HW 05: Difference between revisions

Revision as of 16:40, 17 November 2008

Find the following Fourier Transforms

$F [e^{j ω_{0} t}]$	$= \int_{- \infty}^{\infty} e^{j ω_{0} t} e^{- j ω t} d t$
	$= \int_{- \infty}^{\infty} e^{j (ω_{0} - ω) t} d t$
	$= δ (ω_{0} - ω)$
$F [\cos ω_{0} t]$	$= \int_{- \infty}^{\infty} \frac{e^{j ω_{0} t} + e^{- j ω_{0} t}}{2} e^{- j ω t} d t$
	$= \frac{1}{2} \int_{- \infty}^{\infty} (e^{j ω_{0} t} + e^{- j ω_{0} t}) 2 e^{- j ω t} d t$
	$= \frac{1}{2} \int_{- \infty}^{\infty} 2 e^{j (ω_{0} - ω) t} + 2 e^{- j (ω_{0} + ω) t} d t$
	$= \int_{- \infty}^{\infty} e^{j (ω_{0} - ω) t} + e^{- j (ω_{0} + ω) t}$
	$= δ (ω_{0} - ω) + δ (ω_{0} + ω)$

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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>
 |}