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In this letter, we derive the scaling laws of the sum rate for fading multiple-input multiple-output Gaussian broadcast channels using time sharing to the strongest user, dirty-paper coding (DPC), and beamforming, when the number of users (receivers) n is large. Throughout the letter, we assume a fix average transmit power and consider a block-fading Rayleigh channel. First, we show that for a system with M transmit antennas and users equipped with N antennas, the sum rate scales like MloglognN for DPC, and beamforming when M is fixed and for any N (either growing to infinity or not). On the other hand, when both M and N are fixed, the sum rate of time sharing to the strongest user scales like min(M,N)loglogn. Therefore, the asymptotic gain of DPC over time sharing for the sum rate is (M/min(M,N)) when M and N are fixed. It is also shown that if M grows as logn, the sum rate of DPC and beamforming will grow linearly in M, but with different constant multiplicative factors. In this region, the sum-rate capacity of time -sharing scales like Nloglogn
Sharif et al. (Mon,) studied this question.