  The ensemble properties of Random Vector Quantization (RVQ) codebooks for
limited-feedback beamforming in multi-input multi-output (MIMO) systems are
studied with the metrics of interest being the received SNR loss and mutual
information loss, both relative to a perfect channel state information (CSI)
benchmark. The simplest case of unskewed codebooks is studied in the correlated
MIMO setting and these loss metrics are computed as a function of the number of
bits of feedback ($B$), transmit antenna dimension ($N_t$), and spatial
correlation. In particular, it is established that: i) the loss metrics are a
product of two components -- a quantization component and a channel-dependent
component; ii) the quantization component, which is also common to analysis of
channels with independent and identically distributed (i.i.d.) fading, decays
as $B$ increases at the rate $2^{-B/(N_t-1)}$; iii) the channel-dependent
component reflects the condition number of the channel. Further, the precise
connection between the received SNR loss and the squared singular values of the
channel is shown to be a Schur-convex majorization relationship. Finally, the
ensemble properties of skewed codebooks that are generated by skewing RVQ
codebooks with an appropriately designed fixed skewing matrix are studied.
Based on an estimate of the loss expression for skewed codebooks, it is
established that the optimal skewing matrix is critically dependent on the
condition numbers of the effective channel (product of the true channel and the
skewing matrix) and the skewing matrix.
