Fractional Least Mean Square Research Articles

Identification of Hammerstein nonlinear ARMAX systems using nonlinear adaptive algorithms

In this study, a novel adaptive strategy is designed based on fractional least mean square (LMS) algorithm for parameter estimation of Hammerstein nonlinear autoregressive moving average system with exogenous noise (HN-ARMAX). The design scheme consists of parameterization of HN-ARMAX systems to obtain linear-in-parameter models and to use fractional LMS algorithm for adapting unknown parameter vectors. The performance analysis of the proposed method is carried out based on convergence to the desired values of HN-ARMAX systems, and comparison is made with state-of-the-art kernel LMS and Volterra LMS algorithms. The consistency in terms of accuracy and convergence is established through the results of statistical analysis based on sufficient large number of independent runs rather than single successful run of the algorithm. The performance of proposed scheme is superior due to its strong mathematical foundations, nonlinear weight updating mechanism and more convergence controlling variables but at the cost of bit more computational requirements.

Two-stage fractional least mean square identification algorithm for parameter estimation of CARMA systems

In the present study, single and two-stage least mean square (LMS) adaptive strategies based on fractional signal processing are developed for parameter estimation of controlled autoregressive moving average (CARMA) systems. The main idea is to use fractional LMS identification (FLMSI) and two-stage FLMSI (TS-FLMSI) algorithms for CARMA model that is decomposed into a system and noise models. The performance analyses for both proposed FLMSI and TS-FLMSI schemes are conducted based on adapting the prior known design parameters of the system and comparing the results with standard adaptive algorithms. The accuracy and convergence of the design schemes are verified and validated through the results of statistical analyses based on sufficient number of independent runs to adapt CARMA system. Comparative studies established the dominance of single and two-stage fractional adaptive algorithms over other counterpart in term of model accuracy and reliability in case of different scenarios based on variant signal to noise ratios and step size parameters.

A modified fractional least mean square algorithm for chaotic and nonstationary time series prediction

A method of modifying the architecture of fractional least mean square (FLMS) algorithm is presented to work with nonlinear time series prediction. Here we incorporate an adjustable gain parameter in the weight adaptation equation of the original FLMS algorithm and absorb the gamma function in the fractional step size parameter. This approach provides an interesting achievement in the performance of the filter in terms of handling the nonlinear problems with less computational burden by avoiding the evaluation of complex gamma function. We call this new algorithm as the modified fractional least mean square (MFLMS) algorithm. The predictive performance for the nonlinear Mackey glass chaotic time series is observed and evaluated using the classical LMS, FLMS, kernel LMS, and proposed MFLMS adaptive filters. The simulation results for the time series with and without noise confirm the superiority and improvement in the prediction capability of the proposed MFLMS predictor over its counterparts.