10/09 - Fourier Transform: Difference between revisions
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|<math>=X(f)\,\!</math> | |<math>=X(f)\,\!</math> | ||
|<math>=\int_{-\infty}^{\infty} x(t) e^{-j2\pi ft}dt</math> | |<math>=\int_{-\infty}^{\infty} x(t) e^{-j2\pi ft}dt</math> | ||
|<math>=\left \langle x(t) \mid e^{j2\pi ft}\right \ | |<math>=\left \langle x(t) \mid e^{j2\pi ft}\right \rangle_t</math> | ||
|- | |- | ||
|<math>F^{-1}[x(t)]\,\!</math> | |<math>F^{-1}[x(t)]\,\!</math> | ||
|<math>=x(t)\,\!</math> | |<math>=x(t)\,\!</math> | ||
|<math>=\int_{-\infty}^{\infty} X(f) e^{j2\pi ft} | |<math>=\int_{-\infty}^{\infty} X(f) e^{j2\pi ft}df</math> | ||
|<math>=\left \langle X(f) \mid e^{-j2\pi ft}\right \ | |<math>=\left \langle X(f) \mid e^{-j2\pi ft}\right \rangle_f</math> | ||
|} | |} | ||
==Examples== | ==Examples== | ||
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|<math>F^{-1}[F[x(t)]]\,\!</math> | |<math>F^{-1}[F[x(t)]]\,\!</math> | ||
|<math>=</math> | |<math>=\int_{-\infty}^{\infty}\left [ \int_{-\infty}^{\infty} x(t) e^{-j2\pi ft}dt\right ]e^{j2\pi ft} df</math> | ||
|- | |- | ||
| | | | ||
|<math>= | |<math>=\int_{-\infty}^{\infty}X(f) e^{j2\pi ft} df</math> | ||
|- | |||
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|<math>=x(t)\,\!</math> | |||
|} | |} |
Revision as of 15:10, 17 November 2008
Assuming the function is perodic with the period T | ||
Fourier Transform
Remember from 10/02 - Fourier Series
If we let
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