A discrete memoryless source emits binary equiprobable symbols at a rate of $1000$ symbols per second. The symbols from a one second interval are grouped into pairs and sent over a bandlimited channel using a standard $4-$QAM signal set, whose constellation is shown below. The modulation uses a signal interval $0\mathrm{.}002$ and pulse $p(t)=sinc(\frac{t}{T}).$

Figure $4$: Constellation of $4-$QAM.

$($a$)(5$ pts$)$ Suppose that a sample sequence $u_1,$dots,$u_{500}$ of transmitted signals includes $115$ appearances of $a_1=(-\frac{d}{2},-\frac{d}{2}),130$ appearances of $a_2=(-\frac{d}{2},\frac{d}{2}),120$ appearances of $a_4=(\frac{d}{2},-\frac{d}{2}),$ and $135$ appearances of $a_3=(\frac{d}{2},\frac{d}{2}).$ Find the energy in the corresponding transmitted waveform $u(t)={\displaystyle \underset{k=1}{\overset{500}{}}}{u}_{k}p(t-kT)$ as a function of $d.$

$($b$)(5$ pts$)$ What is the bandwidth of the waveform $u(t)$ in part $($a$)?$

$($c$)(5pts)$ Find $E[{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}{U}^2(t)dt]$ where $U(t)$ is the random waveform ${\displaystyle \underset{k=1}{\overset{500}{}}}{U}_{k}p(t-kT).$