Line 39: Line 39:
 
|<math>\mathfrak{F}(g(t - a)) = e^{-i2\pi fa}*G(f) </math>
 
|<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>\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>\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/>

Revision as of 21:36, 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) $ $ \int_{-\infty}^\infty g(t-a)e^{-2\pi ft}dt $

$ \int_{-\infty}^\infty g(u)e^{-i2\pi f(u+a)}du $
$ \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/> |- } $

Alumni Liaison

has a message for current ECE438 students.

Sean Hu, ECE PhD 2009