For convenience of reference, we refer to the detector (5.1), (5.8), and (5.10) as the symbol-level, bit-level, and nonsymmetric bit-level detector, respectively. Moreover, the bit-level detector with the proposed ER estimator is denoted as ER-based bit-level detector. The simulated performance curve of ER-based bit-level detector is obtained by sequentially applying the proposed method. The simulations are terminated when 500 errors occurs.
First, Fig. 5.6 illustrates the performance curves with three links and QPSK modula-tion. Clearly, the proposed bit-level detector and symbol-level detector have no difference in BER, showing that the approximation induces negligible performance loss. In addi-tion, the ER-based bit-level detector has the identical performance as bit-level detector.
This indicates that the ER-based bit-level detector can be applied in practice with low computational complexity but high performance.
To investigate the effect of higher order modulation and orthogonal one, we also consider 16-QAM modulation and 4-FSK in three links sensor network in the same figure (Fig. 5.6). Although there is a performance gap between the performance curve of nonsymmetric bit-level and of symbol-level detector for 16-QAM, this gap is smaller than 0.5 dB and is insignificant. Notice that the bit-level and nonsymmetric bit-level detector have no observable difference in performance. Hence, we only consider the bit-level detector with blind ER estimator (ER-based bit-level detector). Importantly, the proposed ER-based bit-level detector for 16-QAM and ER-based detector for 4-FSK have insignificant performance loss with low complexity; hence, they can be implemented in wireless sensor networks.
We consider a 3-link wireless sensor network employing 16-QAM modulation and 4-FSK in Fig. 5.7 and 5.8, respectively. To compare the performance under different link qualities, we adopt the normalized MSEE as metric. Normalized MSEE is defined by MSEEkj/(ekj)2, where j and k denote the jth link and kth bit, respectively. As can be seen, the higher the link quality (Eb/N0), the worse the performance, which is the motivation of the proposed noise-enhanced estimator.Although the analytic results (5.27) is almost identical to the simulation results for 4-FSK, the analytic results (5.19) for 16-QAM is a lower bound performance because of the model mismatch. Nevertheless, the gap between the analytic and simulation results is small.
Fig. 5.9 shows the stochastic resonance phenomenon where we consider a 3-link sensor network with 16-QAM modulation. The MSEE reduction ratio (γ) performance is defined by M SEEM SEEn
o, where M SEEn and M SEEo are the mean square estimation error of noise-enhanced and direct estimator (5.18). As shown in [42], the MSEE reduction ratio γ provides the insight about the reduction ratio of requried sample size for a given normalized MSEE. The smaller the γ is, the smaller the required sample size for
noise-0 5 10 15 20 25
Figure 5.6: Bit (symbol) error rate performance of various detectors with QPSK/16-QAM (MFSK) modulation and Gray mapping labelling. The qualities of the three links for QPSK modulation are denoted by
(Eb
and (SNR,SNR+3,SNR+6) are the link qualities for QAM and 4-FSK, respectively.
enhanced estimator is. We plots the MSEE reduction ratio as a function of the first bit’s NEb
0
(w), the ENb
0 after noise injection, at sample size N = 5000 in Fig. 5.9. Noise are also injected into the other two links such that all links have the same NEb
0
(w). These two curves reveal that 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. For example, the MSEE reduction of the e13 is about 0.002 when ENb
0
(w) = 10 dB. This indicates that the required number of samples for a given precision can be reduced more than 500 times [40]. We also notice that the improvement is more impressive when the true ER becomes smaller, which is consistent
1000 2000 3000 4000 5000 1E-3
0.01 0.1
The first bit of the third link Simulation
Analysis
The second bit of the first link Simulation
Analysis
Normalized MSEE
Sample size (N)
The first bit of the first link Simulation
Analysis
Figure 5.7: Normalized MSEE performance of blind bit-level estimator in a 16-QAM-based three link wireless sensor network. The qualities of these three links are 10, 15, and 20 (dB), respectively.
with what the importance sampling theory has predicted. Finally, we can observe that the noise benefit interval (NBI) is quite wide. For instance, the NBI is from 0 to 20 dB for e11 if we define the noise benefit interval as the interval that γ < 0.1. This observation implies that the noise-enhanced estimator is robust to the noise variance estimate error.
Actually, the MSEE reduction ratio behavior depends on the noise injection strat-egy. To show the possibility of existence of stochastic resonance phenomenon for other strategy, we consider a strategy keeping the difference of these three links’ qualities.
This strategy is equivalent to add the noise with the same noise intensity to all three links. The simulation results are shown in Fig. 5.10 for 16-QAM and 5.11 for 4-FSK.
As can be seen, we have the similar conclusions: the existence of stochastic resonance
1000 2000 3000 4000 5000 1E-3
0.01 0.1
The second link Simulation Analysis
The thrid link Simulation Analysis
The first link Simulation Analysis
Normalized MMSE
Sample size (N)
Figure 5.8: Normalized MSEE performance of blind symbol ER estimator in a 4-FSK-based three link wireless sensor network. The qualities of these three links are 15, 20, and 25 (dB), respectively.
phenomenon and robustness to the estimation error of noise variance.
