信噪比盲估计方法。首先,利用信道实时最大时延粗估计值确定无符号间干扰的数据区间;其次,基于JTIDS信号脉冲时隙结构中静默时间的保护间隔特性估计噪声功率;最后,根据选定区间数据的自相关函数估计接收信号的信号功率,从而估计出接收链路的信噪比。仿真实验结果表明,取单时隙样本长度时,在巡航、起降两种空中平台状态下,提出估计方法在较宽信噪比范围内具有良好的均方误差估计性能,可满足JTIDS系统的信噪比估计需求。     

相干光通信中相位估计与信道均衡的联合使用

在使用高阶相位调制的100Gbit/s及以上速率的相干光通信系统中,激光器的相位噪声和光纤传输中的色散效应是影响系统性能的主要因素。为了补偿色散及相位噪声的影响,文章提出一种将电信道均衡与相位估计模块在电域联合使用的方法,相位估计在电信道均衡之后完成,其结果同时用于电信道均衡器的抽头系数更新。该方法可以有效降低对电信道均衡中迭代步长的要求,有助于系统获得更低的误码率和更稳定的误码性能,并且能够支持的传输距离也会增加。     

隐含训练序列信道估计中的功率分配

与传统时分发送训练序列的信道估计算法相比,隐含训练序列信道估计算法将训练序列与信息序列直接相加后通过天线发送,从而节约了信道带宽。然而,在天线发送总功率一定时,训练序列的功率越大,信息序列的功率便越小,从而导致信道均衡器的信噪比减小。本文研究了基于MIMO系统的隐含训练序列信道估计算法,分析了信道均衡器信噪比与训练序列功率的关系,并根据均衡器信噪比最大原则推导出训练序列与信息序列的最佳功率分配。分析和仿真结果表明:在训练序列的最佳功率点上,信道均衡器的信噪比最高;随着接收天线信噪比的增加,训练序列的最佳功率增大。     

Analysis and optimization of pilot symbol-assisted Rake receivers for DS-CDMA systems

The effect of imperfect channel estimation (CE) on the performance of pilot-symbol-assisted modulation (PSAM) and MRC Rake reception over time- or frequency-selective fading channels with either a uniform power delay profile (UPDP) or a nonuniform power delay profile (NPDP) is investigated. For time-selective channels, a Wiener filter or linear minimum mean square error (LMMSE) filter for CE is considered, and a closed-form asymptotic expression for the mean square error (MSE) when the number of pilots used for CE approaches infinity is derived. In high signal-to-noise ratio (SNR), the MSE becomes independent of the channel Doppler spectrum. A characteristic function method is used to derive new closed-form expressions for the bit error rate (BER) of Rake receivers in UPDP and NPDP channels. The results are extended to two-dimensional (2-D) Rake receivers. The pilot-symbol spacing and pilot-to-data power ratio are optimized by minimizing the BER. For UPDP channels, elegant results are obtained in the asymptotic case. Furthermore, robust spacing design criteria are derived for the maximum Doppler frequency.     

Robust MSE equalizer design for MIMO communication systems in the presence of model uncertainties

This paper considers a robust mean-square-error (MSE) equalizer design problem for multiple-input multiple-output (MIMO) communication systems with imperfect channel and noise information at the receiver. When the channel state information (CSI) and the noise covariance are known exactly at the receiver, a minimum-mean-square-error (MMSE) equalizer can be employed to estimate the transmitted signal. However, in actual systems, it is necessary to take into account channel and noise estimation errors. We consider here a worst-case equalizer design problem where the goal is to find the equalizer minimizing the equalization MSE for the least favorable channel model within a neighborhood of the estimated model. The neighborhood is formed by placing a bound on the Kullback-Leibler (KL) divergence between the actual and estimated channel models. Lagrangian optimization is used to convert this min-max problem into a convex min-min problem over a convex domain, which is solved by interchanging the minimization order. The robust MSE equalizer and associated least favorable channel model can then be obtained by solving numerically a scalar convex minimization problem. Simulation results are presented to demonstrate the MSE and bit error rate (BER) performance of robust equalizers when applied to the least favorable channel model.     

Blind NLLS Carrier Frequency-Offset Estimation for QAM, PSK, and PAM Modulations: Performance at Low SNR

We address the problem of blind carrier frequency-offset (CFO) estimation in quadrature amplitude modulation, phase-shift keying, and pulse amplitude modulation communications systems. We study the performance of a standard CFO estimate, which consists of first raising the received signal to the$M$th power, where$M$is an integer depending on the type and size of the symbol constellation, and then applying the nonlinear least squares (NLLS) estimation approach. At low signal-to noise ratio (SNR), the NLLS method fails to provide an accurate CFO estimate because of the presence of outliers. In this letter, we derive an approximate closed-form expression for the outlier probability. This enables us to predict the mean-square error (MSE) on CFO estimation for all SNR values. For a given SNR, the new results also give insight into the minimum number of samples required in the CFO estimation procedure, in order to ensure that the MSE on estimation is not significantly affected by the outliers.     

