We study a multiple-antenna system where the transmitter is equipped with quantized information about instantaneous channel realizations. Assuming that the transmitter uses the quantized information for beamforming, we derive a universal lower bound on the outage probability for any finite set of beamformers. The universal lower bound provides a concise characterization of the gain with each additional bit of feedback information regarding the channel. Using the bound, it is shown that finite information systems approach the perfect information case as (t-1)2/sup -B/t-1/, where B is the number of feedback bits and t is the number of transmit antennas. The geometrical bounding technique, used in the proof of the lower bound, also leads to a design criterion for good beamformers, whose outage performance approaches the lower bound. The design criterion minimizes the maximum inner product between any two beamforming vectors in the beamformer codebook, and is equivalent to the problem of designing unitary space-time codes under certain conditions. Finally, we show that good beamformers are good packings of two-dimensional subspaces in a 2t-dimensional real Grassmannian manifold with chordal distance as the metric.

Space time codes have been proposed in the literature as an efficient means for improving the data rates over fading channels with multiple transmit antennas. In particular, the rank and the determinant of code difference matrices have been shown to be important in the design of space time codes for fading channels. In the present work, we investigate the problem of maximizing the coding gain of space-time codes, given by the minimum of the determinants of all the code difference matrices. We rely on equality of the singular values of the code difference matrices as a necessary and sufficient condition for obtaining the optimal coding gain. Finally, we discuss the construction of trellis codes and present simulation results.