Table of Fourier Transform Properties: Difference between revisions

Latest revision as of 22:45, 1 December 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 $\ h(x)=f(ax)$ , then ${\hat {h}}(\xi )={\frac {1}{\|a\|}}{\hat {f}}\left({\frac {\xi }{a}}\right)$ . If a=−1, then the time-reversal property states: if $\ h(x)=f(-x)$ , then ${\hat {h}}(\xi )={\hat {f}}(-\xi )$ .
Linearity (Victor Shepherd)	${\mathcal {F}}\{ax(t)+by(t)\}=a{F}\{x(t)\}+b{F}\{y(t)\}$

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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]]) || 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)=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>.
+|-
+| Linearity ([[Shepherd,Victor|Victor Shepherd]]) || <math>\mathcal{F}\{ax(t) + by(t)\} = a{F}\{x(t)\} + b{F}\{y(t)\}</math>
 |}