  In this paper, we design the precoding matrices at the base station side and
the combining matrices at the user terminal side for initial downlink
synchronization in millimeter wave massive multiple-input multiple-output
systems. First, we demonstrate two basic requirements for the precoding and
combining matrices, including that all the entries therein should have constant
amplitude under the implementation architecture constraint, and the average
transmission power over the total K time slots taking for synchronization
should be constant for any spatial direction. Then, we derive the optimal
synchronization detector based on generalized likelihood ratio test. By
utilizing this detector, we analyze the effect of the precoding and combining
matrices to the missed detection probability and the false alarm probability,
respectively, and present the corresponding conditions that should be
satisfied. It is shown that, both of the precoding and combining matrices
should guarantee the perfect omnidirectional coverage at each time slot, i.e.,
the average transmission power at each time slot is constant for any spatial
direction, which is more strict than the second basic requirement mentioned
above. We also show that such omnidirectional precoding matrices and
omnidirectional combining matrices exist only when both of the number of
transmit streams and the number of receive streams are equal to or greater than
two. In this case, we propose to utilize Golay complementary pairs and
Golay-Hadamard matrices to design the precoding and combining matrices.
Simulation results verify the effectiveness of the propose approach.
