Exercise: Sawtooth Wave Fourier Transform - Revision history

Cdxskier at 04:37, 18 January 2010

2010-01-18T04:37:30Z

← Older revision		Revision as of 21:37, 17 January 2010
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	==Read By==		==Read By==
			[[Lau, Chris\|Christopher Garrison Lau I]]
	==Reviewed By==		==Reviewed By==
	Colby Fullerton		Colby Fullerton
	*[[Vier, Michael \| Michael Vier]]		*[[Vier, Michael \| Michael Vier]]

John.hawkins: /* Solution Graphs */

2010-01-15T11:22:53Z

Solution Graphs

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	[[Image:Error.jpg\|thumb\|800px\|center]]		[[Image:Error.jpg\|thumb\|800px\|center]]

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			[[Image:Sawtooth.gif\|thumb\|800px\|Solution as number of terms varies from 1 to 30.\|center]]

	==Author==		==Author==

Michaelvier at 18:33, 14 January 2010

2010-01-14T18:33:31Z

← Older revision		Revision as of 11:33, 14 January 2010
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	==Reviewed By==		==Reviewed By==
	Colby Fullerton		Colby Fullerton
			*[[Vier, Michael \| Michael Vier]]

John.hawkins: Excercise: Sawtooth Wave Fourier Transform moved to Exercise: Sawtooth Wave Fourier Transform: Misspelled title.

2010-01-13T08:38:04Z

Excercise: Sawtooth Wave Fourier Transform moved to Exercise: Sawtooth Wave Fourier Transform: Misspelled title.

← Older revision	Revision as of 01:38, 13 January 2010
(No difference)

Colby.fullerton: /* Reviewed By */

2010-01-13T05:02:22Z

Reviewed By

← Older revision		Revision as of 22:02, 12 January 2010
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	==Reviewed By==		==Reviewed By==
			Colby Fullerton

John.hawkins at 03:51, 13 January 2010

2010-01-13T03:51:20Z

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	<math>2\left[\left(\frac{1}{2\pi n}\sin 2\pi n-0\right)-\left[-\left(\frac{1}{2\pi n}\right)^2\cos 2\pi nt\right]_0^1\right]</math>		<math>=2\left[\left(\frac{1}{2\pi n}\sin 2\pi n-0\right)-\left[-\left(\frac{1}{2\pi n}\right)^2\cos 2\pi nt\right]_0^1\right]</math>

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John.hawkins at 02:57, 13 January 2010

2010-01-13T02:57:47Z

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	The figures below graph the first few iterations of ~~this~~ solution. The first graph shows the solution truncated after the first 100 terms of the infinite sum, as well as each of the contributing sine waves with offset. The second figure shows the function truncated after 1, 3, 5, 10, 50, and 100 terms. The last figure shows the Error between the Fourier Series truncated after the first 100 terms and the function itself. These figures were constructed using the following matlab code: [[SawToothFourier]].		==Solution Graphs==
			The figures below graph the first few iterations of the above solution. The first graph shows the solution truncated after the first 100 terms of the infinite sum, as well as each of the contributing sine waves with offset. The second figure shows the function truncated after 1, 3, 5, 10, 50, and 100 terms. The last figure shows the Error between the Fourier Series truncated after the first 100 terms and the function itself. These figures were constructed using the following matlab code: [[SawToothFourier]].

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John.hawkins at 00:56, 13 January 2010

2010-01-13T00:56:38Z

← Older revision		Revision as of 17:56, 12 January 2010
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	The figures below graph the first few iterations of this solution. The first graph shows the solution truncated after the first 100 terms of the infinite sum, as well as each of the contributing sine waves with offset. The second figure shows the function truncated after 1, 3, 5, 10, 50, and 100 terms. These figures were constructed using the following matlab code: [[SawToothFourier]].		The figures below graph the first few iterations of this solution. The first graph shows the solution truncated after the first 100 terms of the infinite sum, as well as each of the contributing sine waves with offset. The second figure shows the function truncated after 1, 3, 5, 10, 50, and 100 terms. The last figure shows the Error between the Fourier Series truncated after the first 100 terms and the function itself. These figures were constructed using the following matlab code: [[SawToothFourier]].

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	[[Image:First_n_Terms.jpg\|thumb\|800px\|center]]		[[Image:First_n_Terms.jpg\|thumb\|800px\|center]]

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			[[Image:Error.jpg\|thumb\|800px\|center]]

	==Author==		==Author==

John.hawkins at 21:40, 12 January 2010

2010-01-12T21:40:17Z

← Older revision		Revision as of 14:40, 12 January 2010
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	The figures below graph the first few iterations of this solution. The first graph shows the solution truncated after the first 100 terms of the infinite sum, as well as each of the contributing sine waves with offset. The second figure shows the function truncated after 1, 3, 5, 10, 50, and 100 terms.		The figures below graph the first few iterations of this solution. The first graph shows the solution truncated after the first 100 terms of the infinite sum, as well as each of the contributing sine waves with offset. The second figure shows the function truncated after 1, 3, 5, 10, 50, and 100 terms. These figures were constructed using the following matlab code: [[SawToothFourier]].

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John.hawkins at 20:35, 12 January 2010

2010-01-12T20:35:05Z

← Older revision		Revision as of 13:35, 12 January 2010
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	<center><math> x(t)=\sum^\infty_{n=0} \left[a_n\cos\frac{2\pi nt}{T}+b_n\sin\frac{2\pi nt}{T}\right]</math></center>		<center><math> x(t)=\frac{a_0}{2}+\sum^\infty_{n=1} \left[a_n\cos\frac{2\pi nt}{T}+b_n\sin\frac{2\pi nt}{T}\right]</math></center>

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	For the sawtooth function given, we note that <math>T=1</math>, and an obvious choice for <math>c</math> is 0 since this allows us to reduce the equation to <math>x(t)=t</math>. It remains, then, only to find the expression for <math>a_n</math> and <math>b_n</math>. We proceed first to find <math>b_n</math>. ~~For <math>b_n</math> we can ignore the case when <math>n=0</math> because <math>\sin\,\! 0=0</math>. Hence, we proceed for <math>n=1,2,3\dots</math>:~~		For the sawtooth function given, we note that <math>T=1</math>, and an obvious choice for <math>c</math> is 0 since this allows us to reduce the equation to <math>x(t)=t</math>. It remains, then, only to find the expression for <math>a_n</math> and <math>b_n</math>. We proceed first to find <math>b_n</math>.

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	</center>		</center>

	Now, for <math>a_n</math> we must consider the case when <math>n=0</math>.		Now, for <math>a_n</math> we must consider the case when <math>n=0</math> separately.

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	<math>a_0=\int_0^1t\ dt=~~\frac{~~t^2~~}{2}~~\bigg\|_0^1=~~\frac{~~1~~}{2}~~</math>		<math>a_0=\frac{2}{1}\int_0^1t\ dt=t^2\bigg\|_0^1=1</math>

	</center>		</center>