Impact of Estimation Uncertainty

Amongst range-based network wide algorithms, there are ones that rely not only on the anchor locations and distances between targets and other nodes , but also the iteratively obtained target location estimates which are cooperatively shared at each iteration. As one of such approaches we can refer to the well-known MPLA (which is categorized as a Bayesian method) in which targets iteratively estimate their own location relying on the location estimates of, as well as distances measurements to, all connected neighbours [2, 16, 18, 19].

It is clear that the derived performance limit of such schemes must also dependent on the uncertainty on the exchanged node location estimates. Therefore, in this section we are going to address this problem for MPLA accordingly and therefore deriving a more realistic performance limit approximation.

The FIM corresponding to network-wide approach derived in equation (3.40) makes use only of the error distributions of measured distances. This implies that the corre-sponding CRLB applies only to partial- or non-cooperative range-based localization algorithms, in which location estimates of all targets are obtained simultaneously, without relying on any prior target location information.

In range-based MPLAs, on the other hand, targets repeatedly exchange both ranging information and their own location estimates [2, 16–19]. This indicates that the ultimate performance of such schemes must also be dependent on the uncertainty on the exchanged target location estimates, which in turn are lower bounded by the CRLB.

In other words, the fundamental performance limit of distributed MPLAs can only be truly captured by the iterated posterior CRLB [100]. The problem with such an approach, however, is that only numerical results can be expressed, such that direct insight on the nature of the cooperation is hard to to be obtained.

Given ranging between two arbitrary targets θ_tand θ_n, there exist two measurements d_tn and d_nt, where for a given ranging model, the ranging errors are equal, or in other words σ_tn² = σ²_nt. Let σ_x:n² and σ_y:n² be the variances of the location estimate errors of the target at θ_n, along the x and y dimensions, respectively, such that the combined error standard deviation along an unknown direction can be approximated by the quantity

σ_tn^∗ ≈q

σ²_tn+¹₂(σ_x:n² + σ²_y:n), as illustrated in Figure 3.7. Therefore between a pair of targets t and n, with t, n≤ nT, one can then consider that such additional uncertainty is transferred to the estimates of d_tn and d_nt.

Note, however, that for high SNR values, the only difference between distance estimates d_tn and d_nt is the position estimation uncertainty of the targets θ_t and θ_n. Therefore, the information terms corresponding to dtn and dnt in the likelihood function can be assumed as uncorrelated. In other words, the corresponding joint likelihood function can be approximated by a product of two independent likelihood functions. Hence the FIM in Equation (3.40) is modified to

F^H_Θ

However for low SNR values, the target location uncertainties are relatively small compared to the ranging errors and therefore, the pair of information terms related to d_tn and d_nt are highly correlated. Consequently, under low SNR, the corresponding joint likelihood function for each pair of targets t and n can be approximated by the largest term between the two, corresponding to the smallest standard deviation, i.e., min{σtn^∗ , σ_nt^∗ }. Therefore the FIM expression becomes

Proceeding in this manner, besides the FIM itself, the inverse of F_ΘT is of interest, as per equation (3.41). That can certainly be obtained numerically from equation (3.40), but in order to retain analytical insight let us consider the following pair of equalities relating the trace and eigenvalues of a matrix such that

tr (F_ΘT) =

where Υ⁰_i= Υ_i/T . Using the geometric-arithmetic mean inequality we can obtain

Finally, combining the relations in (3.58) and the derived inequality in (3.60) we obtain tr(F⁻¹_Θ

T)

ηn_T ≥ ηn_T

tr(F_ΘT), (3.61)

which holds as an equality if and only if λi = λj for∀i, j. From inequality (3.61) one concludes that in networks where all localization errors are similar – which tends to be the case in cooperative systems – the average CRLB can be well approximated by ηnT/tr (FΘ_T), such that the trace of the FIM itself is of great relevance.

Thanks to the structure revealed in equations (3.36) through (3.40), we obtain

tr (FΘ_T) =

Ftn · ¯dtn. Furthermore we highlight the contributions of different structural components of the network in above equation.

Let us now turn our attention to the effect of the uncertainty of target locations themselves onto the cooperation that takes place in MPLAs. To this end, let us consider a 10m×100m space equipped with 4 anchors placed at every corner and 50 targets randomly distributed along the area. The ranging error variances are updated iteratively in order to account for the target location estimate uncertainty where the combined error variances are used to compute the FIMs for high and low SNR scenarios.

Note that the FIM computed here are only an approximation and therefore the inverse of these values cannot be referred to as the CRLB, but rather an approximation for lower bound on the covariance matrix.

✓

✓

x:n

d

Figure 3.7: Illustration of ranging process in the presence of target location uncertainty.

0 0.05 0.1 0.15 0.2 0.25 0.3 10⁻⁴

10⁻³ 10⁻² 10⁻¹ 10⁰ 10¹

Performance of Cooperative Network Localization: Impact of Noise (10m×100m scenario with nT = 50 target nodes, nA= 4 anchor nodes and α = 1.8)

ηnT/tr(FΘT)

σ²₀

Multihopping Cooperation Message Passing Cooperation No iterations

Iterative high SNR10 Iterative low SNR10

Figure 3.8: Impact of location uncertainty on the CRLB of distributed MPLA, as a function of reference ranging estimation errors.

The results after iterations are shown in Figure 3.8. It can be appreciated that when the neighbour location uncertainty is taken into account, the approximation term ηn_T/tr (F_ΘT) for both high and low SNRs comes closer to the dashed curve, which is obtained by evaluating the right hand side of inequality (3.61) using a minorized version of tr (FΘ_T), namely

tr (F_ΘT) =

n_T

X

t=1 N

X

n=nT+1

u_tnu^T_tn

| {z }

target to anchor

, (3.63)

which differs from equation (3.62) only in that the term related to the cooperation amongst targets has been removed.

The dashed curve in Figure 3.8 relates to a network in which the target nodes are only able to communicate with anchor nodes in their neighbourhood. In other words, the figure shows that a good amount of the “cooperation gain” one would expect to obtain from the message-passing strategy of MPLAs is actually diluted by the uncertainty remaining in the exchanged target location estimates, leading to poor results than expected. That is, the analysis indicates that the informational component of the cooperation amongst target nodes in fully cooperative localization (including MPLAs) has fundamentally little impact onto the achievable performance of cooperative network localization systems.

Moreover we do emphasise on the fact that, to the best of our knowledge, previous efforts to obtain CRLB (see for instance [83] and [101], as well as references thereby) for fully cooperative wireless localization systems failed to account for the uncertainty of target locations in the estimation process, which explains the typically loose comparison between analytical and empirical results. To elaborate, as briefly mentioned earlier, the result of such omission is that the derived bounds actually represent the achievable errors of partially cooperative approaches, in which the location of all targets are computed simultaneously without requiring iterative distribution of previous location estimates amongst targets themselves.