If u$($t$)$ is a real valued periodic signal with fundamental period T $,$ u$($t$)$ can be represented

as

u$($t$)=1$

T

X

k$=$

ckejkot $=1$

T co $+2$

T

X

k$=1$

$|$ck$|$ cos $($kot $+$ck$)$

where

ck $=$

Z

T

u$($ $)$e$$jko d

o $=2$

T $.$

Consider the following periodic signal u$($t$)$ for t in $($$,$ $)$

u$($t$)=$

$($

sin t$,0$ t $<1$

$0,1$ t $<2.-4-3-2-101234$

Time $($sec$)$

$-0\mathrm{.}5$

$0$

$0\mathrm{.}5$

$1$

$1\mathrm{.}5$

u$($t$)$

$($a$)(0\mathrm{.}5$pt$)$ What are the fundamental period T $($in sec$)$ and the fundamental frequency

o $($in rad$/$sec$)$ of u$($t$)?$

$($b$)(0\mathrm{.}5$pt$)$ Compute the Fourier series coefficient for k $=0.$

$($c$)(1$pt$)$ Compute the Fourier series coefficient for k $=1.$ Hint: sin t $=$ ejt $$ e$$jt

$2$j $.$

$2$

$($d$)(2$pt$)$ Show that the Fourier series coefficients ck for k $=0,1$ is given by

ck $=$

$$

$$

$$

$2$

$(1$ k$2),$ k $=2,4,6,...$

$0,$ k $=3,5,7,...$

$($e$)(2$pt$)$ Show that the Fourier series representation of u$($t$)$ can be expressed as

uF S $($t$)=1$

$+1$

$2$ sin t $2$

X

n$=1$

$1$

$4$n$21$ cos $(2$nt$).$