Blind MMSE Equalization of FIR/IIR Channels Using Oversampling and Multichannel Linear Prediction

A linear‐prediction‐based blind equalization algorithm for single‐input single‐output (SISO) finite impulse response/infinite impulse response (FIR/IIR) channels is proposed. The new algorithm is based on second‐order statistics, and it does not require channel order estimation. By oversampling the channel output, the SISO channel model is converted to a special single‐input multiple‐output (SIMO) model. Two forward linear predictors with consecutive prediction delays are applied to the subchannel outputs of the SIMO model. It is demonstrated that the partial parameters of the SIMO model can be estimated from the difference between the prediction errors when the length of the predictors is sufficiently large. The sufficient filter length for achieving the optimal prediction is also derived. Based on the estimated parameters, both batch and adaptive minimum‐mean‐square‐error equalizers are developed. The performance of the proposed equalizers is evaluated by computer simulations and compared with existing algorithms.     

Adaptive blind channel estimation by least squares smoothing

A least squares smoothing (LSS) approach is presented for the blind estimation of single-input multiple-output (SIMO) finite impulse response systems. By exploiting the isomorphic relation between the input and output subspaces, this geometrical approach identifies the channel from a specially formed least squares smoothing error of the channel output. LSS has the finite sample convergence property, i.e., in the absence of noise, the channel is estimated perfectly with only a finite number of data samples. Referred to as the adaptive least squares smoothing (A-LSS) algorithm, the adaptive implementation has a high convergence rate and low computation cost with no matrix operations. A-LSS is order recursive and is implemented in part using a lattice filter. It has the advantage that when the channel order varies, channel estimates can be obtained without structural change of the implementation. For uncorrelated input sequence, the proposed algorithm performs direct deconvolution as a by-product     

A blind multichannel identification algorithm robust to orderoverestimation

Active research in blind single input multiple output (SIMO) channel identification has led to a variety of second-order statistics-based algorithms, particularly the subspace (SS) and the linear prediction (LP) approaches. The SS algorithm shows good performance when the channel output is corrupted by noise and available for a finite time duration. However, its performance is subject to exact knowledge of the channel order, which is not guaranteed by current order detection techniques. On the other hand, the linear prediction algorithm is sensitive to observation noise, whereas its robustness to channel order overestimation is not always verified when the channel statistics are estimated. We propose a new second-order statistics-based blind channel identification algorithm that is truly robust to channel order overestimation, i.e., it is able to accurately estimate the channel impulse response from a finite number of noisy channel measurements when the assumed order is arbitrarily greater than the exact channel order. Another interesting feature is that the identification performance can be enhanced by increasing a certain smoothing factor. Moreover, the proposed algorithm proves to clearly outperform the LP algorithm. These facts are justified theoretically and verified through simulations     

Blind estimation of output labels of SIMO channels based on a novelclustering algorithm

This paper addresses the problem of data detection for digital communications employing space diversity reception, where the system model contains a single-input multiple-output (SIMO) vector channel. The received vector corrupted by additive white Gaussian noise (AWGN) is modeled as the noisy output of a finite state vector Markov source. Subsequently, a discretely valued basis of minimal dimensionality is identified for the input space. Estimation of the output labels associated with this basis allows labeling of the state transition diagram. For this purpose, certain identifiable characteristics of the output sequences of the Markov source are used to classify its states and generate an initial codebook for a vector quantizer used to restore the output level     

一种基于子空间分解的IIR信道盲辩识算法

本文证明了对IIR信道输出进行过采样 (采样率是输入码率的整数倍 )可以转化成单输入多输出的多信道模型 ,并且不同的信道有相同的AR系数 .基于这一特性本文提出一种基于子空间分解的信道参数盲辩识方法 ,即不同信道的MA系数可以由输出信号的噪声子空间唯一确定 ,而AR系数则可以通过求解YW方程得到     

Stack filters and selection probabilities

Based on the fact that the output of a given stack filter can be determined if the ranks of the samples in the input window are known and that this output always equals one of the samples in the input window, rank and sample selection probabilities are defined. The output distribution of the stack filter of size N with independent identically distributed (i.i.d.) inputs can be expressed as a weighted sum of the ith, i=1, 2, ..., N order statistics, where the rank selection probabilities are the weights. The sample selection probabilities equal the impulse response coefficients of a finite impulse response (FIR) filter whose output spectrum is closest, of all linear filters, to that of the stack filter for i.i.d. Gaussian inputs. Results are also derived for correlated inputs. Robustness and detail preserving properties of stack filters are related to the selection probabilities. Other statistical properties are also derived. Finally, methods to compute the selection probabilities of the stack filter from its positive Boolean function and the selection probabilities of the weighted median filter from its weights are given in detail     

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.     

