  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.
