  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.
