Simulation Results
6.2 Spatially correlated wireless link noises
Within a WSN, the information from sensors is often gathered inside a small area. Thus, the adjacent link noises may be correlated with each other.
Hence, in this section, spatially correlated wireless link noises are considered instead of the AWGN, where the channel covariance matrix is devised of the form:
With the help of the training sequence, channel estimation and equal-ization at the fusion becomes possible. By adopting the ML estimator in (4.1), our results show that the longer the training sequence, the better the system performance. When there are several faulty sensors, the system per-formance with training becomes even better than the system perper-formance with known covariance matrix, since the training procedure also helps the fusion adapt the sensor fault situation. Specifically, it can be seen from Figs. 6.13–6.16, the misclassification probability decreases as the number of training vectors increases from 15 to 35.
−5 −4 −3 −2 −1 0 1 2 3 4 5 0
0.05 0.1 0.15 0.2 0.25 0.3
CSNR(dB)
Probability of misclassification
4x10 WSN system with LO = 0.1 Spatially Correlated Channel (OSNR = 10dB) (stuck−at−1 fault)
no fault, 15 training no fault, 25 training no fault, 35 training no fault, known covariance matrix 1 fault, 15 training 1 fault, 25 training 1 fault, 35 training 1 fault, known covariance matrix 2 fault, 15 training 2 fault, 25 training 2 fault, 35 training 2 fault, known covariance matrix
Figure 6.13: Performance of the MED rule under ρ = 0.1 spatially correlated channel at OSNR=10 dB for stuck-at-1 faults when 4×10 code matrix is employed.
−5 −4 −3 −2 −1 0 1 2 3 4 5 0
0.05 0.1 0.15 0.2 0.25 0.3
CSNR(dB)
Probability of misclassification
4x10 WSN system with LO = 0.1 Spatially Correlated Channel (OSNR = 10dB) (random fault)
no fault, 15 training no fault, 25 training no fault, 35 training no fault, known covariance matrix 1 fault, 15 training 1 fault, 25 training 1 fault, 35 training 1 fault, known covariance matrix 2 fault, 15 training 2 fault, 25 training 2 fault, 35 training 2 fault, known covariance matrix
Figure 6.14: Performance of the MED rule under ρ = 0.1 spatially correlated channel at OSNR=10 dB for random faults when 4× 10 code matrix is employed.
−2 0 2 4 6 8 10 12 14 0
0.1 0.2 0.3 0.4 0.5 0.6
OSNR(dB)
Probability of misclassification
4x10 WSN system with LO = 0.1 Spatially Correlated Channel (CSNR = 10dB) (stuck−at−1 fault)
no fault, 15 training no fault, 25 training no fault, 35 training no fault, known covariance matrix 1 fault, 15 training 1 fault, 25 training 1 fault, 35 training 1 fault, known covariance matrix 2 fault, 15 training 2 fault, 25 training 2 fault, 35 training 2 fault, known covariance matrix
Figure 6.15: Performance of the MED rule under ρ = 0.1 spatially correlated channel at CSNR=10 dB for stuck-at-1 faults when 4× 10 code matrix is employed.
−2 0 2 4 6 8 10 12 14 0
0.1 0.2 0.3 0.4 0.5 0.6
OSNR(dB)
Probability of misclassification
4x10 WSN system with LO = 0.1 Spatially Correlated Channel (CSNR = 10dB) (random fault)
no fault, 15 training no fault, 25 training no fault, 35 training no fault, known covariance matrix 1 fault, 15 training 1 fault, 25 training 1 fault, 35 training 1 fault, known covariance matrix 2 fault, 15 training 2 fault, 25 training 2 fault, 35 training 2 fault, known covariance matrix
Figure 6.16: Performance of the MED rule under ρ = 0.1 spatially correlated channel at CSNR=10 dB for random faults when 4× 10 code matrix is employed.
−5 −4 −3 −2 −1 0 1 2 3 4 5 0
0.05 0.1 0.15 0.2 0.25 0.3
CSNR(dB)
Probability of misclassification
4x10 WSN system with LO = 0.9 Spatially Correlated Channel (OSNR = 10dB) (stuck−at−1 fault)
no fault, 15 training no fault, 25 training no fault, 35 training no fault, known covariance matrix 1 fault, 15 training 1 fault, 25 training 1 fault, 35 training 1 fault, known covariance matrix 2 fault, 15 training 2 fault, 25 training 2 fault, 35 training 2 fault, known covariance matrix
Figure 6.17: Performance of the MED rule under ρ = 0.9 spatially correlated channel at OSNR=10 dB for stuck-at-1 faults when 4×10 code matrix is employed.
