10/10,13,16,17 - Fourier Transform Properties: Difference between revisions
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===Relation to the Fourier Series=== |
===Relation to the Fourier Series=== |
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|<math>x(t)\,\!</math> |
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|<math>=x(t+T)\,\!</math> |
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|<math>=\sum_{n=-\infty}^{\infty}\alpha_n e^{j\,2\pi n\,t/T}</math> |
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|<math>=\underbrace{\sum_{m=-\infty}^{-1}\alpha_m e^{j\,2\pi m\,t/T}}_{Negative frequencies}+\alpha_0+\sum_{n=1}^{\infty}\alpha_n e^{j\,2\pi n\,t/T}</math> |
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|<math>=\sum_{n=1}^{\infty}\alpha_{-n} e^{-j\,2\pi n\,t/T}+\alpha_0+\sum_{n=1}^{\infty}\alpha_n e^{j\,2\pi n\,t/T}</math> |
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|Let <math>n=-m\,\!</math> and reverse the order of summation |
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|<math>=\alpha_0+2\Re\left[\sum_{n=1}^{\infty}\alpha_n e^{j\,2\pi n\,t/T}\right]</math> |
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|Assume that <math>x(t) \in \Re</math> |
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*Does the server reset every hour? |
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*How can we assume that the answer exists in the real domain? |
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Remember from [[10/02 - Fourier Series]] that <math> \alpha_n = \frac{1}{T}\int_{-T/2}^{T/2} x(t) e^{-j\,2\,\pi \,n\,t/T}\, dt</math> |
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*<math>\alpha_n=\left| \alpha_n \right|\,e^{j\theta}</math>? |
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*Rest of page |
Revision as of 06:20, 24 November 2008
Properties of the Fourier Transform
Linearity
Time Invariance (Delay)
Let and | ||
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 not both at the same time
- Then how can you use the Fourier Transform, but can't build up to it?
Input | LTI System | Output | Reason |
Given | |||
Proportionality | |||
Superposition | |||
Time Invariance | |||
Proportionality | |||
Superposition |
- Having trouble seeing
- Since we were dealing in the frequency domain, is that the reason why multiplying one side did not result in a convolution on the other?
The Game (Time Domain??)
Input | LTI System | Output | Reason |
Proportionality | |||
Why d lambda instead of dt? | |||
Proportionality, Why isn't this a convolution? | |||
Superposition, Not X(f_0)H(f_0)? |
Relation to the Fourier Series
Let and reverse the order of summation | ||
Assume that |
- Does the server reset every hour?
- How can we assume that the answer exists in the real domain?
Remember from 10/02 - Fourier Series that
- ?
- Rest of page