Let $x(t)$ be the triangular wave depicted in Fig. $9\mathrm{.}1$

$8$

$($a$)$ Find the FS coefficients $x[k].$

$($b$)$ Show that the FS representation for $x(t)$ can be expressed in the form

$x(t)={\displaystyle \underset{k=0}{\overset{}{}}}B[k]cos(k{}_{o}t)$

$($c$)$ Define the J$-$term partial$-$sum approximation to $x(t)$ as

hat$(x{)}_j(t)={\displaystyle \underset{k=0}{\overset{1}{}}}B[k]cos(k{}_{o}t)$

Use MATLAB to evaluate and plot one period of the $J$ th term in this sum and hat$(x{)}_j(t)$ for $J=1,3,7,29,$ and $99.$