Fourier Series and Transform - Revision history

Csalais3: /* Transitioning to the Fourier Transform */

2024-12-08T23:36:41Z

Transitioning to the Fourier Transform

← Older revision		Revision as of 19:36, 8 December 2024
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	<br><br><math> \text{1) Since the frequencies are discrete multiples of } \frac{1}{T}\text{, as } T \Rightarrow \infty \text{ the spacing between the frequencies becomes infinitesimally small, which leads to a continuous spectrum of frequencies.} </math><br><br>		<br><br><math> \text{1) Since the frequencies are discrete multiples of } \frac{1}{T}\text{, as } T \Rightarrow \infty \text{ the spacing between the frequencies becomes infinitesimally small, which leads to a continuous spectrum of frequencies.} </math><br><br>
	<br><br><math>\text{2)Since the summation over the discrete frequencies } n \text{ transitions into an integral over a continuous frequency variable } \omega</math><br><br>		<br><br><math>\text{2)Since the summation over the discrete frequencies } n \text{ transitions into an integral over a continuous frequency variable } \omega</math><br><br>
	<br><br><math>\text{3) We denote the Fourier coefficients becoming a continuous function of } \omega \text{ as} \hat{f}(\omega) T \Rightarrow \infty ~~</math>~~ \text{ the frequency becomes continuous as well,}</math><br><br>		<br><br><math>\text{3) We denote the Fourier coefficients becoming a continuous function of } \omega \text{ as} \hat{f}(\omega) T \Rightarrow \infty \text{ the frequency becomes continuous as well,}</math><br><br>

	The result of this transformation is the Fourier Transform, which allows for non-periodic functions:		The result of this transformation is the Fourier Transform, which allows for non-periodic functions:

Csalais3: /* History */

2024-12-08T23:35:34Z

History

← Older revision		Revision as of 19:35, 8 December 2024
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	==History==		==History==
	In 1807, Joseph Fourier published his groundbreaking work “On the Propagation of Heat in Solid Bodies”, which addressed the problem of head distribution in solids. In this paper, he introduced the idea of being able to decompose complex periodic functions into simpler trigonometric components. He then proved that any periodic signal or waveform could be expressed by a sum of sinusoidal functions of differing frequencies and amplitudes, which is now known as the Fourier Series Expansion. While this solved the heat flow problem, he unknowingly laid the foundation for one of the most powerful and influential tools in the history of mathematics. Fourier Analysis was not initially well received, it was in this memoir that he laid the groundwork for such analysis. He read it to the Paris Institute and substantials in academia were not impressed. Namely. Lagrange and Laplace objected to the idea. Fourier, however, did win a prize from the Paris Institute as the result of submitting this paper to a competition. It is important to note that he was one of only two submissions in this competition.		In 1807, Joseph Fourier published his groundbreaking work “On the Propagation of Heat in Solid Bodies”, which addressed the problem of head distribution in solids. In this paper, he introduced the idea of being able to decompose complex periodic functions into simpler trigonometric components. He then proved that any periodic signal or waveform could be expressed by a sum of sinusoidal functions of differing frequencies and amplitudes, which is now known as the Fourier Series Expansion. While this solved the heat flow problem, he unknowingly laid the foundation for one of the most powerful and influential tools in the history of mathematics. Fourier Analysis was not initially well received, it was in this memoir that he laid the groundwork for such analysis. He read it to the Paris Institute and substantials in academia were not impressed. Namely. Lagrange and Laplace objected to the idea. Fourier, however, did win a prize from the Paris Institute as the result of submitting this paper to a competition. It is important to note that he was one of only two submissions in this competition.

