Difference between revisions of "Fourier Transform"

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=[[Fourier transform]]=
{{WikiEntry|key=Fourier Transform|qCode=6520159}}
{{WikiEntry|key=Fourier Transform|qCode=6520159}}


[[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]] <ref> {{:Video/Introduction to the Fourier Transform}}</ref> 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 :
Example for Fourier transform:
<math>\hat{f}(\xi) = \int_{-\infty}^{\infty} f(x)\ e^{-i 2\pi \xi x}\,dx,\quad \forall\ \xi \in \mathbb R.</math>
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.


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 equation <math> f(\omega)= \int f(x) e^{-i \omega x } dx </math>
<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>
exponential term <math>e^{i \omega x} = \cos ({\omega x}) + i\sin ({\omega x}),</math>
<ref>{{:Video/What is convolution? This is the easiest way to understand}}</ref>
 
<ref>{{:Video/Fourier Transform Equation Explained}}</ref>
*<math>f(\omega) </math> is the subject of the equation
*<math>f(x)</math> is the time function we calculating the Fourier Series for.
*<math>i</math> represents imaginary numbers,
*<math>e^{i \omega x}</math> is a exponential term
*<math>\int</math> is a integral
 
-From [[Video/The Fourier Series and Fourier Transform Demystified | The Fourier Series and Fourier Transform Demystified]]<ref>{{:Video/The Fourier Series and Fourier Transform Demystified}}</ref>


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




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Latest revision as of 06:26, 2 August 2022

Fourier Transform(Q6520159)

Fourier Transform [1] 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[2]



References

  1. ↑ Douglas, Brian (Jan 11, 2013). Introduction to the Fourier Transform. local page: Brian Douglas. 
  2. ↑ Tan-Holmes, Jade (Jun 30, 2022). The Fourier Series and Fourier Transform Demystified. local page: Up and Atom. 

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