1 Introduction↩︎

Low-altitude applications such as logistics delivery, urban inspection, emergency rescue, and airspace monitoring are driving the large-scale deployment of uncrewed aerial vehicles (UAVs) in low-altitude wireless networks (LAWN) [1], [2]. As the traffic density and dynamics of low-altitude airspace continue to increase, unauthorized UAVs, birds, or other non-cooperative aerial targets may randomly appear in the forward flight region of a UAV and directly threaten flight safety. Therefore, safe UAV flight relies not only on reliable communication and self-state awareness, but also on timely detection of potential risk targets ahead of the flight direction [3]–[5].

Ensuring safe UAV flight requires cellular infrastructure to provide not only reliable communication but also sensing capability for aerial targets and the surrounding environment. Integrated sensing and communication (ISAC) enables information transmission and low-altitude sensing within the same network by sharing spectrum, hardware, and waveform resources [6]–[9]. However, in low-altitude scenarios, single ground base station (GBS) ISAC is susceptible to limited observation perspectives, blockage, multipath propagation, and link-quality fluctuations, which restrict sensing accuracy and service reliability. In contrast, networked ISAC exploits the cooperation among multiple GBSs to provide spatial diversity and multi-view observations, thereby improving the reliability of target localization, velocity estimation, and trajectory tracking. This makes networked ISAC a promising approach for jointly supporting UAV communication, state estimation, and forward risk detection [10], [11].

Recent studies on networked ISAC mainly focused on cooperative sensing implementation, network-level performance analysis, and communication-sensing resource competition and optimization. For cooperative sensing implementation, Zhang et al. [12] proposed a two-stage localization scheme that integrates signal-level bistatic measurement extraction with data-level measurement association to achieve high-accuracy multi-target localization. Wang et al. [13] reduced the fronthaul overhead of cooperative sensing through distributed compression and quantization, while improving the estimation performance of target position and velocity. Liu et al. [14], [15] systematically investigated key technologies for cooperative ISAC, including synchronization, strong-and-weak path effect mitigation, and interference suppression, and demonstrated the role of cooperative sensing in improving sensing coverage, localization accuracy, and detection performance. Despite these advances, most existing implementations assume predefined sensing objects or regions, and do not explicitly model the UAV forward region of interest (ROI) as a safety-critical region requiring reliable target detection.

For network-level performance analysis, Meng et al. characterized the scaling-law relationship between the cooperative cluster size and the sensing Cramér-Rao lower bound (CRLB) [16], [17], analyzed the impact of centralized and distributed antenna architectures on the communication-sensing performance boundary [18], and further clarified the influence of GBSs topology on regional sensing accuracy [19]. These studies reveal how cooperative scale, antenna architecture, and network deployment affect networked ISAC performance. However, their metrics mainly quantify estimation accuracy, sensing quality, or performance boundaries, and do not directly measure the probability that hazardous targets in the UAV forward ROI are missed.

For communication-sensing resource optimization, Cheng et al. [20] jointly optimized BS beamforming and UAV trajectory to balance UAV communication requirements and cooperative sensing coverage over the target airspace. Li et al. [21] jointly optimized UAV trajectory, communication and sensing power allocation, and dynamic time-division ratio to improve system communication performance while satisfying the sensing mutual information constraint. Gu et al. [22] jointly optimized node selection and coordinated transmit beamforming to reduce the sensing CRLB under communication rate constraints. These designs mainly optimize communication rate, sensing accuracy, or coverage performance, but do not account for the detection reliability of the UAV forward ROI in resource allocation.

In summary, existing studies on networked ISAC have improved cooperative communication and sensing performance. However, reliable detection of non-cooperative targets in the UAV’s dynamic forward ROI has not been sufficiently studied, although it is important for flight safety. Therefore, forward-ROI detection reliability should be further considered in networked ISAC design and incorporated into performance analysis and resource optimization.

Motivated by the above considerations, this paper investigates a networked-ISAC-based forward detection design, where multiple cooperative GBSs jointly support UAV communication, cooperative state estimation, and non-cooperative target detection within the forward ROI, with the main contributions summarized as follows:

• A networked ISAC forward detection model is developed for safe low-altitude UAV flight, supporting UAV communication, state estimation, and forward-ROI detection. The forward ROI is dynamically determined by the UAV velocity and safe braking distance, and is voxelized for per-voxel target-presence decisions of non-cooperative targets. The CRLB for UAV state estimation and the forward-ROI miss-detection probability are further derived to quantify the state-estimation error bound and forward-detection reliability, respectively.

• Scaling-law analysis is established for UAV state estimation and forward-ROI detection under random GBS deployment and non-cooperative target occupancy. In the large-cooperation regime, the UAV state-estimation CRLB scales with the cooperative GBS number \(J\) as \(\ln^{-2}J\). In the high sensing signal-to-interference-plus-noise ratio (SINR) regime, the forward-ROI miss-detection probability follows an exponential-form scaling law, whose exponent scales with \(J\), the non-cooperative target density \(\lambda_{t}\), and the forward safety distance \(D_{f}\) as \(\lambda_{t}D_{f}\ln^{-2}J\). These scaling laws provide guidance for selecting the cooperative GBS scale for both UAV state estimation and forward detection.

• We propose a safety-aware forward detection resource optimization scheme that explicitly incorporates the forward-ROI miss-detection probability into the networked ISAC optimization framework. By jointly configuring the sensing pilot ratio, transmit power, and beam direction, the proposed scheme balances the UAV state-estimation CRLB and forward detection reliability under the communication-rate constraint. Furthermore, the communication-estimation-detection Pareto boundary is characterized to reveal the resource tradeoff among communication rate, state-estimation accuracy, and forward detection reliability.

• Simulation results show that, compared with the baseline scheme that does not consider forward detection, the proposed scheme reduces the average miss-detection probability and the corresponding sensing-induced collision risk by \(17.05\%\), while introducing only limited UAV state-estimation performance degradation, reflected by a \(14.82\%\) increase in the average CRLB.

The remainder of this paper is organized as follows. Section 2 presents the networked ISAC system model. Section 3 describes the processing procedures for UAV state estimation and forward-ROI detection. Section 4 derives the corresponding performance metrics and scaling laws. Section 5 formulates the safety-aware resource optimization problem. Section 6 validates the analytical results and evaluates the proposed design through simulations. Finally, Section 7 concludes the paper.

Notation: The superscripts \((\cdot)^T\), \((\cdot)^H\), and \((\cdot)^{-1}\) denote the transpose, Hermitian transpose, and matrix inverse, respectively. \(\|\cdot\|_2\) denotes the Euclidean norm, \(|\cdot|\) denotes the cardinality of a set or the absolute value of a scalar, and \(\operatorname{tr}(\cdot)\) denotes the trace operator. \(\otimes\) denotes the Kronecker product, and \(\Re\{\cdot\}\) denotes the real part. \(\mathbb{R}^n\) and \(\mathbb{C}^{m\times n}\) denote the \(n\)-dimensional real space and the space of \(m\times n\) complex matrices, respectively. \(\mathbf{I}_n\) and \(\mathbf{0}_n\) denote the \(n\times n\) identity and zero matrices, respectively. \(\mathbb{E}[\cdot]\) and \(\Pr(\cdot)\) denote expectation and probability, respectively. \(\mathcal{CN}(\cdot,\cdot)\) and \(\mathcal{N}(\cdot,\cdot)\) denote complex and real Gaussian distributions, respectively. \(\mathrm{diag}(\cdot)\) and \(\mathrm{rect}(\cdot)\) denote the diagonal matrix operator and rectangular window function, respectively.

2 sec:System32Model↩︎

In this section, we present the networked ISAC system model, including spatial geometry, transmit signal, communication, and sensing models.

2.1 Geometric and Non-Cooperative Target Occupancy Model↩︎

As shown in Fig. 1, we consider a low-altitude networked ISAC scenario where multiple cooperative GBSs support a UAV flying along a predefined virtual aerial corridor [23]. The UAV is equipped with an omnidirectional transceiver, while the cooperative GBSs jointly provide communication, UAV state estimation, and forward-region sensing. In the forward region, non-cooperative airborne targets, such as unauthorized UAVs, birds, or other flying objects, may randomly appear and threaten flight safety.

For sequential UAV state estimation, the continuous UAV trajectory is sampled with sensing interval \(\Delta t\) and represented in discrete time slots. At time slot \(k\), the UAV state is defined as \(\mathbf{x}_{k}^{u}=[(\mathbf{p}_{k}^{u})^T,(\mathbf{v}_{k}^{u})^T]^T\in\mathbb{R}^{6}\), where \(\mathbf{p}_{k}^{u}=[x_{k}^{u},y_{k}^{u},z_{k}^{u}]^T\) and \(\mathbf{v}_{k}^{u}=[v_{x,k},v_{y,k},v_{z,k}]^T\) denote the UAV position and velocity, respectively. For notational simplicity, the time-slot index \(k\) is omitted when a generic time slot is considered.

2.1.1 Forward ROI↩︎

The forward ROI is determined by the flight-safety requirement of the UAV. Let \(\mathbf{e}_{v}=\mathbf{v}^{u}/\|\mathbf{v}^{u}\|_2\) denote the unit vector along the UAV velocity direction. A local right-handed orthonormal basis \(\{\mathbf{e}_{v},\mathbf{e}_{\perp,1},\mathbf{e}_{\perp,2}\}\) is constructed at the UAV, where \(\mathbf{e}_{\perp,1}\) and \(\mathbf{e}_{\perp,2}\) denote the lateral and normal directions spanning the plane perpendicular to \(\mathbf{e}_{v}\). Accordingly, the cuboidal forward ROI is defined as \[\label{eq:sensing95region} \begin{align} \mathcal{V} = \Big\{ \mathbf{p}\in\mathbb{R}^{3}\;\Big|\;& 0\leq(\mathbf{p}-\mathbf{p}^{u})^{T}\mathbf{e}_{v}\leq D_f,\\ & \left|(\mathbf{p}-\mathbf{p}^{u})^{T}\mathbf{e}_{\perp,1}\right| \leq D_s,\\ & \left|(\mathbf{p}-\mathbf{p}^{u})^{T}\mathbf{e}_{\perp,2}\right| \leq D_s \Big\}. \end{align}\tag{1}\] where \(D_f\) denotes the longitudinal depth, and \(D_s\) denotes the lateral and vertical half-width of the ROI.

The longitudinal depth \(D_f\) is determined by the minimum safety distance required for emergency maneuvering, i.e., \(D_f\geq v^{u}t_r+(v^{u})^2/(2a_{\max})+d_m\), where \(v^{u}=\|\mathbf{v}^{u}\|_2\), \(t_r\), \(a_{\max}\), and \(d_m\) denote the UAV speed, sensing-and-control latency, maximum deceleration capability, and longitudinal safety margin, respectively [24]. The lateral and vertical half-width is set as \(D_s=R_u+r_m\), where \(R_u\) is the effective UAV bounding radius and \(r_m\) is the spatial safety margin. Therefore, \(\mathcal{V}\) represents the safety-critical forward space that should be monitored for safe UAV flight.

