Degrees of Freedom in Underspread MIMO Fading Channels

Raul Etkin
(Professor David N. C. Tse)
(NSF) CCR-01-18784

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.

G. Foschini and M. Gans, "On Limits of Wireless Communications in a Fading Environment When Using Multiple Antennas," Wireless Personal Communications, No. 6, 1998.
E. Telatar, "Capacity of Multi-Antenna Gaussian Channels," European Trans. Telecommunications, Vol. 10, No. 6, 1999.
A. Lapidoth and S. Moser, "Capacity Bounds via Duality with Applications to Multi-Antenna Systems on Flat Fading Channels," Preprint, June 2002.
G. Taricco and M. Elia, "Capacity of Fading Channels with no Side Information," Electronics Letters, Vol. 33, July 1997.

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