5.3 Cooperative Complex Domain Algorithms
5.3.3 Turbo MRC SMDS
In this subsection we consider further improvement of the iterative cooperative MRC-SMDS algorithm presented above by taking advantage also of the minor KT of the complex edge kernel. Straightforwardly, we have
from which we readily obtain
where it is implicitly assumed that initial estimates vAT(0) and vTT(0) are in hand, which in turn can in fact be obtained from equation (5.14).
Similar to the other two iterative variations of MRC-SMDS described earlier, the expression in equation (5.19) is quadratic on both vAT and vTT and hence follows the same convergence behaviour. Unlike the others, however, the iterative algorithm here presented exploits all the extrinsic information contained in the complex edge kernel K, albeit by using only a smaller portion of the latter, thus eliminating a significant amount of redundancy.
Thanks to the cleverly designed iterative structure over this reduced kernel, however, this latest algorithm is capable – as shall soon be shown – of achieving an even better performance than that of the full CD-SMDS algorithm presented in Subsection 4.2.3 at a fraction of the computational cost. Due to these features, we refer to this algorithm as the Turbo MRC-SMDS [136], in allusion to the similar improvement that can be achieved by small engines fitted with a turbocharger. A pseudo-code corresponding to the distance-based version of the Turbo MRC-SMDS algorithm here presented is given in Algorithm 8.
The localization accuracy performances of all the new complex-domain algorithms de-scribed in this section, namely, the closed-form cooperative MRC-SMDS of Subsection 5.3.1, the iterative cooperative SMDS of Subsection 5.3.2, and the Turbo MRC-SMDS of this subsection, are compared against both the new complex-domain full SMDS algorithm of Section 5.2 and the original (real-valued) SMDS of [29,30] in Figure 5.6 and 5.7 for different angle measurement error.
To complete the comparison, the relative complexities of all these algorithms, in terms of the computation time ratio, are plotted in Figure 5.8 , which in turn is built from the data collected in Tables 4.1 and 5.1. The results are obtained for the same simulation scenario described in Subsection 4.2.4, that is, various numbers of target nodes inside a square room with each corner equipped with an anchor, gamma-distributed distance measurements, Tikhonov-distributed AoA measurements (anchor-to-target only) with 90th centered percentile bounding angle given by εθ = 40◦, and computation times obtained from code run on the MathWorks Cloud public platform [131].
Algorithm 8 Distance-based1 Turbo MRC-SMDS Input:
• Anchor-to-target and target-to-target distances: ˜dm
• Anchor-to-target angles: ˜θmp, m < N2A
• The coordinates of all anchors.
Steps:
1. Obtain initial estimate of ˆx via Algorithm 5.
2. Construct ˆv = CMRC· ˆx.
3. Construct vAA, K1, K2, K3, K4 and KT via equation (4.15) and (4.25) using d˜m and ˜θmp where available, or vˆm and ˆv∗p where not.
4. Initialize ˆvAT and vˆTT via equation (5.14).
for n = 1 : Niterations do
5. Update ˆvAT andvˆTT via equation (5.19).
6. Reconstruct the full edge vector ˆv via equation (5.15).
7. Retrieve ˆx via equation (4.27) using CMRC as in (4.35).
Table 5.1: Computation time of new algorithms of Section 5.3
No. Targets MRC-SMDS It. MRC-SMDS Turbo MRC-SMDS
1 26.58× 10−7s 11.86× 10−6s 17.66× 10−6s 5 35.53× 10−7s 21.90× 10−6s 32.37× 10−6s 10 53.80× 10−7s 47.77× 10−6s 69.80× 10−6s 15 95.54× 10−7s 12.02× 10−5s 17.29× 10−5s 20 14.14× 10−6s 25.33× 10−5s 35.68× 10−5s
0 0.5 1 1.5 0.03
0.04 0.05 0.06 0.07 0.08 0.09 0.1
Average Estimation Error vs Ranging Error for Cooperative Algorithms (anchor-to-target angle measurement error εθ= 40◦)
AverageEstimationErrorξ(m)
Ranging Error σd (m)
Original (Real-valued) SMDS New (Complex-domain) Algorithms FullSMDS/CD-SMDS
MRC-SMDS
Iterative MRC-SMDS
Turbo CD-SMDS
Figure 5.6: Localization accuracies achieved by the original full SMDS and the new (complex-domain) cooperative algorithms here presented in Sections 4.2.3 and 5.3 with anchor-to-target angle measurement error εθ = 40◦.
0 0.5 1 1.5 0.06
0.08 0.1 0.12 0.14 0.16 0.18
Average Estimation Error vs Ranging Error for Cooperative Algorithms (anchor-to-target angle measurement error εθ= 70◦)
AverageEstimationErrorξ(m)
Ranging Error σd (m)
Original (Real-valued) SMDS New (Complex-domain) Algorithms
MRC-SMDS
Iterative MRC-SMDS
Turbo CD-SMDS
CD-SMDS
Figure 5.7: Localization accuracies achieved by the original full SMDS and the new (complex-domain) cooperative algorithms here presented in Sections 4.2.3 and 5.3 with anchor-to-target angle measurement error εθ = 70◦.
0 2 4 6 8 10 12 14 16 18 20 0
5 10 15 20 25 30
Computation Time Ratio for Different Number of Targets (σd = 0.7, εθ= 40◦ and 3 iterations)
Computationtimeratio
Number of targets Nt
SMDS time/Iterative MRC-SMDS time SMDS time/Turbo MRC-SMDS time SMDS time/CD-SMDS time
Figure 5.8: Computation time ratio as a function of network size for the original (real-valued) SMDS against the new (complex-domain) cooperative algorithms of Sections 4.2.3 and 5.3.
Figures 5.6, 5.7 and 5.8 are very informative, showing several relevant results. For instance, from Figure 5.6 and 5.7 it is found that when anchors are capable of measuring AoA, CD-SMDS and the original SMDS algorithms perform nearly identically, with the new CD-SMDS still retaining, however, the advantage of having a lower computational complexity, as discussed in Subsection 4.2.4 and shown in Figure 5.8.
Looking at these results in combination with those of Figures 4.2 and 4.3, it can there-fore be concluded that for most practical positioning systems in the two-dimensional space, the CD-SMDS is indicated as a superior replacement to the original SMDS, always offering lower computational complexity in additional to performance improve-ment, in case angle information is either unavailable or unreliable. In turn, if very low computational complexity is required, the closed-form MRC-SMDS algorithm can be selected. But ultimately, the Turbo MRC-SMDS algorithm is found to be the best allrounder choice in the case of systems with anchors capable of AoA estimation, since under these conditions the Turbo MRC-SMDS outperforms both the original and the complex-domain SMDS algorithms in accuracy, with only a slightly higher complexity than that of the CD-SMDS.