Table of Fourier Transform Properties: Difference between revisions

Revision as of 11:46, 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 (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 (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(−x), then  <math>\hat{h}(\xi)=\hat{f}(-\xi)} .

@@ Line 5: / Line 5: @@
 | 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]]) || For a non-zero [[real number]] ''a'', if ''h''(''x'')&nbsp;=&nbsp;''ƒ''(''ax''), then&thinsp; <math>\hat{h}(\xi)=\frac{1}{|a|}\hat{f}\left(\frac{\xi}{a}\right)</math>.&nbsp;&nbsp;&nbsp;&nbsp;  The case ''a''&nbsp;=&nbsp;−1 leads to the ''time-reversal'' property, which states: if ''h''(''x'')&nbsp;=&nbsp;''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>.
 |}