The Fourier Transforms - Revision history

172.20.19.231: /* Shifting the function changes the phase of the spectrum */

2007-10-28T20:44:11Z

Shifting the function changes the phase of the spectrum

← Older revision		Revision as of 13:44, 28 October 2007
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	=== Shifting the function changes the phase of the spectrum ===		==== Shifting the function changes the phase of the spectrum ====

	<math>\mathcal{F}[x(t-a)] = X(t)e^{j2\pi f a}</math>		<math>\mathcal{F}[x(t-a)] = X(t)e^{j2\pi f a}</math>

2007-10-28T20:43:58Z

= Shifting the function changes the phase of the spectrum

← Older revision		Revision as of 13:43, 28 October 2007
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	==== Shifting the function changes the phase of the spectrum ===		=== Shifting the function changes the phase of the spectrum ===

	<math>\mathcal{F}[x(t-a)] = X(t)e^{j2\pi f a}</math>		<math>\mathcal{F}[x(t-a)] = X(t)e^{j2\pi f a}</math>

2007-10-12T18:04:23Z

Symmetries =

← Older revision		Revision as of 11:04, 12 October 2007
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	=== Symmetries ====		=== Symmetries ====
	'''		'''
	* if x(t) is real, then <math> X(-f) = F(t)^*</math>		- if x(t) is real, then <math> X(-f) = F(t)^*</math>

	* if x(t) is imaginary, then <math>X(-f) = -X(f)^*</math>		- if x(t) is imaginary, then <math>X(-f) = -X(f)^*</math>

	* if x(t) is even, then <math>X(-f) = X(f)$</math>		- if x(t) is even, then <math>X(-f) = X(f)$</math>

	* if x(t) is odd, then <math> X(-f) = -X(f)$.</math>'''		- if x(t) is odd, then <math> X(-f) = -X(f)$.</math>'''

2007-10-12T18:04:06Z

Symmetries =

← Older revision		Revision as of 11:04, 12 October 2007
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	=== Symmetries ====		=== Symmetries ====
	'''		'''
	* if x(t) is real, then<math> X(-f) = F(t)^*</math>		* if x(t) is real, then <math> X(-f) = F(t)^*</math>

	* if x(t) is imaginary, then <math>X(-f) = -X(f)^*</math>		* if x(t) is imaginary, then <math>X(-f) = -X(f)^*</math>
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	* if x(t) is even, then <math>X(-f) = X(f)$</math>		* if x(t) is even, then <math>X(-f) = X(f)$</math>

	* if x(t) is odd, then<math> X(-f) = -X(f)$.</math>'''		* if x(t) is odd, then <math> X(-f) = -X(f)$.</math>'''

2007-10-12T18:03:41Z

Symmetries =

← Older revision		Revision as of 11:03, 12 October 2007
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	=== Symmetries ====		=== Symmetries ====
			'''
	* if x(t) is real, then<math> X(-f) = F(t)^*</math>		* if x(t) is real, then<math> X(-f) = F(t)^*</math>

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	* if x(t) is even, then <math>X(-f) = X(f)$</math>		* if x(t) is even, then <math>X(-f) = X(f)$</math>

	* if x(t) is odd, then<math> X(-f) = -X(f)$.</math>		* if x(t) is odd, then<math> X(-f) = -X(f)$.</math>'''

2007-10-12T18:03:00Z

Symmetries =

← Older revision		Revision as of 11:03, 12 October 2007
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	* if x(t) is real, then<math> X(-f) = F(t)^*</math>		* if x(t) is real, then<math> X(-f) = F(t)^*</math>

	* if x(t) is imaginary, then <math>X(-f) = -X(f)^*</math>		* if x(t) is imaginary, then <math>X(-f) = -X(f)^*</math>

	* if x(t) is even, then <math>X(-f) = X(f)$</math>		* if x(t) is even, then <math>X(-f) = X(f)$</math>

	* if x(t) is odd, then<math> X(-f) = -X(f)$.</math>		* if x(t) is odd, then<math> X(-f) = -X(f)$.</math>

2007-10-12T18:01:35Z

Properties of a Fourier Transform:

← Older revision		Revision as of 11:01, 12 October 2007
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	=== Symmetries ====		=== Symmetries ====

	* if f(x) is real, then $F(-~~\omega~~) = F(~~\omega~~)^*$		* if x(t) is real, then<math> X(-f) = F(t)^*</math>
	* if f(x) is imaginary, then $F(-~~\omega~~) = -F(~~\omega~~)^*$		* if x(t) is imaginary, then <math>X(-f) = -X(f)^*</math>
	* if f(x) is even, then $F(-~~\omega~~) = F(~~\omega~~)$		* if x(t) is even, then <math>X(-f) = X(f)$</math>
	* if f(x) is odd, then $F(-~~\omega~~) = -F(~~\omega~~)$.		* if x(t) is odd, then<math> X(-f) = -X(f)$.</math>

2007-10-12T17:57:50Z

Properties of a Fourier Transform:

← Older revision		Revision as of 10:57, 12 October 2007
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	<math>\mathcal{F}[x(a*t)] = \frac{1}{a}X(\frac{f}{a})</math>		<math>\mathcal{F}[x(a*t)] = \frac{1}{a}X(\frac{f}{a})</math>

			=== Symmetries ====

			* if f(x) is real, then $F(-\omega) = F(\omega)^*$
			* if f(x) is imaginary, then $F(-\omega) = -F(\omega)^*$
			* if f(x) is even, then $F(-\omega) = F(\omega)$
			* if f(x) is odd, then $F(-\omega) = -F(\omega)$.

2007-10-12T17:56:33Z

Frequency and amplitude are affected when changing spatial scale inversely

← Older revision		Revision as of 10:56, 12 October 2007
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	<math>\mathcal{F}[x(a*t)] = \frac{1}{a}X(\frac{f}{a})</math>		<math>\mathcal{F}[x(a*t)] = \frac{1}{a}X(\frac{f}{a})</math>


	~~frac{x(t) - x(-t)}{2}</math>~~

2007-10-12T17:56:02Z

= Shifting the function changes the phase of the spectrum

← Older revision		Revision as of 10:56, 12 October 2007
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	==== Shifting the function changes the phase of the spectrum ===		==== Shifting the function changes the phase of the spectrum ===

	<math>\mathcal{F}[x(t-a)] = X(t)e^{-j2\pi f a}</math>		<math>\mathcal{F}[x(t-a)] = X(t)e^{j2\pi f a}</math>

	==== Frequency and amplitude are affected when changing spatial scale inversely ====		==== Frequency and amplitude are affected when changing spatial scale inversely ====