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 $h (x) = (f * g) (x)$ , becomes $\hat{h} (ξ) = \hat{f} (ξ) \cdot \hat{g} (ξ) .$
Scaling (Chris Lau)	Given a, which is non-zero and real, and $h (x) = f (a x)$ , then $\hat{h} (ξ) = \frac{1}{\| a \|} \hat{f} (\frac{ξ}{a})$ . If a=−1, then the time-reversal property states: if $h (x) = f (- x)$ , then $\hat{h} (ξ) = \hat{f} (- ξ)$ .
Linearity (Victor Shepherd)	$ℱ {a x (t) + b y (t)} = a F {x (t)} + b F {y (t)}$

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-Scaling
-For a non-zero real number a, if h(x) = ƒ(ax), then  .     The case a = −1 leads to the time-reversal property, which states: if h(x) = ƒ(−x), then
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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>
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-| 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>
 |}