A Kind Of Fractional Fourier Transform And Property

The fractional Fourier transform (FRFT), as the specific case of fractional operator, its property of multiplicity attracts much attention gradually. It can be seen as extension of the ordinary 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 disposal of signal. 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.The fractional Fourier transform has the property of multiplicity. So far, there exist many types of fractional Fourier transforms such as the standard Chirp-type fractional Fourier transform, the standard Weighted-type fractional Fourier transform, the generalized Chirp-type fractional Fourier transform, the generalized Weighted-type fractional Fourier transform and so on. In fact, the multiplicity is caused by the reason that there are a lot of techniques to fractionalize the eigenvalues of Fourier transform in the procedure of constructing the fractional Fourier transform.The main work of this paper is to propose an approach to the multiplicity of the fractional Fourier transform based on the weighted-type fractional Fourier transform. Firstly, using another variable transformation, the new different weighted coefficients will be obtained. Therefore, by making use of the new weighted coefficients, new fractional Fourier transforms can be constructed. Then, some properties of the new and the existed FRFT will be discussed, such as eigenvalues, recursive relation between the coefficients.