10/09 - Fourier Transform - Revision history

Fonggr: /* Definitions */

2008-12-04T20:49:19Z

Definitions

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	\|<math>=\left \langle x(t) \mid e^{j2\pi ft}\right \rangle_t</math>		\|<math>=\left \langle x(t) \mid e^{j2\pi ft}\right \rangle_t</math>
	\|-		\|-
	\|<math>F^{-1}[x(t)]\,\!</math>		\|<math>F^{-1}[X(f)]\,\!</math>
	\|<math>=x(t)\,\!</math>		\|<math>=x(t)\,\!</math>
	\|<math>=\int_{-\infty}^{\infty} X(f) e^{j2\pi ft}df</math>		\|<math>=\int_{-\infty}^{\infty} X(f) e^{j2\pi ft}df</math>
	\|<math>=\left \langle X(f) \mid e^{-j2\pi ft}\right \rangle_f</math>		\|<math>=\left \langle X(f) \mid e^{-j2\pi ft}\right \rangle_f</math>
	\|}		\|}

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

Fonggr: /* Fourier Transform */

2008-12-04T00:31:51Z

Fourier Transform

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	==Fourier Transform==		==Fourier Transform==
	Remember from [[10/02 - Fourier Series]]		Remember from [[10/02 - Fourier Series]]
	*<math> \~~alpha_m~~ = \frac{1}{T}\int_{-T/2}^{T/2} x(t) e^{-j2\pi mt/T}\, dt</math>		*<math> \alpha_n = \frac{1}{T}\int_{-T/2}^{T/2} x(t) e^{-j\,2\,\pi \,n\,t/T}\, dt</math>
	*<math>x(t) = x(t+T) = \sum_{n=-\infty}^\infty \~~alpha_m~~ e^{j2\pi m/T}</math>		*<math>x(t) = x(t+T) = \sum_{n=-\infty}^\infty \alpha_n e^{j\,2\pi \,n/T}</math>

	If we let <math> T \rightarrow \infty</math>		If we let <math> T \rightarrow \infty</math>

Fonggr: /* More properties of the delta function */

2008-11-18T01:44:01Z

More properties of the delta function

← Older revision		Revision as of 18:44, 17 November 2008
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	<math>\delta(\omega)=\delta(2\pi f)=\frac{1}{2\pi}\delta(f)</math>		<math>\delta(\omega)=\delta(2\pi f)=\frac{1}{2\pi}\delta(f)</math>

			{\| border="0" cellpadding="0" cellspacing="0"
			\|-
			\|<math>\int_{-\infty}^{\infty}\delta(a\,t)\,dt</math>
			\|<math>=\int_{-\infty}^{\infty}\delta(u\,t)\,\frac{du}{\left\|a\right\|}</math>
			\|Let <math>a\,t=u</math> and <math>du=a\,dt</math>
			\|-
			\|
			\|<math>=\frac{1}{\left\|a\right\|}</math>
			\|}

Fonggr: /* Sifting property of the delta function */

2008-11-18T01:37:26Z

Sifting property of the delta function

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	<math>x(t)=\int_{-\infty}^{\infty} x(u)\delta(t-u) du</math>		<math>x(t)=\int_{-\infty}^{\infty} x(u)\delta(t-u) du</math>
	*When dealing with <math>\omega\,\!</math>, it behaves slightly different than dealing with <math>f\,\!</math>. When dealing with <math>=\int_{-\infty}^{\infty}x(\lambda) \left [ \frac{1}{2\pi} \int_{-\infty}^{\infty} e^{j(t-\omega) \lambda}d\omega\right ] d\lambda</math>, note that the delta function is <math>\frac{1}{2\pi} \int_{-\infty}^{\infty} e^{j(t-\omega) \lambda}d\omega</math>. The <math>\frac{1}{2\pi}</math> is tacked onto the front. Thus, when dealing with <math>\omega\,\!</math>, you will often need to multiply it by <math>2\pi\,\!</math> to cancel out the <math>\frac{1}{2\pi}</math>.		*When dealing with <math>\omega\,\!</math>, it behaves slightly different than dealing with <math>f\,\!</math>. When dealing with <math>=\int_{-\infty}^{\infty}x(\lambda) \left [ \frac{1}{2\pi} \int_{-\infty}^{\infty} e^{j(t-\omega) \lambda}d\omega\right ] d\lambda</math>, note that the delta function is <math>\frac{1}{2\pi} \int_{-\infty}^{\infty} e^{j(t-\omega) \lambda}d\omega</math>. The <math>\frac{1}{2\pi}</math> is tacked onto the front. Thus, when dealing with <math>\omega\,\!</math>, you will often need to multiply it by <math>2\pi\,\!</math> to cancel out the <math>\frac{1}{2\pi}</math>.

