UAV Trajectory and Beamforming Optimization for Integrated Periodic Sensing and Communication

Integrated sensing and communication (ISAC) has emerged as a key technology in future wireless networks[1]. However, due to shared spectrum resources and complicated surrounding scatters, it is very challenging to strike a good balance between two potentially conflicting design objectives: high-quality communication service and high-timeliness sensing requirement. Recently, there are some research efforts devoted to unifying sensing and communication to mutually assist each other [2]. For example, appropriate radar beam pattern design with the communication requirement guaranteed and Pareto optimization framework of the radar-communication system were presented in [3] and [4], respectively.

Driven by on-demand deployment and strong line-of-sight (LoS) links promised by unmanned aerial vehicles (UAVs) [5, 6], UAV is expected to be a promising aerial ISAC platform to provide more controllable and balanced integrated service based on the requirements of sensing frequency and communication quality [7]. In [8], a joint beamforming and UAV trajectory optimization algorithm was proposed to improve communication quality while guaranteeing sensing requirement. However, the prior works on ISAC (e.g., [3], [4], [8]) mainly focus on improving the performance when both functionalities working simultaneously during the entire considered period, i.e., sensing is always executed along with the communication. This however, may ignore the practical asymmetric sensing and communication requirements. Specifically, the sensing frequency as another important aspect of ISAC systems is practically determined by the timeliness requirement of specific tasks and has not been taken into account in the literature, e.g., a relatively low/high sensing frequency is preferred for low-speed/high-speed target tracking. As such, always forcing both sensing and communication simultaneously may cause excessive sensing, thereby introducing stronger interference and higher energy consumption. Hence, it is crucial to improve sensing efficiency, especially for power limited UAVs. To address this issue, a more general trade-off between sensing and communication needs to be investigated by taking into account the sensing frequency besides the commonly used sensing power, which thus motivates this work.

In this paper, we study a UAV-enabled ISAC system aimed at sensing targets near the ground while providing multi-cast downlink communication for single-antenna users. In particular, to reflect the practically different sensing frequency and sensing power requirements, we propose a periodic ISAC framework where the sensing is periodically executed along with communication. As an initial study, we consider a system model with one target and one cluster of users only, to draw the fundamental insights of UAV-enabled ISAC systems for the considered periodic sensing scenarios, as shown in Fig. 1 (e.g., pipeline inspection or traffic monitoring). In this work, the UAV trajectory, transmit beamforming, and sensing instant are jointly optimized to maximize the communication performance while guaranteeing the sensing requirement. Unlike traditional ISAC considered in [8], which forces data transmitting and radar sensing simultaneously all the time, the proposed scheme offers another opportunity to balance between practical sensing and communication over time. As a result, both standalone communication and always-sensing are special cases of our considered periodic sensing and communication scenarios. The main contribution in this paper is summarized as follows:

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  To cater for different sensing requirements and avoid excessive energy consumption in practice, we propose an integrated periodic sensing and communication (IPSAC) mechanism to provide a more flexible trade-off between sensing and communication over time. We first prove the structural symmetry between optimal solutions in different ISAC frames without location constraints, thereby reducing the algorithm complexity.

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  We solve the optimal transmit precoder according to the rank-one characteristic and derive the optimal achievable rate in closed-form. Accordingly, an optimal UAV trajectory without location constraints is presented, and a high-quality location-constrained solution is also provided.

As shown in Fig. 1, we consider a UAV-enabled ISAC system where a UAV equipped with a uniform linear array (ULA) of $M$ antennas is deployed to sense one target near the ground while providing communication service for a cluster of single-antenna users (viewed as one user for ease of analysis, or viewed as an access point for sensory data upload). For convenience, a two-dimensional Cartesian coordinate system is adopted where the linear distance of user and target is referred to as the $x$-axis with two points ${\bm{u}}=[0,0]^{T}$ and ${\bm{v}}=[{D},0]^{T}$, respectively. The value of ${\bm{v}}$ is determined based on the specific sensing tasks, which can be set as a sampled position in the region of interest for target detection, or set as a roughly priori-position for target tracking. The UAV is assumed to fly at a constant altitude of $H$ m with a flight duration of $T$ s, and the UAV location is denoted by $x(t)$, where $t\in{\cal{T}}\buildrel\Delta\over{=}[0,T]$.

