Difference between revisions of "Talk:Video/The imaginary number i and the Fourier Transform"

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=Trigonometry=
After watching the video:[[Video/The imaginary number i and the Fourier Transform|The imaginary number i and the Fourier Transform]]<ref name=iAndFT>{{:Video/The imaginary number i and the Fourier Transform}}</ref>.
 
=[[Trigonometry]]=
Fourier Transform and the notion of <math>i</math> ([[imaginary number]]), cannot be separated from [[Trigonometry]]<ref>{{:Book/Divine Proportions}}</ref>.
Fourier Transform and the notion of <math>i</math> ([[imaginary number]]), cannot be separated from [[Trigonometry]]<ref>{{:Book/Divine Proportions}}</ref>.
=Fourier Series=
Some functions are hard to work with, for example, discontinuous and fractal functions. On the other hand, some functions have wonderful properties for example sin, cos, and the linear function. In the Fourier series, we can use functions with wonderful properties to approximate functions that are hard to work with, for example, we use sin and cos to approximate fractal functions. Moreover, we can also approximate the functions by adding up functions together and the Fourier series will tell us what coefficient to use in our combination. However, you can only approximate a function on an interval.
<ref>{{:video/The Fourier Series and Fourier Transform Demystified}}</ref><ref>{{:video/Fourier Series}}</ref><ref>{{:video/Fourier Series introduction}}</ref><ref>{{:video/Intro to FOURIER SERIES: The Big Idea}}</ref><ref>{{:video/Fourier Series}}</ref><ref>{{:video/What is a Fourier Series? (Explained by drawing circles) - Smarter Every Day 205}}</ref>
=Fourier transform=
[[Fourier transform]] is the next level of the Fourier Series. [[Fourier transform]] comes up with a way to approximate functions on the hole real line by using exponentials <math>e^{ikx}</math>.
Fourier transforms integral equation :
<math>\hat{f}(\xi) = \int_{-\infty}^{\infty} f(x)\ e^{-i 2\pi \xi x}\,dx,\quad \forall\ \xi \in \mathbb R.</math>
Example for Fourier transform:
We have a signal called <math>x(t)</math> we will represent it in terms of the time domain. We also can represent it in another way which is called <math>x(f)</math> we will represent it in terms of the frequency domain and This is why we called transformation. Fourier Transform is an equivalent representation of the signal.


Convolution equation :
=[[Fourier Series]]=
<math>(f * g)(t) := \int_{-\infty}^\infty f(\tau) g(t - \tau) \, d\tau.</math>
{{:Fourier Series}}


<ref>{{:video/The imaginary number i and the Fourier Transform}}</ref> <ref>{{:video/}}</ref>
=[[Convolution]]=
{{:Convolution}}


==Fourier Series in human body==
=[[Fourier Transform]]=
Find references and put them here.
{{:Fourier Transform}}


=Possible Works to be done for this page=
=Possible Works to be done for this page=
#Learn to use [[Quote|Blockquote]] and [[Cite]]!
#What did you learn from the video?
#What did you learn from the video?
#What are the links of Fourier Transform with Sine/Cosine, and Exponential?
#How does [[Fourier Transform]] relates to [[Uncertainty Principle]]<ref>{{:Video/Heisenberg's Uncertainty Principle EXPLAINED (for beginners)}}</ref> as often mentioned in [[Quantum mechanics]]?
#How does [[Fourier Transform]] relates to [[Superposition]]<ref>{{:Video/Quantum Superposition, Explained Without Woo Woo}}</ref> as often mentioned in [[Quantum mechanics]]?
#What are the links of [[Fourier Transform]] with [[Sine]]/[[Cosine]], and [[Exponential]] functions?
#What other sources related to Sine/Cosine/Exponent and Fourier Transform you have found and what are their Links?
#What other sources related to Sine/Cosine/Exponent and Fourier Transform you have found and what are their Links?
#Does it relate to Polynomials, Infinite Series, and the eventual implementation of Fast Fourier Transform?
#Does it relate to Polynomials, Infinite Series, and the eventual implementation of Fast Fourier Transform?

Latest revision as of 06:50, 2 August 2022

After watching the video:The imaginary number i and the Fourier Transform[1].

Trigonometry

Fourier Transform and the notion of (imaginary number), cannot be separated from Trigonometry[2].

