Location Systems
4.4 Performance Evaluation
E[∆ˆyLS2 ] = ζ22x23σn22 + ζ32x22σn23 + ζ12(x3− x2)2σ2n1
(x3y2− x2y3)2 . (4.27)
Noted that (4.26) and (4.27) are also derived based on the condition that ζi >> ni. Meanwhile, it is interesting to notice that the variance of LLS estimator calculated from (4.26) and (4.27) will be numerically identical to the L-CRLB computed via (4.13), which will be validated and shown in the following section. Therefore, the LLS estimator will approach its defined lower bound L-CRLB under the situation with smaller measurement noises. Furthermore, the L-CRLB will become the conventional CRLB under specific geometric layout as described in Corollary 4.1. As a result, under specific conditions as stated above, the LLS estimator can be claimed as the best estimator since it can finally reach the theoretical lower bound, i.e., CRLB, for unbiased estimators.
4.4 Performance Evaluation
In order to verify the effectiveness of L-CRLB derived in Subsection 4.2.3, different scenarios are provided in the section to validate the correctness of the formulation. The model for the measurement noise of TOA signal ni as in (2.1) is selected as the Gaussian distribution with zero mean and standard deviation σri, i.e., ni ∼ N(0, σr2i) (i.e., LOS noise model in Section 2.2.1.1).
Subsection 4.4.1presents the contour plots in order to numerically describe the difference between the CRLB and L-CRLB, which also validate the correctness of Lemmas4.1to 4.2and Corollaries 4.1to 4.2. Subsection 4.4.2 simulates the performance of LLS method by comparing the L-CRLB and LLS estimator in the regular BS polygon layout. Subsection 4.4.3 illustrates the performance comparison of LLS estimator under both the IPL and OPL cases. Performance comparison under realistic WSN scenario is described in Subsection4.4.4.
4.4.1 Numerical Validation of CRLB and L-CRLB with a Regular Triangular Layout
Example 4.4 (CRLB and L-CRLB contour). In order to observe the difference between the CRLB and L-CRLB, their corresponding contour plots under the number of BSs N = 3 are shown in Figs.
4.3(a) and 4.3(b), respectively. Note that the three BSs are located at the vertexes of a regular triangular which are denoted with red circles in Figs. 4.3(a) and 4.3(b). The positions of BSs are b1 = [300, 200]T with α1 = 0◦, b2 = [150, 286.6]T with α2 = 120◦, and b3 = [150, 113.4]T with α3 = 240◦. Based on the three BS’s positions, each individual contour point represents the corresponding CRLB or L-CRLB value when the MS is situated at that geographical location. The standard deviation of measurement noises σri is chosen as 1 m for simplicity. It can be observed from Fig. 4.3(a) that there are four minimum points for the CRLB value equal to Cm = 1.33 with MS’s positions as x = [200, 200]T, [100, 200]T, [260, 120]T, and [260, 280]T. The conditions for minimum CRLB can be verified by substituting the corresponding parameters into the condition
1.4
Figure 4.3: (a) CRLB contour under N = 3; (b) L-CRLB contour under N = 3. Red circles denote the positions of BSs.
(4.6). The minimum CRLB value can also be validated to satisfy (4.5), which demonstrates the correctness of Lemma4.1.
On the other hand, by comparing Figs. 4.3(a) and 4.3(b), it is observed that the distribution of L-CRLB is different from that of CRLB. The only minimum L-CRLB value identical to that of the CRLB, i.e., CL,m = Cm= 1.33, is located at the center of regular triangle formed by the three BSs, i.e., x = [200, 200]T. Starting at the MS’s position with minimum L-CRLB, the L-CRLB value will increase in all directions. Except for the minimum L-CRLB at the center of the triangle, the relationship that CL > C can be observed from both Figs. 4.3(a) and 4.3(b). Moreover, the difference between the L-CRLB and CRLB inside the triangle is smaller than that outside of the triangle. The reason can be contributed to the estimation of parameter R by adopting the L-CRLB criterion, which introduces the two terms ε1 and ε2. Owing to the nonlinear behavior of location estimation, the additional consideration of R within the L-CRLB can better characterize the performance of linearized location estimator for the LEP. The correctness of minimum L-CRLB value obtained from Fig. 4.3(b)can also be verified by substituting corresponding parameters into the conditions stated in Lemma 4.2, i.e., the conditions (4.6) and (4.18) can all be satisfied.
