User Manual 4.0 Trigonometric Polynomials and Fourier Series - Historique des versions

Admin le 21 février 2018 à 14:03

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	</syntaxhighlight>		</syntaxhighlight>

	The user must build an instance of the FourierDecompositionEngine (using an [{{~~PathCurrentJavaDoc~~}}/fr/cnes/sirius/patrius/math/analysis/integration/UnivariateIntegrator.html integrator]) :		The user must build an instance of the FourierDecompositionEngine (using an [{{JavaDoc4.0}}/fr/cnes/sirius/patrius/math/analysis/integration/UnivariateIntegrator.html integrator]) :

	<syntaxhighlight lang="java">		<syntaxhighlight lang="java">

Admin : /* Contents */

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Contents

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 == Contents ==
 === Interfaces ===
-|=(% colspan="3" %)Interface|=(% colspan="6" %)Summary|=(% colspan="1" %)Javadoc
-|(% colspan="3" %)'''UnivariateDifferentiableFunction'''|(% colspan="6" %)Extension of UnivariateFunction representing a differentiable univariate function.|[{{PathCurrentJavaDoc}}/fr/cnes/sirius/patrius/math/analysis//differentiation/UnivariateDifferentiableFunction.html ...]
+{| class="wikitable"
-|(% colspan="3" %)'''IntegrableUnivariateFunction'''|(% colspan="6" %)Extension of UnivariateRealFunction representing an integrable univariate function.|[{{PathCurrentJavaDoc}}/fr/cnes/sirius/patrius/math/analysis/IntegrableUnivariateFunction.html ...]
+|-
-|(% colspan="3" %)'''UnivariateFunction'''|(% colspan="6" %)Interface representing an univariate function.|[{{PathCurrentJavaDoc}}/fr/cnes/sirius/patrius/math/analysis/UnivariateFunction.html ...]
+! scope="col"| Interface
-|(% colspan="3" %)'''IFastFourierTransformer'''|(% colspan="6" %)Interface representing a FFT.|[{{PathCurrentJavaDoc}}/fr/cnes/sirius/patrius/math/transform/IFastFourierTransformer.html ...]
+! scope="col"| Summary
 === Classes ===
-|=(% colspan="3" %)Interface|=(% colspan="6" %)Summary|=(% colspan="1" %)Javadoc
-|(% colspan="3" %)'''FourierDecompositionEngine'''|(% colspan="6" %)Decompose a UnivariateFunction as a Fourier Series using TrigonometricPolynomialFunction representation.|[{{PathCurrentJavaDoc}}/fr/cnes/sirius/patrius/math/analysis/polynomials/FourierDecompositionEngine.html ...]
+{| class="wikitable"
-|(% colspan="3" %)'''FourierSeries'''|(% colspan="6" %)This class represents a finite Fourier Series.|[{{PathCurrentJavaDoc}}/fr/cnes/sirius/patrius/math/analysis/polynomials/FourierSeries.html ...]
+|-
-|(% colspan="3" %)'''FourierSeriesApproximation'''|(% colspan="6" %)Holder for a UnivariateFunction and its FourierSeries approximation.|[{{PathCurrentJavaDoc}}/fr/cnes/sirius/patrius/math/analysis/polynomials/FourierSeriesApproximation.html ...]
+! scope="col"| Class
-|(% colspan="3" %)'''TrigonometricPolynomialFunction'''|(% colspan="6" %)This class is the Trigonometric Polynomial Function class.|[{{PathCurrentJavaDoc}}/fr/cnes/sirius/patrius/math/analysis/polynomials/TrigonometricPolynomialFunction.html ...]
+! scope="col"| Summary
-|(% colspan="3" %)'''TrigonometricPolynomialPrimitive'''|(% colspan="6" %)This class represents a Trigonometric Polynomial Primitive.|[{{PathCurrentJavaDoc}}/fr/cnes/sirius/patrius/math/analysis/polynomials/TrigonometricPolynomialPrimitive.html ...]
+! scope="col"| Javadoc
-|(% colspan="3" %)'''AbstractFastFourierTransformer'''|(% colspan="6" %) Abstract class representing a FFT.|[{{PathCurrentJavaDoc}}/fr/cnes/sirius/patrius/math/transform/AbstractFastFourierTransformer.html ...]
+|-
-|(% colspan="3" %)'''FastFourierTransformer'''|(% colspan="6" %) Computes the FFT for real and complex arrays.|[{{PathCurrentJavaDoc}}/fr/cnes/sirius/patrius/math/transform/FastFourierTransformer.html ...]
+|'''FourierDecompositionEngine'''
 [[Category:User_Manual_4.0_Mathematics]]

