To verify our MSEE analysis, we first consider the data-aided point-to-point BPSK communication in Rayleigh fading channel (toy example case), illustrated in Fig. 4.2.
As can be seen, our analysis is consistent with simulation results and there is indeed a optimal injected noise power. Based on Theorem 4.2, the optimal value of a(w) is 1000 = 30 (dB), which is the channel quality. In practice, the noise variance should be estimated with some estimation error. Fortunately, this simulation also indicates that the performance is not sensitive to the optimal value of a(w). This nonsensitive property shows the possibility to implement the noise-enhanced estimator in the real world.
Then, we consider a 3-link wireless sensor network in Fig. 4.3 which shows that, for all three binary modulations considered, the analytic predictions are very close to those obtained by simulations even when the sample size is small, and both give identical results if the sample size is large. Similar performance trend for the ISI-VLA scheme in a BPSK-based single-relay CCN is found in the same figure. The normalized MSEE
0 500 1000 1500 2000
Figure 4.2: Normalized MSEE performance of the noise-enhanced estimator for data-aided point-to-point BPSK communication in Rayleigh fading channel. The channel quality is 30 dB (error rate e = 2.5× 10−4). The sample size is 10000.
performance, E[(be − e)2]/e2, where e is the true ER, of the VLA, VLA-EM, and EISI-VLA estimation schemes for a BFSK-based single-relay CCN network is shown in Fig.
4.4. The VLA-EM scheme refers to a modified version of the EM based estimator of [33], which did not consider the hidden SR link. The modifications are needed to apply a VL for resolving the ambiguity and replace the normalization factor such that the equation for updating the ER estimate for the cascaded link becomes
Q(i+1)k
where Qi’s are defined in chapter 2 with the superscripts denote the associated iteration number. The ISI method injects additional noise to estimate the ERs of the resulting links and then converting them back to besr and berd via the analytic formulas given in Table 3.2. The performance curves clearly demonstrate that the advantage of the VLA-EM scheme against the VLA estimator is negligible while the EISI-VLA scheme far outperforms the other two schemes.
0 1000 2000 3000 4000 5000
0.01 0.1 1 10
3-link/BPSK
3-link/DPSK
3-link/BFSK SD link in CCN
RD link in CCN SR link in CCN
Normalized MSEE
N (sample size)
Figure 4.3: Normalized MSEE performance of the ISI-VLA scheme for (a) various binary modulated 3-link networks (e1 = 0.003, e2 = 0.002, e3 = 0.001; the injected noise power is such that SH SNR=2 for link 1 and e(w)1 = e(w)2 = e(w)3 ) and (2) BPSK-based single-relay CCN (esr = 0.02922, erd = 0.001988, esd = 0.04356, a(v)sd = a(v)rd = 2, a(w)sd = 1 and a(w)rd = 30). For 3-link networks, only the performance of be1 is shown. The analytic predictions (solid curves) for these two scenarios are based on (4.23) and (4.42)–(4.44), respectively.
Fig. 4.5 plots the MSEE reduction ratio as a function of the scaling factor a(w)sd while the other scaling factor a(w)rd is chosen such that e(w)rd = e(w)sd . These curves reveal that
0 1000 2000 3000 4000 5000 1E-3
0.01 0.1 1 10 100 1000
RD
SD
SR SD link
EISI-VLA VLA VLA-EM
RD link EISI-VLA VLA VLA-EM
Normalized MSEE
N (sample size) SR link
EISI-VLA VLA VLA-EM
Figure 4.4: Normalized MSEE performance of VLA, VLA-EM, and EISI-VLA schemes in a BFSK-based single-relay CCN with esr = 0.0127, erd = 5.0711×10−5and esd = 0.0298.
Other parameter values used are: a(v)sd = a(v)rd = 2, e(w)sd = e(w)rd = 0.05 and nvl = 30.
the MSEE performance is improved by injecting proper noise power into the received samples and there is an optimal injected noise power that achieves the maximum MSEE improvement. This phenomena is called the stochastic resonance effect which has been observed in some nonlinear systems; see [6] and reference therein. We also notice that the improvement is more impressive when the true ER becomes smaller, which is consistent with what the IS theory has predicted. The noise benefit interval (NBI), defined as the range of the scaling factor values within which the MSEE reduction ratio is less than 1, is a function of the true esd and erd. As mentioned before, we are not able to derive closed-form expressions for the optimal scaling factors used in a noncoherent network. Nevertheless, extensive simulations suggest that it is a good strategy to make
e(w)sd = e(w)rd ≈ 0.05 if both esd and erd are much smaller than 0.05. As was explained in Chapter 4, because of the availability of improved estimates for esd and erd, the performance of besr is also improved although we do not and could not inject noise into samples received at RNs.
