4.4 Verification of Hamming Distortion Lower Bounds
4.4.2 Numerical Results
Figure 33 shows the error probability lower bounds and the BER versus SNR whenp1, p2and SNRs of the two nodes are set identically; this is referred as the symmetric case.
It can be found that, the BER curves obtained by simulations and the theoretical lower bounds on the Hamming distortion exhibit a similar tendency. The gap between the simulated BER and theoretical lower bound on Hamming distortion is caused by: (i) the derived outer bound is not tight, and thus smaller Hamming distortion is obtained for fixed rates; (ii) the Hamming distortion lower bound is obtained by assuming the optimal source coding rate is adopted based on separability, however, fixed coding rate is used in simulations.
Furthermore, it is clearly found that the error floor of the BER obtained by the sim-ulation and the lower bound on the Hamming distortion based on soft combining match exactly. The reason is that if the SNRs of two nodes are large enough, the distortion lev-elsD1andD2are almost 0, which results in the error floor being determined completely by the error probabilitiesp1andp2. A gap clearly appears between the Hamming dis-tortion lower bounds using the soft combining and optimal decision rules. The reason is twofold: 1) the optimality of the soft combining cannot be guaranteed; 2) optimal decision is derived based on the assumption of the binary rate-distortion function
with-−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 10−4
10−3 10−2 10−1 100
SNR of channel #2 (dB)
BER
Simulation (P=[0.02, 0.08]) Lower bound (soft combining) Lower bound (optimal) Simulation (P=[0.005, 0.005]) Lower bound (soft combining) Lower bound (optimal)
Fig. 35. Asymmetricr1 andr2. The coding ratesr1andr2 are set at 14 and 12, respectively.
The transmit power of two nodes is the same. BPSK is used for both nodes.
out any loss during processing the information. To find a better decision rule than soft combining rule is left as a future study. However, it is clear that the Hamming distortion lower bound deriving from the optimal decision cannot be exceeded.
The impact of the variation of the error probabilitiesp1,p2and the coding ratesri are evaluated in AWGN channels. Fig. 34 shows the results for asymmetricp1andp2 but symmetric SNRs. When the coding rates10r1andr2are set as14and12, respectively, the BER performance shown in Fig. 35 is obtained. We further consider using differ-ent modulation schemes for the nodes to achieve differdiffer-ent rates of the channel code in Fig. 36, where QPSK is used for node 1 and BPSK for node 2. Even in these asym-metric cases, the theoretical lower bounds on the Hamming distortion can still provide us with a useful reference when we evaluate the BER performance of practical systems.
Furthermore, the theoretical lower bounds on the Hamming distortion obtained based on our derived outer bound exhibit similar behaviors to those of the BER curves found by simulations.
10We simply transmit the output of ACC without doping to achieve rate 14. No optimized design of the channel code is considered.
−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 10−4
10−3 10−2 10−1 100
SNR of channel #2 (dB)
BER
Simulation (P=[0.01, 0.01]) Lower bound (soft combining) Lower bound (optimal) Simulation (P=[0.005, 0.005]) Lower bound (soft combining) Lower bound (optimal)
Fig. 36. Asymmetricr1andr2. The coding ratesr1andr2are set at1and12, respectively. The transmit power of two nodes is the same. QPSK is used for node 1 and BPSK for node 2.
−10 −8 −6 −4 −2 0 2 4 6 8 10 12 14
10−4 10−3 10−2 10−1 100
per−node average SNR (dB)
BER
Simulation (P=[0.01, 0.01]) Lower bound (soft combining) Lower bound (optimal) Simulation (P=[0.008, 0.005]) Lower bound (soft combining) Lower bound (optimal)
Fig. 37. BER performances over Rayleigh fading channels. Both nodes use BPSK modula-tion.
In both the symmetric and asymmetric cases, the threshold SNR value at which turbo cliff in the BER obtained by the simulation is around 1.5 dB larger than that observed in the theoretical lower bounds in static AWGN channels. In addition, since the lower bounds on the Hamming distortion plateaus at a certain level even if the power is increased at high SNR regime, increasing the number of nodes is a proper way to improve performance in the practical deployment.
In Fig. 37, the channels between two nodes and the destination experience inde-pendent block Rayleigh fading. Therefore, the instantaneous SNRs of two nodes are different while the average SNRs of the two channels are the same. The lower bounds on the Hamming distortion shown in Fig. 37 are calculated as
D⋆fading=
∫ +∞
0
∫ +∞
0
D⋆(γ1,γ2)·Pr(γ1)·Pr(γ2)dγ1dγ2, (102) whereD⋆(γ1,γ2)is the result of (101), obtained for static AWGN channels. Pr(γi)is the probability density function of the SNRγi, which follows the Rayleigh distribution. We use Monte Carlo method to obtain the lower bounds on the average Hamming distortion D⋆fadinginstead of theoretically calculating (102). In the Rayleigh fading case, the shape of the BER curves and the lower bounds on the Hamming distortion are almost the same.
Two points need to be emphasized here. The analytical solution of (102) is difficult to find, becauseD⋆(γ1,γ2)is obtained by solving the formulated convex optimization usingcvxtool. The other point is that, the outage probability approaches 1 using the definition that the outage event happens when the package cannotlosslesslyrecovered.
Hence, the definition of outage should be changed in this case. We follow the method of using Slepian-Wolf theorem and separability to calculate the outage probabilitypout for the situation that bit error floor is reached [57, Section 4.2], where the definition of outage event is
Outage, D>min{p1,p2} Success, otherwise
(103)
The detail of derivingpoutfor two-node case is shown in Appendix 6. We compare the theoretical outage probability poutand the frame error rate (FER)11, where the results is shown in Fig. 38. The FER performance obtained by practical encoding/decoding algorithm are around 1∼2 dB in average SNR to the theoretical outagepout.
11The frame is error if and only ifD>min{p1,p2}.
−10 −8 −6 −4 −2 0 2 4 6 8 10−3
10−2 10−1 100
per−link average SNR (dB)
FER/Outage
p1 = p 2 = 0.005 Outage p1 = 0.005, p
2 = 0.05 Outage
Fig. 38. Comparison between FER and theoreticalpout.