2.1.2 Voxelized Target Occupancy Model↩︎

To characterize the spatial occupancy of non-cooperative targets within the ROI, we adopt a voxelized representation inspired by point-cloud modeling [25]. Specifically, the ROI \(\mathcal{V}\) is discretized into \(N_v\) cubic voxels with side length \(\Delta d\), i.e., \(\mathcal{V}=\bigcup_{l=1}^{N_v}\mathcal{C}_l\), where \(\mathcal{C}_l\) denotes the \(l\)-th voxel.

The non-cooperative targets are assumed to follow a homogeneous three-dimensional (3D) Poisson point process (PPP) with density \(\lambda_t\) [26], [27]. Accordingly, the number of targets in voxel \(\mathcal{C}_l\) follows \(n_l\sim\mathrm{Poisson}(\lambda_t\Delta d^3)\). The voxel occupancy variable is defined as \(b_l=1\) if \(n_l\ge1\) and \(b_l=0\) otherwise. The occupancy probability is then given by \(q\triangleq\Pr(b_l=1)=\Pr(n_l\geq1)=1-\exp(-\lambda_t\Delta d^3)\), and hence \(b_l\sim\mathrm{Bernoulli}(q)\). The set of occupied voxels is denoted by \(\mathcal{L}=\{l\mid b_l=1,\; l=1,\ldots,N_v\}\), and the number of occupied voxels is \(L=|\mathcal{L}|=\sum_{l=1}^{N_v}b_l\). When the voxelization is sufficiently fine such that \(\lambda_t\Delta d^3\ll1\), the probability of having more than one target in a voxel is negligible², and thus \(L\) approximates the number of non-cooperative targets within the ROI. For an occupied voxel \(\mathcal{C}_l\), the corresponding target is represented by the voxel center with position \(\mathbf{p}_l^t=[x_l^t,y_l^t,z_l^t]^T\), \(l\in\mathcal{L}\).

2.1.3 Cooperative GBS Geometry↩︎

Following the standard stochastic-geometry model for network-level analysis, the ground GBS locations are modeled as a two-dimensional homogeneous PPP \(\Phi_{\mathrm B}\) with density \(\lambda_{\mathrm B}\) [16]. In each service slot, the \(J\) nearest GBSs to the UAV ground projection are selected from \(\Phi_{\mathrm B}\) to form the cooperative cluster \(\mathcal{B}\), with \(|\mathcal{B}|=J\). The selected GBSs are ordered according to their horizontal distances to the UAV projection. The cooperative cluster jointly supports UAV communication, UAV state estimation, and non-cooperative target detection.

For each GBS \(j\in\mathcal{B}\), its position is denoted by \(\mathbf{p}_j^B=[x_j^B,y_j^B,z_j^B]^T\), where \(z_j^B=z^B\) is fixed for all GBSs. Each GBS is equipped with a uniform planar array (UPA) with \(N_t=N_{t,x}N_{t,z}\) transmit antennas and \(N_r=N_{r,x}N_{r,z}\) receive antennas. The antenna spacing is set to \(d=\lambda_c/2\), where \(\lambda_c\) is the carrier wavelength. The distance between GBS \(j\) and the UAV is \(d_j^u=\|\mathbf{p}^{u}-\mathbf{p}_j^B\|_2\). For an occupied voxel \(l\in\mathcal{L}\), the distance between GBS \(j\) and the corresponding non-cooperative target is \(d_{j,l}^t=\|\mathbf{p}_l^t-\mathbf{p}_j^B\|_2\). The zenith and azimuth angles from GBS \(j\) to the UAV are defined as \(\theta_j^u=\arccos((z^u-z_j^B)/d_j^u)\) and \(\phi_j^u=\operatorname{atan2}(y^u-y_j^B,x^u-x_j^B)\), respectively. Similarly, the zenith and azimuth angles from GBS \(j\) to the target represented by occupied voxel \(l\) are given by \(\theta_{j,l}^t=\arccos((z_l^t-z_j^B)/d_{j,l}^t)\) and \(\phi_{j,l}^t=\operatorname{atan2}(y_l^t-y_j^B,x_l^t-x_j^B)\), respectively.

2.2 Transmit Signal Model↩︎

The cooperative GBSs employ a unified orthogonal frequency-division multiplexing (OFDM) frame to support communication, UAV-state sensing, and forward-ROI target sensing. The continuous-time transmit signal of GBS \(j\) consists of a communication component \(\mathbf{s}_j^{\mathrm c}(t)\) and a sensing component \(\mathbf{s}_j^{\mathrm s}(t)\), i.e., \(\mathbf{s}_j(t)=\mathbf{s}_j^{\mathrm c}(t)+\mathbf{s}_j^{\mathrm s}(t)\), which can be written as [28] \[\begin{align} &\mathbf{s}_j(t)=\sum_{n=0}^{N-1}\Bigg( \sum_{m\in\mathcal{M}_{\mathrm c}} \sqrt{p_j^{\mathrm c}}\,\mathbf{w}_j^{\mathrm c}a_{m,n}^{\mathrm c} e^{j2\pi m\Delta f(t-nT_s)} \\ &+ \sum_{m\in\mathcal{M}_{\mathrm s}} \sqrt{p_j^{\mathrm s}}\,\mathbf{w}_j^{\mathrm s}a_{j,m,n}^{\mathrm s} e^{j2\pi m\Delta f(t-nT_s)} \Bigg) \mathrm{rect}\!\left(\frac{t-nT_s}{T_s}\right), \end{align} \label{eq:tx95signal95components}\tag{2}\] where \(a_{m,n}^{\mathrm c}\) denotes the common communication symbol for the UAV with \(\mathbb{E}[|a_{m,n}^{\mathrm c}|^2]=1\), and \(a_{j,m,n}^{\mathrm s}\) denotes the sensing pilot transmitted by GBS \(j\) with \(\mathbb{E}[|a_{j,m,n}^{\mathrm s}|^2]=1\). The beamforming vectors \(\mathbf{w}_j^{\mathrm c},\mathbf{w}_j^{\mathrm s}\in\mathbb{C}^{N_t\times1}\) are used for communication and sensing, respectively, and satisfy \(\|\mathbf{w}_j^{\mathrm c}\|_2=\|\mathbf{w}_j^{\mathrm s}\|_2=1\). \(N\) and \(M\) denote the numbers of OFDM symbols and active subcarriers, respectively. The OFDM symbol duration is \(T_s=T_c+T_{\mathrm{cp}}\), where \(T_c=1/\Delta f\) is the useful symbol duration and \(T_{\mathrm{cp}}\) is the cyclic-prefix duration, and the occupied bandwidth is \(B=M\Delta f\). The active subcarriers are partitioned into a communication set \(\mathcal{M}_{\mathrm c}\) and a sensing set \(\mathcal{M}_{\mathrm s}\), with \(|\mathcal{M}_{\mathrm c}|=(1-\rho_{\mathrm s})M\) and \(|\mathcal{M}_{\mathrm s}|=\rho_{\mathrm s}M\), where where \(\rho_{\mathrm s}\in(0,1)\) denotes the sensing subcarrier ratio. Both UAV-state sensing and voxel-level target sensing are performed on the dedicated sensing subcarriers in \(\mathcal{M}_{\mathrm s}\). Let \(P_j^{\mathrm c}\) and \(P_j^{\mathrm s}\) denote the total transmit powers allocated by GBS \(j\) to \(\mathcal{M}_{\mathrm c}\) and \(\mathcal{M}_{\mathrm s}\), respectively, satisfying \(P_j^{\mathrm c}+P_j^{\mathrm s}\le P_j^{\max}\). Under uniform power allocation within each subcarrier set, the per-subcarrier powers are \(p_j^{\mathrm c}=P_j^{\mathrm c}/|\mathcal{M}_{\mathrm c}|\) and \(p_j^{\mathrm s}=P_j^{\mathrm s}/|\mathcal{M}_{\mathrm s}|\).

2.3 Communication Model↩︎

The GBS-UAV communication link is line-of-sight (LoS)-dominant and is modeled as a Rician channel. For communication resource block \((m,n)\) with \(m\in\mathcal{M}_{\mathrm c}\), the channel vector from GBS \(j\) to the UAV is given by [29] \[\mathbf{h}_{j,m,n}^{c} = g_j^c \sqrt{\beta_j^c} e^{j 2\pi n T_s \nu_j^{c}} e^{-j 2\pi m \Delta f \tau_j^{c}} \mathbf{a}_t^H(\theta_j^u, \phi_j^u), \label{eq:comm95channel}\tag{3}\] where \(g_j^c\) and \(\beta_j^c\) are the small-scale fading coefficient and large-scale channel gain, respectively. The small-scale fading follows \(g_j^c \sim \mathcal{CN}\!\left( \sqrt{K/(K+1)} e^{j\varphi_0}, 1/(K+1) \right)\), where \(K\) is the Rician factor and \(\varphi_0\) is the deterministic initial phase of the LoS path [30], [31]. The large-scale gain is modeled as \(\sqrt{\beta_j^c} = \sqrt{\beta_0^c} (d_j^u)^{-\alpha_c/2} 10^{-\varepsilon_j/20}\), where \(\beta_0^c\) is the reference gain at unit distance, \(\alpha_c\) is the path-loss exponent, and \(\varepsilon_j\sim\mathcal{N}(0,\sigma_{\mathrm{SF}}^2)\) is the shadow fading in dB. In addition, \(\tau_j^{c}=d_j^u/c\) and \(\nu_j^{c}\) denote the one-way delay and Doppler shift, respectively, where \(c\) is the speed of light. The transmit array steering vector is \(\mathbf{a}_t(\theta_j^u,\phi_j^u)=\mathbf{a}_{t,x}(\theta_j^u,\phi_j^u)\otimes\mathbf{a}_{t,z}(\theta_j^u)\in\mathbb{C}^{N_t\times1}\), where \(\mathbf{a}_{t,x}(\theta_j^u,\phi_j^u)=\left[1,e^{j\pi\sin\theta_j^u\sin\phi_j^u},\dots,e^{j\pi(N_{t,x}-1)\sin\theta_j^u\sin\phi_j^u}\right]^T\) and \(\mathbf{a}_{t,z}(\theta_j^u)=\left[1,e^{j\pi\cos\theta_j^u},\dots,e^{j\pi(N_{t,z}-1)\cos\theta_j^u}\right]^T\).

