In this chapter, simulations are conducted to show the effectiveness of the proposed DALE algorithms under different network topologies and MS’s positions. The noise model of RSS measurements will be the same as we described in section 3.1. The slow fading effect will be Gaussian-distributed with mean µ equal to zero and standard deviation σ in the range from 3-6 dB. IEEE 802.11a [29] defined the operation frequency at 5.0GHz, while 2.4GHz in IEEE 802.11b [30]. Nowadays many wireless fidelity (Wi-Fi) devices are mainly operating at frequency band around 2.4-2.5 GHz, so we roughly choose the carrier frequency fc as 2.45 GHz. The remaining common simulation parameters are referred to the 3GPP TR 36.814 report [24] as we mentioned in section 3.1 and are listed in Table 5.1. We will evaluate the effectiveness of the proposed DE mechanism first.
Table 5.1: Simulation parameters
Parameters Values
Path Loss at Reference Point P L0 11.5 dB
Carrier Frequency fc 2.4 GHz
Path Loss Exponent α 4.33
Mean of Slow Fading Effect µ 0 dB Standard Deviation of Slow Fading Effect σ 3-6 dB
Table 5.2: Deployment of BSs and MS
BS1 BS2 BS3 MS
S1 (0, 0) (20, 2) (10, 20) (9, 8)
S2 (0, 0) (20, 2) (10, 20) (3, 1.8) S3 (0, 0) (20, 2) (16, 2.8) (12, 1.4) S4 (0, 0) (20, 2) (16, 2.8) (4, 0.5)
S5 (0, 0) (20, 0.2) (10, 8)+(uBS,x, uBS,y) mean(BS)+(uM S,x, uM S,y)
Then the performance comparison between the proposed DALE algorithm with the other existing location estimation methods are conducted in three types of network layouts.
First, we arrange the three BSs in the layout to be close to a regular triangle. Secondly we deploy the three BSs to be like an irregular triangle. In the third topology, we put the three BSs randomly in the space, trying to obtain an arbitrary triangle as evaluating the performance in random cases. Therefore, 5 scenarios are taken into account and the related deployment parameters is given in the Table 5.2, where uBS,x, uBS,y ∼ U(−6, 6), uM S,x, uM S,y ∼ U(−10, 10) with constraints that the MS must lie in the triangle formed by BSs and the distance between MS and each BS must larger than 3 meters according to the path loss model in [24]. Mean(BS) represents the gravity center (GC) of BSs’ layout.
For the convenience of evaluation, we define PLE’s true value as ˆα and the PLE estimation result of the proposed DE mechanism as ˆα. The PLE estimation error αe is defined as
αe = ˆα− α
For the convenience of evaluation, we define MS’s true position as x = (x, y) and the estimation output of location algorithms as ˆx = (ˆx, ˆy). The location estimation error Le is defined as
Le =∥ˆx − x∥ (5.1)
3 3.5 4 4.5 5 5.5 6
Figure 5.1: The relationship between PLE estimation error and noise standard deviation in a regular triangle
The mean square error (MSE) is thus defined as
M SE = mean(∥ˆx − x∥2) (5.2)
where mean(x) is the sample mean of the elements in the vector x. The value of PLE initial guess ˆα is 4.23, which has a bias of 0.1 from the long-term mean value of true PLE α defined in Table 5.1. A random bias uniformly-distributed with [-0.05, 0.05] is appended to the true PLE α to represent the randomness of the real PLE of each propagation channel. The performance is evaluated with 30000 independent trials. The initial guess will be chosen as an arbitrary point with 1 meter from the MS’s true position for any algorithm that requires that.
