10/02 - Fourier Series - Revision history

Fonggr: /* Fourier Series (as compared to vectors) */

2008-12-05T06:12:10Z

Fourier Series (as compared to vectors)

← Older revision		Revision as of 23:12, 4 December 2008
Line 29:		Line 29:
	\|-		\|-
	\|		\|
	\|<math>=\sum_{n=-\infty}^\infty \alpha_n T \~~delta_{nm}~~ </math>		\|<math>=\sum_{n=-\infty}^\infty \alpha_n T \delta(n-m) </math>
	\|-		\|-
	\|		\|
	\|<math>=T\sum_{n=-\infty}^\infty \alpha_n \~~delta_{nm}~~ </math>		\|<math>=T\sum_{n=-\infty}^\infty \alpha_n \delta(n-m) </math>
	\|-		\|-
	\|		\|

Fonggr: 2/10/08 - Fourier Series moved to 10/02 - Fourier Series

2008-11-12T02:15:52Z

2/10/08 - Fourier Series moved to 10/02 - Fourier Series

← Older revision	Revision as of 19:15, 11 November 2008
(No difference)

Fonggr: /* Fourier Series (as compared to vectors) */

2008-11-11T23:30:03Z

Fourier Series (as compared to vectors)

@@ Line 9: / Line 9: @@
 Like vectors we can change to a new basis function by taking the inner product of <math>x(t)\,\!</math> with the mth basis function. Don't forget that the inner product of two vectors requires a complex conjugate. <math> \vec a \cdot \vec b = \sum_{i=1}^n a_i \cdot b_i^* </math>
-<math>\begin{matrix} x(t) \cdot e^{j2\pi mt/T} = \int_{-T/2}^{T/2} x(t) e^{-j2\pi mt/T}\, dt  \\ \ &
+{| border="0" cellpadding="0" cellspacing="0"
-= & \int_{-T/2}^{T/2}\left (\sum_{n=-\infty}^\infty \alpha_n e^{-j2\pi nt/T} \right ) e^{j2\pi mt/T}\, dt \\ \ &
+|-
-= & \sum_{n=-\infty}^\infty \alpha_n \int_{-T/2}^{T/2}e^{j2\pi nt/T}  e^{-j2\pi mt/T}\, dt \\ \ &
+|<math>x(t) \cdot e^{j2\pi mt/T}</math>
-= & \sum_{n=-\infty}^\infty \alpha_n \int_{-T/2}^{T/2}e^{j2\pi (n-m)t/T}\, dt \\ \end{matrix}</math>
+|<math>=\int_{-T/2}^{T/2} x(t) e^{-j2\pi mt/T}\, dt</math>
 *The integral range reflects the period T as defined at the top of the page. The current range carries over to the Fourier series better than going from 0 to T.
 *When switching the order of integrals and summations, you can "cavalierly" switch the order as long as there aren't summations/integrals to infinity.

Fonggr: /* Fourier Series (as compared to vectors) */

2008-11-11T22:33:36Z

Fourier Series (as compared to vectors)

← Older revision		Revision as of 15:33, 11 November 2008
Line 9:		Line 9:
	Like vectors we can change to a new basis function by taking the inner product of <math>x(t)\,\!</math> with the mth basis function. Don't forget that the inner product of two vectors requires a complex conjugate. <math> \vec a \cdot \vec b = \sum_{i=1}^n a_i \cdot b_i^* </math>		Like vectors we can change to a new basis function by taking the inner product of <math>x(t)\,\!</math> with the mth basis function. Don't forget that the inner product of two vectors requires a complex conjugate. <math> \vec a \cdot \vec b = \sum_{i=1}^n a_i \cdot b_i^* </math>

	:<math> x(t) \cdot e^{j2\pi mt/T} = \int_{-T/2}^{T/2} x(t) e^{-j2\pi mt/T}\, dt = \int_{-T/2}^{T/2}\left (\sum_{n=-\infty}^\infty \alpha_n e^{-j2\pi nt/T} \right ) e^{j2\pi mt/T}\, dt ~~</math>~~		<math>\begin{matrix} x(t) \cdot e^{j2\pi mt/T} = \int_{-T/2}^{T/2} x(t) e^{-j2\pi mt/T}\, dt \\ \ &
			= & \int_{-T/2}^{T/2}\left (\sum_{n=-\infty}^\infty \alpha_n e^{-j2\pi nt/T} \right ) e^{j2\pi mt/T}\, dt \\ \ &
	~~:<math>~~ = \sum_{n=-\infty}^\infty \alpha_n \int_{-T/2}^{T/2}e^{j2\pi nt/T} e^{-j2\pi mt/T}\, dt = \sum_{n=-\infty}^\infty \alpha_n \int_{-T/2}^{T/2}e^{j2\pi (n-m)t/T}\, dt</math>		= & \sum_{n=-\infty}^\infty \alpha_n \int_{-T/2}^{T/2}e^{j2\pi nt/T} e^{-j2\pi mt/T}\, dt \\ \ &
			= & \sum_{n=-\infty}^\infty \alpha_n \int_{-T/2}^{T/2}e^{j2\pi (n-m)t/T}\, dt \\ \end{matrix}</math>

	*The integral range reflects the period T as defined at the top of the page. The current range carries over to the Fourier series better than going from 0 to T.		*The integral range reflects the period T as defined at the top of the page. The current range carries over to the Fourier series better than going from 0 to T.