Fourier Series

Fourier Series(Q179467)

Some functions are hard to work with, for example, discontinuous and fractal functions. On the other hand, some functions have wonderful properties for example sin, cos, and the linear function. In the Fourier series, we can use functions with wonderful properties to approximate functions that are hard to work with, for example, we use sin and cos to approximate fractal functions . Moreover, we can also approximate the functions by adding up functions together and the Fourier series [3][4][5] will tell us what coefficient to use in our combination. However, you can only approximate a function on an interval.


Convolution

Convolution(Q210857)

Convolution is a mathematical operation that expresses the product of two functions. It refers to both the result function and to the process of computing it. After one function is reversed and shifted it could be seen as the integral of the product of the two functions. The following equations is copied from the video: " The Convolution of Two Functions Definition & Properties" [6]

The equation of convolution of of and of :


In this equation, the star between and is not multiplication. The star operator:(), takes two different functions and combines them into one function. The in the equation is just a variable, and the tau were just the dummy variable of integration.


Fourier Transform

Fourier Transform(Q6520159)

Fourier Transform [7] is the next level of the Fourier Series, it comes up with a way to approximate hole functions by using exponentials that means, unlike Fourier Series can only approximate a function on an interval, now we can approximate functions that are infinitely long.


Example for Fourier transform: We have a signal called we will represent it in terms of the time domain. We also can represent it in another way which is called we will represent it in terms of the frequency domain and This is why we called transformation. Fourier Transform is an equivalent representation of the signal.


Fourier Series equation

exponential term

  • is the subject of the equation
  • is the time function we calculating the Fourier Series for.
  • represents imaginary numbers,
  • is a exponential term
  • is a integral

-From The Fourier Series and Fourier Transform Demystified[8]



Possible Works to be done for this page

  1. Learn to use Blockquote and Cite!
  2. What did you learn from the video?
  3. How does Fourier Transform relates to Uncertainty Principle[9] as often mentioned in Quantum mechanics?
  4. How does Fourier Transform relates to Superposition[10] as often mentioned in Quantum mechanics?
  5. What are the links of Fourier Transform with Sine/Cosine, and Exponential functions?
  6. What other sources related to Sine/Cosine/Exponent and Fourier Transform you have found and what are their Links?
  7. Does it relate to Polynomials, Infinite Series, and the eventual implementation of Fast Fourier Transform?
  8. What did you learn that changed the views about certain beliefs that you hold before watching this video?
  9. Who can you ask to learn more about what you wish to know.
  10. Can you ask other people, such as your English teacher to look for help in documenting your observations and thoughts?
  11. Since Fourier Transform relates to adding many waves together, What kind of arithmetic does Fourier Transform can do? How does it relate to numerical arithmetic, or how does the arithmetics of waves differ from the arithmetics of numbers?

--Benkoo (talk) 03:04, 28 July 2022 (UTC)

References

  1. ↑ Newman, Mark (Apr 5, 2022). The imaginary number i and the Fourier Transform. local page: Mark Newman. 
  2. ↑ Wildberger, Norman J. (2005). DIVINE PROPORTIONS : Rational Trigonometry to Universal Geometry. local page: Wild Egg Books. ISBN 0-9757492-0-X. 
  3. ↑ Looking Glass Universe, ed. (Jun 30, 2022). Fourier Series. local page: Looking Glass Universe. 
  4. ↑ Sandlin, Destin (Dec 11, 2018). What is a Fourier Series? (Explained by drawing circles) - Smarter Every Day 205. local page: SmarterEveryDay. 
  5. ↑ Tan-Holmes, Jade (Jun 30, 2022). The Fourier Series and Fourier Transform Demystified. local page: Up and Atom. 
  6. ↑ Bazett, Trefor (Apr 12, 2020). The Convolution of Two Functions Definition & Properties. local page: Dr. Trefor Bazett. 
  7. ↑ Douglas, Brian (Jan 11, 2013). Introduction to the Fourier Transform. local page: Brian Douglas. 
  8. ↑ Tan-Holmes, Jade (Jun 30, 2022). The Fourier Series and Fourier Transform Demystified. local page: Up and Atom. 
  9. ↑ G., Parth (Dec 17, 2018). Heisenberg's Uncertainty Principle EXPLAINED (for beginners). local page: Parth G. 
  10. ↑ Lucid, Nick (Nov 29, 2021). Quantum Superposition, Explained Without Woo Woo. local page: The Science Asylum. 

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