  Block diagonalization is a linear precoding technique for the multiple
antenna broadcast (downlink) channel that involves transmission of multiple
data streams to each receiver such that no multi-user interference is
experienced at any of the receivers. This low-complexity scheme operates only a
few dB away from capacity but requires very accurate channel knowledge at the
transmitter. We consider a limited feedback system where each receiver knows
its channel perfectly, but the transmitter is only provided with a finite
number of channel feedback bits from each receiver. Using a random quantization
argument, we quantify the throughput loss due to imperfect channel knowledge as
a function of the feedback level. The quality of channel knowledge must improve
proportional to the SNR in order to prevent interference-limitations, and we
show that scaling the number of feedback bits linearly with the system SNR is
sufficient to maintain a bounded rate loss. Finally, we compare our
quantization strategy to an analog feedback scheme and show the superiority of
quantized feedback.
