  We consider the quantization of a transmit beamforming vector in multiantenna
channels and of a signature vector in code division multiple access (CDMA)
systems. Assuming perfect channel knowledge, the receiver selects for a
transmitter the vector that maximizes the performance from a random vector
quantization (RVQ) codebook, which consists of independent isotropically
distributed unit-norm vectors. The quantized vector is then relayed to the
transmitter via a rate-limited feedback channel. The RVQ codebook requires an
exhaustive search to locate the selected entry. To reduce the search
complexity, we apply generalized Lloyd or $k$-dimensional (kd)-tree algorithms
to organize RVQ entries into a tree. In examples shown, the search complexity
of tree-structured (TS) RVQ can be a few orders of magnitude less than that of
the unstructured RVQ for the same performance. We also derive the performance
approximation for TS-RVQ in a large system limit, which predicts the
performance of a moderate-size system very well.