−5 −4 −3 −2 −1 0 1 2 3 4 5 0
0.05 0.1 0.15 0.2 0.25 0.3
CSNR(dB)
Probability of misclassification
4x10 WSN system with LO = 0.9 Spatially Correlated Channel (OSNR = 10dB) (random fault)
no fault, 15 training no fault, 25 training no fault, 35 training no fault, known covariance matrix 1 fault, 15 training 1 fault, 25 training 1 fault, 35 training 1 fault, known covariance matrix 2 fault, 15 training 2 fault, 25 training 2 fault, 35 training 2 fault, known covariance matrix
Figure 6.18: Performance of the MED rule under ρ = 0.9 spatially correlated channel at OSNR=10 dB for random faults when 4× 10 code matrix is employed.
−2 0 2 4 6 8 10 12 14 0
0.1 0.2 0.3 0.4 0.5 0.6
OSNR(dB)
Probability of misclassification
4x10 WSN system with LO = 0.9 Spatially Correlated Channel (CSNR = 10dB) (stuck−at−1 fault)
no fault, 15 training no fault, 25 training no fault, 35 training no fault, known covariance matrix 1 fault, 15 training 1 fault, 25 training 1 fault, 35 training 1 fault, known covariance matrix 2 fault, 15 training 2 fault, 25 training 2 fault, 35 training 2 fault, known covariance matrix
Figure 6.19: Performance of the MED rule under ρ = 0.9 spatially correlated channel at CSNR=10 dB for stuck-at-1 faults when 4× 10 code matrix is employed.
−2 0 2 4 6 8 10 12 14 0
0.1 0.2 0.3 0.4 0.5 0.6
OSNR(dB)
Probability of misclassification
4x10 WSN system with LO = 0.9 Spatially Correlated Channel (CSNR = 10dB) (random fault)
no fault, 15 training no fault, 25 training no fault, 35 training no fault, known covariance matrix 1 fault, 15 training 1 fault, 25 training 1 fault, 35 training 1 fault, known covariance matrix 2 fault, 15 training 2 fault, 25 training 2 fault, 35 training 2 fault, known covariance matrix
Figure 6.20: Performance of the MED rule under ρ = 0.9 spatially correlated channel at CSNR=10 dB for random faults when 4× 10 code matrix is employed.
−100 −8 −6 −4 −2 0 2 4 6 8 10
4x10 WSN system with Spatially Correlated Channel (OSNR = 10dB) (fault−free)
LO = 0
Figure 6.21: Performance of the MED rule with perfect channel estimation under spatially correlated channel at OSNR=10 dB in fault-free situation when 4× 10 code matrix is employed.
correlation coefficient ρ = 0.9.
Comparing Fig. 6.13 with Fig. 6.17, we can see that the performances curves corresponding to different channel correlation are distinct. We sum-marize in Figs. 6.21 and 6.22 the system performances corresponding to dif-ferent correlation factors under perfect channel estimation.
In Fig. 6.21, we observe that the performance curve turns flatter as ρ grows. Thus, when channel correlation increases, the system performance becomes irrelevant to the CSNR. This phenomenon can be explained through the extreme case of ρ = 1, in which case the channel link noises are all
−100 −8 −6 −4 −2 0 2 4 6 8 10 0.1
0.2 0.3 0.4 0.5 0.6
OSNR(dB)
Probability of misclassification
4x10 WSN system with Spatially Correlated Channel (CSNR = 10dB) (fault−free)
LO = 0 LO = 0.1 LO = 0.2 LO = 0.3 LO = 0.4 LO = 0.5 LO = 0.6 LO = 0.7 LO = 0.8 LO = 0.9 LO = 1
Figure 6.22: Performance of the MED rule with perfect channel estimation under spatially correlated channel at CSNR=10 dB in fault-free situation when 4× 10 code matrix is employed.
equal. Then, no matter how small the CSNR is, the fusion center can easily identify the noise quantity by receiving multiple sensor signals; hence, the performance has no improvement by increasing the CSNR.