			===More properties of the delta function===
			<math>\delta(a\,t) = \frac{1}{\left \| a \right \|}</math>

			<math>\delta(\omega)=\delta(2\pi f)=\frac{1}{2\pi}\delta(f)</math>

Fonggr at 01:20, 18 November 2008

2008-11-18T01:20:06Z

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	The dirac delta function is defined as any function, denoted as <math> \delta(t-u)\,\!</math>, that works for all variables that makes the following equation true:		The dirac delta function is defined as any function, denoted as <math> \delta(t-u)\,\!</math>, that works for all variables that makes the following equation true:
	<math>x(t)=\int_{-\infty}^{\infty} x(u)\delta(t-u) du</math>		<math>x(t)=\int_{-\infty}^{\infty} x(u)\delta(t-u) du</math>
	*When dealing with <math>\omega\,\!</math>, it behaves slightly different than dealing with <math>f\,\!</math>.		*When dealing with <math>\omega\,\!</math>, it behaves slightly different than dealing with <math>f\,\!</math>. When dealing with <math>=\int_{-\infty}^{\infty}x(\lambda) \left [ \frac{1}{2\pi} \int_{-\infty}^{\infty} e^{j(t-\omega) \lambda}d\omega\right ] d\lambda</math>, note that the delta function is <math>\frac{1}{2\pi} \int_{-\infty}^{\infty} e^{j(t-\omega) \lambda}d\omega</math>. The <math>\frac{1}{2\pi}</math> is tacked onto the front. Thus, when dealing with <math>\omega\,\!</math>, you will often need to multiply it by <math>2\pi\,\!</math> to cancel out the <math>\frac{1}{2\pi}</math>.

Fonggr: /* Sifting property of the delta function */

2008-11-18T01:14:35Z

Sifting property of the delta function

← Older revision		Revision as of 18:14, 17 November 2008
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	===Sifting property of the delta function===		===Sifting property of the delta function===
			The dirac delta function is defined as any function, denoted as <math> \delta(t-u)\,\!</math>, that works for all variables that makes the following equation true:
			<math>x(t)=\int_{-\infty}^{\infty} x(u)\delta(t-u) du</math>
			*When dealing with <math>\omega\,\!</math>, it behaves slightly different than dealing with <math>f\,\!</math>.

Fonggr at 01:09, 18 November 2008

2008-11-18T01:09:05Z

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 |}
 ==Examples==
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 ===Sifting property of the delta function===

Fonggr: /* Examples */

2008-11-18T01:08:11Z

Examples

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	\|<math>=x(t)\,\!</math>		\|<math>=x(t)\,\!</math>
	\|}		\|}

			===Sifting property of the delta function===

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

Fonggr: /* Examples */

2008-11-18T00:53:53Z

Examples

← Older revision		Revision as of 17:53, 17 November 2008
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	\|<math>=\int_{-\infty}^{\infty}x(\lambda) \int_{-\infty}^{\infty}e^{j2\pi f(t-\lambda)} df d\lambda</math>		\|<math>=\int_{-\infty}^{\infty}x(\lambda) \int_{-\infty}^{\infty}e^{j2\pi f(t-\lambda)} df d\lambda</math>
	\|<math>=\int_{-\infty}^{\infty}x(\lambda) \delta(t-\lambda) d\lambda</math>		\|<math>=\int_{-\infty}^{\infty}x(\lambda) \delta(t-\lambda) d\lambda</math>
			\|<math>=x(t)\,\!</math>
			\|-
			\|
			\|<math>=\int_{-\infty}^{\infty}\left [ \int_{-\infty}^{\infty} x(\lambda) e^{-j\omega\lambda}d\lambda\right ]e^{j\omega t} \frac{1}{2\pi}d\omega </math>
			\|<math>=\int_{-\infty}^{\infty}x(\lambda) \left [ \frac{1}{2\pi} \int_{-\infty}^{\infty} e^{j(t-\omega) \lambda}d\omega\right ] d\lambda</math>
			\|<math>=\int_{-\infty}^{\infty}x(\lambda) \delta(t-\omega) d\lambda</math>
	\|<math>=x(t)\,\!</math>		\|<math>=x(t)\,\!</math>
	\|}		\|}

Fonggr: /* Examples */

2008-11-17T22:54:35Z

Examples

← Older revision		Revision as of 15:54, 17 November 2008
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	\|-		\|-
	\|<math>F^{-1}[F[x(t)]]\,\!</math>		\|<math>F^{-1}[F[x(t)]]\,\!</math>
	\|<math>=\int_{-\infty}^{\infty}\left [ \int_{-\infty}^{\infty} x(t) e^{-j2\pi ft}dt\right ]e^{j2\pi ft} df</math>		\|<math>=\int_{-\infty}^{\infty}\left [ \int_{-\infty}^{\infty} x(\lambda) e^{-j2\pi f\lambda}d\lambda\right ]e^{j2\pi ft} df</math>
			\|<math>=\int_{-\infty}^{\infty}X(f) e^{j2\pi ft} df</math>
			\|<math>=x(t)\,\!</math>
	\|-		\|-
	\|		\|
	~~\|<math>=\int_{-\infty}^{\infty}X(f) e^{j2\pi ft} df</math>~~
	\|-
	\|		\|
			\|<math>=\int_{-\infty}^{\infty}x(\lambda) \int_{-\infty}^{\infty}e^{j2\pi f(t-\lambda)} df d\lambda</math>
			\|<math>=\int_{-\infty}^{\infty}x(\lambda) \delta(t-\lambda) d\lambda</math>
	\|<math>=x(t)\,\!</math>		\|<math>=x(t)\,\!</math>
	\|}		\|}