  Training sequences are designed to probe wireless channels in order to obtain
channel state information for block-fading channels. Optimal training sounds
the channel using orthogonal beamforming vectors to find an estimate that
optimizes some cost function, such as mean square error. As the number of
transmit antennas increases, however, the training overhead becomes
significant. This creates a need for alternative channel estimation schemes for
increasingly large transmit arrays. In this work, we relax the orthogonal
restriction on sounding vectors. The use of a feedback channel after each
forward channel use during training enables closed-loop sounding vector design.
A misalignment cost function is introduced, which provides a metric to
sequentially design sounding vectors. In turn, the structure of the sounding
vectors aligns the transmit beamformer with the true channel direction, thereby
increasing beamforming gain. This beam alignment scheme for massive MIMO is
shown to improve beamforming gain over conventional orthogonal training for a
MISO channel.
