Correlation theorem and applications associated with the fractional Fourier transform in polar coordinates

Abstract

The two-dimensional fractional Fourier transform (FrFT) has very important applications in applied mathematics and signal processing. The polar coordinate form can not only enables the definitions of instantaneous amplitude, instantaneous phase, and the instantaneous frequency of a signal, but also extract some features that cannot be directly observed in the real signal. In this paper, we study the problem of correlation theorem and applications in the FrFT domain based on the polar coordinates. First, shift theorem and product theorem associated with the FrFT are exploited. Then, a correlation theorem of the FrFT is achieved according to the shift theorem. Furthermore, the relationship between the product theorem and the correlation theorem for the FrFT is established. Finally, we explored the possible applications of the obtained results of the FrFT on time-frequency representation, equation solving, and fast algorithm.