A Posteriori Updates for Adaptive Filterss

In adaptive FIR lters, the least-mean-square (LMS) adaptive algorithm uses the a priori error signal to update the lter coeecients. In this paper , we study the forms and properties of a pos-teriori adaptive lter updates in a general context. We provide a technique by which the stability of an adaptive lter's coeecient update can be easily analyzed using the relationship between the a priori and a posteriori error signals. Using this knowledge, we then develop methods for choosing the algorithm step size to guarantee the robustness and stability of the system and to provide fast adaptation behavior. Simulations verify the usefulness of a poste-riori{error{based adaptive algorithms for unbiased adaptive IIR ltering. 1. INTRODUCTION The normalized least-mean-square (NLMS) adaptive lter is a useful technique for adjusting the L coeecients of a nite-impulse-response (FIR) lter. The NLMS coeecient updates are w(k + 1) = w(k) + (k)e(k)x(k) (1) e(k) = d(k) ? x T (k)w(k) (2) (k) = 0(k) jjx(k)jj 2 ; (3) where w(k) = w0(k) wL?1(k)] T and x(k) = x(k) x(k ?L+1)] T are the coeecient and input signal vectors at time k, respectively, e(k) is the a priori error signal at time k, jjx(k)jj 2 denotes the L2-norm of the vector x(k), and 0(k) is a step size parameter. The NLMS adaptive lter is a version of the LMS adaptive lter in which the eeective step size is (k) = 0(k)=jjx(k)jj 2. Since the LMS adaptive lter is usually derived and analyzed in a statistical context 1], such a view ignores certain useful stability and robustness properties possessed by the update in (1){(3). In particular, it can be shown that the NLMS adaptive lter is a projection-type update, and its stability and robustness can be guaranteed so long as 0 < 0(k) < 2 2, 3]. Recent techniques relating adaptive ltering algorithms to H 1 stability theory show that the NLMS algorithm possesses a characteristic robustness that is independent of the statistical realizations of the signals in x(k) and d(k) 4]{{6]. Moreover, a deter-ministic view of the NLMS algorithm elucidates the reasons behind the fast convergence behavior of this system 0 over that of the LMS adaptive lter. For example, when d(k) = x T (k)wopt with wopt being an unknown FIR coee-cient vector, then w(k) can be made to converge to wopt in L iterations via (1{(3) for 0(k) = 1, so long as the sequence …