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2018-02-21T08:48:15Z

Javadoc

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	=== Javadoc ===		=== Javadoc ===
	The trigonometric polynomial (and related) objects are available in the package <code>fr.cnes.sirius.patrius.math.analysis.polynomials</code> and the FFT algorithms in <code>fr.cnes.sirius.patrius.math.transform</code>		The trigonometric polynomial (and related) objects are available in the package <code>fr.cnes.sirius.patrius.math.analysis.polynomials</code> and the FFT algorithms in <code>fr.cnes.sirius.patrius.math.transform</code>


	{\| class="wikitable"		{\| class="wikitable"

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Javadoc

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	The trigonometric polynomial (and related) objects are available in the package <code>fr.cnes.sirius.patrius.math.analysis.polynomials</code> and the FFT algorithms in <code>fr.cnes.sirius.patrius.math.transform</code>		The trigonometric polynomial (and related) objects are available in the package <code>fr.cnes.sirius.patrius.math.analysis.polynomials</code> and the FFT algorithms in <code>fr.cnes.sirius.patrius.math.transform</code>

	\|=~~(% colspan~~="3" %)Library~~\|=(% colspan~~="6" %)Javadoc
	\|~~(% colspan="3" %)~~Patrius\|~~(% colspan="6" %)~~[{{~~PathCurrentJavaDoc~~}}/fr/cnes/sirius/patrius/math/analysis/polynomials/package-summary.html Package fr.cnes.sirius.patrius.math.geometry.analysis.polynomials]		{\| class="wikitable"
	\|~~(% colspan="3" %)~~Patrius \|~~(% colspan="6" %)~~[{{~~PathCurrentJavaDoc~~}}/fr/cnes/sirius/patrius/math/transform/package-summary.html Package fr.cnes.sirius.patrius.math.transform]		\|-
			! scope="col"\| Library
			! scope="col"\| Javadoc
			\|-
			\|Patrius
			\|[{{JavaDoc4.0}}/fr/cnes/sirius/patrius/math/analysis/polynomials/package-summary.html Package fr.cnes.sirius.patrius.math.geometry.analysis.polynomials]
			\|-
			\|Patrius
			\|[{{JavaDoc4.0}}/fr/cnes/sirius/patrius/math/transform/package-summary.html Package fr.cnes.sirius.patrius.math.transform]
			\|}

	=== Links ===		=== Links ===

Admin : /* Getting Started */

2018-02-20T16:47:24Z

Getting Started

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	</syntaxhighlight>		</syntaxhighlight>

	<u>Example</u>		'''<u>Example</u>'''

	The following code snippet shows how to decompose a square function of period 2 to the 10th order :		The following code snippet shows how to decompose a square function of period 2 to the 10th order :

Admin le 20 février 2018 à 16:46

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			__NOTOC__

	== Introduction ==		== Introduction ==
	=== Scope ===		=== Scope ===

Admin le 19 février 2018 à 11:08

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	Useful resources for this theme can be found here :		Useful resources for this theme can be found here :

	~~(% style="margin-left:30px;list-style-type:square;" %)~~
	* [http://en.wikipedia.org/wiki/Trigonometric_polynomial Trigonometric Polynomials]		* [http://en.wikipedia.org/wiki/Trigonometric_polynomial Trigonometric Polynomials]
	* [http://en.wikipedia.org/wiki/List_of_trigonometric_identities List of trigonometric identities]		* [http://en.wikipedia.org/wiki/List_of_trigonometric_identities List of trigonometric identities]

Admin le 19 février 2018 à 11:07

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	The function and its approximation are shown hereunder. The parasite undulations are known as the [http://en.wikipedia.org/wiki/Gibbs_phenomenon Gibbs Phenomenon] :		The function and its approximation are shown hereunder. The parasite undulations are known as the [http://en.wikipedia.org/wiki/Gibbs_phenomenon Gibbs Phenomenon] :

	~~(% style="text-align:center" %)~~
	[[File:square10.PNG]]		[[File:square10.PNG\|center]]

