10/09 - Fourier Transform: Difference between revisions
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==Fourier Transform== |
==Fourier Transform== |
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Remember from [[10/02 - Fourier Series]] |
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*<math> \alpha_m = \frac{1}{T}\int_{-T/2}^{T/2} x(t) e^{-j2\pi mt/T}\, dt</math> |
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*<math>x(t) = x(t+T) = \sum_{n=-\infty}^\infty \alpha_m e^{j2\pi m/T}</math> |
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If we let <math> T \rightarrow \infty</math> |
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{| border="0" cellpadding="0" cellspacing="0" |
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|- |
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|<math>\frac{1}{T}</math> |
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|<math>\rightarrow df</math> |
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|- |
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|<math>\frac{n}{T}</math> |
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|<math>\rightarrow f</math> |
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|Remember <math>f=\frac{2\pi n}{T} \,\!</math> |
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|- |
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|<math>T\,\!</math> |
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|<math>\rightarrow \infty</math> |
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|- |
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|<math>\sum_{n=-\infty}^{\infty} \frac{1}{T}</math> |
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|<math>\rightarrow \int_{-\infty}^{\infty}() df</math> |
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Revision as of 14:24, 17 November 2008
Assuming the function is perodic with the period T | ||
Fourier Transform
Remember from 10/02 - Fourier Series
If we let
Remember | ||