In this section we report some simulation results to evaluate the validity of our analytical results.
We consider an adaptive blind two-stage partial HPIC receiver using the LMS algorithm. We utilized the random codes as the spreading codes and the processing gain is set as . Only the AWGN channel was used throughout the simulations. For the first set of simulations, we compared theoretical optimal weights with empirical ones for various( ).
Optimal weights for correct and erroneous decision were considered separately. Note that the channel gain was normalized to unity, i.e., as for all . Thus all weights starting adaptation from $
. Figure 4.2 shows the results for a two-user case, which includes exact analytical optimal weights in (4.93), those obtained using the OWA2 in (4.129), and those obtained empirically. It can be seen that both the exact and approximate optimal weights agree
0 2 4 6 8 10 12
−0.6
−0.4
−0.2 0 0.2 0.4 0.6 0.8 1 1.2
Eb/N 0 wopt
Correct decision
Erroneous decision
Exact Analysis OWA2 Empirical Result
Figure 4.2: Optimal weight comparison for two power-balanced users.
with the empirical ones very well. As depicted in the figure, the optimal weights for correct decision are almost the same as the channel gain, while the weights for erroneous decision is not;
its actual value depends on noise variance. The larger the
ratio, the closer the optimal weight to . We also give optimal weights for 5 and 15 users (with various
) in Fig.
4.3 and Fig. 4.4, respectively. In these figures, the results for the OWA1 (using (4.123)) and the OWA2 (using (4.129)) are shown simultaneously. We can see that although these approximates are performed based on the two-user model, the results are very close to the true optimums.
From Figure 4.2-Figure 4.4, we can observe that when theand the number of users vary, the optimal weights for correct decision keep very close to the channel gains which is one, while those for erroneous decision vary. Also note that the performances of the two approximations are very similar. Since the OWA2 is simpler, it is then desirable to use that whenever necessary.
We next consider the weight convergence of the LMS algorithm. Figure 4.5 presents the analytical mean weights along with the empirical mean weights for a two-user scenario. The powers of these two users are equal and
dB. The normalized step size is chosen as
0 2 4 6 8 10 12
−0.8
−0.6
−0.4
−0.2 0 0.2 0.4 0.6 0.8 1 1.2
Eb/N 0 wopt
Correct decision
Erroneous decision OWA1 OWA2 Empirical Result
Figure 4.3: Optimal weight comparison for five power-balanced users.
. The exact analysis in (4.118) and the WEMA2 in (4.141) with the OWA2 are evaluated. In the figure, we can observe that both analytical results match with the empirical mean weights quit well. Similar comparison for 5 and 15 power-balanced users with
dB and
are also shown in Figure 4.6 and Figure 4.7, respectively. The WEMA1 in (4.132) with the OWA1 is compared to WEMA2 with the OWA2. We can see that the analytical results are more accurate for the 5-user case. For the 15-user case, there is some discrepancy between analytical and empirical results. From above simulation results, we can conclude that the WEMA2 with the OWA2 is suffice to give satisfactory results. This combination will render less computational complexity. The weight error power comparison for the two-user case with
dB and is given in Figure 4.8. It is obvious that the analytic result performs close to simulated results. Also note that the weight error power incurred from correct decision is smaller than that form erroneous decision. This is because the weights for erroneous decision converges slower. The similar phenomenon can be observed when the user number is larger. In Figures 4.9 and 4.10, the weight error power for 5 and 15 users are examined
0 2 4 6 8 10 12
−1
−0.5 0 0.5 1 1.5
Eb/N 0 wopt
Correct decision
Erroneous decision
OWA1 OWA2 Empirical Result
Figure 4.4: Optimal weight comparison for 15 power-balanced users.
(
dB and
). As we can see, the analytic results are still accurate even for the erroneous decision of the 15-user case where only an estimation error about 20% is produced. Finally, we present the results for step size optimization. Figure 4.11 gives the step size minimizing MSE using (4.150). The figure reveals that the analytically optimized step size is more accurate in low capacity systems. This is reasonable since the approximate analysis is based on the single-user and two-user cases. We also give the BER comparison for the second stage output with different user numbers in Figure 4.12. From the figure, we observe that the analytical and empirical results are similar for low to moderate.
0 5 10 15 20 25 30 35 0.4
0.5 0.6 0.7 0.8 0.9 1 1.1
n
weight mean
Correct decision
Erroneous decision
Exact Analysis WEMA1 Empirical Result
Figure 4.5: Weight mean comparison for two power-balanced users (
, and
dB).
0 5 10 15 20 25 30 35
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1
n
weight mean
Correct decision
Erroneous decision
WEMA1 WEMA2 Empirical Result
Figure 4.6: Weight mean comparison for five power-balanced users (
, and
dB).
0 5 10 15 20 25 30 35 0.2
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2
n
weight mean
Correct decision
Erroneous decision
WEMA1 WEMA2 Empirical Result
Figure 4.7: Weight mean comparison for 15 power-balanced users (
, and
dB).
0 5 10 15 20 25 30 35
0 0.2 0.4 0.6 0.8 1 1.2 1.4
n
weight error power
Correct decision Erroneous decision
Approximation Empirical Result
Figure 4.8: Weight error power comparison for two power-balanced users (
, and
dB).
0 5 10 15 20 25 30 35 0
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
n
weight error power
Correct decision Erroneous decision
Approximation Empirical Result
Figure 4.9: Weight error power comparison for five power-balanced users (
, and
dB).
0 5 10 15 20 25 30 35
0 0.5 1 1.5 2 2.5 3
n
weight error power
Correct decision
Erroneous decision Approxitation Empirical Result
Figure 4.10: Weight error power comparison for 15 power-balanced users (
, and
dB).
0 1 2 3 4 5 6 0.005
0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05
Eb/N 0 Simulated,K=5
Theoretical,K=5 Simulated,K=10 Theroetical,K=10 Simulated,K=15 Theroetical,K=15
µ0
Figure 4.11: Optimal step-size comparison for different user numbers.
5 10 15 20
10−3 10−2 10−1
Number of users
BER
Theoretical (3 dB) Empirical (3 dB) Theoretical (7 dB) Empirical (7 dB)
Figure 4.12: Second-stage BER comparison for power-balanced cases (
).