Table of Fourier Transform Properties - Revision history

Victor.shepherd at 05:45, 2 December 2010

2010-12-02T05:45:56Z

← Older revision		Revision as of 22:45, 1 December 2010
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	\| Scaling ([[Christopher Garrison Lau I\|Chris Lau]]) \|\| Given ''a'', which is non-zero and real, and <math>\ h(x)=f(ax) </math>, then <math>\hat{h}(\xi)=\frac{1}{\|a\|}\hat{f}\left(\frac{\xi}{a}\right)</math>. If ''a''=−1, then the time-reversal property states: if <math>\ h(x)=f(-x)</math>, then <math>\hat{h}(\xi)=\hat{f}(-\xi)</math>.		\| Scaling ([[Christopher Garrison Lau I\|Chris Lau]]) \|\| Given ''a'', which is non-zero and real, and <math>\ h(x)=f(ax) </math>, then <math>\hat{h}(\xi)=\frac{1}{\|a\|}\hat{f}\left(\frac{\xi}{a}\right)</math>. If ''a''=−1, then the time-reversal property states: if <math>\ h(x)=f(-x)</math>, then <math>\hat{h}(\xi)=\hat{f}(-\xi)</math>.
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			\| Linearity ([[Shepherd,Victor\|Victor Shepherd]]) \|\| <math>\mathcal{F}\{ax(t) + by(t)\} = a{F}\{x(t)\} + b{F}\{y(t)\}</math>
	\|}		\|}

Cdxskier at 19:49, 26 October 2010

2010-10-26T19:49:23Z

← Older revision		Revision as of 12:49, 26 October 2010
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	\| Convolution ([[Ben Henry]]) \|\| If <math>h(x)=\left(f*g\right)(x)</math>, becomes   <math> \hat{h}(\xi)=\hat{f}(\xi)\cdot \hat{g}(\xi).</math>		\| Convolution ([[Ben Henry]]) \|\| If <math>h(x)=\left(f*g\right)(x)</math>, becomes   <math> \hat{h}(\xi)=\hat{f}(\xi)\cdot \hat{g}(\xi).</math>
	\|-		\|-
	\| Scaling ([[Christopher Garrison Lau I\|Chris Lau]]) \|\| Given ''a'', which is non-zero and real, and <math>\ h(x)=f(ax) </math>, then <math>\hat{h}(\xi)=\frac{1}{\|a\|}\hat{f}\left(\frac{\xi}{a}\right)</math>. If ''a''=−1, then the time-reversal property states: if <math>\ h(x)=ƒ(−x)</math>, then <math>\hat{h}(\xi)=\hat{f}(-\xi)</math>.		\| Scaling ([[Christopher Garrison Lau I\|Chris Lau]]) \|\| Given ''a'', which is non-zero and real, and <math>\ h(x)=f(ax) </math>, then <math>\hat{h}(\xi)=\frac{1}{\|a\|}\hat{f}\left(\frac{\xi}{a}\right)</math>. If ''a''=−1, then the time-reversal property states: if <math>\ h(x)=f(-x)</math>, then <math>\hat{h}(\xi)=\hat{f}(-\xi)</math>.
	\|}		\|}

