# A Joint TDOA-PDOA Localization Approach Using Particle Swarm Optimization

## Summary (2 min read)

### Introduction

- In the first formulation, the authors use only PDOA measurements, whereas in the second formulation, they consider a hybrid cost function using both TDOA and PDOA information.
- A PSO is a multidimensional optimization technique inspired by the behavior of bird flock searching for food [21].
- Section II presents the proposed joint TDOA-PDOA localization algorithm.

### B. The proposed PDOA Cost Function

- Practically, the PDOA of two signals can be estimated directly from the cross power spectrum of the signals [15], [25].
- This method requires the two received signals to be transferred to a central processor.
- Practical ways of estimating the error variances of TDOA and PDOA are using the SNRs [10], [25].
- The function cp(p) is non-convex (at least) due to the nonlinearity of the wrapping operation.
- The proposed PDOA cost function in (6) can be related to the TDOA cost function in (2), also known as Remark 1.

### C. The Proposed Joint Cost Function

- One can consider combining both TDOA and PDOA measurements to form a joint cost function that leverages all the available information.
- Recalling (6) and (7), the authors observe that cp(p) is already weighted by the TDOA error variance σ2t .
- The implication of this weighing is to emphasize the contribution from the more precise measurements, TDOA or PDOA.
- It is also expected to inherit some of the characteristics of the cost function cp(p), especially the large number of local minima.

### D. Optimization

- The authors consider localization based on both cost functions (6) and (9).
- The target location can be estimated by finding the global minimum by solving p̂p = argmin p cp(p), (10) or p̂j = argmin p cj(p).
- A plethora of methods, with well-understood performance characteristics, exist to achieve this task.
- Namely, the authors utilize a quasi-Newton optimizer to solve the optimization problem (2) and initialize the PSO algorithm.
- By using (12) and (13), the authors can set the boundaries of the search space of the PSO.

### E. Summary of the Proposed TDOA-PDOA Localization Approach

- I Using TDOA measurements, calculate the initial location by solving (2) using a quasi-Newton optimizer [12].
- Note that, for Algorithm 1, despite the absence of TDOA from the expression of the cost function, it is still essential to use TDOA information for successful initialization of this algorithm.

### A. Simulation Setup

- For simplicity, the authors consider a 2-D localization problem where the target is on the same plane as the anchors.
- TDOAs are calculated using (1), while wrapped PDOAs are calculated based on (5) for a single acoustic frequency.
- The results, however, can easily be extended to the multi-frequency/multi-carrier case.
- The root mean squared error (RMSE) is adopted as the performance metric.

### B. Results

- The proposed approach yields two algorithms, TDOAPDOA-1 and TDOA-PDOA-2, which are summarized in Section II-E.
- The CRLB for TDOA-based localization (CRLB-T), lower bound for PDOA-based localization (LB-P) [26] and the joint TDOAPDOA lower bound (LB-J) [26] are also used to compare performance.
- The proposed TDOA-PDOA methods provide superior results which stay close to the LB in many cases.
- The RMSE of the proposed algorithms tends to converge to the lower bound as the swarm size increases.
- When σt is small compared to σp, the joint lower bound LB-J is close to the TDOA CRLB.

### IV. CONCLUSION

- A localization approach that jointly uses time-differenceof-arrival (TDOA) and phase-difference-of-arrival (PDOA) measurements has been presented.
- Two cost functions were proposed; one using only PDOA information, and the other using a joint function that takes advantage of both TDOA and PDOA measurements.
- A particle swarm optimizer (PSO) is employed to minimize the proposed cost functions and obtain the final location of the target.
- The proposed methods also stay close to the estimation lower bound for moderate SNR values.
- The performance of the proposed approach depends on the TDOA estimation result, and the PSO swarm size which dictates the computational complexity of the proposed approach.

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2,202 citations

### "A Joint TDOA-PDOA Localization Appr..." refers methods in this paper

...based method can be used to obtain TDOA estimates [8]....

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...Time-difference-of-arrival (TDOA) based localization algorithms [8] are mainly used because no synchronization is required between the target and the anchors....

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1,207 citations

644 citations

### "A Joint TDOA-PDOA Localization Appr..." refers background or methods in this paper

...To avoid picking up a local minimum as the final result, a PSO (particle swarm optimizer) is used [20], [22]....

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...It has the advantages of simple implementation, high-quality solutions to global optima, and quick convergence [22]....

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438 citations

### "A Joint TDOA-PDOA Localization Appr..." refers methods in this paper

...estimate the PDOA is by recording the phase of arrival of the signal at each anchor independently and subtracting the two phases to obtain the PDOA [18], [19]....

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