Revision as of 20:34, 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	$\mathfrak{F}(g(t - a)) = e^{-i2\pi fa}*G(f)$	$\int_{-\infty}^\infty g(t-a)e^{-2\pi ft}dt$ $\int_{-\infty}^\infty g(u)e^{-i2\pi f(u+a)} du$ $e^{-i2\pi fa}\int_{-\infty}^\infty g(u)e^{-i2\pi fu} du <\math><br/> <math>e^{-i2\pi fa}G(f)<\math><br/> \|- }$

@@ Line 40: / Line 40: @@
 |<math>\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>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/>
 |-
 }