On the communication subcarriers, the cooperative GBSs operate in the joint-transmission coordinated multipoint (JT-CoMP) mode and transmit the same communication symbol to the UAV [32]. The synchronization and CSI exchange required for coherent JT-CoMP are assumed to be available. The scalar received signal at the UAV on resource block \((m,n)\) is \[y_{m,n}^c = \left( \sum_{j\in\mathcal{B}} \sqrt{p_j^{\mathrm c}}\, \mathbf{h}_{j,m,n}^{c}\mathbf{w}_{j}^{\mathrm c} \right) a_{m,n}^{\mathrm c} + z_{m,n}^c, \label{eq:comm95received95signal}\tag{4}\] where \(z_{m,n}^c \sim \mathcal{CN}(0,\sigma_c^2)\) is the additive white Gaussian noise at the UAV receiver. Accordingly, the normalized data rate of the UAV is given by \[R_{\mathrm c} = \frac{1}{N M} \sum_{n=0}^{N-1} \sum_{m\in\mathcal{M}_{\mathrm c}} \log_2\!\left( 1+ \frac{ \left| \sum_{j\in\mathcal{B}} \sqrt{p_j^{\mathrm c}}\, \mathbf{h}_{j,m,n}^{c}\mathbf{w}_{j}^{\mathrm c} \right|^2 }{ \sigma_c^2 } \right). \label{eq:comm95rate}\tag{5}\]

2.4 Sensing Model↩︎

Within one observation interval, the sensing signal transmitted by GBS \(i\) is scattered by either the UAV or the non-cooperative targets occupying ROI voxels, and is then received by GBS \(j\), forming a GBS \(i\)-scatterer-GBS \(j\) bistatic sensing link. Accordingly, the sensing channel matrix on sensing subcarrier \(m\in\mathcal{M}_{\mathrm s}\) and OFDM symbol \(n\) is modeled as \[\mathbf{H}_{i,j,m,n}^{s} = \mathbf{H}_{i,j,m,n}^{u,s} + \sum_{l=1}^{N_v} b_l \mathbf{H}_{i,j,l,m,n}^{t,s}, \label{eq:sensing95channel95total}\tag{6}\] where \(\mathbf{H}_{i,j,m,n}^{u,s}\in\mathbb{C}^{N_r\times N_t}\) denotes the sensing channel component of the UAV, and \(\mathbf{H}_{i,j,l,m,n}^{t,s}\in\mathbb{C}^{N_r\times N_t}\) denotes the sensing channel component of the non-cooperative target in occupied voxel \(\mathcal{C}_l\).

For an occupied voxel \(\mathcal{C}_l\), i.e., \(b_l=1\), the corresponding sensing channel component is modeled as \[\begin{align} \mathbf{H}_{i,j,l,m,n}^{t,s} = &\,\zeta_l g_{i,j,l}^{t,s}\sqrt{\beta_{i,j,l}^{t,s}} e^{j2\pi nT_s\nu_{i,j,l}^{t,s}} e^{-j2\pi m\Delta f\tau_{i,j,l}^{t,s}} \\ &\cdot \mathbf{a}_r(\theta_{j,l}^{t},\phi_{j,l}^{t}) \mathbf{a}_t^H(\theta_{i,l}^{t},\phi_{i,l}^{t}), \end{align} \label{eq:sensing95channel95target}\tag{7}\] where \(\zeta_l\) denotes the complex scattering coefficient of the target represented by voxel \(\mathcal{C}_l\), with \(\mathbb{E}[|\zeta_l|^2]=\sigma_{\mathrm{tar}}\), where \(\sigma_{\mathrm{tar}}\) is the average radar cross section (RCS) of a non-cooperative target [33]. \(g_{i,j,l}^{t,s}\) and \(\beta_{i,j,l}^{t,s}\) denote the corresponding small-scale fading coefficient and large-scale channel gain, respectively. The large-scale sensing gain is modeled as \(\sqrt{\beta_{i,j,l}^{t,s}} = \sqrt{\beta_0^{\mathrm s}} (d_{i,l}^{t})^{-\alpha_{\mathrm s}/2} (d_{j,l}^{t})^{-\alpha_{\mathrm s}/2} 10^{-\varepsilon_{i,j,l}^{t,s}/20}\), where \(\beta_0^{\mathrm s}\) denotes the reference sensing gain at unit distance, \(\alpha_{\mathrm s}\) is the sensing path-loss exponent, and \(\varepsilon_{i,j,l}^{t,s}\sim\mathcal{N}(0,\sigma_{\mathrm{SF}}^2)\) denotes the shadow fading in dB. \(\tau_{i,j,l}^{t,s}=(d_{i,l}^{t}+d_{j,l}^{t})/c\) denotes the bistatic propagation delay, while \(\nu_{i,j,l}^{t,s}\) denotes the bistatic Doppler shift induced by the motion of the non-cooperative target. The vectors \(\mathbf{a}_r(\theta_{j,l}^{t},\phi_{j,l}^{t})\) and \(\mathbf{a}_t(\theta_{i,l}^{t},\phi_{i,l}^{t})\) denote the receive and transmit array steering vectors, respectively.

Similarly, the cooperative-UAV sensing channel component is explicitly given by \[\begin{align} \mathbf{H}_{i,j,m,n}^{u,s} = &\,\zeta_k^u g_{i,j,k}^{u,s}\sqrt{\beta_{i,j,k}^{u,s}} e^{j2\pi nT_s\nu_{i,j,k}^{u,s}} e^{-j2\pi m\Delta f\tau_{i,j,k}^{u,s}} \\ &\cdot \mathbf{a}_r(\theta_{j,k}^{u},\phi_{j,k}^{u}) \mathbf{a}_t^H(\theta_{i,k}^{u},\phi_{i,k}^{u}), \end{align} \label{eq:sensing95channel95uav}\tag{8}\] where \(\zeta_k^u\) denotes the effective UAV scattering coefficient, while \(g_{i,j,k}^{u,s}\), \(\beta_{i,j,k}^{u,s}\), \(\tau_{i,j,k}^{u,s}\), and \(\nu_{i,j,k}^{u,s}\) denote the small-scale fading coefficient, large-scale sensing gain, bistatic propagation delay, and bistatic Doppler shift of the UAV-associated bistatic sensing link, respectively.

On the sensing subcarriers, different transmitting GBSs use mutually orthogonal pilot sequences over the sensing resource block. Since the block contains \(|\mathcal{M}_{\mathrm s}|N\) time-frequency resource elements, the orthogonality can be supported when \(J\le|\mathcal{M}_{\mathrm s}|N\), enabling link separation through pilot correlation. After pilot correlation and normalization, receiving GBS \(j\) obtains the equivalent link-separated observation associated with transmitting GBS \(i\) as \[\begin{align} \mathbf{y}_{i,j,m,n}^{s} = & \underbrace{ \sqrt{p_i^{\mathrm s}} \mathbf{H}_{i,j,m,n}^{u,s}\mathbf{w}_{i}^{\mathrm s} }_{\text{cooperative-UAV echo}} + \underbrace{ \sqrt{p_i^{\mathrm s}} \sum_{l=1}^{N_v} b_l\mathbf{H}_{i,j,l,m,n}^{t,s}\mathbf{w}_{i}^{\mathrm s} }_{\text{non-cooperative target echoes}} + \mathbf{z}_{i,j,m,n}^{s}, \end{align} \label{eq:link95separated95signal}\tag{9}\] where \(\mathbf{z}_{i,j,m,n}^{s}\sim\mathcal{CN}(\mathbf{0},\sigma_s^2\mathbf{I}_{N_r})\) denotes the equivalent noise, \(\sigma_s^2\) is the sensing noise power, and \(\mathbf{I}_{N_r}\) is the identity matrix with dimension equal to the number of receive antennas \(N_r\).

3 UAV State Estimation and Forward Detection Processing↩︎

This section presents the signal processing for UAV state estimation and forward-ROI non-cooperative target detection.

3.1 UAV State Estimation↩︎

Since the cooperative UAV is always present whereas non-cooperative targets may randomly appear in the forward ROI, the received sensing signals may contain echoes from different scatterers. Therefore, a two-stage processing method is adopted. Each GBS first extracts candidate delay-Doppler-angle observations from the link-separated sensing signals. Then, the CPU identifies the UAV-echo observations through data association, which are used for state estimation, while the non-UAV echoes are used for forward-ROI detection.

3.1.1 Link-Level Observation Extraction↩︎

At time slot \(k\), each receiving GBS extracts geometric observations from the link-separated sensing signals. Specifically, receiving GBS \(j\) correlates the received sensing signal with the orthogonal sensing pilot of transmitting GBS \(i\), yielding the link-separated sensing signal \(\mathbf{y}_{i,j,m,n,k}^{s}\) associated with link \((i,j)\). Then, a two-dimensional fast Fourier transform (2D-FFT) is applied to \(\mathbf{y}_{i,j,m,n,k}^{s}\) over the sensing-subcarrier and OFDM-symbol dimensions, resulting in the delay-Doppler power response [12] \[\mathcal{D}_{i,j,k}(\tau,\nu) = \left\| \sum_{m\in\mathcal{M}_{\mathrm s}} \sum_{n=0}^{N-1} \mathbf{y}_{i,j,m,n,k}^{s} e^{j2\pi m\Delta f\tau} e^{-j2\pi nT_s\nu} \right\|_2^2 . \label{eq:delay95doppler95response}\tag{10}\]

Peak searching over \(\mathcal{D}_{i,j,k}(\tau,\nu)\) may yield multiple candidate delay-Doppler estimates \(\{(\hat{\tau}_{i,j,q,k},\hat{\nu}_{i,j,q,k})\}_q\), where \(q\) is the candidate index. For each candidate delay-Doppler estimate \((\hat{\tau}_{i,j,q,k},\hat{\nu}_{i,j,q,k})\), the corresponding array-domain observation is formed at receiving GBS \(j\), from which the noise subspace \(\widehat{\mathbf{E}}_{\mathrm n,i,j,q,k}\) is obtained for 2D-MUSIC as \[\mathcal{S}_{i,j,q,k}(\theta,\phi) = \frac{1}{ \mathbf{a}_r^H(\theta,\phi) \widehat{\mathbf{E}}_{\mathrm n,i,j,q,k} \widehat{\mathbf{E}}_{\mathrm n,i,j,q,k}^{H} \mathbf{a}_r(\theta,\phi) }. \label{eq:music95spectrum}\tag{11}\]

By peak searching over the MUSIC spectrum, the AoA elevation and azimuth of the \(q\)-th candidate observation are estimated as \(\hat{\theta}_{i,j,q,k}\) and \(\hat{\phi}_{i,j,q,k}\), respectively. Therefore, the \(q\)-th candidate geometric observation on link \((i,j)\) is expressed as \(\hat{\boldsymbol{\xi}}_{i,j,q,k}=(\hat{\tau}_{i,j,q,k},\hat{\nu}_{i,j,q,k},\hat{\theta}_{i,j,q,k},\hat{\phi}_{i,j,q,k})^T\).

