The Radio Environment Map (REM), or spectrum map, functions as a comprehensive database for electromagnetic information and has become a cornerstone technology for electromagnetic environment awareness. It provides crucial data support for applications such as spectrum resource allocation, interference source localization, unmanned drone communication link optimization, and dynamic spectrum access by visualizing parameters like signal strength and source locations within a monitored area. However, generating a high-fidelity REM in scenarios with unknown transmitter locations and impractical fixed sensor deployment—common in post-disaster or remote areas—poses a significant challenge. Unmanned aerial vehicles (UAVs), with their superior mobility, emerge as an ideal platform for such spectrum sensing tasks. The core contradiction lies in achieving high-precision reconstruction under the stringent constraints of limited unmanned drone endurance and onboard computational resources, which necessitates extremely low sampling rates. Traditional REM generation techniques, whether model-based, data-driven, or hybrid, struggle to balance low-cost sampling with high-accuracy reconstruction effectively in such sparse, unknown-source environments for unmanned aerial vehicles.
This paper proposes a lightweight unmanned drone sampling method based on compressive sensing (CS) to address this challenge. The core idea is to intelligently plan the UAV’s flight trajectory to collect a minimal set of high-value measurements, which are then used to reconstruct the full REM. The method integrates the physical sampling process with the construction of the CS measurement matrix. A Two-Stage Maxmin deployment strategy is designed to optimize sampling point locations without prior source information, ensuring both global coverage and enhanced sampling in signal-dense “hotspot” regions. Furthermore, to overcome the unmanned aerial vehicle’s endurance limit, a Greedy+2-opt hybrid algorithm is employed for rapid and efficient path planning. The main contributions are: 1) A framework for constructing the CS measurement matrix directly from the unmanned drone’s sampling trajectory. 2) A hotspot-aware Two-Stage Maxmin deployment strategy that dynamically allocates sampling resources. 3) A lightweight path planning solution that balances tour length and computational cost for the unmanned aerial vehicle platform.
1. System Model and Problem Formulation
The target scenario involves a wide-area, complex electromagnetic environment with multiple RF sources at unknown locations. Deploying fixed sensors or ground vehicles is deemed infeasible. An unmanned drone is tasked with reconstructing the REM over a region R. The UAV has limited flight time T_max, translating to a maximum number of sampling points M, which is much smaller than the number of discrete grid cells N^2 in the area (i.e., M << N^2). The goal is to design a sampling and reconstruction method that yields an accurate REM under this low-sampling-rate constraint. The problem decomposes into three sub-problems: measurement matrix construction, adaptive sampling deployment, and lightweight sampling path planning.
1.1 Sparse Signal Representation
The REM over the 2D area is denoted as a matrix S, discretized into N × N grid cells. Let the vectorized signal be s ∈ R^{N^2}. We assume the signal is sparse in a basis Ψ constructed from a physical propagation model, leading to s = Ψ x, where x ∈ R^{N^2} is a K-sparse coefficient vector (K is the number of sources). This sparsity is the foundation for applying compressive sensing with an unmanned drone.
Consider K unknown transmitters with power P_k and location (x_k, y_k). The signal strength from source k at grid point i with location (u_i, v_i) is modeled by a distance-based attenuation:
$$ p_{i,k} = \frac{P_k}{d_{i,k}^\alpha} $$
where d_{i,k} = \sqrt{(u_i - x_k)^2 + (v_i - y_k)^2} is the Euclidean distance and α is the path loss exponent. The total signal strength at point i is s_i = \sum_{k=1}^{K} p_{i,k} + n_i, with n_i as noise. The sparse basis matrix Ψ is constructed such that its (i, j)-th element represents the attenuation from a hypothetical source at grid j to the observation point i:
$$ \Psi_{i,j} = \frac{1}{d_{i,j}^\alpha} \quad \text{with} \quad \Psi_{i,i} = 1 $$
Thus, the original signal relationship is s = Ψ x + n, where the non-zero elements of x correspond to the source locations and their powers.
1.2 Measurement Matrix Construction from Unmanned Drone Trajectory
The unmanned drone’s sampling path defines the CS measurement matrix. Let the UAV visit M distinct grid locations along its trajectory. The measurement process for a single sample at time step t at position pos_t = (u_t, v_t) is:
$$ y_t = s(pos_t) + e_t = \sum_{k=1}^{K} \frac{P_k}{d(pos_t, source_k)^\alpha} + e_t $$
where e_t is measurement noise. The overall measurement vector y ∈ R^M is related to the sparse signal by y = Φ s = Φ Ψ x + e. The measurement matrix Φ ∈ R^{M × N^2} is a binary selection matrix constructed from the unmanned drone’s path:
$$ \Phi_{t, i} = \begin{cases}
1, & \text{if the UAV samples at grid index } i \text{ at time } t \\
0, & \text{otherwise}
\end{cases} $$
This construction directly links the physical path of the unmanned aerial vehicle to the mathematical framework of CS. The reconstruction problem is then to recover the sparse vector x (and hence the REM s) from the underdetermined system y = A x + e, where A = Φ Ψ is the combined sensing matrix.
