Relations between chirp transform and Fresnel diffraction, Wigner distribution function and a fast algorithm for chirp transform

• Published in: Chinese physics B Vol. 19; no. 7; pp. 238 - 244
• Main Author: 石鹏 曹国威 李永平
• Format: Journal Article
• Language: English
• Published: IOP Publishing 01.07.2010
• Subjects: Algorithms; Chirp; Fourier transforms; Fresnel diffraction; Mathematical analysis; Mathematical models; Transforms; Wigner distribution; Wigner分布函数; 分数阶傅立叶变换; 分数阶傅里叶变换; 快速傅立叶变换算法; 快速算法; 菲涅耳衍射; 魏格纳分布函数
• ISSN: 1674-1056; 2058-3834
• DOI: 10.1088/1674-1056/19/7/074201

Two physical interpretations of chirp transform related to Fresnel diffraction and Wigner distribution function are given. The chirp transform can be regarded as a Fresnel diffraction observed on a spherical tangent to the diffraction plane, or a rotation and stretching transformation of the Wigner distribution function space. A general fast algorithm for the numerical calculation of chirp transform is developed by employing two fast Fourier transform algorithms. The algorithm, by which a good evaluation can be achieved, unifies the calculations of Fresnel diffraction, arbitrary fractional- order Fourier transforms and other scalar diffraction systems. The algorithm is used to calculate the Fourier transform of a Gaussian function and the Fourier transform, the Fresnel transform, the Fractional-order Fourier transforms of a rectangle function to evaluate the performance of this algorithm. The calculated results are in good agreement with the analytical results, both in the amplitude and phase.