Fourier transform

A Fourier transform is an integral transform, typically from a time dimension to a frequency dimension or back.

Theory

Given some function f(t), we would like to decompose it into its constituent frequencies. Sine functions and Cosine functions of various frequency are orthoganal to each other and therefor can form an orthoganal basis for another function. One of the simplest of all Fourier Transforms is the transform of the Gausian bell curve. The transform of a gausian is an other gausian. This is the only function that is its own transform for Fourier transform. Otherwise, narrow functions transform to spread out functions and visa-versa.

$F(w)=\int \limits _{-\infty }^{\infty }f(t)\ e^{-i2\pi wt}\,dt,$

Here, F(w) is the Frequency domain representation of the function f(t). The function f(t) has been transformed into F(w) with the help of the integral.

Applications

Applications include the processing of audio signals or video images. One wonderful property of the fourier transform is that it changes convolution into multiplication. Suppose we have two functions: g(t) and h(t) that we wish to convolve. We wish to solve for k(t). We can transform g(t) and h(t) to G(w) and H(w).

   k(t)  =   g(t) *  h(t)
 F(k(t)) = F(g(t)) F(h(t))    Take the Fourier transform, F, of both sides.
   K(t)  =   G(w)    H(w)     Take the product of G(w) and H(w).
 f(K(t)) =                    Take the inverse transform, f, of both sides.
   k(t)  = f(G(w)    H(w))    The value of k(t) is the inverse fourier transform
                              of the product of G(t) and H(t).