Wavelet-based filters for accurate computation of derivatives

Abstract

Let f ( x ) f(x) be a smooth function whose derivative of a given order must be computed. The signal f ( x ) f(x) is affected by two kinds of perturbation. The perturbation caused by the presence of the machine epsilon ϵ M \epsilon _M of the computer may be considered to be an extremely high-frequency noise of very small amplitude. The way to minimize its effect consists of choosing an appropriate value for the step size of the difference quotient. The second perturbation, caused by the presence of noise, requires first the signal to be treated in some way. It is the purpose of this work to construct a wavelet-based band-pass filter that deals with the two cited perturbations simultaneously. In effect our wavelet acts like a “smoothed difference quotient" whose stepsize is of the same order as that of the usual difference quotient. Moreover the wavelet effectively removes the noise and computes the derivative with an accuracy equal to the one obtained by the corresponding difference quotient in the absence of noise.