HW 05: Difference between revisions
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|<math>F[\sin{\omega_0 t}]\,\!</math> |
|<math>F[\sin{\omega_0 t}]\,\!</math> |
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|<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>=\frac{1}{2j}\int_{-\infty}^{\infty}\left (e^{j\omega_0 t} - e^{-j\omega_0 t}\right )2je^{-j \omega t} dt</math> |
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|<math>=\int_{-\infty}^{\infty} \left (e^{j(\omega_0-\omega) t} - e^{-j(\omega_0+\omega) t}\right ) dt</math> |
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|<math>=2\pi\left [ \frac{1}{2\pi}\int_{-\infty}^{\infty} \left (e^{j(\omega_0-\omega) t} - e^{-j(\omega_0+\omega) t}\right )\,dt\right]</math> |
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|<math>=2\pi\delta(\omega_0-\omega) - 2\pi\delta(\omega_0+\omega)\,\!</math> |
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Revision as of 00:37, 18 November 2008
Find the following Fourier Transforms
![{\displaystyle F[e^{j\omega _{0}t}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/297667fd9533ec3aa3ad2923bfd68cb0d030c276)
![{\displaystyle F[\cos {\omega _{0}t}]\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d26d3f72149a74e0d94cfbd8edce6910fb100d8)
![{\displaystyle F[\sum _{-\infty }^{\infty }\alpha _{n}e^{j2\pi nt/T}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a016e88402c31d492838b6621da234c13829c741)
![{\displaystyle F[\sin {\omega _{0}t}]\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/42480573dd54907d94eebd1a92d716484d6ac65e)
![{\displaystyle =2\pi \left[{\frac {1}{2\pi }}\int _{-\infty }^{\infty }e^{j(\omega _{0}-\omega )t}dt\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3375b8f226843adcfbf9a87db8eae8c4db3fe3da)
![{\displaystyle ={\frac {1}{2}}\int _{-\infty }^{\infty }\left[2e^{j(\omega _{0}-\omega )t}+2e^{-j(\omega _{0}+\omega )t}\right]dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b13d6e5bb32432ba86365d6d5c617a4f7dde0f5)
![{\displaystyle =2\pi \left[{\frac {1}{2\pi }}\int _{-\infty }^{\infty }\left(e^{j(\omega _{0}-\omega )t}+e^{-j(\omega _{0}+\omega )t}\right)\,dt\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/add800a301730a483938a49f3c98e23b3a6e5a93)
![{\displaystyle =2\pi \left[{\frac {1}{2\pi }}\int _{-\infty }^{\infty }\left(e^{j(\omega _{0}-\omega )t}-e^{-j(\omega _{0}+\omega )t}\right)\,dt\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e8f770b898f736dfa665f686ea0c3a3a8b32e168)
