# Enhancing PDOP for indoor localisation by considering the effects of multipaths with virtual beacons

Elizabeth Colin, Laurie Conteville, Antoine Covet

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Elizabeth Colin, Laurie Conteville, Antoine Covet. Enhancing PDOP for indoor localisation by considering the effects of multipaths with virtual beacons. 14th edition of the International Conference on Indoor Positioning and Indoor Navigation (IPIN), Hong Kong Polytechnic University, Oct 2024, Hong Kong, China. hal-05286512

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Submitted on 27 Sep 2025

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# Enhancing PDOP for indoor localisation by considering the effects of multipaths with virtual beacons

Antoine Covet, Elizabeth Colin and Laurie Conteville

Efrei Research Lab, Efrei Paris Panthéon Assas Université, Villejuif, France

**Abstract**

The challenge addressed in this paper is to develop a method for optimizing beacon placement in indoor environments to minimize PDOP by effectively utilizing multipath signals. Traditional approaches often neglect the potential benefits of multipath, focusing instead on mitigating their effects. Our approach aims to integrate these multipath signals into the positioning algorithm to enhance the overall accuracy.

**Keywords**

Indoor localisation, PDOP, multipath, ray-tracing

## 1. Introduction

Accurate positioning within indoor environments presents a significant challenge, primarily due to the complexity and variability of signal propagation. Traditional GNSS-based systems are often inadequate for indoor applications, leading to the development of alternative methods to enhance positioning accuracy. One of the critical metrics used to evaluate the precision of positioning systems is the Position Dilution of Precision (PDOP). PDOP is a measure of the geometrical strength of satellite or beacon configurations, influencing the overall accuracy of the positioning solution.

The primary challenge in indoor positioning is the presence of multipath effects, where signals reflect off walls and other objects before reaching the receiver. These reflections can cause significant errors in position estimates if not considered properly. These multipath signals can be exploited to improve positioning accuracy rather than being treated solely as a source of error.

In complex indoor environments, the placement of beacons significantly impacts the PDOP. Incorrect placement can lead to high PDOP values, resulting in poor positioning accuracy. Therefore, optimizing beacon placement is crucial for minimizing PDOP and enhancing the performance of indoor positioning systems. This optimization involves not only considering the direct line-of-sight (LOS) signals but also leveraging the multipath signals to improve the geometric configuration of the beacons.

The challenge addressed in this paper is to develop a method for optimizing beacon placement in indoor environments to minimize PDOP by effectively utilizing multipath signals. Traditional approaches often neglect the potential benefits of multipath, focusing instead on mitigating their effects. Our approach aims to integrate these multipath signals into the positioning algorithm to enhance the overall accuracy.

The objective of this study is to improve indoor positioning accuracy by developing a theoretical framework and practical algorithms for optimizing beacon placement with a focus on minimizing PDOP. This involves:

* Developing mathematical models that incorporate both direct and reflected signal paths.

* Treating reflected paths as virtual beacons to utilize multipath information.

* Optimizing the spatial arrangement of beacons to reduce PDOP.

This work is motivated by the need for highly accurate indoor positioning systems in various applications such as emergency response, asset tracking, and navigation within large buildings. By

<figure><img src="image_2.png" /></figure>

<figure><img src="image_3.png" /></figure>

<figure><img src="image_4.png" /></figure>
transforming the traditionally detrimental multipath effects into beneficial components of the positioning system, we aim to significantly enhance the performance of indoor localization solutions.

This paper proposes a novel approach to minimize PDOP by leveraging multipath signals to optimize beacon placement. Through detailed mathematical modeling and optimization techniques, we seek to provide a robust framework that improves the accuracy and reliability of indoor positioning systems.

This paper is organized as follows, section two presents the works related mainly to PDOP in an indoor environment. Section three presents the theoretical basis that support the simulations presented in section four. Section five summarizes the results and proposes future perspectives.

## 2. Related Works

The study of improving Position Dilution of Precision (PDOP) using multipath effects in indoor environments has seen significant advancements through various methodologies focused on beacon placement, geometric optimization, and multipath signal exploitation.

