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

Ensemble Properties of RVQ-Based Limited-Feedback Beamforming Codebooks