Cdxskier at 19:48, 26 October 2010

2010-10-26T19:48:42Z

← Older revision		Revision as of 12:48, 26 October 2010
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	\| Convolution ([[Ben Henry]]) \|\| If <math>h(x)=\left(f*g\right)(x)</math>, becomes   <math> \hat{h}(\xi)=\hat{f}(\xi)\cdot \hat{g}(\xi).</math>		\| Convolution ([[Ben Henry]]) \|\| If <math>h(x)=\left(f*g\right)(x)</math>, becomes   <math> \hat{h}(\xi)=\hat{f}(\xi)\cdot \hat{g}(\xi).</math>
	\|-		\|-
	\| Scaling ([[Christopher Garrison Lau I\|Chris Lau]]) \|\| Given ''a'', which is non-zero and real, and <math>\ h(x)=ƒ(ax) </math>, then <math>\hat{h}(\xi)=\frac{1}{\|a\|}\hat{f}\left(\frac{\xi}{a}\right)</math>. If ''a''=−1, then the time-reversal property states: if <math>\ h(x)=ƒ(−x)</math>, then <math>\hat{h}(\xi)=\hat{f}(-\xi)</math>.		\| Scaling ([[Christopher Garrison Lau I\|Chris Lau]]) \|\| Given ''a'', which is non-zero and real, and <math>\ h(x)=f(ax) </math>, then <math>\hat{h}(\xi)=\frac{1}{\|a\|}\hat{f}\left(\frac{\xi}{a}\right)</math>. If ''a''=−1, then the time-reversal property states: if <math>\ h(x)=ƒ(−x)</math>, then <math>\hat{h}(\xi)=\hat{f}(-\xi)</math>.
	\|}		\|}

Cdxskier at 19:48, 26 October 2010

2010-10-26T19:48:21Z

← Older revision		Revision as of 12:48, 26 October 2010
Line 5:		Line 5:
	\| Convolution ([[Ben Henry]]) \|\| If <math>h(x)=\left(f*g\right)(x)</math>, becomes   <math> \hat{h}(\xi)=\hat{f}(\xi)\cdot \hat{g}(\xi).</math>		\| Convolution ([[Ben Henry]]) \|\| If <math>h(x)=\left(f*g\right)(x)</math>, becomes   <math> \hat{h}(\xi)=\hat{f}(\xi)\cdot \hat{g}(\xi).</math>
	\|-		\|-
	\| Scaling ([[Christopher Garrison Lau I\|Chris Lau]]) \|\| Given ''a'', which is non-zero and real, and <math>\ h(x)=ƒ(ax) </math>, then <math>\hat{h}(\xi)=\frac{1}{\|a\|}\hat{f}\left(\frac{\xi}{a}\right)</math>. If ''a''=−1, then the time-reversal property states: if <math> \ h(x)=ƒ(−x), then~~ ~~ <math>\hat{h}(\xi)=\hat{f}(-\xi)</math>.		\| Scaling ([[Christopher Garrison Lau I\|Chris Lau]]) \|\| Given ''a'', which is non-zero and real, and <math>\ h(x)=ƒ(ax) </math>, then <math>\hat{h}(\xi)=\frac{1}{\|a\|}\hat{f}\left(\frac{\xi}{a}\right)</math>. If ''a''=−1, then the time-reversal property states: if <math>\ h(x)=ƒ(−x)</math>, then <math>\hat{h}(\xi)=\hat{f}(-\xi)</math>.
	\|}		\|}

Cdxskier at 19:47, 26 October 2010

2010-10-26T19:47:49Z

← Older revision		Revision as of 12:47, 26 October 2010
Line 5:		Line 5:
	\| Convolution ([[Ben Henry]]) \|\| If <math>h(x)=\left(f*g\right)(x)</math>, becomes   <math> \hat{h}(\xi)=\hat{f}(\xi)\cdot \hat{g}(\xi).</math>		\| Convolution ([[Ben Henry]]) \|\| If <math>h(x)=\left(f*g\right)(x)</math>, becomes   <math> \hat{h}(\xi)=\hat{f}(\xi)\cdot \hat{g}(\xi).</math>
	\|-		\|-
	\| Scaling ([[Christopher Garrison Lau I\|Chris Lau]]) \|\| Given ''a'', which is non-zero and real, and <math>\ h(x)=ƒ(ax), then <math>\hat{h}(\xi)=\frac{1}{\|a\|}\hat{f}\left(\frac{\xi}{a}\right)</math>. If ''a''~~ ~~=~~ ~~−1, then the time-reversal property states: if <math> \ h(x)=ƒ(−x), then  <math>\hat{h}(\xi)=\hat{f}(-\xi)</math>.		\| Scaling ([[Christopher Garrison Lau I\|Chris Lau]]) \|\| Given ''a'', which is non-zero and real, and <math>\ h(x)=ƒ(ax) </math>, then <math>\hat{h}(\xi)=\frac{1}{\|a\|}\hat{f}\left(\frac{\xi}{a}\right)</math>. If ''a''=−1, then the time-reversal property states: if <math> \ h(x)=ƒ(−x), then  <math>\hat{h}(\xi)=\hat{f}(-\xi)</math>.
	\|}		\|}

