(New page: Category:2010 Fall ECE 438 Boutin ---- == Solution to Q1 of Week 14 Quiz Pool == ---- ---- Back to Lab Week 14 Quiz Pool Back to [[ECE438_Lab_Fall_2010|ECE 4...) |
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| + | Using the definition of the CSFT,<br/> | ||
| + | <math> | ||
| + | \begin{align} | ||
| + | F(u,v) &= \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f(x,y)e^{-j2\pi (ux+vy)}dxdy \\ | ||
| + | F(u,0) &= \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f(x,y)e^{-j2\pi (ux)}dxdy \\ | ||
| + | &= \int_{-\infty}^{\infty} \left( \int_{-\infty}^{\infty}f(x,y) dy \right) e^{-j2\pi ux}dx \\ | ||
| + | &= \int_{-\infty}^{\infty} p(x) e^{-j2\pi ux}dx \\ | ||
| + | &= P(u) \\ | ||
| + | \end{align} | ||
| + | </math> | ||
| + | |||
| + | so F(u,0) is the same as P(u) which is the CTFT of the function p(x). | ||
| + | |||
| + | Credit: Prof. Bouman | ||
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Latest revision as of 08:15, 28 November 2010
Solution to Q1 of Week 14 Quiz Pool
Using the definition of the CSFT,
$ \begin{align} F(u,v) &= \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f(x,y)e^{-j2\pi (ux+vy)}dxdy \\ F(u,0) &= \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f(x,y)e^{-j2\pi (ux)}dxdy \\ &= \int_{-\infty}^{\infty} \left( \int_{-\infty}^{\infty}f(x,y) dy \right) e^{-j2\pi ux}dx \\ &= \int_{-\infty}^{\infty} p(x) e^{-j2\pi ux}dx \\ &= P(u) \\ \end{align} $
so F(u,0) is the same as P(u) which is the CTFT of the function p(x).
Credit: Prof. Bouman
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