A
&
A
(5.49)
§ 5.4 Simulation Results
A. Parameter optimization
In this section, we will report simulation results to demonstrate the effectiveness of the pro-posed algorithm. We have used random codes of length 31 as spreading sequences. Partial HPIC receivers up to five stages are considered. First, we determine the optimal parameters for each receiver in order to obtain the best system performance. We let the user number be ,
dB, and power was balanced. For the conventional partial HPIC, we have empirically found the optimal PCFs for stage 2 to 5 as $. As to the adaptive blind partial HPIC, the normalized step sizes, defined as
, are
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.006
0.008 0.01 0.012 0.014 0.016 0.018 0.02 0.022
ξf(2)
BER
ξs(2)=0.6 ξs(2)=0.8 ξs(2)=1.0 ξs(2)=1.2 ξs(2)=1.4 ξs(2)=1.6
Figure 5.6: Second stage parameter optimization fo the proposed algorithm (
).
. For the proposed algorithm, we have additional two parameters
and
. To simplify the problem, we do not perform weight selection and determine
first. Figure 5.5 shows the BER performance vs.
andfor the second stage output. From the figure, we can see that the optimal step size is around that is larger than the step size used in the in the conventional approach. This is because the weight post filtering operation removes some weight noise for users regarded reliable. The weight variance is then decreased and the resulting interference cancellation is more accurate. Thus, a larger step size is permitted for faster convergence. We then incorporate the weight selection operation into the parameter optimization. The result is shown in Fig. 5.6. Here, we let the step size be fixed as . In the figure, we can observe that the optimal parameter setting is and
. Note that the system performance is not sensitive for higher
values. This is because most reli-able weights have been selected during weight selection. The theoretical BER in (5.49) for the proposed algorithm is also evaluated in Figure 5.7. We can observe that our analysis resembles performance trend as the simulated results; however, there exhibits some gaps in between. The inaccuracy may be due to the Gaussian assumption used in the calculation. Optimal
parame-0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.006
0.008 0.01 0.012 0.014 0.016 0.018 0.02
ξf(2)
Simulated,ξs(2)=0.8 Simulated,ξs(2)=1.2 Simulated,ξs(2)=1.6 Theoretical,ξs(2)=0.8 Theroetical,ξs(2)=1.2 Theroetical,ξs(2)=1.6
BER
Figure 5.7: Second stage BER performance for the proposed algorithm ( ,
, and
dB).
ters found in a specific stage may not be optimal for all the stages. However, the optimization will be cumbersome. For simplicity, we will use the parameters found in the second stage for all stages. The superscript on parameters for denoting the stage number is then omitted in the sequel. In the fading channels; however, those parameters should be tuned again to obtain the best performance.
B. Performance comparison
In the following, we present the performance comparison for various multiuser receivers which include the conventional matched filter, the non-adaptive partial HPIC (referred to as PHPIC), the adaptive blind partial HPIC (referred to as the APHPIC), the proposed algorithm, and the GGS algorithm. The GGS algorithm serves as a post-processor for both the APHPIC and the proposed algorithm. Note that we let the GGS only perform one iteration (one bit correction) in each cancellation stage. Figure 5.8 expresses the second stage performance of the proposed algorithm and other methods vs. different user numbers (
dB). We
5 10 15 20 25 30 10−4
10−3 10−2 10−1 100
Number of users
BER
Conv. receiver PHPIC APHPIC APHPIC+GGS Proposed Proposed+GGS
Figure 5.8: Second stage BER performance comparison vs. user numbers (
,
,
, and
dB).
can find that the conventional matched filter receiver perform worst due to the heavy MAI. The PHPIC performs worse than the APHPIC and the proposed algorithm. The combined APHPIC and GGS receivers provides more performance improvement in light loading condition. This is because the GGS algorithm performs at most one bit correction in one stage; it is more effective for low error rate scenario. The proposed algorithm is better than the APHPIC and the post GGS processing enhances the proposed algorithm in all cases. We also show the performance for higher stage processing in Figures 5.9-5.11. As we can see, the GGS algorithm gives less and less improvement as the stage number increases. All adaptive partial HPIC receivers perform close to the single user bound when the number of users is small. However, adaptive receivers degrade in heavy loading scenarios. If we want to further improve the performance, we have to increase the adaptation length and decrease the step size. In such a way, we can reduce the weight variance from inaccurate interference cancellation. We then compare the proposed algorithm with other methods under different
(ten users). Fig. 5.12 and 5.13 show the results for the second and the fifth stage, respectively. We observe that the performance
5 10 15 20 25 30 10−4
10−3 10−2 10−1 100
Number of users
BER
Conv. receiver PHPIC APHPIC APHPIC+GGS Proposed Proposed+GGS Singleuser bound
Figure 5.9: Third stage BER performance comparison vs. user numbers (the parameter setting is the same as that in Figure 5.8 for all stages).
of the proposed algorithm is close to the single user bound when the number stage is five and
is low to median. We also compare the system performance under the power-imbalanced scenario. The user powers are equally distributed and the power ratio between the strongest and weakest users is dB. In Fig.5.14 and Fig.5.15, we present the BER performance for the weakest user in the second and fifth stage. It can be seen that the proposed algorithm provides a significant performance gain in the fifth stage, especially when the user number is large.
