Table of Fourier Transform Properties: Difference between revisions

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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>
| 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>
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| 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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Revision as of 12:46, 26 October 2010

Fourier Transform Properties
Property (contributor) Expanation
Convolution (Ben Henry) If , becomes  
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&thinsp; <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&thinsp; <math>\hat{h}(\xi)=\hat{f}(-\xi)} .