3.1.2 CPU-Level Observation Fusion↩︎

The extracted candidate observations \(\{\hat{\boldsymbol{\xi}}_{i,j,q,k}\}\) are uploaded to the CPU. For the UAV state \(\mathbf{x}_k^u=[(\mathbf{p}_k^u)^T,(\mathbf{v}_k^u)^T]^T\), which consists of the position \(\mathbf{p}_k^u\) and velocity \(\mathbf{v}_k^u\), the geometric observation mapping corresponding to link \((i,j)\) is written as [34] \[\mathbf{f}_{i,j}(\mathbf{x}_{k}^{u}) = \begin{bmatrix} \dfrac{\|\mathbf{p}_{k}^{u}-\mathbf{p}_{i}^{B}\|_2+\|\mathbf{p}_{k}^{u}-\mathbf{p}_{j}^{B}\|_2}{c} \\[0.8ex] \dfrac{1}{\lambda_c} \left( \dfrac{(\mathbf{p}_{k}^{u}-\mathbf{p}_{i}^{B})^T\mathbf{v}_{k}^{u}}{\|\mathbf{p}_{k}^{u}-\mathbf{p}_{i}^{B}\|_2} + \dfrac{(\mathbf{p}_{k}^{u}-\mathbf{p}_{j}^{B})^T\mathbf{v}_{k}^{u}}{\|\mathbf{p}_{k}^{u}-\mathbf{p}_{j}^{B}\|_2} \right) \\[1.0ex] \arccos\!\left(\dfrac{z_k^u-z_j^B}{\|\mathbf{p}_{k}^{u}-\mathbf{p}_{j}^{B}\|_2}\right) \\[1.0ex] \operatorname{atan2}\!\left(y_k^u-y_j^B,x_k^u-x_j^B\right) \end{bmatrix}.\]

At time slot \(k\), given the posterior state estimate \(\hat{\mathbf{x}}_{k-1|k-1}^{u}\) and the posterior state error covariance matrix \(\mathbf{P}_{k-1|k-1}\) from the previous time slot, the CPU first obtains the predicted state and covariance as [35] \[\begin{align} \hat{\mathbf{x}}_{k|k-1}^{u} &= \mathbf{F}\hat{\mathbf{x}}_{k-1|k-1}^{u},\\ \mathbf{P}_{k|k-1} &= \mathbf{F}\mathbf{P}_{k-1|k-1}\mathbf{F}^{T} + \mathbf{Q}_{k}, \end{align}\] where \(\mathbf{F}=\begin{bmatrix}\mathbf{I}_3&\Delta t\mathbf{I}_3\\ \mathbf{0}_3&\mathbf{I}_3\end{bmatrix}\) is the state transition matrix, \(\Delta t\) is the sensing interval, and \(\mathbf{Q}_k=\mathrm{diag}(\sigma_p^2\mathbf{I}_3,\sigma_v^2\mathbf{I}_3)\) is the process noise covariance matrix. The UAV-associated candidate on link \((i,j)\) is selected via nearest-neighbor association as \(\hat{\boldsymbol{\xi}}_{i,j,k}^{u}=\hat{\boldsymbol{\xi}}_{i,j,q_{i,j,k}^{\star},k}\), where \(q_{i,j,k}^{\star}=\arg\min_q\left\|\hat{\boldsymbol{\xi}}_{i,j,q,k}-\mathbf{f}_{i,j}(\hat{\mathbf{x}}_{k|k-1}^{u})\right\|_2^2\).

For multi-link fusion, the UAV-echo observations from different links are stacked in the same link order, yielding the joint observation model \[\hat{\boldsymbol{\xi}}_{k}^{u} = \mathbf{f}(\mathbf{x}_k^u) + \mathbf{e}_k^u ,\] where \(\hat{\boldsymbol{\xi}}_{k}^{u}=[\{(\hat{\boldsymbol{\xi}}_{i,j,k}^{u})^T\}_{i,j\in\mathcal{B}}]^T\) and \(\mathbf{f}(\mathbf{x}_k^u)=[\{(\mathbf{f}_{i,j}(\mathbf{x}_k^u))^T\}_{i,j\in\mathcal{B}}]^T\) denote the stacked multi-link UAV-echo observation and the joint geometric observation mapping, respectively. The vector \(\mathbf{e}_k^u\) denotes the stacked observation error, whose covariance under the independent residual-error approximation across links is \(\mathbf{C}_{\xi,k}^{u}=\mathrm{diag}(\{\sigma_{\tau_{i,j}}^2,\sigma_{\nu_{i,j}}^2,\sigma_{\theta_{i,j}}^2,\sigma_{\phi_{i,j}}^2\}_{i,j\in\mathcal{B}})\).

Finally, the CPU adopts the extended Kalman filter (EKF) for state updating [5]. The posterior UAV state estimate is updated as \[\hat{\mathbf{x}}_{k|k}^{u} = \hat{\mathbf{x}}_{k|k-1}^{u} + \mathbf{K}_{k} \left( \hat{\boldsymbol{\xi}}_{k}^{u} - \mathbf{f}(\hat{\mathbf{x}}_{k|k-1}^{u}) \right), \label{eq:uav95state95update}\tag{12}\] where \(\mathbf{K}_{k}=\mathbf{P}_{k|k-1}\mathbf{G}_{k}^{T}\left(\mathbf{G}_{k}\mathbf{P}_{k|k-1}\mathbf{G}_{k}^{T}+\mathbf{C}_{\xi,k}^{u}\right)^{-1}\) is the Kalman gain, and \(\mathbf{G}_{k}=\left.\frac{\partial\mathbf{f}(\mathbf{x})}{\partial\mathbf{x}}\right|_{\mathbf{x}=\hat{\mathbf{x}}_{k|k-1}^{u}}\) is the Jacobian of \(\mathbf{f}(\cdot)\) at \(\hat{\mathbf{x}}_{k|k-1}^{u}\). The posterior covariance matrix is updated as \(\mathbf{P}_{k|k}=(\mathbf{I}_6-\mathbf{K}_k\mathbf{G}_k)\mathbf{P}_{k|k-1}\).

3.2 ROI Determination and Voxel-Level Non-Cooperative Target Detection↩︎

Based on the posterior UAV state estimate \(\hat{\mathbf{x}}_{k|k}^{u}\), the forward ROI is determined and discretized into \(N_{v,k}\) voxels \(\mathcal{C}_{l,k}\) with side length \(\Delta d\). Then, the reconstructed UAV echo component is suppressed from the link-separated observation \(\mathbf{y}_{i,j,m,n,k}^{s}\), yielding the residual observation \(\bar{\mathbf{y}}_{i,j,m,n,k}^{s}\), i.e., \(\bar{\mathbf{y}}_{i,j,m,n,k}^{s}=\mathbf{y}_{i,j,m,n,k}^{s}-\sqrt{p_{i,k}^{\mathrm s}}\widehat{\mathbf{H}}_{i,j,m,n,k}^{u,s}\mathbf{w}_{i,k}^{\mathrm s}\), where \(\widehat{\mathbf{H}}_{i,j,m,n,k}^{u,s}\) denotes the reconstructed UAV sensing channel matrix.

At time slot \(k\), each receiving GBS performs For voxel \(\mathcal{C}_{l,k}\), the equivalent scalar observation is given by \[\begin{align} y_{l,k} &= \frac{1}{J\sqrt{|\mathcal{M}_{\mathrm s}|N}} \sum_{i\in\mathcal{B}} \sum_{j\in\mathcal{B}} \sum_{m\in\mathcal{M}_{\mathrm s}} \sum_{n=0}^{N-1} e^{-j\psi_{i,j,l,k}^{\mathrm{cal}}} (\tilde{\mathbf{v}}_{j,l,k})^H \\ &\quad \times \bar{\mathbf{y}}_{i,j,m,n,k}^{s} e^{j2\pi m\Delta f\tilde{\tau}_{i,j,l,k}} e^{-j2\pi nT_s\tilde{\nu}_{i,j,l,k}} , \end{align}\] where \(\tilde{\tau}_{i,j,l,k}\) is the bistatic delay of link \((i,j)\) determined by the voxel center. \(\tilde{\nu}_{i,j,l,k}\) is the Doppler compensation term, which is set to zero under the quasi-static approximation since only target existence is considered. The receive matched-filter vector is \(\tilde{\mathbf{v}}_{j,l,k}=\mathbf{a}_r(\tilde{\theta}_{j,l,k}^{\mathrm{AoA}},\tilde{\phi}_{j,l,k}^{\mathrm{AoA}})/\|\mathbf{a}_r(\tilde{\theta}_{j,l,k}^{\mathrm{AoA}},\tilde{\phi}_{j,l,k}^{\mathrm{AoA}})\|_2\), where \(\tilde{\theta}_{j,l,k}^{\mathrm{AoA}}\) and \(\tilde{\phi}_{j,l,k}^{\mathrm{AoA}}\) denote the AoA of the voxel center with respect to receiving GBS \(j\), and \(\psi_{i,j,l,k}^{\mathrm{cal}}\) is the deterministic phase offset obtained from link phase calibration.

Furthermore, the occupancy detection of voxel \(\mathcal{C}_{l,k}\) is modeled as a binary hypothesis test [36] \[\begin{cases} \mathcal{H}_{0,l,k}: & y_{l,k} = I_{l,k} + z_{l,k}, \\ \mathcal{H}_{1,l,k}: & y_{l,k} = s_{l,k} + I_{l,k} + z_{l,k}, \end{cases} \label{eq:voxel95hypothesis95test}\tag{13}\] where \(\mathcal{H}_{0,l,k}\) and \(\mathcal{H}_{1,l,k}\) denote the absence and presence of a non-cooperative target in voxel \(\mathcal{C}_{l,k}\), respectively. \(s_{l,k}\), \(I_{l,k}\), and \(z_{l,k}\) denote the equivalent target echo, residual interference, and thermal noise after voxel-level extraction, respectively, with \(I_{l,k}\) including interference from the cooperative-UAV and other occupied-voxel echoes.

Based on 13 , a square-law energy detector is adopted with test statistic \(T_{l,k}=|y_{l,k}|^2\). The voxel-level decision is made according to \(T_{l,k} \mathop{\gtrless}_{\mathcal{H}_{0,l,k}}^{\mathcal{H}_{1,l,k}} \Gamma_{\mathrm{th},l,k}\), where \(T_{l,k}\ge\Gamma_{\mathrm{th},l,k}\) indicates that voxel \(\mathcal{C}_{l,k}\) is occupied; otherwise, it is declared empty [37].

