4.2 Simulation Results
4.2.2 Imperfect relay error detection
In this subsection, we use simulation to investigate performance of the system under im-perfect relay error detection, including the PEP with ML decoder when there exist some harmful relays, for which we are incapable of carrying out an analytical solution. In Fig.
5, we present the block error rate under p0|1 = 0 and different positive p1|0. We can see
−5 0 5 10 15 20 25
Comparison of two decoders under 1% PDP
SNR(dB)
Comparison of two decoders under 10% PDP
block error rate
SNR(dB)
subpot, 10%
ML, 10%
Figure 4.3: The block error rate versus SNR of different PDP under different decoders.
that even with p0|1 = 0, it still has error floor for all cases, which corresponds to Fact 1, fixed positive p1|0 leads to zero diversity. However, we may observe that increase the relay number can lower the error floor efficiently. It directly comes from BER −→ pR1|0 as SN R−→ ∞.
Then, in Fig. 6, we present the block error rate under some particular relay status with two different decoders used in receiver. The number of relay is set to be R = 4.
In the legend, the status of relays is indicated as “1” for useful relay, “0” for useless relay and “2” for harmful relay. For example, “ML1112” represent that three relays are useful and one is harmful with receiver employing ML decoder of equation (3.33). And
“subopt1102” stands for two useful relays, one useless and one harmful with receiver use the suboptimum correlator-like decoder. As we can see in Fig. 6, no matter how many useful relays is currently in the system, the suboptimum receiver has a severe error floor
0 5 10 15 20 25 30 10−4
10−3 10−2 10−1
BER under different positive P
0|1 with R=2, 4
Figure 4.4: The block error rate versus SNR of two decoder under existence of harmful relay.
when there exist any harmful relay. While for ML decoder, there is no error floor for both
“1102” and “1112”. And from the figure, we see the diversity that ML decoder could retrieve is of order 2 for “1102” and of order 4 for “1112”. We might guess that the diversity would be
L× (#(useful relay) − #(harmful relay)) (4.1)
for using ML decoder at receiver. And we can also see that the BER of the two decoders has only minor gap at low SNR but it turn to be significant very soon when we increase the SNR. This again demonstrate the importance of the diversity.
To have a realistic scenario, we further simulate the system for the relay employing
0 5 10 15 20 25 30 10−6
10−5 10−4 10−3 10−2 10−1 100
Performance of two decoder under existence of harmful relay with R=4
SNR(dB)
block error rate
R4ML1102 R4ML1112 R4subopt1102 R4subopt1112
Figure 4.5: The block error rate versus SNR of two decoder under existence of harmful relay.
no error detection, i.e., all relays are always active. This scenario is simpler in imple-mentation since the relay do not need any control protocol. It corresponds to p1|0 = 0 and p0|1 = 1 in our model. We simulate it under both number of relay R = 4 and R = 2. In Fig. 6, “R2nocontrolsubopt” stands for R = 2 and that suboptimum cor-relator decoder is used. Similar for other legend. From the figure, we can observe that the achieved diversity order is 2 for suboptimum decoder for both R = 2 and R = 4.
It verified Proposition 3 given in Sec. 3.2, i.e., correlator-like decoder can retrieve only diversity of order min{L + L01, RL10, RL} = L = 2 under detection accuracy p0|1 = positive constant .
= P−0 and p1|0 = 0 .
= P−∞. In such case, as we can see, in-creasing number of relay has only limited help and has no increase in diversity if we
em-ploying the correlator-like decoder at receiver. On the other hand, ML decoder achieves diversity 2 and 4 for R = 2 and R = 4 respectively, as presented in Fig. 7. Comparing withFact 2 in Sec. 3.2, diversity under fixed positive p0|1 do not exceed R2 L, we can see that ML decoder achieves R2 L in the presented cases. It could be guessed that the maximum diversity under fixed positive p0|1 and p1|0 = 0 would indeed achieve R2 L by an ML decoder. Actually, we have following conjecture,
Conjecture 4 For p0|1 .
This conjecture can be proved by a similar argument of finding the worst diversity in summand of (3.30) and solving the minimization, which is
min
k10+k01≤R k10≥0,k01≥0
k10L10+ k01(L + L01) + max{R − k10− 2k01, 0}L,
provided the guess (4.1) is true. Finally note that Conjecture 4 covers both Fact 1 and Fact 2, and as mentioned, the result of ML decoder in Fig. 7 also conforms to it.
−5 0 5 10 15 10−8
10−7 10−6 10−5 10−4 10−3 10−2 10−1 100
Performance of system without relay error detecion for two decoders on R=2,4
SNR(dB)
block error rate
R2nocontrolsubopt R2nocontrolML R4nocontrolsubopt R4nocontrolML
Figure 4.6: The block error rate versus SNR of two decoder under no control protocol.
Chapter 5
Conclusion and Future Work
5.1 Concluding Remarks
In this work, we analyzed the diversity order of noncoherent cooperative SFC in a two-hop wireless relay network with the DAF protocol. We considered a realistic scenario that each relay node may fail to correctly decode the message from the source node, thus modeling the uncertainty naturally arisen in wireless relay networks. We first analyzed the system under a usual assumption of perfect relay censoring and discussed partial knowl-edge receiver and completely noncoherent receiver. For partial knowlknowl-edge receiver, we analyzed the PEP based on the ML decoding rule and establish the code design crite-ria. For completely noncoherent receiver, we proposed a maximum diversity achieving suboptimum correlator-like decoder, and showed that the maximum achievable diversity using the suboptimum decoder is the same as that of the partial knowledge ML decoder.
We justified that the diversity gain of the non-coherent cooperative SFC in the relay net-work under perfect relay error detection is, on the average, equal to the product of the total number of cooperating relays and the channel order in the relay-destination link, a result identical to that of the non-coherent SFC in MIMO-OFDM systems. This provided us with the insight that the noncoherent virtual MIMO networks could potentially offer as good performance, in terms of diversity order, as that promised by the conventional
noncoherent MIMO networks. Furthermore, we explored the impact of imperfect cen-soring and showed that it has significant influence on the achievable diversity, especially when there exists any “harmful” relays. We concluded that, in a DAF relay system, it is crucial to carefully design the relay censoring schemes in order to maintain the achievable diversity that potentially existed in the system.