View source for Table of CT Fourier series coefficients and properties (include some computations and proofs if you are really brave) - Rhea

= Table of CT Fourier Series Coefficients and Properties =

== Some Fourier series ==


{| border="1" class="wikitable"
|-
! Function 
! Fourier Series 
! Coefficients
|-
| <math>sin(w_0t)</math> 
| <math>\frac{1}{2j}e^{jw_0t}-\frac{1}{2j}e^{-jw_0t}</math> 
| <math>a_1=\frac{1}{2j}, a_{-1}=\frac{-1}{2j}, a_k=0 \mbox{ for } k \ne 1,-1</math>
|-
| <math>cos(w_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>
|-
| 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 ==



{| border="1" class="wikitable"
|-
! Property 
! Periodic Signal 
! Fourier Series Coefficients
|-
|  
| 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>

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[[2011_Spring_ECE_301_Boutin|Back to ECE301 Spring 2011 Prof. Boutin]]

--[[User:Cmcmican|Cmcmican]] 19:02, 28 March 2011 (UTC)