Linear Time Invariant System - Revision history

Frohro: /* Eigenfunctions and Eigenvalues of an LTI Systems */

2010-01-07T05:40:48Z

Eigenfunctions and Eigenvalues of an LTI Systems

← Older revision		Revision as of 22:40, 6 January 2010
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	It is an interesting exercise to show that <math>e^{j\omega t}</math> are eigenfunctions of any LTI system. The eigenvalues are <math>H(\omega)</math>.		It is an interesting exercise to show that <math>e^{j\omega t}</math> are eigenfunctions of any LTI system. The eigenvalues are <math>H(\omega)</math>.


	{\| class="wikitable" border="1"		{\| class="wikitable" border="1"

Frohro: /* Eigenfunctions and Eigenvalues of an LTI Systems */

2010-01-07T05:40:20Z

Eigenfunctions and Eigenvalues of an LTI Systems

← Older revision		Revision as of 22:40, 6 January 2010
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	\| <math>e^{j\omega t} </math> \|\| <math>\int_{-\infty}^{\infty} e^{j \omega \lambda} h(t- \lambda ) d \lambda = \int_{-\infty}^{\infty} e^{j \omega (t- \lambda)} h(\lambda ) d \lambda = e^{j \omega t} \int_{-\infty}^{\infty} e^{-j \omega \lambda} h(\lambda ) d \lambda = e^{j \omega t} H(\omega)</math> \|\| Applying the line above to <math>e^{j \omega t}</math>		\| <math>e^{j\omega t} </math> \|\| <math>\int_{-\infty}^{\infty} e^{j \omega \lambda} h(t- \lambda ) d \lambda = \int_{-\infty}^{\infty} e^{j \omega (t- \lambda)} h(\lambda ) d \lambda = e^{j \omega t} \int_{-\infty}^{\infty} e^{-j \omega \lambda} h(\lambda ) d \lambda = e^{j \omega t} H(\omega)</math> \|\| Applying the line above to <math>e^{j \omega t}</math>
	\|}		\|}

			Note that the last line is obtained by doing a change of variables, then recognizing the Fourier Transform.

Frohro: /* Eigenfunctions and Eigenvalues of an LTI Systems */

2010-01-07T05:39:03Z

Eigenfunctions and Eigenvalues of an LTI Systems

← Older revision		Revision as of 22:39, 6 January 2010
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	\| <math>x(t) = \int_{-\infty}^{\infty} x(\lambda) \delta (t- \lambda ) d \lambda</math> \|\| <math>\int_{-\infty}^{\infty} x(\lambda) h(t- \lambda ) d \lambda = y(t)</math> \|\| Superposition		\| <math>x(t) = \int_{-\infty}^{\infty} x(\lambda) \delta (t- \lambda ) d \lambda</math> \|\| <math>\int_{-\infty}^{\infty} x(\lambda) h(t- \lambda ) d \lambda = y(t)</math> \|\| Superposition
	\|-		\|-
	\| <math>e^{j\omega t} </math> \|\| <math>\int_{-\infty}^{\infty} e^{j \omega \lambda} h(t- \lambda ) d \lambda = \int_{-\infty}^{\infty} e^{j \omega (t- \lambda)} h(\lambda ) d \lambda = e^{j \omega t} \int_{-\infty}^{\infty} e^{-j \omega \lambda} h(\lambda ) d \lambda = e^{j \omega t} H(w)</math>		\| <math>e^{j\omega t} </math> \|\| <math>\int_{-\infty}^{\infty} e^{j \omega \lambda} h(t- \lambda ) d \lambda = \int_{-\infty}^{\infty} e^{j \omega (t- \lambda)} h(\lambda ) d \lambda = e^{j \omega t} \int_{-\infty}^{\infty} e^{-j \omega \lambda} h(\lambda ) d \lambda = e^{j \omega t} H(\omega)</math> \|\| Applying the line above to <math>e^{j \omega t}</math>
	\|}		\|}

Frohro: /* Eigenfunctions and Eigenvalues of an LTI Systems */

2010-01-07T05:37:39Z

Eigenfunctions and Eigenvalues of an LTI Systems

← Older revision		Revision as of 22:37, 6 January 2010
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	===Eigenfunctions and Eigenvalues of an LTI Systems===		===Eigenfunctions and Eigenvalues of an LTI Systems===

	It is an interesting exercise to show that <math>e^{j\omega t}</math> are eigenfunctions of any LTI system. The eigenvalues are <math>\omega</math>.		It is an interesting exercise to show that <math>e^{j\omega t}</math> are eigenfunctions of any LTI system. The eigenvalues are <math>H(\omega)</math>.


