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

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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

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. [3][4][5][6][7][8][9][10]

Convolution

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.

[11] [12] [13]

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 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 :

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.

Convolution equation : Cite error: Invalid <ref> tag; invalid names, e.g. too many [14] [15] [16] [17] [18]

Fourier Series in human body

Find references and put them here.

Possible Works to be done for this page

  1. Learn to use Blockquote and Cite!
  2. What did you learn from the video?
  3. What are the links of Fourier Transform with Sine/Cosine, and Exponential functions?
  4. What other sources related to Sine/Cosine/Exponent and Fourier Transform you have found and what are their Links?
  5. Does it relate to Polynomials, Infinite Series, and the eventual implementation of Fast Fourier Transform?
  6. What did you learn that changed the views about certain beliefs that you hold before watching this video?
  7. Who can you ask to learn more about what you wish to know.
  8. Can you ask other people, such as your English teacher to look for help in documenting your observations and thoughts?
  9. 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. ↑ Tan-Holmes, Jade (Jun 30, 2022). The Fourier Series and Fourier Transform Demystified. local page: Up and Atom. 
  4. ↑ Strang, Gilbert (May 7, 2016). MIT OpenCourseWare, ed. Fourier Series. local page: MIT OpenCourseWare. 
  5. ↑ Khan, Salman (Aug 4, 2016). Khan Academy, ed. Fourier Series introduction. local page: Khan Academy. 
  6. ↑ Video/Intro to FOURIER SERIES: The Big Idea
  7. ↑ Strang, Gilbert (May 7, 2016). MIT OpenCourseWare, ed. Fourier Series. local page: MIT OpenCourseWare. 
  8. ↑ Sandlin, Destin (Dec 11, 2018). What is a Fourier Series? (Explained by drawing circles) - Smarter Every Day 205. local page: SmarterEveryDay. 
  9. ↑ Hernandez, Saul (Aug 7, 2011). Fourier Series by Saul Hernandez. local page: Saul Hernandez. 
  10. ↑ Newman, Mark (May 19, 2020). The birth of the Fourier Series. local page: Mark Newman. 
  11. ↑ Bazett, Trefor (Apr 12, 2020). The Convolution of Two Functions Definition & Properties. local page: Dr. Trefor Bazett. 
  12. ↑ SigFyg, ed. (Apr 12, 2020). Convolutions are not Convoluted. local page: SigFyg. 
  13. ↑ Introducing Convolutions: Intuition + Convolution Theorem
  14. ↑ Discretised, ed. (Aug 25, 2020). What is convolution? This is the easiest way to understand. local page: Discretised. 
  15. ↑ Collings, Iain (Sep 9, 2019). What is convolution? This is the easiest way to understand. local page: Iain Explains Signals, Systems, and Digital Comms. 
  16. ↑ Douglas, Brian (Jan 11, 2013). Introduction to the Fourier Transform (Part 1). local page: Brian Douglas. 
  17. ↑ Douglas, Brian (Aug 25, 2020). Introduction to the Fourier Transform (Part 2). local page: Brian Douglas. 
  18. ↑ Khutoryansky, Eugene (Aug 25, 2020). Fourier Transform, Fourier Series, and frequency spectrum. local page: Physics Videos by Eugene Khutoryansky. 

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