By comparing the results from Figs. 4.3(a) and 4.3(b), Corollaries4.1 to 4.2and Examples 4.2to 4.3can all be validated by substituting the corresponding numerical values. ⋄
4.4.2 Performance Validation of LLS Estimator with a Regular BS Polygon Layout
In this subsection, the performance of LLS estimator is simulated to further validate the rela-tionship between the estimator and the lower bound.
4.4. Performance Evaluation
Table 4.1: Simulation Parameters Number of BSs i-th BS’s Coordinate bi in meter
3 BSs 0◦: [300, 200]T 120◦: [150, 286.6]T 240◦: [150, 113.4]T 4 BSs 0◦: [300, 200]T 90◦: [200, 300]T 180◦: [100, 200]T
270◦: [200, 100]T
5 BSs 0◦: [300, 200]T 72◦: [230.9, 295.1]T 144◦: [119.1, 258.8]T 216◦: [119.1, 141.2]T 288◦: [230.9, 104.9]T
6 BSs 0◦: [300, 200]T 60◦: [250, 286.6]T 120◦: [150, 286.6]T 180◦: [100, 200]T 240◦: [150, 113.4]T 300◦: [250, 113.4]T 7 BSs 0◦: [300, 200]T 51.4◦: [262.3, 278.2]T 102.8◦: [177.7, 297.5]T
154.3◦: [109.9, 243.4]T 205.7◦: [109.9, 156.6]T 257.1◦: [177.7, 102.5]T 308.6◦: [262.3, 121.8]T
8 BSs 0◦: [300, 200]T 45◦: [270.7, 270.7]T 90◦: [200, 300]T 135◦: [129.3, 270.7]T 180◦: [100, 200]T 225◦: [129.3, 129.3]T 270◦: [200, 100]T 315◦: [270.7, 129.3]T
Example 4.5 (A Regular BS Polygon Layout at N = 3). Fig. 4.4 illustrates the performance comparison under different noise standard deviations in the regular triangular layout, i.e., N = 3.
The coordinates of the 3 BSs are listed in the 3 BS case of Table 4.1, and the MS is located at the coordinate x = [200, 200]T. Moreover, Fig. 4.5 shows the performance comparison between different numbers BSs of regular BS polygon layout where the MS lies at x = [200, 200]T and the standard deviation of measurement noise is equal to 10 m. The BS’s coordinates correspond to different numbers of BSs’ layout are listed in Table 4.1. Regarding the comparison metrics, instead of showing the variances, the root mean square error (RMSE) is obtained in order to clearly illustrate the difference between different curves, i.e., RMSE = hPNr
i=1kx − ˆx(i)k2/Nri1/2
, where Nr= 10, 000 indicates the number of simulation runs. As for the curves within Figs. 4.4and 4.5, the LLS estimator denotes the RMSE of ˆxLS acquired from (2.12) by simulating the Gaussian noises with corresponding noise standard deviations. Since the CRLB and the L-CRLB represent the variance of an unbiased estimator, both the CRLB and L-CRLB curves are obtained by taking the square root in order to compared with the RMSE values of LLS estimator. Furthermore, the curve of LLS standard deviation in Fig. 4.4is obtained as the square root of LLS variance derived from (4.26) and (4.27).
