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A [https://www.projectrhea.org/learning/slectures.php slecture] by [[ECE]] student JOE BLO
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A [https://www.projectrhea.org/learning/slectures.php slecture] by [[ECE]] student Dauren Nurmaganbetov
  
 
Partly based on the [[2014_Fall_ECE_438_Boutin|ECE438 Fall 2014 lecture]] material of [[user:mboutin|Prof. Mireille Boutin]].  
 
Partly based on the [[2014_Fall_ECE_438_Boutin|ECE438 Fall 2014 lecture]] material of [[user:mboutin|Prof. Mireille Boutin]].  
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#Conclusion
 
#Conclusion
 
#References
 
#References
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----
 
==Introduction==
 
==Introduction==
 
In my slecture I will explain Fourier transform as a function of frequency ω versus Fourier transform as a function of frequency f (in hertz).
 
In my slecture I will explain Fourier transform as a function of frequency ω versus Fourier transform as a function of frequency f (in hertz).
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----
 
==Theory==
 
==Theory==
 
* Review of formulas used in ECE 301
 
* Review of formulas used in ECE 301
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{|
 
{|
 
|-
 
|-
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| <math>\mathcal{X}(\omega)=\mathcal{F}(x(t))=\int_{-\infty}^{\infty} x(t) e^{-i\omega t} dt</math>
 
| <math>\mathcal{X}(\omega)=\mathcal{F}(x(t))=\int_{-\infty}^{\infty} x(t) e^{-i\omega t} dt</math>
 
|-
 
|-
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| align="right" style="padding-right: 1em;" |  Inverse DT Fourier Transform  
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| align="right" style="padding-right: 1em;" |  Inverse Fourier Transform  
 
| <math>\, x(t)=\mathcal{F}^{-1}(\mathcal{X}(\omega))=\frac{1}{2\pi} \int_{-\infty}^{\infty}\mathcal{X}(\omega)e^{i\omega t} d \omega\,</math>
 
| <math>\, x(t)=\mathcal{F}^{-1}(\mathcal{X}(\omega))=\frac{1}{2\pi} \int_{-\infty}^{\infty}\mathcal{X}(\omega)e^{i\omega t} d \omega\,</math>
 
|}
 
|}
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* Review of formulas used in ECE 438.  
 
* Review of formulas used in ECE 438.  
 
{|
 
{|
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| <math>X(f)=\mathcal{F}(x(t))=\int_{-\infty}^{\infty} x(t) e^{-i2\pi ft} dt</math>
 
| <math>X(f)=\mathcal{F}(x(t))=\int_{-\infty}^{\infty} x(t) e^{-i2\pi ft} dt</math>
 
|-
 
|-
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| align="right" style="padding-right: 1em;" | Inverse DT Fourier Transform  
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| align="right" style="padding-right: 1em;" | Inverse Fourier Transform  
 
| <math>\, x(t)=\mathcal{F}^{-1}(X(f))=\int_{-\infty}^{\infty}X(f)e^{i2\pi ft} df \,</math>
 
| <math>\, x(t)=\mathcal{F}^{-1}(X(f))=\int_{-\infty}^{\infty}X(f)e^{i2\pi ft} df \,</math>
 
|}
 
|}
  
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*This is a [[Main_Page| link to a Rhea page]]
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*For more formulas see the [[CTFourierTransformPairsCollectedfromECE301withomega|table of CT Fourier transform pairs and properties]]
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==Example==
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----
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#
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==Examples==
 +
1)
 
{|
 
{|
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| align="right" style="padding-right: 1em;" | <br>
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| <math> x(t) \  </math> 
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| <math>\longrightarrow</math>
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| <math> \mathcal{X}(\omega) </math>
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|
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|-
 
| align="right" style="padding-right: 1em;" |  
 
| align="right" style="padding-right: 1em;" |  
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| align="right" style="padding-right: 1em;" | Let's find CTFT of a
 
 
| <math>\cos(\omega_0 t) \ </math>  
 
| <math>\cos(\omega_0 t) \ </math>  
 
|  
 
|  
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|in two different ways. First way is to change FT pair and make change of variables.
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| <math> \pi \left[\delta (\omega - \omega_0) + \delta (\omega + \omega_0)\right] \ </math>
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|-
 
|}
 
|}
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Let's compute FT of a cosine in two different ways:
 +
 +
First way is by changing FT pair and changing of variable
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 +
Let
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<math>\, \mathcal\omega={2\pi}f</math> ,  <math>\, \mathcal\omega_0={2\pi}f_0</math>
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Also recall that
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<math> \displaystyle\delta(\alpha f)=\frac{1}{\alpha}\delta(f)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;for\;\;\alpha>0</math>
  