Cdxskier at 19:47, 26 October 2010

2010-10-26T19:47:20Z

← Older revision		Revision as of 12:47, 26 October 2010
Line 5:		Line 5:
	\| Convolution ([[Ben Henry]]) \|\| If <math>h(x)=\left(f*g\right)(x)</math>, becomes   <math> \hat{h}(\xi)=\hat{f}(\xi)\cdot \hat{g}(\xi).</math>		\| Convolution ([[Ben Henry]]) \|\| If <math>h(x)=\left(f*g\right)(x)</math>, becomes   <math> \hat{h}(\xi)=\hat{f}(\xi)\cdot \hat{g}(\xi).</math>
	\|-		\|-
	\| Scaling ([[Christopher Garrison Lau I\|Chris Lau]]) \|\| Given ''a'', which is non-zero and real, and <math> \ h(x)=ƒ(ax), then~~ ~~ <math>\hat{h}(\xi)=\frac{1}{\|a\|}\hat{f}\left(\frac{\xi}{a}\right)</math>.~~    ~~ If ''a'' = −1, then the time-reversal property states: if <math> \ h(x)=ƒ(−x), then  <math>\hat{h}(\xi)=\hat{f}(-\xi)</math>.		\| Scaling ([[Christopher Garrison Lau I\|Chris Lau]]) \|\| Given ''a'', which is non-zero and real, and <math>\ h(x)=ƒ(ax), then <math>\hat{h}(\xi)=\frac{1}{\|a\|}\hat{f}\left(\frac{\xi}{a}\right)</math>. If ''a'' = −1, then the time-reversal property states: if <math> \ h(x)=ƒ(−x), then  <math>\hat{h}(\xi)=\hat{f}(-\xi)</math>.
	\|}		\|}

Cdxskier at 19:46, 26 October 2010

2010-10-26T19:46:32Z

← Older revision		Revision as of 12:46, 26 October 2010
Line 5:		Line 5:
	\| Convolution ([[Ben Henry]]) \|\| If <math>h(x)=\left(f*g\right)(x)</math>, becomes   <math> \hat{h}(\xi)=\hat{f}(\xi)\cdot \hat{g}(\xi).</math>		\| Convolution ([[Ben Henry]]) \|\| If <math>h(x)=\left(f*g\right)(x)</math>, becomes   <math> \hat{h}(\xi)=\hat{f}(\xi)\cdot \hat{g}(\xi).</math>
	\|-		\|-
	\| Scaling ([[Christopher Garrison Lau I\|Chris Lau]]) \|\| Given ''a'', which is non-zero and real, and <math> / h(x)=ƒ(ax), then  <math>\hat{h}(\xi)=\frac{1}{\|a\|}\hat{f}\left(\frac{\xi}{a}\right)</math>.     If ''a'' = −1, then the time-reversal property states: if <math> / h(x)=ƒ(−x), then  <math>\hat{h}(\xi)=\hat{f}(-\xi)</math>.		\| Scaling ([[Christopher Garrison Lau I\|Chris Lau]]) \|\| Given ''a'', which is non-zero and real, and <math> \ h(x)=ƒ(ax), then  <math>\hat{h}(\xi)=\frac{1}{\|a\|}\hat{f}\left(\frac{\xi}{a}\right)</math>.     If ''a'' = −1, then the time-reversal property states: if <math> \ h(x)=ƒ(−x), then  <math>\hat{h}(\xi)=\hat{f}(-\xi)</math>.
	\|}		\|}

