Algorithm for Computing Fractional Fourier Transform

The Fractional Fourier Transform (FrFT) is a generalization of the classical Fourier transform that enables continuous angular transformations between the time and frequency domains. It finds extensive applications in signal processing, image analysis, and quantum mechanics.

### Algorithmic Approach Signal Decomposition: First, the input signal is decomposed into a joint time-frequency representation, typically using Short-Time Fourier Transform (STFT) or Wavelet Transform as preprocessing steps. In MATLAB implementation, this can be achieved using spectrogram() or cwt() functions with appropriate windowing parameters. Rotation Operation: The core of FrFT involves constructing a rotation operator that rotates the signal in the time-frequency plane by an arbitrary angle α. The rotation matrix can be implemented using eigen decomposition of the Fourier operator or through linear canonical transformation matrices. Kernel Computation: The FrFT kernel is constructed using Hermite polynomials or chirp functions, ensuring the transform satisfies additivity and rotation properties. Code implementation often involves calculating weighted sums of Hermite-Gaussian functions or using quadratic phase terms with fractional orders. Numerical Implementation: Efficient computation is achieved through discrete sampling and matrix operations. Common methods include Discrete Fractional Fourier Transform (DFrFT) implemented via eigenvalue decomposition of the DFT matrix, or FFT-based approximate algorithms using interpolation and chirp multiplication techniques.

### Application Extensions Signal Denoising: FrFT provides more flexible signal-noise separation capabilities than traditional Fourier transform. Implementation involves optimizing the fractional order parameter to maximize signal concentration in specific time-frequency regions. Time-Frequency Analysis: Particularly effective for non-stationary signal processing, where it outperforms conventional Fourier transform in characterizing local signal properties. Code implementation includes adaptive order selection algorithms based on signal characteristics. Optical Imaging: Used in optical systems to simulate fractional diffraction effects of lenses, with implementations involving phase mask multiplication and propagation kernel calculations for image quality optimization.

By adjusting the transform order parameter, this algorithm adapts to various analytical requirements, representing a significant extension to the Fourier transform family with versatile implementation approaches.