Compressed Sensing-Based Spectrum Map Sampling Using Drone Technology

In modern wireless environments, the ability to obtain accurate spatial-spectral information is critical for dynamic spectrum access, interference management, and mission planning. Spectrum maps, also known as radio environment maps (REMs), provide a comprehensive spatial representation of signal strength across a geographic area. However, constructing high-fidelity REMs in complex, infrastructure-limited regions—such as post-disaster zones or remote areas—remains a formidable challenge. Traditional fixed sensor networks are often infeasible due to cost, deployment difficulties, or physical damage. In this context, drone technology emerges as a flexible and cost-effective solution for aerial spectrum sensing. Unmanned aerial vehicles (UAVs) can be rapidly deployed, cover extensive areas, and operate where ground access is restricted. Yet, the inherent limitations of drone technology—finite battery life, limited payload capacity, and constrained onboard computation—demand lightweight yet effective sampling strategies. This work addresses the core trade-off between low-cost sampling and high-accuracy reconstruction for REM generation using drone technology. We propose a compressed sensing (CS) based framework that integrates optimized sampling point deployment with efficient path planning, enabling high-quality map reconstruction with minimal sampling effort. Our contributions include: (i) a measurement matrix formulation that directly maps UAV sampling positions to CS observations; (ii) a two-stage Maxmin deployment strategy that first ensures global coverage and then adaptively focuses on high-value hot-spot regions, all without prior source location information; (iii) a lightweight Greedy+2-opt hybrid path planning algorithm that significantly reduces the flight distance while preserving computational efficiency. Experimental results demonstrate that our approach achieves superior signal reconstruction accuracy, source localization precision, and signal strength estimation compared to conventional deployment methods, especially at low sampling rates. By leveraging drone technology, our method provides a practical, scalable, and high-performance solution for spectrum map generation in challenging environments.

The remainder of this paper is organized as follows. Section II presents the system model and problem formulation. Section III details the measurement matrix construction and the two-stage Maxmin deployment strategy. Section IV describes the lightweight path optimization. Section V provides extensive simulation results and analysis. Section VI concludes the paper with insights and future work.

System Model and Problem Formulation

We consider a typical scenario where a number of unknown radio frequency (RF) sources are distributed over a region of interest. The region is discretized into N grid cells, each representing a spatial location. The spatial-spectral map x ∈ ℝ^N denotes the received signal strength (RSS) values at all grid points. The primary objective is to reconstruct x with high fidelity using only a small number of M measurements collected by a single UAV flying through the area. This reconstruction must be achieved without any prior information about the number, positions, or transmit powers of the sources. The core challenge is threefold: (1) constructing a measurement matrix that enables reliable CS reconstruction, (2) designing a sampling strategy that balances global coverage and local precision, and (3) planning a flight path that minimizes energy consumption while visiting all selected sampling points.

Sparse representation is the cornerstone of CS. The RSS map x is assumed to be K-sparse in some basis, meaning it can be represented as a linear combination of only K basis vectors. In typical electromagnetic environments, sources are sparsely distributed, making the map itself sparse in the spatial domain when the propagation model is incorporated. We adopt a distance-based propagation model to construct the sparsifying dictionary. Given K sources with unknown positions p_i and powers P_i, the RSS at a location q from source i is expressed as:

$$P(\mathbf{q})_i = P_i \cdot \|\mathbf{q} – \mathbf{p}_i\|_2^{-\alpha}$$

where α is the path loss exponent (commonly between 2 and 4). The total RSS at q is the sum over all sources plus noise. In CS, we define the dictionary matrix Ψ ∈ ℝ^N×N where each column corresponds to the spatial influence of a potential source location. Specifically, the (i, j)-th entry of Ψ is the RSS contribution at grid point i from a unit-power source at grid point j, following the same distance decay law. Then the true map x can be expressed as x = Ψs, where s is the sparse coefficient vector containing the transmit powers at the source grid points. The goal of CS is to recover s from a small number of measurements y = Φx + n = ΦΨs + n, where Φ ∈ ℝ^M×N is the measurement matrix determined by the UAV sampling positions.

