Outage Capacity with two-bit channel feedback for a two-transmit and single receive antenna system

Channel information at the transmitter is known to significantly increase the information throughput. However, practical constraints in the opposite link limit the possible amount of feedback. In this paper we describe a way to calculate the capacity of a simple, yet relevant feedback scheme. In particular, we consider the case of a two-antenna transmitter and a single antenna receiver, and limit the amount of feedback to 2 bits. This case is of practical interest because it is also included in the UMTS standards. We determine the transmission weights maximizing the outage mutual information, and find them to be a combination of beamforming and orthogonal coding. Further, the case of the feedback bits having finite and generally unequal probabilities of error is also considered. In the latter case, the optimal relative phase between the two transmitting antennas deviates from their symmetric $pm , pi pm ,pi/4$ value. This effect is important, if the bits are fed back sequentially and thus their reliability is not the same.