HW 05: Difference between revisions
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Find the following Fourier Transforms |
Find the following Fourier Transforms |
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*<math>F[e^{j \omega_0 t}]</math> |
*<math>F\left[e^{j \omega_0 t}\right]</math> |
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*<math>F[\cos {\omega_0 t}]\,\!</math> |
*<math>F\left[\cos {\omega_0 t}\right]\,\!</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> |
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*<math>F[\sin{\omega_0 t}]\,\!</math> |
*<math>F\left[\sin{\omega_0 t}\right]\,\!</math> |
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==Solutions== |
==Solutions== |
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{| 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> |
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|<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> |
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|<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>=\frac{1}{2}\int_{-\infty}^{\infty}\left (e^{j\omega_0 t} + e^{-j\omega_0 t}\right ) |
|<math>=\frac{1}{2}\int_{-\infty}^{\infty}\left (e^{j\omega_0 t} + e^{-j\omega_0 t}\right )e^{-j \omega t} dt</math> |
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|<math>=\frac{1}{2}\int_{-\infty}^{\infty} |
|<math>=\frac{1}{2}\int_{-\infty}^{\infty} \left [e^{j(\omega_0-\omega) t} + e^{-j(\omega_0+\omega) t}\right ] dt</math> |
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|<math>=\int_{-\infty}^{\infty} e^{j(\omega_0-\omega) t} + e^{-j(\omega_0+\omega) t}</math> |
|<math>=\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>=\delta(\omega_0-\omega) + \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> |
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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 )e^{-j \omega t} dt</math> |
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|<math>=\frac{1}{2j}\int_{-\infty}^{\infty} \left (e^{j(\omega_0-\omega) t} - e^{-j(\omega_0+\omega) t}\right ) dt</math> |
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|<math>=\frac{\pi}{j}\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>=-j\pi\delta(\omega_0-\omega) + j\pi\delta(\omega_0+\omega)\,\!</math> |
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|<math>F\left[\sum_{-\infty}^{\infty}\alpha_n e^{j2\pi nt/T}\right]</math> |
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|<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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|<math>=\sum_{-\infty}^{\infty}\alpha_n \left (\int_{-\infty}^{\infty} e^{j2\pi nt/T} e^{-j2\pi ft}dt\right )</math> |
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|<math>=\sum_{-\infty}^{\infty}\alpha_n \left (\int_{-\infty}^{\infty} e^{j2\pi t (\frac{n}{T}-f)} dt\right )</math> |
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|<math>=\sum_{-\infty}^{\infty}\alpha_n \delta\left(\frac{n}{T}-f\right) </math> |
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Latest revision as of 20:27, 23 November 2008
Find the following Fourier Transforms
![{\displaystyle F\left[e^{j\omega _{0}t}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4160b35ebd3b877b57188379113a2a61b48ca874)
![{\displaystyle F\left[\cos {\omega _{0}t}\right]\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a6e612e45a5f9e57270d09c5e76d13f1d9aef197)
![{\displaystyle F\left[\sum _{-\infty }^{\infty }\alpha _{n}e^{j2\pi nt/T}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8036420493818391105fcd6f13cd1e6a3ff950a9)
![{\displaystyle F\left[\sin {\omega _{0}t}\right]\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/79d9c2ce63f20aeda29d4b9a887f786bd8b788b6)
![{\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[e^{j(\omega _{0}-\omega )t}+e^{-j(\omega _{0}+\omega )t}\right]dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4762e54cbb3ec772bf15a506b0cd065addb46daa)
![{\displaystyle =\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/d239c2743ae7ffcb4d7354f71725c75b2093e891)
![{\displaystyle ={\frac {\pi }{j}}\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/9f75b4cea2498d8d3457db89b60ba12ea0740c3d)
