View source for Talk:Video/The imaginary number i and the Fourier Transform

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 Series}}

{{:Convolution}}

=[[Fourier transform]]=
[[Fourier transform]] is the next level of the Fourier Series, it comes up with a way to approximate hole functions by using exponentials <math>e^{ikx}</math> that means, unlike Fourier Series can only approximate a function on an interval, now we can approximate functions that are infinitely long.


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 :
<math>(f * g)(t) := \int_{-\infty}^\infty f(\tau) g(t - \tau) \, d\tau.</math>
<ref extends=iAndFT>starting at [https://youtu.be/Ea4GIkjCJs8?t=569 9' 29" of the video]</ref>
<ref>{{:Video/What is convolution? This is the easiest way to understand}}</ref>
<ref>{{:Video/Fourier Transform Equation Explained}}</ref>

<ref> {{:Video/Introduction to the Fourier Transform}}</ref>
<ref>{{:Video/Fourier Transform, Fourier Series, and frequency spectrum}}</ref>


=Possible Works to be done for this page=
#Learn to use [[Quote|Blockquote]] and [[Cite]]!
#What did you learn from the video?
#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?
#Does it relate to Polynomials, Infinite Series, and the eventual implementation of Fast Fourier Transform?
#What did you learn that changed the views about certain beliefs that you hold before watching this video?
#Who can you ask to learn more about what you wish to know.
#Can you ask other people, such as your English teacher to look for help in documenting your observations and thoughts?
#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?

--[[User:Benkoo|Benkoo]] ([[User talk:Benkoo|talk]]) 03:04, 28 July 2022 (UTC)
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