HW 05: Difference between revisions

Revision as of 17:03, 23 November 2008

Find the following Fourier Transforms

$F [e^{j ω_{0} t}]$
$F [\cos ω_{0} t]$
$F [\sum_{- \infty}^{\infty} α_{n} e^{j 2 π n t / T}]$
$F [\sin ω_{0} t]$

Solutions

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

Is the last problem done correctly?

@@ Line 24: / Line 24: @@
 |-
 |
-|<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}\left (e^{j\omega_0 t} + e^{-j\omega_0 t}\right )e^{-j \omega t} dt</math>
 |-
 |
-|<math>=\frac{1}{2}\int_{-\infty}^{\infty} \left [2e^{j(\omega_0-\omega) t} + 2e^{-j(\omega_0+\omega) t}\right ] dt</math>
+|<math>=\frac{1}{2}\int_{-\infty}^{\infty} \left [e^{j(\omega_0-\omega) t} + e^{-j(\omega_0+\omega) t}\right ] dt</math>
 |-
 |
-|<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>
+|<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>
 |-
 |
-|<math>=2\pi\delta(\omega_0-\omega) + 2\pi\delta(\omega_0+\omega)\,\!</math>
+|<math>=\pi\delta(\omega_0-\omega) + \pi\delta(\omega_0+\omega)\,\!</math>
 |-
 |<math>F[\sin{\omega_0 t}]\,\!</math>
@@ Line 39: / Line 39: @@
 |-
 |
-|<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>
+|<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>
 |-
 |
-|<math>=\int_{-\infty}^{\infty} \left (e^{j(\omega_0-\omega) t} - e^{-j(\omega_0+\omega) t}\right ) dt</math>
+|<math>=\frac{1}{2j}\int_{-\infty}^{\infty} \left (e^{j(\omega_0-\omega) t} - e^{-j(\omega_0+\omega) t}\right ) dt</math>
 |-
 |
-|<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>
+|<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>
 |-
 |
-|<math>=2\pi\delta(\omega_0-\omega) - 2\pi\delta(\omega_0+\omega)\,\!</math>
+|<math>=-j\pi\delta(\omega_0-\omega) + j\pi\delta(\omega_0+\omega)\,\!</math>
 |-
 |<math>F[\sum_{-\infty}^{\infty}\alpha_n e^{j2\pi nt/T}]</math>

HW 05: Difference between revisions

Revision as of 17:03, 23 November 2008

Solutions

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