A Comprehensive Comparative Study of Two-Dimensional and Three-Dimensional Fractional Fourier Transforms

The Fractional Fourier Transform (FrFT) has established itself as a fundamental mathematical tool in signal processing, optics, and communications, with applications spanning filter design, signal recovery, phase retrieval, pattern recognition, and image encryption. While the one-dimensional FrFT is theoretically well-understood, modern applications increasingly involve multidimensional data, particularly images (2D) and medical/geophysical volumes (3D). This paper presents a comprehensive comparative study of separable two-dimensional and three-dimensional Fractional Fourier Transforms. We provide complete mathematical proofs of fundamental properties including linearity, index additivity, shift and modulation theorems, Parseval’s relation, eigenfunction expansions, and convolution structures for both 2D and 3D cases. Through systematic comparison, we demonstrate that while separable FrFTs preserve many desirable properties in higher dimensions, they are inherently limited to separable phase space rotations. We establish the relationship between separable multidimensional FrFTs and more general non-separable linear canonical transforms, identifying scenarios where separable transforms suffice and where genuinely coupled transforms become necessary.