Something interesting from class - HW2 - Revision history

Max.Woesner at 15:46, 8 October 2009

2009-10-08T15:46:47Z

← Older revision		Revision as of 08:46, 8 October 2009
Line 1:		Line 1:
	== Max Woesner ==		== Max Woesner ==
			[[Max Woesner\|Back to my Home Page]]

	=== Homework #2 - Something interesting from class ===		=== Homework #2 - Something interesting from class ===

Max.Woesner at 03:36, 7 October 2009

2009-10-07T03:36:57Z

← Older revision		Revision as of 20:36, 6 October 2009
Line 84:		Line 84:

	Let <math>\lambda\ = t-t_0</math>, so <math>t_0 = t-\lambda\ </math> and <math>dt_0 = -d\lambda\ </math><br>		Let <math>\lambda\ = t-t_0</math>, so <math>t_0 = t-\lambda\ </math> and <math>dt_0 = -d\lambda\ </math><br>
	Therefore <math>\int_{-\infty}^{\infty} e^{j2\pi ft_0}h(t-t_0)dt_0 = \int_{-\infty}^{\infty} h(\lambda)e^{j2\pi f(t-\lambda)}(-d\lambda) = e^{j2\pi ft}\int_{-\infty}^{\infty} h(\lambda)e^{-j2\pi f\lambda}d\lambda\!</math><br>		Therefore <math>\int_{-\infty}^{\infty} e^{j2\pi ft_0}h(t-t_0)dt_0 = \int_{+\infty}^{-\infty} h(\lambda)e^{j2\pi f(t-\lambda)}(-d\lambda) = e^{j2\pi ft}\int_{-\infty}^{\infty} h(\lambda)e^{-j2\pi f\lambda}d\lambda\!</math><br>
	This tells us that <math>e^{j2\pi ft}\!</math> is the eigenfunction and <math>\int_{-\infty}^{\infty} h(\lambda)e^{-j2\pi f\lambda}d\lambda\!</math> is the eigenvalue of <b>all linear time invariant systems.</b><br>		This tells us that <math>e^{j2\pi ft}\!</math> is the eigenfunction and <math>\int_{-\infty}^{\infty} h(\lambda)e^{-j2\pi f\lambda}d\lambda\!</math> is the eigenvalue of <b>all linear time invariant systems.</b><br>
	This amazing conclusion makes solving linear time invariant systems (the only systems we are really able to solve) so much simpler that we usually approximate real-world nonlinear problems as linear systems so we can solve them.<br>		This amazing conclusion makes solving linear time invariant systems (the only systems we are really able to solve) so much simpler that we usually approximate real-world nonlinear problems as linear systems so we can solve them.<br>

Max.Woesner: /* Homework #2 - Something interesting from class */

2009-10-07T03:29:48Z

Homework #2 - Something interesting from class

← Older revision		Revision as of 20:29, 6 October 2009
Line 4:		Line 4:
	=== Homework #2 - Something interesting from class ===		=== Homework #2 - Something interesting from class ===

	This ~~page~~ is ~~under development~~		<br>
			The <b>Linear Time Invariant System Game</b> can be used to help us understand the impulse response of a linear time invariant system.

			<table border=1>
			<tr>
			<td><b>Input<math>-----\longrightarrow\,\!</math></b></td>
			<td><b>Linear Time Invariant System</b></td>
			<td><b><math>\longrightarrow\,\!</math>Output</b></td>
			<td><b>Reason</b></td>
			</tr>
			<tr>
			<td><math>\delta (t) \!</math></td>
			<td><math>-------\longrightarrow\,\!</math></td>
			<td><math>h(t)\!</math></td>
			<td>Given</td>
			</tr>
			<tr>
			<td><math>\delta (t-t_0)\!</math></td>
			<td><math>-------\longrightarrow\,\!</math></td>
			<td><math>h(t-t_0)\!</math></td>
			<td>Time Invariance</td>
			</tr>
			<tr>
			<td><math>x(t_0)\delta (t-t_0)\!</math></td>
			<td><math>-------\longrightarrow\,\!</math></td>
			<td><math>x(t_0)h(t-t_0)\!</math></td>
			<td>Proportionality</td>
			<tr>
			<td><math>\int_{-\infty}^{\infty} x(t_0)\delta\ (t-t_0)dt_0\!</math></td>
			<td><math>-------\longrightarrow\,\!</math></td>
			<td><math>\int_{-\infty}^{\infty} x(t_0)h(t-t_0)dt_0\!</math></td>
			<td>Superposition</td>
			</tr>

			</table>
			<br>
			where <math>\int_{-\infty}^{\infty} x(t_0)\delta\ (t-t_0)dt_0 = x(t)\!</math> <b> for any <math> x(t)\!</math></b> and <math>\int_{-\infty}^{\infty} x(t_0)h(t-t_0)dt_0\!</math> is the <b> convolution integral.</b><br>
			We can expand the game further.

