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| Linearity | | Linearity | ||
| − | |<math>\mathfrak{F} | + | |<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 20:15, 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} $ |