Cdxskier at 19:46, 26 October 2010

2010-10-26T19:46:13Z

← Older revision		Revision as of 12:46, 26 October 2010
Line 5:		Line 5:
	\| Convolution ([[Ben Henry]]) \|\| If <math>h(x)=\left(f*g\right)(x)</math>, becomes   <math> \hat{h}(\xi)=\hat{f}(\xi)\cdot \hat{g}(\xi).</math>		\| Convolution ([[Ben Henry]]) \|\| If <math>h(x)=\left(f*g\right)(x)</math>, becomes   <math> \hat{h}(\xi)=\hat{f}(\xi)\cdot \hat{g}(\xi).</math>
	\|-		\|-
	\| Scaling ([[Christopher Garrison Lau I\|Chris Lau]]) \|\| ~~For~~ a non-zero [[real ~~number]] ''a''~~, ~~if ''~~h''(''x'')~~ ~~=~~ ''~~ƒ''(''ax''), then  <math>\hat{h}(\xi)=\frac{1}{\|a\|}\hat{f}\left(\frac{\xi}{a}\right)</math>.     ~~The case~~ ''a'' = −1 ~~leads to~~ the ''time-reversal'' property~~, which~~ states: if ''h''(''x'')~~ ~~=~~ ''~~ƒ''(~~−''x''~~), then  <math>\hat{h}(\xi)=\hat{f}(-\xi)</math>.		\| Scaling ([[Christopher Garrison Lau I\|Chris Lau]]) \|\| Given ''a'', which is non-zero and real, and <math> / h(x)=ƒ(ax), then  <math>\hat{h}(\xi)=\frac{1}{\|a\|}\hat{f}\left(\frac{\xi}{a}\right)</math>.     If ''a'' = −1, then the time-reversal property states: if <math> / h(x)=ƒ(−x), then  <math>\hat{h}(\xi)=\hat{f}(-\xi)</math>.
	\|}		\|}

Cdxskier at 19:42, 26 October 2010

2010-10-26T19:42:19Z

← Older revision		Revision as of 12:42, 26 October 2010
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	~~Scaling~~
	~~For a non-zero real number a, if h(x) = ƒ(ax), then . The case a = −1 leads to the time-reversal property, which states: if h(x) = ƒ(−x), then~~

	{\| class="wikitable" border="1"		{\| class="wikitable" border="1"
	\|+ Fourier Transform Properties		\|+ Fourier Transform Properties

Cdxskier: Created page with ' Scaling For a non-zero real number a, if h(x) = ƒ(ax), then . The case a = −1 leads to the time-reversal property, which states: if h(x) = ƒ(−x), then {| class=…'

2010-10-26T19:42:06Z

Created page with ' Scaling For a non-zero real number a, if h(x) = ƒ(ax), then . The case a = −1 leads to the time-reversal property, which states: if h(x) = ƒ(−x), then {| class=…'

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Scaling
For a non-zero real number a, if h(x) = ƒ(ax), then . The case a = −1 leads to the time-reversal property, which states: if h(x) = ƒ(−x), then

{| class="wikitable" border="1"
|+ Fourier Transform Properties
! Property (contributor) !! Expanation
|-
| Convolution ([[Ben Henry]]) || If <math>h(x)=\left(f*g\right)(x)</math>, becomes   <math> \hat{h}(\xi)=\hat{f}(\xi)\cdot \hat{g}(\xi).</math>
|-
| Scaling ([[Christopher Garrison Lau I|Chris Lau]]) || For a non-zero [[real number]] ''a'', if ''h''(''x'') = ''ƒ''(''ax''), then  <math>\hat{h}(\xi)=\frac{1}{|a|}\hat{f}\left(\frac{\xi}{a}\right)</math>.     The case ''a'' = −1 leads to the ''time-reversal'' property, which states: if ''h''(''x'') = ''ƒ''(−''x''), then  <math>\hat{h}(\xi)=\hat{f}(-\xi)</math>.
|}