4 Performance Metrics and Scaling-Law Analysis↩︎

To characterize the UAV state-estimation accuracy and forward safety-sensing reliability, this section derives the CRLB, the forward-ROI miss-detection probability, and their corresponding scaling laws.

4.1 CRLB for UAV State Estimation↩︎

This subsection derives the conditional CRLB based on the UAV-associated observations selected in subsection 3.1.2. The UAV scattering coefficient is assumed to be calibrated or pre-estimated and treated as known in the geometric-parameter FIM derivation. From 9 , the UAV-associated observation on link \((i,j)\) is denoted by \(\mathbf{y}_{i,j,m,n,k}^{u,s}\), with noiseless mean vector is given by \[\boldsymbol{\mu}_{i,j,m,n,k}^{u,s} = \alpha_{i,j,k}^{u} \mathbf{a}_r(\theta_{i,j,k}^{u},\phi_{i,j,k}^{u}) e^{-j2\pi m\Delta f\tau_{i,j,k}^{u,s}} e^{j2\pi nT_s\nu_{i,j,k}^{u,s}}, \label{eq:uav95mean95vector}\tag{14}\] where \(\alpha_{i,j,k}^{u} \triangleq \sqrt{p_{i,k}^{\mathrm s}}\, \zeta_k^u g_{i,j,k}^{u,s}\sqrt{\beta_{i,j,k}^{u,s}} \mathbf{a}_t^H(\theta_{i,k}^{u},\phi_{i,k}^{u}) \mathbf{w}_{i,k}^{\mathrm s}\) denotes the effective complex coefficient of the cooperative-UAV echo on link \((i,j)\). Since the sensing pilot symbol has been removed through pilot correlation and normalization in the link separation process, \(\alpha_{i,j,k}^{u}\) does not contain the pilot symbol. The residual term \(\mathbf{y}_{i,j,m,n,k}^{u,s}-\boldsymbol{\mu}_{i,j,m,n,k}^{u,s}\) is modeled as an equivalent circularly symmetric complex Gaussian disturbance with covariance \(\sigma_s^2\mathbf{I}\), accounting for thermal noise and residual non-UAV echo interference.

Let \(\boldsymbol{\xi}_{i,j,k}^{u}=(\tau_{i,j,k}^{u,s},\nu_{i,j,k}^{u,s},\theta_{i,j,k}^{u},\phi_{i,j,k}^{u})^T\) denote the local geometric observation parameter vector, and let \(\mathbf{Y}_{i,j,k}^{u}=\{\mathbf{y}_{i,j,m,n,k}^{u,s}\}_{m\in\mathcal{M}_{\mathrm s},\,n=0,\ldots,N-1}\) denote the corresponding UAV-associated observation set. Conditioned on \(\{\alpha_{i,j,k}^{u}\}\), the log-likelihood function is given by \[\begin{align} \ln p(\mathbf{Y}_{i,j,k}^{u}\mid \boldsymbol{\xi}_{i,j,k}^{u}) &= - \frac{1}{\sigma_s^2} \sum_{n=0}^{N-1} \sum_{m\in\mathcal{M}_{\mathrm s}} \left\| \mathbf{y}_{i,j,m,n,k}^{u,s} - \boldsymbol{\mu}_{i,j,m,n,k}^{u,s} \right\|^2 \\ &\quad + C_{i,j,k}, \end{align} \label{eq:loglikelihood95xi}\tag{15}\] where \(C_{i,j,k}\) is independent of \(\boldsymbol{\xi}_{i,j,k}^{u}\). The conditional Fisher information matrix (FIM) for \(\boldsymbol{\xi}_{i,j,k}^{u}\) is expressed as \[\mathbf{J}_{\boldsymbol{\xi}_{i,j,k}^{u}} = \frac{2}{\sigma_s^2} \sum_{n=0}^{N-1} \sum_{m\in\mathcal{M}_{\mathrm s}} \Re\! \left\{ \left( \frac{\partial \boldsymbol{\mu}_{i,j,m,n,k}^{u,s}}{\partial \boldsymbol{\xi}_{i,j,k}^{u}} \right)^H \left( \frac{\partial \boldsymbol{\mu}_{i,j,m,n,k}^{u,s}}{\partial \boldsymbol{\xi}_{i,j,k}^{u}} \right) \right\}. \label{eq:fim95xi95def}\tag{16}\]

The FIM associated with bistatic sensing link \((i,j)\) is obtained by evaluating the derivatives of the observation mean vector with respect to \(\boldsymbol{\xi}_{i,j,k}^{\mathrm{u}}\), as given in 2.

None

where \(\eta_{i,j,k} = \frac{ |\zeta_k^u|^{2} |g_{i,j,k}^{u,\mathrm{s}}|^{2} \beta_{i,j,k}^{u,\mathrm{s}} P_{i,k}^{\mathrm{s}}N }{\sigma_s^{2}} \left| \mathbf{a}_t^H(\theta_{i,k}^{u},\phi_{i,k}^{u}) \mathbf{w}_{i,k}^{\mathrm{s}} \right|^{2}\).

From the preceding FIM derivation, \(\mathbf{J}_{\boldsymbol{\xi}_{i,j,k}^{u}}\) characterizes the information of link \((i,j)\) with respect to the geometric observation parameters \(\boldsymbol{\xi}_{i,j,k}^{u}\). Since \(\boldsymbol{\xi}_{i,j,k}^{u}\) is determined by the UAV state \(\mathbf{x}_k^u\) through the nonlinear mapping \(\boldsymbol{\xi}_{i,j,k}^{u}=\mathbf{f}_{i,j}(\mathbf{x}_k^u)\), the equivalent FIM contribution of link \((i,j)\) to the UAV state is obtained by the chain rule as [38] \[\mathbf{J}_{i,j,k}(\mathbf{x}_k^{u}) = \mathbf{G}_{i,j,k}^{T} \mathbf{J}_{\boldsymbol{\xi}_{i,j,k}^{u}} \mathbf{G}_{i,j,k}, \label{eq:equivalent95fim95state}\tag{17}\] where \(\mathbf{G}_{i,j,k}=\partial \mathbf{f}_{i,j}(\mathbf{x}_k^u)/\partial \mathbf{x}_k^u\) is the Jacobian matrix of the geometric observation mapping with respect to the UAV state.

Since the link-level observations are obtained after transmit-receive link separation and are treated as conditionally independent, the global observation FIM for UAV state estimation is given by [21] \[\mathbf{J}_{\mathrm{obs},k}(\mathbf{x}_k^{u}) = \sum_{i\in\mathcal{B}} \sum_{j\in\mathcal{B}} \mathbf{J}_{i,j,k}(\mathbf{x}_k^{u}). \label{eq:crlb95fim95global}\tag{18}\]

The matrix CRLB for UAV state estimation is \(\mathbf{C}_{\mathrm{sec:CRLB},k}=\mathbf{J}_{\mathrm{obs},k}^{-1}(\mathbf{x}_k^u)\). The scalar position-estimation CRLB is defined as \[\mathrm{sec:CRLB}_{p,k} = \operatorname{tr} \left( \left[ \mathbf{C}_{\mathrm{sec:CRLB},k} \right]_{1:3,1:3} \right), \label{eq:position95crlb}\tag{19}\] where \([\cdot]_{1:3,1:3}\) denotes the submatrix corresponding to the position components. The velocity-estimation CRLB can be obtained similarly from the submatrix \([\mathbf{C}_{\mathrm{sec:CRLB},k}]_{4:6,4:6}\).

4.2 Scaling Law of UAV State Estimation↩︎

This subsection analyzes the large-scale behavior of the UAV position-estimation CRLB \(\mathrm{sec:CRLB}_{p,k}\) under random GBS deployment. From 2, the geometric-parameter FIM of link \((i,j)\) consists of the delay, Doppler, and AoA information terms, and is weighted by the effective echo-strength factor \(\eta_{i,j,k}\). Further combined with 17 , these geometric-parameter information terms are mapped into the UAV state domain through the geometric mapping, yielding the state-domain FIM contribution of link \((i,j)\). According to the definition of \(\eta_{i,j,k}\), the distance-dependent term is the bistatic large-scale sensing gain \(\beta_{i,j,k}^{u,\mathrm s}\), while the remaining factors, including the sensing power, UAV scattering strength, AoA derivatives, and geometric mapping, are absorbed into a positive coefficient \(\Omega_k>0\). After multi-link FIM accumulation, the UAV position-estimation CRLB can be approximated as \[\mathrm{sec:CRLB}_{p,k} \approx \left( \Omega_k \sum_{i\in\mathcal{B}} \sum_{j\in\mathcal{B}} \beta_{i,j,k}^{u,\mathrm s} \right)^{-1}. \label{eq:large95scale95crlb95beta}\tag{20}\]

Under the free-space-dominated sensing propagation condition, i.e., the sensing path-loss exponent is \(\alpha_{\mathrm s}=2\), the bistatic large-scale sensing gain can be approximated as \(\beta_{i,j,k}^{u,\mathrm s}\approx(d_{i,k}^{u})^{-2}(d_{j,k}^{u})^{-2}\). Hence, we have \(\sum_{i\in\mathcal{B}}\sum_{j\in\mathcal{B}}\beta_{i,j,k}^{u,\mathrm s}\approx(\sum_{n=1}^{J}(d_{n,k}^{u})^{-2})^2\), where \(d_{n,k}^{u}\) denotes the three-dimensional distance between the UAV and the \(n\)-th nearest GBS. Assuming that the cooperative GBSs have the same deployment height \(z^B\), we define the relative height difference as \(\Delta z_k\triangleq z_k^u-z^B\). For a two-dimensional homogeneous PPP of GBSs, the squared horizontal distance from the \(n\)-th nearest GBS to the UAV projection can be approximated by \(n/(\pi\lambda_{\mathrm B})\), yielding \(d_{n,k}^{u}\approx\sqrt{\Delta z_k^2+n/(\pi\lambda_{\mathrm B})}\). Therefore, for any finite cooperative cluster size \(J\ge1\), the discrete large-scale CRLB approximation is given by \[\mathrm{sec:CRLB}_{p,k} \approx \left[ \Omega_k \left( \sum_{n=1}^{J} \left( \Delta z_k^2+ \frac{n}{\pi\lambda_{\mathrm B}} \right)^{-1} \right)^2 \right]^{-1}. \label{eq:large95scale95crlb95discrete}\tag{21}\]

When \(J\) is large, the distance accumulation term can be approximated by the midpoint-corrected integral, i.e., \(\sum_{n=1}^{J}(\Delta z_k^2+n/(\pi\lambda_{\mathrm B}))^{-1} \approx \pi\lambda_{\mathrm B} \ln(1+J/(\pi\lambda_{\mathrm B}\Delta z_k^2+\frac{1}{2}))\). Thus, the closed-form CRLB approximation becomes \[\mathrm{sec:CRLB}_{p,k} \approx \frac{1}{ \Omega_k\pi^2\lambda_{\mathrm B}^2 \ln^2 \left( 1+ \frac{J}{\pi\lambda_{\mathrm B}\Delta z_k^2+\frac{1}{2}} \right) }. \label{eq:large95scale95crlb95closed95form}\tag{22}\]

Proposition 1 shows that, with respect to the cooperative GBS number \(J\), the UAV state-estimation CRLB follows the scaling law \(\ln^{-2}(1+c_0J)\), where \(c_0=1/(\lambda_{\mathrm B}\Delta z_k^2+1)\). When the cooperative cluster size is sufficiently large, i.e., \(J\gg \lambda_{\mathrm B}\Delta z_k^2+1\)³, this CRLB scaling law can be approximated by \(\ln^{-2}J\).

