On MIMO capacity in the ultra-wideband regime

We consider multiple antenna communication over a wideband, non-coherent Rayleigh block fading channel. As the available power is spread over a large bandwidth, the signal-to-noise ratio (SNR) per degree of freedom is low for these channels. While the capacity of multiple antenna channels in the high SNR regime has recently become well understood, the low SNR regime has been scantily studied. We explore the interaction among transmit antennas, coherence time, SNR and receive antennas for time-varying channels at low SNR, with no channel state information at either the transmitter or receiver. We obtain a necessary criterion the coherence length of the non-coherent channel must satisfy, for a certain rate to be achievable. This derivation also tells us how large the coherence length necessarily needs to be, for the non-coherent channel to have a "near" coherent performance.