$(6$ points$)$ Question $3-2$: We want to find the optimal power allocations on the $3$ available channels, i$.$e$.,$ we want to find the optimal values for $P_1,P_2$ and $P_3$ such that the capacity is maximized. The optimal values for the power allocations are given by the waterfilling solution which is

$)$

$(u-N_i$

Assume that the noise powers are given as $N_1=6,N_2=20$ and $N_3=36.$ Assume that the total power constraint is $P=34.$ Find the optimal power allocations $P_1,P_2$ and $P_3$ Hint: the value of

$u$ should be no larger than $40.$

Step $1.$ Assume the largest possible value for

$u,$ and find the corresponding power allocations $P_1,P_2$ and $P_3.$ Is this the optimal power allocation? Why?

Step $2.$ Find the optimal power allocation. Hint: try to find the optimal value of

$u.$ You can search for this value of

$u$ using binary search which works as follows. If the value you used in Step $1$ is too large, divide it by half and repeat. If it is too small, add to it half its value and repeat. Keep doing this until you find the right

$u.$