Figure 4.5: MSEE reduction ratio (γ) performance of the ISI-VLA estimator with BFSK modulation and a(v)sd = a(v)rd = 2. Part (a) is obtained by assuming dsr = 5, SH-SNR=25 dB with the path loss exponent = 2 (which leads to esr = 0.0016, erd = 0.0016, esd = 0.0062). Part (b) assumes that dsr= 8, SH-SNR=18 dB with path loss exponent = 4 so that esr = 0.0127, erd = 5.0711× 10−5, esd = 0.0298. The MSEE reduction ratio of the RD link is not shown in part (b) as it is relatively small (∼ O(10−3)).
Although proper noise-injection does improve the convergence rate performance, in some cases such as those shown in Fig. 4.5, the improvement is not quite as significant as one wishes. The MSEE reduction ratio can be further improved by the enhanced ISI-VLA estimator as is shown in Fig. 4.6 where the simulation conditions are identical
to those assumed in Fig. 4.5(b). As expected, the performance is improved with the increase of nvl and the improvement is much more impressive when the true ER is small:
the required sample size reduction is more than 10 times for the SD link and is greater than 8000 times for the RD link when nvl = 30. Another benefit of using multiple VLs is that the NBI becomes larger as nvl increases.
0 1000 2000 3000 4000 5000
1E-4 1E-3 0.01 0.1 1
SR-link
RD-link
SD-link
γ
N (sample size)
nvl=10 nvl=20 nvl=30
Figure 4.6: MSEE reduction ratio behavior of the EISI-VLA estimator for BFSK based CCN with different nvl. Other system parameter values are the same as those of Fig.
4.5(b).
Chapter 5
Data fusion and blind multiple error rate estimation in a non-binary
modulation based wireless sensor network
For optimal M -hypothesis detection in a parallel system, the performance of the sensor nodes must be available at the fusion center (FC). Such information can be obtained by the LJW blind estimator [34] based on a multinomial distribution model with pa-rameters related to each links’ ERs. To get the estimates, we need to solve a nonlinear optimization problem. As shown in section 5.2, this algorithm is not feasible for large M due to the prohibitively high computational complexity. To approximate the optimal detector, we propose a suboptimal detector based on bit-level representation and a cor-responding blind estimator to estimate the error rate of sensor nodes in section 5.3. The complexity of our estimator is much lower than that of LJW as we are able to obtain a closed-form salutation instead of employing an iterative algorithm for solving a nonlin-ear optimization. To further improve the convergence rate, we propose a noise-enhanced estimator in section 5.5. Simulation results show that the proposed suboptimal detector using the proposed blind estimator render negligible performance loss with respect to that of the optimal detector. A stochastic resonance phenomenon is observed in the estimator’s mean square estimation error performance.
5.1 System model and optimal detector
The parallel sensing system illustrated in Fig. 5.1 consists of one source, L sensors, and one FC. In a wireless sensor network, source transmits a signal from M candidate signals {s0, s1,· · · , sM−1} with a prior probability P(Hi) =P(si is transmitted). Each sensor detects and forwards its decision to the FC which then determines which hypothesis (Hi) is true based on the signals forwarded by the sensors. In this dissertation, we assume that either the sensing (source-sensor) channels or the reporting (sensors-FC) channels is error-free. Such an assumption losses no generality as each combined source-sensor-FC link can be modeled as an equivalent composite channel [34]. The optimal fusion rule depends on the parameters of the equivalent channels which can be estimated by the method described below.
˦˸́̆̂̅ʳ˄
˦̂̈̅˶˸ ˙̈̆˼̂́ʳ
˶˸́̇˸̅
˦˸́̆̂̅ʳ˅
˦˸́̆̂̅ʳL
Figure 5.1: Parallel distributed detection system
Denote by byj ∈ {di, i = 0,· · · , M − 1} the jth sensor’s hard decision and define the event probability eikj
eikj = P(byj = dk|Hi), j = 1,· · · , L, i, k = 1, · · · , M, i ̸= k
Assuming P(Hi) = 1/M , i = 0,· · · , M − 1 and the sensor-FC links are noiseless, we
have the optimal data fusion rule [34]