Leune et al. [1] address the optimization of wireless locating in complex environments by strategically placing anchor nodes using evolutionary algorithms. By simulating different geometries, they identify configurations that minimize PDOP, enhancing positioning precision. This foundational work sets the stage for further exploration into geometric optimization techniques that directly impact PDOP.

Building on this, Löffler et al. [2] emphasize the importance of beacon distribution within indoor environments. Using constraint programming, their research ensures that beacons are optimally placed to guarantee accurate trilateration. This method not only reduces PDOP but also improves the overall reliability of indoor positioning systems by ensuring consistent coverage.

Kammoun [3] extends this understanding by providing a comprehensive theoretical framework for optimizing beacon placement specifically to minimize PDOP. His thesis highlights the significance of considering both direct and reflected signal paths (multipath) in PDOP calculations. By incorporating these factors, Kammoun demonstrates substantial improvements in indoor positioning accuracy, offering a detailed methodology for integrating multipath into optimization processes.

Further advancing the discussion, Yun and Iskander [4] explore the use of ray tracing for radio propagation modeling. Their work emphasizes the importance of detailed geometric modeling to simulate signal propagation, including reflections and diffractions. Ray tracing provides a robust method to incorporate multipath information accurately, significantly enhancing PDOP and positioning accuracy in complex indoor settings.

These studies collectively advance the understanding of beacon placement strategies, geometric optimization, and theoretical frameworks for PDOP improvement in indoor environments. By building on these foundational works, our research aims to refine these techniques further, focusing on the effective utilization of multipath signals to achieve higher positioning accuracy. This approach not only mitigates the negative impacts of multipath but also leverages them as a beneficial factor, ultimately leading to more reliable and precise indoor positioning solutions.

## 3. Enhanced PDOP with virtual beacons

The Dilution Of Precision (DOP) criteria determines the error due to the geometry of satellite or beacons. In this paper, we use the Position Dilution Of Precision (PDOP), to calculate it we must get the positions of the emitters (satellites or beacons) and the receiver (The target we want to locate).

We define G as the matrix of geometry, we use it to compute the PDOP. G considers the positions of the beacons normalized by the distance between it and the receiver.
$$ G = \begin{bmatrix} \frac{x_1 - x_r}{d_1} & \frac{y_1 - y_r}{d_1} \\ \frac{x_2 - x_r}{d_2} & \frac{y_2 - y_r}{d_2} \\ \vdots & \vdots \\ \frac{x_n - x_r}{d_n} & \frac{y_n - y_r}{d_n} \end{bmatrix} \qquad (1) $$

With:

* $x_i, y_i$ the coordinates of the beacon $i$

* $x_r, y_r$ the coordinates of the receiver

* $d_i$ the distance between the beacon $i$ and the receiver

$$ d_i = \sqrt{(x_i - x_r) + (y_i - y_r)} \qquad (2) $$

$$ PDOP(G) = \sqrt{tr((G^T G)^{-1})} \qquad (3) $$

This criterion has many downsides when applied to indoor positioning. Indeed, the signal sent by the beacons is reflected on the walls around. The objective is then to include in the computation of the PDOP the influence of these multipaths, typical in indoor environments.

In order to include the influence of multipaths in the computation of the PDOP, we model the environment in two dimensions (2D), and the rays sent by the emitter. The environment is modeled considering the walls, the direct ray and the reflected rays. To model the latter, we create a virtual emitter that represents the ray after the reflection like shown in the figure 1:

<figure><img src="image_5.png" /><figcaption>**Figure 1:** Creation of virtual beacons (gray stars) that represents the multipaths that reach the receiver (red dot)</figcaption></figure>

These virtual beacons are wall-symmetric to the original beacon, for a one-reflection multipath. For a n-reflections multipath, the virtual beacons are wall-symmetric to the (n-1)-rebound multipath. As an example, in the figure 1, the virtual gray beacon on the right is left wall-symmetric to the one on the right, therefore, he represents the multipath of a ray that rebounds one the right wall, then on the left wall, and finally reaches the receiver.

