Transform of Fourier

In analysis, the transformation of Fourier is an analog of the theory of the Fourier series for the nonperiodic functions, and makes it possible to associate a frequency spectrum to them. One then seeks to obtain the expression of the function as “summons infinite” goniometrical functions of all frequencies who form his spectrum. Such a summation will thus be presented in the form of integral. The Analyze nonstandard makes it possible to present it in the form of a series and justifies the intuitive point of view. Series and transformation of Fourier constitute the two basic tools of the harmonic Analyze.

The transformed of Fourier $\backslash \; mathcal\; \{F\}$ is an operation which transforms an integrable function into another function, describing the frequency spectrum of F . If F is an integrable function, its transform of Fourier is the function F ( F ) and given by the formula

$F\; (F):\; S\; \backslash \; mapsto\; \backslash \; hat\; \{F\}\; (S)\; =\; \backslash \; int\_\; \{-\; \backslash \; infty\}\; ^\; \{+\; \backslash \; infty\}\; F\; (X)\; \backslash ,\; e^\; \{-\; I\; sx\}\; \backslash ,\; dx$

The starting whole is the whole of the integrable functions F of a real variable X. The whole of arrival is the whole of the functions F (F) of a real variable S. Concrètement when this transformation is used into treatment of the signal, one says that X is the variable time, that F is in the temporal field , that S is the frequency and that F is in the frequential field .

The formula known as of transformation of Fourier opposite, noted operation TF ^-1, which is that makes it possible (under conditions) to find F starting from the spectrum:

$f\; (X)\; =\; \{1\; \backslash \; over\; 2\; \backslash \; pi\}\; \backslash ,\; \backslash \; int\_\; \{-\; \backslash \; infty\}\; ^\; \{+\; \backslash \; infty\}\; F\; (W)\; \backslash ,\; e^\; \{iwx\}\; \backslash ,\; dw$

In physics, the transformation of Fourier makes it possible to determine the spectrum of a signal. The phenomena of Diffraction give an indication of the dual space of the network, they are a kind of “machine with natural transformation of Fourier”.

The most natural framework to define the transforms of Fourier is that of the integrable functions. However, of many operations (derivations, transformed of Fourier reverses) cannot be written in any general information. One owes with Plancherel the introduction of the transformation of Fourier for the functions of summable Carré, for which the formula of inversion is true. Then the theory of the distributions of Schwartz made it possible to find a framework adapted perfectly.

Transformation of Fourier for the integrable functions

If F is a integrable function on $\backslash \; mathbb\; \{R\}$, its transformed of Fourier is given by the formula

$F\; (S)\; =\; \backslash \; hat\; \{F\}\; (S)\; =\; \backslash \; int\_\; \{-\; \backslash \; infty\}\; ^\; \{+\; \backslash \; infty\}\; F\; (X)\; \backslash ,\; e^\; \{-\; I\; S\; X\}\; \backslash ,\; dx$

But one can also use this formula

$F\; (S)\; =\; \{1\; \backslash \; over\; 2\; \backslash \; pi\}\; \backslash ,\; \backslash \; int\_\; \{-\; \backslash \; infty\}\; ^\; \{+\; \backslash \; infty\}\; F\; (X)\; \backslash ,\; e^\; \{-\; isx\}\; \backslash ,\; dx$

F is also sometimes noted $\backslash \; mathcal\; \{F\}\; \backslash \; \{F\; \backslash \}$ or TF (ƒ).

The transformed of Fourier spreads with many groups, one can quote the abelian groups locally compact (cf Dualité of Pontryagin) or more simply the abelian groups finished (cf harmonic Analyze on an abelian group finished). The base used is not any more those of the imaginary exponential functions but the elements of the dual group.

Properties

• this transformation is linear

• the transform of Fourier of F is a continuous function, of null limit ad infinitum (Théorème of Riemann-Lebesgue), in particular limited by

$\backslash |\backslash \; hat\; \{F\}\; \backslash |\_\; \backslash \; infty\; \backslash \; Leq\; \backslash |F\; \backslash |\_1$

• by Changement of variable one finds formulas interesting when a translation is carried out, dilation of the graph of F
• the transform of Fourier of a Gaussienne is Gaussian.
• one can try to apply a theorem of Dérivation under integral: if the function G (X) =-ixf (X) is it also integrable, then the derivative of $\backslash \; hat\; \{F\}$ is the transform of Fourier of G .
• if F is derivable, of null limit ad infinitum, and if the derivative of F is integrable, then $\backslash \; hat\; \{f\text{'}\}\; (S)\; =is\; \backslash \; hat\; \{F\}\; (S)$ is the transform of Fourier of derived from F .

