Finite Fourier transform for functions with restricted support

Finite Fourier Transform, Circulant Matrices, and the Fast Fourier Transform

Suppose we have a function s(t) that measures the sound level at time t of an analog audio signal. We assume that s(t) is piecewise-continuous and of finite duration: s(t) = 0 when t is outside some interval a ≤ t ≤ b. Make a change of variable x = (t− a)/(b− a) and set f(x) = s(t). Then 0 ≤ x ≤ 1 when a ≤ t ≤ b, and f(x) is a piecewise continuous function of x. We convert f(x) into a digital s...

Compatible with the Finite Fourier Transform

Let Z/n denote the integers mod n and let % denote the finite Fourier transform on L\Z/n). We let ©2$), » F operate on ©2L2(Z/n) and show that ©2L2(Z/n) can be given a graded algebra structure (with no zero divisors) such that f(/g) ■" ^U)^(g). We do this by establishing a natural isomorphism with the algebra of theta functions with period i. In addition, we find all algebra structures on 02L2(...

Computing with the Finite Fourier Transform Is

Apodizing functions for Fourier transform spectroscopy.

Apodizing functions are used in Fourier transform spectroscopy (FTS) to reduce the magnitude of the sidelobes in the instrumental line shape (ILS), which are a direct result of the finite maximum optical path difference in the measured interferogram. Three apodizing functions, which are considered optimal in the sense of producing the smallest loss in spectral resolution for a given reduction i...

The (finite field) Fast Fourier Transform

There are numerous directions from which one can approach the subject of the fast Fourier Transform (FFT). It can be explained via numerous connections to convolution, signal processing, and various other properties and applications of the algorithm. We (along with Geddes/Czapor/Labahn) take a rather simple view from the algebraic manipulation standpoint. As will be apparent shortly, we relate ...