Fourier transform

A Fourier transform is an integral transform, typically from a time domain to a frequency domain. It is named in honor of Joseph Fourier. The operation of transforming back from the frequency domain to the time domain is called the inverse Fourier transform.

Theory

Given some complex-valued function $\scriptstyle f\,:\,\mathbb {R} \rightarrow \mathbb {C}$, we would like to decompose it into its constituent frequencies. The main idea is to employ sine and cosine functions of a continuous range of frequencies (in fact from $\scriptstyle -\infty$ to $\scriptstyle \infty$) as a continuum of bases with which to "expand" a given function. Formally speaking, a Fourier transform can be thought of as the "Fourier series" of function with an infinite period, and this was the conceptual idea that lead to the rigorous definition and theory the Fourier transform that is known today. The Fourier transform of a function $\scriptstyle f$, often denoted with a capital F or as $\scriptstyle {\mathcal {F}}[f]$ (i.e., $\scriptstyle F\,=\,{\mathcal {F}}[f]$) is another function $\scriptstyle {\mathcal {F}}[f]\,:\,\mathbb {R} \rightarrow \mathbb {C}$ defined by:

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

assuming the integral is well defined and exists for the given time domain function $\scriptstyle f$ for almost all $w$ (i.e., except for $\scriptstyle w$ in a subset of $\scriptstyle \mathbb {R}$ of (Lebesgue) measure zero). Here, $\scriptstyle {\mathcal {F}}[f]$ is the frequency domain representation of the function $\scriptstyle f(t)$. The function $f$ has been transformed into $\scriptstyle F$ with the above integral.

One of the simplest of all Fourier transforms is the transform of the Gaussian bell curve $\scriptstyle g\,=\,c\,\exp(-{\frac {1}{2}}{\frac {(x-a)^{2}}{b}})$, where a, b and c are real parameters with $\scriptstyle b>0$. The transform of a Gaussian is an other Gaussian. This is the only function that is its own transform for Fourier transform. Otherwise, narrow functions transform to spread out functions and vice versa.

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 (the convolution operation is often denoted by *). 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).