Measurement matrix construction via UAV sampling is a key enabler of drone technology for CS. The UAV collects a measurement at each of its M visited locations. Let the set of sampling positions be ℙ = {p₁, …, p_M}. We define the measurement matrix Φ such that its j-th row corresponds to the j-th sampling point: the (j, k)-th entry is 1 if the j-th measurement is taken at grid k, and 0 otherwise. In other words, each row has exactly one nonzero entry. This “point-sampling” structure is natural for UAV-based sensing. The resulting measurement matrix is a row selection matrix (or its binary version) that picks the RSS values at the visited grid points. The quality of CS reconstruction depends critically on the properties of ΦΨ. To ensure stable recovery, the sensing matrix A = ΦΨ should satisfy the restricted isometry property (RIP) or have low mutual coherence. By carefully designing the sampling positions, we can influence the structure of Φ and thus improve reconstruction performance. Our work focuses on optimizing these positions to achieve both low mutual coherence and high information capture.

We formulate the overall problem as follows: given a limited sampling budget M (much smaller than N), find a set of sampling points ℙ and a flight path that visits all points such that (1) the reconstruction error of the REM from CS is minimized, and (2) the total flight distance is as short as possible. This problem is tackled in two stages: first we determine the sampling points using a two-stage Maxmin deployment, then we plan the path using a Greedy+2-opt algorithm.

Two-Stage Maxmin Deployment for Drone-Based Sampling

The core idea of our sampling strategy is to first achieve uniform coverage of the entire region while maximizing spatial diversity, and then to concentrate additional sampling resources on promising hotspot areas (i.e., regions containing strong signal sources). This two-stage approach is motivated by the need to balance two conflicting goals: avoiding blind spots in the global map, and capturing fine-grained information near sources where signal gradients are steep.

Stage 1: Global Exploration using Greedy Maxmin. The Maxmin criterion aims to select a set of points that maximizes the minimum pairwise distance among them. This is analogous to the “maximin” optimal design in spatial statistics. Since exact optimization is NP-hard, we use a greedy algorithm that iteratively adds the point farthest from the existing set. Starting with a randomly chosen initial point, at each step we evaluate all candidate grid points. For each candidate, we compute its Euclidean distance to the nearest already selected point, and then pick the candidate with the largest such distance. This process continues until M₁ points are selected. The pseudo-code is given in Algorithm 1. This stage ensures that the entire region is uniformly covered, avoiding large gaps that would lead to poor interpolation or CS recovery. The resulting point set has low spatial correlation, which benefits the mutual coherence of the sensing matrix.

Stage 2: Hotspot Refinement. After performing the global measurements at the M₁ points, we can estimate a coarse initial map. However, due to the low sampling rate, this initial map is still inaccurate, especially near sources. To refine the map, we use the measurements from Stage 1 to identify potential hotspot regions. First, we construct a Voronoi tessellation using the Stage 1 sampling points as generators. Each cell contains points that are closer to its generator than to any other Stage-1 point. We then compute the average RSS within each cell based on the single measurement at its generator. Cells with average RSS above a threshold τ are marked as hotspot cells. Adjacent hotspot cells are merged into continuous hotspot regions. The threshold is chosen adaptively: τ = μ + 0.5σ, where μ and σ are the mean and standard deviation of the RSSs measured at Stage-1 points. This heuristic effectively separates high-signal areas from background noise.

Once hotspot regions {R₁, …, R_H} are identified, we need to allocate the remaining M₂ samples among them. The allocation should prioritize regions that are both large and have strong signals, as these contribute most to the overall reconstruction error. We define a value function for each region:

$$V(R_h) = \text{Area}(R_h) \cdot \bar{P}(R_h)$$

where Area(R_h) is the number of grid cells in region h, and P̄(R_h) is the average RSS of the measurements in that region (interpolated from Stage 1). The number of additional samples for region h is then:

$$m_h = \text{round}\left( M_2 \cdot \frac{V(R_h)}{\sum_{j=1}^{H} V(R_j)} \right)$$

Within each hotspot region, we again apply the greedy Maxmin algorithm to place m_h points, ensuring that even within a hotspot, points are spread out to maximize spatial information gain. This second stage effectively boosts the sampling density in critical areas without wasting samples on regions that are already well-characterized or irrelevant. The complete two-stage deployment is summarized in Algorithm 2.