4.3 forward-ROI Miss-Detection Probability for Non-Cooperative Targets↩︎

Based on the voxel-level binary hypothesis testing model in Section 3.2, we derive the forward-ROI miss-detection probability for non-cooperative targets. For voxel \(\mathcal{C}_{l,k}\) under test, under \(\mathcal{H}_{0,l,k}\), the observation contains only the equivalent interference and noise, i.e., \(y_{l,k}=I_{l,k}+z_{l,k}\). Let \(\tilde{\sigma}_{s,l,k}^2\triangleq \mathbb{E}[|I_{l,k}+z_{l,k}|^2]\) denote the equivalent interference-plus-noise power after voxel-level observation extraction. Then, \(y_{l,k}\sim\mathcal{CN}(0,\tilde{\sigma}_{s,l,k}^2)\), and the test statistic \(T_{l,k}=|y_{l,k}|^2\) follows an exponential distribution with mean \(\tilde{\sigma}_{s,l,k}^2\). For a prescribed false-alarm probability \(\mathbb{P}_{\mathrm{fa}}\), the detection threshold \(\Gamma_{\mathrm{th},l,k}\) satisfies [39] \[\begin{align} \mathbb{P}_{\mathrm{fa}} &= \mathbb{P}(T_{l,k}\ge\Gamma_{\mathrm{th},l,k}\mid\mathcal{H}_{0,l,k}) \\ &= \int_{\Gamma_{\mathrm{th},l,k}}^{\infty} \frac{1}{\tilde{\sigma}_{s,l,k}^2} \exp\!\left(-\frac{x}{\tilde{\sigma}_{s,l,k}^2}\right)\mathrm{d}x = \exp\!\left(-\frac{\Gamma_{\mathrm{th},l,k}}{\tilde{\sigma}_{s,l,k}^2}\right). \end{align} \label{eq:false95alarm95probability}\tag{23}\]

Therefore, \(\Gamma_{\mathrm{th},l,k}=-\tilde{\sigma}_{s,l,k}^2\ln(\mathbb{P}_{\mathrm{fa}})\). Under \(\mathcal{H}_{1,l,k}\), the observation further contains the equivalent target echo from voxel \(\mathcal{C}_{l,k}\). According to the non-cooperative target sensing channel model in 7 , the voxel-level equivalent target echo is modeled as \(s_{l,k}\sim\mathcal{CN}(0,P_{l,k}^{t})\), where \(P_{l,k}^{t}\triangleq\mathbb{E}[|s_{l,k}|^2]\) denotes the average target echo power after voxel-level observation extraction. Assuming that \(s_{l,k}\) is independent of \(I_{l,k}+z_{l,k}\), we have \(y_{l,k}\sim\mathcal{CN}(0,P_{l,k}^{t}+\tilde{\sigma}_{s,l,k}^2)\). The voxel-level sensing signal-to-interference-plus-noise ratio (SINR) is defined as \(\gamma_{l,k}^{s}=P_{l,k}^{t}/\tilde{\sigma}_{s,l,k}^2\).

Under \(\mathcal{H}_{1,l,k}\), \(T_{l,k}\) is exponentially distributed with mean \(P_{l,k}^{t}+\tilde{\sigma}_{s,l,k}^2\). Therefore, the conditional detection probability is given by \[\begin{align} \mathbb{P}_{\mathrm{d},l,k} &= \mathbb{P}(T_{l,k}\ge\Gamma_{\mathrm{th},l,k}\mid\mathcal{H}_{1,l,k}) \\ &= \int_{\Gamma_{\mathrm{th},l,k}}^{\infty} \frac{1}{P_{l,k}^{t}+\tilde{\sigma}_{s,l,k}^2} \exp\!\left( -\frac{x}{P_{l,k}^{t}+\tilde{\sigma}_{s,l,k}^2} \right)\mathrm{d}x \\ &= \exp\!\left( -\frac{\Gamma_{\mathrm{th},l,k}}{P_{l,k}^{t}+\tilde{\sigma}_{s,l,k}^2} \right). \end{align} \label{eq:voxel95detection95probability}\tag{24}\]

Using \(\Gamma_{\mathrm{th},l,k}=-\tilde{\sigma}_{s,l,k}^2\ln(\mathbb{P}_{\mathrm{fa}})\) and \(\gamma_{l,k}^s=P_{l,k}^{t}/\tilde{\sigma}_{s,l,k}^2\) in 24 , we obtain \(\mathbb{P}_{\mathrm{d},l,k}=\mathbb{P}_{\mathrm{fa}}^{1/(1+\gamma_{l,k}^s)}\). Assuming independent voxel occupancies with prior occupancy probability \(q_k\), the missed-detection probability of the ROI is expressed as \[\mathbb{P}_{\mathrm{md},\mathcal{V}_k} = 1- \prod_{l=1}^{N_{v,k}} \left[ 1- q_k \left( 1- \mathbb{P}_{\mathrm{fa}}^{\frac{1}{1+\gamma_{l,k}^s}} \right) \right]. \label{eq:region95miss95detection95probability}\tag{25}\]

4.4 Scaling Law of forward-ROI Miss-Detection Probability↩︎

This subsection analyzes the large-scale behavior of the forward-ROI miss-detection probability under random GBS deployment and random target occupancy. To extract a region-level scaling relation, we characterize the detection performance of a typical voxel by the ROI-average sensing SINR, defined as \(\bar{\gamma}_k^s\triangleq (1/N_{v,k})\sum_{l=1}^{N_{v,k}}\gamma_{l,k}^s\). By adopting the same ordered-distance and closed-form distance-accumulation approximation as in 22 , and by absorbing the non-distance-dominant factors, including target scattering strength, sensing power, beam gain, and equivalent interference-plus-noise power, into a positive coefficient \(\Xi_k>0\), the ROI-average sensing SINR can be approximated as \[\bar{\gamma}_{k}^{s} \approx \Xi_k\pi^2\lambda_{\mathrm B}^2 \ln^2\!\left( 1+ \frac{J}{\pi\lambda_{\mathrm B}\Delta z_k^2+\frac{1}{2}} \right). \label{eq:average95sensing95sinr95scaling}\tag{26}\]

Using the average sensing SINR \(\bar{\gamma}_k^s\) as a representative voxel-level SINR within the ROI, the voxel-dependent detection term in 25 is approximated by a common term. The forward-ROI miss-detection probability is approximated as \[\mathbb{P}_{\mathrm{md},\mathcal{V}_k} \approx 1- \left[ 1- q_k \left( 1- \mathbb{P}_{\mathrm{fa}}^{1/(1+\bar{\gamma}_k^s)} \right) \right]^{N_{v,k}}. \label{eq:regional95md95average}\tag{27}\]

Since \(0\le 1-\mathbb{P}_{\mathrm{fa}}^{1/(1+\bar{\gamma}*k^s)}\le1\), the sparse occupancy condition \(q_k\ll1\) ensures that \(q_k\bigl(1-\mathbb{P}*{\mathrm{fa}}^{1/(1+\bar{\gamma}_k^s)}\bigr)\ll1\). Applying \((1-x)^N\approx\exp(-Nx)\) for small \(x\) to 27 , the miss-detection probability is approximated as \[\mathbb{P}_{\mathrm{md},\mathcal{V}_k} \approx 1- \exp\left[ -q_kN_{v,k} \left( 1- \mathbb{P}_{\mathrm{fa}}^{1/(1+\bar{\gamma}_k^s)} \right) \right]. \label{eq:regional95md95sparse}\tag{28}\]

According to the voxel occupancy model, the single-voxel occupancy probability is \(q_k=1-\exp(-\lambda_{t,k}\Delta d^3)\), where \(\lambda_{t,k}\Delta d^3\) represents the average number of targets in one voxel. When the voxelization is sufficiently fine such that \(\lambda_{t,k}\Delta d^3\ll1\), we have \(q_k\approx\lambda_{t,k}\Delta d^3\). Combining this with the ROI voxelization relation \(N_{v,k}\approx|\mathcal{V}_k|/\Delta d^3\), we obtain \(q_kN_{v,k}\approx\lambda_{t,k}|\mathcal{V}_k|\). Substituting \(q_kN_{v,k}\approx\lambda_{t,k}|\mathcal{V}_k|\) into 28 , the forward-ROI miss-detection probability is approximated by \[\mathbb{P}_{\mathrm{md},\mathcal{V}_k} \approx 1- \exp\!\left[ -\lambda_{t,k}|\mathcal{V}_k| \left( 1- \mathbb{P}_{\mathrm{fa}}^{\frac{1}{1+\bar{\gamma}_k^s}} \right) \right], \label{eq:regional95md95scaling95closed95form}\tag{29}\] where \(|\mathcal{V}_k|\approx4D_s^2D_{f,k}\), \(D_{f,k}=v_k^u t_r+(v_k^u)^2/(2a_{\max})+d_m\).

When the average sensing SINR is sufficiently high, i.e., \(\bar{\gamma}_k^s\gg-\ln\mathbb{P}_{\mathrm{fa}}\), the voxel-level miss-detection term can be approximated as \(1-\mathbb{P}_{\mathrm{fa}}^{1/(1+\bar{\gamma}_k^s)} \approx -\ln(\mathbb{P}_{\mathrm{fa}})/\bar{\gamma}_k^s\). Substituting this approximation into 29 , and further using the average sensing-SINR scaling in 26 and \(|\mathcal{V}_k|\approx4D_s^2D_{f,k}\), where \(4D_s^2\) is absorbed into \(\Theta(\cdot)\), gives the following proposition.

