HW 05: Difference between revisions

Latest revision as of 21:27, 23 November 2008

Find the following Fourier Transforms

$F\left[e^{j\omega _{0}t}\right]$
$F\left[\cos {\omega _{0}t}\right]\,\!$
$F\left[\sum _{-\infty }^{\infty }\alpha _{n}e^{j2\pi nt/T}\right]$
$F\left[\sin {\omega _{0}t}\right]\,\!$

Solutions

$F\left[e^{j\omega _{0}t}\right]$	$=\int _{-\infty }^{\infty }e^{j\omega _{0}t}e^{-j\omega t}dt$
	$=\int _{-\infty }^{\infty }e^{j(\omega _{0}-\omega )t}dt$
	$=2\pi \left[{\frac {1}{2\pi }}\int _{-\infty }^{\infty }e^{j(\omega _{0}-\omega )t}dt\right]$
	$=2\pi \delta (\omega _{0}-\omega )\,\!$
$F\left[\cos {\omega _{0}t}\right]\,\!$	$=\int _{-\infty }^{\infty }{\frac {e^{j\omega _{0}t}+e^{-j\omega _{0}t}}{2}}e^{-j\omega t}dt$
	$={\frac {1}{2}}\int _{-\infty }^{\infty }\left(e^{j\omega _{0}t}+e^{-j\omega _{0}t}\right)e^{-j\omega t}dt$
	$={\frac {1}{2}}\int _{-\infty }^{\infty }\left[e^{j(\omega _{0}-\omega )t}+e^{-j(\omega _{0}+\omega )t}\right]dt$
	$=\pi \left[{\frac {1}{2\pi }}\int _{-\infty }^{\infty }\left(e^{j(\omega _{0}-\omega )t}+e^{-j(\omega _{0}+\omega )t}\right)\,dt\right]$
	$=\pi \delta (\omega _{0}-\omega )+\pi \delta (\omega _{0}+\omega )\,\!$
$F\left[\sin {\omega _{0}t}\right]\,\!$	$=\int _{-\infty }^{\infty }{\frac {e^{j\omega _{0}t}-e^{-j\omega _{0}t}}{2j}}e^{-j\omega t}dt$
	$={\frac {1}{2j}}\int _{-\infty }^{\infty }\left(e^{j\omega _{0}t}-e^{-j\omega _{0}t}\right)e^{-j\omega t}dt$
	$={\frac {1}{2j}}\int _{-\infty }^{\infty }\left(e^{j(\omega _{0}-\omega )t}-e^{-j(\omega _{0}+\omega )t}\right)dt$
	$={\frac {\pi }{j}}\left[{\frac {1}{2\pi }}\int _{-\infty }^{\infty }\left(e^{j(\omega _{0}-\omega )t}-e^{-j(\omega _{0}+\omega )t}\right)\,dt\right]$
	$=-j\pi \delta (\omega _{0}-\omega )+j\pi \delta (\omega _{0}+\omega )\,\!$
$F\left[\sum _{-\infty }^{\infty }\alpha _{n}e^{j2\pi nt/T}\right]$	$=\int _{-\infty }^{\infty }\left(\sum _{-\infty }^{\infty }\alpha _{n}e^{j2\pi nt/T}\right)e^{-j\omega t}dt$
	$=\sum _{-\infty }^{\infty }\alpha _{n}\left(\int _{-\infty }^{\infty }e^{j2\pi nt/T}e^{-j2\pi ft}dt\right)$
	$=\sum _{-\infty }^{\infty }\alpha _{n}\left(\int _{-\infty }^{\infty }e^{j2\pi t({\frac {n}{T}}-f)}dt\right)$
	$=\sum _{-\infty }^{\infty }\alpha _{n}\delta \left({\frac {n}{T}}-f\right)$

@@ Line 1: / Line 1: @@
 Find the following Fourier Transforms
-*<math>F[e^{j \omega_0 t}]</math>
+*<math>F\left[e^{j \omega_0 t}\right]</math>
-*<math>F[\cos {\omega_0 t}]\,\!</math>
+*<math>F\left[\cos {\omega_0 t}\right]\,\!</math>
-*<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>
-*<math>F[\sin{\omega_0 t}]\,\!</math>
+*<math>F\left[\sin{\omega_0 t}\right]\,\!</math>
 ==Solutions==
 {| border="0" cellpadding="0" cellspacing="0"
 |-
-|<math>F[e^{j \omega_0 t}]</math>
+|<math>F\left[e^{j \omega_0 t}\right]</math>
 |<math>=\int_{-\infty}^{\infty} e^{j \omega_0 t} e^{-j \omega t}dt</math>
 |-
@@ Line 20: / Line 20: @@
 |<math>=2\pi \delta(\omega_0-\omega)\,\!</math>
 |-
-|<math>F[\cos {\omega_0 t}]\,\!</math>
+|<math>F\left[\cos {\omega_0 t}\right]\,\!</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}\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>
+|<math>F\left[\sin{\omega_0 t}\right]\,\!</math>
 |<math>=\int_{-\infty}^{\infty}\frac{e^{j\omega_0 t} - e^{-j\omega_0 t}}{2j} e^{-j \omega t}dt</math>
 |-
 |
-|<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>
+|<math>F\left[\sum_{-\infty}^{\infty}\alpha_n e^{j2\pi nt/T}\right]</math>
 |<math>=\int_{-\infty}^{\infty} \left (\sum_{-\infty}^{\infty}\alpha_n e^{j2\pi nt/T} \right )e^{-j \omega t}dt</math>
 |-
@@ Line 57: / Line 57: @@
 |-
 |
-||<math>=\sum_{-\infty}^{\infty}\alpha_n \left (\int_{-\infty}^{\infty} e^{j2\pi t (\frac{n}{T}-f)} dt\right )</math>
+|<math>=\sum_{-\infty}^{\infty}\alpha_n \left (\int_{-\infty}^{\infty} e^{j2\pi t (\frac{n}{T}-f)} dt\right )</math>
+|-
+|
+|<math>=\sum_{-\infty}^{\infty}\alpha_n \delta\left(\frac{n}{T}-f\right) </math>
 |}
-*Think something went wrong here.

HW 05: Difference between revisions

Latest revision as of 21:27, 23 November 2008

Solutions

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