	The Fourier Series was useful; however, it only worked well for periodic signals. Unfortunately most real-world signals were non-periodic. This limitation drove the development of something more general, the Fourier Transform. In the late 19th and early 20th centuries, mathematicians like Henri Lebesgue, Baptiste Joseph Fourier, and Carl Gustav Jacobi expanded the idea into continuous form. The Fourier Transform was able to handle non-periodic function, making it applicable to a wider range of problems in physics and engineering, and serves to be one of the greatest discoveries in mathematics.		The Fourier Series was useful; however, it only worked well for periodic signals. Unfortunately most real-world signals were non-periodic. This limitation drove the development of something more general, the Fourier Transform. In the late 19th and early 20th centuries, mathematicians like Henri Lebesgue, Baptiste Joseph Fourier, and Carl Gustav Jacobi expanded the idea into continuous form. The Fourier Transform was able to handle non-periodic function, making it applicable to a wider range of problems in physics and engineering, and serves to be one of the greatest discoveries in mathematics.

Csalais3: /* History */ second paragraph

2024-12-08T23:35:24Z

History: second paragraph

← Older revision		Revision as of 19:35, 8 December 2024
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	==History==		==History==
	In 1807, Joseph Fourier published his groundbreaking work “On the Propagation of Heat in Solid Bodies”, which addressed the problem of head distribution in solids. In this paper, he introduced the idea of being able to decompose complex periodic functions into simpler trigonometric components. He then proved that any periodic signal or waveform could be expressed by a sum of sinusoidal functions of differing frequencies and amplitudes, which is now known as the Fourier Series Expansion. While this solved the heat flow problem, he unknowingly laid the foundation for one of the most powerful and influential tools in the history of mathematics. Fourier Analysis was not initially well received, it was in this memoir that he laid the groundwork for such analysis. He read it to the Paris Institute and substantials in academia were not impressed. Namely. Lagrange and Laplace objected to the idea. Fourier, however, did win a prize from the Paris Institute as the result of submitting this paper to a competition. It is important to note that he was one of only two submissions in this competition.		In 1807, Joseph Fourier published his groundbreaking work “On the Propagation of Heat in Solid Bodies”, which addressed the problem of head distribution in solids. In this paper, he introduced the idea of being able to decompose complex periodic functions into simpler trigonometric components. He then proved that any periodic signal or waveform could be expressed by a sum of sinusoidal functions of differing frequencies and amplitudes, which is now known as the Fourier Series Expansion. While this solved the heat flow problem, he unknowingly laid the foundation for one of the most powerful and influential tools in the history of mathematics. Fourier Analysis was not initially well received, it was in this memoir that he laid the groundwork for such analysis. He read it to the Paris Institute and substantials in academia were not impressed. Namely. Lagrange and Laplace objected to the idea. Fourier, however, did win a prize from the Paris Institute as the result of submitting this paper to a competition. It is important to note that he was one of only two submissions in this competition.
			The Fourier Series was useful; however, it only worked well for periodic signals. Unfortunately most real-world signals were non-periodic. This limitation drove the development of something more general, the Fourier Transform. In the late 19th and early 20th centuries, mathematicians like Henri Lebesgue, Baptiste Joseph Fourier, and Carl Gustav Jacobi expanded the idea into continuous form. The Fourier Transform was able to handle non-periodic function, making it applicable to a wider range of problems in physics and engineering, and serves to be one of the greatest discoveries in mathematics.

	==Connectedness==		==Connectedness==

Csalais3: /* Transitioning to the Fourier Transform */ small fix

2024-12-08T23:29:20Z

Transitioning to the Fourier Transform: small fix

← Older revision		Revision as of 19:29, 8 December 2024
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	<br><br><math>\text{For non-periodic functions, the period } T \Rightarrow \infty \text{. This implies that:} </math><br><br>		<br><br><math>\text{For non-periodic functions, the period } T \Rightarrow \infty \text{. This implies that:} </math><br><br>
	<br><br><math>\text{1) Since the frequencies are discrete multiples of } \frac{1}{T}\text{, as } T \Rightarrow \infty ~~</math>~~ \text{ the spacing between the frequencies becomes infinitesimally small, which leads to a continuous spectrum of frequencies.}</math><br><br>
			<br><br><math> \text{1) Since the frequencies are discrete multiples of } \frac{1}{T}\text{, as } T \Rightarrow \infty \text{ the spacing between the frequencies becomes infinitesimally small, which leads to a continuous spectrum of frequencies.} </math><br><br>
	<br><br><math>\text{2)Since the summation over the discrete frequencies } n \text{ transitions into an integral over a continuous frequency variable } \omega</math><br><br>		<br><br><math>\text{2)Since the summation over the discrete frequencies } n \text{ transitions into an integral over a continuous frequency variable } \omega</math><br><br>

