In this section, we use numerical simulations to verify the analytical result established in Section 2.2. In all simulations, the random parameters, θ, nl,k, and ν, are zero-mean Gaussian. And we assume that σ2θ = σ2ν = 1. The observation gains fl,k are assumed to be uniformly distributed in the interval [0.5, 1]. The channel gains are taken as cgd−3.5, where d is uniformly drawn from the interval [1, 10] and cg = 22.6 is a normalization constant to make E[gl,n] = 1 as in [26].
Simulation 2.1 - Effects of N : In this simulation, we demonstrate the effect of the number of transmitters. In particular, we consider the effect of different numbers of transmitters N , where N ≤ K, in the full collaboration case. We set K = 10 and σ2n = 0.4. In Figure 2.2, we plot the average MSE versus N with power levels P = 0 dB, 5 dB, and 10 dB. We note that as N increases, the MSEs decrease; also, we see that large power levels result in smaller MSEs.
1 2 3 4 5 6 7 8 9 10 0.1
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
N
Average MSE
Full collaboration: σθ2=1,σn2=0.4,σν2=1,K=10
P=0 dB P=5 dB P=10 dB
Figure 2.2: MSE of full collaboration case with different numbers of transmitters
Since the MSE decreases as N increases, in all the simulations to follow, the number of sensors and the number of transmitters are set equal.
Simulation 2.2 - Effects of Observation Noise Powers: In this simulation, we compare the MSE of the full collaboration case to that of the non-collaboration case for different observation noise powers. Figure 2.3 shows the average MSE versus P for the full col-laboration and non-colcol-laboration cases with different observation noises, σn2 = 0.4 and σ2n= 0.8. We set K = 20. For σn2 = 0.4, the case with full collaboration performs better than the non-collaboration case. Moreover, as the transmitted power increases, the MSEs for both two cases decrease. In fact, from (2.11), these two cases approach identical MSE as P → ∞. We also see that the MSE of the case with σ2n = 0.4 is smaller than that of the case with σn2 = 0.8, that is, a large signal to noise ratio results in a good performance.
Simulation 2.3 - Comparison between Full Collaboration and Non-collaboration: In this simulation, we compare the full collaboration case with two non-collaboration cases,
0 5 10 15 20 25 30 10−2
10−1 100
P (dB)
Average MSE
σθ2=1,σν2=1,K=20
Full collaboration: σn2=0.4 Full collaboration: σn2=0.8 Non−collaboration: σn2=0.4 Non−collaboration: σn2=0.8
Figure 2.3: MSE of full collaboration and non-collaboration cases with different noise power levels
which, respectively, use the optimal power allocation scheme and the equal power al-location scheme. Note that the equal power alal-location scheme is to set the amplification gains as ak = p
P/K, 1≤ k ≤ K. We set K = 50 and σ2n = 0.4. Figure 2.4 shows that optimal power allocation improves performance over equal power allocation for the non-collaboration case. In addition, the reduction in MSE by full non-collaboration with optimal power allocation is about 10 dB compared with the equal power allocation scheme.
Simulation 2.4 - Effects of Number of Sensors in Each Cluster: In this simulation, we demonstrate the effect of the number of sensors in each cluster. Specifically, we fix the number of sensors in the network and consider two multiple cluster cases: in case 1, each cluster consists of 4 sensors, and in case 2, each cluster consists of 8 sensors. Hence for a fixed number of sensors K, case 1 has K/4 clusters and case 2 has K/8 clusters. We compare their performance with the full collaboration and non-collaboration cases. We set P = 0 dB and σ2n= 0.4. Figure 2.5 shows that MSE of case 2 is less than that of case 1
0 5 10 15 20 25 30 10−2
10−1 100
P (dB)
Average MSE
σθ2=1,σn2=0.4,σν2=1,K=50
Full collaboration
Non−collaboration:Optimal Non−collaboration:Equal
Figure 2.4: MSE of full collaboration and non-collaboration cases with different power levels
0 20 40 60 80 100 120
10−3 10−2 10−1 100
K
Average MSE
σθ2=1,σn2=0.4,σν2=1,P=0dB
Full collaboration Multiple Cluster: K
l=8 Multiple Cluster: K
l=4 Non−collaboration
Figure 2.5: MSEs for Kl = 1, 4, 8, and K with different number of sensors
since for a fixed K, case 2 has a smaller number of clusters and thus more collaboration among sensors. That the full collaboration has the lowest MSE and the non-collaboration case has the highest MSE is as predicted by (2.16).
