The convolution theorem for the continuous wavelet tranform

Abstract

We study the application of the continuous wavelet transform to perform signal 1ltering processes. We 1rst show that the convolution and correlation of two wavelet functions satisfy the required admissibility and regularity conditions. By using these new wavelet functions to analyze both convolutions and correlations, respectively, we derive convolution and correlation theorems for the continuous wavelet transform and show them to be similar to that of other joint spatial/spatial–frequency or time/frequency representations. We then investigate the e5ect of multiplying the continuous wavelet transform of a given signal by a related transfer function and show how to perform spatially variant 1ltering operations in the wavelet domain. Finally, we present numerical examples showing the usefulness of applying the convolution theorem for the continuous wavelet transform to perform signal restoration in the presence of additive noise.