Research On The Performance Of Leaky Algorithm

The performance analysis of the adaptive filter mainly detects the transient,steady state and tracking performance of the adaptive algorithm,where the transient performance gives the convergence speed and stability information of the adaptive algorithm;the steady state performance provides the steady state mean square error information;Tracking performance shows the adaptability of the adaptive algorithm in a non-stationary environment.Since the introduction of the leakage factor in conventional adaptive algorithms can alleviate the weight drift problem of conventional adaptive filters,many scholars have studied leaky adaptive filters and analyzed their performance.However,no matter whether it is the research of leaky adaptive algorithm or its performance analysis is only for real number system,there is no literature about complex leaky adaptive algorithm and its performance analysis in complex number system.Therefore,for the real and complex leaky adaptive algorithm,this paper proposes a unified theoretical to analyze the transient performance,steady-state performance and tracking performance.The main work of this article are as follows:(1)The weighted energy conservation relationship is derived based on the stochastic gradient iteration of the leaky adaptive algorithm,which is used as the starting point.The transient performance of the leaky algorithm self-contained nonlinear estimation error function is analyzed using the real and complex Taylor expansion and the separation,assumption principle.State equations and mean performance expressions describing transient performance under white input and related input data are derived respectively.In addition,the general state equations and mean performance expressions are applied to the leaky least mean p-order(LLMP)algorithm and the leaky least mean mixed norm(LLMMN)algorithm.A comparison chart of the theoretical and simulated values of instantaneous mean square deviation and mean square error of the complex LLMP and the complex LLMMN algorithm in a Gaussian noise environment is given.(2)Based on the analysis results of the transient performance,solve the limit of the transient performance to obtain the steady-state performance,and give the unary quadratic equation with the steady-state mean square error.The coefficients of each term of the steady-state mean square error in the unary quadratic equation corresponding to the LLMP and LLMMN algorithm are given.The theoretical value of the steady-state mean square error is obtained by solving the unary quartic equation using a computer.The simulation experiment compares the theoretical value with the simulated value of the steady-state mean square error,and verifies the accuracy of the derivation result.(3)Taking the energy conservation relationship of the leaky adaptive algorithm as the starting point,the nonlinear estimation error function of the algorithm is subjected to Taylor expansion in real and complex forms,and the optimal step size expression and the unary quantity with steady state tracking mean square error are derived.The coefficients of each term of the steady-state tracking mean square error in the unary quadratic equation corresponding to the leaky least mean p-order algorithm and the leaky least mean mixed norm algorithm are given.The theoretical value of the steady-state tracking mean square error is obtained by solving the unary quartic equation using a computer.The simulation experiment compares the theoretical value with the simulated value of the steady state tracking mean square error,and verifies the accuracy of the derivation result.