	*When switching the order of integrals and summations, you can "cavalierly" switch the order as long as there aren't summations/integrals to infinity.		*When switching the order of integrals and summations, you can "cavalierly" switch the order as long as there aren't summations/integrals to infinity.

Fonggr: /* Fourier Series (as compared to vectors) */

2008-11-11T00:40:46Z

Fourier Series (as compared to vectors)

← Older revision		Revision as of 17:40, 10 November 2008
Line 9:		Line 9:
	Like vectors we can change to a new basis function by taking the inner product of <math>x(t)\,\!</math> with the mth basis function. Don't forget that the inner product of two vectors requires a complex conjugate. <math> \vec a \cdot \vec b = \sum_{i=1}^n a_i \cdot b_i^* </math>		Like vectors we can change to a new basis function by taking the inner product of <math>x(t)\,\!</math> with the mth basis function. Don't forget that the inner product of two vectors requires a complex conjugate. <math> \vec a \cdot \vec b = \sum_{i=1}^n a_i \cdot b_i^* </math>

	<math> x(t) \cdot e^{j2\pi mt/T} = \int_{-T/2}^{T/2} x(t) e^{-j2\pi mt/T}\, dt = \int_{-T/2}^{T/2}\left (\sum_{n=-\infty}^\infty \alpha_n e^{-j2\pi nt/T} \right ) e^{j2\pi mt/T}\, dt = \sum_{n=-\infty}^\infty \alpha_n \int_{-T/2}^{T/2}e^{j2\pi nt/T} e^{-j2\pi mt/T}\, dt = \sum_{n=-\infty}^\infty \alpha_n \int_{-T/2}^{T/2}e^{j2\pi (n-m)t/T}\, dt</math>		:<math> x(t) \cdot e^{j2\pi mt/T} = \int_{-T/2}^{T/2} x(t) e^{-j2\pi mt/T}\, dt = \int_{-T/2}^{T/2}\left (\sum_{n=-\infty}^\infty \alpha_n e^{-j2\pi nt/T} \right ) e^{j2\pi mt/T}\, dt </math>

			:<math> = \sum_{n=-\infty}^\infty \alpha_n \int_{-T/2}^{T/2}e^{j2\pi nt/T} e^{-j2\pi mt/T}\, dt = \sum_{n=-\infty}^\infty \alpha_n \int_{-T/2}^{T/2}e^{j2\pi (n-m)t/T}\, dt</math>

	*The integral range reflects the period T as defined at the top of the page. The current range carries over to the Fourier series better than going from 0 to T.		*The integral range reflects the period T as defined at the top of the page. The current range carries over to the Fourier series better than going from 0 to T.

	*When switching the order of integrals and summations, you can "cavalierly" switch the order as long as there aren't summations/integrals to infinity.		*When switching the order of integrals and summations, you can "cavalierly" switch the order as long as there aren't summations/integrals to infinity.

Fonggr: /* Fourier Series (as compared to vectors) */

2008-11-11T00:39:43Z

Fourier Series (as compared to vectors)

← Older revision		Revision as of 17:39, 10 November 2008
Line 9:		Line 9:
	Like vectors we can change to a new basis function by taking the inner product of <math>x(t)\,\!</math> with the mth basis function. Don't forget that the inner product of two vectors requires a complex conjugate. <math> \vec a \cdot \vec b = \sum_{i=1}^n a_i \cdot b_i^* </math>		Like vectors we can change to a new basis function by taking the inner product of <math>x(t)\,\!</math> with the mth basis function. Don't forget that the inner product of two vectors requires a complex conjugate. <math> \vec a \cdot \vec b = \sum_{i=1}^n a_i \cdot b_i^* </math>