2. Hotspot-Aware Two-Stage Maxmin Deployment
The quality of the CS reconstruction heavily depends on the properties of matrix A, which in turn depends on the sampling locations defined by Φ. A naive random or uniform deployment of the limited M points may be suboptimal. We propose a Two-Stage Maxmin strategy to deploy the unmanned drone’s sampling points intelligently without prior source information.
2.1 Stage 1: Global Exploration via Greedy Maxmin
The first stage allocates M_1 = βM points (0 < β < 1) to achieve global coverage and avoid sampling blind spots. The Maxmin criterion is used: it selects points to maximize the minimum distance between any two deployed points. This spreads points evenly across the region R. A greedy algorithm efficiently approximates the solution.
| Algorithm 1: Greedy Maxmin Exploration Deployment |
|---|
Input: Region R, Number of points M_1, Candidate grid set COutput: Stage-1 sampling set S_11: Initialize S_1 = {}2: Randomly select first point p_1 from C; S_1.add(p_1); C.remove(p_1)3: for j = 2 to M_1 do4: best_dist = -1; best_point = null5: for each candidate point c in C do6: min_d = min_{p ∈ S_1} distance(c, p)7: if min_d > best_dist then8: best_dist = min_d; best_point = c9: end if 10: end for 11: S_1.add(best_point); C.remove(best_point)12: end for 13: return S_1
|
2.2 Stage 2: Hotspot-Focused Fine Deployment
After Stage 1, the unmanned drone collects measurements at points in S_1. These measurements are used to identify potential hotspot regions without any prior model. First, the region R is partitioned into Voronoi cells based on S_1. Each cell V_i is associated with a measurement value y_i. A cell is flagged as a “hotspot cell” if its measured power exceeds a dynamic threshold, e.g., y_i > \bar{y} + γ σ_y, where \bar{y} and σ_y are the mean and standard deviation of all Stage-1 measurements, and γ is a parameter. Adjacent hotspot cells are merged to form contiguous hotspot regions {H_1, H_2, ..., H_L}.
The remaining M_2 = M - M_1 sampling points are allocated to these hotspot regions proportionally to their assessed “value.” The value of a hotspot region H_l is defined as V_l = |H_l| * \bar{y}_l, considering both its area |H_l| and the average signal strength \bar{y}_l within it. The number of points allocated to region l is:
$$ m_l = \left\lfloor M_2 \cdot \frac{V_l}{\sum_{j=1}^{L} V_j} \right\rceil $$
Within each hotspot region H_l, the allocated m_l points are deployed using the Maxmin criterion again (Algorithm 1 applied to H_l), ensuring they are well-spread within the region to capture local features effectively. This two-stage approach allows the unmanned aerial vehicle to adapt its sampling density, focusing resources on areas with stronger or more dynamic signals, which are critical for accurate source localization and power estimation.
| Algorithm 2: Two-Stage Hotspot-Aware Deployment |
|---|
Input: Total points M, Ratio βOutput: Complete sampling set S1: M_1 = round(βM)2: S_1 = Greedy_Maxmin(R, M_1) // Stage 1: Global Exploration3: UAV samples at S_1, obtains measurements Y_14: Build Voronoi diagram for R using S_1 as generators.5: Identify hotspot cells and merge into regions {H_1,..., H_L} based on Y_1.6: Calculate value V_l for each hotspot region H_l.7: M_2 = M - M_18: Allocate m_l points to each H_l proportionally to V_l.9: for each hotspot region H_l do10: S_{2,l} = Greedy_Maxmin(H_l, m_l) // Stage 2: Fine Deployment11: end for 12: S = S_1 ∪ (∪_{l=1}^{L} S_{2,l})13: return S
|
3. Lightweight Sampling Path Planning for Unmanned Drone
Once the set S of M sampling points is determined, the unmanned drone needs to visit them in an order that minimizes the total travel distance (or time) to conserve energy. This is essentially a Traveling Salesman Problem (TSP) on the grid. Given the limited onboard computational capacity of a typical unmanned aerial vehicle, we employ a hybrid Greedy+2-opt algorithm for rapid and effective path planning.