Then, we calculate the power received by the receiver from the real and the virtual beacons, using the free space propagation model [5]:

$$ P_r = R^n P_t \left( \frac{\lambda}{4\pi d} \right)^2 \qquad (4) $$

With:

* $P_r$ as the received power

* $P_t$ as the transmitted power

* $\lambda$ as the wavelength
* d as the distance travelled by the signal

* R as the coefficient of reflection

* n as the number of reflections

These representations of virtual beacons allow us to precise the measure of PDOP, including the effects of multipaths. We consider the virtual beacons in the PDOP calculation as shown in (5).

$$G_v = \begin{bmatrix} \frac{x_1-x_r}{d_1} & \frac{y_1-y_r}{d_1} \\ \frac{x_2-x_r}{d_2} & \frac{y_2-y_r}{d_2} \\ \vdots & \vdots \\ \frac{x_n-x_r}{d_n} & \frac{y_n-y_r}{d_n} \\ \frac{x_{v1}-x_r}{d_{v1}} & \frac{y_{v1}-y_r}{d_{v1}} \\ \vdots & \vdots \\ \frac{x_{vm}-x_r}{d_{vm}} & \frac{y_{vm}-y_r}{d_{vm}} \end{bmatrix} \quad (5)$$

With:

* $x_{vi}, y_{vi}$ the coordinates of the virtual beacon i

* m the number of virtual beacons

* $d_{vi}$ the distance between the virtual beacon i and the receiver

There are an infinite number of reflections, then, there are an infinite number of these virtual beacons. We consider only the ones for which the power, when reaching the receiver, is higher than the latter's sensitivity.

Then:

Let $P \in \mathbb{R}^n$ the vector of the power received by the receiver from each beacon (real or not) and k the threshold representing the receiver's sensitivity.

$$G_v' = G_v \setminus \{G_{v_i} | P_i < k\} \quad (6)$$

$$PDOP_v = PDOP(G_v') \quad (7)$$

With $PDOP_v$ the PDOP considering the real and virtual beacons

## 4. PDOP simulations in 2D

To simulate, we will consider a room of 10 meters by 10 meters, with 4 emitters and one receiver, all the simulations will be made in a 2D environment, the sensitivity of the receiver is 10⁻⁵ W, the power emitted from the beacons is 1mW, the frequency of the emitted signal is 2.45GHz, and the walls are made of white plaster. its coefficient of reflection is 0,4571 at 2,4GHz, according [6].

To start off, we consider the evolution of the PDOP in the 2D space, with beacons in the middle of each wall.

$$Beacon\_coordinates = \begin{bmatrix} B_1 \\ B_2 \\ B_3 \\ B_4 \end{bmatrix} = \begin{bmatrix} 0 & 5 \\ 5 & 0 \\ 10 & 5 \\ 5 & 10 \end{bmatrix}$$

Then, in figure 1 (b), we determine the position of the virtual beacons as explained in section III for one reflection.
<figure><img src="image_6.png" /><figcaption>Figure 1 (a): Position of each beacon (blue circles)</figcaption></figure>

**Figure 1 (b):** Position of each virtual beacons (1 reflection) (the receptor is at the coordinates 5,5). In red are the multipath rays from each beacon to the receptor.

If we calculate the PDOP for case (a):

$$PDOP = 1.1180$$

For case (b), considering the virtual beacons, we get:

$$PDOP_v = 0,6455$$

We see an improvement of the PDOP because we take into account the multipaths when using the virtual beacons.

If we increase the number of reflections to about seven (for reasonable calculation time), we observe a drastic decrease of the PDOP from 0,6455 to 0.1403, due to the even greater number of virtual beacons considered in the second calculation.

<figure><img src="image_7.png" /><figcaption>Figure 2: Position of each virtual beacons (blue circles) for a receiver's position of [5,5]</figcaption></figure>

$$PDOP_v' = 0.1403$$

PDOP_v' is the PDOP where we consider all the virtual and real beacons simulated.

Afterwards, we do not consider beacons that are so far away that the power received is below the sensitivity threshold.
<figure><img src="image_8.png" /><figcaption>Figure 3: Position of each virtual beacons (blue circles), and in green the ones in range to be considered part of the PDOP computation for a receiver position at (5,5)</figcaption></figure>

Then, we can calculate the PDOP:

$$PDOP_v = 0,4226$$

To compute the PDOP<sub>v</sub> a reduced number of virtual beacons are valid. It comes down to decreasing the number of considered reflections. In this case, only three reflections are needed.

