AbstractThe logarithmic adaptive digital filter (ADF) estimates the unknown system in which the transfer function is represented by a rational function of z. It consists of an ADF to estimate the logarithm of the denominator polynomial of the transfer function and a transversal ADF to estimate the numerator polynomial. Compared to the conventional transversal ADF, it is known that the number of taps can be reduced in this kind of ADF.Studies have been made concerning the adaptive algorithm of the logarithmic ADF and the convergence condition. A problem is that the correlation matrix is a function of the input and the output signals of ADF. This increases the convergence time, compared to the transversal ADF, when the input signal is white.This paper discusses the method of the improvement of the convergence speed and proposes the adaptive algorithm for the logarithmic ADF based on the LMS/Newton method. The successive computation of the inverse matrix of the correlation matrix (Jacobian) as well as the adaptive algorithm are derived. In contrast to the algorithm for the transversal ADF, it is shown that the square term of the estimation error is included in the update term of the tap vector.Finally, the convergence condition for the proposed algorithm is derived. The convergence speed, the adaptive algorithm and the convergence condition are examined by a computer simulation.