Difference between revisions of "Fourier Transform"

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[[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 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.


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.


Fourier transforms integral equation :
Fourier Series equation <math> f(\omega)= \int f(x) e^{-i \omega x } dx </math> <ref>{{:Video/The Fourier Series and Fourier Transform Demystified}}</ref>
<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:
Eulers formula <math>e^{ix} = \cos {\omega x} + i\sin {\omega x},</math> </math> <ref>{{:Video/The Fourier Series and Fourier Transform Demystified}}</ref>
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>starting at [https://youtu.be/Ea4GIkjCJs8?t=569 9' 29" of the video]</ref>
<ref>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/What is convolution? This is the easiest way to understand}}</ref>

Revision as of 13:59, 30 July 2022

Fourier Transform(Q6520159)

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.

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 [1]

Eulers formula </math> [2]

[3] [4] [5]

[6] [7]


References

  1. ↑ Tan-Holmes, Jade (Jun 30, 2022). The Fourier Series and Fourier Transform Demystified. local page: Up and Atom. 
  2. ↑ Tan-Holmes, Jade (Jun 30, 2022). The Fourier Series and Fourier Transform Demystified. local page: Up and Atom. 
  3. ↑ starting at 9' 29" of the video
  4. ↑ Discretised, ed. (Aug 25, 2020). What is convolution? This is the easiest way to understand. local page: Discretised. 
  5. ↑ Collings, Iain (Sep 9, 2019). What is convolution? This is the easiest way to understand. local page: Iain Explains Signals, Systems, and Digital Comms. 
  6. ↑ Douglas, Brian (Jan 11, 2013). Introduction to the Fourier Transform. local page: Brian Douglas. 
  7. ↑ Khutoryansky, Eugene (Aug 25, 2020). Fourier Transform, Fourier Series, and frequency spectrum. local page: Physics Videos by Eugene Khutoryansky. 

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