Optimal Shape-Gain Quantization for Multiuser MIMO Systems with Linear   Precoding
  This paper studies the optimal bit allocation for shape-gain vector quantization of wireless channels in multiuser (MU) multiple-input multiple-output (MIMO) downlink systems based on linear precoding. Our design minimizes the mean squared-error between the original and quantized channels through optimal bit allocation across shape (direction) and gain (magnitude) for a fixed feedback overhead per user. This is shown to significantly reduce the quantization error, which in turn, decreases the MU interference. This paper makes three main contributions: first, we focus on channel gain quantization and derive the quantization distortion, based on a Euclidean distance measure, corresponding to singular values of a MIMO channel. Second, we show that the Euclidean distance-based distortion of a unit norm complex channel, due to shape quantization, is proportional to \frac{2^{-2Bs}}{2M-1}, where, Bs is the number of shape quantization bits and M is the number of transmit antennas. Finally, we show that for channels in complex space and allowing for a large feedback overhead, the number of direction quantization bits should be approximately (2M - 1) times the number of channel magnitude quantization bits. 
Muhammad Nazmul Islam, Raviraj Adve, Behrouz Khoshnevis
cs.IT math.IT