Fractionalize The Kernelof The Fourier Transform

The fractional Fourier transform (FRFT) is the generalized version of the classical Fourier transform, when the fractional order gradually increases from 0 to 1, the fractional Fourier transform of signal can offer more time-frequency representation than the classical Fourier transform, and can provide extensive optional space for signal processing. Especially in the study of optical information processing, optical fractional Fourier transform provides the ability of disposing non-focal-plane of signal, which brings optical information processing much convenience and expands optical application to a new field.In general, the fractional Fourier transform can be classified into two kinds, namely, the classical fractional Fourier transform and the weighted fractional Fourier transform. The other types of fractional Fourier transform are developed on the base of the theories of the two fractional Fourier transforms. Because the kernel of the fractional Fourier transform completely determines the operator itself, the full analysis and research to the kernel of the fractional Fourier transform has a great impact on the analysis and study of the operator self.The main work of this paper is to study the structure of the fractional Fourier transform kernel. According to the standard chirp-type fractional Fourier transform kernel and the standard weighted-type fractional Fourier transform kernel proposed in the literature. I derive the structure of the fractional Fourier transform kernels with periods which are divided by 4 and discuss the relationship between them and the nature of weighted coefficients. Furthermore, the period is extended to one which is divided by any natural number. I get the structure of the fractional Fourier transform kernels and prove the relationship between them and the nature of weighted coefficients. Finally, a general conclusion is get.