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	\| <math>x(t) = \int_{-\infty}^{\infty} x(\lambda) \delta (t- \lambda ) d \lambda</math> \|\| <math>\int_{-\infty}^{\infty} x(\lambda) h(t- \lambda ) d \lambda = y(t)</math> \|\| Superposition		\| <math>x(t) = \int_{-\infty}^{\infty} x(\lambda) \delta (t- \lambda ) d \lambda</math> \|\| <math>\int_{-\infty}^{\infty} x(\lambda) h(t- \lambda ) d \lambda = y(t)</math> \|\| Superposition
	\|-		\|-
	\| <math>e^{j\omega t} </math> \|\| <math>\int_{-\infty}^{\infty} e^{j \omega \lambda} h(t- \lambda ) d \lambda = \int_{-\infty}^{\infty} e^{j \omega (t- \lambda)} h(\lambda ) d \lambda = e^{j \omega t} \int_{-\infty}^{\infty} e^{-j \omega \lambda} h(\lambda ) d \lambda </math>		\| <math>e^{j\omega t} </math> \|\| <math>\int_{-\infty}^{\infty} e^{j \omega \lambda} h(t- \lambda ) d \lambda = \int_{-\infty}^{\infty} e^{j \omega (t- \lambda)} h(\lambda ) d \lambda = e^{j \omega t} \int_{-\infty}^{\infty} e^{-j \omega \lambda} h(\lambda ) d \lambda = e^{j \omega t} H(w)</math>
	\|}		\|}

Frohro: /* Eigenfunctions and Eigenvalues of an LTI Systems */

2010-01-07T05:36:36Z

Eigenfunctions and Eigenvalues of an LTI Systems

← Older revision		Revision as of 22:36, 6 January 2010
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	\| <math>x(t) = \int_{-\infty}^{\infty} x(\lambda) \delta (t- \lambda ) d \lambda</math> \|\| <math>\int_{-\infty}^{\infty} x(\lambda) h(t- \lambda ) d \lambda = y(t)</math> \|\| Superposition		\| <math>x(t) = \int_{-\infty}^{\infty} x(\lambda) \delta (t- \lambda ) d \lambda</math> \|\| <math>\int_{-\infty}^{\infty} x(\lambda) h(t- \lambda ) d \lambda = y(t)</math> \|\| Superposition
	\|-		\|-
	\| <math>e^{j\omega t} </math> \|\| <math>\int_{-\infty}^{\infty} e^{j \omega \lambda} h(t- \lambda ) d \lambda = \int_{-\infty}^{\infty} e^{j \omega (t- \lambda)} h(\lambda ) d \lambda</math>		\| <math>e^{j\omega t} </math> \|\| <math>\int_{-\infty}^{\infty} e^{j \omega \lambda} h(t- \lambda ) d \lambda = \int_{-\infty}^{\infty} e^{j \omega (t- \lambda)} h(\lambda ) d \lambda = e^{j \omega t} \int_{-\infty}^{\infty} e^{-j \omega \lambda} h(\lambda ) d \lambda </math>
	\|}		\|}

Frohro: /* Eigenfunctions and Eigenvalues of an LTI Systems */

2010-01-07T05:35:17Z

Eigenfunctions and Eigenvalues of an LTI Systems

← Older revision		Revision as of 22:35, 6 January 2010
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	\| <math>x(t) = \int_{-\infty}^{\infty} x(\lambda) \delta (t- \lambda ) d \lambda</math> \|\| <math>\int_{-\infty}^{\infty} x(\lambda) h(t- \lambda ) d \lambda = y(t)</math> \|\| Superposition		\| <math>x(t) = \int_{-\infty}^{\infty} x(\lambda) \delta (t- \lambda ) d \lambda</math> \|\| <math>\int_{-\infty}^{\infty} x(\lambda) h(t- \lambda ) d \lambda = y(t)</math> \|\| Superposition
	\|-		\|-
	\| <math>e^{j\omega t} </math> \|\| <math>\int_{-\infty}^{\infty} e^{j \omega \lambda} h(t- \lambda ) d \lambda</math>		\| <math>e^{j\omega t} </math> \|\| <math>\int_{-\infty}^{\infty} e^{j \omega \lambda} h(t- \lambda ) d \lambda = \int_{-\infty}^{\infty} e^{j \omega (t- \lambda)} h(\lambda ) d \lambda</math>
	\|}		\|}

