| Line 13: | Line 13: | ||
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]]. | ||
</center> | </center> | ||
| − | |||
---- | ---- | ||
==OUTLINE== | ==OUTLINE== | ||
| Line 27: | Line 26: | ||
==Theory== | ==Theory== | ||
* Review of formulas used in ECE 301 | * Review of formulas used in ECE 301 | ||
| − | |||
{| | {| | ||
|- | |- | ||
| Line 36: | Line 34: | ||
| <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> | ||
|} | |} | ||
| − | |||
| − | |||
* Review of formulas used in ECE 438. | * Review of formulas used in ECE 438. | ||
{| | {| | ||
| Line 59: | Line 55: | ||
Also recall that | Also recall that | ||
<math> \displaystyle\delta(\alpha f)=\frac{1}{\alpha}\delta(f)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;for\;\;\alpha>0</math> | <math> \displaystyle\delta(\alpha f)=\frac{1}{\alpha}\delta(f)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;for\;\;\alpha>0</math> | ||
| − | |||
{| | {| | ||
|- | |- | ||
| Line 74: | Line 69: | ||
|- | |- | ||
|} | |} | ||
| − | |||
{| | {| | ||
|- | |- | ||
| Line 83: | Line 77: | ||
|<math>X(f)= \frac{1}{2}\left[\delta (f - f_0) + \delta (f + f_0)\right] \ </math> | |<math>X(f)= \frac{1}{2}\left[\delta (f - f_0) + \delta (f + f_0)\right] \ </math> | ||
|} | |} | ||
| − | |||
Second way is by direct using CTFT formula | Second way is by direct using CTFT formula | ||
{| | {| | ||
| Line 105: | Line 98: | ||
{| | {| | ||
|- | |- | ||
| + | | align="right" style="padding-right: 1em;" | | ||
| <math>X(f)=e^{-i2\pi ft} = e^{-i\omega ft} </math> | | <math>X(f)=e^{-i2\pi ft} = e^{-i\omega ft} </math> | ||
|} | |} | ||
| − | |||
| − | |||
| − | |||
==Conclusion== | ==Conclusion== | ||
Revision as of 16:15, 18 September 2014
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.
Contents
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
| 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 \, $ |
- For more formulas see the link to a Rhea page
Examples
1) 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(t) \ $ | $ \longrightarrow $ | $ \mathcal{X}(\omega) $ | |
| $ \cos(\omega_0 t) \ $ | $ \pi \left[\delta (\omega - \omega_0) + \delta (\omega + \omega_0)\right] \ $ |
| $ 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
2) Let's compute 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 $ |
| CT Fourier Transform | $ X(f)=\mathcal{F}(\delta (t-t_0))=\int_{-\infty}^{\infty} \delta (t-t_0) e^{-i2\pi ft} dt $ |
| $ X(f)=e^{-i2\pi ft} = e^{-i\omega ft} $ |
Conclusion
References
[1].Mireille Boutin, "ECE438 Digital Signal Processing with Applications," Purdue University August 26,2009
Post your slecture material here. Guidelines:
- 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.
- Rephrase the material in your own way, in your own words, based on Prof. Boutin's lecture material.
- Feel free to add your own examples or your own material.
- Focus on the clarity of your explanation. It must be clear, easily understandable.
- Type text using wikitext markup language. Do not post a pdf. Do not upload a word file.
- Type all equations using latex code between <math> </math> tags.
- You may include graphs, pictures, animated graphics, etc.
- You may include links to other Project Rhea pages.
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.
(create a question page and put a link below)
Questions and comments
If you have any questions, comments, etc. please post them on this page.
