Effect of Finite Rate Feedback on CDMA Signature Optimization and MIMO Beamforming Vector Selection

We analyze the effect of finite rate feedback on CDMA (code-division multiple access) signature optimization and MIMO (multi-input-multi-output) beamforming vector selection. In CDMA signature optimization, for a particular user, the receiver selects a signature vector from a codebook to best avoid interference from other users, and then feeds the corresponding index back to the specified user. For MIMO beamforming vector selection, the receiver chooses a beamforming vector from a given codebook to maximize throughput, and feeds back the corresponding index to the transmitter. These two problems are dual: both can be modeled as selecting a unit norm vector from a finite size codebook to "match" a randomly generated Gaussian matrix. In signature optimization, the least match is required while the maximum match is preferred for beamforming selection. Assuming that the feedback link is rate limited, our main result is an exact asymptotic performance formulae where the length of the signature/beamforming vector, the dimensions of interference/channel matrix, and the feedback rate approach infinity with constant ratios. The proof rests on a large deviation principle over a random matrix ensemble. Further, we show that random codebooks generated from the isotropic distritution are asymptotically optimal not only on average, but also with probability one.