	<br><br><math>\text{3) We denote the Fourier coefficients becoming a continuous function of } \omega \text{ as} \hat{f}(\omega) T \Rightarrow \infty </math> \text{ the frequency becomes continuous as well,}</math><br><br>		<br><br><math>\text{3) We denote the Fourier coefficients becoming a continuous function of } \omega \text{ as} \hat{f}(\omega) T \Rightarrow \infty </math> \text{ the frequency becomes continuous as well,}</math><br><br>

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	<br><br><math> \hat{f}(\omega) = \int^{\infty}_{-\infty}{f(t)e^{\frac{i2\pi n t}{T}}dt} </math><br><br>		<br><br><math> \hat{f}(\omega) = \int^{\infty}_{-\infty}{f(t)e^{\frac{i2\pi n t}{T}}dt} </math><br><br>


	==Fourier Transform==		==Fourier Transform==

Csalais3: added a transition section

2024-12-08T23:26:34Z

added a transition section

@@ Line 59: / Line 59: @@
 <math>F(x)=\sum_{n=1}^{\infty}{(-2\frac{(-1)^nL}{\pi n})\sin(\frac{n\pi x}{L})}</math><br><br>
 ==Fourier Transform==
@@ Line 101: / Line 122: @@
 ==History==
 In 1807, Joseph Fourier published his groundbreaking work “On the Propagation of Heat in Solid Bodies”, which addressed the problem of head distribution in solids. In this paper, he introduced the idea of being able to decompose complex periodic functions into simpler trigonometric components. He then proved that any periodic signal or waveform could be expressed by a sum of sinusoidal functions of differing frequencies and amplitudes, which is now known as the Fourier Series Expansion. While this solved the heat flow problem, he unknowingly laid the foundation for one of the most powerful and influential tools in the history of mathematics. Fourier Analysis was not initially well received, it was in this memoir that he laid the groundwork for such analysis. He read it to the Paris Institute and substantials in academia were not impressed. Namely. Lagrange and Laplace objected to the idea. Fourier, however, did win a prize from the Paris Institute as the result of submitting this paper to a competition. It is important to note that he was one of only two submissions in this competition.
 ==Connectedness==

Csalais3: /* History */ small fix

2024-12-08T23:24:54Z

History: small fix

← Older revision		Revision as of 19:24, 8 December 2024
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	In 1807, Joseph Fourier published his groundbreaking work “On the Propagation of Heat in Solid Bodies”, which addressed the problem of head distribution in solids. In this paper, he introduced the idea of being able to decompose complex periodic functions into simpler trigonometric components. He then proved that any periodic signal or waveform could be expressed by a sum of sinusoidal functions of differing frequencies and amplitudes, which is now known as the Fourier Series Expansion. While this solved the heat flow problem, he unknowingly laid the foundation for one of the most powerful and influential tools in the history of mathematics. Fourier Analysis was not initially well received, it was in this memoir that he laid the groundwork for such analysis. He read it to the Paris Institute and substantials in academia were not impressed. Namely. Lagrange and Laplace objected to the idea. Fourier, however, did win a prize from the Paris Institute as the result of submitting this paper to a competition. It is important to note that he was one of only two submissions in this competition.		In 1807, Joseph Fourier published his groundbreaking work “On the Propagation of Heat in Solid Bodies”, which addressed the problem of head distribution in solids. In this paper, he introduced the idea of being able to decompose complex periodic functions into simpler trigonometric components. He then proved that any periodic signal or waveform could be expressed by a sum of sinusoidal functions of differing frequencies and amplitudes, which is now known as the Fourier Series Expansion. While this solved the heat flow problem, he unknowingly laid the foundation for one of the most powerful and influential tools in the history of mathematics. Fourier Analysis was not initially well received, it was in this memoir that he laid the groundwork for such analysis. He read it to the Paris Institute and substantials in academia were not impressed. Namely. Lagrange and Laplace objected to the idea. Fourier, however, did win a prize from the Paris Institute as the result of submitting this paper to a competition. It is important to note that he was one of only two submissions in this competition.

