channel - Advances in Electronics and Telecommunications

STAMATIOU et al.: SPATIAL MULTIPLEXING IN RANDOM WIRELESS NETWORKS 11
Transm. cap. (nats/symbol/m 2 )
0.0003
0.00025
0.0002
0.00015
N = 8
0.0001
5e-05
N = 4
DB, eq. (43)
Sim, DB
ZF, eq. (32)
Sim, ZF
VB, eq. (36)
Sim, VB
0
1 2 3 4 5 6 7 8
Fig. 4. Transmission capacity vs. M for N = 4, 8 (ǫ = 0.1, R = 20 m,
b = 3, θ = 6 dB). The optimal number of streams for DBLAST is 3 when
N = 4, and 6 when N = 8. These numbers are in accordance with (45). In
the case of ZF, the optimal number of streams is 1 when N = 4, and 3 when
N = 8. These numbers are also in accordance with (35). To avoid cluttering
the figure, DBLAST is denoted as DB and VBLAST as VB.
Transm. cap. (nats/symbol/m 2 )
0.0004
0.00035
0.0003
0.00025
0.0002
M
N = 8
0.00015
0.0001
5e-05
N = 4
DB, eq. (43)
Sim, DB
ZF, eq. (32)
Sim, ZF
VB, eq. (36)
Sim, VB
1 2 3 4 5 6 7 8
Fig. 5. Transmission capacity vs. M for N = 4, 8 (ǫ = 0.1, R = 20 m,
b = 4, θ = 6 dB). To avoid cluttering the figure, DBLAST is denoted as DB
and VBLAST as VB.
In Fig. 3, the theoretical and simulated success probability
of ZF and DBLAST are plotted vs. the density of the PPP for
a system where M = 3 antennas are employed in each TX.
The agreement between theory and simulation is once again
very satisfactory, which, in the case of DBLAST, confirms the
validity of the approximations in Section IV-C.
Fig. 4 shows the dependence of the transmission capacity
on the number of transmitted streams for the three MIMO
techniques considered in Section IV. The total number of
antennas takes two values N = 4, 8, the propagation exponent
is b = 3 and a constraint ǫ = 0.1 is placed on the outage
probability. The DBLAST transmission scheme results in
higher transmission capacity compared to VBLAST or simple
ZF. Moreover, the gain between VBLAST and simple ZF is
marginal, which is attributed to the fact that, with VBLAST,
the maximum contention density is still determined by the
subchannel with the smallest diversity order.
M
Transm. cap. (nats/symbol/m 2 )
4e-05
3.5e-05
3e-05
2.5e-05
2e-05
1.5e-05
1e-05
5e-06
DBLAST
Sim, DBLAST
ZF
Sim, ZF
ZF, M = N
Sim, ZF, M = N
MRC
Sim, MRC
0
1 2 3 4 5 6 7 8
Fig. 6. Transmission capacity vs. N (ǫ = 0.01, R = 20 m, b = 4,
θ = 6 dB). If all TX antennas are activated, the transmission capacity of ZF
is lower than the transmission capacity of MRC, even though the scaling in
both cases is Θ( √ N). Setting M = N and b = 4 in (32) we can also see
that TCzf TCǫ
ǫ ≈ , which is confirmed by the plot.
Γ(1/2)
Optimal M
8
7
6
5
4
3
2
1
DBLAST
ZF
1 2 3 4 5 6 7 8
N
Fig. 7. Optimal number of streams vs. N (ǫ = 0.01, R = 20 m, b = 4,
θ = 6 dB).
In Fig. 5, the propagation exponent takes the value b = 4.
As predicted in Section IV-C, activating all the TX antennas
maximizes the transmission capacity for DBLAST. In the case
of ZF, the optimal number of streams is dictated by (35).
Overall, in both Fig. 4 and Fig. 5, the agreement between
theory and simulation is satisfactory.
In Fig. 6 and Fig. 7, the transmission capacity and the
respective - optimal - number of streams are plotted vs. N for
DBLAST, ZF and MRC and an outage probability constraint
ǫ = 0.01. As predicted in Section IV, in the case of DBLAST
and ZF, and optimally selected M, the transmission capacity
scales linearly in N, while, in the case of MRC, it scales as
N α = √ N. At N ≥ 3, DBLAST provides a capacity gain of
approximately 1.4 compared to ZF, which is in agreement with
the √ 2 gain predicted at the end of Section IV-C. Moreover,
it is observed that, for N ≤ 3, MRC and ZF result in
approximately the same transmission capacity.
N