in both noise models is a function of the geometry of the AN and the TN, it is obvious that certain AN locations would offer better accuracy than others. These optimal AN locations are discussed in the next section.
5.4 Optimal Anchor Positions
The estimation of different TN positions is subject to different accuracies. The aim is to find AN locations that would give us an overall best accuracy for all target positions. Thus the ANs that offers the minimum of the mean CRB are chosen. In the following subsection, these optimal AN locations are discussed.
5.4.1 Optimal anchor positions for aNm
Trilateration in a 2-D case requires a minimum of three ANs. Individual distance between each AN and the TN is represented by a circle or line of position (LoP). The point of intersection of these circles is the TN location. In order to get an insight on how the lower bound is affected by the relative angle between the target and the AN node, the CRB for every point in a 10 × 10 2-D plane is calculated for fixed AN positions. Furthermore, in order to achieve the AN positions that give the minimum mean CRB, all the combinations of ANs are taken. i.e.
Crn= n!
r!(n − r)!, (5.19)
where n is the dimension of the area and r is the number of ANs. The mean CRB is given by
mean CRB = 1 n2 n2 X i,j=1 CRBi,j
5.4 Optimal Anchor Positions
for i = 1, ..., n and j = 1, ..., n. Where CRBi,j is the CRB at the (xi, yj) TN
position. Hence the mean CRB is the average CRB of the CRBs taken at all TN locations.
Figure 5.3: Optimal AN positions and corresponding CRB for aNm.
As an example, in the present case, an area of 10 × 10 is taken for case (b1) in Fig. 5.3, where 3 ANs are employed, a total combination of C3100 =161,700 AN positions are obtained. It is well known that when all ANs are placed along the same line (x or y coordinates being the same), then the variance of the CRB rises to infinity and in such cases positioning algorithms such as the LLS fail to estimate the TN’s coordinates. Thus in order to avoid this problem, all the collinear AN positions are not considered in the simulations. The number of such combination is given by x0∗ Cx0
r + y0∗ Cy
0
r , where x0, y0 represent the lengths of x
and y coordinates respectively. For case (b1) in Fig. 5.3, a total of 2,400 collinear AN positions are avoided. Fig. 5.3 shows the optimal AN positions for 3-8 ANs. The contour plots Fig. 5.3 (b1-b6) are obtained for a constant standard deviation for all cases i.e. σi = σ = 2. It is observed that when only 3 ANs are placed in a
square area, the highest accuracy in the estimated location is achieved when the trio is placed at the corners of an equilateral triangle. This triangle is of maximum size as 2 ANs are placed at the corners of one side of the square area while the 3rd
5.4 Optimal Anchor Positions
AN is placed at the centre of the opposite side. It is also noted that the bound increases as the TN goes near any of the AN nodes. The best location for 4 ANs is at the corners of the square area while the best location for an additional 5th AN is the centre of the area. Similarly such symmetrical AN locations are
exhibited in Fig. 5.3 (b4-b6) where 6, 7 and 8 ANs are used. The white points in the figures show the AN locations where the TN placement is avoided. It should be noted that these configurations are independent of rotation i.e. the same results are obtained if the entire set of ANs are simultaneously rotated clockwise or counter-clock wise by 900or 1800. Fig. 5.3a displays the mean CRB
as a function of variance. It is noted that as the number of ANs increase the variance effect on the mean CRB becomes smaller. Fig. 5.4 (b1-b6) illustrates the AN locations which exhibits the worst localisation accuracy and which gives the maximum mean CRB. It is observed that the variance of the estimator is the highest if all the ANs are placed in the same corner of a square area. It is also seen in 5.4a that the improvement in performance is negligible if the number of ANs is increased from 5 to 8 for such a poor network geometry. Furthermore, it is evident from both Fig. 5.3 and Fig. 5.4 that when the minimum 3 ANs are placed optimally (with mean CRB = 1.5063 and 9.0379 for σ2= 1 and 6), it
outperforms a poor deployment of 8 ANs (mean CRB = 13.562 and 81.375 for
σ2= 1 and 6).
5.4.2 Optimal anchor positions for mNm
The plot in Fig. 5.5a illustrates the mNm mean CRB as a function of the number of ANs placed at the optimal positions for the aNm. The contour plot for κ = 0.001 [50] and α = 4 is given in Fig. 5.5 (b1-b6). The mean CRB for the mNm is lower with the aNm for ANs 5 and more. However this is not true for all values
5.4 Optimal Anchor Positions
κ of and α. The optimal AN placement for mNm is different than the aNm, as
shown in Fig. 5.6 (a1-a3) for 3, 4 and 5 ANs. However, these optimal placements depend on the actual scale of the area, the constant κ and α. The mNm CRB becomes lower as values of κ and α are decreased. While both CRBs are almost identical for α = 2. In a dense urban environments where α = 4 to 5 or highly cluttered indoor scenarios, the mNm is a more suitable noise model. In such cases, results from Fig. 5.6 suggest that it is not always optimal to place the AN nodes at the corners as in Fig. 5.3. In fact, the optimality cannot even be guaranteed by the AN placement in Fig. 5.6 as they are for a particular dimension and for an assumed value of κ and α. Thus, for the mNm case, the values of κ and α need to be obtained experimentally before AN deployment. Finally, the worst AN placement for mNm is similar as that for aNm i.e. all ANs are placed at one corner of the area.
5.5 Performance of Linear Least Squares (LLS) Method at Optimal Anchor