	The Fourier Series seemed to only work well for periodic signals; however, most of the signals found in the real world were non-periodic. This limitation drove the development of its more generalized form, the Fourier Transform. The Fourier Series was expanded into its continuous form as ~~follows:~~		The Fourier Series seemed to only work well for periodic signals; however, most of the signals found in the real world were non-periodic. This limitation drove the development of its more generalized form, the Fourier Transform. The Fourier Series was expanded into its continuous form as shown earlier.

	==Connectedness==		==Connectedness==

Csalais3: /* History */ added to section

2024-12-08T22:29:14Z

History: added to section

← Older revision		Revision as of 18:29, 8 December 2024
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	==History==		==History==
	~~Fourier Analysis was not initially well received.~~ In 1807, Joseph Fourier published ~~''On~~ the Propagation of heat in ~~solid bodies''~~. It was in this memoir that he laid the groundwork for such analysis. He read it to the Paris Institute and substantials in academia were not impressed. Namely. Lagrange and Laplace objected to the idea. Fourier, however, did win a prize from the Paris Institute as the result of submitting this paper to a competition. It is important to note that he was one of only two submissions in this competition.		In 1807, Joseph Fourier published his groundbreaking work “On the Propagation of Heat in Solid Bodies”, which addressed the problem of head distribution in solids. In this paper, he introduced the idea of being able to decompose complex periodic functions into simpler trigonometric components. He then proved that any periodic signal or waveform could be expressed by a sum of sinusoidal functions of differing frequencies and amplitudes, which is now known as the Fourier Series Expansion. While this solved the heat flow problem, he unknowingly laid the foundation for one of the most powerful and influential tools in the history of mathematics. Fourier Analysis was not initially well received, it was in this memoir that he laid the groundwork for such analysis. He read it to the Paris Institute and substantials in academia were not impressed. Namely. Lagrange and Laplace objected to the idea. Fourier, however, did win a prize from the Paris Institute as the result of submitting this paper to a competition. It is important to note that he was one of only two submissions in this competition.

			The Fourier Series seemed to only work well for periodic signals; however, most of the signals found in the real world were non-periodic. This limitation drove the development of its more generalized form, the Fourier Transform. The Fourier Series was expanded into its continuous form as follows:

	==Connectedness==		==Connectedness==

Csalais3: Added a visualization section with a link to the voice decomposition github repo

2024-12-08T22:26:42Z

Added a visualization section with a link to the voice decomposition github repo

← Older revision		Revision as of 18:26, 8 December 2024
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	'''Claimed - ~~Eric Carder~~ Fall ~~2022~~''' <br>		'''Claimed - Christian Salais Fall 2024''' <br>
	==Fourier Series==		==Fourier Series==
	A Fourier Series is an expansion of trigonometric functions to model periodic functions. This method proves useful in the study of harmonic systems as the analysis in a more familiar domain may be simpler than in its original domain. It has a variety of applications ranging from signal processing to quantum mechanics. The Fourier Series is defined as: <br><math>f(x)=\sum_{n=1}^{\infty}{a_n\cos{(\frac{n\pi x}{L}})}+\sum_{n=1}^{\infty}{b_n\sin{(\frac{n\pi x}{L}})}</math><br> for <math>x\in[-L,L]</math>		A Fourier Series is an expansion of trigonometric functions to model periodic functions. This method proves useful in the study of harmonic systems as the analysis in a more familiar domain may be simpler than in its original domain. It has a variety of applications ranging from signal processing to quantum mechanics. The Fourier Series is defined as: <br><math>f(x)=\sum_{n=1}^{\infty}{a_n\cos{(\frac{n\pi x}{L}})}+\sum_{n=1}^{\infty}{b_n\sin{(\frac{n\pi x}{L}})}</math><br> for <math>x\in[-L,L]</math>
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	Or inversely:<br><br>		Or inversely:<br><br>
	<math>x(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}{X(\omega)e^{jwt}dw}</math>		<math>x(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}{X(\omega)e^{jwt}dw}</math>

			==Visualization==
			The Fourier Transform can be used continuously such that it can be used to transform sound waves into the frequency domain and even further, decompose them into their sinusoidal components. The following link leads to a GitHub repository that visualizes a live audio feed into the aforementioned frequency domain and sinusoidal components.

