Two dimensional beamforming: an optimal solution for joint multiple array transmit and receive problem

For the radio channel the channel impulse response spreads around the transmitter and the receiver. The transmission and reception through such channels can be considered as a 2D spatial problem with the transmission processing matrix and receiving processing matrix unknown. We model the mobile radio fading channel by three blocks: transmit angle spread matrix, fading path gain matrix and receive angle spread matrix. We derive the optimal solutions in the MMSE sense with a fixed or snap-shot of a fading channel impulse response for two scenarios: hot spot (multiple mobiles are closely located in one location) and multi-beam transmission (one mobile transmits with multiple transmission streams intending to achieve higher transmission rate).