Adaptive Algorithm For System Identification In The Orthogonal Transform Domain With Generalized Structures

In the system identification,the adaptive filter is often employed to model the impulse response of the unknown linear system to deal with the time-varying statistical property of the signal as well as to reduce the computational complexity.In this sense,the performance of system identification often depends on the performance of the adaptive filter,which is decided by the structure and the algorithm of the latter.The least mean square(LMS)algorithm based on the transversal filter structure is widely studied owing to its simplicity,robustness and ease of use,and it is always one of the focuses in the field of adaptive filter theory.However the convergence rate of the LMS algorithm is inversely related to the correlation of the input,i.e.,the convergence rate decreases as the eigenvalue spread of the input autocorrelation matrix increases,which,to some extent,limits the application of LMS algorithm in the correlated input case.How to solve this problem motivates the people to propose the modified LMS algorithms such as short-time transform domain(STFT)LMS algorithm,transform domain(TD)LMS algorithm,fractional tap-length(FT)LMS algorithm and so on.The STFTLMS algorithm,TDLMS algorithm and FTLMS algorithm have developed into different research fields for their special properties,but their core is still the LMS algorithm.To explore the relation of these modified LMS algorithms and the traditional LMS algorithm,the generalized structure of orthogonal transform domain LMS algorithm is introduced.These algorithms,as the special cases in the generalized structure,make use of proper strategies with proper parameters to improve the convergence rate of the LMS algorithm.However,it often depends on the prior knowledge of the system or the experience of the user to ensure these parameters be suitable to balance the contradiction between the convergence rate and steady-state mean square error.To avoid this,the STFTLMS algorithm,TDLMS algorithm and FTLMS algorithm are modified by means of designing the adaptive strategies for these parameters on the background of system identification.The main contents are as follows.Firstly,the theory with the emphasis on the threshold parameters of the tap-length in the FTLMS algorithm is improved.By this theory,we provide a convincible mathematical analysis justifying the advantage of the variable tap-length adaptive step size(TASS)over the fixed one in the tap-length convergence rate and small steady-state fluctuation.This analysis result also motivates us to establish a straightforward but effective strategy to vary the TASS.Then from the viewpoint of threshold parameters,we further discuss the reasons of unexpected change of the steady-sate tap-length caused by the noise.To maintain the steady-state tap-length stable,a simple hybrid algorithm is proposed which incorporates the idea of limiting amplitude and the convex combination scheme.The adaptation rules for tap-length and tap-vector in the FTLMS algorithm are decoupled,so we propose a novel algorithm where TASS and weight adaptive step size can be adjusted adaptively to improve the convergence performance of the tap-length and mean square error.Secondly,we study two convex combination schemes for the TDLMS algorithm,one is carried out in one adaptive filter and the other is between two adaptive filters.The convergence performance of the former is analyzed and then two adaptive strategies of the mixing parameter of the convex combination scheme are designed by the output error and the gradient vector,respectively,to achieve the fast convergence rate and low steadystate mean square error.To further the improvement of convergence rate,the convex combination of two adaptive filters is employed.The theoretical excess mean square error(EMSE)model of the component filters and cross-EMSE model between them are derived and verified by the simulation.Thirdly,the novel STFTLMS algorithms are developed where number of the crossband filters or number of the cross-multiplicative transfer functions in the subband can be controlled adaptively.The algorithm with adaptive number of crossband filters in the subband can be implemented with the multi-filter structure or the single-filter structure.For the former,the estimation errors of three transversal adaptive filters serve as the measurement criteria to adjust the number of crossband filters by some strategy.The segmented error is introduced in the algorithm with single-filter structure,and then the number of crossband filters is adjusted by comparing the ratio of the estimation error and segmented error with the presupposed threshold parameters.To reduce the computational burden,the scheme of the single-filter structure is used to control the number of cross-multiplicative transfer functions.Finally,a large amount of simulation is conducted to verify that the proposed algorithms can improve the convergence rate as well as maintain low steady-state mean square error.These algorithms are then used to model the impulse response of acoustic echo path for the acoustic echo cancellation.The practical values of these algorithms are evaluated in terms of echo return loss enhancement(ERLE)level and running time.