New insights into the normalization of the least mean fourth algorithm

The available normalization of the least mean fourth algorithm is investigated. It is shown that that normalization does not protect the algorithm from divergence when the input power of the adaptive filter increases. The reason of this drawback is that the normalization is done by dividing the weight vector update term by the squared norm of the regressor, while the update term is a fourth order polynomial in the regressor. The paper presents a normalized LMF algorithm that is based on dividing the weight vector update term by the fourth power of the norm of the regressor. This normalization protects the algorithm from divergence when the input power increases. An approximate stability step-size bound of the proposed algorithm is derived. The step-size bound depends on the weight initialization, while it does not depend on the input power of the adaptive filter for non-small signal-to-noise ratio. Simulation results support the analytical results of the paper.     

Matched median filter for detecting QAM signals

In this paper, the complex matched median filter (MMF) is developed for QAM signal detection. It is shown that the MMF is robust against impulsive type noise. By combining the MMF and the linear matched filter (LMF), an extended class of matched filters is introduced. These filters combine the desirable properties of MMF and LMF and behave well in varying noise environments. Computer simulations demonstrate that the proposed detectors give a much smaller symbol error probability than the LMF when the noise has an impulsive component and produces only a slight performance degradation in the case of pure Gaussian noise     

Stochastic analysis of the least mean fourth algorithm for non-stationary white Gaussian inputs

This paper studies the stochastic behavior of the least mean fourth (LMF) algorithm for a system identification framework when the input signal is a non-stationary white Gaussian process. The unknown system is modeled by the standard random-walk model. A theory is developed which is based upon the instantaneous average power and the instantaneous average squared power in the adaptive filter taps. A recursion is derived for the instantaneous mean square deviation of the LMF algorithm. This recursion yields interesting results about the transient and steady-state behaviors of the algorithm with time-varying input power. The theory is supported by Monte Carlo simulations for sinusoidal input power variations.     

A Mean-Square Stability Analysis of the Least Mean Fourth Adaptive Algorithm

This paper presents a new convergence analysis of the least mean fourth (LMF) adaptive algorithm, in the mean square sense. The analysis improves previous results, in that it is valid for non-Gaussian noise distributions and explicitly shows the dependence of algorithm stability on the initial conditions of the weights. Analytical expressions are derived presenting the relationship between the step size, the initial weight error vector, and mean-square stability. The analysis assumes a white zero-mean Gaussian reference signal and an independent, identically distributed (i.i.d.) measurement noise with any even probability density function (pdf). It has been shown by Nascimento and Bermudez ["Probability of Divergence for the Least-Mean Fourth (LMF) Algorithm," IEEE Transactions on Signal Processing, vol 54, no. 4, pp. 1376-1385, Apr. 2006] that the LMF algorithm is not mean-square stable for reference signals whose pdfs have infinite support. However, the probability of divergence as a function of the step size value tends to rise abruptly only when it moves past a given threshold. Our analysis provides a simple (and yet precise) estimate of the region of quick rise in the probability of divergence. Hence, the present analysis is useful for predicting algorithm instability in most practical applications.     

An improved statistical analysis of the least mean fourth (LMF) adaptive algorithm

The paper presents an improved statistical analysis of the least mean fourth (LMF) adaptive algorithm behavior for a stationary Gaussian input. The analysis improves previous results in that higher order moments of the weight error vector are not neglected and that it is not restricted to a specific noise distribution. The analysis is based on the independence theory and assumes reasonably slow learning and a large number of adaptive filter coefficients. A new analytical model is derived, which is able to predict the algorithm behavior accurately, both during transient and in steady-state, for small step sizes and long impulse responses. The new model is valid for any zero-mean symmetric noise density function and for any signal-to-noise ratio (SNR). Computer simulations illustrate the accuracy of the new model in predicting the algorithm behavior in several different situations.     

