Comparison
Comparison between analysis and simulation is presented in this section. Figure 4.2 and 4.3 show the comparison between analytical and simulated results for a 15-user system. For simplicity, the simulated system is assumed to be chip-synchronous, i.e., all path delays are assumed to be multiples of Tc. However, the system is asynchronous on the symbol level. Perfect slow
power control is assumed in the sense that Pk = PLl=1k Pk,l, the average
received power, is equal for all users. Different paths are assumed to have equal gain and the channel coefficients are normalized so that each user has unity gain, i.e., Pk,1 = Pk,2 =· · · = Pk,Lk and Pk =
PLk
4.4 Analytical Results and Performance Comparison 61
number of multipath channels Lk is set to be 4, (Lk= L = 4) for all k. The
simulated PIC performance in Figure 4.2.b) and 4.3.b) is derived assuming perfect knowledge of the complex channel gains, e.g., the genie-aided case. We observe that the analysis obtained by (4.6) is more accurate for the genie-aided PIC and the analysis obtained by the approximation expressed by (4.7) is more accurate for the PIC scheme with channel estimation (CE). Since channel information has to be estimated in reality, the genie-aided case is not much of practical interest, we therefore focus on the PIC with CE and use (4.7) for the following analysis and comparisons.
Figures 4.4 – 4.7 show the comparison between analytical and simu- lated results for different number of users. The simulated curves precisely match the theoretical ones for the first noncoherent stage, which proves that Gaussian approximation is accurate to model MAI and ISI sequences as well as the elements of each interference sequence in long-code systems. The analysis starts to deviate slightly from simulations, but is still fairly accurate, after the first noncoherent stage. The theoretical analysis is a little pessimistic when the system is too lightly loaded, and a little opti- mistic when the system is too heavily loaded. From both simulation and analysis, one can observe that it takes PIC more stages to converge as K increases (the system becomes more heavily loaded). Seven stages (exclud- ing the first noncoherent stage) ought be enough for the system to reach convergence in any case.
The convergence property of MAI elimination is studied analytically in Figure 4.8 for a 18-user system which is almost fully loaded considering the spreading factor equals 64/3. We find that the variance of MAI approaches its limit quickly. Only 5 stages are required to achieve the desired perfor- mance. The interference can be more thoroughly removed at each iteration at high SNRs. A reasonable level of SNR therefore needs to be maintained in order to benefit from the PIC iteration process. The plot also indicates that MAI cannot be totally removed, thus the receiver cannot achieve sin- gle user performance in a heavily loaded system. Those findings agree with previous analysis and simulation results.
System capacity is illustrated in Figure 4.9 by plotting BER as a func- tion of the number of users using both analytical and simulated results. It is clearly shown that analysis is in close agreement with simulation for BER above 10−4. However, the analysis tends to over-estimate the MAI when
the number of users is very small. Conversely, the MAI is under-estimated when there are too many active users. Compared with the topmost curve which represents the first noncoherent stage, the subsequent PIC stages significantly increase system capacity and BER performance as indicated by both analysis and simulation.
In Figure 4.10, we analyze the PIC with different degree of diversity (different number of paths). It can be seen that the system performance degrades for the first stage as the degree of diversity increases. The rea-
4 6 8 10 12 14 16
10−4
10−3
10−2
10−1
Signal to Noise Ratio E
b/N0 [dB]
BER
M = 8, N = 64, K = 15, L = 4, fdT = 0.01
Simulated results Analytical results
(a) Analysis vs. PIC with CE.
4 6 8 10 12 14 16
10−4
10−3
10−2
10−1
Signal to Noise Ratio E
b/N0 [dB]
BER
M = 8, N = 64, K = 15, L = 4, fdT = 0.01
Simulated results (genie−aided) Analytical results
(b) Analysis vs. genie-aided PIC.
Figure 4.2:Analysis vs. simulation. Analytical BER is derived by (4.6). The number of users is K = 15. Topmost curve represents noncoherent first stage and the second curve from top represents the first stage PIC, the bottommost curve represents the 7th stage PIC.
