Fast-Fourier-Transform-based Direct Integration Algorithm for the Linear Canonical Transform

The linear canonical transform(LCT) is a parameterized linear integral transform, which is the general case of many well-known transforms such as the Fourier transform(FT), the fractional Fourier transform(FRT) and the Fresnel transform(FST). These integral transforms are of great importance in wave propagation problems because they are the solutions of the wave equation under a variety of circumstances. In optics, the LCT can be used to model paraxial free space propagation and other quadratic phase systems such as lens and graded-index media. A number of algorithms have been presented to fast compute the LCT. When they are used to compute the LCT, the sampling period in the transform domain is dependent on that in the signal domain. This drawback limits their applicability in some cases such as color digital holography. In this paper, a Fast-Fourier-Transform-based Direct Integration algorithm(FFT-DI) for the LCT is presented. The FFT-DI is a fast computational method of the Direct Integration(DI) for the LCT. It removes the dependency of the sampling period in the transform domain on that in the signal domain. Simulations and experimental results are presented to validate this idea.