Modeling the amplitude statistics of ultrasonic images

In this paper, a new statistical model for representing the amplitude statistics of ultrasonic images is presented. The model is called the Rician inverse Gaussian (RiIG) distribution, due to the fact that it is constructed as a mixture of the Rice distribution and the Inverse Gaussian distribution. The probability density function (pdf) of the RiIG model is given in closed form as a function of three parameters. Some theoretical background on this new model is discussed, and an iterative algorithm for estimating its parameters from data is given. Then, the appropriateness of the RiIG distribution as a model for the amplitude statistics of medical ultrasound images is experimentally studied. It is shown that the new distribution can fit to the various shapes of local histograms of linearly scaled ultrasound data better than existing models. A log-likelihood cross-validation comparison of the predictive performance of the RiIG, the K, and the generalized Nakagami models turns out in favor of the new model. Furthermore, a maximum a posteriori (MAP) filter is developed based on the RiIG distribution. Experimental studies show that the RiIG MAP filter has excellent filtering performance in the sense that it smooths homogeneous regions, and at the same time preserves details.     

A noise constrained least mean fourth (NCLMF) adaptive algorithm

The learning speed of an adaptive algorithm can be improved by properly constraining the cost function of the adaptive algorithm. In this work, a noise-constrained least mean fourth (NCLMF) adaptive algorithm is proposed. The NCLMF algorithm is obtained by constraining the cost function of the standard LMF algorithm to the fourth-order moment of the additive noise. The NCLMF algorithm can be seen as a variable step-size LMF algorithm. The main aim of this work is to derive the NCLMF adaptive algorithm, analyze its convergence behavior, and assess its performance in different noise environments. Furthermore, the analysis of the proposed NCLMF algorithm is carried out using the concept of energy conservation. Finally, a number of simulation results are carried out to corroborate the theoretical findings, and as expected, improved performance is obtained through the use of this technique over the traditional LMF algorithm.     

A general class of nonlinear normalized adaptive filteringalgorithms

The normalized least mean square (NLMS) algorithm is an important variant of the classical LMS algorithm for adaptive linear filtering. It possesses many advantages over the LMS algorithm, including having a faster convergence and providing for an automatic time-varying choice of the LMS stepsize parameter that affects the stability, steady-state mean square error (MSE), and convergence speed of the algorithm. An auxiliary fixed step-size that is often introduced in the NLMS algorithm has the advantage that its stability region (step-size range for algorithm stability) is independent of the signal statistics. In this paper, we generalize the NLMS algorithm by deriving a class of nonlinear normalized LMS-type (NLMS-type) algorithms that are applicable to a wide variety of nonlinear filter structures. We obtain a general nonlinear NLMS-type algorithm by choosing an optimal time-varying step-size that minimizes the next-step MSE at each iteration of the general nonlinear LMS-type algorithm. As in the linear case, we introduce a dimensionless auxiliary step-size whose stability range is independent of the signal statistics. The stability region could therefore be determined empirically for any given nonlinear filter type. We present computer simulations of these algorithms for two specific nonlinear filter structures: Volterra filters and the previously proposed class of Myriad filters. These simulations indicate that the NLMS-type algorithms, in general, converge faster than their LMS-type counterparts     

Convergence analysis of constrained joint adaptation in recording channels

Partial response (PR) equalization employing the linearly constrained least-mean-square (LCLMS) adaptive algorithm is widely used for jointly designing equalizer and PR target in recording channels. However, there is no literature on its convergence analysis. Further, existing analyses of the least-mean-square (LMS) algorithm assume that the input signals are jointly Gaussian, an assumption that is invalid for PR equalization with binary input. In this paper, we present a novel method to analyze the convergence of the LCLMS algorithm, without the Gaussian assumption. Our approach accommodates distinct step sizes for equalizer and PR target. It is shown that the step-size range required to guarantee stability of LCLMS with binary data is larger than that with Gaussian data. The analytical results are corroborated by extensive simulation studies.     

Convergence analysis of the adaptive lattice filter for a mixed Gaussian input sequence

The convergence performance of the adaptive lattice filter (ALF) using the stochastic gradient algorithm is measured by the convergence speed and estimated error variance of the PARCOR coefficient. The convergence properties of the ALF are analysed when the filter input has a Gaussian mixture distribution. First, theoretical expressions for the convergence rate and asymptotic error variance of the PARCOR coefficient are derived, and then the theoretical expressions are compared for single and mixed Gaussian input sequences. It is shown that the convergence performance of the ALF improves as the distribution of the input signal approaches a single Gaussian distribution.     

