Deployment Optimization of Dual-functional UAVs for Integrated Localization and Communication

Modern 5G communication networks with high speed and low latency are expected to cover most urban areas. New applications enabled by 5G such as remote healthcare, Internet of Vehicles, smart homes, industrial control, and environmental monitoring will bring unprecedented intelligent service experience to citizens [1, 2, 3]. However, urban 5G networks are under the risk of regional service interruption in case of earthquakes, floods or other emergency accidents[4]. Besides, in remote areas such as mountainous environments and wild fields, it is difficult to guarantee smooth communication when performing terrain survey, reconnaissance and rescue operations due to the high deployment cost and insufficient coverage of communications infrastructure. Recently, unmanned aerial vehicle (UAV) technology has emerged as an important solution to assist communication in emergency, where UAV-mounted aerial base stations (BSs) can be flexibly deployed to cover ground users with high-speed and reliable line-of-sight (LoS) communication services[5].

Besides reliable communication services, ground terminals such as rescue devices also demand accurate positioning services in emergency scenarios. The global navigation satellite system (GNSS) can provide satisfactory positioning services in open-sky environments [6]. However, under severe obstructions and scattering in mountainous or disaster environments, the positioning accuracy of GNSS plummets dramatically (e.g., from 5-10 meters root mean square error to tens of meters) and the positioning service suffers from frequent interruptions[7]. Besides, the GNSS has relatively low positioning accuracy in vertical dimension, and thus it cannot provide accurate three-dimensional (3D) localization services required in many emergency scenarios[8]. In addition to GNSS, existing terrestrial cellular networks have independent localization capabilities, where terrestrial BSs can serve as anchor nodes (ANs) to support a variety of localization methods such as observed time difference of arrival (OTDoA) positioning [9]. Achieving high 3D localization accuracy requires not only sufficient number of ANs, but also spatial diversity in the deployment of ANs[10]. For instance, to attain high vertical positioning accuracy, the anchors should differ noticeably in amplitude. When the ground ANs are co-planar, UAVs can be quickly and flexibly deployed as temporary aerial ANs to improve the overall localization accuracy, especially in the vertical dimension. Existing UAV deployment solutions mostly use UAVs for either communication or positioning purpose, and lack efficient coordination with terrestrial networks. As a result, a large number of UAVs are needed to ensure both communication and positioning performance in a large area. In practice, this could lead to large networking delay, high control complexity and signalling overhead.

To minimize the UAV deployment cost of emergency networks, we propose a dual-functional UAV paradigm that utilizes UAVs as both air-based ANs and BSs to provide integrated localization and communication (ILAC) services to ground UEs. Specifically, we consider to deploy multiple dual-functional UAVs working collaboratively with terrestrial BSs to extend the communication and localization service coverage. The key design problem lies in the potential conflicting effects of UAV positions on the communication and localization performance of ground UEs. Intuitively, as an aerial BS, a UAV needs to be deployed close to the ground UEs for stronger communication link quality; as an aerial AN, however, such a close-to-ground UAV deployment may cause large vertical localization error due to its similar altitude with terrestrial anchors. To meet the diverse service requirements of ground UEs, the deployment of UAVs should coordinate with terrestrial BSs to achieve a balanced communication and localization performance.

To this end, this paper aims to answer the following two key questions:

Q1: For a UE at a given target location, how to determine the position of a single dual-functional UAV in an analytical form, such that the UAV and ground BSs can collaboratively satisfy the communication and localization performance requirements of the UE.

Q2: For a set of target UEs distributed over a large area, how to deploy minimum number of dual-functional UAVs within a short computational time, such that the integrated air-ground network can meet the communication and localization performance requirements of all the UEs.

To understand the technical challenges of the above questions, we review in the following some related work using UAVs to provide communication and localization services.

In this paper, we study the optimal deployment problem of dual-functional UAVs that operate collaboratively with ground BSs to provide both communication and localization services to ground UEs. In particular, we are interested in deploying minimal UAVs to satisfy the communication and localization performance requirements of UEs at a set of target locations. Our contributions are summarized as follows.

