Difference between revisions of "Fourier transformation"
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| − | The '''Fourier transformation''' (often called the '''Fourier transform''' or '''Fourier integral''') is an invertible [[integral]] transformation that decomposes a square integrable, | + | The '''Fourier transformation''' (often called the '''Fourier transform''' or '''Fourier integral''') is an invertible [[integral]] transformation that decomposes a square integrable, piecewise continuous functions on a [[topological group]] into a linear combination of basis elements. It can be thought of as the ultimate extension of the [[Fourier series]], in which the interval of periodicity becomes infinitely long and the "coefficients" infinitely close together, becoming a function instead of an infinite series. |
Often, functions which are difficult to analyze in one topological group become much easier to analyze when transformed to another topological group. | Often, functions which are difficult to analyze in one topological group become much easier to analyze when transformed to another topological group. | ||
| − | The formulas usually favored by mathematicians are the "normalized" form. Given a function f( | + | The formulas usually favored by mathematicians are the "normalized" form. Given a function <math>f(t)</math> defined on the entire real line, its Fourier transform <math>g(\omega)</math> is given by: |
| − | :<math>g( | + | :<math>\mathcal{F}[f(t)](\omega) = g(\omega) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(t) e^{-i \omega t}\, dt</math> |
The inverse transform, that recovers the original function, is: | The inverse transform, that recovers the original function, is: | ||
| − | :<math>f( | + | :<math>\mathcal{F}^{-1}[g(\omega)](t) = f(t) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} g(\omega) e^{i \omega t}\, d \omega</math> |
The constants in front of the integrals are arbitrary, so long as their product is <math>1/2\pi</math>. In order to make the forward and inverse transforms as similar as possible, an oft-used convention is to set them both equal to <math>1/\sqrt{2\pi}</math>. | The constants in front of the integrals are arbitrary, so long as their product is <math>1/2\pi</math>. In order to make the forward and inverse transforms as similar as possible, an oft-used convention is to set them both equal to <math>1/\sqrt{2\pi}</math>. | ||
| + | The Fourier transform of a function <math>f(t)</math> is often denoted with a hat: <math>\hat f(\omega)</math> | ||
==Discrete Fourier transformation== | ==Discrete Fourier transformation== | ||
Latest revision as of 21:50, November 23, 2016
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d}{dx} \sin x=?\,} | This article/section deals with mathematical concepts appropriate for late high school or early college. |
The Fourier transformation (often called the Fourier transform or Fourier integral) is an invertible integral transformation that decomposes a square integrable, piecewise continuous functions on a topological group into a linear combination of basis elements. It can be thought of as the ultimate extension of the Fourier series, in which the interval of periodicity becomes infinitely long and the "coefficients" infinitely close together, becoming a function instead of an infinite series.
Often, functions which are difficult to analyze in one topological group become much easier to analyze when transformed to another topological group.
The formulas usually favored by mathematicians are the "normalized" form. Given a function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(t)} defined on the entire real line, its Fourier transform Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(\omega)} is given by:
=g(\omega )={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }f(t)e^{-i\omega t}\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1bddd5f43e2f6728086a88f6cf924f198363b604)
The inverse transform, that recovers the original function, is:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{F}^{-1}[g(\omega)](t) = f(t) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} g(\omega) e^{i \omega t}\, d \omega}
The constants in front of the integrals are arbitrary, so long as their product is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1/2\pi} . In order to make the forward and inverse transforms as similar as possible, an oft-used convention is to set them both equal to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1/\sqrt{2\pi}} . The Fourier transform of a function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(t)} is often denoted with a hat: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat f(\omega)}
Discrete Fourier transformation
Discrete Fourier transformations are defined on discrete topological groups, and the integral is replaced by summation.