Proposition 2 shows that, in the high-sensing-SINR regime, the forward-ROI miss-detection probability has an exponential scaling form, whose exponent with respect to the cooperative GBS number \(J\) follows the scaling law \(\lambda_{t,k}D_{f,k}\ln^{-2}(1+c_0J)\), where \(c_0=1/(\lambda_{\mathrm B}\Delta z_k^2+1)\). When the cooperative cluster size is sufficiently large, i.e., \(J\gg\lambda_{\mathrm B}\Delta z_k^2+1\), this scaling law can be approximated by \(\lambda_{t,k}D_{f,k}\ln^{-2}J\). Moreover, since \(D_{f,k}=v_k^u t_r+(v_k^u)^2/(2a_{\max})+d_m\), a higher UAV speed leads to a larger forward safety distance and thus increases the forward-ROI miss-detection probability.

5 Safety-Aware Forward-Detection Resource Optimization↩︎

In the considered networked ISAC system, communication, UAV state estimation, and forward-ROI target detection share limited subcarrier, power, and beam resources. If the resource configuration mainly favors communication or UAV state estimation, the sensing capability required for forward detection may become insufficient, making it difficult to reliably detect non-cooperative targets within the forward ROI. Therefore, this section formulates a forward detection resource optimization problem that balances UAV state-estimation accuracy and forward-detection reliability.

The optimization is performed within the cooperative cluster \(\mathcal{B}\), which consists of the \(J\) GBSs nearest to the predicted UAV ground projection. The sensing pilot ratio, transmit power allocation, and beam selection are jointly configured. To avoid the complexity of high-dimensional continuous beamforming optimization, the CPU constructs finite beam codebooks before resource configuration. According to the predicted UAV state \(\hat{\mathbf{x}}_{k|k-1}^{u}\) and its prediction uncertainty, the CPU first determines the UAV angular support \(\mathcal{A}_{j,k}^{u}\) at GBS \(j\). Then, the communication codebook is constructed from the discretized angular set \(\widetilde{\mathcal{A}}_{j,k}^{\mathrm c}\) of \(\mathcal{A}_{j,k}^{u}\) as \[\mathcal{W}_{j,k}^{\mathrm c} = \left\{ \frac{\mathbf{a}_t(\theta,\phi)}{\|\mathbf{a}_t(\theta,\phi)\|_2} \mid (\theta,\phi)\in\widetilde{\mathcal{A}}_{j,k}^{\mathrm c} \right\}. \label{eq:comm95beam95codebook}\tag{30}\]

For sensing, the sensing codebook of GBS \(j\) consists of a UAV-sensing sub-codebook and an ROI-sensing sub-codebook. Specifically, \(\mathcal{W}_{j,k}^{u,\mathrm s}\) is constructed from the UAV angular support \(\mathcal{A}_{j,k}^{u}\), while \(\mathcal{W}_{j,k}^{\mathcal{V},\mathrm s}\) is constructed from the transmit angular support \(\mathcal{A}_{j,k}^{\mathcal{V}}\) subtended by the forward ROI \(\mathcal{V}_k\) at GBS \(j\). Both sensing sub-codebooks are generated using normalized steering vectors in the same manner as 30 . Hence, the sensing codebook is \(\mathcal{W}_{j,k}^{\mathrm s} = \mathcal{W}_{j,k}^{u,\mathrm s} \cup \mathcal{W}_{j,k}^{\mathcal{V},\mathrm s}\). The resource configuration at time slot \(k\) is written as \[\boldsymbol{\chi}_k = \left\{ \rho_{\mathrm s,k}, \left\{ P_{j,k}^{\mathrm c}, P_{j,k}^{\mathrm s}, \mathbf{w}_{j,k}^{\mathrm c}, \mathbf{w}_{j,k}^{\mathrm s} \right\}_{j\in\mathcal{B}} \right\}. \label{eq:resource95configuration}\tag{31}\]

To ensure integer numbers of both communication and sensing subcarriers and avoid an empty resource partition, the sensing pilot ratio \(\rho_{\mathrm{s},k}\) is selected from the finite set \(\mathcal{G}_\rho=\{M_s/M\mid M_s\in\mathbb{Z},\;0<M_s<M\}\). Accordingly, the feasible resource set is defined as \[\mathcal{X}_k(J) = \left\{ \boldsymbol{\chi}_k \;\middle|\; \begin{array}{l} \rho_{\mathrm s,k}\in\mathcal{G}_\rho,\\ P_{j,k}^{\mathrm c}\ge0,\; P_{j,k}^{\mathrm s}\ge0,\\ P_{j,k}^{\mathrm c}+P_{j,k}^{\mathrm s}\le P_j^{\max},\\ \mathbf{w}_{j,k}^{\mathrm c}\in\mathcal{W}_{j,k}^{\mathrm c},\; \mathbf{w}_{j,k}^{\mathrm s}\in\mathcal{W}_{j,k}^{\mathrm s},\\ \forall j\in\mathcal{B}. \end{array} \right\}. \label{eq:feasible95resource95set}\tag{32}\]

Accordingly, the achievable performance set at time slot \(k\) is defined as \(\mathcal{P}_k(J) = \left\{ \left( R_k(\boldsymbol{\chi}_k), \mathrm{sec:CRLB}_{p,k}(\boldsymbol{\chi}_k), \mathbb{P}_{\mathrm{md},\mathcal{V}_k}(\boldsymbol{\chi}_k) \right) \mid \boldsymbol{\chi}_k\in\mathcal{X}_k(J) \right\}\). For two feasible configurations \(\boldsymbol{\chi}_a\) and \(\boldsymbol{\chi}_b\), \(\boldsymbol{\chi}_a\) is said to dominate \(\boldsymbol{\chi}_b\) if \(R_k(\boldsymbol{\chi}_a)\ge R_k(\boldsymbol{\chi}_b)\), \(\mathrm{sec:CRLB}_{p,k}(\boldsymbol{\chi}_a)\le\mathrm{sec:CRLB}_{p,k}(\boldsymbol{\chi}_b)\), and \(\mathbb{P}_{\mathrm{md},\mathcal{V}_k}(\boldsymbol{\chi}_a)\le \mathbb{P}_{\mathrm{md},\mathcal{V}_k}(\boldsymbol{\chi}_b)\), with at least one strict inequality. Therefore, the non-dominated points in \(\mathcal{P}_k(J)\) form the Pareto boundary among communication, UAV state estimation, and forward-ROI detection.

Based on the above, to ensure communication performance while improving sensing reliability, we impose the communication rate as a constraint and jointly minimize the UAV state-estimation CRLB and the forward-ROI miss-detection probability. The safety-aware forward detection resource optimization problem \(\mathbf{P}\) is formulated as \[\tag{33} \begin{align} \min_{\boldsymbol{\chi}_k\in\mathcal{X}_k(J)}\quad & \varpi \frac{\mathbb{P}_{\mathrm{md},\mathcal{V}_k}(\boldsymbol{\chi}_k)}{\mathbb{P}_{\mathrm{md},\mathrm{ref}}}+ (1-\varpi) \frac{\mathrm{sec:CRLB}_{p,k}(\boldsymbol{\chi}_k)}{\mathrm{sec:CRLB}_{p,\mathrm{ref}}} \tag{34} \\ \mathrm{s.t.}\quad & R_k(\boldsymbol{\chi}_k)\ge R_{\min,k}, \tag{35} \\ & R_k(\boldsymbol{\chi}_k)+\eta_{\mathrm{fh}}J\le C_{\mathrm{fh},k}, \tag{36} \end{align}\] where \(\varpi\in[0,1]\) is the forward-detection weighting factor, and \(\mathrm{sec:CRLB}_{p,\mathrm{ref}}\) and \(\mathbb{P}_{\mathrm{md},\mathrm{ref}}\) are normalization factors. Constraints 35 and 36 represent the communication-rate requirement and the fronthaul-capacity limitation, respectively, where \(R_{\min,k}\) is the minimum communication-rate requirement, \(C_{\mathrm{fh},k}\) is the fronthaul capacity, and \(\eta_{\mathrm{fh}}J\) characterizes the cooperation overhead caused by synchronization, signaling exchange, CSI sharing, and sensing-data forwarding.

Since \(\rho_{\mathrm s,k}\), \(\mathbf{w}_{j,k}^{\mathrm c}\), and \(\mathbf{w}_{j,k}^{\mathrm s}\) are selected from finite candidate sets, and the power variables can be discretized over predefined grids, \(\mathbf{P}\) can be solved by finite grid search. For the given cooperative cluster \(\mathcal{B}\), we enumerate the communication beams, sensing beams, sensing subcarrier ratios, and power allocation candidates, and then select the feasible configuration with the minimum objective value among those satisfying the rate and fronthaul constraints.

6 sec:Performance32Evaluation↩︎

6.1 Simulation Parameter Settings↩︎

This section presents numerical simulations to validate the analytical results and evaluate the proposed detection-aware ISAC resource allocation method. For scaling law validation, the finite-sum expressions and closed-form approximations are compared with Monte Carlo results under random GBS deployments, where the GBS locations follow a two-dimensional homogeneous PPP. For resource allocation, a representative time slot is considered, and the optimal configuration is obtained by searching over a predefined finite resource set. The main simulation parameters are summarized in Table 1.

6.2 Scaling Law Validation↩︎

Fig. 3 validates the derived scaling law expressions for UAV state estimation and regional non-cooperative target detection. First, Figs. 3 (a) and (b) show the scalar CRLB metric versus the cooperative GBS number \(J\) and the GBS density \(\lambda_{\mathrm B}\), respectively. It can be observed that the CRLB decreases as either \(J\) or \(\lambda_{\mathrm B}\) increases, and the Monte Carlo simulations closely match the finite-sum expression in 21 . Meanwhile, the UAV-GBS scaling law in 22 also agrees well with the finite-sum expression, especially for moderate and large \(J\) or \(\lambda_{\mathrm B}\). These results show that the finite-sum approximation based on the average ordered distances of PPP-distributed GBSs can accurately characterize the average CRLB scaling behavior, while the midpoint-corrected closed-form approximation effectively captures its main trend. In contrast, the terrestrial-network scaling law [16] underestimates the CRLB, since it ignores the UAV-GBS height difference and thus overestimates the UAV state-estimation accuracy.

Second, Figs. 3 (c) and (d) present the forward-ROI miss-detection probability versus the cooperative GBS number \(J\) and the GBS deployment density \(\lambda_{\mathrm B}\), respectively. The miss-detection probability decreases as \(J\) or \(\lambda_{\mathrm B}\) increases, and the Monte Carlo simulations, finite-sum expression, and UAV-GBS scaling law are closely aligned. This is because a larger cooperative cluster or a denser GBS deployment increases the accumulated bistatic sensing gain, thereby improving the average sensing SINR. These results validate the exponential-form regional miss-detection approximation in 29 , which is obtained from the sparse occupancy model and the average sensing SINR scaling. They also indicate that the same distance-accumulation mechanism governing the CRLB scaling can characterize the regional detection performance.

