General consideration of an algorithm and a convergence condition of nonlinear adaptive digital filter

Abstract

Adaptive digital filters (ADF) estimating the transfer function of the unknown system can be classified as linear ADFs and nonlinear ADFs. In the linear ADF, the output is the inner product of the tap vector and the tap input vector unrelated to the tap vector. The transversal type ADF is a typical example. Other ADFs are nonlinear ones, among which the recursive ADF and logarithmic ADF are known. Previously, most discussions of nonlinear ADFs have been concerned with the recursive ADF. However, the transfer function H(z) of the nonlinear ADF is in general H (z) = Φ (G(z)) in which G(z) is the transfer function of a linear ADF. In the recursive ADF, Φ (G(z)) = 1/(1 − G(z)), which is a special case of Φ (G(z)). In this paper, the adaptive algorithm and its convergence condition are considered for a nonlinear ADF with a general transfer function H(z) = Φ (G(z)). The adaptive algorithm is based on the orthogonality principle and has a correlation between the tap input signal vector and the estimated error used as the updating vector. For this adaptive algorithm, the constraint condition on the transfer function of the unknown system needed for the tap vector to converge to that of the unknown system is derived. This condition can in general be derived from the condition under which the Hesse matrix becomes positive definite at the convergence value of the tap vector. Further, this condition is evaluated in detail for a specific example of Φ (G(z)). © 1999 Scripta Technica, Electron Comm Jpn Pt 3, 82(11): 9–17, 1999