The objective of the DE mechanism is to estimate the PLE and the distance with RSS measurements and the estimation performance is investigated under different scenarios defined in Table 5.2. The initial guesses of the PLE and distance for the TSE-based DE mechanism are obtained as described in Section 4.1. Fig. 5.1 and Fig. 5.2 are the
3 3.5 4 4.5 5 5.5 6 0.7
0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6
STD (dB)
Location Estimation Error (m)
DALE (S1) DALE (S2) DALE w/o DE (S1) DALE w/o DE (S2)
Figure 5.2: The relationship between location estimation error and noise standard deviation in a regular triangle
estimation errors of PLE and distance in scenarios S1 and S2 with respect to different noise standard deviations. It is noted that the scenario S1 represents a network topology with the BSs locate as a regular triangle whose center the MS lies at, while the scenario S2 has the MS locates somewhere apart from the center of the network. We evaluate the performance of our proposed DALE algorithm with the schemes of DALE without the DE mechanism (DALE w/o DE) to see whether the DALE algorithm have any improve by performing the DE mechanism. In Fig. 5.1 we can see that for those schemes with DE mechanism, that is, DALE (S1) and DALE (S2), have less PLE estimation error, while the schemes without DE mechanism have a constant bias because they do not have any mechanism to estimate the PLE. On the other hand, in Fig. 5.2, we can make an intuitive conclusion that since DALE algorithm has the correction of PLE and thus it has less location estimation error in both scenario S1 and S2. Moreover, we notice that the PLE estimation in S2 is better than that in S2. This makes the location estimation error in S2 is less than that in S1. We thus make a brief conclusion that the better PLE estimation one have, the better location estimation results can be obtained.
3 3.5 4 4.5 5 5.5 6
Figure 5.3: The relationship between PLE estimation error and noise standard deviation in an irregular triangle
Figure 5.4: The relationship between location estimation error and noise standard deviation in an irregular triangle
3 3.5 4 4.5 5 5.5 6 0.096
0.0965 0.097 0.0975 0.098 0.0985 0.099 0.0995 0.1
STD (dB)
PLE Error
DALE (S5) DALE w/o DE (S5)
Figure 5.5: The relationship between PLE estimation error and noise standard deviation in an arbitrary triangle
The performance of the DE mechanism in the scenarios S3 and S4 is studied in Fig.
5.3 and Fig. 5.4. In Fig. 5.3 we can discover that DALE (S3) and DALE (S4) have less PLE estimation error than the scheme without DE mechanism which has a constant bias. Observing from Fig. 5.4 we notice again that the one with better PLE estimation will has less location estimation error in both S3 and S4. Since the PLE estimation in S4 is better than that in S3, it makes better location estimation in S4 than in S3.
Therefore, it is revealed again that better PLE estimation is coupled with better distance estimation. In the scenario S5, a random network topology is applied as addressed in Table 5.1. According to the path loss model in TR 36.814 [24], the distance between the MS and any BS should be larger than 3 meters. The Fig. 5.3 and Fig. 5.4 represent the estimation performance of the DE mechanism in S5. The DALE has the PLE with less bias, thus its location estimation error is less than the DALE without DE scheme. By the general case, we can confirm the effectiveness of the DE mechanism can be ensured.
The location errors of the DALE algorithm and many other methods are also evaluated in the scenarios mentioned above. The performance of our proposed DALE algorithm
3 3.5 4 4.5 5 5.5 6
Figure 5.6: The relationship between location estimation error and noise standard deviation in an arbitrary triangle
Figure 5.7: The comparison of location estimation error under the fading noise with 4 dB standard deviation in scenario S1 and S2
is compared with the schemes of DALE without the DE mechanism (DALE w/o DE), DALE without the design of weighting (DALE w/o W), TSLS algorithm [19] as well as TSE algorithm [18]. It is noted that the proposed DALE in scenario 1 is close to other algorithms, especially the DALE without DE scheme. This is due to the perfect layout of BSs that reduces the effect of noise. However, DALE (S1) is still the one with the best performance. In scenario S2, the MS moves toward one corner of the regular triangle, making the MS become closer to one of the MS than scenario S1. This increase the chance of having bad performance for all the algorithms because the shortest distance between MS and BSs become very short. Any noise with extreme value could cause a significant ratio of change in the shortest path, which could lead to a severe error in the calculation results. However, we can still discover that our proposed DALE (S2) have the best performance in Fig. 5.7. Without any weighting, the DALE w/o W scheme could be even worse than the TSLS and TSE that have taken noise STD as weighting design.
For our proposed DALE with moderate weighting, it turns into the best one. Moreover, it is not hard to discover that the gain of DE is tiny in these two scenario. This could be not extremely bad layout that makes the gain of RA mechanism to be tiny.
In equations of (5.1) and (5.2), we think that the location estimation error and MSE has a kind of positive correlation. For the clearness of illustration, we eliminates the DALE w/o DE and DALE w/o W schemes in our MSE Figures since they do not have better performance than DALE in the CDF of location estimation error. In other words, we will evaluate the performance of our proposed DALE algorithm with TSLS algorithm, and TSE algorithm. In Fig. 5.8, we compare the MSE of the algorithms we just mentioned.