The algorithm works in two phases. First, a feasible tour is quickly constructed using the Nearest Neighbor (Greedy) heuristic, starting from the UAV’s initial depot. Second, the tour is iteratively improved using the 2-opt local search, which systematically swaps pairs of edges in the tour if it reduces the total length. This hybrid approach offers a good trade-off between solution quality and computational cost, suitable for real-time planning on an unmanned drone platform.
| Algorithm 3: Greedy+2-opt Path Planning |
|---|
Input: Sampling point set S (including depot), Max iterations I_{max}Output: Optimized tour sequence π1: // Phase 1: Greedy Construction 2: Start at depot p_0; π = [p_0]; unvisited = S \ {p_0}3: while unvisited is not empty do4: current = last(π)5: Find p_{next} ∈ unvisited with min distance to current6: Append p_{next} to π; remove p_{next} from unvisited7: end while 8: Return to depot; append p_0 to π9: // Phase 2: 2-opt Local Optimization 10: improved = true; iter = 011: while improved and iter < I_{max} do12: improved = false13: for i = 1 to |π|-3 and j = i+2 to |π|-1 do14: Compute gain if edges (i,i+1) and (j,j+1) are swapped. 15: if gain > 0 then 16: Reverse the segment π[i+1 ... j] in the tour.17: improved = true18: end if 19: end for 20: iter = iter + 121: end while 22: return π
|
4. Experimental Results and Discussion
We conduct simulations to evaluate the proposed method. The monitored area is 128m×128m, discretized into a 32×32 grid (N²=1024). Three signal sources with random locations and a transmit power of 30 dBm are placed. The path loss exponent α is set to 2 (free-space). The LASSO algorithm is used for CS reconstruction. The proposed Two-Stage Maxmin (TS-Maxmin) method is compared against four baseline deployment strategies for the unmanned drone: Random, Uniform, single-stage Maxmin, and a Hybrid Uniform+Maxmin. Sampling rate is defined as M/N². Results are averaged over 100 Monte Carlo runs.
4.1 Signal Recovery Performance
The global reconstruction accuracy is measured by Root Mean Square Error (RMSE): RMSE = \sqrt{ \frac{1}{N^2} \sum_{i=1}^{N^2} (s_i - \hat{s}_i)^2 }. Figure below shows that the proposed TS-Maxmin method achieves the lowest RMSE across all sampling rates, with a particularly significant advantage at very low rates (e.g., <5%). This confirms that the hotspot-aware deployment provides more informative measurements for the CS recovery process.
4.2 Source Localization and Strength Estimation
We also evaluate the accuracy in estimating the source parameters. The Average Source Location Error (ASLE) is defined as the mean Euclidean distance between true and estimated source positions. The Average Source Strength Error (ASSE) is the mean absolute error in estimated power. The proposed method consistently outperforms the baselines, especially at low sampling rates.
| Sampling Rate | Method | RMSE (dB) | ASLE (m) | ASSE (dBm) |
|---|---|---|---|---|
| 5% (M=51) | Random | 4.82 | 12.7 | 5.1 |
| Uniform | 4.35 | 11.2 | 4.6 | |
| Maxmin | 3.98 | 10.5 | 4.3 | |
| Hybrid | 3.71 | 9.8 | 4.0 | |
| TS-Maxmin (Proposed) | 3.22 | 8.1 | 3.4 | |
| 10% (M=102) | Random | 3.45 | 9.3 | 3.9 |
| Uniform | 3.10 | 8.5 | 3.5 | |
| Maxmin | 2.88 | 7.9 | 3.3 | |
| Hybrid | 2.65 | 7.2 | 3.0 | |
| TS-Maxmin (Proposed) | 2.30 | 6.0 | 2.5 |
4.3 Sensitivity Analysis and Path Planning Cost
The parameter β, controlling the proportion of points in Stage 1, is crucial. An analysis shows that a β value around 0.3 provides the best balance between global exploration and hotspot focus for the unmanned drone across different sampling rates and source configurations. Using a β too small leads to poor hotspot identification, while a β too large degenerates the strategy to a near-single-stage Maxmin.
For path planning, we compare the Greedy+2-opt hybrid against pure Greedy and pure 2-opt algorithms. The hybrid approach reduces the tour length by approximately 15% compared to the pure Greedy solution, while its computation time is about 75% less than that of running a full 2-opt search from a random initial tour. This demonstrates its suitability for the computational constraints of an unmanned aerial vehicle.
| Algorithm | Average Tour Length (m) | Relative to Greedy | Average Runtime (ms) |
|---|---|---|---|
| Greedy (Nearest Neighbor) | 1420 | 100% | < 10 |
| 2-opt (from Random Tour) | 1250 | 88% | ~320 |
| Greedy+2-opt (Proposed) | 1210 | 85% | ~80 |
5. Conclusion
This paper presented a compressive sensing-based framework for unmanned drone spectrum mapping in environments with unknown transmitters. The key innovation is the integration of a hotspot-aware Two-Stage Maxmin deployment strategy with the CS measurement model, enabling high-quality signal reconstruction from very few measurements collected by the unmanned aerial vehicle. A lightweight Greedy+2-opt path planner ensures efficient data collection within the drone’s endurance limits. Simulation results demonstrate that the proposed method significantly outperforms traditional deployment strategies in terms of global reconstruction error, source localization accuracy, and source power estimation at low sampling rates.
The current work assumes a simplified propagation model and static sources. Future work will focus on extending the framework to more complex and dynamic electromagnetic environments. This includes investigating adaptive mechanisms to estimate path loss parameters online and exploring collaborative sampling strategies for multiple unmanned drones to enhance coverage and robustness further.