FT - Revision history

Harrde: /* Fourier Transform Derivation */

2007-11-13T01:33:48Z

Fourier Transform Derivation

← Older revision		Revision as of 18:33, 12 November 2007
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	<math> \sum _{n=-\infty}^\infty \alpha_n e^{j2 \pi nt/T} </math> -> <math> \int_{-\infty}^\infty \alpha_n e^{j2 \pi ft}df </math>		<math> \sum _{n=-\infty}^\infty \alpha_n e^{j2 \pi nt/T} </math> -> <math> \int_{-\infty}^\infty \alpha_n e^{j2 \pi ft}df </math>

	<math> \alpha_n </math> -> <math> X(f) = \int_{-\infty} ^{\infty} x(u) e^{-j2 \pi ft} dt </math>		<math> \alpha_n </math> -> <math> X(f) = \int_{-\infty} ^{\infty} x(t) e^{-j2 \pi ft} dt </math>

	Now we have:		Now we have:

2007-11-13T01:33:17Z

Fourier Transform Derivation

← Older revision		Revision as of 18:33, 12 November 2007
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	<math> \sum _{n=-\infty}^\infty \alpha_n e^{j2 \pi nt/T} </math> -> <math> \int_{-\infty}^\infty \alpha_n e^{j2 \pi ft}df </math>		<math> \sum _{n=-\infty}^\infty \alpha_n e^{j2 \pi nt/T} </math> -> <math> \int_{-\infty}^\infty \alpha_n e^{j2 \pi ft}df </math>

	<math> \alpha_n </math> -> <math> X(f) = \int_{-\infty} ^{\infty} x(u) e^{-j2 \pi ft} du </math>		<math> \alpha_n </math> -> <math> X(f) = \int_{-\infty} ^{\infty} x(u) e^{-j2 \pi ft} dt </math>

	Now we have:		Now we have:

2007-10-12T06:21:33Z

Fourier Transform Identities

← Older revision		Revision as of 23:21, 11 October 2007
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	<math> \int_{-\infty}^\infty X(f) e^{j2 \pi ft}df = x(t) = \mathcal{F}^{-1}[X(f)] </math>		<math> \int_{-\infty}^\infty X(f) e^{j2 \pi ft}df = x(t) = \mathcal{F}^{-1}[X(f)] </math>

	===Fourier Transform Identities===		===Some Fourier Transform Identities===

	=====Differentiation=====		=====Differentiation=====

2007-10-12T06:21:07Z

Fourier Transform Identities

← Older revision		Revision as of 23:21, 11 October 2007
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	<math> = \mathcal{F}^{-1}[e^{-j 2 \pi f t_o}X(f)] </math>		<math> = \mathcal{F}^{-1}[e^{-j 2 \pi f t_o}X(f)] </math>

	~~=====Frequency Shift =====~~
	~~Given: <math>X(f) = \mathcal{F}^{-1}[X(f)] = \int_{-\infty}^\infty X(f) e^{j 2 \pi f t}df </math>~~

	~~<math>X(f-f_o) = \int_{-\infty}^\infty X(f) e^{j 2 \pi (f-f_o) t}df = \mathcal{F}[e^{j 2 \pi f_o t}x(t)]</math>~~

2007-10-12T06:20:10Z

← Older revision		Revision as of 23:20, 11 October 2007
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	<math> \int_{-\infty}^\infty X(f) e^{j2 \pi ft}df = x(t) = \mathcal{F}^{-1}[X(f)] </math>		<math> \int_{-\infty}^\infty X(f) e^{j2 \pi ft}df = x(t) = \mathcal{F}^{-1}[X(f)] </math>

			===Fourier Transform Identities===

			=====Differentiation=====
			<math>\frac{dx}{dt} = \int_{-\infty}^\infty j 2 \pi X(f) e^{j 2 \pi f t} df = \mathcal{F}^{-1}[j 2 \pi f X(f)]</math>

			=====Time Shift=====
			<math>x(t-t_o) = \int_{-\infty}^\infty j 2 \pi X(f) e^{j 2 \pi f(t-t_o)} df</math>

			<math> = \int_{-\infty}^\infty e^{-j 2 \pi f t_o} X(f) e^{j 2 \pi f t} df</math>

			<math> = \mathcal{F}^{-1}[e^{-j 2 \pi f t_o}X(f)] </math>

			=====Frequency Shift =====
			Given: <math>X(f) = \mathcal{F}^{-1}[X(f)] = \int_{-\infty}^\infty X(f) e^{j 2 \pi f t}df </math>

			<math>X(f-f_o) = \int_{-\infty}^\infty X(f) e^{j 2 \pi (f-f_o) t}df = \mathcal{F}[e^{j 2 \pi f_o t}x(t)]</math>

2007-10-12T06:19:31Z

Fourier Transform

← Older revision		Revision as of 23:19, 11 October 2007
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	<math> \sum_{n=-\infty}^\infty \alpha_n e^{j2 \pi nt/T} </math>, where <math> \alpha_n = 1/T \int_{-T/2} ^{T/2} x(t) e^{-j2 \pi nt/T} dt </math>, and can be used to represent periodic functions.		<math> \sum_{n=-\infty}^\infty \alpha_n e^{j2 \pi nt/T} </math>, where <math> \alpha_n = 1/T \int_{-T/2} ^{T/2} x(t) e^{-j2 \pi nt/T} dt </math>, and can be used to represent periodic functions.