Similar to what we saw in Fig. 5.7, in scenario S1 all the algorithms have the performance closed to each other, while in S2, the bad layout formed by the BSs make all the algorithms to have worse estimation results.
In Fig. 5.9, we evaluate the positioning performance in the scenario S3 and S4. From
3 3.5 4 4.5 5 5.5 6
Figure 5.8: The comparison of mean squared error in the scenarios S1 and S2
0 0.5 1 1.5 2 2.5 3 3.5
Figure 5.9: The comparison of location estimation error under the fading noise with 4 dB standard deviation in scenario S3 and S4
3 3.5 4 4.5 5 5.5 6 100
101
STD (dB) MSE (m2)
DALE (S3) DALE (S4) TSLS (S3) TSLS (S4) TSE (S3) TSE (S4)
Figure 5.10: The comparison of mean squared error in the scenarios S3 and S4
Fig. 5.9 we can discover that for the real lines, our proposed DALE algorithm as well as other two schemes hold the position of the best three performance. The TSE (S3) seems to be better than TSLS (S3) in a significant gain. This might be due to TSE have a good initial guess that is close to the MS’s true position. The DALE w/o W (S3) scheme here also performs better than TSE (S3) with a gain. This might be due to the layout of BSs is an irregular triangle. For other algorithms it is easy to obtain a location estimation ˆx out of the triangle, while our proposed DALE algorithm have the mechanism of RA that can form some additional constrains that reduce the effect of abnormal layout and noises.
And we can see that the consideration of weighting and DE mechanism both have some gains. If focusing on dashed lines, we can discover similar phenomenons that happened in scenario 3. However, the gain of DE mechanism in S4 is greater than that in S3, which implies that it works better in worse layouts. After all, it seems that Fig. 5.7 and Fig.
5.9 imply our proposed DALE algorithm works well in any circumstances and has most gain than any other algorithm at the worst case.
In Fig. 5.10 the MSE performance in the scenarios S3 and S4 is conducted. As the
0 0.5 1 1.5 2 2.5 3 3.5 4
Figure 5.11: The comparison of location estimation error under the fading noise with 4 dB standard deviation in the scenario S5
layout becomes even irregular, all the algorithm have worse performances than that in Fig.
5.8. However, the proposed DALE can still perform well than any other algorithms. This confirms that due to the RA mechanism, the proposed DALE algorithm can eliminate chances of having extremely bad estimations and thus holds the performance.
In Fig. 5.11, the comparison of location estimation is evaluated in the scenario S5.
The performance is in consist with what we have discovered and discussed in scenario S1 to S4. Through the observation of location estimation error’s CDF from scenario 1 to 5 with the STD of noise in 4 dB, we can make a brief conclusion that our proposed DALE algorithm is better than other two control group scheme, that is, DALE without DE mechanism and DALE without the design of weighting.
As the scenario S5 represents a general case of the network topology, in Fig. 5.12 it can be observed that the DALE w/o W scheme performs better than TSLS and TSE, which shall be the gain of having RA mechanism. And next we discover that DALE w/o DE scheme have a gain of taking the noise variance into consideration to design the weighting. Moreover, we can see the gain of DE mechanism from the performance of
3 3.5 4 4.5 5 5.5 6 100.2
100.3 100.4 100.5 100.6 100.7 100.8 100.9
STD (dB) MSE (m2)
DALE (S5) DALE w/o DE (S5) DALE w/o W (S5) TSLS (S5) TSE (S5)
Figure 5.12: Scenario 5’s MSE with respect to standard deviation
proposed DALE.
Chapter 6 Conclusion
An efficient diversity-augmented location estimation (DALE) algorithm based on the mea-surement of RSS is proposed in this thesis. The DALE scheme inherits the merits of the GALE algorithm and enhances the conventional TSLS algorithm by imposing additional virtual spatial constraints within its formulation. By using the proposed DALE algo-rithm, the computational efficiency acquired from the TSLS method is preserved, and the requirement of hardware’s transmitting and measurement capability is lowered to a moderate level by adopting RSS as the measurement source. Higher location estimation accuracy for the MS is also achieved. Moreover, the jittering of PLE can be reduced by adopting the proposed DE mechanism. The proposed RA mechanism can effectively deal with various geometric layouts between the MS and its associated BSs. It is shown in the simulation results that the proposed DALE algorithm provides better position location estimate comparing with other existing methods.
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