HW 05: Difference between revisions

Revision as of 00:43, 18 November 2008

Find the following Fourier Transforms

$F [e^{j ω_{0} t}]$	$= \int_{- \infty}^{\infty} e^{j ω_{0} t} e^{- j ω t} d t$
	$= \int_{- \infty}^{\infty} e^{j (ω_{0} - ω) t} d t$
	$= 2 π [\frac{1}{2 π} \int_{- \infty}^{\infty} e^{j (ω_{0} - ω) t} d t]$
	$= 2 π δ (ω_{0} - ω)$
$F [\cos ω_{0} t]$	$= \int_{- \infty}^{\infty} \frac{e^{j ω_{0} t} + e^{- j ω_{0} t}}{2} e^{- j ω t} d t$
	$= \frac{1}{2} \int_{- \infty}^{\infty} (e^{j ω_{0} t} + e^{- j ω_{0} t}) 2 e^{- j ω t} d t$
	$= \frac{1}{2} \int_{- \infty}^{\infty} [2 e^{j (ω_{0} - ω) t} + 2 e^{- j (ω_{0} + ω) t}] d t$
	$= 2 π [\frac{1}{2 π} \int_{- \infty}^{\infty} (e^{j (ω_{0} - ω) t} + e^{- j (ω_{0} + ω) t}) d t]$
	$= 2 π δ (ω_{0} - ω) + 2 π δ (ω_{0} + ω)$
$F [\sin ω_{0} t]$	$= \int_{- \infty}^{\infty} \frac{e^{j ω_{0} t} - e^{- j ω_{0} t}}{2 j} e^{- j ω t} d t$
	$= \frac{1}{2 j} \int_{- \infty}^{\infty} (e^{j ω_{0} t} - e^{- j ω_{0} t}) 2 j e^{- j ω t} d t$
	$= \int_{- \infty}^{\infty} (e^{j (ω_{0} - ω) t} - e^{- j (ω_{0} + ω) t}) d t$
	$= 2 π [\frac{1}{2 π} \int_{- \infty}^{\infty} (e^{j (ω_{0} - ω) t} - e^{- j (ω_{0} + ω) t}) d t]$
	$= 2 π δ (ω_{0} - ω) - 2 π δ (ω_{0} + ω)$
$F [\sum_{- \infty}^{\infty} α_{n} e^{j 2 π n t / T}]$	$= \int_{- \infty}^{\infty} (\sum_{- \infty}^{\infty} α_{n} e^{j 2 π n t / T}) e^{- j ω t} d t$
	$= \sum_{- \infty}^{\infty} α_{n} (\int_{- \infty}^{\infty} e^{j 2 π n t / T} e^{- j 2 π f t} d t)$
	$= \sum_{- \infty}^{\infty} α_{n} (\int_{- \infty}^{\infty} e^{j 2 π t (\frac{n}{T} - f)} d t)$

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 |
 |<math>=2\pi\delta(\omega_0-\omega) - 2\pi\delta(\omega_0+\omega)\,\!</math>
+|-
+|<math>F[\sum_{-\infty}^{\infty}\alpha_n e^{j2\pi nt/T}]</math>
+|<math>=\int_{-\infty}^{\infty} \left (\sum_{-\infty}^{\infty}\alpha_n e^{j2\pi nt/T} \right )e^{-j \omega t}dt</math>
+|-
+|
+|<math>=\sum_{-\infty}^{\infty}\alpha_n \left (\int_{-\infty}^{\infty} e^{j2\pi nt/T} e^{-j2\pi ft}dt\right )</math>
+|-
+|
+||<math>=\sum_{-\infty}^{\infty}\alpha_n \left (\int_{-\infty}^{\infty} e^{j2\pi t (\frac{n}{T}-f)} dt\right )</math>
 |}
+*Think something went wrong here.