| Line 10: | Line 10: | ||
|- | |- | ||
! Function | ! Function | ||
| − | + | ! Fourier Series | |
| − | ! Fourier Series | + | ! Coefficients |
|- | |- | ||
| <math>sin(w_0t)</math> | | <math>sin(w_0t)</math> | ||
| Line 20: | Line 20: | ||
| <math>\frac{1}{2}e^{jw_0t}+\frac{1}{2}e^{-jw_0t}</math> | | <math>\frac{1}{2}e^{jw_0t}+\frac{1}{2}e^{-jw_0t}</math> | ||
| <math>a_1=\frac{1}{2}, a_{-1}=\frac{1}{2}, a_k=0 \mbox{ for } k \ne 1,-1</math> | | <math>a_1=\frac{1}{2}, a_{-1}=\frac{1}{2}, a_k=0 \mbox{ for } k \ne 1,-1</math> | ||
| + | |- | ||
| + | | periodic square wave | ||
| + | <math>x(t)=\begin{cases} | ||
| + | 1, & \mbox{if }t<T_1 \\ | ||
| + | 0, & \mbox{if }T_1<t<T/2 | ||
| + | \end{cases}</math> | ||
| + | |||
| + | where T is the period and <math>2T_1</math> is the width of the pulse | ||
| + | | <math>\sum_{k=1}^N k^2 a_k e^{jk(\frac{2\pi}{T})t}</math> | ||
| + | (just the normal formula) | ||
| + | | <math>a_k = \frac{2sin(k\omega_0T_1)}{k\omega_0T_1}</math> | ||
|} | |} | ||
Revision as of 14:09, 16 February 2011
Table of CT Fourier series coefficients and properties
- THIS PAGE IS STILL UNDER CONSTRUCTION
- You may want to switch the titles of the second and third column. -pm
Some Fourier series
| Function | Fourier Series | Coefficients |
|---|---|---|
| $ sin(w_0t) $ | $ \frac{1}{2j}e^{jw_0t}-\frac{1}{2j}e^{-jw_0t} $ | $ a_1=\frac{1}{2j}, a_{-1}=\frac{-1}{2j}, a_k=0 \mbox{ for } k \ne 1,-1 $ |
| $ cos(w_0t) $ | $ \frac{1}{2}e^{jw_0t}+\frac{1}{2}e^{-jw_0t} $ | $ a_1=\frac{1}{2}, a_{-1}=\frac{1}{2}, a_k=0 \mbox{ for } k \ne 1,-1 $ |
| periodic square wave
$ x(t)=\begin{cases} 1, & \mbox{if }t<T_1 \\ 0, & \mbox{if }T_1<t<T/2 \end{cases} $ where T is the period and $ 2T_1 $ is the width of the pulse |
$ \sum_{k=1}^N k^2 a_k e^{jk(\frac{2\pi}{T})t} $
(just the normal formula) |
$ a_k = \frac{2sin(k\omega_0T_1)}{k\omega_0T_1} $ |
Properties of CT Fourier systems
| Property | Periodic Signal | Fourier Series Coefficients |
|---|---|---|
