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== Some Fourier series == | == Some Fourier series == | ||
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== Properties of CT Fourier systems == | == Properties of CT Fourier systems == | ||
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| − | | | + | | x(t), y(t) are periodic with period T |
| − | | | + | | <math>a_k</math> for x(t) and <math>b_k</math> for y(t) |
| + | |- | ||
| + | | Linearity | ||
| + | | <math>Ax(t)+By(t)</math> | ||
| + | | <math>Aa_k+Bb_k</math> | ||
| + | |- | ||
| + | | Time Shifting | ||
| + | | <math>x(t-t_0)</math> | ||
| + | | <math>e^{-j k \omega_0 t_0}a_k = e^{-j k \frac{2\pi}{T}t_0}a_k</math> | ||
| + | |- | ||
| + | | Frequency Shifting | ||
| + | | <math>e^{jM\omega_0t}x(t) = e^{jM\frac{2\pi}{T}t}x(t)</math> | ||
| + | | <math>a_k-M</math> | ||
| + | |- | ||
| + | | Conjugation | ||
| + | | <math>x^*(t)</math> | ||
| + | | <math>a^*_{(-k)}</math> | ||
| + | |- | ||
| + | | Time Reversal | ||
| + | | <math>x(-t)</math> | ||
| + | | <math>a_{(-k)}</math> | ||
| + | |- | ||
| + | | Time scaling | ||
| + | | <math>x(ct), c < 0,</math> periodic with period T/c | ||
| + | | <math>a_k</math> | ||
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Revision as of 09:40, 7 March 2011
Table of CT Fourier series coefficients and properties
- THIS PAGE IS STILL UNDER CONSTRUCTION
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 |
|---|---|---|
| x(t), y(t) are periodic with period T | $ a_k $ for x(t) and $ b_k $ for y(t) | |
| Linearity | $ Ax(t)+By(t) $ | $ Aa_k+Bb_k $ |
| Time Shifting | $ x(t-t_0) $ | $ e^{-j k \omega_0 t_0}a_k = e^{-j k \frac{2\pi}{T}t_0}a_k $ |
| Frequency Shifting | $ e^{jM\omega_0t}x(t) = e^{jM\frac{2\pi}{T}t}x(t) $ | $ a_k-M $ |
| Conjugation | $ x^*(t) $ | $ a^*_{(-k)} $ |
| Time Reversal | $ x(-t) $ | $ a_{(-k)} $ |
| Time scaling | $ x(ct), c < 0, $ periodic with period T/c | $ a_k $ |
