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We propose and study a class of transmit beamforming techniques for systems with multiple transmit and multiple receive antennas with a per-antenna transmit power constraint. The per-antenna transmit power constraint is more realistic than the widely used total (across all transmit antennas) power constraint, since in practice each transmit antenna is driven by a separate power amplifier with a maximum power rating. Under the per-antenna power constraint, from an implementation perspective, it becomes desirable to vary only the phases (as opposed to both power and phase variation) of the signals departing from the transmit antennas. We name this class of techniques generalized co-phasing and formulate an optimization problem to calculate the transmit antenna phases. Furthermore, we propose five heuristic algorithms to solve the optimization problem. All the proposed algorithms except one are optimal for the case of two transmit antennas and an arbitrary number of receive antennas. For an arbitrary number of transmit and receive antennas, simulations indicate that the proposed algorithms perform very close to the optimal solution calculated through an exhaustive search of all possible transmit phases.