In our proposed IPSAC mechanism, as shown in Fig. 1, the sensing task should be performed at least once in each ISAC frame, and its length is denoted by $T_{f}$, which is set according to the requirement of task execution frequency. The total frame number $L$ is set as an integer for ease of analysis ($L\cdot T_{f}=T$), and the index of ISAC frame is denoted by $l\in{\cal{L}}=\{1,\cdots,L\}$. The sensing time in the $l$th ISAC frame is given by ${\cal{T}}_{l}\buildrel\Delta\over{=}[t_{l},t_{l}+\tau_{0}]$, where $\tau_{0}$ is the sensing period. The minimum sensing period is practically determined by radar bandwidth, waveform, and so on [2]. It is assumed that the UAV is hovering during detecting or tracking the target, since the sensing period is generally small and the fixed UAV location can avoid the introduction of complex Doppler shift due to the movement of the UAV.

The communication links between the UAV and the user are assumed to be dominated by the line-of-sight (LoS) component [5]. Hence, it is assumed that the aerial-ground channel follows the free-space path loss model, and the channel power gain from the UAV to the user can be given by

where $\beta_{0}$ represents the channel power at the reference distance 1 m. The transmit array response vector of the UAV toward the user’s location ${\bm{u}}$ is expressed as

where $d$ is the half-wavelength antenna spacing, $\lambda$ is the carrier wavelength, $\theta_{1}$ denotes the elevation angle of the geographical path connecting the UAV to the user, and $\sin(\theta_{1}(x(t),\bm{u}))=H/\sqrt{x(t)^{2}+H^{2}}$. Therefore, the baseband equivalent channel from the UAV to the user can be given by

For practical implementation, the linear precoding is adopted at the UAV-enabled ISAC system. The transmitted signal from the UAV is given by ${\bm{z}}(t)={{\bm{w}}_{c}}(t){s_{c}}(t)$, where ${s}_{c}$ and ${\bm{w}}_{c}(t)\in\mathbb{C}^{M\times 1}$ are the information-bearing signal to the user and its corresponding transmit precoder, ${s}_{c}\sim\mathcal{C}\mathcal{N}(0,1)$, and $\mathbb{E}(|s_{c}|^{2})=1$. Then, the received signal at user is

where $n_{c}(t)\sim{\cal{CN}}(0,\sigma^{2})$ denotes the additive white Gaussian noise (AWGN) at the user’s receiver. Accordingly, the signal-to-noise-ratio (SNR) of the user is given by ${\gamma}(t)=|{{\bm{h}}_{c}(x(t)){{\bm{w}}_{c}}(t)}|^{2}/\sigma^{2},\forall t\in{\cal{T}}$. As a result, the achievable rate of the user at time $t$ is ${R}(t)={\log_{2}(1+{\gamma}(t))}$. In our designed ISAC system, the reflected communication signals from the objects of interest can also be analyzed for radar sensing [2], and thus, the communication signals $s_{c}(t)$ is also exploited for sensing performance. The transmit beam pattern gain from the UAV to the target’s location ${\bm{v}}$ can be expressed as

Our objective is to maximize the achievable rate, subject to the beam pattern gain constraint and the maximum transmit power constraint. The corresponding optimization problem can be formulated as

In (P1), (7a) represents the beam pattern gain should be no less than the threshold ${\tilde{\Gamma}}$ under the given sensing frequency and the total power constraint is given in (7b). The UAV speed constraint, the initial and final location constraints are given in (7c) and (7d), respectively. Problem (P1) is difficult to solve optimally, since it involves an infinite number of variables $\{x(t)\}$, and the constraints in (7a) and objective function are non-convex due to the non-concavity of $\Gamma\left(x(t_{l}),\bm{v}\right)$ and $R(t)$.

