The optimal power allocation with full CSI in the symmetric uplink with the assumption that there is always a unique user with the strongest channel at any one time. This assumption holds with probability 1 when the fading distributions are continuous. Moreover, under this assumption, the solution is unique. This is in contrast to the uplink AWGN channel where there is a continuum of solutions that achieves the optimal sum rate, of which only one is orthogonal. We will see in this exercise that transmitting to only one user at a time is not necessarily the unique optimal solution even for fading channels, if the fading distribution is discrete (to model measurement realities, such as the feedback of a finite number of rate levels).
Consider the full CSI two-user uplink with identical, independent, stationary and ergodic flat fading processes for the two users. The stationary distribution of the flat fading for both of the users takes one of just two values: channel amplitude is either at 0 or at 1 (with equal probability). Both of the users are individually average power constrained (by P¯ ).
Calculate explicitly all the optimal joint power allocation and decoding policies to maximize the sum rate. Is the optimal solution unique? Hint: Clearly there is no benefit by allocating power to a user whose channel is fully faded (the zero amplitude state).