Algebraic structures for $\bigoplus \sum \sb{n\geq 1}L\sp{2}(Z/n)$\ compatible with the finite Fourier transform

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

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...

The Poisson sum formulae associated with the fractional Fourier transform

The theorem of sampling formulae has been deduced for band-limited or time-limited signals in the fractional Fourier domain by different authors. Even though the properties and applications of these formulae have been studied extensively in the literature, none of the research papers throw light on the Poisson sum formula and non-band-limited signals associated with the fractional Fourier trans...

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 ...