Study On Fractional Order LMS Adaptive Filtering Algorithm

Thanks to the strong capabilities on signal processing and being implemented in engineering practice easily in the modern production and our daily life,the adaptive filtering algorithms have attracted many scientists' and engineers' attention in the last a half century.An amount of theoretical results have been developed and applied in various fields,such as,digital communications,automatic control,seismic detection and biomedicine,etc.The LMS algorithm,one of the most famous adaptive filtering algorithms,has attracted a lot of researcher's attention because of its many compelling advantages such as the favourable stable performance,simple structure,and easy to be realized etc.Some existing results show that the fractional order theory can improve the convergence performance of LMS algorithm.As the extension and promotion of conventional integer order counterpart,frac-tional order calculus was born more than 300 years ago.It has attracted many schol-ars' research interests and has been widely used in physics,chemistry,biology and electronics etc.,and obtained better performance than the conventional integer order counterparts.However,in the discrete time domain,the corresponding fractional order sum and difference have gradually gained attention in recent decades,and some related research is not sufficient.Some existing results have shown that the discrete fractional orders sum and difference can be used in signal processing,image encryption,and oth-er fields,which can achieve incomparable effects against the conventional integer order method.Therefore,whether for the algorithm theory research or for the application in the engineering,the combination of the LMS algorithm with the fractional theory is very significant.Firstly,this thesis focuses on a class of more general linear discrete fractional order systems,whose order a is been extended in the interval(0,2).The stability and time domain response characteristics of the discrete fractional order systems are analyzed in detail,and a wider and more general system stability condition is obtained.On this basis,it is rigorously demonstrated that when ??(0,1),the system output converges to the stable point monotonically and asymptotically,and for the ? ?(1,2)case,the system output converges to the stable point asymptotically with overshoot.In addition,the traditional gradient method is extended to the fractional case,and the initial value is designed as variable.The proposed method can remove the problem that the existing gradient method cannot converge to the exact extreme value.Secondly,by viewing iterative way of traditional integer order LMS adaptive fil-tering algorithm as the first order difference,and extending the difference order to the interval(0,2),the fractional order iterative order based LMS filtering algorithm is pro-posed.For the purpose of analysing the performance of the fractional order LMS al-gorithm,the algorithm is transformed into a discrete fractional order system.Then one deduced the relationship between the convergence performance and the step size(?)and the update order(?):the larger value of the ? or a is,the faster the algorithm con-vergence speed with larger steady state misadjustment can be obtained;for ? ?(0,1)case,the weight vector converges to the exact one monotonically and asymptotically,and the weight vector converges to the exact weight vector asymptotically with over-shoot when ? fall within the interval(1,2).In order to modified convergence perfor-mance,an iterative order hybrid switching method is proposed.Afterwards,based on the fractional gradient descent method,the traditional integer order LMS adaptive filtering algorithm is extended to a more general case,namely,the fractional order gradient based LMS adaptive filtering algorithm.The convergence performance of this class of fractional order LMS algorithm is related with the step size(?),the gradient order(?),and the of the initial value length(K):the larger ?or ? is,the faster the algorithm converges can be obtained,but bring a larger steady state misadjustment.Based on this performance,an novel algorithm based varying gradient order and varying length of the initial value is developed,which can remove the contradiction between convergence rapidity and convergence accuracy and improve the convergence performance of the algorithm.Finally,in order to illustrate the validity and practicability of the proposed method,the fractional order LMS algorithm is applied to identify the polynomial nonlinear Hammerstein systems and arbitrary nonlinear Hammerstein systems.For the identi-fication of polynomial nonlinear Hammerstein system,the proposed LMS algorithm based on fractional order gradient is combined with multi-innovation,and the conver-gence performance is analyzed in detail.Then a improved algorithm is achieved,whose performance is better than the conventional integer order one.For the identification of arbitrary nonlinear Hammerstein systems,the cubic spline interpolation method is in-troduced for the fractional iteration order based LMS algorithm and can obtain better performance than the conventional counterpart.