Difference between revisions of "Fourier Transform"

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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.
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.


<H1>You must list out <math>i,k,x,t,f</math> individually, and state individually what they are explicitly.</H1>
For example:


where:
Fourier Series equation <math> f(\omega)= \int f(x) e^{-i \omega x } dx </math>
*<math>i</math> represents imaginary numbers,
*<math>k</math> is so and so,
*<math>x(t)</math> is the strength of the signal over time,
*<math>t</math> represents time,
*<math>f(x)</math> or <math>x(f)</math> represents the frequency function....


<H1>This must apply to all equations ...</H1>
exponential term <math>e^{i \omega x} = \cos ({\omega x}) + i\sin ({\omega x}),</math>


*<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


Fourier Series equation <math> f(\omega)= \int f(x) e^{-i \omega x } dx </math>
-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>


Eulers formula <math>e^{i \omega x} = \cos ({\omega x}) + i\sin ({\omega x}),</math>
-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>


In this equation f(x) is the time function we're calculating the Fourier series for. Then we times it by exponential<math>e^{i \omega x}</math>.


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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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