高斯混合多伯努利滤波器基于柯西施瓦兹散度的传感器控制方法

针对多目标跟踪中的传感器控制问题,本文基于有限集统计（FISST）理论,利用高斯混合多伯努利滤波器研究并提出相应的传感器控制策略.首先,文中给出容积卡尔曼高斯混合势均衡多目标多伯努利滤波器（CK-GMCBMeMBerF）的实现形式,并提取高斯混合分量近似多伯努利密度.然后,研究两个高斯混合之间的柯西施瓦兹（Cauchy-Schwarz）散度的求取,推导多目标概率密度变化所对应的信息增益,并以此为基础提出相应的传感器控制策略.此外,结合CK-GMCBMeMBer,详细推导了目标势的后验期望（PENT）准则的高斯混合（GM）实现过程,以GM-PENT作为评价准则进行相应的传感器控制方法的研究.最后,仿真实验验证了所提算法的有效性.     

On the convergence behavior of the LMS and the normalized LMSalgorithms

It is shown that the normalized least mean square (NLMS) algorithm is a potentially faster converging algorithm compared to the LMS algorithm where the design of the adaptive filter is based on the usually quite limited knowledge of its input signal statistics. A very simple model for the input signal vectors that greatly simplifies analysis of the convergence behavior of the LMS and NLMS algorithms is proposed. Using this model, answers can be obtained to questions for which no answers are currently available using other (perhaps more realistic) models. Examples are given to illustrate that even quantitatively, the answers obtained can be good approximations. It is emphasized that the convergence of the NLMS algorithm can be speeded up significantly by employing a time-varying step size. The optimal step-size sequence can be specified a priori for the case of a white input signal with arbitrary distribution     

Stochastic gradient adaptation under general error criteria

Examines a family of adaptive filter algorithms of the form W_k+1=W_k+μf(d_k-W_k^t X_k)X_k in which f(·) is a memoryless odd-symmetric nonlinearity acting upon the error. Such algorithms are a generalization of the least-mean-square (LMS) adaptive filtering algorithm for even-symmetric error criteria. For this algorithm family, the authors derive general expressions for the mean and mean-square convergence of the filter coefficients For both arbitrary stochastic input data and Gaussian input data. They then provide methods for optimizing the nonlinearity to minimize the algorithm misadjustment for a given convergence rate. Using the calculus of variations, it is shown that the optimum nonlinearity to minimize misadjustment near convergence under slow adaptation conditions is independent of the statistics of the input data and can be expressed as -p'(x)/p(x), where p(x) is the probability density function of the uncorrelated plant noise. For faster adaptation under the white Gaussian input and noise assumptions, the nonlinearity is shown to be x/{1+μλx²/σ_k ²}, where λ is the input signal power and σ_k² is the conditional error power. Thus, the optimum stochastic gradient error criterion for Gaussian noise is not mean-square. It is shown that the equations governing the convergence of the nonlinear algorithm are exactly those which describe the behavior of the optimum scalar data nonlinear adaptive algorithm for white Gaussian input. Simulations verify the results for a host of noise interferences and indicate the improvement using non-mean-square error criteria     

认知无线电中基于高斯混合概率假设密度滤波的主用户跟踪算法

基于随机集的高斯混合概率假设密度滤波算法是一种典型的多目标跟踪算法,可以在目标数目未知的情况下进行多目标跟踪,但是该算法要求已知目标的起始位置,在很多情况下,目标的起始位置信息是无法获得的。本文针对这一问题,提出了改进的高斯混合概率假设密度滤波算法,并将本文算法应用于认知无线电系统的主用户跟踪问题。该算法利用双向预测的方式对检测结果进行估计,即使用正向预测算法来估计现存主用户的位置,然后采用后向预测算法来搜索新生的主用户并估计出新生主用户的位置。本文算法的主要优点是在主用户的数目、出现的时间和起始位置均未知的情况下仍可以有效的跟踪目标。最后,通过仿真对本文算法的性能进行了分析。仿真结果表明,本文算法在误检率较高的情况下可以准确地跟踪主用户。      

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.     

