Optimal transmission rate for MISO channels with joint sum and per-antenna power constraints

We consider multiple-input single-output (MISO) Gaussian channels with joint sum and per-antenna power constraints. A closed-form solution of the optimal beamforming vector is derived which achieves the maximal transmission rate. The result shows that if the sum power constraint only optimal power allocation violates a per-antenna power constraint then the joint power constraint optimal power allocation is at the intersection of the sum power constraint and the per-antenna power constraints.