	== Contents ==		== Contents ==
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	\|(% colspan="3" %)'''AbstractFastFourierTransformer'''\|(% colspan="6" %) Abstract class representing a FFT.\|[{{PathCurrentJavaDoc}}/fr/cnes/sirius/patrius/math/transform/AbstractFastFourierTransformer.html ...]		\|(% colspan="3" %)'''AbstractFastFourierTransformer'''\|(% colspan="6" %) Abstract class representing a FFT.\|[{{PathCurrentJavaDoc}}/fr/cnes/sirius/patrius/math/transform/AbstractFastFourierTransformer.html ...]
	\|(% colspan="3" %)'''FastFourierTransformer'''\|(% colspan="6" %) Computes the FFT for real and complex arrays.\|[{{PathCurrentJavaDoc}}/fr/cnes/sirius/patrius/math/transform/FastFourierTransformer.html ...]		\|(% colspan="3" %)'''FastFourierTransformer'''\|(% colspan="6" %) Computes the FFT for real and complex arrays.\|[{{PathCurrentJavaDoc}}/fr/cnes/sirius/patrius/math/transform/FastFourierTransformer.html ...]

			[[Category:User_Manual_4.0_Mathematics]]

Admin : /* Fast Fourier Transforms */

2018-02-19T11:06:57Z

Fast Fourier Transforms

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Fourier Series

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	Let <math>f</math> be a periodic, real variable, real function with period <math>T</math>. The ''partial sums of the Fourier Series'' of <math>f</math> are given by		Let <math>f</math> be a periodic, real variable, real function with period <math>T</math>. The ''partial sums of the Fourier Series'' of <math>f</math> are given by

	<math>~~\begin{math}~~		<math>
	\forall N \in \mathbb{R}^{+*}, \, \left(S_Nf\right)(x) = a_0 + \sum_{n=1}^N\left(a_n\cos\left(n t {2\pi\over T}\right) + b_n\sin\left(n t {2\pi\over T}\right)\right)		\forall N \in \mathbb{R}^{+*}, \, \left(S_Nf\right)(x) = a_0 + \sum_{n=1}^N\left(a_n\cos\left(n t {2\pi\over T}\right) + b_n\sin\left(n t {2\pi\over T}\right)\right)
	~~\end{math}~~</math>		</math>

	The Fourier coefficients (n > 0) of <math>f</math> are given by :		The Fourier coefficients (n > 0) of <math>f</math> are given by :

	<math>~~\begin{math}~~		<math>
	a_0 = {1\over T}\int_{-T/2}^{T/2} \! f(t) \, \mathrm{d}t		a_0 = {1\over T}\int_{-T/2}^{T/2} \! f(t) \, \mathrm{d}t
	~~\end{math}~~</math>		</math>

	<math>~~\begin{math}~~		<math>
	a_n = {2\over T}\int_{-T/2}^{T/2} \! f(t) \cos\left(n t {2\pi\over T}\right) \, \mathrm{d}t		a_n = {2\over T}\int_{-T/2}^{T/2} \! f(t) \cos\left(n t {2\pi\over T}\right) \, \mathrm{d}t
	~~\end{math}~~</math>		</math>

	<math>~~\begin{math}~~		<math>
	b_n = {2\over T}\int_{-T/2}^{T/2} \! f(t) \sin\left(n t {2\pi\over T}\right) \, \mathrm{d}t		b_n = {2\over T}\int_{-T/2}^{T/2} \! f(t) \sin\left(n t {2\pi\over T}\right) \, \mathrm{d}t
	~~\end{math}~~</math>		</math>

	Classes that allow decomposing functions into finite Fourier series are described.		Classes that allow decomposing functions into finite Fourier series are described.


	=== Fast Fourier Transforms ===		=== Fast Fourier Transforms ===

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	=== Fast Fourier Transforms ===		=== Fast Fourier Transforms ===
	A fast Fourier transform (FFT) is an algorithm to compute the discrete Fourier transform (DFT) and its inverse. Fourier analysis converts time (or space) to frequency and vice versa. An FFT rapidly computes such transformations by factorizing the DFT matrix into a product of sparse (mostly zero) factors.		A fast Fourier transform (FFT) is an algorithm to compute the discrete Fourier transform (DFT) and its inverse. Fourier analysis converts time (or space) to frequency and vice versa. An FFT rapidly computes such transformations by factorizing the DFT matrix into a product of sparse (mostly zero) factors.<br>
	Let <math>( X_0, ..., X_{N-1})</math> be complex numbers. The DFT is defined by the formula		Let <math>( X_0, ..., X_{N-1})</math> be complex numbers. The DFT is defined by the formula<br>
	<math>X_k = \sum^{N-1}_{n=0} x_n e^{-2i\pi k n/N}, \quad k=\left\{0, ..., N-1\right\}.</math>		<math>X_k = \sum^{N-1}_{n=0} x_n e^{-2i\pi k n/N}, \quad k=\left\{0, ..., N-1\right\}.</math>

