Wavelet-based filters for accurate computation of derivatives

Abstract

Abstract. Let f (x) be a smooth function whose derivative of a given order must be computed. The signal f (x) is affected by two kinds of perturbation. The perturbation caused by the presence of the machine 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.

A new rotating machinery fault diagnosis method based on improved local mean decomposition

“…The first product function PF 1 (t) can be constructed by the envelope function a 1 (t) and the purely frequency modulated signal s 1n (t), which can be written as PF 1 (t) = a 1 (t)s 1n (t) (8) In theory, PF 1 (t) contains the highest frequency oscillations of the signal x(t), whose instantaneous amplitude (IA) is exactly the envelope signal a 1 (t) and instantaneous frequency (IF) is calculated by the purely frequency modulated signal s 1n (t), it can be expressed as…”

“…Wavelet transform (WT) can decompose multiscales into several scale time-frequency components, which has ability of processing the non-stationary and nonlinear signals, it has been widely used to diagnose the rotating machine. In fact, WT is essentially an adjustable window Fourier transform, which doesn't have the nature of self-adaptive feature [7,8].…”

“…, pre_PF j i (t), j = 1, .. . , n according to the condition (II) of OPF and set the pre_OPF j i (t) as OPF i (t) (8). Then the residual r i (t) = x(t) − OPF i (t) and take the r i (t) as the original signal.…”

Investigation and Performance Enhancement of the Empirical Mode Decomposition Method Based on a Heuristic Search Optimization Approach

“…Second, it is well known that differentiation can result in noise amplification. Thus, differentiation has to be accompanied with some form of appropriate filtering, with Savitzky-Golay smoothing filter [18] and specialized wavelet techniques [19] being possible candidates. Additionally, the adoption of adaptivesmoothing splines [20] can be an extension of EMD in the presence of noise.…”