| (3 intermediate revisions by one other user not shown) | |||
| Line 1: | Line 1: | ||
| − | + | [[Category:bonus point project]] | |
| − | + | = Table of CT Fourier Series Coefficients and Properties = | |
| − | |||
== Some Fourier series == | == Some Fourier series == | ||
| Line 10: | Line 9: | ||
|- | |- | ||
! Function | ! Function | ||
| − | + | ! Fourier Series | |
| − | ! Fourier Series | + | ! Coefficients |
|- | |- | ||
| <math>sin(w_0t)</math> | | <math>sin(w_0t)</math> | ||
| Line 20: | Line 19: | ||
| <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> | ||
|} | |} | ||
== Properties of CT Fourier systems == | == Properties of CT Fourier systems == | ||
| + | |||
| + | |||
{| border="1" class="wikitable" | {| border="1" class="wikitable" | ||
| Line 31: | Line 43: | ||
|- | |- | ||
| | | | ||
| − | | | + | | 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> | ||
| + | |- | ||
| + | | Multiplication | ||
| + | | <math>x(t)y(t)</math> | ||
| + | | <math>\sum_{l=-\infty}^\infty a_l b_{k-l}</math> | ||
| + | |- | ||
| + | | Differentiation | ||
| + | | <math>\frac{dx(t)}{dt}</math> | ||
| + | | <math>jk\omega_0a_k=jk\frac{2\pi}{T}a_k</math> | ||
| + | |- | ||
| + | | Real and Even Signals | ||
| + | | <math class="inline">x(t)</math> real and even | ||
| + | | <math class="inline">a_k</math> real and even | ||
| + | |- | ||
| + | | Real and Odd Signals | ||
| + | | <math class="inline">x(t)</math> real and odd | ||
| + | | <math class="inline">a_k</math> purely imaginary and odd | ||
|} | |} | ||
| + | |||
| + | |||
| + | == Parseval's Relation == | ||
| + | |||
| + | <math>\frac{1}{T}\int_T \Big| x(t) \Big| ^2 dt = \sum_{k=-\infty}^\infty \Big| a_k \Big| ^2</math> | ||
---- | ---- | ||
[[2011_Spring_ECE_301_Boutin|Back to ECE301 Spring 2011 Prof. Boutin]] | [[2011_Spring_ECE_301_Boutin|Back to ECE301 Spring 2011 Prof. Boutin]] | ||
| + | |||
| + | --[[User:Cmcmican|Cmcmican]] 19:02, 28 March 2011 (UTC) | ||
Latest revision as of 09:52, 6 May 2012
Contents
Table of CT Fourier Series Coefficients and Properties
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 $ |
| Multiplication | $ x(t)y(t) $ | $ \sum_{l=-\infty}^\infty a_l b_{k-l} $ |
| Differentiation | $ \frac{dx(t)}{dt} $ | $ jk\omega_0a_k=jk\frac{2\pi}{T}a_k $ |
| Real and Even Signals | $ x(t) $ real and even | $ a_k $ real and even |
| Real and Odd Signals | $ x(t) $ real and odd | $ a_k $ purely imaginary and odd |
Parseval's Relation
$ \frac{1}{T}\int_T \Big| x(t) \Big| ^2 dt = \sum_{k=-\infty}^\infty \Big| a_k \Big| ^2 $
Back to ECE301 Spring 2011 Prof. Boutin
--Cmcmican 19:02, 28 March 2011 (UTC)