自适应非线性GM-PHD滤波及在无源跟踪中的应用

首先针对无源传感器目标跟踪中的非线性问题,将高斯-厄米特求积分规则运用于高斯混合概率假设密度滤波,提出一种求积分卡尔曼概率假设密度滤波。其次,针对未知时变过程噪声,将基于极大后验估计原理的噪声估计器运用到概率假设密度滤波中,同时依据目标状态一步预测与状态滤波结果之间的残差,提出一种对滤波发散情况判断和抑制的算法。最后通过无源传感器双站跟踪仿真表明:相较于已有的非线性高斯混合概率假设密度滤波,所提算法有更高的精度,并且在未知时变噪声环境中具有较好跟踪效果。     

A piloted adaptive notch filter

In the implementation of an adaptive notch filter using the least mean squares (LMS) algorithm, the zero of the filter is steered toward the input sinusoid based on the gradient information. The convergent may be speeded up if a larger step size is used when the zero of the notch filter is far away from the frequency of the input sinusoid. The gradient provides information on the direction where the zero should be steered but does not provide information on the distance between the zero and the frequency of the sinusoid. Conventional variable step-size algorithms determine the step size based on a (linear/nonlinear) weighted average of the gradient estimate at several sampling instances (time domain averaging). In this paper, we propose a new method for extracting information on the distance between the frequency of the input sinusoid and the zero of the notch. We use three (or more) notches, namely, a main notch and two (or more) pilot notches implemented with minimal additional cost. The pilot notches are used to analyze the gradient estimates at the same sampling instance but at several frequency points as the main notch. Simulation results show that our new piloted notch technique is significantly superior to step-size determination based on a time-averaging technique. Novel theoretical analysis is presented. Our method can be used in conjunction with most existing algorithms to determine the step size.     

基于GM-PHD的目标空间分布感知算法

对边扫描边跟踪的认知雷达,目标的空间分布特性是实现对雷达信号控制的重要依据之一。本文介绍了一种基于高斯混合概率假设密度（GM-PHD）滤波算法的目标空间分布感知方法,利用该算法,可同时实现多目标高虚警环境下的目标数目和目标空间位置以及运动状态的估计。该算法实际是一种对标准GM-PHD滤波器的改进算法,能在新生目标强度未知的情况下完成对新生目标的检测跟踪。实验表明该算法不仅能在未知新生目标强度的情况下检测并跟踪新生目标,且在新生目标速度较大的情况下,该算法对新生目标的检测性能优于标准GM-PHD滤波器。     

基于衰减记忆高斯和滤波的星间精密测距技术

星间精密测距是导航星座实现自主导航的核心技术。针对导航星座中码测量值精度低但无整周模糊度,载波相位测量值精度高但存在整周模糊度的特点,该文根据贝叶斯递推原理提出了一种衰减记忆高斯和滤波(Fading Memory Gaussian Sum Filter, FMGSF)的伪距估计方法。该方法用高斯和形式近似表示系统后验概率密度,并根据卡尔曼滤波原理来更新高斯项的均值和方差,同时引入衰减记忆因子克服由于模型失配导致的滤波结果发散问题,利用重采样解决由于载波相位测量值不确定导致的算法复杂度增加问题。理论分析和仿真结果表明,该文提出的方法不仅能够克服周跳对伪距估计的影响,而且可以获得更好的测距精度。     

LMS和归一化LMS算法收敛门限与步长的确定

从LMS算法失调量的准确表达式出发,根据输入信号特征值分布重新研究了LMS,归一化LMS(Normalized LMS,NLMS)算法收敛的必要条件,推导出LMS和NLMS 算法收敛的步长门限,并分析了输入信号特征值分布、滤波器阶数对算法收敛步长门限的影响,推导出满足性能失调下步长的自适应计算公式,减小了应用 LMS,NLMS算法时步长选取的盲目性,与已有的算法相比,具有计算简单、实用、自适应性能强,同时可获得满意失调量的特点,计算机模拟结果表明该方法的正确性。