Frohro: /* Eigenfunctions and Eigenvalues of an LTI Systems */

2010-01-07T05:34:08Z

Eigenfunctions and Eigenvalues of an LTI Systems

← Older revision		Revision as of 22:34, 6 January 2010
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	\| <math>x(t) = \int_{-\infty}^{\infty} x(\lambda) \delta (t- \lambda ) d \lambda</math> \|\| <math>\int_{-\infty}^{\infty} x(\lambda) h(t- \lambda ) d \lambda = y(t)</math> \|\| Superposition		\| <math>x(t) = \int_{-\infty}^{\infty} x(\lambda) \delta (t- \lambda ) d \lambda</math> \|\| <math>\int_{-\infty}^{\infty} x(\lambda) h(t- \lambda ) d \lambda = y(t)</math> \|\| Superposition
	\|-		\|-
	\| <math>e^{j\omega t} </math> \|\| <math>\int_{-\infty}^{\infty} e^{j \~~omeag~~ \lambda} h(t- \lambda ) d \lambda</math>		\| <math>e^{j\omega t} </math> \|\| <math>\int_{-\infty}^{\infty} e^{j \omega \lambda} h(t- \lambda ) d \lambda</math>
	\|}		\|}

Frohro: /* Eigenfunctions and Eigenvalues of an LTI Systems */

2010-01-07T05:33:30Z

Eigenfunctions and Eigenvalues of an LTI Systems

← Older revision		Revision as of 22:33, 6 January 2010
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	\| <math>x(\lambda) \delta (t- \lambda )</math> \|\| <math>x(\lambda) h(t-\lambda )</math> \|\| Proportionality		\| <math>x(\lambda) \delta (t- \lambda )</math> \|\| <math>x(\lambda) h(t-\lambda )</math> \|\| Proportionality
	\|-		\|-
	\| <math>\int_{-\infty}^{\infty} x(\lambda) \delta (t- \lambda ) d \lambda</math> \|\| <math>\int_{-\infty}^{\infty} x(\lambda) h(t- \lambda ) d \lambda~~</math> \|\| Superposition~~		\| <math>x(t) = \int_{-\infty}^{\infty} x(\lambda) \delta (t- \lambda ) d \lambda</math> \|\| <math>\int_{-\infty}^{\infty} x(\lambda) h(t- \lambda ) d \lambda = y(t)</math> \|\| Superposition
	\|}

	~~<math>\int_-\infty^\infty x(\lambda) \delta~~ (t~~- \lambda~~ ) ~~d \lambda~~</math>

	{\| ~~class="wikitable" border="1"~~
	\|~~+ caption~~
	~~! heading !! heading~~
	\|-		\|-
	\| ~~cell~~ \|\| ~~cell~~		\| <math>e^{j\omega t} </math> \|\| <math>\int_{-\infty}^{\infty} e^{j \omeag \lambda} h(t- \lambda ) d \lambda</math>
	\|-
	~~\| cell \|\| cell~~
	\|}		\|}

Frohro: /* Eigenfunctions and Eigenvalues of an LTI Systems */

2010-01-07T05:30:11Z

Eigenfunctions and Eigenvalues of an LTI Systems

← Older revision		Revision as of 22:30, 6 January 2010
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	\| <math>x(\lambda) \delta (t- \lambda )</math> \|\| <math>x(\lambda) h(t-\lambda )</math> \|\| Proportionality		\| <math>x(\lambda) \delta (t- \lambda )</math> \|\| <math>x(\lambda) h(t-\lambda )</math> \|\| Proportionality
	\|-		\|-
	\| <math>\int_-\infty^\infty x(\lambda) \delta (t- \lambda ) d \lambda</math> \|\| <math>\int_-\infty^\infty x(\lambda) h(t- \lambda ) d \lambda</math> \|\| Superposition		\| <math>\int_{-\infty}^{\infty} x(\lambda) \delta (t- \lambda ) d \lambda</math> \|\| <math>\int_{-\infty}^{\infty} x(\lambda) h(t- \lambda ) d \lambda</math> \|\| Superposition
	\|}		\|}

Frohro: /* Eigenfunctions and Eigenvalues of an LTI Systems */

2010-01-07T05:28:13Z

Eigenfunctions and Eigenvalues of an LTI Systems

← Older revision		Revision as of 22:28, 6 January 2010
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	\|}		\|}

			<math>\int_-\infty^\infty x(\lambda) \delta (t- \lambda ) d \lambda</math>

	{\| class="wikitable" border="1"		{\| class="wikitable" border="1"