It can be observed from Fig. 4.4 that the CRLB, L-CRLB, and LLS standard deviation can achieve the same values in the regular triangular layout. The reason for the CRLB and L-CRLB to possess the same value is identical to the conditions as stated in Lemmas4.1and 4.2. As described in Section 2.3, the LLS standard deviation is numerically validated in this figure to be identical to the square roots of CRLB and L-CRLB under the condition ζi >> ni. The performance of LLS estimator obtained from simulations can also approach both lower bounds, i.e., the CRLB and L-CRLB, under the cases with smaller measurement noises. This result demonstrates that the LLS estimator can be considered as an efficient estimator for the LEP and L-LEP under smaller
0 5 10 15 20 25 0
5 10 15 20 25 30
Standard Deviation of Measurement Noise (m)
Root Mean Square Error (m)
CRLB L−CRLB
LLS standard deviation LLS estimator
Figure 4.4: Performance comparison for location estimation with MS at the center of a regular triangle formed by 3 BSs as listed in Table 4.1: RMSE vs. standard deviation of measurement noise. The CRLB, L-CRLB, and LLS standard deviation achieve the same values.
measurement noises. On the other hand, as the noise becomes larger which disobeys the relationship ζi >> ni, it is observed that the RMSE of LLS estimator will be slightly higher than that obtained
from the L-CRLB. ⋄
Example 4.6 (A Regular BS Polygon Layout at different number of BSs). Fig. 4.5 validates the performance of LLS estimator in the regular BS polygon layouts under different numbers of BSs. In order to observe the difference between the LLS estimator and the L-CRLB, the error confidential level (δ) is defined as the difference between the RMSE of LLS estimator (RLS) and the square root of L-CRLB (RL), i.e., δ = |RLS − RL|/RL. In Fig. 4.5, the error confidential levels can be obtained as δ = [0.67, 0.53, 0.20, 0.27, 0.41, 0.17]% under the number of BSs equal to [3, 4, 5, 6, 7, 8].
It is observed that the LLS estimator can closely approach both of the lower bounds CRLB and
L-CRLB under different numbers of available BSs. ⋄
4.4.3 Performance Comparison of LLS Estimation with IPL and OPL
In the subsection, the IPL and OPL which achieve the same CRLB value are adopted to validate the correctness of Lemma4.3.
Example 4.7 (IPL and OPL Comparision). The MS is placed at the position x = [200, 200]T m, and the distances from all the BSs to the MS are designed to be equal to 100 m. The angle set for the IPL is assigned as {0◦, 70◦, 240◦}, and that for the OPL is {0◦, 60◦, 70◦}. That is, the three BSs of IPL is placed at [300, 200]T, [234.2, 294]T, and [150, 113.4]T, and that for the OPL is located at [300, 200]T, [250, 286.6]T, and [234.2, 294]T. It is noted that the square roots of CRLB for both the IPL and the OPL are obtained to have the same value as 1.34.
4.4. Performance Evaluation
3 4 5 6 7 8
5 6 7 8 9 10 11 12
Number of BSs
Root Mean Square Error (m)
CRLB L−CRLB LLS estimator
Figure 4.5: Performance comparison for location estimation with MS at the center of a regular BS polygon formed by the BSs as listed in Table 4.1: RMSE vs. the number of BSs. The standard deviation of measurement noise is 10 m. The CRLB and L-CRLB achieve the same values.
0 5 10 15 20 25
0 5 10 15 20 25 30 35 40
Std of Measurement Noise (m)
Root Mean Square Error (m)
CRLB: in L−CRLB: in LLS estimator: in
0 5 10 15 20 25
0 50 100 150 200 250 300 350 400
Std of Measurement Noise (m) CRLB: out L−CRLB: out LLS estimator: out
Figure 4.6: Performance comparison for location estimation under 3 BSs triangular layout: RMSE vs.
standard deviation of measurement noise. Left plot: the MS is located inside the triangle formed by BSs;
Right plot: the MS is outside of the triangle formed by BSs. The CRLB values are the same in both plots.
100 150 200 250 300 100
120 140 160 180 200 220 240 260 280 300
x−axis (m)
y−axis (m)
3 BSs 4 BSs 5 BSs 6 BSs 7 BSs 8 BSs MS
Figure 4.7: Layout of the WSN scenario formed by different numbers of BSs. The MS lies in a rectan-gular room with both x-coordinates and y-coordinates uniformly distributed between [100, 300]. The BS’s coordinates are listed in Table4.1.
