User Manual 3.3 Trigonometric Polynomials and Fourier Series - Historique des versions

Admin le 27 février 2018 à 14:22

2018-02-27T14:22:05Z

← Version précédente		Version du 27 février 2018 à 14:22
Ligne 1 :		Ligne 1 :
			__NOTOC__

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

Admin le 27 février 2018 à 14:21

2018-02-27T14:21:50Z

← Version précédente		Version du 27 février 2018 à 14:21
Ligne 262 :		Ligne 262 :
	== [[File:lightBulb.png]] Tips & Tricks ==		== [[File:lightBulb.png]] Tips & Tricks ==
	* <code>a</code> and <code>b</code> arrays must have the same lengths!		* <code>a</code> and <code>b</code> arrays must have the same lengths!

			[[Category:User_Manual_3.3_Mathematics]]

Admin : /* Contents */

2018-02-27T14:14:50Z

Contents

@@ Ligne 93 : / Ligne 93 : @@
 == Contents ==
 === Interfaces ===
-|=(% colspan="3" %)Interface|=(% colspan="6" %)Summary|=(% colspan="1" %)Javadoc
+{| class="wikitable"
-|(% colspan="3" %)'''DifferentiableUnivariateFunction'''|(% colspan="6" %)Extension of UnivariateFunction representing a differentiable univariate function.|[{{JavaDoc3.3}}/org/apache/commons/math3/analysis/DifferentiableUnivariateFunction.html ...]
+|-
-|(% colspan="3" %)'''IntegrableUnivariateFunction'''|(% colspan="6" %)Extension of UnivariateRealFunction representing an integrable univariate function.|[{{JavaDoc3.3}}/org/apache/commons/math3/analysis/IntegrableUnivariateFunction.html ...]
+! scope="col"| Interface
-|(% colspan="3" %)'''DifferentiableIntegrableUnivariateFunction'''|(% colspan="6" %)Extension of UnivariateRealFunction representing a differentiable and integrable univariate function.|[{{JavaDoc3.3}}/org/apache/commons/math3/analysis/DifferentiableIntegrableUnivariateFunction.html ...]
+! scope="col"| Summary
-|(% colspan="3" %)'''UnivariateFunction'''|(% colspan="6" %)Interface representing an univariate function.|[{{JavaDoc3.3}}/org/apache/commons/math3/analysis/UnivariateFunction.html ...]
+! scope="col"| Javadoc
-|(% colspan="3" %)'''IFastFourierTransformer'''|(% colspan="6" %)Interface representing a FFT.|[{{JavaDoc3.3}}/org/apache/commons/math3/transform/IFastFourierTransformer.html ...]
+|-
 === Classes ===
-|=(% colspan="3" %)Interface|=(% colspan="6" %)Summary|=(% colspan="1" %)Javadoc
+{| class="wikitable"
-|(% colspan="3" %)'''FourierDecompositionEngine'''|(% colspan="6" %)Decompose a UnivariateFunction as a Fourier Series using TrigonometricPolynomialFunction representation.|[{{JavaDoc3.3}}/org/apache/commons/math3/analysis/polynomials/FourierDecompositionEngine.html ...]
+|-
-|(% colspan="3" %)'''FourierSeries'''|(% colspan="6" %)This class represents a finite Fourier Series.|[{{JavaDoc3.3}}/org/apache/commons/math3/analysis/polynomials/FourierSeries.html ...]
+! scope="col"| Class
-|(% colspan="3" %)'''FourierSeriesApproximation'''|(% colspan="6" %)Holder for a UnivariateFunction and its FourierSeries approximation.|[{{JavaDoc3.3}}/org/apache/commons/math3/analysis/polynomials/FourierSeriesApproximation.html ...]
+! scope="col"| Summary
-|(% colspan="3" %)'''TrigonometricPolynomialFunction'''|(% colspan="6" %)This class is the Trigonometric Polynomial Function class.|[{{JavaDoc3.3}}/org/apache/commons/math3/analysis/polynomials/TrigonometricPolynomialFunction.html ...]
+! scope="col"| Javadoc
-|(% colspan="3" %)'''TrigonometricPolynomialPrimitive'''|(% colspan="6" %)This class represents a Trigonometric Polynomial Primitive.|[{{JavaDoc3.3}}/org/apache/commons/math3/analysis/polynomials/TrigonometricPolynomialPrimitive.html ...]
+|-
-|(% colspan="3" %)'''AbstractFastFourierTransformer'''|(% colspan="6" %) Abstract class representing a FFT.|[{{JavaDoc3.3}}/org/apache/commons/math3/transform/AbstractFastFourierTransformer.html ...]
+|'''FourierDecompositionEngine'''
-|(% colspan="3" %)'''FastFourierTransformer'''|(% colspan="6" %) Computes the FFT for real and complex arrays.|[{{JavaDoc3.3}}/org/apache/commons/math3/transform/FastFourierTransformer.html ...]
+|Decompose a UnivariateFunction as a Fourier Series using TrigonometricPolynomialFunction representation.
 === Usage of a trigonometric polynomial ===
@@ Ligne 206 : / Ligne 252 : @@
 [[File:square10.PNG|center]]
 == Tutorials ==