Lightweight Path Planning Using Greedy+2-Opt

After obtaining the complete set of sampling points (Stage 1 + Stage 2), the UAV must physically visit each point to collect RSS measurements. Since the UAV’s battery endurance is limited, the flight path should be as short as possible. This is essentially a traveling salesman problem (TSP) defined on the set of M waypoints plus the starting location. Solving the TSP exactly via dynamic programming is computationally prohibitive (O(M!)). Heuristics like genetic algorithms or ant colony optimization can yield near-optimal solutions but require heavy computation and parameter tuning, which is undesirable for onboard real-time operation. To address this, we propose a lightweight hybrid algorithm that combines the speed of greedy construction with the local improvement ability of the 2-opt heuristic.

Algorithm 3: Greedy+2-opt path optimization. In the first phase, we build an initial tour using the nearest-neighbor (greedy) rule: start from the UAV’s base (or first point), repeatedly go to the closest unvisited point, and finally return to the start. This produces a feasible loop with complexity O(M²). In the second phase, we apply the 2-opt heuristic: for each pair of non-adjacent edges in the current tour, we attempt to replace them with two new edges that cross-connect the paths. If the total distance decreases, the tour is updated. We iterate this process until no improvement is found or a maximum number of iterations is reached. The 2-opt step has a worst-case complexity of O(M³) per iteration, but in practice it converges within a few iterations because the initial greedy path is already reasonable. The overall computational cost is dominated by the greedy phase, making it suitable for onboard implementation on resource-constrained drones.

We also analyze the impact of path length on total mission time. Assuming a typical drone speed of 10 m/s and step size of 4 m per grid, the total flight time for M = 80 points using Greedy+2-opt is about 278 × 4 / 10 = 111.2 seconds (plus measurement dwell time). This is well within the endurance of most consumer drones, allowing for additional operations such as data transmission or repeated missions.

Experimental Results and Discussion

To validate the effectiveness of the proposed methodology, we conducted extensive simulations using Monte Carlo experiments. The region size is set to 128 m × 128 m, discretized into 32 × 32 grid cells (4 m resolution). Three RF sources are randomly placed in each trial, each with 30 dBm transmit power and a path loss exponent α = 2. Gaussian measurement noise with standard deviation 1 dBm is added. We compare our two-stage Maxmin deployment against four baseline strategies: (1) Random: M points uniformly randomly chosen; (2) Uniform: points placed on a regular grid; (3) Maxmin: one-stage greedy Maxmin deployment using all M samples; (4) Uniform+Maxmin Hybrid: half of the samples placed uniformly, half via Maxmin (without hotspot adaptation). The reconstruction algorithm used is LASSO (ℓ₁ regularization) solved via the alternating direction method of multipliers (ADMM). We report the global root-mean-square error (RMSE), average source localization error, and average source power estimation error as functions of the sampling ratio M/N.

Global RMSE comparison. The RMSE is defined as:

$$\text{RMSE} = \sqrt{\frac{1}{N}\sum_{i=1}^N (\hat{x}_i – x_i)^2}$$

Figure 5 shows the RMSE across different sampling ratios from 1% to 20%. At very low sampling ratios (e.g., 5%), our two-stage Maxmin method achieves an RMSE of 2.1 dB, while Random and Uniform exceed 3.5 dB. Even at 10%, the proposed method maintains a 25% reduction in RMSE compared to the best baseline (one-stage Maxmin). The advantage stems from the hotspot refinement: by allocating extra samples near the sources, the reconstruction is more accurate in high-gradient areas, which dominate the total error.