	===Fourier Transform===		===Fourier Transform Derivation===
	Now from the Fourier Series we want to derive the Fourier Transform, which is very similar except that it can be used for nonperiodic funtions. This can be done by simply letting the period <math> T </math> go to infinity.		Now from the Fourier Series we want to derive the Fourier Transform, which is very similar except that it can be used for nonperiodic funtions. This can be done by simply letting the period <math> T </math> go to infinity.

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	and		and

	<math> \int_{-\infty}^\infty X(f) e^{j2 \pi ft}df = \mathcal{F}^{-1}[X(f)] </math>		<math> \int_{-\infty}^\infty X(f) e^{j2 \pi ft}df = x(t) = \mathcal{F}^{-1}[X(f)] </math>

2007-10-12T06:17:10Z

Fourier Transform

← Older revision		Revision as of 23:17, 11 October 2007
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	Now we have:		Now we have:

	<math> x(t) = \int_{-\infty}^\infty (\int_{-\infty} ^{\infty} x(u) e^{-j2 \pi ft} du) e^{j2 \pi ft}df </math> -> <math> x(t) = \mathcal{F}^{-1}[X(t)] </math>		<math> x(t) = \int_{-\infty}^\infty (\int_{-\infty} ^{\infty} x(u) e^{-j2 \pi ft} du) e^{j2 \pi ft}df </math> -> <math> x(t) = \mathcal{F}^{-1}[X(f)] </math>

	Where <math> \int_{-\infty} ^{\infty} x(u) e^{-j2 \pi ft} du = X(t) = \mathcal{F}[x(t)] </math>		Where <math> \int_{-\infty} ^{\infty} x(u) e^{-j2 \pi ft} du = X(f) = \mathcal{F}[x(t)] </math>

	and		and

	<math> ~~x(t) =~~ \int_{-\infty}^\infty e^{j2 \pi ft}df </math>		<math> \int_{-\infty}^\infty X(f) e^{j2 \pi ft}df = \mathcal{F}^{-1}[X(f)] </math>

2007-10-12T06:13:24Z

Fourier Transform

← Older revision		Revision as of 23:13, 11 October 2007
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	Where <math> \int_{-\infty} ^{\infty} x(u) e^{-j2 \pi ft} du = X(t) = \mathcal{F}[x(t)] </math>		Where <math> \int_{-\infty} ^{\infty} x(u) e^{-j2 \pi ft} du = X(t) = \mathcal{F}[x(t)] </math>

			and

			<math> x(t) = \int_{-\infty}^\infty e^{j2 \pi ft}df </math>

2007-10-12T06:11:08Z

Fourier Transform

← Older revision		Revision as of 23:11, 11 October 2007
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	<math> x(t) = \int_{-\infty}^\infty (\int_{-\infty} ^{\infty} x(u) e^{-j2 \pi ft} du) e^{j2 \pi ft}df </math> -> <math> x(t) = \mathcal{F}^{-1}[X(t)] </math>		<math> x(t) = \int_{-\infty}^\infty (\int_{-\infty} ^{\infty} x(u) e^{-j2 \pi ft} du) e^{j2 \pi ft}df </math> -> <math> x(t) = \mathcal{F}^{-1}[X(t)] </math>

	Where <math> ~~X(t) = \mathcal{F}[x(t)] =~~ \int_{-\infty} ^{\infty} x(u) e^{-j2 \pi ft} du) e^{~~j2 \pi ft~~} </math>		Where <math> \int_{-\infty} ^{\infty} x(u) e^{-j2 \pi ft} du = X(t) = \mathcal{F}[x(t)] </math>

2007-10-12T06:09:48Z

Fourier Transform

← Older revision		Revision as of 23:09, 11 October 2007
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	<math> x(t) = \int_{-\infty}^\infty (\int_{-\infty} ^{\infty} x(u) e^{-j2 \pi ft} du) e^{j2 \pi ft}df </math> -> <math> x(t) = \mathcal{F}^{-1}[X(t)] </math>		<math> x(t) = \int_{-\infty}^\infty (\int_{-\infty} ^{\infty} x(u) e^{-j2 \pi ft} du) e^{j2 \pi ft}df </math> -> <math> x(t) = \mathcal{F}^{-1}[X(t)] </math>

	Where <math> X(t) = \mathcal{F}[x(t)] = \int_{-\infty} ^{\infty} x(u) e^{-j2 \pi ft} du) e^{j2 \pi ft </math>		Where <math> X(t) = \mathcal{F}[x(t)] = \int_{-\infty} ^{\infty} x(u) e^{-j2 \pi ft} du) e^{j2 \pi ft} </math>