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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle -\infty} to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle f } , often denoted with a capital F or as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \mathcal{F}[f] } (i.e., Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle F\,=\,\mathcal{F}[f] } ) is another function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \mathcal{F}[f]\,:\,\mathbb{R}\rightarrow \mathbb{C}} defined by:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{F}[f](w) = \int \limits _{-\infty}^{\infty} f(t)\ e^{-i 2\pi w t}\,dt, }

assuming the integral is well defined and exists for the given time domain function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle f} for almost all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w} (i.e., except for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle w} in a subset of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \mathbb{R}} of (Lebesgue) measure zero). Here, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \mathcal{F}[f]} is the frequency domain representation of the function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle f(t)} . The function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} has been transformed into Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle F} with the above integral.

One of the simplest of all Fourier transforms is the transform of the Gaussian bell curve Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle g\,=\,c\, \exp(-\frac{1}{2}\frac{(x-a)^2}{b})} , where a, b and c are real parameters with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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).