Admin : /* Fast Fourier Transforms */

2018-02-27T14:07:53Z

Fast Fourier Transforms

← Version précédente		Version du 27 février 2018 à 14:07
Ligne 68 :		Ligne 68 :

	=== 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
	<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 <code>FastFourierTransformer.java</code> owns two methods		The class <code>FastFourierTransformer.java</code> owns two methods<br>
	<code>transform(final double[] f, final TransformType type)</code>		<code>transform(final double[] f, final TransformType type)</code><br>
	<code>transform(final Complex[] f, final TransformType type)</code>		<code>transform(final Complex[] f, final TransformType type)</code><br>
	where <code>TransformType</code> 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.

Ligne 81 :		Ligne 81 :
	==== 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.

Admin : /* Fast Fourier Transforms */

2018-02-27T14:07:17Z

Fast Fourier Transforms

← Version précédente		Version du 27 février 2018 à 14:07
Ligne 70 :		Ligne 70 :
	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.
	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
	<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 <code>FastFourierTransformer.java</code> owns two methods		The class <code>FastFourierTransformer.java</code> owns two methods

Admin : /* Fourier Series */

2018-02-27T14:05:39Z

Fourier Series

← Version précédente		Version du 27 février 2018 à 14:05
Ligne 47 :		Ligne 47 :
	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 ===

Admin : /* Links */

2018-02-27T14:04:52Z

Links

← Version précédente		Version du 27 février 2018 à 14:04
Ligne 23 :		Ligne 23 :
	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 27 février 2018 à 14:04

2018-02-27T14:04:26Z

← Version précédente		Version du 27 février 2018 à 14:04
Ligne 8 :		Ligne 8 :
	The trigonometric polynomial (and related) objects are available in the package <code>org.commons.math3.analysis.polynomials</code> in the Commons Math Library (and Commons Math add-ons) and the FFT algorithms in<code>org.commons.math3.transform</code>		The trigonometric polynomial (and related) objects are available in the package <code>org.commons.math3.analysis.polynomials</code> in the Commons Math Library (and Commons Math add-ons) and the FFT algorithms in<code>org.commons.math3.transform</code>

	\|=~~(% colspan~~="3" %)Library~~\|=(% colspan~~="6" %)Javadoc		{\| class="wikitable"
	\|~~(% colspan="3" %)~~Commons Math add-ons\|~~(% colspan="6" %)~~[{{JavaDoc3.3}}/org/apache/commons/math3/analysis/polynomials/package-summary.html Package org.commons.math3.geometry.analysis.polynomials]		\|-
	\|~~(% colspan="3" %)~~Commons Math \|~~(% colspan="6" %)~~[{{JavaDoc3.3}}/org/apache/commons/math3/transform/package-summary.html Package org.commons.math3.transform]		! scope="col"\| Library
			! scope="col"\| Javadoc
			\|-
			\|Commons Math add-ons
			\|[{{JavaDoc3.3}}/org/apache/commons/math3/analysis/polynomials/package-summary.html Package org.commons.math3.geometry.analysis.polynomials]
			\|-
			\|Commons Math
			\|[{{JavaDoc3.3}}/org/apache/commons/math3/transform/package-summary.html Package org.commons.math3.transform]
			\|}

	=== Links ===		=== Links ===
Ligne 199 :		Ligne 207 :
	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\|center]]
	[[File:square10.PNG]]

Admin : Admin a déplacé la page Trigonometric Polynomials and Fourier Series vers User Manual 3.3 Trigonometric Polynomials and Fourier Series sans laisser de redirection

2018-02-27T14:02:43Z

Admin a déplacé la page Trigonometric Polynomials and Fourier Series vers User Manual 3.3 Trigonometric Polynomials and Fourier Series sans laisser de redirection

← Version précédente		Version du 27 février 2018 à 14:02
(Aucune différence)

Admin : 1 révision importée

2018-02-27T14:01:56Z

1 révision importée

← Version précédente		Version du 27 février 2018 à 14:01
(Aucune différence)

User Manual 3.3 Trigonometric Polynomials and Fourier Series - Historique des versions

Admin le 27 février 2018 à 14:22

Admin le 27 février 2018 à 14:21

Admin : /* Contents */

Admin : /* Fast Fourier Transforms */

Admin : /* Fast Fourier Transforms */

Admin : /* Fourier Series */

Admin : /* Links */

Admin le 27 février 2018 à 14:04

Admin : Admin a déplacé la page Trigonometric Polynomials and Fourier Series vers User Manual 3.3 Trigonometric Polynomials and Fourier Series sans laisser de redirection

Admin : 1 révision importée

Admin : /* Contents */

Admin : /* Fast Fourier Transforms */

Admin : /* Fast Fourier Transforms */

Admin : /* Fourier Series */

Admin : /* Links */

Admin : Admin a déplacé la page Trigonometric Polynomials and Fourier Series vers User Manual 3.3 Trigonometric Polynomials and Fourier Series sans laisser de redirection

Admin : 1 révision importée