 
{|
 
{|
 
|-
 
|-
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| align="right" style="padding-right: 1em;" |
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|<math>X(f)=\mathcal{X}({2\pi}f)=\pi \left[\delta ({2\pi}f - {2\pi}f_0) + \delta ( {2\pi}f+ {2\pi}f_0)\right] \  </math>
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| align="right" style="padding-right: 1em;" | Let
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|-
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|<math>\, \mathcal\omega={2\pi}f</math>
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|<math>X(f)= \pi \left[\frac{1}{2\pi }\delta (f - f_0) + \frac{1}{2\pi }\delta (f + f_0)\right] \ </math>  
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|   and
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|-
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| <math>\, \mathcal\omega_0={2\pi}f_0</math>
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|<math>X(f)= \frac{1}{2}\left[\delta (f - f_0) + \delta (f + f_0)\right] \ </math>  
 
|}
 
|}
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Second way is by direct using CTFT formula
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{|
 +
|-
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<math>X(f)= \frac{1}{2} \left[\delta (f - \frac{\omega_0}{2\pi}) + \delta (f + \frac{\omega_0}{2\pi})\right] \ </math>
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|}
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2) Let's find CTFT of a shifted unit impulse:
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<math>\delta (t-t_0)\ </math>
  
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Keep in mind that:
 
{|
 
{|
 
|-
 
|-
 
| align="right" style="padding-right: 1em;" |  CT Fourier Transform  
 
| align="right" style="padding-right: 1em;" |  CT Fourier Transform  
 
| <math>X(f)=\mathcal{F}(x(t))=\int_{-\infty}^{\infty} x(t) e^{-i2\pi ft} dt</math>
 
| <math>X(f)=\mathcal{F}(x(t))=\int_{-\infty}^{\infty} x(t) e^{-i2\pi ft} dt</math>
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|}
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{|
 
|-
 
|-
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| align="right" style="padding-right: 1em;" |  From above equation
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| <math>X(f)=\mathcal{F}(\delta (t-t_0))=\int_{-\infty}^{\infty} \delta (t-t_0) e^{-i2\pi ft} dt</math>
 
|}
 
|}
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{|
 
{|
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|-
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|-  
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| align="right" style="padding-right: 1em;" | <br>
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| align="right" style="padding-right: 1em;" |   Thus we get
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| align="right" style="padding-right: 1em;" |
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|  <math>X(f)=e^{-i2\pi ft_0} = e^{-i\omega t_0} </math>
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| <math> x(t) \ </math>
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| <math>\longrightarrow</math>
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| <math> \mathcal{X}(\omega) </math>
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|
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|
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|-
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| align="right" style="padding-right: 1em;" |
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| align="right" style="padding-right: 1em;" | CTFT of a cosine
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| <math>\cos(\omega_0 t) \ </math>
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|
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| <math> \pi \left[\delta (\omega - \omega_0) + \delta (\omega + \omega_0)\right] \ </math>  
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|
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|
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|-
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|}
 
|}
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#THIS IS THE SECONF ITEM
 
 
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Post your slecture material here. Guidelines:
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==Conclusion==
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*If you wish to post your slecture anonymously, please contact your instructor to get an anonymous login. Otherwise, you will be identifiable through your Purdue CAREER account, and thus you will NOT be anonymous.
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Observe that the expressions for the FT are different because we used change of variables.  
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*Rephrase the material in your own way, in your own words, based on Prof. Boutin's lecture material.
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*Feel free to add your own examples or your own material.
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*Focus on the clarity of your explanation. It must be clear, easily understandable.
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*Type text using wikitext markup language. Do not post a pdf. Do not upload a word file.
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*Type all equations using latex code between <nowiki> <math> </math> </nowiki> tags.
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*You may include graphs, pictures, animated graphics, etc.
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*You may include links to other [https://www.projectrhea.org/learning/about_Rhea.php Project Rhea] pages.  
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IMPORTANT: DO NOT PLAGIARIZE. If you use other material than Prof. Boutin's lecture material, you must cite your sources. Do not copy text word for word from another source; rephrase everything using your own words. Similarly for graphs, illustrations, pictures, etc. Make your own! Do not copy them from other sources.
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Also notice that one can transform one expression into  the other using the scaling property of the Dirac delta
 
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==References==
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[1].Mireille Boutin, "ECE438 Digital Signal Processing with Applications," Purdue University August 26,2009
 
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(create a question page and put a link below)
 
 
==[[Slecture_Fourier_transform_w_f_ECE438_review|Questions and comments]]==
 
==[[Slecture_Fourier_transform_w_f_ECE438_review|Questions and comments]]==
  
 
If you have any questions, comments, etc. please post them on [[Slecture_Fourier_transform_w_f_ECE438_review|this page]].
 
