HW 06 - Revision history

Fonggr: /* Proof */

2008-12-03T23:38:42Z

Proof

← Older revision		Revision as of 16:38, 3 December 2008
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	\|-		\|-
	\|		\|
	\|<math>=\int_{-\infty}^{\infty}\,o(t)\,\left[\cos(2\,\pi\,f\,t)+j\,\sin(2\,\pi\,f\,t)\right]</math>		\|<math>=\int_{-\infty}^{\infty}\,o(t)\,\left[\cos(2\,\pi\,f\,t)+j\,\sin(2\,\pi\,f\,t)\right]\,dt</math>
	\|Euler's identity		\|Euler's identity
	\|-		\|-
	\|		\|
	\|<math>=\int_{-\infty}^{\infty}\,o(t)\,j\,\sin(2\,\pi\,f\,t)</math>		\|<math>=\int_{-\infty}^{\infty}\,o(t)\,j\,\sin(2\,\pi\,f\,t)\,dt</math>
	\|Even function integrates out over symmetric limits		\|Even function integrates out over symmetric limits
	\|-		\|-
	\|		\|
	\|<math>=\mbox{Im }e(t) \mbox{ and an Im }o(f)\,\!</math>		\|<math>=\int_{-\infty}^{\infty}\,\left[\mbox{Im }e(t) \mbox{ and an Im }o(f)\right]\,dt</math>
			\|-
			\|
			\|<math>=\mbox{Im }o(f)\,\!</math>
			\|Time integrates out
	\|}		\|}
	*The odd function of time has no component (ie. 0) of frequency. Thus it is an even function in frequency.		*The odd function of time has no component (ie. 0) of frequency. Thus it is an even function in frequency.

2008-12-03T23:26:08Z

Solution

← Older revision		Revision as of 16:26, 3 December 2008
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	\|}		\|}

	==Solution==		==Correct Solution==
	{\| border="0" cellpadding="0" cellspacing="0"		{\| border="0" cellpadding="0" cellspacing="0"
	\|-		\|-

2008-12-03T23:25:53Z

Proof

← Older revision		Revision as of 16:25, 3 December 2008
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	\|-		\|-
	\|		\|
	\|<math>=~~Imaginary Even function of Time & Imaginary Odd function of Frequency~~ </math>		\|<math>=\mbox{Im }e(t) \mbox{ and an Im }o(f)\,\!</math>
	\|
	\|}		\|}
	*The odd function of time has no component (ie. 0) of frequency. Thus it is an even function in frequency.		*The odd function of time has no component (ie. 0) of frequency. Thus it is an even function in frequency.

	===Functions===		===Functions===
	EvenEven=Even		EvenEven=Even

2008-12-03T23:21:17Z

Background

@@ Line 3: / Line 3: @@
 ==Background==
-This is the incorrect solution derived in class. Cosine is incorrect, because an odd function of time, <math>\sgn(t)\,\!</math>,should map to
+This is the incorrect solution derived in class. Cosine is incorrect, because a real odd function of time, <math>\sgn(t)\,\!</math>,should map to an imaginary odd function of frequency.
 ==Incorrect Solution derived in class==

2008-12-03T23:12:04Z

Solution

← Older revision		Revision as of 16:12, 3 December 2008
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	This is the incorrect solution derived in class. Cosine is incorrect, because an odd function of time, <math>\sgn(t)\,\!</math>,should map to		This is the incorrect solution derived in class. Cosine is incorrect, because an odd function of time, <math>\sgn(t)\,\!</math>,should map to

	==Solution==		==Incorrect Solution derived in class==

	{\| border="0" cellpadding="0" cellspacing="0"		{\| border="0" cellpadding="0" cellspacing="0"

2008-12-02T21:00:13Z

Problem

@@ Line 1: / Line 1: @@
 ==Problem==
 Figure out why <math>\int_{0}^{\infty} \cos(2\pi\,f\,u)\,du</math> seems to equal an imaginary odd function of frequency, but there is no j.
 ==Solution==

2008-12-02T20:53:10Z

← Older revision		Revision as of 13:53, 2 December 2008
Line 1:		Line 1:
			==Problem==
			Figure out why <math>\int_{0}^{\infty} \cos(2\pi\,f\,u)\,du</math> seems to equal an imaginary odd function of frequency, but there is no j.

			==Solution==
	{\| border="0" cellpadding="0" cellspacing="0"		{\| border="0" cellpadding="0" cellspacing="0"
	\|-		\|-