  In this paper, we consider a multiple-input multiple-output broadcast channel
with limited feedback where all users share the feedback rates. Firstly, we
find the optimal feedback rate sharing strategy using zero-forcing transmission
scheme at the transmitter and random vector quantization at each user. We
mathematically prove that equal sharing of sum feedback size among all users is
the optimal strategy in the low signal-to-noise ratio (SNR) region, while
allocating whole feedback size to a single user is the optimal strategy in the
high SNR region. For the mid-SNR region, we propose a simple numerical method
to find the optimal feedback rate sharing strategy based on our analysis and
show that the equal allocation of sum feedback rate to a partial number of
users is the optimal strategy. It is also shown that the proposed simple
numerical method can be applicable to finding the optimal feedback rate sharing
strategy when different path losses of the users are taken into account. We
show that our proposed feedback rate sharing scheme can be extended to the
system with stream control and is still useful for the systems with other
techniques such as regularized zero-forcing and spherical cap codebook.
