Local Frequencies in Superoscillatory Phenomena

Abstract

Superoscillations correspond to a non-linear phenomenon theoretically addressed by Aharonov in 1991. The resulting waves or functions have the particularity of being of limited bandwidth and contain faster amplitude variations than that corresponding to the fastest components obtained applying the Fourier transform. Also, the amplitude developed in the region where it occurs is small, since it decreases exponentially. These characteristics prevent its determination using the Fourier transform since it is not a stationary phenomenon. With this perspective, we have tested other methods for determining these features, such as wavelet transforms and Hilbert-Huang transform. Wavelet transforms can capture both low- and high-frequency components of the signal. The Hilbert-Huang transform allows the decomposing of a signal into the so-called intrinsic mode functions (IMF) together with a trend, and obtaining instantaneous frequencies. We also proposed a methodology using Gabor-adaptive windows to perform detection. Finally, filtering results were added using a multiresolution analysis decomposition that allows separating the super-oscillatory part of one and therefore localizes the oscillations in time, that is, local frequencies.