10/3,6 - The Game - Revision history

Fonggr at 07:04, 14 November 2008

2008-11-14T07:04:18Z

← Older revision		Revision as of 00:04, 14 November 2008
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	\|-		\|-
	\|		\|
	\|<math>=\left \langle h \mid e^{j \omega_n u} \right \rangle e^{j \omega_n t}</math>		\|<math>=\left \langle h(u) \mid e^{j \omega_n u} \right \rangle e^{j \omega_n t}</math>
	\|Different notation		\|Different notation
	\|-		\|-
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	\|-		\|-
	\|		\|
	\|<math></math>		\|<math>=\sum_{n=-\infty}^{\infty} \frac {1}{T} \left \langle x(t) \mid e^{j\omega_n t}\right \rangle
	\|		\left \langle h(u) \mid e^{j \omega_n u}\right \rangle e^{j \omega_n t}</math>
			\|Different notation
	\|}		\|}

Fonggr: /* Example 2 */

2008-11-14T06:40:04Z

Example 2

← Older revision		Revision as of 23:40, 13 November 2008
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	\|		\|
	\|<math>~~=n^2 + 2n + 1~~</math>		\|<math></math>
	\|		\|
	\|}		\|}

Fonggr: /* Example 2 */

2008-11-13T01:53:13Z

Example 2

← Older revision		Revision as of 18:53, 12 November 2008
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	==Example 2==		==Example 2==
	Let <math> x(t) = x(t+T)=\sum_{n=-\infty}^{\infty} \alpha_n e^{j2\pi nt/T} = \sum_{n=-\infty}^{\infty} \alpha_n e^{j \omega_n t/T}</math>		Let <math> x(t) = x(t+T)=\sum_{n=-\infty}^{\infty} \alpha_n e^{j2\pi nt/T} = \sum_{n=-\infty}^{\infty} \alpha_n e^{j \omega_n t}</math>

	{\| border="0" cellpadding="0" cellspacing="0"		{\| border="0" cellpadding="0" cellspacing="0"
	\|-		\|-
	\|<math>\sum_{n=-\infty}^{\infty} \alpha_n e^{j2\~~pi nt/T~~}</math>		\|<math>\sum_{n=-\infty}^{\infty} \alpha_n e^{j \omega_n t}</math>
	\|<math>=\sum_{n=-\infty}^{\infty} \alpha_n H(2\~~pi n/T~~)e^{j2\~~pi nt/T~~}</math>		\|<math>=\sum_{n=-\infty}^{\infty} \alpha_n H(\omega_n)e^{j \omega_n t}</math>
	\|From Example 1		\|From Example 1
	\|-		\|-

Fonggr: /* Example 2 */

2008-11-13T01:51:15Z

Example 2

← Older revision		Revision as of 18:51, 12 November 2008
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	==Example 2==		==Example 2==
	Let <math> x(t) = x(t+T)=\sum_{n=-\infty}^{\infty} \alpha_n e^{j2\pi nt/T} </math>		Let <math> x(t) = x(t+T)=\sum_{n=-\infty}^{\infty} \alpha_n e^{j2\pi nt/T} = \sum_{n=-\infty}^{\infty} \alpha_n e^{j \omega_n t/T}</math>

