10/10,13,16,17 - Fourier Transform Properties: Difference between revisions
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|<math>= |
|<math>=\frac{1}{2}\left[\int_{0}^{-\infty} 1\cdot e^{-j\,2\,\pi\,f\,t}\,dt+\int_{0}^{\infty} 1\cdot e^{-j\,2\,\pi\,f\,t}\,dt\right]</math> |
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|<math>=\int_{0}^{\infty} |
|<math>=\underbrace{\frac{1}{2}\int_{0}^{\infty} -e^{j\,2\,\pi\,f\,u}\,du}_{\begin{matrix}u=-t \\ du=-dt\end{matrix}}+\underbrace{\frac{1}{2}\int_{0}^{\infty} e^{-j\,2\,\pi\,f\,u}\,du}_{\begin{matrix}u=t \\ du=dt\end{matrix}}</math> |
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|<math>=\int_{0}^{\infty} \ |
|<math>=\int_{0}^{\infty} \frac{-e^{j\,2\,\pi\,f\,u} + e^{-j\,2\,\pi\,f\,u}}{2}\,du</math> |
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|<math>\ne \int_{0}^{\infty} \cos(2\,\pi\,f\,u)\,du</math> |
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Revision as of 19:14, 1 December 2008
Properties of the Fourier Transform
Linearity
Time Invariance (Delay)
Let and | ||
Why isn't this |
Frequency Shifting
Double Sideband Modulation
Differentiation in Time
Thus is a linear filter with transfer function |
The Game (frequency domain)
- You can play the game in the frequency or time domain, but it's not advisable to play it in both at same time
Input | LTI System | Output | Reason |
Given | |||
Proportionality | |||
Superposition | |||
Time Invariance | |||
Proportionality | |||
Superposition |
- Having trouble seeing
The Game (Time Domain??)
Input | LTI System | Output | Reason |
Proportionality | |||
from 10/3,6 - The Game | |||
Proportionality | |||
Superposition |
Relation to the Fourier Series
Let and reverse the order of summation | ||
Note that is the complex conjugate of | ||
- How can we assume that the answer exists in the real domain?
Remember from 10/02 - Fourier Series that
- ?
- Rest of page
Building up to
Euler's Identity | ||
Real odd function of t | ||
& | ||
= Real odd. Integrates out over symmetric limits. | ||
Imaginary Odd function of | ||
Real even function of t | ||
& | ||
= Real odd. Integrates out over symmetric limits. | ||
Real Even function of |
Definitions
Can't x(t) have parts that aren't even or odd? You can break any function down into a Taylor series. There are even and odd powers in the series. | ||