| Line 33: | Line 33: | ||
| 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 \frac{1-\cos(4 \pi t)}{2} dt \\ | + | |<math>\mathfrak{F}(c_1g(t) + c_2h(t) = \int_{-\infty}^\infty \frac{1-\cos(4 \pi t)}{2} dt + \int_{-\infty}^\infty \frac{1-\cos(4 \pi t)}{2} 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 20:16, 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 \frac{1-\cos(4 \pi t)}{2} dt + \int_{-\infty}^\infty \frac{1-\cos(4 \pi t)}{2} 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} $ |