			[[https://github.com/Csalais3/Live-Voice-Decomposition\| Voice Decomposition]]

	==Fourier Transform in Quantum Mechanics==		==Fourier Transform in Quantum Mechanics==
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	<math>A(k)=\frac{1}{\sqrt{2\pi}}\bigg[\frac{1}{k_0-k}\sin(\frac{L}{2}(k_0-k))+\frac{1}{k_0+k}\sin(\frac{L}{2}(k_0+k))\bigg]</math>		<math>A(k)=\frac{1}{\sqrt{2\pi}}\bigg[\frac{1}{k_0-k}\sin(\frac{L}{2}(k_0-k))+\frac{1}{k_0+k}\sin(\frac{L}{2}(k_0+k))\bigg]</math>

	==History==		==History==
	Fourier Analysis was not initially well received. In 1807, Joseph Fourier published ''On the Propagation of heat in solid bodies''. It was in this memoir that he laid the groundwork for such analysis. He read it to the Paris Institute and substantials in academia were not impressed. Namely. Lagrange and Laplace objected to the idea. Fourier, however, did win a prize from the Paris Institute as the result of submitting this paper to a competition. It is important to note that he was one of only two submissions in this competition.		Fourier Analysis was not initially well received. In 1807, Joseph Fourier published ''On the Propagation of heat in solid bodies''. It was in this memoir that he laid the groundwork for such analysis. He read it to the Paris Institute and substantials in academia were not impressed. Namely. Lagrange and Laplace objected to the idea. Fourier, however, did win a prize from the Paris Institute as the result of submitting this paper to a competition. It is important to note that he was one of only two submissions in this competition.

	==Connectedness==		==Connectedness==
	Fourier Series and Transforms are applicable on a variety of frontiers. They have proven themselves useful in Modern Physics, but there is a good chance anyone in a mathematics-heavy STEM field will encounter Fourier Series and Transforms as they are heavily prevalent in other areas of science and technology. Even if these aren't useful in a given person's industry, it is still useful to understand them and their importance, as they involve a clever manipulation of information. This type of analysis is largely prevalent in signal processing: the interpretation and modification of signals.		Fourier Series and Transforms are applicable on a variety of frontiers. They have proven themselves useful in Modern Physics, but there is a good chance anyone in a mathematics-heavy STEM field will encounter Fourier Series and Transforms as they are heavily prevalent in other areas of science and technology. Even if these aren't useful in a given person's industry, it is still useful to understand them and their importance, as they involve a clever manipulation of information. This type of analysis is largely prevalent in signal processing: the interpretation and modification of signals.

