, and dB); The optimal PCFs for the partial HPIC were obtained by trial and error and those for the SPIC were obtained from (3.27).
Note that the first terms in (3.65)-(3.68) are those in (3.32) and (3.34)-(3.36) which correspond to the optimal PCFs in an AWGN channel. Other terms are due to the multipath channel effect.
It is evident to see that if3 , and the metrics above are then degenerated to (3.32) and (3.34)-(3.36).
In prior sections optimal PCFs for different scenarios are derived under the assumption of static channels. The received user amplitudes are regarded known and to be varying slowly. The extension to fading channels is straightforward. The derivation is summarized in Appendix C.
§ 3.5 Simulation Results
A. Performance comparison for various partial PIC structures
1 2 3 4 5 6 10−5
10−4 10−3 10−2 10−1 100
∆γ
BER
Decoupled structure Coupled structure
user #1
user #2
user #3
Figure 3.3: Performance comparison for the coupled and decoupled structures (three users with
,, and 8 dB); The optimal PCFs for the coupled structure were obtained by trial and error, and those for the decoupled structure were obtained from (3.38).
In this section we provide simulation results to verify the validity of our derived PCFs. Before we do that, we give some comparison results to justify the PIC structure we considered. First, we compare the performance of a partial SPIC and that of a partial HPIC. We used equicorrelated codes of length (
) as spreading codes. Let be 8 dB ( ), and assume a perfect power control scenario. It is straightforward to see that in the perfect power control case, optimal PCFs are equal for the coupled and decoupled structures. Figure 3.2 shows the bit error rate (BER) performance versus the number of users. Here, optimal PCFs for the partial HPIC were determined empirically (trial and error with a resolution of 0.01).
Surprisingly, we found that the optimal partial SPIC outperformed the optimal partial HPIC.
This result differs from the result given in [56] where the full SPIC was found to be inferior to the full HPIC.
In the second set of simulations, we compared the performance of the coupled and
decou-0 2 4 6 8 10 12 14 16 10−4
10−3 10−2 10−1 100
Eb/N 0
BER
Empirical (K=10) Theoretical (K=10) Empirical (K=20) Theoretical (K=20)
Figure 3.4: BER of the partial SPIC detector versus
(aperiodic AWGN channels, and power balanced).
pled structures (using a partial SPIC). As mentioned above, optimal PCFs are equal for both structures under perfect power control. Thus, we compared their performance in an imperfect power control scenario. The optimal PCFs for the coupled structure were determined empiri-cally. Let the number of users be three and the spreading code be aperiodic (of length 31). We assumed that the third user had a fixed
of 8 dB, and the other two users had variable
values ofand dB, respectively. Figure 3.3 shows the BER performance versus for these two structures. As we can see, both structures performed similarly.
B. Validity of derived PCFs
In this paragraph, we report simulation results demonstrating the accuracy of our theoretical solutions. A two-stage decoupled partial SPIC was considered. For the simulations conducted, we used Gold codes for periodic code systems and random codes for aperiodic code systems.
Figure 3.4 gives the empirical and theoretical BERs for the optimal partial SPIC detector (with the aperiodic code scenario). This figure shows the validity of the Gaussian assumption used in
5 10 15 20 25 30 0.4
0.45 0.5 0.55 0.6 0.65 0.7 0.75
K
PCF
Empirical Theoretical
Figure 3.5: Optimal PCF versus number of users (Gold codes, asynchronous AWGN channels,
dB, and power balanced).
our derivation. As we can see, when the number of users was smaller and
was higher, the Gaussian approximation was less valid. Figure 3.5 shows the optimal PCFs in (3.27) and the empirical optimal PCFs versus the number of users. The channel here was an asynchronous AWGN channel, the spreading codes were periodic, and
was 8 dB for each user. In the figure, it can be seen that the theoretical optimal PCFs were very close to the empirical ones in all cases. We then considered optimal PCFs for a multipath channel. The multipath channel assumed was a two-ray channel with the transfer function4
(for all users). Theoretical optimal PCFs derived in (3.56) were compared with empirical PCFs and the results are shown in Fig. 3.6. We can observe that the theoretical results also matched with the empirical ones satisfactorily. Note that when the number of users was smaller, the theoretical values were less accurate. This was because when the user number is small, the Gaussian approximation in (3.30) is less valid. This was also consistent with the result observed in Fig.