Third, Figs. 3 (e) and (f) show the miss-detection probability versus the non-cooperative target density \(\lambda_t\) and the UAV speed \(v^u\), respectively. The miss-detection probability increases with either \(\lambda_t\) or \(v^u\). This is because a larger \(\lambda_t\) increases the expected number of occupied voxels in the ROI, while a higher \(v^u\) enlarges the forward safety-critical ROI through the braking-distance constraint. As a result, both parameters increase the effective potential target scale \(\lambda_t|\mathcal{V}|\), which is consistent with the dependence in 29 .

In summary, the simulation results verify two key observations. First, compared with the terrestrial-network scaling law [16], the scaling law derived in this paper explicitly retains the UAV-GBS height difference \(\Delta z_k\), thereby providing a more accurate characterization of the variation of the UAV state-estimation CRLB under the 3D UAV-GBS geometry. Second, the miss-detection probability is jointly governed by the average sensing SINR and the product \(\lambda_t|\mathcal{V}|\), which respectively characterize the voxel-level detection capability and the expected number of non-cooperative targets in the forward ROI.

6.3 Performance Tradeoff Evaluation↩︎

Figs. 4 and 5 evaluate the capability of the proposed resource optimization formulation to characterize the tradeoff among communication, UAV state estimation, and forward-ROI target detection.

First, Fig. 4 presents the 3D performance region achieved by different feasible resource configurations. The feasible points form a clear Pareto boundary in the space of average data rate \(R\), state-estimation CRLB, and miss-detection probability \(\mathbb{P}_{\mathrm{md},\mathcal{V}}\). The boundary extends toward the desirable direction of higher data rate, lower CRLB, and lower miss-detection probability, indicating that the three metrics are intrinsically coupled. This coupling arises from the shared use of subcarrier, power, and beam-direction resources. Communication-oriented configurations improve \(R\) by allocating more resources to communication transmission or by selecting beams with higher communication gain, but they reduce the resources available for sensing and may increase the CRLB or \(\mathbb{P}_{\mathrm{md},\mathcal{V}}\). Conversely, sensing-oriented configurations improve UAV state estimation and forward-ROI detection at the cost of communication performance. Therefore, Fig. 4 reveals the resource-transfer cost among communication, state estimation, and detection tasks.

Second, Fig. 5 shows the two-dimensional projections of the above performance region. In Fig. 5 (a), the minimum achievable CRLB generally increases with \(R\), showing that improving communication performance degrades UAV state-estimation accuracy due to reduced sensing resources. In Fig. 5 (b), \(\mathbb{P}_{\mathrm{md},\mathcal{V}}\) also increases with \(R\), indicating a tradeoff between communication and forward-ROI detection. The increase is mild in the low-to-moderate rate region, where communication and detection can still be jointly coordinated. In contrast, in the high-rate region, \(\mathbb{P}_{\mathrm{md},\mathcal{V}}\) rises more rapidly and gradually approaches the saturation level \(1-\exp(-\lambda_t|\mathcal{V}|)\), implying that overly communication-oriented resource allocation can degrade forward-detection reliability. Further, Fig. 5 (c) shows the tradeoff between the CRLB and \(\mathbb{P}_{\mathrm{md},\mathcal{V}}\). The Pareto boundary lies toward the lower-left region, since both metrics should be minimized. This projection reveals the competition within sensing resources: UAV state estimation favors beams focused on the UAV angular region, whereas forward-ROI detection requires effective illumination of the ROI.

From Figs. 4 and 5, we observe that communication, UAV state estimation, and forward-ROI detection are subject to a tradeoff under shared resource constraints. Increasing the communication rate reduces the resources available for sensing, thereby increasing both the state-estimation CRLB and the miss-detection probability. Meanwhile, state estimation focuses on UAV sensing, whereas forward-ROI detection focuses on detection over the forward region; hence, the two sensing tasks still need to be balanced in resource configuration.

6.4 Performance of Forward Detection Resource Optimization↩︎

To validate the necessity of incorporating the forward-ROI miss-detection probability \(\mathbb{P}_{\mathrm{md},\mathcal{V}}\) into resource optimization, we compare the proposed networked-ISAC-based forward detection resource optimization scheme with a baseline scheme that does not consider forward detection, as shown in Fig. 6. As the minimum communication-rate requirement \(R_{\min}\) increases, both the CRLB and \(\mathbb{P}_{\mathrm{md},\mathcal{V}}\) increase for the two schemes. This is because stricter communication requirements shift more subcarrier, power, and beam resources toward communication transmission, thereby reducing the resources available for sensing. Compared with the baseline scheme, the proposed scheme reduces the average miss-detection probability \(\mathbb{P}_{\mathrm{md},\mathcal{V}}\) by 17.05%, at the cost of a 14.82% increase in the average CRLB. According to Remark 2, the reduction in \(\mathbb{P}_{\mathrm{md},\mathcal{V}}\) also indicates a corresponding reduction in the sensing-induced collision risk. It is evident that incorporating \(\mathbb{P}_{\mathrm{md},\mathcal{V}}\) into resource optimization improves forward-ROI detection reliability and reduces sensing-induced collision risk under a controllable loss of UAV state-estimation accuracy.

6.5 Impact of Cooperative Cluster Size on Forward Detection Resource Optimization↩︎

Based on the resource optimization problem formulated in Section 5, Fig. 7 shows the UAV state-estimation CRLB and the forward-ROI miss-detection probability \(P_{\mathrm{md},\mathcal{V}}\) versus the minimum data-rate requirement \(R_{\min}\) under different cooperative cluster sizes \(J\). First, as \(R_{\min}\) increases, the CRLB increases significantly for all values of \(J\), while \(\mathbb{P}_{\mathrm{md},\mathcal{V}}\) also increases and gradually approaches saturation. This is because a stricter communication-rate constraint allocates more subcarrier, power, and beam resources to communication transmission, thereby reducing the sensing resources available for UAV state estimation and forward-ROI detection.

Secondly, as the cooperative cluster size \(J\) increases, the CRLB and \(\mathbb{P}_{\mathrm{md},\mathcal{V}}\) curves shift downward. Under the same \(R_{\min}\), a larger \(J\) corresponds to a lower CRLB and a smaller \(\mathbb{P}_{\mathrm{md},\mathcal{V}}\). For example, when \(R_{\min}=0.5\) bit/s/Hz, compared with the single-GBS case with \(J=1\), the configuration with \(J=3\) reduces the CRLB and \(\mathbb{P}_{\mathrm{md},\mathcal{V}}\) by approximately 82.2% and 37.2%, respectively. This is because multi-GBS cooperation increases the number of bistatic links and provides spatial observation and signal aggregation gains.

Thirdly, in the high-\(R_{\min}\) region, the curves of \(\mathbb{P}_{\mathrm{md},\mathcal{V}}\) under different values of \(J\) become closer, indicating that the forward-detection gain brought by additional cooperative GBSs becomes limited when sensing resources are strongly constrained by the communication-rate requirement.

Overall, these results show that the cooperative cluster size significantly affects both the CRLB and the forward-ROI miss-detection probability under communication constraints, highlighting its importance in cooperative cluster design.

6.6 Weight Sensitivity of Forward Detection Resource Optimization↩︎

This subsection investigates the impact of the detection weight \(\varpi\) on the proposed forward detection resource optimization scheme. Fig. 8 shows the resulting UAV state-estimation CRLB and the forward-ROI miss-detection probability \(P_{\mathrm{md},\mathcal{V}}\) under a fixed communication-rate constraint.

When \(\varpi \leq 0.2\), the CRLB and \(P_{\mathrm{md},\mathcal{V}}\) remain almost unchanged. This is because the sensing pilot ratio, power allocation, and beam directions are selected from finite candidate sets, and a small change in the weight is insufficient to alter the optimal resource configuration. As \(\varpi\) increases, the weight of the forward-ROI detection term in the objective function increases, and the resource configuration gradually shifts toward detection performance, leading to a decrease in \(P_{\mathrm{md},\mathcal{V}}\) and an increase in the CRLB. When \(\varpi\) approaches 1, the decrease in \(P_{\mathrm{md},\mathcal{V}}\) tends to saturate, whereas the CRLB degrades significantly, indicating that further increasing the detection weight trades a large state-estimation performance loss for limited detection gain.

These results show that \(\varpi\) controls the resource allocation preference between UAV state estimation and forward-ROI detection. A moderate detection weight provides a favorable operating point, where the miss-detection probability can be effectively reduced with limited state-estimation loss. Therefore, \(\varpi\) should be adaptively selected according to the UAV state-estimation accuracy requirement and the forward-risk level in practical deployment.

7 sec:Conclusion↩︎

To address the insufficient detection reliability of non-cooperative targets in the UAV forward region, this paper proposed a networked-ISAC-based forward detection design for safe UAV flight. A motion-dependent forward ROI was constructed and voxelized for target-existence decisions, based on which the UAV state-estimation CRLB and forward-ROI miss-detection probability were derived and analyzed. A safety-aware resource optimization scheme was then formulated to balance UAV state estimation and forward detection under the communication-rate constraint. Simulation results showed that the proposed scheme reduced the average miss-detection probability and the corresponding sensing-induced collision risk by \(17.05\%\) compared with the baseline without forward detection, while introducing only limited UAV state-estimation performance degradation, reflected by a \(14.82\%\) increase in the average CRLB.

References↩︎

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1. Jingli Li, Yiyan Ma, Wei Chen, Guoyu Ma, Mi Yang, Yunlong Lu, and Zhangdui Zhong are with the School of Electronic and Information Engineering, Beijing Jiaotong University, Beijing 100044, China. Weijie Yuan is with the School of System Design and Intelligent Manufacturing, Southern University of Science and Technology, Shenzhen 518055, China. Tongyang Xu is with the Department of Electronic and Electrical Engineering, University College London, WC1E 7JE London, U.K. Qingqing Cheng is with the School of Electrical Engineering and Robotics, Queensland University of Technology, Brisbane, QLD 4000, Australia. Wenwei Yue is with the State Key Laboratory of Integrated Services Networks, Xidian University, Xi’an, Shaanxi 710071, China.(Corresponding authors: mayiyan@bjtu.edu.cn.)↩︎

2. Since \(n_l\sim\mathrm{Poisson}(\lambda_t\Delta d^3)\), we have \(\Pr(n_l=1)=\lambda_t\Delta d^3 e^{-\lambda_t\Delta d^3}\) and \(\Pr(n_l\ge2)=1-e^{-\lambda_t\Delta d^3}(1+\lambda_t\Delta d^3)\). When \(\lambda_t\Delta d^3\ll1\), \(\frac{\Pr(n_l\ge2)}{\Pr(n_l=1)}\approx\frac{\lambda_t\Delta d^3}{2}\ll1\).↩︎

3. In practice, the maximum cooperative cluster size is limited by the available orthogonal resources and the fronthaul capacity.↩︎