# Best linear unbiased estimator approach for time-of-arrival based localisation

Abstract: A common technique for source localisation is to utilise the time-of-arrival (TOA) measurements between the source and several spatially separated sensors. The TOA information defines a set of circular equations from which the source position can be calculated with the knowledge of the sensor positions. Apart from nonlinear optimisation, least squares calibration (LSC) and linear least squares (LLS) are two computationally simple positioning alternatives which reorganise the circular equations into a unique and non-unique set of linear equations, respectively. As the LSC and LLS algorithms employ standard least squares (LS), an obvious improvement is to utilise weighted LS estimation. In the paper, it is proved that the best linear unbiased estimator (BLUE) version of the LLS algorithm will give identical estimation performance as long as the linear equations correspond to the independent set. The equivalence of the BLUE-LLS approach and the BLUE variant of the LSC method is analysed. Simulation results are also included to show the comparative performance of the BLUE-LSC, BLUE-LLS, LSC, LLS and constrained weighted LSC methods with Crame-r-Rao lower bound.

## Summary (2 min read)

### 1 Introduction

- Source localization using measurements from an array of spatially separated sensors has been an important problem in radar, sonar, global positioning system [1], mobile communications [2], multimedia [3] and wireless sensor networks [4].
- For two-dimensional positioning, each noise-free TOA provides a circle centered at the sensor on which the source must lie.
- It is computationally intensive and sufficiently precise initial estimates are required to obtain the global solution.
- Simulation results are included in Section 3 to evaluate the estimation performance of the BLUE-LSC and BLUE-LLS algorithms by comparing with the LSC, LLS and constrained weighted LSC [14] methods as well as verify their theoretical development.
- Finally, conclusions are drawn in Section 4.

### 2 Best Linear Unbiased Estimator based Positioning

- The authors first present the signal model for TOA-based localization.
- The BLUE-LSC and BLUE-LLS algorithms are then devised from the LSC and LLS formulations, respectively.
- Their relationship, estimation performance and computational complexity are also provided.
- For simplicity, the authors assume line-of-sight propagation between the source and all sensors such that each ni is a zero-mean white process with known variance σ2i [14].

### 2.1 BLUE-LSC Algorithm

- BLUE [12] is a linear estimator which is unbiased and has minimum variance among all other linear estimators.
- It is suitable for practical implementation as only the mean and covariance of the data are required and complete knowledge of the probability density function is not necessary.
- The BLUE version of the LSC estimator is derived as follows.
- Since {di} are unknown, they will be substituted by {ri} in practice.

### 2.2 BLUE-LLS Algorithm

- The estimator of (9) has minimum variance according to the data model of (8).
- This is analogous to TOA-based and TDOA-based positioning where the former estimation performance bound is lower than that of the latter if the TDOAs are obtained from substraction between the TOAs [15]-[16].
- In the following, the authors will prove that as long as the (M − 1) equations belong to the independent set, the BLUE-LLS estimator performance will agree with the covariance matrices given by (6) and (12).
- Their suboptimality is then illustrated by contrasting with the CRLB.

### 2.3 Relationship and Performance

- From (17), (20) and (22), the authors easily see that the estimation performance of the BLUE-LLS and BLUELSC algorithms is essentially identical.
- Assuming that {ni} are Gaussian distributed, comparison of (12) and the CRLB for positioning is made as follows.
- Denote the corresponding Fisher information matrix by D−1, which has the form of [14].

### 2.4 Complexity Analysis

- Finally, the computational complexity of the linear equation based algorithms is investigated.
- The numbers of multiplications and additions, denoted by M and A , respectively, required in the BLUELSC, LSC, BLUE-LLS and LLS algorithms are provided in Table I which clearly shows the calculation breakdown.
- Note that the Gaussian elimination is employed for performing the matrix inverse operation.
- Excluding the computationally extensive task of solving the Lagrange multiplier corresponding to the constraint of x2 + y2 = R, the CWLSC method needs (16M + 24) multiplications and (10M +7) additions.
- The authors see that the former is preferable because it is more computationally attractive.

### 3 Numerical Examples

- Computer simulations have been conducted to evaluate the performance of the BLUE-LSC and BLUELLS algorithms by comparing with the LLS, LSC and CWLSC [14] algorithms as well as CRLB.
- It is seen that the CWLSC scheme has the best estimation performance as its MSPE attains the CRLB when the average noise power is less than 70 dBm2 where m is referenced to one meter or σ = 103.5m.
- The theoretical development of (6) or (11) is again confirmed for sufficiently small noise conditions and the suboptimality as well as equivalence of the BLUE-LSC and BLUE-LLS methods are demonstrated.
- As a result, the estimation performance of the methods differs at each trial because the positioning accuracy varies with the relative geometry between the source and sensors.

### 4 Conclusion

- Best linear unbiased estimator (BLUE) versions of the least squares calibration (LSC) and linear least squares (LLS) time-of-arrival based positioning algorithms have been examined.
- It is proved that various realizations of the BLUE-LLS approach are indifferent as long as the equations which correspond to the independent set are employed, and their estimation performance is identical to that of the BLUE-LSC algorithm.
- In spite of the suboptimality of the BLUE approach, its estimation accuracy can be close to Cramér-Rao lower bound particularly when the source is located inside the region bounded by sensor coordinates.
- Furthermore, the computational requirement of the BLUE-LSC algorithm is similar to that of the standard LSC and LLS methods and is significantly less than that of the constrained weighted LSC estimator which provides optimal positioning accuracy for sufficiently small noise conditions.

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### Cites background or methods from "Best linear unbiased estimator appr..."

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