	<math> x(t) \cdot e^{j2\pi mt/T} = \int_{-T/2}^{T/2} x(t) e^{j2\pi mt/T}\, dt = \int_{-T/2}^{T/2}\left (\sum_{n=-\infty}^\infty \alpha_n e^{j2\pi nt/T} \right ) e^{j2\pi mt/T}\, dt</math>		<math> x(t) \cdot e^{j2\pi mt/T} = \int_{-T/2}^{T/2} x(t) e^{-j2\pi mt/T}\, dt = \int_{-T/2}^{T/2}\left (\sum_{n=-\infty}^\infty \alpha_n e^{-j2\pi nt/T} \right ) e^{j2\pi mt/T}\, dt = \sum_{n=-\infty}^\infty \alpha_n \int_{-T/2}^{T/2}e^{j2\pi nt/T} e^{-j2\pi mt/T}\, dt = \sum_{n=-\infty}^\infty \alpha_n \int_{-T/2}^{T/2}e^{j2\pi (n-m)t/T}\, dt</math>

	*The integral range reflects the period T as defined at the top of the page. The current range carries over to the Fourier series better than going from 0 to T.		*The integral range reflects the period T as defined at the top of the page. The current range carries over to the Fourier series better than going from 0 to T.

	*When switching the order of integrals and summations, you can "cavalierly" switch the order as long as there aren't summations/integrals to infinity.		*When switching the order of integrals and summations, you can "cavalierly" switch the order as long as there aren't summations/integrals to infinity.

Fonggr: /* Fourier Series (as compared to vectors) */

2008-11-11T00:33:38Z

Fourier Series (as compared to vectors)

← Older revision		Revision as of 17:33, 10 November 2008
Line 7:		Line 7:
	**x(t) must be bounded		**x(t) must be bounded

	Like vectors we can change to a new basis function by taking the inner product of <math>x(t)\,\!</math> with the mth basis function.		Like vectors we can change to a new basis function by taking the inner product of <math>x(t)\,\!</math> with the mth basis function. Don't forget that the inner product of two vectors requires a complex conjugate. <math> \vec a \cdot \vec b = \sum_{i=1}^n a_i \cdot b_i^* </math>

			<math> x(t) \cdot e^{j2\pi mt/T} = \int_{-T/2}^{T/2} x(t) e^{j2\pi mt/T}\, dt = \int_{-T/2}^{T/2}\left (\sum_{n=-\infty}^\infty \alpha_n e^{j2\pi nt/T} \right ) e^{j2\pi mt/T}\, dt</math>

			*The integral range reflects the period T as defined at the top of the page. The current range carries over to the Fourier series better than going from 0 to T.

			*When switching the order of integrals and summations, you can "cavalierly" switch the order as long as there aren't summations/integrals to infinity.

Fonggr: /* Fourier Series (as compared to vectors) */

2008-11-11T00:13:50Z

Fourier Series (as compared to vectors)

← Older revision		Revision as of 17:13, 10 November 2008
Line 6:		Line 6:
	**x(t) must be absolutely integrable over a period		**x(t) must be absolutely integrable over a period
	**x(t) must be bounded		**x(t) must be bounded

			Like vectors we can change to a new basis function by taking the inner product of <math>x(t)\,\!</math> with the mth basis function.

Fonggr: New page: ==Fourier Series (as compared to vectors)== If a function is periodic, $x(t) = x(t+T) \,\!$ , and it meets the Dirichlet conditions, then we can write it as $x(t) = \sum_{...$

2008-11-09T20:34:03Z

New page: ==Fourier Series (as compared to vectors)== If a function is periodic, <math> x(t) = x(t+T) \,\!</math>, and it meets the Dirichlet conditions, then we can write it as <math> x(t) = \sum_{...

New page

==Fourier Series (as compared to vectors)==
If a function is periodic, <math> x(t) = x(t+T) \,\!</math>, and it meets the Dirichlet conditions, then we can write it as <math> x(t) = \sum_{n=-\infty}^\infty \alpha_n e^{j2\pi nt/T}</math>
*Dirichlet conditions
**x(t) must have a finite number of extrema in any given interval
**x(t) must have a finite number of discontinuities in any given interval
**x(t) must be absolutely integrable over a period
**x(t) must be bounded

10/02 - Fourier Series - Revision history

Fonggr: /* Fourier Series (as compared to vectors) */

Fonggr: 2/10/08 - Fourier Series moved to 10/02 - Fourier Series

Fonggr: /* Fourier Series (as compared to vectors) */

Fonggr: /* Fourier Series (as compared to vectors) */

Fonggr: /* Fourier Series (as compared to vectors) */

Fonggr: /* Fourier Series (as compared to vectors) */

Fonggr: /* Fourier Series (as compared to vectors) */

Fonggr: /* Fourier Series (as compared to vectors) */

Fonggr: New page: ==Fourier Series (as compared to vectors)== If a function is periodic, x(t) = x(t+T) \,\!, and it meets the Dirichlet conditions, then we can write it as x(t) = \sum_{...

Fonggr: New page: ==Fourier Series (as compared to vectors)== If a function is periodic, $x(t) = x(t+T) \,\!$ , and it meets the Dirichlet conditions, then we can write it as $x(t) = \sum_{...$