Adaptive Quantization on a Grassmann-Manifold for Limited Feedback Beamforming Systems

In this paper we examine delay limited adaptive quantization on the Grassmann manifold of 1-dimensional subspaces in n-dimensional space. Due to strict delay limits, vector quantization over multiple time instants cannot be applied to exploit the temporal correlation of the source signal. Instead, a vector predictive quantizer is proposed that combines prediction and differential quantization algorithms to achieve an efficient quantization of the correlated Grassmannian source. The proposed predictor is based on adaptive finite impulse response filters to adjust to the temporal statistics of the source signal. It is shown that the prediction error in the tangent space associated with the Grassmann manifold behaves approximately Gaussian, provided its norm is sufficiently small. The proposed quantization algorithm is applied to channel state information quantization in multi-user beamforming wireless communication systems. Large throughput gains are demonstrated in comparison to memoryless quantization, due to reduced multi-user interference.