Fractional-Fourier-transform calculation through the fast-Fourier-transform algorithm

Area of Science:

• Signal Processing
• Applied Mathematics
• Optics

Background:

• The fractional Fourier transform (FRT) is a generalization of the Fourier transform with applications in signal processing and optics.
• Efficient computation of the FRT is crucial for its practical implementation.

Purpose of the Study:

• To present a novel method for calculating the fractional Fourier transform (FRT).
• To leverage the fast Fourier transform (FFT) algorithm for efficient FRT computation.
• To analyze the validity and complexity of the proposed FRT calculation method.

Main Methods:

• The proposed method utilizes a cascade of two fast Fourier transform (FFT) algorithms.
• The computational complexity of the method is analyzed and compared to the standard FFT.
• The method's validity is established for fractional orders ranging from -1 to 1.

Main Results:

• A computationally efficient method for calculating the FRT is demonstrated.
• The method exhibits the same computational complexity as the fast Fourier transform (FFT) algorithm.
• Scaling factors relevant to FRT and Fresnel diffraction calculations using FFT are discussed.

Conclusions:

• The presented method provides an efficient way to compute the fractional Fourier transform (FRT) using readily available fast Fourier transform (FFT) algorithms.
• The technique is applicable across a wide range of fractional orders, enhancing its utility in various scientific domains.
• Understanding scaling factors is essential for accurate FRT and Fresnel diffraction analysis via FFT.