3.4.
5 10 15 20 25 30 0.2
0.3 0.4 0.5 0.6 0.7 0.8 0.9
K
PCF
Empirical Theoretical
Figure 3.6: Optimal PCF versus number of users (aperiodic codes, multipath channels,
dB, and power balanced).
C. BER performance comparison
In what follows, we report the BER performance for various SPIC detectors. Figure 3.7 gives the performance comparison for an optimal two-stage partial SPIC, a conventional matched-filter receiver, a two-stage full SPIC, and a three-stage full SPIC. The spreading codes were periodic and the channel was an asynchronous AWGN channel. Also,
was 10 dB and perfect power control was assumed. From the figure, we can see that the optimal two-stage partial SPIC outperformed others in all cases. The two-stage and three-stage SPIC receivers performed even worse than the conventional matched-filter receiver when the number of users was large. The optimal two-stage partial SPIC always performed better than the matched-filter receiver. Finally, Figure 3.8 shows the performance comparison for the detectors considered above in the multipath channel. The simulation setup was identical to that in the previous cases except that the spreading code was aperiodic. The PCFs for the optimal two-stage partial SPIC
5 10 15 20 25 30 10−4
10−3 10−2 10−1 100
K
BER
Matched filter 2−stage partial SPIC 2−stage full SPIC 3−stage full SPIC
Figure 3.7: BER versus number of users (Gold codes, asynchronous AWGN channels,
dB, and power balanced).
were calculated using (3.56). As in the AWGN channel, the optimal two-stage partial SPIC outperformed other types of detectors.
D. Effect of imperfect parameter estimation
In the optimal PCF formulation, we assumed that the required parameters are perfectly known. In practice, this may not be always possible. Some parameters will have to be esti-mated for time-varying channels which may introduce errors. The main parameters we need to know are the channel responses and the noise variance. Once the channel responses are known,
’s,’s and’s can be calculated accordingly. We modeled the channel estimation error as follows. Let
)
be the-th path channel of User, and
, where
was the estimated channel response, was the actual response, and was a Gaussian ran-dom variable denoting the estimation error. We first let the noise variance be exactly known and varied the channel estimation error. The performance impact is shown in Fig. 3.9. The result corresponds to the case in which the user number was six, the spreading code was aperiodic, the
5 10 15 20 25 30 10−3
10−2 10−1 100
K
BER
Rake receiver 2−stage partial SPIC 2−stage full SPIC 3−stage full SPIC
Figure 3.8: BER versus number of users (aperiodic spreading codes, multipath channels,
dB, and power balanced).
channel was the multipath channel, and
was 10 dB. In the figure,denotes the standard deviation of (same for all’s and’s). Since the matched-filter and the full SPIC receivers do not rely on channel information, the channel estimation error had no influence on their per-formance. (The BER variations in Fig. 3.9 were due to the random data used in different runs).
As we can see, the partial SPIC performance was not affected until
. Note that the magnitude of the main path was 0.762. Thus, the estimation error was quite large in this case.
The second case we considered was noise variance estimation error. The simulation setup was identical to the previous one. We let the channel responses be known and varied noise variance from to , where was the actual noise variance. We found that the optimal SPIC performance was almost unaffected. Thus, we conclude that the optimal partial SPIC has good immunity to parameter estimation errors.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 10−3
10−2 10−1
σg
BER
Rake receiver 2−stage partial SPIC 2−stage full SPIC 3−stage full SPIC
Figure 3.9: BER with channel estimation error (aperiodic spreading codes, multipath channels,
=6, dB, and power balanced).