1 Introduction
The Fifth Generation (5G) of mobile networks is expected to deliver a wide range of location-based services [
1]. To pave the way for those services, a myriad of precise positioning techniques have been introduced in the literature, the majority of which rely on the cooperation between the Access Points (APs) serving the Mobile Users (MUs) [
2]. In particular, to estimate the location, these techniques capitalize on the time measurements carried out between the agents, i.e., MUs and APs, requiring them to have a common time base [
3]. Therefore, for the cooperative approaches to function, the APs need to be accurately synchronized among each other as well as with MUs [
4,
5].
Considerable effort has been made to design fast, continuous, and precise synchronization algorithms across different networks, from Wireless Sensor Networks (WSNs) to wireless communication networks [
6]. Generally, state-of-the-art synchronization algorithms adopt two main macroscopic approaches: (a) designing a
network-wide synchronization algorithm from scratch [
7‐
10], and (b) employing the existing
pairwise synchronization protocols in a structural manner, e.g., layer-by-layer pairwise synchronization [
11‐
13].
Network-wide synchronization in WSNs has been addressed in [
7,
9,
10] by employing the Belief Propagation (BP) algorithm. Typically, BP runs on a Factor Graph (FG) corresponding to the network and calculates the marginals at each node by iteratively exchanging beliefs between neighboring nodes [
14]. The algorithm is advantageous in the sense that it is fully distributed and estimates the clock offset and skew with high accuracy. However, the time required to compute the pairwise conditional
probability distribution functions (pdfs) needed for FG, and then conducting the iterative message passing, can be considered as a potential drawback rendering its practical applicability limited.
Pairwise synchronization is mostly conducted by exchanging time-stamps between the nodes using the Precision Time Protocol (PTP) [
15]. To perform network synchronization in a layer-by-layer manner, PTP is then combined with the Best Master Clock Algorithm (BMCA), whose role is to determine the Master Node (MN) in the network. While this combination operates sufficiently robust in tree-structured networks with medium time-sensitivity (sub-
\(\mu\)s range), BMCA’s poor performance in networks with mesh topology on one hand, and uncertainty in time-stamping on the other hand, render the algorithm futile in highly time-sensitive (sub-hundred ns range) loopy networks.
Despite the attempts in [
11,
16] to address the time-stamping uncertainty (or error) by the virtue of Kalman filtering, this approach is not optimal in the Bayesian sense since all the information available from time-stamps is not utilized. Instead, the Bayesian Recursive Filtering (BRF) utilized in [
17] can be employed to capture all the available information in time-stamps, thereby optimally rectifying the time-stamping error. We have already revealed the outstanding merit of BRF in the mitigation of time-stamping error in [
18].
Although all aforementioned techniques have made invaluable contribution, none of them alone can be expected to meet the global and local time precision aimed by 5G for accurate localization [
19]. Instead, a combination of these algorithms is more likely to deliver a superior performance owing to diverse network typologies (mesh, tree, or a combination thereof) [
16]. In particular, to successfully achieve precise network synchronization, it is suggested by [
20] that the architecture of a large-scale network should consist of common synchronization areas and multiple synchronization domains. Therefore, equipping networks with different synchronization algorithms (or a combination thereof) appears to be a balanced approach, whereby each domain can, based on its topology and capabilities, leverage the most suitable algorithm. In this manner, it is easier to satisfy the requirement of the relative time error in the synchronization domains while keeping the absolute time error low. This is particularly of interest in applications where ultra-high time accuracy is required in a specific synchronization domain, e.g., positioning services.
In [
16], we have introduced and thoroughly described the idea of hybrid synchronization, whereby clock offset can be precisely estimated and correspondingly corrected. The extension to incorporate the clock skew estimation was proposed in [
18]. In this paper, we expand on [
16,
18] and design a hybrid synchronization algorithm based on
asymmetric time-stamp exchange to allow for accurate localization [
21]. The merit of asymmetric time-stamp for localization has been revealed in [
3,
22,
23]. Furthermore, the
fine time measurement standard introduced in [
24] allows for implementation of such time-stamp exchange mechanism. Given that, in addition to analysis of clock offset and skew estimation, we examine the impact of the proposed hybrid approach on a localization algorithm based on the technique presented in [
22].
The contribution of this paper can then be briefly summarized as follows:
-
We present the principles of BP-based network-wide and BRF-based pairwise synchronization based on asymmetric time-stamp exchange.
-
We develop a hybrid statistical synchronization algorithm by combining the two above-mentioned Bayesian approaches.
-
We analyze the performance of the hybrid approach when estimating the clock offset and skew as well as its impact on a localization algorithm.
The rest of this paper is structured as follows: In Sect.
2, the system model is introduced. Section
3 deals with the estimation methods for network-wide, pairwise, and hybrid synchronization. We present and discuss the simulation results in Sect.
4. Section
5 is devoted to the impact of hybrid synchronization on MU localization. Finally, Sect.
6 concludes this work and points to the future work.
1.1 Notation
The boldface capital \({\varvec{A}}\) and lower case \({\varvec{a}}\) letters denote matrices and vectors, respectively. \({\mathbf {a}}(n)\) indicates the nth element of vector \({\mathbf {a}}\). \({\varvec{A}}^T\) represents the transposed of matrix \({\varvec{A}}\). \({\varvec{I}}_N\) denotes a N dimensional identity matrix. \({\mathcal {N}}({\mathbf {x}}|\varvec{\mu }, \varvec{\Sigma })\) indicates a random vector \({\mathbf {x}}\) distributed as Gaussian with mean vector \(\varvec{\mu }\) and covariance matrix \(\varvec{\Sigma }.\) diag\((x_1, \ldots , x_K)\)
denotes a diagonal matrix with the diagonal elements given by \((x_1, \ldots , x_K).\) The symbol \(\thicksim\) stands for “is distributed as,” and the symbol \(\propto\) represents the linear scalar relationship between two functions.
6 Conclusions and future work
We presented two Bayesian approaches toward clock offset and skew estimation in communication networks. In particular, Belief Propagation (BP) was employed to perform high-precision network-wide synchronization, albeit at the cost of a high number of time-stamp exchanges and message passing iterations. Additionally, Bayesian Recursive Filtering (BRF) was leveraged to carry out pairwise synchronization, delivering a superb performance at the edge of the network. Based on these two algorithms, a hybrid Bayesian approach was proposed to not only fulfill a low relative time error at a local level but also to maintain a high synchronization accuracy at a global level. Lastly, we analyzed the impact of the proposed hybrid approach on a joint synchronization and localization (sync&loc) algorithm. Simulation results show that the proposed hybrid approach achieves faster and more frequent synchronization at the cost of only a slight deterioration in performance, i.e., around 3 ns, 0.5 ppm, and 0.1 m in the RMSEs of the clock offset, clock skew, and position, respectively.
Given the promising results, our future work targets the implementation of the hybrid synchronization algorithm presented in this work using Commercial-Off-The-Shelf (COTS) millimeter wave hardware. This would then allow the implementation of the joint synchronization and localization at the edge of the network as well.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.