One can summarize the two last properties: under conditions of existence, the transformation of Fourier exchanges derivation and multiplication by (more or less) ix . It is precisely to free itself from these unpleasant conditions of existence that it will be necessary to widen the class of the functions on which the transformation of Fourier operates.

Inversion of Fourier

If the transform of Fourier of F is itself a integrable function and according to the transform of Fourier whom one uses, the formula of inversion is:

$f\; (X)\; =\; \backslash \; int\_\; \{-\; \backslash \; infty\}\; ^\; \{+\; \backslash \; infty\}\; F\; (S)\; \backslash ,\; e^\; \{isx\}\; \backslash ,\; ds$

or

$f\; (X)\; =\; \{1\; \backslash \; over\; 2\; \backslash \; pi\}\; \backslash ,\; \backslash \; int\_\; \{-\; \backslash \; infty\}\; ^\; \{+\; \backslash \; infty\}\; F\; (S)\; \backslash ,\; e^\; \{isx\}\; \backslash ,\; ds$

This operation of opposite transformation of Fourier has properties similar to the direct transformation, since only the multiplicative coefficient and the change - I become I .

Extension to space $\backslash \; mathbb\; \{R\}\; ^n$

If F is a integrable function on $\backslash \; mathbb\; \{R\}\; ^n$, its transformed of Fourier is given by the formula

$F\; (S)\; =\; \backslash \; hat\; \{F\}\; (S)\; =\; \backslash \; int\; F\; (X)\; \backslash ,\; e^\; \{-\; I\; S\; \backslash \; cdot\; X\}\; \backslash ,\; dx$

The integral is taken on whole space and the point indicates the scalar product between S and X .

If the transform of Fourier of F is itself a integrable function:

$f\; (X)\; =\; \backslash \; int\_\; \{-\; \backslash \; infty\}\; ^\; \{+\; \backslash \; infty\}\; F\; (W)\; \backslash ,\; e^\; \{iwx\}\; \backslash ,\; dw$

Transformation of Fourier for the functions of summable square

The Théorème of Plancherel makes it possible to extend the transformation of Fourier to the functions of summable Carré. One thus places on the space of functions $L^2\; (\backslash \; mathbb\; \{R\})$, provided with his canonical standard. For reasons which will appear clear, one slightly modifies convention on the transform of Fourier in this section.

Either F a summable function of square on $\backslash \; mathbb\; \{R\}$ and or A>0 . One can define the transform of Fourier of the function truncated in has :

$\backslash \; hat\; \{F\}\; \_A\; (S)\; =\; \backslash \; frac1\; \{\backslash \; sqrt\; \{2\; \backslash \; pi\}\}\; \backslash \; int\_\; \{-\; has\}\; ^A\; F\; (X)\; \backslash ,\; e^\; \{-\; I\; sx\}\; \backslash ,\; dx$

Then when has tends towards the infinite one, the functions $\backslash \; hat\; \{F\}\; \_A$ converge on average quadratic towards a function which one notes $\backslash \; hat\; \{F\}$ and which one calls transformed of Fourier (or Fourier-Plancherel) of F .

Moreover the formula of inversion of Fourier is checked: the function $\backslash \; hat\; \{F\}$ is itself of square summable and

$f\; =\; \backslash \; underset\; \{has\; \backslash \; mapsto\; +\; \backslash \; infty\}\; \{\backslash \; lim\; \backslash \; limits\_\; \{\backslash |\backslash ;\; \backslash |\_2\}\}\; \backslash \; left\; \backslash \; frac1\; \{\backslash \; sqrt\; \{2\; \backslash \; pi\}\}\; \backslash ,\; \backslash \; int\_\; \{-\; has\}\; ^\; \{has\}\; \backslash \; hat\; \{F\}\; (W)\; \backslash ,\; e^\; \{iwx\}\; \backslash ,\; dw\; \backslash \; right$

Thus the transformation of Fourier-Plancherel defines an automorphism of the L² space, which is a isometry

$\backslash |F\; \backslash |\_2\; =\; \backslash |\backslash \; hat\; \{F\}\; \backslash |\_2$ In physics, one interprets the $term|\backslash \; hat\; \{F\}\; (W)|^2$ appearing under the integral like a Spectral concentration of power.