基于最大相关熵准则的恒模盲均衡算法

针对恒模算法(constant modulus algorithm, CMA)在脉冲噪声环境下性能退化的问题,本文基于最大相关熵准则(maximum correntropy criterion, MCC)对恒模算法中基于最小均方误差(mean square error, MSE)准则的代价函数进行修正,推导出适用于脉冲噪声环境的基于MCC准则的恒模盲均衡算法(MCC_CMA)。该算法利用通信信号的恒模特性,首先得到发送信号与均衡器输出信号模值的误差信号,再通过使模值误差信号的相关熵最大来获得其迭代误差调节项,避免了传统高阶统计量算法在脉冲噪声环境下性能退化的问题。对高斯噪声以及α-稳定分布和混合高斯分布两种脉冲噪声环境下的信道均衡问题的仿真实验表明,相对于经典的自适应恒模盲均衡算法,MCC_CMA算法不依赖噪声的先验知识就能获得较快的收敛速度、较低的剩余码间干扰和误码率,并且在不同脉冲强度的脉冲噪声环境下都能够得到较好的均衡结果,表明MCC_CMA算法具有很好的鲁棒性。      

Optimization of pump and signal powers for wavelength converters based on FWM in semiconductor optical amplifiers

We investigate the effect of input pump and signal powers on the noise performance and intersymbol interference (ISI) in a frequency-converter based on four-wave mixing (FWM) in a semiconductor optical amplifier. We demonstrate that there is an input pump power at which the noise figure of the frequency converter is a minimum, and a corresponding input signal power for which the output signal-to-noise ratio (SNR) is a maximum. We report bit-error-rate measurements which show that there is a trade-off between maximizing the output SNR, and minimizing intersymbol interference in the SOA. Consequently, the power penalty incurred in the frequency conversion can be minimized by careful selection of the input signal power. We show that power penalties of less than 1 dB are achievable.     

FDD大规模MIMO系统速率最大化导频信号设计方法

该文针对FDD大规模MIMO下行系统,联合考虑信道空间相关性、信道估计精度和波束成型方案对导频信号设计的影响,以最大化下行遍历可达速率为目标,以系统总功耗为约束,建立关于导频信号矩阵的优化模型。由于原始优化问题的目标函数没有闭合形式表达式,利用大维矩阵理论中确定性等价原理可获得其近解析表达式,从而得到了遍历速率关于导频矩阵的显示表达式。基于此,利用主导理论,推导出了最优导频信号所具有的矩阵结构特征。进而,将原优化问题转换为等价的导频序列功率分配问题,再利用拉格朗日对偶法求解得到了最优导频信号的闭合形式解。通过分析,给出了最优导频序列长度的实际值与最大范围。数值仿真结果验证了所推导的遍历速率解析表达式的精确性和有效性,并对比了所提出的导频方案与传统基于最小均方误差准则导频方案在遍历速率方面的性能增益,以及所使用的导频序列长度差异,同时给出了两种方案在导频信号功率分配时的差异性。      

Fractionally spaced equalization using CMA: robustness to channelnoise and lack of disparity

In the noise-free case, the fractionally spaced equalization using constant modulus (FSE-CM) criterion has been studied previously. Its minima were shown to achieve perfect equalization when zero-forcing (ZF) conditions are satisfied and to be able to still achieve fair equalization when there is lack of disparity. However, to our best knowledge, the effect of additive channel noise on the FSE-CM cost-function minima has not been studied. In this paper, we show that the noisy FSE-CM cost function is subject to a smoothing effect with respect to the noise-free cost function, the result of which is a tradeoff between achieving zero forcing and noise enhancement. Furthermore, we give an analytical closed-form expression for the loss of performance due to the noise in terms of input-output mean square error (MSE). Under the ZF conditions, the FSE-CM MSE is shown to be mostly due to output noise enhancement and not to residual intersymbol interference (ISI). When there is lack of disparity, an irreducible amount of ISI appears independently of the algorithm. It is the lower equalizability bound for given channel conditions and equalizer length-the so-called minimum MSE (MMSE). The MMSE lower bound is the sum of the MMSE and of additional MSE mostly due to noise enhancement. Finally, we compare the FSE-CM MSE to this lower bound     

