A hybrid Fourier transform

A Fourier Transform

The harmonic function F exp(j2rvt) plays an important role in science and engineering. It has frequency v and complex amplitude F. Its real part IFIcos(2~vt + arg{F}) is a cosine function with amplitude jF( and phase arg{F}. The variable t usually represents time; the frequency v has units of cycles/s or Hz. The harmonic function is regarded as a building block from which other functions may be...

A New Fourier Transform

In order to define a geometric Fourier transform, one usually works with either `-adic sheaves in characteristic p > 0 or with D-modules in characteristic 0. If one considers `-adic sheaves on the stack quotient of a vector bundle V by the homothety action of Gm, however, Laumon provides a uniform geometric construction of the Fourier transform in any characteristic. The category of sheaves on ...

The Fourier Transform { A

Fourier Transform

Very broadly speaking, the Fourier transform is a systematic way to decompose “generic” functions into a superposition of “symmetric” functions. These symmetric functions are usually quite explicit (such as a trigonometric function sin(nx) or cos(nx)), and are often associated with physical concepts such as frequency or energy. What “symmetric” means here will be left vague, but it will usually...

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...