4.4 Analytical Results and Performance Comparison 63 4 6 8 10 12 14 16 10−5 10−4 10−3 10−2 10−1
Signal to Noise Ratio E
b/N0 [dB]
BER
M = 8, N = 64, K = 15, L = 4, fdT = 0.01
Simulated results Analytical results
(a) Analysis vs. PIC with CE.
4 6 8 10 12 14 16 10−5 10−4 10−3 10−2 10−1
Signal to Noise Ratio E
b/N0 [dB]
BER
M = 8, N = 64, K = 15, L = 4, fdT = 0.01
Simulated results (genie−aided) Analytical results
(b) Analysis vs. genie-aided PIC.
Figure 4.3:Analysis vs. simulation. Analytical BER is derived by (4.7). The number of users is K = 15. Topmost curve represents noncoherent first stage and the second curve from top represents the first stage PIC, the bottommost curve represents the 7th stage PIC.
4 6 8 10 12 14 16 10−5 10−4 10−3 10−2 10−1
Signal to Noise Ratio E
b/N0 [dB]
BER
M = 8, N = 64, K = 6, L = 4, fdT = 0.01
Simulated results Analytical results
Figure 4.4:Analysis vs. simulation, K = 6.
4 6 8 10 12 14 16 10−5 10−4 10−3 10−2 10−1
Signal to Noise Ratio E
b/N0 [dB]
BER
M = 8, N = 64, K = 9, L = 4, fdT = 0.01
Simulated results Analytical results
4.4 Analytical Results and Performance Comparison 65 4 6 8 10 12 14 16 10−5 10−4 10−3 10−2 10−1
Signal to Noise Ratio E
b/N0 [dB]
BER
M = 8, N = 64, K = 12, L = 4, fdT = 0.01
Simulated results Analytical results
Figure 4.6:Analysis vs. simulation, K = 12.
4 6 8 10 12 14 16 10−5 10−4 10−3 10−2 10−1
Signal to Noise Ratio E
b/N0 [dB]
BER
M = 8, N = 64, K = 18, L = 4, fdT = 0.01
Simulated results Analytical results
4 6 8 10 12 14 16 10−2 10−1 100 101 102 103
Signal to Noise Ratio E
b/N0 [dB]
variance of MAI
M = 8, N = 64, K = 18, L = 4, fdT = 0.01
Figure 4.8:Analytical results for MAI variance at each stage of PIC.
4 6 8 10 12 14 16 18 10−5 10−4 10−3 10−2 10−1 Number of users K BER M = 8, N = 64, L = 4, E b/N0 = 17, fdT = 0.01 Simulated results Analytical results
4.4 Analytical Results and Performance Comparison 67
4 6 8 10 12 14 16
10−0.6
10−0.5
Signal to Noise Ratio E
b/N0 [dB] BER M = 8, N = 64, K = 18, L = 4, fdT = 0.01 First MF stage, L=1 First MF stage, L=2 First MF stage, L=3 First MF stage, L=4 First MF stage, L=5 First MF stage, L=6 First MF stage, L=7
(a) Analytical BER for noncoherent first stage.
4 6 8 10 12 14 16 10−6 10−5 10−4 10−3 10−2 10−1
Signal to Noise Ratio E
b/N0 [dB] BER M = 8, N = 64, K = 18, L = 4, fdT = 0.01 7th PIC stage, L=1 7th PIC stage, L=2 7th PIC stage, L=3 7th PIC stage, L=4 7th PIC stage, L=5 7th PIC stage, L=6 7th PIC stage, L=7
(b) Analytical BER for 7thPIC stage.
son is that with a noncoherent MF receiver, the interference is dominant and the multipath combining gain is not sufficient to compensate for the increased interference as the number of paths increases. However, for the following coherent PIC stages, the conclusion is opposite. The interference is effectively removed and the multipath gain becomes dominant. Further- more, the cancellation residual and noise present in the imaginary part of the decision statistic are eliminated. Another discovery is that the first few taps exhibit big performance gain compared to single-path case, while the multipath gain gradually diminishes as the number of paths increases.
