Applications of Wavelet Transforms to the Analysis of Superoscillations

Abstract

The phenomenon of superoscillation is the local oscillation of a band limited function at a frequency ω higher than the band limit. Superoscillations exist during the limited time intervals, and their amplitude is small compared to the signal components with the frequencies inside the bandwidth. For this reason, the wavelet transform is a useful mathematical tool for the quantitative description of the superoscillations. Continuoustime wavelet transform (CWT) of a transient signal ft ðÞis a function of two variables: one of them represents a time shift, and the other one is the scale or dilation variable. As a result, CWT permits the simultaneous analysis of the transient signals both in the time and frequency domain. We show that the superoscillations strongly localized in time and frequency domains can be identified by using CWT analysis. We use CWT with the Mexican hat and Morlet mother wavelets for the theoretical investigation of superoscillation spectral features and time dependence for the first time, to our best knowledge. The results clearly show that the high superoscillation frequencies, time duration, and energy contours can be found by using CWT of the superoscillating signals.

Local Frequencies in Superoscillatory Phenomena

“…When the dilation parameter a and the translation parameter b are discrete and take the form a j ¼ 2 Àj and b jk ¼ k2 Àj (see Eq. ( 17) ) where k and l are integers, the expression for the CWT is a series where the dyadic coefficients dj , k ðÞ correspond to the DWT of xt ðÞ [10].…”