If you have any questions, comments, etc. please post them on [[Slecture_Fourier_transform_w_f_ECE438_review|this page]].
 
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[[2014_Fall_ECE_438_Boutin|Back to ECE438, Fall 2014]]
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[[2014_Fall_ECE_438_Boutin_digital_signal_processing_slectures|Back to ECE438 slectures, Fall 2014]]

Latest revision as of 17:51, 16 March 2015


Fourier transform as a function of frequency ω versus Fourier transform as a function of frequency f

A slecture by ECE student Dauren Nurmaganbetov

Partly based on the ECE438 Fall 2014 lecture material of Prof. Mireille Boutin.

OUTLINE

  1. Introduction
  2. Theory
  3. Examples
  4. Conclusion
  5. References

Introduction

In my slecture I will explain Fourier transform as a function of frequency ω versus Fourier transform as a function of frequency f (in hertz).

Theory

  • Review of formulas used in ECE 301
CT Fourier Transform $ \mathcal{X}(\omega)=\mathcal{F}(x(t))=\int_{-\infty}^{\infty} x(t) e^{-i\omega t} dt $
Inverse Fourier Transform $ \, x(t)=\mathcal{F}^{-1}(\mathcal{X}(\omega))=\frac{1}{2\pi} \int_{-\infty}^{\infty}\mathcal{X}(\omega)e^{i\omega t} d \omega\, $
  • Review of formulas used in ECE 438.
CT Fourier Transform $ X(f)=\mathcal{F}(x(t))=\int_{-\infty}^{\infty} x(t) e^{-i2\pi ft} dt $
Inverse Fourier Transform $ \, x(t)=\mathcal{F}^{-1}(X(f))=\int_{-\infty}^{\infty}X(f)e^{i2\pi ft} df \, $

Examples

1)


$ x(t) \ $ $ \longrightarrow $ $ \mathcal{X}(\omega) $
$ \cos(\omega_0 t) \ $ $ \pi \left[\delta (\omega - \omega_0) + \delta (\omega + \omega_0)\right] \ $

Let's compute FT of a cosine in two different ways:

First way is by changing FT pair and changing of variable

Let 

$ \, \mathcal\omega={2\pi}f $ ,  $ \, \mathcal\omega_0={2\pi}f_0 $

Also recall that 

$  \displaystyle\delta(\alpha f)=\frac{1}{\alpha}\delta(f)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;for\;\;\alpha>0 $
$ X(f)=\mathcal{X}({2\pi}f)=\pi \left[\delta ({2\pi}f - {2\pi}f_0) + \delta ( {2\pi}f+ {2\pi}f_0)\right] \ $
$ X(f)= \pi \left[\frac{1}{2\pi }\delta (f - f_0) + \frac{1}{2\pi }\delta (f + f_0)\right] \ $
$ X(f)= \frac{1}{2}\left[\delta (f - f_0) + \delta (f + f_0)\right] \ $

Second way is by direct using CTFT formula

$ X(f)= \frac{1}{2} \left[\delta (f - \frac{\omega_0}{2\pi}) + \delta (f + \frac{\omega_0}{2\pi})\right] \ $

2) Let's find CTFT of a shifted unit impulse:

$ \delta (t-t_0)\ $

Keep in mind that:

CT Fourier Transform $ X(f)=\mathcal{F}(x(t))=\int_{-\infty}^{\infty} x(t) e^{-i2\pi ft} dt $
From above equation $ X(f)=\mathcal{F}(\delta (t-t_0))=\int_{-\infty}^{\infty} \delta (t-t_0) e^{-i2\pi ft} dt $
Thus we get $ X(f)=e^{-i2\pi ft_0} = e^{-i\omega t_0} $

Conclusion

Observe that the expressions for the FT are different because we used change of variables.

Also notice that one can transform one expression into the other using the scaling property of the Dirac delta

References

[1].Mireille Boutin, "ECE438 Digital Signal Processing with Applications," Purdue University August 26,2009

Questions and comments

If you have any questions, comments, etc. please post them on this page.

Back to ECE438 slectures, Fall 2014

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