Digital computation of the fractional Fourier transform

Abstract

Abstract-An algorithm for efficient and accurate computation of the fractional Fourier transform is given. For signals with time-bandwidth product N , the presented algorithm computes the fractional transform in O( N log N ) time. A definition for the discrete fractional Fourier transform that emerges from our analysis is also discussed.

Fast computation of the ambiguity function and the Wigner distribution on arbitrary line segments

“…In other words, any th-order FrFT of , including the signal itself and its ordinary Fourier transform, has negligible energy outside the symmetric interval . For a signal with approximate time and frequency supports of and , respectively, the required scaling is , where [36]. After the scaling, the double-sided bandwidth of is .…”

“…After the discretization, the obtained form lends itself for an efficient digital computation since the required samples of the FrFT [ ] can be computed using the recently developed fast computation algorithm [36] in flops. The summation in (12) can be recast into a -point discrete Fourier transformation (DFT), which can be computed in flops using the fast Fourier transform algorithm.…”

“…By using the projection-slice theorem given in (25), the nonradial slice of the WD of can be obtained as (34) where is the -Radon projection of the ambiguity function . Since the required -Radon projection satisfies the following FrFT relationship: (35) where , it can be efficiently computed by using the fast FrFT algorithm proposed in [36] and given here as Algorithm 1. The steps of the proposed algorithm are given in Algorithm 3.…”

“…Note that unlike , which is the -Radon Algorithm 1. Fast fractional Fourier transform algorithm proposed in [36].…”

Spectral Analysis of Generalized Triangular and Welch Window Functions using Fractional Fourier Transform

“…FrFT is the generalization of ordinary Fourier transform (FT) that depends on a parameter α and can be interpreted as a rotation by an angle α in the time-frequency plane [2,11,12]. It is represented by R α and α=aπ/2, where α is the angle of rotation and a is the fractional order parameter.…”