This paper is about a new efficient method for the implementation of convolvers and correlators using the Fermat Number Transform (FNT) and the inverse (IFNT). The latter present advantages compared to Inverse Fast Fourier Transform (IFFT). An efficient state space method for implementing the Inverse FNT (IFNT) over rectangular windows is proposed for the cases where there is a large overlap be...

In November of 1979, there appeared a paper in the Bulletin of the AMS by L. Auslander and R. Tolimieri with the delightful title of “Is Computing with the Finite Fourier Transform Pure or Applied Mathematics?” The question was of course rhetorical and one of the main goals of the paper was to show that the finite Fourier transform, and the family of efficient algorithms used to compute it, the...

I. INTRODUCTION THE accurate analysis, design and optimization of guided-wave structures are essential for the development of integrated photonic devices. Of all the methods used to simulate guided-wave propagation in these devices, the beam propagation method (BPM) has been one of the most popular approaches used in the modeling and simulation of electromagnetic wave

We study the Zak transform of totally positive (TP) functions. We use the convergence of the Zak transform of TP functions of finite type to prove that the Zak transforms of all TP functions without Gaussian factor in the Fourier transform have only one zero in their fundamental domain of quasi-periodicity. Our proof is based on complex analysis, especially the Theorem of Hurwitz and some real ...

to date, a number of numerical methods, including the popular finite-difference time domain (fdtd) technique, have been proposed to simulate ground-penetrating radar (gpr) responses. despite having a number of advantages, the finite-difference method also has pitfalls such as being very time consuming in simulating the most common case of media with high dielectric permittivity, causing the for...

Traditional linear signal processing is based on viewing signals as sequences or functions in time that flow in one direction, from past through present into future. Somewhat surprisingly, the assumption that the most basic operation that can be performed on a signal is a time shift, or “delay,” is sufficient to derive many relevant signal processing concepts, including spectrum, Fourier transf...

Let G be a finite group, f: G --+ C a function, and p an ir-reducible representation of G. The Fourier transform is defined as j(p) =E SEG f(s)p(s) . Direct computation for all irreducible representations involvesorder IGI2 operations. We derive fast algorithms and develop them for thesymmetric groupSn. There, (n!)2 is reduced to n(n!)a/2, where a is theconstant for matr...

In recent times, considerable attention was focused on the computation of the eigenvalues and eigenfunctions of the Finite (truncated) Fourier Transform (FFT). An incentive to that a series of papers by Slepian and his collaborators at Bell labs was. The latest results published by Walter and Shen on sampling with the prolate spheroidal functions — the 1D FFT eigenfunctions — will necessary pro...

Pressing questions in cosmology such as the nature of dark matter and dark energy can be addressed using large galaxy surveys, which measure the positions, properties and redshifts of galaxies in order to map the large-scale structure of the Universe. We review the Fourier-Laguerre transform, a novel transform in 3D spherical coordinates which is based on spherical harmonics combined with dampe...