10/10,13,16,17 - Fourier Transform Properties: Difference between revisions
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*Having trouble seeing <math>F\left[x(t)*h(t)\right]=X(f)\cdot H(f)</math> |
*Having trouble seeing <math>F\left[x(t)*h(t)\right]=X(f)\cdot H(f)</math> |
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*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? |
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Now back in the time domain |
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|Input |
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|LTI System |
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|Reason |
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|<math> 1\cdot e^{j\,2\,\pi f_0\,t}</math> |
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|<math> \Longrightarrow </math> |
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|<math> h(t)* e^{j\,2\,\pi f_0\,t}=e^{j\,2\,\pi f_0\,t}*h(t)</math> |
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|Proportionality |
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|<math>\int_{-\infty}^{\infty} h(\lambda)\cdot e^{j\,2\,\pi f_0\,(t-\lambda)}\,dt</math> |
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Revision as of 03:33, 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?
Now back in the time domain
Input | LTI System | Output | Reason |
Proportionality | |||