An error analysis of an approximation method for band‐limited signals using minimum energy signal‐existence of the optimum number of sampling points and its estimation

Abstract

AbstractFor an approximation problem of band‐limited signals, “minimum energy signal” is defined as a band‐limited waveform with minimum energy that takes the specified sample valueas at the given sampling points.By using minimum energy signals, we have already reported a new approximation method for band‐limited signals. In the proposed method, sampling points are selected from a given time interval so as to minimize the square error between the original and the approximation signals. However, when this approximation method is realized with a digital hardware or a digital computer, there are round‐off noises in sample values due to the finite word length of the hardware system. Thus, to employ our proposed approximation method in practice, it is essentially important to analyze the effect of this round‐off noise.In this paper, some properties of minimum energy signals are investigated when the sample values of the signal are quantized to fit a finite word length of the hardware, i.e., a fixed‐point of floating point format. Also, it is shown that an optimum number of sampling points may exist when the proposed approximation method is applied to signals with noise. Moreover, by considering the effect of noise in advance, we propose a method to estimate the optimum number of sampling point as well as to determine the statistically optimum sampling points.