Fractional discreteq-Fourier transforms

Abstract

The discrete Fourier transform (DFT) matrix has a manifold of fractionalizations that depend on the choice of its eigenbases. One prominent basis is that of Mehta functions; here we examine a family of fractionalizations of the DFT stemming from q-extensions of this basis. Although closed expressions are given, many results of our analysis derive from numerical computation and display. Thus we suggest that the account of fractional Fourier transformations applied on signals as presented by other authors—typically of a centred rectangle function—may be biased because the support of the function lies in the central part of the domain only. The phase and amplitude of the whole fractional DFT matrices reveal the location of departures from the continuous kernel of the fractional Fourier integral transform, whose phase and constant amplitude are well known.

SU(2) and SU(1,1) Approaches to Phase Operators and Temporally Stable Phase States: Applications to Mutually Unbiased Bases and Discrete Fourier Transforms

“…Prima facie, it is not clear whether this subset is linearly independent and orthogonal, or not -Mehta [56] left unresolved their orthogonality, which was lately described thoroughly by Ruzzi [57]. The departure from strict orthogonality of the vectors of the Mehta basis was investigated in [58]; the departure is small for low values of n and gradually worsens up to d − 1.…”

“…. , N ϕ k − 1, we denote them by {Υ (ϕ k ,j) (m)}, periodic in m modulo d [46,58]; and we assume that they are complete in C d and thus have a dual basis {Υ (ϕ k ,j) (m)} periodic in m, such that…”