基于输出过采样的IIR信道迫零盲均衡

本文提出一种新的针对单输入单输出IIR信道的盲均衡算法.首先通过对信道输出的过采样建立特殊的多信道模型.对多信道模型的输出应用线性预测,证明了预测误差只包含多信道模型冲激响应在第一个时隙的参数,并给出最佳线性预测器的长度.通过预测误差的协方差矩阵可以求解该冲激响应参数.基于该参数可构造出不同时延的迫零均衡器.仿真结果显示了本文算法的有效性.     

Causal Cubic Splines: Formulations, Interpolation Properties and Implementations

The paper presents two formulations of causal cubic splines with equidistant knots. Both are based on a causal direct B-spline filter with parallel or cascade implementation. In either implementation, the causal part of the impulse response is realized with an efficient infinite-impulse-response (IIR) structure, while only the anticausal part is approximated with a finite-order finite-impulse-response (FIR) filter. Resulting cubic coefficients are computed from the causal B-spline coefficients by using a third-order output FIR filter with either single-input multiple-output (SIMO) or multiple-input multiple-output (MIMO) structure, depending on the chosen formulation of the cubic spline. The paper demonstrates and proves that the properties of the resulting causal splines are quite different, whether they are based on a more popular B-spline formulation, or a bit neglected tridiagonal matrix formulation. It is shown that the proposed low-complexity but accurate causal interpolators can be realized for many practical applications with the delay of only a few samples.      

Digital Laguerre filter design with maximum passband-to-stopband energy ratio subject to peak and Group delay constraints

In this paper, we formulate a general design of transversal filter structures with maximum relative passband-to-stopband energy ratio subject to complex frequency response constraints in the passband and the stopband as well as additional constraints such as constraints. These constraints are important for applications where the suppression of noise at certain frequencies are important. Additional constraints are introduced allowing approximately linear phase and constant group delay in the passband. For a given set of basis functions, the design problem can be formulated as a semi-infinite quadratic optimization problem in the filter coefficients, which are the decision variables to be optimized. In this paper, we focus on the design of digital Laguerre filter and digital finite impulse response (FIR) filter structures. A modified bridging algorithm is developed for searching for the optimum pole of the Laguerre filters. Design examples are given to demonstrate the effectiveness of the proposed algorithm.     

基于均衡代价函数的信道阶数盲估计算法

针对信道阶数估计问题,利用单输入多输出(Single-Input Multiple-Output,SIMO)有限冲激响应(Finite Impulse Response,FIR)信道的结构特点和输入/输出信号的统计特征,提出了一种基于均衡代价函数的信道阶数盲估计算法.首先计算了归一化最小二乘均衡(Normalized Least Squares Equalization,NLSE)代价函数在理想条件下的理论渐近值,并指出其拐点与信道阶数之间的对应关系.然后分析了NLSE代价函数在实际条件下的近似值.最后引入了拐点优化因子,提出了一种基于NLSE代价函数拐点检测的信道阶数估计算法.理论分析和仿真结果表明,在信噪比(Signal-to-Noise Ratio,SNR)较低和信道首尾系数较小的情况下,该算法比现有其它方法具有更强的鲁棒性,可以获得更小的接收信号均衡误差.     

Blind methods of system identification

This paper reviews the basic identifiability conditions and identification methods for blind system identification. This review focuses on the exploitation of the second-order statistics of the system output. The blind methods vary significantly according to the categories of the systems: i.e., single-input, single-output (SISO) systems, single-input, multiple-output (SIMO) systems, or multiple-input, multiple-output (MIMO) systems. For SISO systems, the blind methods require white input and minimum phase frequency response. For SIMO systems, the blind methods can generally yield the exact identification up to a scalar using a finite set of data. For MIMO systems, the blind identifiability conditions and the blind methods are much more involved.     

Energy constrained frequency-domain normalized LMS algorithm for blind channel identification

This paper deals with the blind adaptive identification of single-input multi-output (SIMO) finite impulse response acoustic channels from noise-corrupted observations. The normalized multichannel frequency-domain least-mean-squares (NMCFLMS) algorithm [1] is known to be a very effective and efficient technique for identification of such channels when noise effects can be ignored. It, however, misconverges in presence of noise [2]. In this paper, we present an analysis of noise effects on the NMCFLMS algorithm and propose a novel technique for ameliorating such misconvergence characteristics of the NMCFLMS algorithm for blind channel identification (BCI) with noise by attaching a spectral constraint in the adaptation rule. Experimental results demonstrate that the robustness of the NMCFLMS algorithm for BCI can be significantly improved using such a constraint.     

