  The channel feedback overhead for massive MIMO systems with a large number of
base station (BS) antennas is very high, since the number of feedback bits of
traditional codebooks scales linearly with the number of BS antennas. To reduce
the feedback overhead, an effective codebook based on channel statistics has
been designed, where the required number of feedback bits only scales linearly
with the rank of the channel correlation matrix. However, this attractive
conclusion was only intuitively explained and then verified through simulation
results in the literature, while no rigorous theoretical proof has been
provided. To fill in the gap between the theoretical conclusion and simulation
results, in this paper, we quantitatively analyze the performance of the
channel statistics-based codebook. Specifically, we firstly introduce the rate
gap between the ideal case of perfect channel state information at the
transmitter and the practical case of limited channel feedback, where we find
that the rate gap is dependent on the quantization error of the codebook. Then,
we derive an upper bound of the quantization error, based on which we prove
that the required feedback bits to ensure a constant rate gap only scales
linearly with the rank of the channel correlation matrix. Finally, numerical
results are provided to verify this conclusion. To the best of our knowledge,
our work is the first one to provide a rigorous proof of this conclusion.