A Signal Perturbation Free Whitening-Rotation-Based Semiblind Approach for MIMO Channel Estimation

It was shown in our previous work that, in the noise-free case, the whitening-rotation (WR)-based MIMO channel estimation algorithm is subject to a signal perturbation error, justifying that the WR-based method is efficient only in the low signal-to-noise ratio (SNR) case. In this paper, a very efficient signal-perturbation-free WR-based approach is proposed for semiblind channel estimation of MIMO systems. A novel transmit scheme is developed based on the eigenvalue decomposition of the correlation matrix of the transmitted signal. The new scheme is to send a small volume of data bearing the information of the correlation matrix to the receiver for the cancellation of the signal perturbation error so as to improve the performance of the WR-based method in the case of high SNRs. Then, a perturbation analysis of the proposed WR-based semiblind method with the new transmit scheme is conducted, leading to a closed-form expression for the mean square error (MSE) of the channel estimate. Computer simulations show that the proposed approach significantly outperforms the original WR-based method as well as some other channel estimation methods for all SNR levels.     

WCDMA上行链路一种新的参数优化方法

本文在研究了WCDMA上行链路导频信道辅助相干解调的基础上,提出了参数优化的一种新方法.根据给定信噪比下误码概率最小的准则,本文得到了理论上信道估计器最佳的平均长度和导频信道相对数据信道最优的功率比.     

Optimal Training Signals for MIMO OFDM Channel Estimation in the Presence of Frequency Offset and Phase Noise

We develop robust mean-square error (MSE)-optimal training signal designs for multiple-input multiple-output orthogonal frequency-division multiplexing channel estimation with frequency offset and phase noise (PN), and present analytical and simulation results for the frequency-offset and PN effects on channel estimation. The proposed designs are more advantageous for moderate-to-high values of signal-to-noise ratio (SNR), residual frequency offset, and PN level. At SNR = 10 dB, the normalized MSE reductions of our proposed training signals at normalized frequency offset$vert vvert=0.1,0.5$are about 9 and 19 dB, respectively, for one transmit antenna, and 6 and 11 dB for two transmit antennas.     

The impact of white gaussian noise mismatch on linear and decisionfeedback equalizers

Analyzes the sensitivity of equalizers to white gaussian noise mismatch, which exists whenever the noise power observed during training differs from that observed during detection. A closed-form solution for the mean-square error is obtained for linear and decision feedback equalizers as a function of the channel spectrum and degree of noise mismatch. The results show that increasing equalizer complexity may actually degrade performance in the presence of channel mismatch     

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.     

Consistent normalized least mean square filtering with noisy data matrix

When the ordinary least squares method is applied to the parameter estimation problem with noisy data matrix, it is well-known that the estimates turn out to be biased. While this bias term can be somewhat reduced by the use of models of higher order, or by requiring a high signal-to-noise ratio (SNR), it can never be completely removed. Consistent estimates can be obtained by means of the instrumental variable method (IVM),or the total/data least squares method (TLS/DLS). In the adaptive setting for the such problem, a variety of least-mean-squares (LMS)-type algorithms have been researched rather than their recursive versions of IVM or TLS/DLS that cost considerable computations. Motivated by these observations, we propose a consistent LMS-type algorithm for the data least square estimation problem. This novel approach is based on the geometry of the mean squared error (MSE) function, rendering the step-size normalization and the heuristic filtered estimation of the noise variance, respectively, for fast convergence and robustness to stochastic noise. Monte Carlo simulations of a zero-forcing adaptive finite-impulse-response (FIR) channel equalizer demonstrate the efficacy of our algorithm.     

On the robustness of the finite-length MMSE-DFE with respect to channel and second-order statistics estimation errors

The finite-length minimum mean square error decision-feedback equalizer (MMSE-DFE) is an efficient structure mitigating intersymbol interference (ISI) introduced by practically all communication channels at high-enough symbol rates. The filters constituting the MMSE-DFE, as well as related performance measures, can be computed by assuming perfect knowledge of the channel impulse response and the input and noise second-order statistics (SOS). In practice, we estimate the unknown quantities, and thus, inevitable estimation errors arise. We model the estimation errors as small perturbations, and we derive a second-order approximation to the excess MSE. Furthermore, we derive second-order approximations to the mean excess MSE in terms of the parameter estimation error covariance matrices and simple and informative bounds, revealing the factors that govern the behavior of MMSE-DFE under mismatch. Simulations confirm that the derived second-order approximations provide accurate estimates of the MMSE-DFE performance degradation due to mismatch.