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)	$a_k = \frac{2sin(k\omega_0T_1)}{k\omega_0T_1}$

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

$\frac{1}{T}\int_T \Big| x(t) \Big| ^2 dt = \sum_{k=-\infty}^\infty \Big| a_k \Big| ^2$

--Cmcmican 19:02, 28 March 2011 (UTC)