Revision as of 20:22, 22 April 2018

Table of CT Fourier Series Coefficients and Properties

Function	Fourier Series	Coefficients

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Property Name	Property	Proof
Linearity	$\mathfrak{F}(c_1g(t) + c_2h(t) = c_1G(f) + c_2H(f)$	$\mathfrak{F}(c_1g(t) + c_2h(t) = \int_{-\infty}^\infty c_1g(t) dt + \int_{-\infty}^\infty c_2h(t) dt$ $c_1\int_{-\infty}^\infty g(t)e^{i2\pi ft} dt + c_2 \int_{-\infty}^\infty g(t)e^{i2\pi ft} dt$ <be/> $$ c_1G(f) + c_2H(f) $$

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 ! Proof
 |-
 | 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) = \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\pi ft} dt + c_2 \int_{-\infty}^\infty g(t)e^{i2\pi ft} dt
+<math>c_1\int_{-\infty}^\infty g(t)e^{i2\pi ft} dt + c_2 \int_{-\infty}^\infty g(t)e^{i2\pi ft} dt </math><be/>
+<math>c_1G(f) + c_2H(f)</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 />
-<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 />
 |-
 }