Note that the proposed algorithm makes the performance of the weakest user indistinguishable as compared to that of the single user case when the user number is less than twenty. The reason for this superior performance may be due to the weight selection process where stronger users are almost all recognized and excluded from the training phase. This results is similar to the behavior in SIC, where the most reliable user is first detected and subtracted from the received signal. In the following, we consider the performance of the HPIC receivers under the fading channel environment. Figure 5.16 demonstrates the performance comparison of 5-stage receivers for a single-path rician fading channel. The reflect-to-diffuse ratio was set 7 dB and
5 10 15 20 25 30 10−4
10−3 10−2 10−1 100
Number of users
BER
Conv. receiver PHPIC APHPIC APHPIC+GGS Proposed Proposed+GGS Singleuser bound
Figure 5.10: Fourth stage BER performance comparison vs. user numbers (the parameter set-ting is the same as that in Figure 5.8 for all stages).
dB. Note that the channel gain was constant during a bit interval and varied bit by bit independently. We can observe that the proposed algorithm outperforms the PHPIC and APHPIC receivers. We next use a two-path fading channel; the second path is one chip delay with respect to the main path, and each path gain is Gaussian distributed with zero mean. The result is shown in Figure 5.17. The proposed algorithm still has the best performance. The GGS algorithm is not employed here since it is not suitable for the bit-asynchronous systems.
C. Effect of channel estimation error
In the adaptive HPIC receiver scenario, channel information is necessary for initial setting and for interference cancellation. The proposed algorithm also requires channel information to determine the optimal parameters. All of the simulations conducted above have assumed perfect channel estimation. However, in practice, channel estimation cannot be perfect and its error has to be taken into account. Consider a model for channel estimation error as
(5.50)
5 10 15 20 25 30 10−4
10−3 10−2 10−1 100
Number of users
BER
Conv. receiver PHPIC APHPIC APHPIC+GGS Proposed Proposed+GGS Singleuser bound
Figure 5.11: Fifth stage BER performance comparison vs. user numbers (the parameter setting is the same as that in Figure 5.8 for all stages).
where is a Gaussian distributed random variable with zeros mean and standard deviation
. We then use instead of in the receiver. Figure 5.18 show the simulation results. As seen from the figure, the proposed algorithm always performs better than the APHPIC under different channel estimation error scenarios. Note that the GGS algorithm will fail when the channel estimation error is large. The proposed algorithm is the most robust one among the multiuser receivers compared.
0 1 2 3 4 5 6 7 8 9 10 10−6
10−5 10−4 10−3 10−2 10−1 100
Eb/N0
BER
Conv. receiver PHPIC APHPIC APHPIC+GGS Proposed Proposed+GGS Singleuser bound
Figure 5.12: Second stage BER performance comparison vs.
ratios ( , and the parameter setting is the same as that in Figure 5.8 for all stages).
0 1 2 3 4 5 6 7 8 9 10
10−6 10−5 10−4 10−3 10−2 10−1 100
Eb/N0
BER
Conv. receiver PHPIC APHPIC APHPIC+GGS Proposed Proposed+GGS Singleuser bound
Figure 5.13: Fifth stage BER performance comparison vs. ratios ( , and the parameter setting is the same as that in Fig. 5.8.
5 10 15 20 25 30 10−4
10−3 10−2 10−1 100
Number of users
BER
Conv. receiver PHPIC APHPIC APHPIC+GGS Proposed Proposed+GGS Singleuser bound
Figure 5.14: Second stage BER performance comparison for the weakest user (power-imbalanced, dB, and the parameter setting is the same as that in Fig. 5.8).
5 10 15 20 25 30
10−4 10−3 10−2 10−1 100
Number of users
BER
Conv. receiver PHPIC APHPIC APHPIC+GGS Proposed Proposed+GGS Singleuser bound
Figure 5.15: Fifth stage BER performance comparison for the weakest user (power-imbalanced,
dB and the parameter setting is the same as that in Fig. 5.8).