0 5 10 15 20 25 30 1E-3
0.01 0.1 1
J
(Eb/N0)(w) (dB) Link 1: first bit
Link 1: second bit Link 3: first bit Link 3: second bit
Figure 5.9: MSEE reduction ratio behavior of the noise-enhanced estimator in three links sensor network with 16-QAM modulation and Rayleigh fading channel. The qualities (Eb
N0
)
of these three links are 30, 35, and 40 (dB). Noise are injected into the three links such that all links have the same NEb
0
(w).
-10 0 10 20 30 0.01
0.1 1
γ
(Eb/N0)(w) (dB)
Link 1: first bit Link 1: second bit Link 3: first bit Link 3: second bit
Figure 5.10: MSEE reduction ratio behavior of the noise-enhanced estimator in three links sensor network with 16-QAM modulation and Rayleigh fading channel. The qual-ities
(Eb
N0
)
of these three links are 30, 35, and 40 (dB). We inject noise into the first link ((Eb
N0
)
= 30 (dB) )
such that the quality of the link is ENb
0
(w). We keep the difference of the quality of these three links. That is, the other two links with noise injection have the quality NEb
0
(w)+ 5 and ENb
0
(w)+ 10 (dB).
0 5 10 15 20 25 30 0.01
0.1
γ
SNR(w) (dB)
The first link The second link The thrid link
Figure 5.11: MSEE reduction ratio behavior of the noise-enhanced estimator in three links sensor network with 4-FSK modulation and Rayleigh fading channel. The qualities (Eb
N0
)
of these three links are 30, 35, and 40 (dB). We inject noise into the first link ((Eb
N0
)
= 30 (dB) )
such that the quality of the link is ENb
0
(w). We keep the difference of the quality of these three links. That is, the other two links with noise injection have the quality NEb
0
(w)+ 5 and ENb
0
(w)+ 10 (dB).
Chapter 6
Conclusions and future work
Blind ER estimation is needed for data detection or fusion in wireless relay networks which include sensor networks and cooperative communication networks as subclasses.
Earlier proposals suffer from slow convergence and were unable to estimate the ERs of hidden SR links. Some ambiguity issues associated with cascaded links and the lack of enough links remain unsolved before.
In this dissertation, we first propose noise-enhanced blind ER estimators for binary modulation based wireless relay networks. Noise-enhancement manifests itself in three aspects. Firstly, noise is added to the received samples to create VLs for removing the CSI requirement and resolving the ambiguity associated with an underdetermined system and that due to the symmetric nature of a cascaded link. Secondly, multiple noise-injected VLs are used to reduce the estimation variance and the number of relays needed for estimating ERs. Thirdly, inspired by the IS theory used in computer simulation based ER estimation, noise with proper power is inserted to improve the ER estimator’s convergence performance. The MSEE performance of some special networks is analyzed and both analysis and simulations show that the IS inspired estimator exhibits the so-called stochastic resonance phenomenon which amounts to the effect that injecting noise with a proper power helps improving an estimator’s performance and there exists an optimal injected noise power that offers the best MSEE improvement. Simulation results indicate that the performance of the ML detector using our estimators is very
close to that of the ideal ML detector which knows SR link’s ER perfectly. Moreover, the Monte-Carlo based ISI approach is capable of bringing about several orders of MSEE reduction.
For networks using high order modulation, we find that the optimal symbol-level de-tector is not feasible because of the prohibitive computing load. However, if orthogonal signals such as MFSK is used, the optimal symbol-level detector can be greatly simpli-fied. For general high order modulation based networks, we derive a bit-level detector which requires much smaller number of ER parameters. We propose ERs estimators for the latter two cases. These estimators require low complexity while the existing ERs estimator has to solve a large-scale optimization problem. Simulation results shows that our symbol/bit-level fusion rule using the proposed ER estimator render small per-formance loss (less than 0.5 dB). We also propose noise-enhanced blind ER estimators to improve the MSEE performance for the nonbinary modulation based networks. As expected, simulation results demonstrate that injecting noise with proper power does bring about significant performance improvement and an optimal injected noise power level can be found.
Our work can be extended to deal with applications in distributed source coding [47]. As the noise-enhanced estimator achieves its best performance only if the optimal injected power is known, a more efficient way to find this power level for different sce-narios is needed. In Chapter 5, we have neglected the band-limiting effect and the fusion center receives complete soft outputs [48]. There are cases when only the quantized mea-surements are available at either the sensor nodes or the fusion center and it is desired to have a distributed estimation algorithm [49]. These are some of topics that calls for further investigations. Finally, we also believe that there are many interesting stochastic resonance phenomenon in nonlinear communication systems and networks that deserve much more research efforts to explore their applications.
Appendix A
A.1 Derivation of (2.6)
Given the complete CSI,{hsd, hrd, σ2d, esr} = Icsi and unit transmit powers, Ps = Pr = 1, the conditional joint pdf of the matched filter outputs, ysd, yrd, can be represented as
f (ysd, yrd|Icsi) = C exp
where C is a normalization constant. By removing the terms independent of esr, we obtain