For convenience of reference we refer to the ML detector using the ER estimators pre-sented in Section II as the physical-link-only (PLO) detector and that using a VLA estimator as the VLA detector. The ML detector with perfect CSI is called the ideal detector. Let dsrk, drkd, dsd be the distances of the kth SR, RD links and the SD link and θsrk be the angle between the SD and kth RD links; see Fig. 1.2. Without loss of generality, we use the normalization, dsd = 10 so that
d2srk = d2rkd+ d2sd− 2drkddsdcos θsrk = 100 + d2rkd− 20drkdcos θsrk, (3.13) We assume the path loss model, σij2 ∝ d−αij with the normalization σ2sd = 1 and α > 0.
Denote by σij2 the variance of the Rayleigh faded link gain and dij the distance between node i and node j, (i, j) ∈ {(s, rk), (rk, d), k = 1,· · · , L}. All the simulated performance curves are obtained by sequentially applying the proposed methods, i.e., the estimated ERs are updated sequentially as each new sample becomes available and the updated
estimates are then used for detecting each received bit. As in [25], we define the SH average SNR as the average received SNR for the direct SD link without relaying, ¯γsd. Simulation for a given ¯γsdterminates whenever the number of error events in the detector output exceeds 500. We assume that noise powers at DN and RN are the same, σ2d = σr2, and use the normalization P = Ps+∑L
i=1Pri = 1 such that ¯γsd = 1/σd2. To reduce the complexity of the ML detector, [24] suggested a piecewise linear function to approximate the nonlinearity (2.3). As it causes negligible performance degradation with respect to that of the ML detector so long as esrk < 12, we use the same approximation in our simulation efforts. Fig. 3.2 illustrates the block diagram of the maximum likelihood (ML) detector, where fT(t) is approximated by a piecewise linear function: fT(t; esrk)≈ min(max(t,−T ), T ) and T = ln(1−e
Figure 3.2: Block diagram of the maximum likelihood (ML) detector.
The performance of the PLO and VLA detectors for the simplest case, L = 1 with BPSK modulation, is illustrated in Fig. 3.3. For the PLO detector, only esr is unknown while the VLA detector assumes ERs of other component links are also unavailable and uses a rotation angle θ = 45◦, which is equivalent to injecting noise with a(v)sd = a(v)rd = 2.
The performance of both detectors are found to approach that of the ideal ML detector.
We also investigate the effect of correlated fading on the performance of the VLA detector for DPSK signals and the result is shown in the same figure. Modified Jake’s model [38]
with normalized Doppler frequency J = fdTs = 0.001, where fdand Tsbeing the Doppler frequency and the sampling period, respectively, is used to generate the component link gains, {hsd[n]}, {hsr[n]}, and {hrd[n]}, as a function of sampling epochs. For the DPSK system, we use the noise-injected VLA detector with scaling factors a(v)sd = a(v)rd = 2; see (3.10). Obviously, the performance of the VLA detector is almost the same as that of the ML detector within the range of interest, indicating that the i.i.d. assumption gives accurate ER estimates for moderately correlated fading environments.
Fig. 3.3 also show the performance for the cases of two and four RNs. In the two-relay case, we assume that the PLO detector knows erkd perfectly. Again, both PLO and VLA detectors yield performance almost identical to that of the ML detector. For the four-relay case, we decompose the problem into four single-relay CCN subproblems, each involves only one SRD and the SD links. It can be seen that at the low SH-SNR region (0 ∼ 2 dB), the performance of the VLA detector is slightly worse than that of the optimal detector. This is due to fact that the sample size used is not large enough to offer a very reliable BER estimate. Nevertheless its performance is still superior to that of the MRC detector.
In the communication systems, the transmission are packet-based. To investigate the effect of finite length packet, we first estimate the ERs given two fixed finite samples and simulate the bit error rate performance given the estimation results. If the ERs are small, the ER estimates are quite bad given insufficient samples. Hence, it can be expected that the performance will be degenerated, especially at high SNR region, as shown in Fig. 3.4. Moreover, increasing the sample size can also improve the bit error rate performance due to the better ER estimates.
Finally, we consider the effect of ER estimate error in bit error rate performance.
Fig. 3.6 shows that the distribution of ysd has a peak near 0 while yrd is more smooth.
