FourierTransformsJW - Revision history

Wonoje: /* Fourier Transform */

2005-12-05T03:57:15Z

Fourier Transform

← Older revision		Revision as of 20:57, 4 December 2005
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	<math> = \frac{1}{2}X(f-f_o) + \frac{1}{2}X(f+f_o)</math>		<math> = \frac{1}{2}X(f-f_o) + \frac{1}{2}X(f+f_o)</math>

	~~=====Time Scaling=====~~
	<small>Principle author of this page: Jeffrey Wonoprbowo</small>		<small>Principle author of this page: Jeffrey Wonoprbowo</small>

2005-12-05T03:52:30Z

Modulation

← Older revision		Revision as of 20:52, 4 December 2005
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	=====Modulation=====		=====Modulation=====
	<math>\mathcal{F}[cos(2 \pi f_o t)x(t)] = \int_{\infty}^\infty x(t)cos(2 \pi f_o t) e^{j 2 \pi f t} dt</math>		<math>\mathcal{F}[cos(2 \pi f_o t)x(t)] = \int_{\infty}^\infty x(t)cos(2 \pi f_o t) e^{-j 2 \pi f t} dt</math>

			<math>= \int_{\infty}^\infty \frac{e^{j 2 \pi f_o t} + e^{-j 2 \pi f_o t}}{2} x(t) e^{j 2 \pi f t} dt </math>

			<math> = \frac{1}{2} \int_{\infty}^\infty x(t) e^{-j 2 \pi (f-f_o) t} dt + \frac{1}{2} \int_{\infty}^\infty x(t)e^{-j 2 \pi (f+f_o) t} dt</math>

			<math> = \frac{1}{2}X(f-f_o) + \frac{1}{2}X(f+f_o)</math>

	=====Time Scaling=====		=====Time Scaling=====
	<small>Principle author of this page: Jeffrey Wonoprbowo</small>		<small>Principle author of this page: Jeffrey Wonoprbowo</small>

2005-12-05T03:44:28Z

Fourier Transform

← Older revision		Revision as of 20:44, 4 December 2005
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	===Some Properties of the Fourier Transform===		===Some Properties of the Fourier Transform===
	~~Let~~ <math>x(t) = \mathcal{F}^{-1}[X(f)] = \int_{-\infty}^\infty X(f) e^{j 2 \pi f t} df</math>		Given: <math>x(t) = \mathcal{F}^{-1}[X(f)] = \int_{-\infty}^\infty X(f) e^{j 2 \pi f t} df</math>

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

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

			=====Time Scaling=====
	<small>Principle author of this page: Jeffrey Wonoprbowo</small>		<small>Principle author of this page: Jeffrey Wonoprbowo</small>

2005-12-05T03:37:20Z

Frequency Shift

← Older revision		Revision as of 20:37, 4 December 2005
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	=====Frequency Shift =====		=====Frequency Shift =====
	<math>X(f) = \mathcal{F}^{-1}[X(f)] = \int_{-\infty}^\infty X(f) e^{j 2 \pi f t}df </math		<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 </math>		<math>X(f-f_o) = \int_{-\infty}^\infty X(f) e^{j 2 \pi (f-f_o) t}df </math>

2005-12-05T03:36:50Z

Fourier Transform

← Older revision		Revision as of 20:36, 4 December 2005
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	=====Frequency Shift =====		=====Frequency Shift =====
			<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 </math>


	<small>Principle author of this page: Jeffrey Wonoprbowo</small>		<small>Principle author of this page: Jeffrey Wonoprbowo</small>

	~~</math>~~

2005-12-05T03:33:27Z

Fourier Transform

← Older revision		Revision as of 20:33, 4 December 2005
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	=====Differentiation=====		=====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>		<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 =====

	<small>Principle author of this page: Jeffrey Wonoprbowo</small>		<small>Principle author of this page: Jeffrey Wonoprbowo</small>

	</math>		</math>

2005-12-05T03:28:06Z

Fourier Transform

← Older revision		Revision as of 20:28, 4 December 2005
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	Let <math>x(t) = \mathcal{F}^{-1}[X(f)] = \int_{-\infty}^\infty X(f) e^{j 2 \pi f t} df</math>		Let <math>x(t) = \mathcal{F}^{-1}[X(f)] = \int_{-\infty}^\infty X(f) e^{j 2 \pi f t} df</math>

	====Differentiation====		=====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>\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>

	<small>Principle author of this page: Jeffrey Wonoprbowo</small>		<small>Principle author of this page: Jeffrey Wonoprbowo</small>

	</math>		</math>

2005-12-05T03:27:31Z

Fourier Transform

← Older revision		Revision as of 20:27, 4 December 2005
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	===Some Properties of the Fourier Transform===		===Some Properties of the Fourier Transform===
			Let <math>x(t) = \mathcal{F}^{-1}[X(f)] = \int_{-\infty}^\infty X(f) e^{j 2 \pi f t} df</math>

			====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)]

	<small>Principle author of this page: Jeffrey Wonoprbowo</small>		<small>Principle author of this page: Jeffrey Wonoprbowo</small>

			</math>

2005-12-05T03:21:55Z

Fourier Transform

← Older revision		Revision as of 20:21, 4 December 2005
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	where <math>x(t)</math> is a function of time and <math>X(f)</math> is the Fourier transform of <math>x(t)</math> and is it's Fourier Transform.		where <math>x(t)</math> is a function of time and <math>X(f)</math> is the Fourier transform of <math>x(t)</math> and is it's Fourier Transform.


			===Some Properties of the Fourier Transform===

	<small>Principle author of this page: Jeffrey Wonoprbowo</small>		<small>Principle author of this page: Jeffrey Wonoprbowo</small>

2005-12-05T03:16:05Z

Fourier Transform Defined

← Older revision		Revision as of 20:16, 4 December 2005
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	===Fourier Transform Defined===		===Fourier Transform Defined===
	The Fourier Transform can be defined as ~~follows~~:		The Fourier Transform of a function <math>x(t)</math> can be defined as:

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

	<math> X(f) = \mathcal{F}[x(t)] = \int_{-\infty}^\infty x(t) e^ {-j 2 \pi f t} dt</math>		then the inverse Fourier transform of a function <math>X(f)</math> will be:

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

	where <math>x(t)</math> is a function of time and <math>X(f)</math> is the Fourier transform of <math>x(t)</math> and is it's Fourier Transform.		where <math>x(t)</math> is a function of time and <math>X(f)</math> is the Fourier transform of <math>x(t)</math> and is it's Fourier Transform.

	<small>Principle author of this page: Jeffrey Wonoprbowo</small>		<small>Principle author of this page: Jeffrey Wonoprbowo</small>