Considering the occasional blockage between the UAVs and the ground UEs, we adopt the commonly used probabilistic LoS channel model to determine the large-scale attenuation for the ground-to-air (G2A) links[11]. The probability of geometric LoS between the UAV and UE depends on statistical parameters related to the environment and the elevation angle. Specifically, we denote the LoS probability between the UAV at location $\bm{b}_{u}$ and UE $k$ as $\mathbb{P}(LoS,\theta_{ku})$, which can be approximated as a modified sigmoid function of the following form [11]

where $e_{1}$ and $e_{2}$ are environment-related parameters. $\theta_{ku}$ is the elevation angle given by

and $\theta_{ku}\in[0^{\circ},90^{\circ}]$. Accordingly, the expected channel power gain is equal to

where $\kappa<1$ is the attenuation effect of the NLoS channel, $\alpha\geq 2$ is the path loss exponent and $\gamma_{0}=(\tfrac{4\pi f_{c}}{c})^{2}$ is the reference free space path loss at a distance of 1 m with $f_{c}$ being the carrier frequency of the G2A channel and $c$ being the speed of light. Then, the uplink transmission rate (bits/s) between the user $k$ and the UAV is

where $W_{ku}$ is the communication bandwidth, $P_{k}$ is the transmission power of user $k$, $N_{0}$ is the noise power spectral density and $\alpha$ is the path loss exponent of the G2A channel.

Due to the rich scattering environment in the ground, the ground-to-ground (G2G) channel between the user device and the terrestrial BS can be modeled as a Rayleigh fading channel, which consists of large-scale attenuation and time-varying small-scale fading. We denote the small-scale fading coefficient as $h_{\text{G2G}}(t)$ and $\mathbb{E}[|h_{\text{G2G}}(t)|^{2}]=1$. The instantaneous capacity of the G2G channel is given by

which changes dynamically in the fading channel. Given an outage probability tolerance $\varepsilon$, an achievable transmission rate (bits/s) of the G2G Rayleigh fading channel can be calculated from Jensen’s inequality as

where $F(\cdot)$ is the cumulative distribution function (CDF) of the Rayleigh fading coefficient, which can be obtained from empirical measurements[24]. The power gain of the G2G channel can be derived as

We consider the OTDoA positioning method to localize the ground UEs. To reduce the computational complexity of UE, each UE estimates its 3D location from the reference signals sent by four ANs, including all the three ground BSs and a UAV. We assume that the accurate locations of terrestrial BSs and UAVs are known, since UAVs can use altimeters and satellite systems to locate themselves. Each UE measures the time-of-arrival (ToA) of localization signals from the aerial and terrestrial ANs. The TDoA observations are the time difference between the ToA of a reference node and the ToAs of the remaining ANs. Suppose that all ANs are accurately synchronized in time and BS 1 is the reference node, device $k$ estimates its location $\bm{m}_{k}$ by solving the TDoA equations:

where $c$ is the propagation speed of the signals and $\tau_{n1}$ is the TDoA between the $n$-th anchor and the reference node BS 1. There exist many linear and non-linear estimation algorithms to obtain $\bm{m}_{k}$ from the TDoA equations[25]. In this paper, we are interested in the theoretical bound of localization accuracy regardless of the specific estimation algorithm.

Without loss of generality, we analyze the performance of the OTDoA method by considering the narrowband positioning reference signal (NPRS) in the narrowband IoT (NB-IoT) system. NPRS is an orthogonal frequency division multiplexing (OFDM) based downlink reference signal occupying 180 kHz bandwidth (one LTE resource block). The variance of the ToA measurement using $N_{sub}$ OFDM subframes is expressed as [26]

where $T_{s}$ and SNR are the symbol duration and signal-to-noise ratio of the received signal, respectively. In one localization period, we have $N_{sub}$ OFDM subframes and each subframe have a set $\mathcal{S}=\{1,2,...,S\}$ of symbols containing NPRS signal. In symbol $s$, the subset of subcarriers containing NPRS signal is denoted as $\mathcal{N}_{s}$ and $p_{i}\in[0,1]$ is the relative power weight of each subcarrier $i$. According to (9), the variance of the ToA measurement from UAV $u$ to UE $k$ is given by

where $W$ is the NPRS bandwidth, $N_{0}$ is the noise power density, $P_{u}$ is the transmission power of the UAV, and $g_{ku}$ is the power gain of G2A channel.

In the G2G channel, advanced signal processing algorithms are needed to eliminate the impacts of multi-path and NLoS propagation on the ToA measurement. The random estimation error is commonly modelled as an additional Gaussian noise term with variance $\sigma_{\text{ NLoS}}^{2}$. With the power gain in (7), we derive the variance of the ToA measurements for the G2G channel as

where $P_{n}$ is the transmission power of the BS $n$. Then, the covariance matrix $\mathbf{R}_{\text{TDoA}}$ of the TDoA observations in unit $\text{s}^{2}$ is given by [27]

where var$(x)$ is the variance of variable $x$ and cov$(x,y)$ is the covariance of variables $x$ and $y$.