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

Fonggr: /* Example 1 */

2008-11-13T01:50:46Z

Example 1

← Older revision		Revision as of 18:50, 12 November 2008
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	==Example 1==		==Example 1==
	Let <math>x(t) = e^{j2\pi nt/T} = e^{~~j2\pi~~ \omega_n t}</math>		Let <math>x(t) = e^{j2\pi nt/T} = e^{j\omega_n t}</math>
	{\| border="0" cellpadding="0" cellspacing="0"		{\| border="0" cellpadding="0" cellspacing="0"
	\|-		\|-
	\|<math> e^{~~j2\pi~~ \omega_n t} </math>		\|<math> e^{j\omega_n t} </math>
	\|<math> = \int_{-\infty}^{\infty} e^{~~j2\pi~~ \omega_n \lambda} h(t-\lambda)\, d\lambda </math>		\|<math> = \int_{-\infty}^{\infty} e^{j \omega_n \lambda} h(t-\lambda)\, d\lambda </math>
	\|Let <math> t-\lambda = u \,\!</math> thus <math> du = -d\lambda \,\!</math>		\|Let <math> t-\lambda = u \,\!</math> thus <math> du = -d\lambda \,\!</math>
	\|-		\|-
	\|		\|
	\|<math>= -\int_{\infty}^{-\infty} e^{~~j2\pi~~ \omega_n (t-u)} h(u)\, du</math>		\|<math>= -\int_{\infty}^{-\infty} e^{j \omega_n (t-u)} h(u)\, du</math>
	\|The order of integration switched due to changing from <math>-\lambda = u\,\!</math>		\|The order of integration switched due to changing from <math>-\lambda = u\,\!</math>
	\|-		\|-
	\|		\|
	\|<math>=\underbrace{\left (\int_{-\infty}^{\infty} e^{-~~j2\pi~~ \omega_nu} h(u)\, du \right )}_{eigenvalue} \underbrace{e^{j2\pi \omega_nt}}_{eigenfunction}</math>		\|<math>=\underbrace{\left (\int_{-\infty}^{\infty} e^{-j \omega_nu} h(u)\, du \right )}_{eigenvalue} \underbrace{e^{j2\pi \omega_nt}}_{eigenfunction}</math>
	\|		\|
	\|-		\|-
	\|		\|
	\|<math>=\left \langle h \mid e^{~~j2\pi~~ \omega_n u} \right \rangle e^{~~j2\pi~~ \omega_n t}</math>		\|<math>=\left \langle h \mid e^{j \omega_n u} \right \rangle e^{j \omega_n t}</math>
	\|Different notation		\|Different notation
	\|-		\|-
	\|		\|
	\|<math>=H(\omega_n)e^{~~j2\pi~~ \omega_n t}</math>		\|<math>=H(\omega_n)e^{j \omega_n t}</math>
	\|Different notation		\|Different notation
	\|}		\|}

Fonggr at 01:39, 13 November 2008

2008-11-13T01:39:35Z

← Older revision		Revision as of 18:39, 12 November 2008
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	==Example 2==		==Example 2==
			Let <math> x(t) = x(t+T)=\sum_{n=-\infty}^{\infty} \alpha_n e^{j2\pi nt/T} </math>

			{\| border="0" cellpadding="0" cellspacing="0"
			\|-
			\|<math>\sum_{n=-\infty}^{\infty} \alpha_n e^{j2\pi nt/T}</math>
			\|<math>=\sum_{n=-\infty}^{\infty} \alpha_n H(2\pi n/T)e^{j2\pi nt/T}</math>
			\|From Example 1
			\|-
			\|
			\|<math>=n^2 + 2n + 1</math>
			\|
			\|}

	==Questions==		==Questions==
	*How do eigenfunction and basisfunctions differ?		*How do eigenfunction and basisfunctions differ?
	*Eigenfunctions will "point" in the same direction after going through the LTI system. It may (probably) have a different coefficient however. Very convenient.		*Eigenfunctions will "point" in the same direction after going through the LTI system. It may (probably) have a different coefficient however. Very convenient.

Fonggr: /* Questions */

2008-11-13T01:24:32Z

Questions

← Older revision		Revision as of 18:24, 12 November 2008
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	==Questions==		==Questions==
	*How do eigenfunction and basisfunctions differ?		*How do eigenfunction and basisfunctions differ?
	*Eigenfunctions will "point" in the same direction after going through the LTI system. It may (probably) have a different coefficient however.		*Eigenfunctions will "point" in the same direction after going through the LTI system. It may (probably) have a different coefficient however. Very convenient.

Fonggr: /* Questions */

2008-11-13T01:14:34Z

Questions

← Older revision		Revision as of 18:14, 12 November 2008
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	==Questions==		==Questions==
	*How do eigenfunction and basisfunctions differ?		*How do eigenfunction and basisfunctions differ?
	*~~How do eigenvalues and coefficients differ?~~		*Eigenfunctions will "point" in the same direction after going through the LTI system. It may (probably) have a different coefficient however.

Fonggr: /* Questions */

2008-11-13T00:40:37Z

Questions

← Older revision		Revision as of 17:40, 12 November 2008
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	==Questions==		==Questions==
	How do eigenfunction and basisfunctions differ?		*How do eigenfunction and basisfunctions differ?
	How do eigenvalues and coefficients differ?		*How do eigenvalues and coefficients differ?

Fonggr: /* Example 2 */

2008-11-13T00:40:26Z

Example 2

← Older revision		Revision as of 17:40, 12 November 2008
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	==Example 2==		==Example 2==

			==Questions==
			How do eigenfunction and basisfunctions differ?
			How do eigenvalues and coefficients differ?