Ecarder at 02:10, 15 December 2022

2022-12-15T02:10:04Z

← Older revision		Revision as of 22:10, 14 December 2022
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	<math>A(k)=\frac{1}{\sqrt{2\pi}}\bigg[\frac{1}{k_0-k}\sin(\frac{L}{2}(k_0-k))+\frac{1}{k_0+k}\sin(\frac{L}{2}(k_0+k))\bigg]</math>		<math>A(k)=\frac{1}{\sqrt{2\pi}}\bigg[\frac{1}{k_0-k}\sin(\frac{L}{2}(k_0-k))+\frac{1}{k_0+k}\sin(\frac{L}{2}(k_0+k))\bigg]</math>
			==History==
			Fourier Analysis was not initially well received. In 1807, Joseph Fourier published ''On the Propagation of heat in solid bodies''. It was in this memoir that he laid the groundwork for such analysis. He read it to the Paris Institute and substantials in academia were not impressed. Namely. Lagrange and Laplace objected to the idea. Fourier, however, did win a prize from the Paris Institute as the result of submitting this paper to a competition. It is important to note that he was one of only two submissions in this competition.
	==Connectedness==		==Connectedness==
	Fourier Series and Transforms are applicable on a variety of frontiers. They ~~has~~ proven themselves useful in Modern Physics, but there is a good chance anyone in a mathematics heavy STEM field will encounter Fourier Series and Transforms as they are heavily prevalent in other areas of science and technology. Even if these aren't useful in a given person's industry, it is still useful to understand them and their importance, as they involve a clever manipulation of information.		Fourier Series and Transforms are applicable on a variety of frontiers. They have proven themselves useful in Modern Physics, but there is a good chance anyone in a mathematics-heavy STEM field will encounter Fourier Series and Transforms as they are heavily prevalent in other areas of science and technology. Even if these aren't useful in a given person's industry, it is still useful to understand them and their importance, as they involve a clever manipulation of information. This type of analysis is largely prevalent in signal processing: the interpretation and modification of signals.
	==References==		==References==
	Branson, J. (2013, April 22). Position Space and Momentum Space. Position space and Momentum Space. Retrieved 2022, from https://quantummechanics.ucsd.edu/ph130a/130_notes/node82.html <br><br>		Branson, J. (2013, April 22). Position Space and Momentum Space. Position space and Momentum Space. Retrieved 2022, from https://quantummechanics.ucsd.edu/ph130a/130_notes/node82.html <br><br>
	Cheever, E. (2022). Introduction to the Fourier transform. Linear Physical Systems - Erik Cheever. Retrieved 2022, from https://lpsa.swarthmore.edu/Fourier/Xforms/FXformIntro.html <br> <br>		Cheever, E. (2022). Introduction to the Fourier transform. Linear Physical Systems - Erik Cheever. Retrieved 2022, from https://lpsa.swarthmore.edu/Fourier/Xforms/FXformIntro.html <br> <br>
	Dawkins, P. (2022). Fourier Series. Differential equations - fourier series. Retrieved 2022, from https://tutorial.math.lamar.edu/classes/de/fourierseries.aspx <br><br>		Dawkins, P. (2022). Fourier Series. Differential equations - fourier series. Retrieved 2022, from https://tutorial.math.lamar.edu/classes/de/fourierseries.aspx <br><br>
			Chodos, A. (Ed.). (2010, March). This Month in physics history. This Month in Physics History. Retrieved 2022, from https://www.aps.org/publications/apsnews/201003/physicshistory.cfm <br><br>
	Greco, E. (2022). GPS Week 7. Modern Physics. Retrieved 2022.		Greco, E. (2022). GPS Week 7. Modern Physics. Retrieved 2022.

Ecarder at 17:26, 7 December 2022

2022-12-07T17:26:48Z

← Older revision		Revision as of 13:26, 7 December 2022
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			'''Claimed - Eric Carder Fall 2022''' <br>
	==Fourier Series==		==Fourier Series==
	A Fourier Series is an expansion of trigonometric functions to model periodic functions. This method proves useful in the study of harmonic systems as the analysis in a more familiar domain may be simpler than in its original domain. It has a variety of applications ranging from signal processing to quantum mechanics. The Fourier Series is defined as: <br><math>f(x)=\sum_{n=1}^{\infty}{a_n\cos{(\frac{n\pi x}{L}})}+\sum_{n=1}^{\infty}{b_n\sin{(\frac{n\pi x}{L}})}</math><br> for <math>x\in[-L,L]</math>		A Fourier Series is an expansion of trigonometric functions to model periodic functions. This method proves useful in the study of harmonic systems as the analysis in a more familiar domain may be simpler than in its original domain. It has a variety of applications ranging from signal processing to quantum mechanics. The Fourier Series is defined as: <br><math>f(x)=\sum_{n=1}^{\infty}{a_n\cos{(\frac{n\pi x}{L}})}+\sum_{n=1}^{\infty}{b_n\sin{(\frac{n\pi x}{L}})}</math><br> for <math>x\in[-L,L]</math>