Table of Fourier Transform Properties: Difference between revisions

Revision as of 11:48, 26 October 2010

Fourier Transform Properties
Property (contributor)	Expanation
Convolution (Ben Henry)	If $h(x)=\left(f*g\right)(x)$ , becomes ${\hat {h}}(\xi )={\hat {f}}(\xi )\cdot {\hat {g}}(\xi ).$
Scaling (Chris Lau)	Given a, which is non-zero and real, and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ h(x)=ƒ(ax) } , then ${\hat {h}}(\xi )={\frac {1}{\|a\|}}{\hat {f}}\left({\frac {\xi }{a}}\right)$ . If a=−1, then the time-reversal property states: if Failed to parse (syntax error): {\displaystyle \ h(x)=ƒ(−x)} , then ${\hat {h}}(\xi )={\hat {f}}(-\xi )$ .

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 | Convolution ([[Ben Henry]]) || If <math>h(x)=\left(f*g\right)(x)</math>, becomes &thinsp; <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&thinsp; <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>.
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