基于反幂法和卡尔曼滤波的自适应语音去混响方法

噪声鲁棒的自适应语音信号去混响是现代语音信号处理的重要研究内容,其困难在于语音信号的非白性、非平稳性及房间的超长冲激响应特性。针对单输入多输出(SIMO)麦克风阵列系统获取的多路混响语音信号,提出了一种新的去混响算法。首先通过相关法时延估计对SIMO混响语音信号进行时延对齐;其次在保持SIMO系统输出信号间交叉关联关系(cross relation)基础上对混响语音信号进行预白化处理;最后把交叉关联关系、用于矩阵最小特征向量计算的反幂法与卡尔曼滤波解卷积方法有机结合,实现了SIMO混响语音信号的实时自适应去混响。仿真与实验研究表明,本方法对混响语音信号去混响效果明显,同时具有较好的抗噪声性能。      

On the determination of the optimal pole position of Laguerrefilters

As is known in the literature, at each stationary point of the squared error (SE) curve of a Laguerre filter at least one of a pair of certain Laguerre coefficients (weights) vanishes. The author describes a very efficient way to compute the derivatives of each one of these coefficients with respect to the pole position of the Laguerre filter. The knowledge of these derivatives makes possible the computation of high-order approximations to the coefficients in question, such as truncated Taylor series and Pade approximants. The zeros of these approximations are usually good estimates of the location of the stationary points of the SE curve nearer to the center of the approximation. In this way, the position of these stationary points, in particular local minima, can be estimated without resorting to a numerical search algorithm. Both continuous time and discrete time Laguerre filters are discussed, excited either by an impulse or by an arbitrary signal. The authors illustrate the main results of the paper with a numerical example     

Subspace based blind identification of LTI FIR MIMO systems and equalization of finite memory SIMO Volterra systems

In this paper a novel algorithm based on subspace projections is developed for the blind kernel identification of LTI FIR multiple input multiple output (MIMO) systems, as well as blind equalization of finite memory SIMO Volterra systems. In addition, for Volterra systems, the algorithm computes the memory lengths of the nonlinearities involved. Simulations in the context of blind channel equalization and identification demonstrate good performance in comparison to existing schemes.     

An FIR Channel Estimation Filter with Robustness to Channel Mismatch Condition

This letter proposes a finite impulse response (FIR) channel estimation filter that has robustness against the channel mismatch due to the FIR structure. The channel impulse response is described with a complex state space model and then estimated from received data on the recent time interval. Numerical results show that the FIR channel estimation filter can provide more robust performance than conventional Kalman IIR filters when channel model parameters are not correct.     

Blind Deconvolution of Multi-Input Single-Output Systems With Binary Sources

The problem of blind source separation for multi-input single-output (MISO) systems with binary inputs is treated in this paper. Our approach exploits the constellation properties of the successor values for each output sample. In the absence of noise, the successors of each output value form a characteristic finite set of clusters (successor constellation). The shape of this constellation is invariant of the predecessor value and it only depends on the last filter tap. Consequently, the localization of the successors constellation can lead to the removal of the last filter tap, thus reducing the length of the filter—a process we call channel deflation. Based on the successor observation clustering (SOC), we develop two algorithms for blind source separation—SOC-1 and SOC-2—differing mainly on the required size of the data set. Furthermore, the treatment of the system in the presence of noise is described using data clustering and data correction. The problem of noise is attacked using a statistical-mode-based method. Moreover, we correct the problem of misclassified observations using an iterative scheme based on the Viterbi algorithm for the decoding of a hidden Markov model (HMM).     

Blind channel estimation for single‐input multiple‐output OFDM systems: zero padding based or cyclic prefix based?

Orthogonal frequency division multiplexing (OFDM) transmission equipped with multiple receive antennas constitutes a single‐input multiple‐output (SIMO) OFDM system. SIMO‐OFDM systems have been widely used in wireless communications. Compared to those approaches using training sequences, blind channel estimation methods for SIMO‐OFDM systems have the advantage of saving bandwidth and improving energy efficiency and system throughput. As far as blind channel identification is concerned, it is known that zero padding (ZP)‐based single‐input single‐output (SISO)‐OFDM systems have desirable features compared to conventional cyclic prefix (CP)‐based SISO‐OFDM systems. However, it is yet unknown whether ZP‐ or CP‐based SIMO‐OFDM systems are favourable for blind channel estimation. To investigate this problem, we first propose a short‐data effective method for blind channel estimation for ZP‐based SIMO‐OFDM systems. Then we analyse a number of issues surrounding blind channel estimation for ZP‐ and CP‐based SIMO‐OFDM systems. The issues brought up in the paper have not been discussed in the existing research. The significance of our investigation is that it provides a deep insight into blind channel estimation for ZP‐ and CP‐based SIMO‐OFDM systems. Copyright © 2011 John Wiley & Sons, Ltd.     

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.