	The class <~~syntaxhighlight~~>FastFourierTransformer.java</~~syntaxhighlight~~> owns two methods		The class <code>FastFourierTransformer.java</code> owns two methods<br>
	<~~syntaxhighlight~~>transform(final double[] f, final TransformType type)</~~syntaxhighlight~~>		<code>transform(final double[] f, final TransformType type)</code><br>
	<~~syntaxhighlight~~>transform(final Complex[] f, final TransformType type)</~~syntaxhighlight~~>		<code>transform(final Complex[] f, final TransformType type)</code><br>
	where <~~syntaxhighlight~~>TransformType</~~syntaxhighlight~~> is an enumeration that defines the type of transform which is to be computed. It can be FORWARD or INVERSE.		where <code>TransformType</code> is an enumeration that defines the type of transform which is to be computed. It can be FORWARD or INVERSE.

	Depending on the size, this method will use a radix-2 algorithm if the size is a power-of-two, and the Bluestein algorithm otherwise. Both algorithms are explained above.		Depending on the size, this method will use a radix-2 algorithm if the size is a power-of-two, and the Bluestein algorithm otherwise. Both algorithms are explained above.
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	==== The Cooley–Tukey & radix-2 the algorithm ====		==== The Cooley–Tukey & radix-2 the algorithm ====

	By far the most commonly used FFT algorithm. This is a divide and conquer algorithm that recursively breaks down a DFT of any composite size <math>N = N_1N_2</math> into many smaller DFTs of sizes <math>N_1</math> and <math>N_2</math>, along with <math>O(N)</math> multiplications by complex roots of unity.		By far the most commonly used FFT algorithm. This is a divide and conquer algorithm that recursively breaks down a DFT of any composite size <math>N = N_1N_2</math> into many smaller DFTs of sizes <math>N_1</math> and <math>N_2</math>, along with <math>O(N)</math> multiplications by complex roots of unity.<br>
	The radix-2 algorithme is the most common use of FFT algorithm and is based on the Cooley–Tukey algorithm. It divides the transform into two pieces of size <math>N/2</math> at each step, and is therefore limited to power-of-two sizes.		The radix-2 algorithme is the most common use of FFT algorithm and is based on the Cooley–Tukey algorithm. It divides the transform into two pieces of size <math>N/2</math> at each step, and is therefore limited to power-of-two sizes.

	==== The Bluestein algorithm ====		==== The Bluestein algorithm ====

	It is commonly called the chirp z-transform algorithm. It computes the discrete Fourier transform (DFT) of arbitrary sizes (including prime sizes) by re-expressing the DFT as a convolution. For more information, see [http://en.wikipedia.org/wiki/Bluestein%27s_FFT_algorithm]		It is commonly called the chirp z-transform algorithm. It computes the discrete Fourier transform (DFT) of arbitrary sizes (including prime sizes) by re-expressing the DFT as a convolution. For more information, see [http://en.wikipedia.org/wiki/Bluestein%27s_FFT_algorithm http://en.wikipedia.org/wiki/Bluestein%27s_FFT_algorithm]

	== Getting Started ==		== Getting Started ==

User Manual 4.0 Trigonometric Polynomials and Fourier Series - Historique des versions

Admin le 21 février 2018 à 14:03

Admin : /* Contents */

Admin : /* Javadoc */

Admin : /* Javadoc */

Admin : /* Getting Started */

Admin le 20 février 2018 à 16:46

Admin le 19 février 2018 à 11:08

Admin le 19 février 2018 à 11:07

Admin : /* Fast Fourier Transforms */

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Admin : /* Contents */

Admin : /* Javadoc */

Admin : /* Javadoc */

Admin : /* Getting Started */

Admin : /* Fast Fourier Transforms */

Admin : /* Fourier Series */