$($a$)$ Consider the transmit pulse $g_{Tx}(t)=sinc(\frac{t}{T})sinc(\frac{t}{2}T).$ Draw the frequency spectrum of $g_{Tx}(t).$ Explain why $g_{Tx}(t)$ satisfies the Nyquist criterion at symbol rate $\frac{1}{T}?$

$(8\%)$

$($b$)$ If $g_{Tx}(t)$ is used for Nyquist signalling with $8-$PSK at $6$Mbit$\frac{s}{s},$ what is the signal bandwidth?

$(5\%)$

$($c$)$ For the three $8-$ary constellations shown in Fig. $1,$ if they have the same $E_s,$ which constellation do you expect to have the smallest symbol error probability over a high SNR AWGN channel $?$ Please explain your answer. $($Hint: express the minimum distance of each constellation in terms of $E_s)$

$(8\%)$

$($d$)$ For each constellation in Fig. $1,$ determine whether you can label the constellation points using Gray code. If so$,$ draw the constellation and the bit labels in your answer book. If not, explain why Gray code cannot be found.

$(4\%)$