In addition, there are two distinct behaviors in the relationship between channel correlation ρ and system performance. At low CSNR, the system performance improves as ρ increases, while at high CSNR, the system perfor-mance is worse for smaller ρ. The former case can be justified by the same reason stated in the previous paragraph. A possible reason for the latter case is that the DCFECC code matrix adopted here is searched under the uncorrelated channel assumption, namely, ρ = 0, and the same code may not be optimal when a non-zero ρ is considered. Similar trends on how the system performance varies with ρ can be observed in Fig. 6.22.
Repeating the above simulations with 3× 10 and 5 × 10 code matrices under ρ = 0.1 and 0.9 yields Figs. 6.23–6.38, from which the same conclusion can be drawn.
−5 −4 −3 −2 −1 0 1 2 3 4 5 0
0.05 0.1 0.15 0.2 0.25
CSNR(dB)
Probability of misclassification
3x10 WSN system with LO = 0.1 Spatially Correlated Channel (OSNR = 10dB) (stuck−at−1 fault)
no fault, 15 training no fault, 25 training no fault, 35 training no fault, known covariance matrix 1 fault, 15 training 1 fault, 25 training 1 fault, 35 training 1 fault, known covariance matrix 2 fault, 15 training 2 fault, 25 training 2 fault, 35 training 2 fault, known covariance matrix
Figure 6.23: Performance of the MED rule under ρ = 0.1 spatially correlated channel at OSNR=10 dB for stuck-at-1 faults when 3×10 code matrix is employed.
−5 −4 −3 −2 −1 0 1 2 3 4 5 0
0.05 0.1 0.15 0.2 0.25
CSNR(dB)
Probability of misclassification
3x10 WSN system with LO = 0.1 Spatially Correlated Channel (OSNR = 10dB) (random fault)
no fault, 15 training no fault, 25 training no fault, 35 training no fault, known covariance matrix 1 fault, 15 training 1 fault, 25 training 1 fault, 35 training 1 fault, known covariance matrix 2 fault, 15 training 2 fault, 25 training 2 fault, 35 training 2 fault, known covariance matrix
Figure 6.24: Performance of the MED rule under ρ = 0.1 spatially correlated channel at OSNR=10 dB for random faults when 3× 10 code matrix is employed.
−2 0 2 4 6 8 10 12 14 0
0.1 0.2 0.3 0.4 0.5
OSNR(dB)
Probability of misclassification
3x10 WSN system with LO = 0.1 Spatially Correlated Channel (CSNR = 10dB) (stuck−at−1 fault)
no fault, 15 training no fault, 25 training no fault, 35 training no fault, known covariance matrix 1 fault, 15 training 1 fault, 25 training 1 fault, 35 training 1 fault, known covariance matrix 2 fault, 15 training 2 fault, 25 training 2 fault, 35 training 2 fault, known covariance matrix
Figure 6.25: Performance of the MED rule under ρ = 0.1 spatially correlated channel at CSNR=10 dB for stuck-at-1 faults when 3× 10 code matrix is employed.
−2 0 2 4 6 8 10 12 14 0
0.1 0.2 0.3 0.4 0.5
OSNR(dB)
Probability of misclassification
3x10 WSN system with LO = 0.1 Spatially Correlated Channel (CSNR = 10dB) (random fault)
no fault, 15 training no fault, 25 training no fault, 35 training no fault, known covariance matrix 1 fault, 15 training 1 fault, 25 training 1 fault, 35 training 1 fault, known covariance matrix 2 fault, 15 training 2 fault, 25 training 2 fault, 35 training 2 fault, known covariance matrix
Figure 6.26: Performance of the MED rule under ρ = 0.1 spatially correlated channel at CSNR=10 dB for random faults when 3× 10 code matrix is employed.
−5 −4 −3 −2 −1 0 1 2 3 4 5 0
0.05 0.1 0.15 0.2 0.25
CSNR(dB)
Probability of misclassification
3x10 WSN system with LO = 0.9 Spatially Correlated Channel (OSNR = 10dB) (stuck−at−1 fault)
no fault, 15 training no fault, 25 training no fault, 35 training no fault, known covariance matrix 1 fault, 15 training 1 fault, 25 training 1 fault, 35 training 1 fault, known covariance matrix 2 fault, 15 training 2 fault, 25 training 2 fault, 35 training 2 fault, known covariance matrix
Figure 6.27: Performance of the MED rule under ρ = 0.9 spatially correlated channel at OSNR=10 dB for stuck-at-1 faults when 3×10 code matrix is employed.
