Influence of time constant on coefficients estimation error derived from a lowpass filter expression for learning identification algorithm
Kensaku Fujii, Yoshihiro Sakai, Juro Ohga · Electronics and Communications in Japan (Part III Fundamental Electronic Science) · 1991
Abstract This paper shows that the learning identification (the normalized LMS) algorithm can be expressed by a first‐order IIR low‐pass filter having rapidly varying coefficients as a function of output signal to be sent to an unknown system to be estimated. Based on the expression, this paper studies the estimation error of adaptive filters obtained by the learning identification algorithm. It is pointed out that the mean of estimation error given by the conventional analysis corresponds to the output fluctuation which can be computed by replacing varying coefficients of the lowpass filter with its arithmetic mean. However, the output fluctuation is affected by the time constant specified by the coefficients of the lowpass filter and it has a characteristic that it increases rapidly when time constant is short and decreases slowly when it is long. This means that the actual estimation error becomes larger than the result given by the conventional analysis. According to the analysis shown in this paper, it can be seen that the estimation error consists of elements for which the portion increased by the effect of time constant become fewer anti‐proportionally to the number of the tap, and those elements which increase at a constant rate independently of the number of the tap. It was shown also by simulation using white noise that the difference between the estimation error obtained in actual estimation operation which occurs when the number of the tap is small and the result given by the conventional analysis can be explained well by the analysis by simulation using white noise.