In this section, we obtain the optimal transmit precoder by SDR technique together with eigenvalue decomposition, and provide a closed-form user SNR for trajectory optimizing.

In this section, numerical results are provided for characterizing the performance of the proposed IPSAC mechanism. The required parameters are set as follows unless specified otherwise: $P_{\max}=0.1$ W, $\beta_{0}=-30$ dB, $\sigma^{2}=-100$ dB, $\tilde{\Gamma}=6\times e^{-5}$, $T=500$ s, $T_{f}=5$ s, $\tau_{0}=0.1$ s, $M=10$, $d={\lambda\mathord{\left/{\vphantom{\lambda 2}}\right.\kern-1.2pt}2}$, $V_{\max}=30$ m/s, $H=50$ m, $D=400$ m, and $x_{I}=x_{F}=D$. Our proposed schemes for problem (SP2) and (P3) denoted as 1): "Upper bound" and 2): "Proposed" in Fig. 2, respectively, are compared to two benchmarks. 3): "Time division": The transmit precoder is set as the MRT to the user and target in a time-division manner; 4): "Optimal precoder only": The transmit precoder is obtained by our proposed method. Both benchmarks assume that the UAV performs sensing tasks at the beginning of each ISAC frame, and flies from the initial location to the user during the first-half ISAC frame and then flies to the target during the second half-frame, respectively.

Figs. 2(a)-2(b) illustrate the fundamental trade-off among sensing power requirement, sensing frequency, and achievable rate for the considered system. It is observed from Fig. 2(a) that the achievable rate gain achieved by our proposed scheme over the "optimal precoder only" scheme increases as the sensing frequency decreases, as the UAV has more non-sensing time to adjust its trajectory for communication performance improvement. Moreover, the achievable rate of our proposed scheme under high sensing frequency achieves significant improvement as compared to that of the "Time division" scheme, since more sensing time can be fully utilized for data transmission. In particular, the performance of our proposed method is very close to the upper bound of (P1), which is obtained by solving (SP2) based on Lemma 2, thereby illustrating the near-optimality of the proposed solution. Besides, when the sensing frequency increases, the achievable rate with a higher beam pattern gain threshold ${\tilde{\Gamma}}$ degrades faster as compared to that with a lower threshold. The main reason is that a higher beam pattern gain threshold forces the UAV to perform sensing tasks at a location closer to the target, thereby resulting in increasing path loss within pure communication duration. Similarly, it can be seen from Fig. 2(b) that as the beam pattern gain threshold ${\tilde{\Gamma}}$ increases, the achievable rate under higher sensing frequency degrades faster as compared to that with a higher one. In Fig. 2(c), it is shown that for our proposed scheme, the achievable rates under different ${\tilde{\Gamma}}$ are almost equal when the distance $D$ is less than 100 m, i.e., a better communication performance can be achieved when sensing the target closer to the user (receiver). In Fig. 3, the achievable rate gain achieved by our proposed scheme over the "optimal precoder only" scheme increases as the maximum flight speed $V_{\max}$ increases, since the UAV could obtain better channel gain within a shorter flying time.

This letter studied the achievable rate maximization problem in a UAV-enabled ISAC system. In particular, it was shown that there exists a novel structure-symmetry characteristic between optimal solutions in different ISAC frames without location constraints. Furthermore, a near-optimal location-constrained solution was achieved based on the derived closed-form achievable rate. Numerical results verified that the proposed scheme is able to drastically enlarge sensing and communication performance trade-off of UAV-enable ISAC systems. The more general 3D UAV trajectory optimization problems for multi-UAV ISAC scenarios are worthwhile future works.