Digital computation of the weighted-type fractional Fourier transform

Abstract

The paper reveals the time-frequency symmetric property of the weighted-type fractional Fourier transform (WFRFT) by investigating the original definition of the WFRFT, and proposes a discrete algorithm of the WFRFT based on the weighted discrete Fourier transform (WDFT) algorithm with constraint conditions of the definition of the WFRFT and time-domain sampling. When the WDFT is considered in digital computation of the WFRFT, the Fourier transform in the definition of the WFRFT should be defined in frequency (Hz) but not angular frequency (rad/s). The sampling period Δt and sampling duration T should satisfy \(\Delta t = {T \mathord{\left/ {\vphantom {T N}} \right. \kern-\nulldelimiterspace} N} = {1 \mathord{\left/ {\vphantom {1 {\sqrt N }}} \right. \kern-\nulldelimiterspace} {\sqrt N }}\) when N-point DFT is utilized. Since Hermite-Gaussian functions are the best known eigenfunctions of the fractional Fourier transform (FRFT), digital computation based on eigendecomposition is also carried out as the additional verification and validation for the WFRFT calculation.