HW 05: Difference between revisions
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{| border="0" cellpadding="0" cellspacing="0" | {| border="0" cellpadding="0" cellspacing="0" | ||
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|<math>F[e^{j \omega_0 t}]</math> | |<math>F\left[e^{j \omega_0 t}\right]</math> | ||
|<math>=\int_{-\infty}^{\infty} e^{j \omega_0 t} e^{-j \omega t}dt</math> | |<math>=\int_{-\infty}^{\infty} e^{j \omega_0 t} e^{-j \omega t}dt</math> | ||
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|<math>=2\pi \delta(\omega_0-\omega)\,\!</math> | |<math>=2\pi \delta(\omega_0-\omega)\,\!</math> | ||
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|<math>F[\cos {\omega_0 t}]\,\!</math> | |<math>F\left[\cos {\omega_0 t}\right]\,\!</math> | ||
|<math>=\int_{-\infty}^{\infty}\frac{e^{j\omega_0 t} + e^{-j\omega_0 t}}{2} e^{-j \omega t}dt</math> | |<math>=\int_{-\infty}^{\infty}\frac{e^{j\omega_0 t} + e^{-j\omega_0 t}}{2} e^{-j \omega t}dt</math> | ||
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|<math>=\pi\delta(\omega_0-\omega) + \pi\delta(\omega_0+\omega)\,\!</math> | |<math>=\pi\delta(\omega_0-\omega) + \pi\delta(\omega_0+\omega)\,\!</math> | ||
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|<math>F[\sin{\omega_0 t}]\,\!</math> | |<math>F\left[\sin{\omega_0 t}\right]\,\!</math> | ||
|<math>=\int_{-\infty}^{\infty}\frac{e^{j\omega_0 t} - e^{-j\omega_0 t}}{2j} e^{-j \omega t}dt</math> | |<math>=\int_{-\infty}^{\infty}\frac{e^{j\omega_0 t} - e^{-j\omega_0 t}}{2j} e^{-j \omega t}dt</math> | ||
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|<math>=-j\pi\delta(\omega_0-\omega) + j\pi\delta(\omega_0+\omega)\,\!</math> | |<math>=-j\pi\delta(\omega_0-\omega) + j\pi\delta(\omega_0+\omega)\,\!</math> | ||
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|<math>F[\sum_{-\infty}^{\infty}\alpha_n e^{j2\pi nt/T}]</math> | |<math>F\left[\sum_{-\infty}^{\infty}\alpha_n e^{j2\pi nt/T}\right]</math> | ||
|<math>=\int_{-\infty}^{\infty} \left (\sum_{-\infty}^{\infty}\alpha_n e^{j2\pi nt/T} \right )e^{-j \omega t}dt</math> | |<math>=\int_{-\infty}^{\infty} \left (\sum_{-\infty}^{\infty}\alpha_n e^{j2\pi nt/T} \right )e^{-j \omega t}dt</math> | ||
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Revision as of 21:27, 23 November 2008
Find the following Fourier Transforms