A Semidefinite Relaxation Method for Elliptical Location
Abstract
:1. Introduction
- We present the negative log-likelihood (NLL) function of the elliptical location estimation problem with multiple transmitters and multiple receivers in a concise form, and then derive the Cramér–Rao bound (CRB) of the problem. This is the theoretical basis for the feasibility analysis of the problem.
- We identify the NLL function minimization as a quadratic optimization problem with multiple quadratic equality/inequality constraints and design its semidefinite relaxation.
- We find the dual problem of the primal SDP problem and give the pseudo-code of the interior-point primal-dual path-following algorithm. With these results, engineers can design customized algorithms themselves, instead of relying on codes provided by mathematicians that are not optimized for a specific problem.
- We theoretically analyze the polynomial time complexity of the algorithm and experimentally confirm that the proposed estimator attains the CRB approximately.
- We cautiously verify the deployability of the algorithm by setting the measurement noise levels far higher than what can be achieved by real systems. Simulation experiments with extreme parameter settings are alternative to the costly field tests.
2. Problem Formulation and Data Model
3. Cramér–Rao Bound
4. Design of the Estimator
4.1. Formulation of MLE as a Quadratic Optimization Problem
4.2. Some Equivalent Transformation
4.3. Semidefinite Relaxation
4.4. Semidefinite Programming Solver
Algorithm 1 Primal-dual path-following interior-point method. |
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4.5. Complexity Analysis
5. Results and Discussion
- To ascertain the statistical performance of the proposed SDR estimator, we compared it with several closed-form methods recently proposed, i.e., Yang’s TS-WLS method [21] and Amiri’s CWLS method [20]. For a given estimator of the unknown parameter , its performance is measured by the empirical bias and RMSE, which are defined as follows.
- To measure the tightness of the relaxed optimization model in (16), we recorded the condition numbers of the optimal solutions of and in (17) at different noise levels. The motivation is that if the condition number of a matrix is greater than , then it can be approximated as a rank-1 matrix [44].
- To verify whether the SDR solution needs post-processing, we checked the performance improvement gifted by post-processing using local search under different noise conditions. We used the SDR solution as the initial point for the commercial optimization solver MATLAB function lsqnonlin, and the algorithm for lsqnonlin was set as ‘trust-region-reflective‘. Our estimator with post-processing is simply referred to as SDR-LS.
5.1. Localization Scenarios
5.2. Visualization of Cramér–Rao Bound and Selection of Testing Points
5.3. Results on the Two Representative Test Points
5.3.1. Results on Easier Test Point
5.3.2. Results on Harder Test Point
6. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Abbreviations
AEL | Asynchronous Elliptical Location |
AOA | Angle of Arrival |
CRB | Cramér–Rao Bound |
CWLS | Constrained Weighted Least-squares |
FDOA | Frequency Difference of Arrival |
MLE | Maximum Likelihood Estimator |
MSE | Mean Square Error |
NLL | Negative Log-likelihood |
RMSE | Root-mean-square Error |
RSS | Received Signal Strength |
SDP | Semidefinite Programming |
SDR | Semidefinite Relaxation |
SEL | Synchronous Elliptical Location |
SI | Spherical-interpolation |
SX | Spherical-intersection |
TDOA | Time Difference of Arrival |
TOA | Time of Arrival |
TS-WLS | Two-stage Weighted least-squares |
WLS | Weighted Least-squares |
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Transmitter/Receiver No. | Position Coordinates (m) |
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Wang, X.; Ding, Y.; Yang, L. A Semidefinite Relaxation Method for Elliptical Location. Electronics 2020, 9, 128. https://doi.org/10.3390/electronics9010128
Wang X, Ding Y, Yang L. A Semidefinite Relaxation Method for Elliptical Location. Electronics. 2020; 9(1):128. https://doi.org/10.3390/electronics9010128
Chicago/Turabian StyleWang, Xin, Ying Ding, and Le Yang. 2020. "A Semidefinite Relaxation Method for Elliptical Location" Electronics 9, no. 1: 128. https://doi.org/10.3390/electronics9010128
APA StyleWang, X., Ding, Y., & Yang, L. (2020). A Semidefinite Relaxation Method for Elliptical Location. Electronics, 9(1), 128. https://doi.org/10.3390/electronics9010128