The downlink problem in a wireless system with multiple transmit antennas and multiple users can be modeled by a multiple input multiple output Gaussian broadcast channel. The capacity region of this channel is still an open problem. Caire and Shamai  applied a technique known as "dirty-paper coding," due to Costa , to find an achievable region. It was later shown  that the maximum sum rate of this region is also the maximum possible sum rate. It is widely believed that this achievable region is in fact optimal. We attempt to show that this conjecture is true at high SNRs. We hope that solving this problem would give insights into the general problem.
MIMO channels with perfect channel state information (CSI) at the receiver and Rayleigh flat faded channel gains i.i.d. across antenna pairs have a capacity that grows as min(nt, nr) log SNR for large SNR, where nt and nr are the number of transmit and receive antennas, and SNR is the signal to noise ratio [1,2]. The parameter min(nt, nr) can be interpreted as the number of degrees of freedom (d.o.f.) of the channel: the dimension of the space over which communication can take place.
In high mobility applications the perfect CSI assumption may not be reasonable. If one relaxes this assumption the channel uncertainty at high SNR may have a significant impact on performance. This leads to the question: what is the high SNR capacity of time-varying fading channels without the prior assumption of CSI? Recent results indicate that the first order term in the high SNR capacity expansion is log log SNR regardless of nt and nr [3,4]. This implies that at sufficiently large SNR the benefit of having multiple transmit and receive antennas appears only as a second order effect, and hence the increase in the number of d.o.f. has minimal impact.
The above result is obtained by keeping the channel variation process fixed while taking the SNR to infinity, so its regime of validity corresponds to the case of a noise level much smaller than the channel variation between samples. However, typical wireless channels are underspread, which means that this variation is small. In this work we consider the capacity of underspread MIMO fading channels without CSI when the SNR goes to infinity while the channel variation between samples goes to zero simultaneously. We define three regimes of operation based on the relationship between the SNR and the channel variation. We show that in the first two regimes the capacity is proportional to the degrees of freedom in the channel and argue that most practical systems operate in these two regimes. This suggests that in underspread fading channels, multiple antennas provide significant gains and the concept of degrees of freedom is a useful measure of that performance gain, even without the assumption of perfect CSI.