10/09 - Fourier Transform: Difference between revisions
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|<math>=\int_{-\infty}^{\infty}x(\lambda) \int_{-\infty}^{\infty}e^{j2\pi f(t-\lambda)} df d\lambda</math> | |<math>=\int_{-\infty}^{\infty}x(\lambda) \int_{-\infty}^{\infty}e^{j2\pi f(t-\lambda)} df d\lambda</math> | ||
|<math>=\int_{-\infty}^{\infty}x(\lambda) \delta(t-\lambda) d\lambda</math> | |<math>=\int_{-\infty}^{\infty}x(\lambda) \delta(t-\lambda) d\lambda</math> | ||
|<math>=x(t)\,\!</math> | |||
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|<math>=\int_{-\infty}^{\infty}\left [ \int_{-\infty}^{\infty} x(\lambda) e^{-j\omega\lambda}d\lambda\right ]e^{j\omega t} \frac{1}{2\pi}d\omega </math> | |||
|<math>=\int_{-\infty}^{\infty}x(\lambda) \left [ \frac{1}{2\pi} \int_{-\infty}^{\infty} e^{j(t-\omega) \lambda}d\omega\right ] d\lambda</math> | |||
|<math>=\int_{-\infty}^{\infty}x(\lambda) \delta(t-\omega) d\lambda</math> | |||
|<math>=x(t)\,\!</math> | |<math>=x(t)\,\!</math> | ||
|} | |} |
Revision as of 17:53, 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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