Mean-Square Stability Analysis of a Normalized Least Mean Fourth Algorithm for a Markov Plant

The problem of distributed estimation is considered for distributed network. A diffusion model on adaptive affine projection sign algorithm based on least mean fourth algorithm is introduced according to an Adapt-Then-Combine (ATC) design for the distributed framework. Least mean fourth (LMF) algorithm is defined by taking the fourth power of error to against the non-Gaussian environment. An adaptive affine projection sign algorithm based on LMF algorithm is proposed firstly. Then, an ATC diffusion model on adaptive affine projection sign algorithm based on LMF algorithm is presented in terms of the low complexity of sign approach with the order of adaptation and combination, named as ATC-DAPS-LMF. Numerical simulations verify that the proposed algorithm can perform significantly to the conventional affine projection sign algorithm.

Adaptive equalization is undoubtedly one of the cornerstones of digital communication as it allows for fast and reliable data transmission over practical channels that are non-ideal and that are plagued with serious problems such as signal dispersion and non-Gaussian interferences. In such cases, the performance of the well-known least-mean square (LMS) algorithm or that of any of its variants remains sub-optimal at best. As a remedy to this, the least-mean fourth (LMF) algorithm was later proposed but its high computational load was found to be a constraining factor in its practical implementation, particularly for real-time applications. Inspired from a previous work done on the LMS [1], we recently proposed a new algorithm [2], termed LMF-PTQ, combining the power of the LMF with the implementational advantages of the power-of-two quantization scheme. In this work, we demonstrate, through an extensive simulation study, the effectiveness of our proposed LMF-PTQ algorithm in the equalization of a time-varying channel in a CDMA system which suffers from both signal dispersion and non-Gaussian disturbances. The results obtained so far are very encouraging and provide us with ample enthusiasm to investigate the proposed algorithm further.

A simplified forms of adaptive set-membership affine projection based on least mean fourth algorithm is proposed. Least mean fourth (LMF) algorithm is defined by taking the fourth power of error to against the non-Gaussian environment. A set-membership adaptive filtering is described firstly. Then, an affine projection algorithm is based on LMF algorithm. A simplified set-membership affine projection algorithm based on LMF is introduced. Adaptive threshold parameter and learning rate are presented for improvement of convergence rate. Numerical simulations show that the proposed algorithm can provide significantly to the conventional affine projection algorithm.

Least-Mean Square algorithm, or simply the LMS, is generally one of the most commonly used techniques for optimized solution in adaptive schemes especially in the Gaussian environment. However, least mean fourth (LMF) algorithm and its variants, such as normalized LMF (NLMF), perform better in the non-Gaussian environment. The conventional LMF algorithms usually diverge in non-Gaussian environment with dynamic input. Conventionally, regularization is archived by using a small constant compensation term in the denominator of the learning rate to protect the algorithm from divergence. This paper introduces an efficient time-varying regularized normalized least mean fourth (R-NLMF) algorithm. In the proposed algorithm, the regularization term is made time varying and gradient adaptive according to steepest descent approach. Thus the proposed algorithm adapts its learning rate according to the environment and the input signal dynamics. A similar approach, namely Generalized Normalized Gradient Descent (GNGD) algorithm, has been previously applied to the normalized least mean-square (NLMS) algorithm in the Gaussian environment. However, due to its dependence on NLMS, the GNGD algorithm performance degrades in non-Gaussian environment. In order to overcome this problem, an efficient regularized NLMF algorithm for non-Gaussian environment is proposed. The algorithm shows promising results and achieves faster convergence while maintaining lesser steady-state miss-adjustment. Simulations are carried out to support the theoretical development.

In this work, the convergence analysis of the variable weight mixed-norm least-mean squares (LMS)-least mean-fourth (LMF) adaptive algorithm is derived. The proposed algorithm minimizes an objective function defined as a weighted sum of the LMS and LMF cost functions, where the weighting factor is time varying and adapts itself to allow the algorithm to keep track of the variations in the environment. As a by-product of this novel approach, new necessary and sufficient conditions for the LMF algorithm have been derived. Furthermore, a more general expression for the excess steady-state error for the LMF algorithm has been derived.

A modified least mean fourth (LMF) adaptive algorithm applicable to non-stationary signals is presented. The performance of the proposed algorithm is studied by simulation for non-stationarities in bandwidth, centre frequency and gain of a stochastic signal. These non-stationarities are in the form of linear, sinusoidal and jump variations of the parameters. The proposed LMF adaptation is found to have better parameter tracking capability than the LMS adaptation for the same speed of convergence.

