10/10,13,16,17 - Fourier Transform Properties: Difference between revisions
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===The Game (Time Domain)=== |
===The Game (Time Domain??)=== |
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|<math>\int_{-\infty}^{\infty} h(\lambda)\cdot e^{j\,2\,\pi f_0\,(t-\lambda)}\, |
|<math>\int_{-\infty}^{\infty} h(\lambda)\cdot e^{j\,2\,\pi f_0\,(t-\lambda)}\,d\lambda</math> |
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|Why d lambda instead of dt? |
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|<math>e^{j\,2\,\pi f_0\,\lambda}\int_{-\infty}^{\infty} h(\lambda)\cdot e^{-j\,2\,\pi f_0\,\lambda}\,d\lambda</math> |
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|<math>e^{j\,2\,\pi f_0\,t}\,H(f_0)</math> |
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|<math> X(f_0)\cdot e^{j\,2\,\pi f_0\,t}</math> |
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|<math> \Longrightarrow </math> |
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|<math>X(f_0)\cdot e^{j\,2\,\pi f_0\,t}\,H(f_0)</math> |
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|Proportionality, Why isn't this a convolution? |
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|<math> \int_{-\infty}^{\infty}X(f_0)\cdot e^{j\,2\,\pi f_0\,t}\,d f_0=x(t)</math> |
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|<math> \Longrightarrow </math> |
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|<math>\int_{-\infty}^{\infty}X(f_0)H(f_0)\cdot e^{j\,2\,\pi f_0\,t}\,d f_0=F^{-1}\left[X(f)H(f)\right]</math> |
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|Superposition, Not X(f_0)H(f_0)? |
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Revision as of 04:05, 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)? |