			<table border=1>
			<tr>
			<td><b>Input<math>-----\longrightarrow\,\!</math></b></td>
			<td><b>Linear Time Invariant System</b></td>
			<td><b><math>\longrightarrow\,\!</math>Output</b></td>
			<td><b>Reason</b></td>
			</tr>
			<tr>
			<td><math>\delta (t) \!</math></td>
			<td><math>-------\longrightarrow\,\!</math></td>
			<td><math>h(t)\!</math></td>
			<td>Given</td>
			</tr>
			<tr>
			<td><math>\delta (t-t_0)\!</math></td>
			<td><math>-------\longrightarrow\,\!</math></td>
			<td><math>h(t-t_0)\!</math></td>
			<td>Time Invariance</td>
			</tr>
			<tr>
			<td><math>x(t_0)\delta\ (t-t_0)\!</math></td>
			<td><math>-------\longrightarrow\,\!</math></td>
			<td><math>x(t_0)h(t-t_0)\!</math></td>
			<td>Proportionality</td>
			<tr>
			<td><math>x(t)\!</math></td>
			<td><math>-------\longrightarrow\,\!</math></td>
			<td><math>\int_{-\infty}^{\infty} x(t_0)h(t-t_0)dt_0\!</math></td>
			<td>Superposition</td>
			</tr>
			<tr>
			<td><math>e^{j2\pi ft}\!</math></td>
			<td><math>-------\longrightarrow\,\!</math></td>
			<td><math>\int_{-\infty}^{\infty} e^{j2\pi ft_0}h(t-t_0)dt_0\!</math></td>
			<td>Superposition</td>
			</tr>

			</table>
			<br>

			Let <math>\lambda\ = t-t_0</math>, so <math>t_0 = t-\lambda\ </math> and <math>dt_0 = -d\lambda\ </math><br>
			Therefore <math>\int_{-\infty}^{\infty} e^{j2\pi ft_0}h(t-t_0)dt_0 = \int_{-\infty}^{\infty} h(\lambda)e^{j2\pi f(t-\lambda)}(-d\lambda) = e^{j2\pi ft}\int_{-\infty}^{\infty} h(\lambda)e^{-j2\pi f\lambda}d\lambda\!</math><br>
			This tells us that <math>e^{j2\pi ft}\!</math> is the eigenfunction and <math>\int_{-\infty}^{\infty} h(\lambda)e^{-j2\pi f\lambda}d\lambda\!</math> is the eigenvalue of <b>all linear time invariant systems.</b><br>
			This amazing conclusion makes solving linear time invariant systems (the only systems we are really able to solve) so much simpler that we usually approximate real-world nonlinear problems as linear systems so we can solve them.<br>

Max.Woesner at 18:34, 5 October 2009

2009-10-05T18:34:32Z

← Older revision		Revision as of 11:34, 5 October 2009
Line 5:		Line 5:

	This page is under development		This page is under development

	~~==== The Linear Time Invariant System Game ====~~

Max.Woesner at 18:34, 5 October 2009

2009-10-05T18:34:21Z

← Older revision		Revision as of 11:34, 5 October 2009
Line 5:		Line 5:

	This page is under development		This page is under development

			==== The Linear Time Invariant System Game ====

Max.Woesner at 16:40, 5 October 2009

2009-10-05T16:40:13Z

← Older revision		Revision as of 09:40, 5 October 2009
Line 4:		Line 4:
	=== Homework #2 - Something interesting from class ===		=== Homework #2 - Something interesting from class ===

	~~More to come~~		This page is under development

Max.Woesner: New page: == Max Woesner == === Homework #2 - Something interesting from class === More to come

2009-10-02T01:31:26Z

New page: == Max Woesner == === Homework #2 - Something interesting from class === More to come

New page

== Max Woesner ==

=== Homework #2 - Something interesting from class ===

More to come