$4\mathrm{.}28.($a$)$ Let $x(t)$ have the Fourier transform $x(j),$ and let $p(t)$ be periodic with fing mental frequency ${}_0$ and Fourier series representation

$p(t)={\displaystyle \underset{n=-}{\overset{+}{}}}{a}_n{e}^{jn{}_{0}t}$

Determine an expression for the Fourier transform of

$y(t)=x(t)p(t)$

$($b$)$ Suppose that $x(j)$ is as depicted in Figure $P4\mathrm{.}28(a).$ Sketch the spectum $y(t)$ in eq$.($P$4\mathrm{.}28-1)$ for each of the following choices of $p(t)$ :

$($i$)p(t)=cos(\frac{t}{2})$

$($ii$)p(t)=cost$

$($iii$)p(t)=cos2t$

$($iv$)p(t)=(sint)(sin2t)$

$($v$)p(t)=cos2t-cost$

$($vi$)p(t)={\displaystyle \underset{n=-}{\overset{+}{}}}(t-n)$

$($vii$)p(t)={\displaystyle \underset{n=-}{\overset{+}{}}}(t-2n)$

$($viii$)p(t)={\displaystyle \underset{n=-}{\overset{+}{}}}(t-4n)$

$($ix$)p(t)={\displaystyle \underset{n=-}{\overset{+}{}}}(t-2n)-\frac{1}{2}{\displaystyle \underset{n=-}{\overset{+}{}}}(t-n)$

$($x$)p(t)=$ the periodic square wave shown in Figure P$4\mathrm{.}28($b$).$

Figure P$4\mathrm{.}28$

Only ix and x