  This paper considers the jointly optimal pilot and data power allocation in
single-cell uplink massive multiple-input-multiple-output (MIMO) systems. Using
the spectral efficiency (SE) as performance metric and setting a total energy
budget per coherence interval, the power control is formulated as optimization
problems for two different objective functions: the weighted minimum SE among
the users and the weighted sum SE. A closed form solution for the optimal
length of the pilot sequence is derived. The optimal power control policy for
the former problem is found by solving a simple equation with a single
variable. Utilizing the special structure arising from imperfect channel
estimation, a convex reformulation is found to solve the latter problem to
global optimality in polynomial time. The gain of the optimal joint power
control is theoretically justified, and is proved to be large in the low SNR
regime. Simulation results also show the advantage of optimizing the power
control over both pilot and data power, as compared to the cases of using full
power and of only optimizing the data powers as done in previous work.