−5 −4 −3 −2 −1 0 1 2 3 4 5 0
0.05 0.1 0.15 0.2 0.25
CSNR(dB)
Probability of misclassification
3x10 WSN system with LO = 0.9 Spatially Correlated Channel (OSNR = 10dB) (random fault)
no fault, 15 training no fault, 25 training no fault, 35 training no fault, known covariance matrix 1 fault, 15 training 1 fault, 25 training 1 fault, 35 training 1 fault, known covariance matrix 2 fault, 15 training 2 fault, 25 training 2 fault, 35 training 2 fault, known covariance matrix
Figure 6.28: Performance of the MED rule under ρ = 0.9 spatially correlated channel at OSNR=10 dB for random faults when 3× 10 code matrix is employed.
−2 0 2 4 6 8 10 12 14 0
0.1 0.2 0.3 0.4 0.5
OSNR(dB)
Probability of misclassification
3x10 WSN system with LO = 0.9 Spatially Correlated Channel (CSNR = 10dB) (stuck−at−1 fault)
no fault, 15 training no fault, 25 training no fault, 35 training no fault, known covariance matrix 1 fault, 15 training 1 fault, 25 training 1 fault, 35 training 1 fault, known covariance matrix 2 fault, 15 training 2 fault, 25 training 2 fault, 35 training 2 fault, known covariance matrix
Figure 6.29: Performance of the MED rule under ρ = 0.9 spatially correlated channel at CSNR=10 dB for stuck-at-1 faults when 3× 10 code matrix is employed.
−2 0 2 4 6 8 10 12 14 0
0.1 0.2 0.3 0.4 0.5
OSNR(dB)
Probability of misclassification
3x10 WSN system with LO = 0.9 Spatially Correlated Channel (CSNR = 10dB) (random fault)
no fault, 15 training no fault, 25 training no fault, 35 training no fault, known covariance matrix 1 fault, 15 training 1 fault, 25 training 1 fault, 35 training 1 fault, known covariance matrix 2 fault, 15 training 2 fault, 25 training 2 fault, 35 training 2 fault, known covariance matrix
Figure 6.30: Performance of the MED rule under ρ = 0.9 spatially correlated channel at CSNR=10 dB for random faults when 3× 10 code matrix is employed.
−5 −4 −3 −2 −1 0 1 2 3 4 5 0
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
CSNR(dB)
Probability of misclassification
5x10 WSN system with LO = 0.1 Spatially Correlated Channel (OSNR = 10dB) (stuck−at−1 fault)
no fault, 15 training no fault, 25 training no fault, 35 training no fault, known covariance matrix 1 fault, 15 training 1 fault, 25 training 1 fault, 35 training 1 fault, known covariance matrix 2 fault, 15 training 2 fault, 25 training 2 fault, 35 training 2 fault, known covariance matrix
Figure 6.31: Performance of the MED rule under ρ = 0.1 spatially correlated channel at OSNR=10 dB for stuck-at-1 faults when 5×10 code matrix is employed.
−5 −4 −3 −2 −1 0 1 2 3 4 5 0
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
CSNR(dB)
Probability of misclassification
5x10 WSN system with LO = 0.1 Spatially Correlated Channel (OSNR = 10dB) (random fault)
no fault, 15 training no fault, 25 training no fault, 35 training no fault, known covariance matrix 1 fault, 15 training 1 fault, 25 training 1 fault, 35 training 1 fault, known covariance matrix 2 fault, 15 training 2 fault, 25 training 2 fault, 35 training 2 fault, known covariance matrix
Figure 6.32: Performance of the MED rule under ρ = 0.1 spatially correlated channel at OSNR=10 dB for random faults when 5× 10 code matrix is employed.