The left subplot of Fig. 4.6 shows the performance of LLS estimator comparing with both CRLB and L-CRLB under the IPL; while the right subplot of Fig. 4.6corresponds to that for the OPL. In order to clearly show the difference between these curves, different scales are utilized in both plots. It can be observed that the performance of LLS estimator still matches that of the L-CRLB under the cases with smaller measurement noises for both plots, which again shows that the L-CRLB can closely characterize the behaviors of LLS estimator. However, the difference between the L-CRLB and the CRLB in the OPL is comparably larger than the IPL case, which validates the correctness of Lemma4.3. Therefore, it is concluded that the LLS estimator can provide better performance in the IPL compared to the OPL even though both layouts result in same value of
CRLB. ⋄
4.4.4 Performance Comparison of LLS Estimation in a WSN Scenario
In order to consider more realistic environments, Fig. 4.7 illustrates the simulation scenarios of a WSN with the coordinates of both the MSs and BSs under different BS polygon layouts. The MS’s positions are placed at 100 different locations uniformly distributed in a two-dimensional rectangular plane with both the x-coordinate and y-coordinate in the region of [100, 300] m. The coordinates of BSs for different deployments of BS polygon are listed in Table 4.1. According to the random deployment, some of the MSs will be located in the IPL and the others are in the OPL.
Example 4.8 (A WSN Scenario). Based on the WSN setup, Fig. 4.8 shows the performance evaluation of LLS estimator in comparison with both CRLB and L-CRLB under different numbers of BSs. Noted that each RMSE value of LLS estimator in Fig. 4.8 is obtained by averaging the
4.5. Concluding Remarks
Root Mean Square Error (m)
3 4 5 6 7 8
Figure 4.8: Performance comparison for location estimation with uniformly distributed MS and the BS’s coordinates listed in Table4.1: RMSE vs. number of BSs. The standard deviation of measurement noise is 10 m. Left plot: the MS is located inside the BS polygon; Right plot: the MS is situated outside of the BS polygon.
RMSE value from 100 different MS’s positions, where the RMSE of each MS’s position is simulated with 100 times, i.e., Nr = 100. Both the CRLB and L-CRLB are also obtained by averaging the corresponding values from 100 different MS’s positions. The simulation scenario is regarded as a more generic case since the distances from the MS to BSs will not be the same in every simulation point. Comparing with the CRLB, it can be observed that the proposed L-CRLB can better characterize the simulation results for LLS estimator under both the IPL and OPL cases.
Moreover, the LLS estimator can still provide better performance within the IPL in comparison with that in the OPL. This conclusion is both validated via theoretical proof in Lemma 4.3 and simulation results in Fig. 4.8. Furthermore, the LLS performance under the IPL can be compatible with the LLS performance under the OPL which utilizes one more measurement. For example, the RMSE of the 4BSs case under the IPL is 10.7 m while the RMSE of the 5BSs case under the OPL is 10.65 m. The more number of BSs utilized in a location estimate requires more communication overheads. As a consequence, from the study of geometric effect of LLS estimator, it is suggested that the OPL should be avoided for increasing the precision of MS’s location estimation. This conclusion will be valuable for either cellular networks or WSNs while conducting the deployment
of BSs or implementing a BS selection algorithm. ⋄
4.5 Concluding Remarks
This chapter derives the linearized location estimation problem based Cram`er-Rao lower bound (L-CRLB) which provides the analytical form to discuss the geometric effect for the linear least
square (LLS) estimator. The geometric properties and the relationships between the L-CRLB and conventional CRLB are obtained with theoretical proofs. It is validated in the simulations that the L-CRLB can provide the tight lower bound for the LLS estimator, especially under the situations with smaller measurement noises. Moreover, the proposed L-CRLB can be utilized to describe the performance difference of an LLS estimator under different geometric layouts. The MS locates inside a BS-constrained geometry will provide higher estimation accuracy comparing with the case that the MS is situated outside of the BS-confined geometry layout.