View source for Slecture Fourier transform w f ECE438 - Rhea

[[Category:slecture]]
[[Category:ECE438Fall2014Boutin]] 
[[Category:ECE]]
[[Category:ECE438]]
[[Category:signal processing]]  

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Fourier transform as a function of frequency ω versus Fourier transform as a function of frequency f 
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A [https://www.projectrhea.org/learning/slectures.php slecture] by [[ECE]] student JOE BLO

Partly based on the [[2014_Fall_ECE_438_Boutin|ECE438 Fall 2014 lecture]] material of [[user:mboutin|Prof. Mireille Boutin]]. 
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==OUTLINE==
#Introduction
#Theory
#Examples
#Conclusion
#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

{|
|-
| align="right" style="padding-right: 1em;" | CT Fourier Transform 
| <math>\mathcal{X}(\omega)=\mathcal{F}(x(t))=\int_{-\infty}^{\infty} x(t) e^{-i\omega t} dt</math>
|-
| align="right" style="padding-right: 1em;" |  Inverse DT 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>
|}


* Review of formulas used in ECE 438. 
{|
|-
| 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>
|-
| align="right" style="padding-right: 1em;" | Inverse DT Fourier Transform 
| <math>\, x(t)=\mathcal{F}^{-1}(X(f))=\int_{-\infty}^{\infty}X(f)e^{i2\pi ft} df \,</math>
|}

*This is a [[Main_Page| link to a Rhea page]]
==Example==
#
{|
| align="right" style="padding-right: 1em;" | 
| align="right" style="padding-right: 1em;" | Let's find CTFT of a 
| <math>\cos(\omega_0 t) \ </math> 
| 
|in two different ways
|}

{|
|-
| align="right" style="padding-right: 1em;" | 
| align="right" style="padding-right: 1em;" | Let 
|<math>\, \mathcal\omega={2\pi}f</math>
|   and 
|  <math>\, \mathcal\omega_0={2\pi}f_0</math>
|}


{|
|-
| align="right" style="padding-right: 1em;" | <br> 
| align="right" style="padding-right: 1em;" | 
| <math> x(t) \  </math>  
| <math>\longrightarrow</math> 
| <math> \mathcal{X}(\omega) </math> 
| 
|
|-
| align="right" style="padding-right: 1em;" | 
| align="right" style="padding-right: 1em;" | CTFT of a cosine 
| <math>\cos(\omega_0 t) \ </math> 
| 
| <math> \pi \left[\delta (\omega - \omega_0) + \delta (\omega + \omega_0)\right] \ </math> 
| 
| 
|-
|}
#THIS IS THE SECONF ITEM
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==[[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]].
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