For the two$-$user SISO uplink $($multiple$-$access channel$)Y=x_1+x_2+Z,$ the

sum rate is constrained by the mutual information of signal $(x_1,x_2)$ and $Y$ as

$ZY=x_1+x_2\frac{1}{2}N(0,1)ZPlog\frac{1+2P}{2}(x_1,x_2)R_1+R_2$

$(a)(5$ points$)$ Assume the noise $Zisabsent$ and both user signals have binary

alphabets. $In$ this binary and noiseless uplink $Y=x_1+x_2,$ please find the

maximum sum rate when both input $is$ Bernoulli$\frac{1}{2.}$

$(b)$ Now $we$ remove the assumption $in(a)$ and focus $on$ uplink with Gaussian

$N(0,1)$ noise $Z$ also user signals have individual but same power constraints $P.$

$In$ this uplink, users send independent information and cannot cooperate $in$ the

encoding; the maximum sum rate $islog\frac{1+2P}{2},$ obtained when $(x_1,x_2)$ are

jointly Gaussian. Now $if$ they could cooperate, what $is$ the maximum sum rate,

still assuming individual but same power constraints $Pon$ two users?