HW 05: Difference between revisions

Revision as of 16:40, 17 November 2008

Find the following Fourier Transforms

$F[e^{j\omega _{0}t}]$	$=\int _{-\infty }^{\infty }e^{j\omega _{0}t}e^{-j\omega t}dt$
	$=\int _{-\infty }^{\infty }e^{j(\omega _{0}-\omega )t}dt$
	$=\delta (\omega _{0}-\omega )\,\!$
$F[\cos {\omega _{0}t}]\,\!$	$=\int _{-\infty }^{\infty }{\frac {e^{j\omega _{0}t}+e^{-j\omega _{0}t}}{2}}e^{-j\omega t}dt$
	$={\frac {1}{2}}\int _{-\infty }^{\infty }\left(e^{j\omega _{0}t}+e^{-j\omega _{0}t}\right)2e^{-j\omega t}dt$
	$={\frac {1}{2}}\int _{-\infty }^{\infty }2e^{j(\omega _{0}-\omega )t}+2e^{-j(\omega _{0}+\omega )t}dt$
	$=\int _{-\infty }^{\infty }e^{j(\omega _{0}-\omega )t}+e^{-j(\omega _{0}+\omega )t}$
	$=\delta (\omega _{0}-\omega )+\delta (\omega _{0}+\omega )\,\!$

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 |
 |<math>=\delta(\omega_0-\omega)\,\!</math>
+|-
+|<math>F[\cos {\omega_0 t}]\,\!</math>
+|<math>=\int_{-\infty}^{\infty}\frac{e^{j\omega_0 t} + e^{-j\omega_0 t}}{2} e^{-j \omega t}dt</math>
+|-
+|
+|<math>=\frac{1}{2}\int_{-\infty}^{\infty}\left (e^{j\omega_0 t} + e^{-j\omega_0 t}\right )2e^{-j \omega t} dt</math>
+|-
+|
+|<math>=\frac{1}{2}\int_{-\infty}^{\infty} 2e^{j(\omega_0-\omega) t} + 2e^{-j(\omega_0+\omega) t} dt</math>
+|-
+|
+|<math>=\int_{-\infty}^{\infty} e^{j(\omega_0-\omega) t} + e^{-j(\omega_0+\omega) t}</math>
+|-
+|
+|<math>=\delta(\omega_0-\omega) + \delta(\omega_0+\omega)\,\!</math>
 |}