HW 05 - Revision history

Fonggr at 04:27, 24 November 2008

2008-11-24T04:27:46Z

← Older revision		Revision as of 21:27, 23 November 2008
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	Find the following Fourier Transforms		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==		==Solutions==

Fonggr: /* Solutions */

2008-11-24T04:27:11Z

Solutions

← Older revision		Revision as of 21:27, 23 November 2008
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	{\| border="0" cellpadding="0" cellspacing="0"		{\| 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>		\|<math>=\int_{-\infty}^{\infty} e^{j \omega_0 t} e^{-j \omega t}dt</math>
	\|-		\|-
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	\|<math>=2\pi \delta(\omega_0-\omega)\,\!</math>		\|<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>=\int_{-\infty}^{\infty}\frac{e^{j\omega_0 t} + e^{-j\omega_0 t}}{2} e^{-j \omega t}dt</math>
	\|-		\|-
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	\|<math>=\pi\delta(\omega_0-\omega) + \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>=\int_{-\infty}^{\infty}\frac{e^{j\omega_0 t} - e^{-j\omega_0 t}}{2j} e^{-j \omega t}dt</math>
	\|-		\|-
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	\|<math>=-j\pi\delta(\omega_0-\omega) + j\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>		\|<math>=\int_{-\infty}^{\infty} \left (\sum_{-\infty}^{\infty}\alpha_n e^{j2\pi nt/T} \right )e^{-j \omega t}dt</math>
	\|-		\|-

Fonggr: /* Solutions */

2008-11-24T04:25:57Z

Solutions

← Older revision		Revision as of 21:25, 23 November 2008
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	\|-		\|-
	\|		\|
	\|<math>=\sum_{-\infty}^{\infty}\alpha_n \~~delta_{~~\frac{n}{T}-f~~} </math>~~		\|<math>=\sum_{-\infty}^{\infty}\alpha_n \delta\left(\frac{n}{T}-f\right) </math>
	\|-
	\|
	~~\|<math>=\alpha_{fT}\,~~\!</math>
	\|}		\|}
	~~Is the last problem done correctly?~~

Fonggr: /* Solutions */

2008-11-24T00:03:31Z

Solutions

← Older revision		Revision as of 17:03, 23 November 2008
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	\|-		\|-
	\|		\|
	\|<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[\sin{\omega_0 t}]\,\!</math>
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	\|-		\|-
	\|		\|
	\|<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[\sum_{-\infty}^{\infty}\alpha_n e^{j2\pi nt/T}]</math>

Fonggr: /* Solutions */

2008-11-22T09:15:11Z

Solutions

← Older revision		Revision as of 02:15, 22 November 2008
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	\|<math>=\alpha_{fT}\,\!</math>		\|<math>=\alpha_{fT}\,\!</math>
	\|}		\|}
			Is the last problem done correctly?

Fonggr: /* Solutions */

2008-11-22T00:11:32Z

Solutions

← Older revision		Revision as of 17:11, 21 November 2008
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	\|-		\|-
	\|		\|
	\|\|<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_{\frac{n}{T}-f} </math>
			\|-
			\|
			\|<math>=\alpha_{fT}\,\!</math>
	\|}		\|}
	*Think something went wrong here.

Fonggr at 07:43, 18 November 2008

2008-11-18T07:43:13Z

@@ Line 49: / Line 49: @@
+|
 |<math>=2\pi\delta(\omega_0-\omega) - 2\pi\delta(\omega_0+\omega)\,\!</math>
 |}

Fonggr: /* Solutions */

2008-11-18T07:37:10Z

Solutions

@@ Line 37: / Line 37: @@
 |<math>F[\sin{\omega_0 t}]\,\!</math>
 |<math>=\int_{-\infty}^{\infty}\frac{e^{j\omega_0 t} - e^{-j\omega_0 t}}{2j} e^{-j \omega t}dt</math>
 |}

Fonggr: /* Solutions */

2008-11-18T07:25:28Z

Solutions

← Older revision		Revision as of 00:25, 18 November 2008
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	\|-		\|-
	\|		\|
	\|<math>=\frac{1}{2}\int_{-\infty}^{\infty} 2e^{j(\omega_0-\omega) t} + 2e^{-j(\omega_0+\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>=\int_{-\infty}^{\infty} e^{j(\omega_0-\omega) t} + e^{-j(\omega_0+\omega) t}</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>=\delta(\omega_0-\omega) + \delta(\omega_0+\omega)\,\!</math>		\|<math>=2\pi\delta(\omega_0-\omega) + 2\pi\delta(\omega_0+\omega)\,\!</math>
	\|-		\|-
	\|<math>F[\sin{\omega_0 t}]\,\!</math>		\|<math>F[\sin{\omega_0 t}]\,\!</math>
	\|<math>=\int_{-\infty}^{\infty}\frac{e^{j\omega_0 t} - e^{-j\omega_0 t}}{2j} e^{-j \omega t}dt</math>		\|<math>=\int_{-\infty}^{\infty}\frac{e^{j\omega_0 t} - e^{-j\omega_0 t}}{2j} e^{-j \omega t}dt</math>
	\|}		\|}

Fonggr: /* Solutions */

2008-11-18T01:21:59Z

Solutions

← Older revision		Revision as of 18:21, 17 November 2008
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	\|-		\|-
	\|		\|
	\|<math>=\delta(\omega_0-\omega)\,\!</math>		\|<math>=2\pi\left [ \frac{1}{2\pi}\int_{-\infty}^{\infty} e^{j (\omega_0-\omega) t}dt \right ]</math>
			\|-
			\|
			\|<math>=2\pi \delta(\omega_0-\omega)\,\!</math>
	\|-		\|-
	\|<math>F[\cos {\omega_0 t}]\,\!</math>		\|<math>F[\cos {\omega_0 t}]\,\!</math>