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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}(c_1g(t) + c_2h(t) = \int_{-\infty}^\infty c_1G(f) dt + \int_{-\infty}^\infty c_2H(f) dt \\
+
|<math>\mathfrak{F}(c_1g(t) + c_2h(t) = \int_{-\infty}^\infty c_1g(t) dt + \int_{-\infty}^\infty c_2h(t) dt </math><br/>
 +
<math>c_1\int_{-\infty}^\infty g(t)e^{i2\pift} dt + c_2 \int_{-\infty}^\infty g(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>\mathfrak{F}(ax_{1}[n] + bx_{2}[n]) = \sum_{n=-\infty}^{\infty}[ax_{1}[n] + bx_{2}[n]]e^{-j\omega n}</math><br />

Revision as of 21:19, 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\pift} dt + c_2 \int_{-\infty}^\infty g(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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