The Fractional Fourier Transforms And Their Relationships

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.The fractional Fourier transform has the property of multiplicity. In fact, diversity is the number operator of any fractional intrinsic nature of the diversity on the fractional Fourier transform, from the many proposed in the literature the definition of fractional Fourier transform can be seen by. 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 fra- ctional 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 research work is sampled by the sampling theorem and to analyze the relationship between standards and standards-weighted chirp fractional Fourier transform relationship between the fractional Fourier transform, the standard non-standard chip fractional Fourier transform and fractional Fourier chirp transform the relationship between the non-standard non-standard chirp fractional Fourier transform and the weighted relationship between the fractional Fourier transform.