0 5 10 15 20 25 30 1E-6
1E-5 1E-4 1E-3 0.01 0.1
1-relay/DPSK 1-relay/BPSK
2-relay BPSK 4-relay/BPSK
Bit error rate
Single-hop average SNR (dB)
Figure 3.3: Bit error rate performance of the ML (solid curves), MRC (◦), PLO (△) and VLA (∇) detectors. The following system parameter values are used. (i) single-relay system: Ps = Pr = 0.5P , dsr/dsd = 0.8, θsr = 0◦, and avsd = avrd = 2 (single-relay), (ii) 2-relay system: dr1d= dsr2 = 7/10, θsr1 = θsr2 = 0◦, Ps = 0.5P, Pr1 = Pr2 = 0.25P , and θ = 45◦, (iii) 4-relay system: dsd = 10, dsr1 = 5, θsr1 = 45◦, dsr2 = 6, θsr2 = 30◦, dsr3 = 4, θsr3 = 60◦, dsr4 = 5, θsr1 = 0◦, Pp = 0.5P, Pri = 0.125P , for i = 1, 2, 3, 4 and θ = 30◦.
If ysr is negative and near 0, then high value of yrd will be underestimated when besr is overestimated (see Fig. 3.5). In this case, the probability that ysr+fT(yrd) < 0 increases, yielding more errors. On the other hand, if ysr is positive and near 0, then high value of yrd will be overestimated when besr is underestimated as the case of error-free. Hence, the error event also occurs more frequently. Nevertheless, if the ER estimate error is not large, then there is not a great difference between the nonlinear function with ER estimate and that with true ER, as shown in Fig. 3.5. Consequently, the performance is not degenerated significantly.
0 2 4 6 8 10 12 14 1E-4
1E-3 0.01 0.1
BER
SH-SNR (dB) PLO-1000
PLO-6144 VLA-1000 VLA-6144 ML
MRC
Figure 3.4: Bit error rate performance of the ML, MRC, PLO, and VLA detectors.
The following system parameter values are used. Single-relay system: Ps = Pr = 0.5P , dsr/dsd = 0., θsr = 45◦, and avsd = avrd = 2.
-10 -5 0 5 10 -10
-5 0 5 10
esr
2esr Error-free
10esr
Nonlinear function output
yrd
Figure 3.5: The nonlinear function (2.3) under various ERs. The system parameter values are Ps = Pr= 0.5P , dsr/dsd = 0., θsr = 45◦ and SNR= 8 dB (esr = 0.0049).
-10 -8 -6 -4 -2 0 2 4 6 8 10 -0.005
0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045 0.050 0.055 0.060 0.065 0.070
ysd
yrd
Probability density function
ysd/yrd
Figure 3.6: The probability density functions of ysd and yrd given x = 1 in single-relay system. The system parameter values are Ps = Pr = 0.5P , dsr/dsd = 0., θsr = 45◦ and SNR= 8 dB.
Chapter 4
Noise-Enhanced ER Estimations
4.1 Convergence consideration and a simple vari-ance reduction method
It is easy to see that, like the estimator for the SMP p defined in Chapter 2, bpsr, bp(vs)r, bps(vr), and bp(vs)(vr) converge in probability. As the proposed estimators are continuous functions of these estimates, the continuous mapping theorem [39] implies that the estimators {besr,berd,besd} converge in probability as well and their variances depend on those of the SMP estimators. The latter are all derived from the same compare-and-count process which is similar to that used in simulation-based ER estimations [40]. The main difference is that, for the latter, the desired detector output is known perfectly and one has complete information and control of the operating average SNR and the link output statistic. In contrast, our scheme can only rely on blind counting without a pilot sequence and the link statistic is either unavailable or only partially known. Both estimation methods, however, have the same order of convergence rate and require a large number of samples to obtain a reliable estimate if the true ER is small; see Lemma 2.1 and [40].
A straightforward approach to improve the convergence performance is to use mul-tiple VLs, i.e., we add nvl− 1 virtual RD and/or SD links with the same noise power.
Each VL renders a set of new estimates and the final estimates are obtained by taking average of the nvl estimates. This method is called the enhanced-VLA (EVLA)
estima-tor which yields a reduced variance for a given sample size, or equivalently, achieves the same variance as that of the original (nvl = 1) estimator with a smaller sample size.