The Fisher information matrix (FIM) quantifies the amount of information about the unknown parameter that measurement vector carries. The FIM for OTDoA positioning measurements in (8) is defined as

where $\mathbf{R}=c^{2}\cdot\mathbf{R}_{\text{TDoA}}$ is the covariance matrix of OTDoA positioning in unit $\text{m}^{2}$, and $\mathbf{H}$ is the Jacobian matrix of the TDoA equations

and the elements in the first and second rows of $\mathbf{H}$ can be derived as

with $n=2,3$. The elements in the third row of $\mathbf{H}$ can be obtained from (15) by replacing $\bm{b}_{n}$ with $\bm{b}_{u}$. There are several localization performance metrics defined based on the FIM. A common metric of localization accuracy is CRLB, which represents the theoretical lower bound of mean square estimation error. CRLB can be expressed as a function of UE location and UAV location $L(\bm{m}_{k},\bm{b}_{u})=\text{Tr}\left(\mathbf{F}^{-1}\right)$, where Tr($\cdot$) means the trace of matrix. However, it is difficult to derive an analytical expression of the feasible UAV location with CRLB. In this paper, we adopt the D-optimality criterion to analyze the localization performance, which is a well-known performance metric of localization accuracy [28]. With the D-optimality metric, we will be able to find intuitive and meaningful geometrical characterizations of UAV deployment in Section IV. Specifically, D-optimality seeks to maximize the following opt-$D$ value

The determinant of $\mathbf{F}^{-1}$ is inversely proportional to the uncertainty area of an unbiased location estimation [29]. Intuitively, a D-optimality design minimizes the uncertainty ellipsoid, or equivalently, minimize the average location estimation error.

To facilitate the analysis, we propose in this subsection an approximation of the opt-$D$ value for a tagged UE $k$. For simplicity, we drop the index $k$ and denote $\sigma_{nk}$ as $\sigma_{n}$. We calculate the determinant of the covariance matrix

where $D_{1}\triangleq c^{6}\sigma_{1}^{2}\sigma_{2}^{2}\sigma_{3}^{2},D_{2}\triangleq c^{6}(\sigma_{1}^{2}\sigma_{3}^{2}+\sigma_{1}^{2}\sigma_{2}^{2}+\sigma_{2}^{2}\sigma_{3}^{2})$. Supposing that $D_{1}\gg D_{2}\cdot\tfrac{\psi}{\text{SNR}_{u}}$, we can ignore the second term and approximate $\det(\mathbf{R})$ using $D_{1}$. Then, we denote the corresponding value as opt-$D_{1}$

In contrast, when $D_{1}\ll D_{2}\cdot\tfrac{\psi}{\text{SNR}_{u}}$, we ignore the constant term $D_{1}$ and approximate $\det(\mathbf{R})$ using the second term. We denote this approximation as opt-$D_{2}$, where

In the following lemma, we show the condition when opt-$D_{1}$ is an accurate approximation.

In practice, the random estimation error $\sigma_{\text{ NLoS}}^{2}$ could be larger than the noise variance of G2A channel $\sigma_{u}^{2}$, and thus the condition for Lemma 1 is easily satisfied. In Fig. 4(a), we compare the two approximations and the opt-$D$ value with different ratios $\frac{\text{SNR}_{u}}{\text{SNR}_{n}}\in[10^{-2},10]$, and depict the values in a log scale. We observe that opt-$D_{1}$ is much more accurate than opt-$D_{2}$ within the considered ratio range. In Fig. 4(b), the opt-$D$ and opt-$D_{1}$ values are almost on top with each other when the ratio is greater than 1, and the gap is less that 2% when the ratio is larger than 0.1. Therefore, we can safely adopt opt-$D_{1}$ as an accurate approximation in the following analysis.

Firstly, we define the unit vector from the $u$-th UAV to a location point $\bm{m}=[x_{0},y_{0},h_{0}]$ as $\bm{q}_{u}$, and that from ground AN $n$ to $\bm{m}$ as $\bm{q}_{n}$, where

Then, $\mathbf{H}$ can be represented by the unit vectors

and we explicitly express $\det(\mathbf{H})$ using the elements in (15) such that

where the coefficients $\alpha_{n}$ for $n=1,2,3$ are derived using (24)

The localization requirement $\text{opt-}D\geq\epsilon$ with opt-$D_{1}$ approximation in (19) becomes

Based on the value of $\det(\mathbf{H})$, we derive the implicit geometry of (30) in two cases.