−2 0 2 4 6 8 10 12 14 0
0.1 0.2 0.3 0.4 0.5 0.6
OSNR(dB)
Probability of misclassification
5x10 WSN system with LO = 0.1 Spatially Correlated Channel (CSNR = 10dB) (stuck−at−1 fault)
no fault, 15 training no fault, 25 training no fault, 35 training no fault, known covariance matrix 1 fault, 15 training 1 fault, 25 training 1 fault, 35 training 1 fault, known covariance matrix 2 fault, 15 training 2 fault, 25 training 2 fault, 35 training 2 fault, known covariance matrix
Figure 6.33: Performance of the MED rule under ρ = 0.1 spatially correlated channel at CSNR=10 dB for stuck-at-1 faults when 5× 10 code matrix is employed.
−2 0 2 4 6 8 10 12 14 0
0.1 0.2 0.3 0.4 0.5 0.6
OSNR(dB)
Probability of misclassification
5x10 WSN system with LO = 0.1 Spatially Correlated Channel (CSNR = 10dB) (random fault)
no fault, 15 training no fault, 25 training no fault, 35 training no fault, known covariance matrix 1 fault, 15 training 1 fault, 25 training 1 fault, 35 training 1 fault, known covariance matrix 2 fault, 15 training 2 fault, 25 training 2 fault, 35 training 2 fault, known covariance matrix
Figure 6.34: Performance of the MED rule under ρ = 0.1 spatially correlated channel at CSNR=10 dB for random faults when 5× 10 code matrix is employed.
−5 −4 −3 −2 −1 0 1 2 3 4 5 0
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
CSNR(dB)
Probability of misclassification
5x10 WSN system with LO = 0.9 Spatially Correlated Channel (OSNR = 10dB) (stuck−at−1 fault)
no fault, 15 training no fault, 25 training no fault, 35 training no fault, known covariance matrix 1 fault, 15 training 1 fault, 25 training 1 fault, 35 training 1 fault, known covariance matrix 2 fault, 15 training 2 fault, 25 training 2 fault, 35 training 2 fault, known covariance matrix
Figure 6.35: Performance of the MED rule under ρ = 0.9 spatially correlated channel at OSNR=10 dB for stuck-at-1 faults when 5×10 code matrix is employed.
−5 −4 −3 −2 −1 0 1 2 3 4 5 0
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
CSNR(dB)
Probability of misclassification
5x10 WSN system with LO = 0.9 Spatially Correlated Channel (OSNR = 10dB) (random fault)
no fault, 15 training no fault, 25 training no fault, 35 training no fault, known covariance matrix 1 fault, 15 training 1 fault, 25 training 1 fault, 35 training 1 fault, known covariance matrix 2 fault, 15 training 2 fault, 25 training 2 fault, 35 training 2 fault, known covariance matrix
Figure 6.36: Performance of the MED rule under ρ = 0.9 spatially correlated channel at OSNR=10 dB for random faults when 5× 10 code matrix is employed.
−2 0 2 4 6 8 10 12 14 0
0.1 0.2 0.3 0.4 0.5 0.6
OSNR(dB)
Probability of misclassification
5x10 WSN system with LO = 0.9 Spatially Correlated Channel (CSNR = 10dB) (stuck−at−1 fault)
no fault, 15 training no fault, 25 training no fault, 35 training no fault, known covariance matrix 1 fault, 15 training 1 fault, 25 training 1 fault, 35 training 1 fault, known covariance matrix 2 fault, 15 training 2 fault, 25 training 2 fault, 35 training 2 fault, known covariance matrix
Figure 6.37: Performance of the MED rule under ρ = 0.9 spatially correlated channel at CSNR=10 dB for stuck-at-1 faults when 5× 10 code matrix is employed.
−2 0 2 4 6 8 10 12 14 0
0.1 0.2 0.3 0.4 0.5 0.6
OSNR(dB)
Probability of misclassification
5x10 WSN system with LO = 0.9 Spatially Correlated Channel (CSNR = 10dB) (random fault)
no fault, 15 training no fault, 25 training no fault, 35 training no fault, known covariance matrix 1 fault, 15 training 1 fault, 25 training 1 fault, 35 training 1 fault, known covariance matrix 2 fault, 15 training 2 fault, 25 training 2 fault, 35 training 2 fault, known covariance matrix
Figure 6.38: Performance of the MED rule under ρ = 0.9 spatially correlated channel at CSNR=10 dB for random faults when 5× 10 code matrix is employed.