Module frft 

Fractional Fourier Transform module

The Fractional Fourier Transform (FrFT) is a generalization of the standard Fourier transform, allowing for transformation at arbitrary angles in the time-frequency plane. It provides a continuous transformation between the time and frequency domains.

Mathematical Definition

The continuous Fractional Fourier Transform of order α for a signal f(t) is defined as:

F_α(u) = ∫ f(t) K_α(t, u) dt

where K_α(t, u) is the transformation kernel:

K_α(t, u) = √(1-j cot(α)) * exp(j π (t² cot(α) - 2tu csc(α) + u² cot(α)))

Special Cases

• α = 0: Identity transform (returns the input signal)
• α = 1: Standard Fourier transform
• α = 2: Time reversal (f(t) → f(-t))
• α = 3: Inverse Fourier transform
• α = 4: Identity transform (cycles back to original)

Implementation

This implementation uses an efficient algorithm based on the FFT, with special handling for the cases where α is close to 0, 1, 2, or 3.

Numerical Stability

Important: The current implementation has known numerical stability issues, particularly with the additivity property. See FRFT_NUMERICAL_ISSUES.md for detailed information about these limitations and proposed solutions.

Functions

frft

Computes the Fractional Fourier Transform of order alpha.

frft_bandwidth_saturated_simd

Bandwidth-saturated SIMD implementation of Fractional Fourier Transform

frft_complex

Special implementation for Complex64 input to avoid conversion issues.

frft_dft

Computes the Fractional Fourier Transform using DFT eigenvector decomposition

frft_near_special_case_bandwidth_saturated_simd

Bandwidth-saturated SIMD implementation of near special case handling

frft_stable

Computes the Fractional Fourier Transform using the Ozaktas-Kutay algorithm

frft_stable_bandwidth_saturated_simd

High-performance stable FrFT with bandwidth-saturated SIMD optimizations