Adaptive Opportunistic Beamforming in Ricean Fading Channels - Adaptive Signal Processing in Wireless Communications - page 182

for the slow fading state of any individual user k. Then, almost surely, we have

() M bf

lim KT = R (7.9)

k k

M →∞

for all k. T ( M ) is the average throughput of user k in a system with M users and R bf is the
instantaneous data rate that user k achieves when it is in the beamforming configuration,
i.e. when its instantaneous SNR is

∑

N

2

”

P h nk ρ .

, k

n = 1

Proof: See appendix A in [ 13 ].

When the number of users is sufficiently large, theorem 7.1 implies that, with very
high probability, one user is selected when it is in its beamforming configuration and
every user is allocated an equal amount of time. The stationary distributions of α n ( t ) and
ϕ n ( t ) required by this theorem are given in closed form in [13]. In a slow Rayleigh fading
environment with ten antennas at the base station, the throughput of one specific user is
given in Figure 7.4 . The proportional fair scheduling algorithm is employed. One can see
that the throughput converges to that of coherent transmit beamforming asymptotically
with the number of users. Figure 7.5 demonstrates the total throughput of all users.

7.3.2 Fast Fading
We have seen that opportunistic beamforming can considerably improve the perfor-
mance when the underlying channels are slow fading. But if the underlying channels
are already fast fading, can opportunistic beamforming improve the performance? The
performance gain of opportunistic beamforming is from the randomization of α n ( t ) and
ϕ n ( t ), which makes the dynamic range of the channel fluctuation larger and the variation
rate of the channel faster. Therefore, if the dynamic range of the equivalent channel ˜ k ( t )
becomes larger after using opportunistic beamforming, the system will achieve better
performance.
We first consider the independent Rayleigh fading environment, where h n,k ( t ) are i.i.d
circularly symmetric complex Gaussian random variables with zero mean. In this case,
it is easy to see from (7.4) that the distribution of ˜ k ( t ) is exactly the same as that of
the h n,k ( t ). The use of α n ( t ) and ϕ n ( t ) neither makes the dynamic range of the channel
fluctuation larger nor makes the variation rate of the channel faster. Therefore, in an
independent fast Rayleigh fading environment, opportunistic beamforming is not able
to improve the performance.
In contrast, when the underlying channels are Ricean fading channels, opportunis-
tic beamforming can bring considerable performance gain. The Ricean channel can be
modeled as [19]