If we change the position of the receiver to [9,8]:

<figure><img src="image_9.png" /><figcaption>Figure 4: Position of each virtual beacons (blue circles), and in red the ones in range to be considered part of the PDOP computation for a position of (9,8)</figcaption></figure>

$$PDOP_v = 0,3799$$

We observe that the geometry of the valid virtual beacons is different, resulting in a different PDOP<sub>v</sub>.

We then compute the PDOP for a 10*10 meters square area, with a step of 1 meter as shown in figure 6.
<figure><img src="image_10.png" /><figcaption>Figure 5: The stars represent every position taken by the receiver.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Figure 6: Evolution of the $PDOP_v$ for each position in a 10\*10 2D area</figcaption></figure>

We see in the figure 7 that, since the geometry of the beacons is symmetrical as seen in figure 1, the distribution of the $PDOP_v$ is also symmetrical.

We also find that the $PDOP_v$ is very low, between 0.423 and 0.373, lower than PDOP when we consider real beacons only as shown in the figure 8, where the PDOP varies between 1.118 and 5.177. $PDOP_v$ is better than PDOP because we consider multipaths, which is equivalent to increasing the number of beacons.

<figure><img src="image_12.png" /><figcaption>Figure 7: Evolution of the PDOP for each position in a 10\*10 2D space without virtual beacons</figcaption></figure>
Including multipaths in the calculations of the PDOP improves it. PDOP from 1.118 to 5.177 versus PDOP<sub>v</sub> from 0.423 to 0.373, due to a greater number of beacons considered. Therefore, the inclusion of these virtual beacons takes advantage of the multipaths to improve the indoor localization.

# 5. Conclusion and future perspective

In this paper, we present a new method of calculating an enhanced PDOP that includes the multipath through virtual beacons that represent reflected rays. Using this method, we observe a significant improvement when compared to the classical PDOP.

There are, though, a few ways the simulation can be improved upon. For now, this method was only applied to very simple architectures, a square. In future works, we aim to extend this approach to more realistic environments with Non-Line of Sight (NLOS).

Additionally, our current simulations are limited to a 2D area. An extension to 3D will enable a more comprehensive evaluation of the enhanced PDOP method's performance and robustness.

In summary, the current implementation of our method shows promising improvements in PDOP calculation. By addressing it, we aim to develop a solution for indoor positioning that effectively leverages multipath signals to improve accuracy and reliability.

# References

[1] Tilman Leune, Thorsten Wehs, Manuel Janssen, Carsten Koch, Gerd von Cölln, Optimization of wireless locating in complex environments by placement of anchor nodes with evolutionary algorithms, in: 2013 IEEE 18th Conference on Emerging Technologies & Factory Automation (ETFA), IEEE, Cagliari, Italy, DOI: 10.1109/ETFA.2013.6648068

[2] Sven Löffler, Franz Kroll, Ilja Becker, Petra Hofstedt, Optimal Beacon Placement for Indoor Positioning Using Constraint Programming, in: 2022 IEEE/ACS 19th International Conference on Computer Systems and Applications (AICCSA), IEEE, Abu Dhabi, United Arab Emirates, DOI: 10.1109/AICCSA56895.2022.10017575

[3] S. Kammoun, Géolocalisation à l'intérieur d'un bâtiment pour terminaux mobiles, Ph.D. thesis, Télécom ParisTech, Paris, 2016. NNT : 2016ENST0041

[4] ZHENGQING YUN, MAGDY F. ISKANDER, Ray Tracing for Radio Propagation Modeling: Principles and Applications, IEEE Access (2015) 1089 - 1100. DOI: 10.1109/ACCESS.2015.2453991

[5] H.T. Friis, A Note on a Simple Transmission Formula, Proceedings of the IRE, Volume 34, issue 5 (1946). DOI: 10.1109/JRPROC.1946.234568

[6] Kamel Haddadi; Tuami Lasri, Geometrical Optics-Based Model for Dielectric Constant and Loss Tangent Free-Space Measurement, IEEE Transactions on Instrumentation and Measurement (2014) 1818 – 1823. DOI: 10.1109/TIM.2013.2297811