Parseval-Like Identity of Wavelet Transform and Its Applications to Noise Reduction

Abstract

In this paper we examine the noise-reducing property of continuous wavelet transform, and propose a concept of optimal wavelets in the sense of complete noise reduction for separable noises in the time-scale domain. We first discuss about the wavelet transform of stochastic processes, focusing mainly on certain properties of it, such as the stationarity preserving property and Parseval-like identity. We then show that the mean square power of separable noises in the time-scale domain is possible to be zero by inverse wavelet transform if the basic wavelet φ (t) is chosen properly. A numerical example is given for illustration.