The definition of the transformation of Fourier-Plancherel is compatible with the usual definition of the transform of Fourier of the integrable functions . Indeed, One can show that the application $\backslash \; mathcal\; F:\; L\_2\; \backslash \; mapsto\; L\_2$ prolongs the application which has a function $f$, integrable, associates its transform of Fourier. One places oneself then on space $L\_1\; \backslash \; course\; L\_2\; \backslash ,$ on which the transform of Fourier is well defined and who is dense in $L\_2\; \backslash ,$. Like $L\_2\; \backslash ,$ is a space of Banach, one with the unicity of $\backslash \; mathcal\; F$.

Bond with the product of convolution

The transforms of Fourier have very interesting properties been dependant on the Produit convolution.

As follows:

• $\backslash \; widehat\; \{(f*g)\}(T)\; =\; \backslash \; widehat\; F\; (T)\; \backslash \; cdot\; \backslash \; widehat\; G\; (T)$

• If $F,\; G\; \backslash \; in\; L\_1\; (\backslash \; mathbb\; R),\; f*g\; \backslash \; in\; L\_1\; (\backslash \; mathbb\; R)$ and $\backslash |f*g\; \backslash |\_\; \{L\_1\}\; \backslash \; the\; \backslash |F\; \backslash |\_\{L\_1\}\; \backslash cdot\; \backslash |G\; \backslash |\_\; \{L\_1\}$
• If $F\; \backslash \; in\; L\_1\; (\backslash \; mathbb\; R),\; G\; \backslash \; in\; L\_2\; (\backslash \; mathbb\; R),\; f*g\; \backslash \; in\; L\_2\; (\backslash \; mathbb\; R)$ and $\backslash |f*g\; \backslash |\_\; \{L\_2\}\; \backslash \; the\; \backslash |F\; \backslash |\_\{L\_1\}\; \backslash cdot\; \backslash |G\; \backslash |\_\{L\_2\}$

Transformation of Fourier for the moderate distributions

Bonds with other transformations

Bond between transformation of Fourier and transformation of Laplace

If one notes $\backslash \; mathcal\; \{L\}$ the Transformée of Laplace, then

$\backslash \; mathcal\; \{F\}\; \backslash \; \{F\; \backslash \}\; (S)\; =\; \backslash \; mathcal\; \{L\}\; \backslash \; \{f^+\; \backslash \}\; (2i\; \backslash \; pi\; S)\; +\; \backslash \; mathcal\; \{L\}\; \backslash \; \{f^-\; \backslash \}\; (-\; 2i\; \backslash \; pi\; S)$

where the functions $f^+\; (T)$ and $f^-\; (T)$ are defined by:

$f^+\; (T)\; =\; F\; (T)$ if T ≥ 0 and 0 if not.

$f^-\; (T)\; =\; F\; (-\; T)$ if T ≥ 0 and 0 if not.

Parallels with Fourier series

Parallel formal

The transform of Fourier is defined in a similar way: the variable of integration T is replaced by N Δ T , N being an indication of summation, and the integral by the sum.

$X\; (F)\; =\; \backslash \; Delta\; T\; \backslash \; sum\_\; \{n=-\; \backslash \; infty\}\; ^\; \backslash \; infty\; X\; (N)\; e^\; \{-\; i2\; \backslash \; pi\; fn\; \backslash \; Delta\; T\}$ ↔ $x\; (T)\; =\; \backslash \; int\_\; \{f\_e\}\; X\; (F)\; e^\; \{i2\; \backslash \; pi\; ft\}\; df$

One will find some remarks on this subject in spectral Analyze.

Transform

The following standardized variables are used:

$F=\; \{F\; \backslash \; over\; f\_e\}\; =f\; \backslash \; Delta\; T\; =\; F|\_\; \{\backslash \; Delta\; t=1\}$, $\backslash \; Omega\; =e\; \backslash \; pi\; F=2\; \backslash \; pi\; F\; \backslash \; Delta\; t=\; \backslash \; Omega\; \backslash \; delta\; T|\_\; \{\backslash \; Delta\; t=1\}$

References

• Jean-Michel Bony, Systems design course , Editions of the Polytechnic school

• Srishti D. Chatterji Polytechnic Systems design course , Presses and French Academics 1998

See too

• Spectral concentration of power

• Produces convolution
• Fast Fourier transform
• discrete Transformation of Fourier
• Transformation of Laplace

External bond

• Course of license treating the Analysis of Fourier (and also spaces of Hilbert)

Be-X-old: ПераўтварэннеФур'е Simple: Fourier transformation

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