0 10 20 30 40 50
10−3 10−2 10−1
L
Average MSE
σθ2=1,σn2=0.4,σν2=1,K l=10
P = 0 dB P = 10 dB Lower bound
Figure 2.6: MSEs with different number of clusters
Simulation 2.5 - Effects of Cluster Number: In this simulation, we fix the number of sensors Kl = 10 in each cluster and investigate the relationship between the MSE and the number of cluster. Hence, as the number of clusters is L, the sensors in the network are 10× L. Figure 2.6 shows the MSE as a function of L with, respectively, P = 0 dB, P = 10 dB, and P =∞, which is equivalent to the performance lower bound as shown in (2.11). From this figure, we see that when the number of sensors in each cluster is fixed, the MSE decreases as L increases. In addition, compared the MSEs of P = 0 dB to that of P = 10 dB, we see that the performance gain for the increase of power approaches a constant as L > 10.
Simulation 2.6 - Comparison with Orthogonal MAC Model [24]: In this simulation, we
θ
Cluster 1 f1
Cluster 2 f2
Cluster L fL
▽ G1
L ν1
y1
▽ G2
L ν2
y2
▽ GL
L νL
yL
FC
θˆFigure 2.7: Cluster-based sensor network with orthogonal MAC
0 20 40 60 80 100 120
10−3 10−2 10−1 100
L
Average MSE
σθ2=1,σn2=1,σν2=1,K l=3
Coherent MAC Orthogonal MAC
Figure 2.8: Comparison of the coherent MAC model to that of the orthogonal MAC model
compare the proposed scheme to that in [24], in which the orthogonal MAC model is taken into account. Figure 2.7 shows the model in [24]: the measurement vector for the lth clus-ter is xl = flθ + nl which then transmits to the lth receiver through a diagonal channel gain matrix Gl after multiplying by an amplification matrix Al; at the FC, the received signal vector from the lth cluster is yl= GlAlxl+ νl, 1≤ l ≤ L, where the additive noise νl is assumed to be E[νl] = 0, E[νlνT
l ] = σν2IKl, and E[νlνT
j ] = 0Kl×Kl for j 6= l. After collecting L signal vectors at the FC, the LMMSE fusion rule is used for estimating the source signal. The performance comparison for the orthogonal and coherent MAC models is plotted in Figure 2.8, in which we take P = 10 dB, Kl = 3 for all clusters, and σn2 = 1.
We see that the MSE of the coherent MAC model performs better than that of the or-thogonal MAC model. This is because by using the oror-thogonal MAC model, the number of receiver noises increases as the number of clusters increases. However, by using the coherent MAC model, there is only one receiver noise regardless of the number of clusters.
Table 2.1: Different number of sensors in clusters
K = 30 P = 0 dB
Number of entries 252 226 218 218 200 184 184 166 162 124
MSE (JM) 0.0566 0.0603 0.0617 0.0635 0.0642 0.0659 0.0687 0.0690 0.0704 0.0790
Simulation 2.7 - Relation between Collboration: In this simulation, we see quantitatively the relation between collaboration and MSE for the coherent MAC model, we consider a network with 30 sensors and P = 0 dB. We perform 10 simulations with the number of cluster ranging from 4 to 9. The number of sensors in each cluster is randomly chosen from 1 to 10. In each case, we count the total number of entries in the amplification ma-trices Al. For example, in the first case there are 4 clusters, the numbers of sensors in the clusters are respectively 9, 9, 9, and 3, and the number of entries is 92+ 92+ 92+ 32 = 252.
Table 1 shows the number of cluster, the number of entries, and the corresponding MSE
for each case. From the table, we see that the MSE decreases as the number of entries increases. For comparison, the MSE for the two special cases are respectively JN = 0.1689 and JC = 0.0392.
A Brief Summary and Discussion: We study optimal collaboration for distributed esti-mation in cluster-based wireless sensor network. We show that the optimal amplification matrix of each cluster is a rank one matrix obtained as a scaled outer product of the observation gain and the channel gain vectors. We also show that with optimal collab-oration matrices, the performance of the collabcollab-oration case is better than that of the non-collaboration case. For a fixed number of sensors in the network, we demonstrate, through simulation results, that the amount of improvement is closely related to the amount of collaboration.