Revision as of 20:42, 22 April 2018

Table of CT Fourier Series Coefficients and Properties

Function	Fourier Series	Coefficients

}

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

@@ Line 38: / Line 38: @@
 |Time Shifting
 |<math>\mathfrak{F}(g(t - a)) = e^{-i2\pi fa}*G(f) </math>
-|<math>\int_{-\infty}^\infty g(t-a)e^{-2\pi ft}dt </math><br/>
+|<math>\mathfrak{F}(g(t - a)) = \int_{-\infty}^\infty g(t-a)e^{-2\pi ft}dt </math><br/>
-<math>\int_{-\infty}^\infty g(u)e^{-i2\pi f(u+a)}du </math><br/>
+<math>=\int_{-\infty}^\infty g(u)e^{-i2\pi f(u+a)}du </math><br/>
-<math>e^{-i2\pi fa}\int_{-\infty}^\infty g(u)e^{-i2\pi fu}du </math><br/>
+<math>=e^{-i2\pi fa}\int_{-\infty}^\infty g(u)e^{-i2\pi fu}du </math><br/>
-<math>e^{-i2\pi fa} G(f)</math><br/>
+<math>=e^{-i2\pi fa} G(f)</math><br/>
+|Time Scaling
+|<math>\mathfrak{F}(g(ct)) = |fraq{G(\fraq{f}{c}}{|c|} </math><br/>
+|<math>\mathfrak{F}(g(ct)) = \int_{-\infty}^\infty g(ct)e^{-i2\pi ft}dt </math><br/>
 |-
 }