5 10 15 20 25 30 10−3
10−2 10−1 100
Number of users
BER
Conv. receiver PHPIC APHPIC APHPIC+GGS Proposed Proposed+GGS Singleuser bound
Figure 5.16: Fifth stage BER performance comparison for single-path rician fading channels ( dB, , and for all stages).
5 10 15 20 25 30
10−4 10−3 10−2 10−1 100
Number of users
BER
Conv. receiver PHPIC APHPIC Proposed
Singleuser bound(estimated)
Figure 5.17: Fifth stage BER performance comparison for two-ray multipath fading channels ( dB, , and for all stages).
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 10−3
10−2 10−1 100
σa/a k
BER
Conv. receiver PHPIC APHPIC APHPIC+GGS Proposed Proposed+GGS
Figure 5.18: Fifth stage BER performance comparison vs. channel estimation errors ( ,
dB, , , and for all stages).
Chapter 6 Conclusions
In DS-CDMA communication systems, MAI is considered as the main factor limiting the sys-tem performance. Among many multiuser detection schemes, the PIC receiver is considered as a simple yet effective approach. It has been shown that the performance of the PIC can be fur-ther improved if interference is not fully cancelled. The performance of a partial PIC depends heavily on the PCFs. Thus, how to determine PCFs optimally is of great concern.
In this dissertation, we have studied two types of partial PICs. Using the BER criterion, We first develop a two-stage decoupled partial SPIC and derive a set of closed-form solutions for optimal PCFs. These PCFs are useful for periodic and aperiodic spreading codes in additive white Gaussian noise channels, and for those in multipath channels. Simulation results show that the derived optimal PCFs agree closely with empirical optimal PCFs. The optimal two-stage partial SPIC outperforms a conventional matched filter detector, a two-two-stage full SPIC detector, and even a three-stage full SPIC. Simulations have also shown that the derived de-coupled partial SPIC performs similarly to the optimal two-stage partial SPIC with de-coupled structure. We have also shown that the derived PCFs are not sensitive to parameter estimation errors. It can be noted that the optimal PCFs for aperiodic spreading code systems in AWGN channels have a simple expression. This will be a great advantage for real-world applications since the optimal PCFs can be calculated efficiently on-line in a time-varying environment.
We then conduct performance analysis for a two-stage adaptive blind partial HPIC receiver in the AWGN channel. We first derive the analytical result for the optimal weight, the adapted weight mean, and the adapted weight variance in a single-user case. Then, we derive the op-timal weights and adapted weight mean for a two-user case. Finally, we extend the result to a general-user case. With the results obtained above, we are able to derive the formula for the output MSE and the BER. Using the output MSE criterion, the optimal step size can then be obtained. Simulation results show that the analytical results are accurate. In the final part of the dissertation, we propose an improved adaptive blind partial HPIC receiver. The main idea is to reduce weight variance introduced by the LMS algorithm so as to reduce the output MSE.
We use two approaches; the first is to reduce the number of adapted weights and the second is further process the convergent weights. To implement these ideas, we propose the weight se-lection and weight post filtering schemes. Simulation results show that the proposed algorithm outperforms the conventional adaptive approach in all scenarios. In power-imbalanced systems, the proposed algorithm can approach the optimum performance. We also derive analytical re-sults for the proposed algorithm which include output MSE and BER. It has been shown that the analysis results are reasonably accurate.
In concluding this dissertation, we suggest some topics for further research. The optimal PCFs derived for the SPIC in the multipath scenarios are complicated and not suitable for real-time calculations. We then need a simpler approximate expression. Also, we are mainly con-cerned with BPSK modulation. Note that the same result can be extended to accommodate QAM modulation. In this case, however, we have to take the interference between inphase and quadrature components into account. It turns out that for the inphase or quadrature component of one user, we may treat the number of interfering users as .
In the analysis of adaptive blind HPIC, we do not derive the weight variance for the two-user case. As an alternative, we use the result from the single user to perform-user approximation.
This contribute resultant inaccuracy significantly. Since we use the two-user result in weight mean analysis, the analytical weight mean is more accurate than the analytical weight variance.
The other problem is that we do not consider the multipath scenario. It seems that we can extend our results to the scenario; however, the derivation may become much more complex.
The proposed improved adaptive algorithm has not taken full information we have. The weight selection process only consider the two-stepsize case. It can be expected that a continu-ous step size will give even better result. Also, we have not considered the initial value problem.
If the first stage decision is likely to be erroneous, the initial should be close to zero. On the other hands, it should be close to . The weight post filtering does not achieve its optimal performance either. As we mentioned, the optimal filtering function consists of a hyperbolic tangent function. The parameters of the function should depend on the weight variance. So, it will be different stage by stage. The information we have is the channel gain which is. Whether or not the processing schemes mentioned above can fully explore the information de-serves further study.