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| Linearity   
 
| Linearity   
|<math>\mathfrak{F}{(c_1g(t) + c_2h(t)} = c_1G(f) + c_2H(f)</math>
+
|<math>\mathfrak{F}(c_1g(t) + c_2h(t) = c_1G(f) + c_2H(f)</math>
|<math>\mathfrak{F}(ax_{1}[n] + bx_{2}[n]) = \sum_{n=-\infty}^{\infty}[ax_{1}[n] + bx_{2}[n]]e^{-j\omega n}</math><br />
+
|<math>\mathfrak{F}(c_1g(t) + c_2h(t) = \int {-\infty}^\infty \c_1g(t)e^{-i2\pift} dt \\
 +
 
 +
<math>\mathfrak{F}(ax_{1}[n] + bx_{2}[n]) = \sum_{n=-\infty}^{\infty}[ax_{1}[n] + bx_{2}[n]]e^{-j\omega n}</math><br />
 
<math>\sum_{n=-\infty}^{\infty}ax_{1}[n]e^{-j\omega n} + \sum_{n=-\infty}^{\infty}bx_{2}[n]e^{-j\omega n}</math><br />
 
<math>\sum_{n=-\infty}^{\infty}ax_{1}[n]e^{-j\omega n} + \sum_{n=-\infty}^{\infty}bx_{2}[n]e^{-j\omega n}</math><br />
 
<math>=a\chi_{1}(\omega) + b\chi_{2}(\omega) </math> <br />________________________________<br />
 
<math>=a\chi_{1}(\omega) + b\chi_{2}(\omega) </math> <br />________________________________<br />
 
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|-
 
}
 
}

Revision as of 21:15, 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)e^{-i2\pift} dt \\ <math>\mathfrak{F}(ax_{1}[n] + bx_{2}[n]) = \sum_{n=-\infty}^{\infty}[ax_{1}[n] + bx_{2}[n]]e^{-j\omega n} $

$ \sum_{n=-\infty}^{\infty}ax_{1}[n]e^{-j\omega n} + \sum_{n=-\infty}^{\infty}bx_{2}[n]e^{-j\omega n} $
$ =a\chi_{1}(\omega) + b\chi_{2}(\omega) $
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