curve matches the step-like nature of the snapshot trajectory, as expected. That is, the curves are optimized for the points at which the decoding trajectory is expected to make contact. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 IAIC, IETD I IC E , I T D A Turbo Decoder (P∗∞) Interference Canceller (P∗∞) Turbo Decoder (P∗4) Interference Canceller (P∗4) Snapshot Trajectories (P∗4)
Figure 5.7: EXIT chart for the unconstrained (P∗
∞) and 4-iteration (P∗4) power optimized IMUD receivers, with snapshot trajectories in the complexity constrained system, for
KT = 60 users (N = 30) and no power groups at γb= 0dB.
5.4
Simulation Results
Since the EXIT functions described in Section 4.8 assume a large-scale system (K, N → ∞) such that that PDF of MAI is approximately Gaussian, and the block length is finite, we expect some performance differences between this system and the asymptotic performance predicted. However, we have observed a good match when the number of users K >20. In Fig. 5.8, a snapshot trajectory for a system withK = [20,10,5] users and SF ofN = 20 at γb = 2dB (¯γb = 2.85dB) is shown. The snapshot closely matches the EXIT chart, demonstrating that KT = 35 users is sufficiently large for the MAI to have a Gaussian distribution.
Two systems with KT = 60 users and spreading factor N = 30 were simulated, first with equal power (i.e. un-optimized) then with the optimized power levels for NK = 3 power groups from Section 5.2.2. Note that the power optimization algorithm and thresholds are defined such that all user groups achieve the target BER. Recall that unless otherwise specified,Pref =P1 and the average SNR ¯γb is calculated using (4.19), which is used to compare systems with different power profiles P. Unless specified otherwise, all BER values are the system average, calculated as in (4.35).
5.4.1 Equal Power System
Consider a heavily loaded (K= [60],P = [1], N = 30) equal power system. EXIT chart analysis in Fig. 5.9 shows the convergence threshold (dashed line) occurs at an SNR of
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 IAIC, IETD I IC E , I T D A Turbo Decoder Interference Canceller Snapshot Trajectory
Figure 5.8: EXIT Chart for power optimized unequal power IMUD receiver with snapshot trajectory, K= [20,10,5], N = 20,P∗
∞= [1,1.465,1.795] atγb = 2dB (¯γb = 2.845dB).
γb= ¯γb = 9.15dB and the 4-iteration threshold (dot-dashed line) at 17dB. We observe that the EXIT characteristics of the TD cause the bottleneck in this equal power system. The receiver should exhibit a sharp drop in BER over iterations once decoding has progressed through the narrow tunnel. However, in this case the receiver fails to converge, as shown by the solid line in Fig. 5.10 which presents the BER performance of the equal power system withid= 6 andir = 4. Also shown in Fig. 5.10 is the BER performance predicted through EXIT chart analysis (dashed line). Clearly EXIT analysis has overestimated the performance of the equal power system, which is interference limited. Referring to (4.10), we note that for high system load (K/N) the interference term is dominant and increasing the SNR has a decreasing effect on the system performance, as is evident from Fig. 5.10. This is the reason for the absence of a waterfall region, where a small increase in SNR causes a sharp drop in BER. Furthermore, the tunnel between the EXIT functions in Fig. 5.9 is narrow and the analysis is therefore very sensitive. A small overestimation of the TD performance on the first iteration will be amplified at later iterations and cause a relatively large error.
5.4.2 Unequal Power Optimized System
A turbo coded unequal power CDMA system was simulated with K= [20,20,20] users, SFN = 30 and optimized power profileP∗
∞= [1,1.5381,2.3917]. Note that the KT = 60 users are arbitrarily divided into NK = 3 groups. The performance gains are potentially greater whenNK = 60, i.e. the power of every user is individually optimized. However, for simplicity and in order to maintain the complexity of the scheduling algorithm presented in Chapter 6 to reasonable level for simulations we restrict NK to three.
According to EXIT chart analysis in Fig. 5.2 the convergence threshold of this system is atγb = 1.06dB (average SNR ¯γb= 2.95dB) and the 4-iteration threshold is atγb= 4.05dB.
5.4 Simulation Results 93 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 IAIC, IETD I IC E , I T D A Turbo Decoder Interference Canceler (9.15dB) Interference Canceler (17dB)
Figure 5.9: EXIT chart for equal power IMUD receiver with K= 60, N = 30 system at
γb= 9.15 and 17dB. 14 15 16 17 18 19 20 10−6 10−5 10−4 10−3 10−2 10−1 100 SNRγb(dB) BER Simulation
EXIT Chart Analysis
Figure 5.10: BER vs SNR for equal power system withK= 60,N = 30 system.
We simulated the system over a range of SNR in the region of the 4-iteration threshold. Using (4.19), the average SNR at the 4-iteration threshold is ¯γb = 5.93dB, while the equal power system (with an identical load) failed to converge in 4 receiver iterations. The full decoding schedule was arbitrarily set as all groups running 6 TD iterations (id = 6) and 4 receiver iterations (ir = 4). The corresponding EXIT chart snapshot trajectories are shown in Fig. 5.2 atγb= 4.05dB. The snapshot trajectories closely match the EXIT chart analysis.
BER performance is plotted versus SNR in Fig. 5.11, for EXIT chart analysis (dashed line) and simulation (solid line). Observe that the simulation results quite closely match
EXIT chart analysis. Importantly, the convergence threshold is accurately predicted by EXIT chart analysis.
3.6 3.65 3.7 3.75 3.8 3.85 3.9 3.95 4 10−4 10−3 10−2 10−1 SNRγb(dB) BER
EXIT Chart Analysis Simulation
Figure 5.11: BER performance of unequal power CDMA system K = [20,20,20], P∗ ∞ =
[1,1.5381,2.3917] and N = 30 for IMUD receiver following the dynamic, static and full decoding schedules.
5.4.3 4-Iteration Power Optimized System
Fig. 5.12 compares the BER performance from EXIT chart analysis with simulation of the 4-iteration optimized system, with 3 equal sized power groups (K= [20,20,20], N = 30). We observe that the EXIT chart analysis slightly overestimates the BER, however importantly the convergence threshold is within a decibel of the prediction.
Several snapshot trajectories are presented in Fig. 5.7 for the complexity constrained power optimized system. The trajectories exhibit a good match with the EXIT functions and all users achieve the target BER within 4 receiver iterations as expected.