STEADY-STATE ANALYSIS OF THE NORMALIZED LEAST MEAN FOURTH ALGORITHM WITHOUT THE INDEPENDENCE AND SMALL STEP SIZE ASSUMPTIONS Muhammad Moinuddin and Azzedine Zerguine Electrical Engineering Department King Fahd University of Petroleum & Minerals Dhahran, 31261, Saudi Arabia. E-mail :{moinudin, azzedine}@kfupm.edu.sa ABSTRACT In this work, the steady-state analysis of the Normalized Least Mean Fourth (NLMF) algorithm under very weak assumptions is investigated. No restrictions are made on the dependence between input successive regressors, the dependence among input regres- sor elements, the length of the adaptive filter, the distribution of noise and the filter input. Moreover, in our approach, there is no restriction made on the step size value and therefore the analy- sis holds for all the values of the step size in the range where the NLMF algorithm is stable. The analysis is based on the effective weight deviation vector performance measure [1]. This vector is the component of weight deviation vector in the direction of the in- put regressor. The asymptotic time-averaged convergence for the mean square effective weight deviation, the mean absolute excess estimation error, and the mean square excess estimation error for the NLMF algorithm are derived. Finally, a number of simulation results are carried out to corroborate the theoretical findings. Index Terms — Adaptive filters, NLMF algorithm, Conver- gence Analysis. 1. INTRODUCTION The Normalized Least Mean Fourth (NLMF) algorithm [2] is the normalized version of the Least Mean Fourth (LMF) algorithm [3]. The analysis of the NLMF becomes difficult because of the nor- malization term. Therefore, until now, the analysis of the NLMF algorithm is carried out using some strong assumptions [4, 5], for example, using the independence assumption [6] or the long fil- ter assumption [7]. Recently a new performance measure, the ef- fective weight deviation vector, is introduced for the convergence analysis of the NLMS algorithm [1]. This vector is the compo- nent of weight deviation vector in the direction of input regres- sor vector. It is shown that the effective weight deviation is the only component that contributes to the excess estimation error [1]. Therefore, the analysis based on the study of this component can give more insight on the performance of the adaptive algorithm. In this work, we have used the framework of [1] for the analysis of the NLMF algorithm using the concept of the effective weight deviation vector. The main contribution of this paper is a rigorous convergence analysis of the NLMF algorithm that has the following advantages: (1) it holds for arbitrary dependence among successive regressor vectors, (2) it holds for arbitrary dependence among the elements of regressor vector, (3) this analysis is not restricted to the class of long filters, (4) it holds for arbitrary distributions of the filter input and the noise, and (5) it holds for all the values of the step size in the range that insures the stability of the NLMF algorithm. The paper is organized as follows. After introducing the sys- tem model in the following subsection, a brief overview the newly introduced performance measure is presented in Section 3. In Section 4, asymptotic time-averaged convergence analysis for the mean square effective weight deviation, the mean absolute excess estimation error, and the mean square excess estimation error of the NLMF algorithm is carried out. Simulation results are pre- sented to validate the theoretical findings in Section 5 and paper is ended with concluding remarks in Section 6. 2. SYSTEM MODEL Consider the case of adaptive plant identification problem [6, 7]. The output y k of the plant is given by y k = c T x k + η k , (1) where c =[c1,c2,...,cN ] T (2) is the vector of the unknown system, and x k =[x 1,k ,x 2,k ,...,x N,k ] T (3) is the input data vector at time k, η k is the plant noise, N is the number of plant parameters, and [·] T is the transpose operation. The inputs x 1,k ,x 2,k ,..., and x N,k may be successive samples of same signal, such as in the case of adaptive echo cancelation [8] and adaptive line enhancement [9]. They may also be the in- stantaneous output of N parallel sensors, such as in the case of adaptive beamforming [6]. The identification of the plant is made by an adaptive FIR filter whose weight vector w k , assumed of di- mension N , is adapted on the basis of error e k given by e k = y k − w T k x k . (4) The adaptation algorithm considered in this paper is NLMF algo- rithm [4] described by w k+1 = w k + μ ||x k || 2 x k e 3 k , (5) where μ> 0 is the algorithm step size and the norm of a vector x is defined as ||x|| 2 ≡ x T x. The error e k can be decomposed to two terms which are the plant noise η k and the excess estimation error ε k defined by ε k = e k − η k . (6) 3097 978-1-4244-2354-5/09/$25.00 ©2009 IEEE ICASSP 2009