This paper presents a new statistical analysis of the least mean fourth (LMF) adaptive algorithm behavior. Nonlinear recursive equations are derived which predict the behavior of the first and second order moments of the adaptive weights for Gaussian inputs. These recursions can be used to predict the mean square error (MSE) behavior. The new model improves the available models in that it predicts both the transient and steady-state behaviors for measurement noise having any zero-mean probability density function (pdf) and for any signal-to-noise ratio. This is important because the LMF algorithm is known to outperform the LMS algorithm for non-Gaussian noise distributions and for large signal-to-noise ratios. In addition, the new model explicitly shows the dependence of the algorithm's dynamics on the initial weight conditions. Computer simulations illustrate the accuracy of the new model in predicting the algorithm behavior.

An adaptive filtering algorithm that minimizes the variance on the squared error subject to a constraint on the mean-squared error (MSE) is proposed. For the adaptive-linear-combiner signal-processing configuration, this algorithm, called the LVCMS algorithm (for least variance subject to a constraint on mean squared error) has a gradient that is a weighted linear combination of the least-mean-square (LMS) and least-mean-fourth (LMF) algorithm gradients. Theoretical expressions are derived for the LVCMS algorithm convergence factor and misadjustment, and comparisons are made with the LMS and LMF adaptive rules for Gaussian, Laplacian, and uniform plant noise nd driving term distributions. Performance comparisons of the LVCMS, LMS, and LMF algorithms based on Monte Carlo simulation studies indicate that the LVCMS adaptation rule not only can yield a small variance of the squared error but it also produces favorable values of the mean-squared error and the mean-squared coefficient error. >

Mean-square stability of the Normalized Least-Mean Fourth algorithm for white Gaussian inputs

Adaptive filtering under multi-peak noise

In this paper, we provide some insights into the convergence and steady-state behaviors of the sign-error least mean fourth (SLMF), sign-sign least mean fourth (SSLMF), normalized sign-error least mean fourth (NSLMF), and normalized sign-sign least mean fourth (NSSLMF) algorithms for both cases of real- and complex-valued data. Moreover, we also report the equivalence algorithms of the block-based normalized sign-error least mean fourth (BBNSLMF) and block-based normalized sign-sign least mean fourth (BBNSSLMF) algorithms. Finally, simulations are conducted for both cases of real- and complex-valued data to provide us with more insights into the performance of the SLMF, SSLMF, NSLMF, and NSSLMF algorithms.

This dissertation presents a new model for two adaptive filtering algorithms based on fourth-order moments: the least-mean fourth (LMF) and the leastmean mixed-norm (LMMN) algorithms. The novelty of the new model is its emphasis on computing the probability of a reasonable performance of a single realization of the algorithm (in this case, convergence), instead of looking for average performance indices such as mean-square error.

The paper provides new results concerning the tracking performance of the least mean fourth algorithm in comparison with that of the least mean square algorithm. The analysis is done in the context of tracking a Markov plant with a white Gaussian input. The comparison is done in terms of the minimum mean square deviation attained by each algorithm over the stability range of its step-size. Gaussian, uniform, and binary distributions of the plant noise are considered. Conditions that make one algorithm outperform the other are determined. Analytical results are supported by simulations.

In this work, a family of normalized least mean fourth algorithms is presented. Unlike the LMF algorithm, the convergence behavior of these algorithms is independent of the input data correlation statistics. Moreover, the tracking analysis of these algorithms is carried out in the presence of two sources of nonstationarites (carrier frequency offset between transmitter and receiver, and random variations in the environment). Simulation results agree very closely with theory and an enhancement in performance is obtained through the use of these techniques over the LMF algorithm. the step size.

Six one–dimensional models, based on the Ito–type stochastic equation, are presented and compared. Four of these take into account up to the fourth order moment of vertical velocity fluctuations, and two up to the third order moment. Four models make use of a bi–Gaussian probability density function (PDF) and the other two are based on a Gram–Charlier series expansion truncated to the third or fourth order. All the models were run with a parameterisation of input turbulence (i.e. w2, w3, and τ profiles). Concerning the fourth order moment w4, two different parameterisations were considered. Comparisons are made between ground–level concentrations, plume height and plume width observed in the Willis and Deardorff water tank experiments and those predicted by the different models here considered. The goal of this study was to find the models that give greater confidence in their applicability in dispersion studies and to verify the importance of considering the fourth order moment. The main conclusions are: simulation results largely depend on the turbulence parameterisation chosen; the Gram–Charlier PDF gives the best agreement with observations; some combinations of models and turbulence parameterisations perform well in simulating the shape of the ground–level concentration (g.1.c.) trend but fail in correctly simulating the form of the plume (plume height and vertical width); in the case of the Gram–Charlier PDF, the fourth order model reproduced the vertical plume width better than the third order one, whereas the two schemes yielded similar g.1.c. distributions.