Source localization accuracy. We define the average localization error as:

$$\text{LocErr} = \frac{1}{K}\sum_{i=1}^K \|\hat{\mathbf{p}}_i – \mathbf{p}_i\|_2$$

where K = 3. Figure 6 plots the localization error versus sampling ratio. At 5% sampling, the proposed method yields a localization error of 11.2 m, compared to 20.3 m for Random and 16.4 m for Uniform. At 10%, our error drops to 6.5 m versus 9.8 m for the best baseline. This dramatic improvement is due to the fact that hotspot refinement directly increases the density of measurements in the vicinity of the true source positions, enabling the CS reconstruction to precisely locate the sparse coefficients in the dictionary.

Source power estimation. The average power estimation error is:

$$\text{PowErr} = \frac{1}{K}\sum_{i=1}^K |\hat{P}_i – P_i|$$

Figure 7 shows that the proposed method consistently estimates source powers within 2 dB of the true values even at low sampling ratios, while random deployment exceeds 5 dB. The two-stage approach leverages the fact that the sparse coefficients s are more accurately recovered when the sensing matrix has low coherence; the hotspot-adapted sampling reduces mutual coherence between columns corresponding to potential source locations.

Sensitivity analysis on exploration ratio γ. We evaluated γ ∈ {0.1, 0.2, 0.3, 0.4, 0.5} at sampling ratios 5%, 10%, and 15%. Figures 8–10 (not shown due to space constraints) indicate that γ = 0.3 consistently achieves the best trade-off. A lower γ (e.g., 0.1) leaves insufficient global exploration, leading to undetected hotspots and large blind areas. A higher γ (e.g., 0.5) dedicates too many samples to global coverage, leaving too few for hotspot refinement, thus missing critical local details. This analysis confirms that the 30% exploration / 70% refinement split is near-optimal for typical source densities.

Path planning efficiency. Table 3 already summarizes the path length and computation time. Figure 11 (omitted) shows the trend: for M from 20 to 100, Greedy+2-opt maintains a path length within 2% of the exact optimal while requiring less than 0.5 seconds, compared to several seconds for genetic algorithm and prohibitive time for exact. This makes it highly suitable for drone technology where onboard real-time path generation is required immediately after sampling point determination.

Overall, the experimental results demonstrate that the proposed framework, combining two-stage Maxmin sampling with lightweight path planning, significantly outperforms conventional methods in terms of REM reconstruction accuracy, source localization, and operational efficiency. The success is attributable to the synergy between compressed sensing theory and practical drone technology: the measurement matrix derived from carefully chosen sampling positions exhibits low mutual coherence, while the path planning ensures the sampling mission is completed within the drone’s energy budget.

Conclusion and Future Work

In this paper, we have presented a novel compressed sensing based spectrum map sampling method leveraging drone technology. The key innovations include a two-stage Maxmin deployment strategy that first ensures uniform global coverage and then adaptively concentrates measurements on hotspot regions, as well as a lightweight Greedy+2-opt path planning algorithm that enables real-time, energy-efficient flight trajectories. Extensive simulations confirm that our approach achieves substantially lower reconstruction error, more accurate source localization, and better signal strength estimation compared to conventional random, uniform, or single-stage Maxmin deployments, especially under low sampling rates. The optimal allocation of sampling resources (30% exploration, 70% refinement) provides robust performance across different source configurations.

The use of drone technology in this context demonstrates its potential for spectrum monitoring in remote, hazard-prone, or infrastructure-deficient environments. By integrating CS principles with pragmatic UAV constraints, we bridge the gap between theoretical sampling theory and field-deployable systems. Future work will focus on several directions: (i) extending the approach to dynamic environments where sources may be mobile or intermittently active; (ii) incorporating multi-UAV cooperation to cover larger areas and reduce mission time; (iii) validating the method with real-world drone hardware and spectrum measurements to account for practical effects such as multipath fading and interference; (iv) exploring deep learning based reconstruction that can further improve accuracy when combined with our optimized sampling points. We believe that the synergy between compressed sensing and drone technology will continue to yield powerful solutions for spectrum awareness and electromagnetic situational understanding.

As drone technology evolves, the ability to autonomously execute intelligent sampling missions will become increasingly critical. Our contributions provide a foundational framework that can be integrated into future autonomous spectrum patrol systems, enabling real-time environmental cognition with minimal human intervention.