| Line 71: | Line 71: | ||
|- | |- | ||
| align="right" style="padding-right: 1em;" | CT Fourier Transform | | align="right" style="padding-right: 1em;" | CT Fourier Transform | ||
| − | | <math>X(f)=\mathcal{X}({2\pi}f)=\pi \left[\delta (\omega - \omega) + \delta (\omega + \ | + | | <math>X(f)=\mathcal{X}({2\pi}f)=\pi \left[\delta (\omega - \omega) + \delta (\omega + \omega)\right] \ </math> |
|- | |- | ||
|} | |} | ||
Revision as of 10:49, 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 JOE BLO
Partly based on the ECE438 Fall 2014 lecture material of Prof. Mireille Boutin.
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 DT 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 DT Fourier Transform | $ \, x(t)=\mathcal{F}^{-1}(X(f))=\int_{-\infty}^{\infty}X(f)e^{i2\pi ft} df \, $ |
- This is a link to a Rhea page
Example
| Let's find CTFT of a | $ \cos(\omega_0 t) \ $ | in two different ways. First way is to change FT pair and make change of variables. |
| Let | $ \, \mathcal\omega={2\pi}f $ | and | $ \, \mathcal\omega_0={2\pi}f_0 $ |
| CT Fourier Transform | $ X(f)=\mathcal{X}({2\pi}f)=\pi \left[\delta (\omega - \omega) + \delta (\omega + \omega)\right] \ $ |
| |
$ x(t) \ $ | $ \longrightarrow $ | $ \mathcal{X}(\omega) $ | |||
| CTFT of a cosine | $ \cos(\omega_0 t) \ $ | $ \pi \left[\delta (\omega - \omega_0) + \delta (\omega + \omega_0)\right] \ $ |
- THIS IS THE SECONF ITEM
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.