Ideal power control is assumed in the above discussion. The near-far robustness of the PIC algorithm is analytically examined in Figure 4.11 by plotting the resulting BER as a function of near-far ratio, which refers to the difference between the power of each of interfering user (it is assumed that P2= P3=· · · = Pm=· · · = PK), and the power of the desired user P1
(the first user is the user of interest). From Figure 4.11.a), we see that the PIC scheme in general is not sensitive to the variations in the interfering signal strengths and is near-far resistant. The only exception is for the single-path system in severe near far situation (when Pm− P1 > 10 dB,
i.e., the desired user is much weaker than the other interfering users), the system performance degrades. This concurs with the results shown in [36]. Figure 4.11.b) shows that the near-far robustness of the PIC scheme comes from interference cancellation process. The initial few stages do exhibit some degree of near-far problem, which will gradually vanish as the iteration goes on and the system reaches convergence. The rationale is that the error probability for strong interfering users is very low due to their high signal power level, we therefore have better chance to make correct cancellation and cancel their contributions, which greatly alleviates the near-far effect. The performance of the PIC algorithm in presence of unequal power among different diversity branches is studied in Figure 4.12 for a 4-path channel. We use the analytical results (4.17) derived in Section 4.3 as well as its approximation expressed by (4.7) and (4.18). In this test, power control is assumed so that the average received power is equal for all users and each user has unity gain. However, the power difference between different paths is set to be ∆Pk,l= Pk,4− Pk,3= Pk,3− Pk,2= Pk,2− Pk,1 = 0, 3, 6
dB, respectively. Figure 4.12 shows that the PIC works the best when all the branches have equal power, i.e., when ∆Pk,l= 0. The bigger deviation
in power, the worse performance it gets.
The above analysis and comparisons are based on the assumption of chip synchronism. The system performance is compared between a chip- synchronous system and a chip-asynchronous system in Figure 4.13 with analytical approach. As expected, the latter system poses less interference and therefore has better performance. Chip synchronism does represent a worse-case interference scenario as stated in [25].
4.4 Analytical Results and Performance Comparison 69 −5 0 5 10 15 10−3 10−2 10−1 Near−Far Ratio P m − P1 [dB] BER M = 8, N = 64, K = 12 , E b/N0 = 14, fdT = 0.01 7th PIC stage, L=1 7th PIC stage, L=2 7th PIC stage, L=3 7th PIC stage, L=4
(a) Analytical BER vs. NFR with different number of paths.
−5 0 5 10 15 10−2 10−1 Near−Far Ratio P m − P1 [dB] BER M = 8, N = 64, K = 12, L = 2 , E b/N0 = 14, fdT = 0.01
(b) Analytical BER vs. NFR at different stages of PIC.
4 6 8 10 12 14 16
10−4
10−3
10−2
10−1
Signal to Noise Ratio E
b/N0 [dB] BER M = 8, N = 64, K = 18, L = 4, fdT = 0.01 PSfrag replacements MF, ∆Pk,l = 0 dB MF, ∆Pk,l = 3 dB MF, ∆Pk,l = 6 dB PIC, ∆Pk,l = 0 dB PIC, ∆Pk,l = 3 dB PIC, ∆Pk,l = 6 dB
(a) BER derived by (4.17).
4 6 8 10 12 14 16
10−4
10−3
10−2
10−1
Signal to Noise Ratio E
b/N0 [dB] BER M = 8, N = 64, K = 18, L = 4, fdT = 0.01 PSfrag replacements MF, ∆Pk,l = 0 dB MF, ∆Pk,l = 3 dB MF, ∆Pk,l = 6 dB PIC, ∆Pk,l = 0 dB PIC, ∆Pk,l = 3 dB PIC, ∆Pk,l = 6 dB
(b) BER derived by (4.7) and (4.18).
Figure 4.12: PIC performance for unequal power diversity branches. MF curves represent the first noncoherent stage. PIC curves are plotted for the 7thstage.
4.5 Conclusions 71