Fractional Fourier Transform Function

The Fractional Fourier Transform (FRFT) serves as a generalized form of the conventional Fourier transform, introducing a fractional order parameter to enable more flexible analysis of time-frequency characteristics in signals. When implementing this function in MATLAB, it can be encapsulated into a reusable module suitable for non-stationary signal processing, image encryption, and related applications.

The core algorithm relies on discrete sampling and linear algebraic operations: first performing Hermite interpolation on the input signal to construct a transformation kernel matrix of specific fractional order, then executing the transform through matrix multiplication. MATLAB's vectorization capabilities significantly enhance computational efficiency, while implementation should carefully handle complex-valued signals and boundary conditions for the fractional order parameter. Key functions would involve interp1 for interpolation and matrix multiplication operators for the transform computation.

Typical applications include time-frequency localization in radar signals and solving blind source separation problems in convolution systems. Compared to traditional FFT, the fractional order characteristic provides superior performance when analyzing linear frequency-modulated signals, allowing gradual transition from time domain to frequency domain through order adjustment. The implementation requires only specifying the signal vector and fractional order parameter to obtain transform results, typically structured as frft_result = frft_function(input_signal, fractional_order).