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|Time Scaling  
 
|Time Scaling  
|<math>\mathfrak{F}(g(ct)) = \frac{G(\frac{f}{c}}{|c|} </math><br/>
+
|<math>\mathfrak{F}(g(ct)) = \frac{G(\frac{f}{c})}{|c|} </math><br/>
 
|<math>\mathfrak{F}(g(ct)) = \int_{-\infty}^\infty g(ct)e^{-i2\pi ft}dt </math><br/>
 
|<math>\mathfrak{F}(g(ct)) = \int_{-\infty}^\infty g(ct)e^{-i2\pi ft}dt </math><br/>
 +
subtitute : u = ct, du = cdt <br/>
 +
<math> \mathfrak{F}(g(ct)) = \int_{-c\infty}^{c\infty} \frac{g(u)}{c}e^{-i2\pi f\frac{u}{c}}du </math><br/>
 +
if c is greater than 0: then no signs change.
 +
if c is less than 0: the integration must be flipped as well as the negative from the c so you still get the same equation. therefore the absolute value of c is obtained.
 
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Revision as of 21:49, 22 April 2018


Table of CT Fourier Series Coefficients and Properties

Fourier series Coefficients

Function Fourier Series Coefficients


Properties of CT Fourier systems

}
Property Name Property Proof
Linearity $ \mathfrak{F}(c_1g(t) + c_2h(t) = c_1G(f) + c_2H(f) $ $ \mathfrak{F}(c_1g(t) + c_2h(t) = \int_{-\infty}^\infty c_1g(t) dt + \int_{-\infty}^\infty c_2h(t) dt $

$ =c_1\int_{-\infty}^\infty g(t)e^{i2\pi ft} dt + c_2 \int_{-\infty}^\infty g(t)e^{i2\pi ft} dt $
$ =c_1G(f) + c_2H(f) $

Time Shifting $ \mathfrak{F}(g(t - a)) = e^{-i2\pi fa}*G(f) $ $ \mathfrak{F}(g(t - a)) = \int_{-\infty}^\infty g(t-a)e^{-2\pi ft}dt $

$ =\int_{-\infty}^\infty g(u)e^{-i2\pi f(u+a)}du $
$ =e^{-i2\pi fa}\int_{-\infty}^\infty g(u)e^{-i2\pi fu}du $
$ =e^{-i2\pi fa} G(f) $

Time Scaling $ \mathfrak{F}(g(ct)) = \frac{G(\frac{f}{c})}{|c|} $
$ \mathfrak{F}(g(ct)) = \int_{-\infty}^\infty g(ct)e^{-i2\pi ft}dt $

subtitute : u = ct, du = cdt
$ \mathfrak{F}(g(ct)) = \int_{-c\infty}^{c\infty} \frac{g(u)}{c}e^{-i2\pi f\frac{u}{c}}du $
if c is greater than 0: then no signs change. if c is less than 0: the integration must be flipped as well as the negative from the c so you still get the same equation. therefore the absolute value of c is obtained.

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Basic linear algebra uncovers and clarifies very important geometry and algebra.

Dr. Paul Garrett