Table of Fourier Transform Properties: Difference between revisions

Revision as of 12:46, 26 October 2010

Fourier Transform Properties
Property (contributor)	Expanation
Convolution (Ben Henry)	If $h (x) = (f * g) (x)$ , becomes $\hat{h} (ξ) = \hat{f} (ξ) \cdot \hat{g} (ξ) .$
Scaling (Chris Lau)	Given a, which is non-zero and real, and Failed to parse (syntax error): {\displaystyle \ h(x)=ƒ(ax), then  <math>\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  <math>\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), then&thinsp; <math>\hat{h}(\xi)=\frac{1}{|a|}\hat{f}\left(\frac{\xi}{a}\right)</math>.&nbsp;&nbsp;&nbsp;&nbsp;  If ''a''&nbsp;=&nbsp;−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), then&thinsp; <math>\hat{h}(\xi)=\frac{1}{|a|}\hat{f}\left(\frac{\xi}{a}\right)</math>.&nbsp;&nbsp;&nbsp;&nbsp;  If ''a''&nbsp;=&nbsp;−1, then the time-reversal property states: if <math> \ h(x)=ƒ(−x), then&thinsp; <math>\hat{h}(\xi)=\hat{f}(-\xi)</math>.
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