In recent years, the proliferation of unmanned aerial vehicles (UAVs), or drones, has been exponential. The strategic application of various drones in modern warfare has profoundly altered combat paradigms. Concurrently, in the civilian sector, the concept of the “low-altitude economy” is gaining significant traction, encompassing scenarios such as drone-based logistics, urban management, and emergency response, and is rapidly scaling up. This surge inevitably leads to a corresponding exponential increase in the pressure for low-altitude security and airspace management. Effective governance, ensuring not only that drones “fly well” but also that they are “managed well,” has become paramount. Within this context, high-precision and robust direction-finding (DF) technology for UAV drones emerges as a critical breakthrough to address low-altitude security challenges.
In the realm of radio-frequency-based DF, subspace methods like the Multiple Signal Classification (MUSIC) algorithm are classical “full-array” techniques known for high resolution. However, their performance heavily relies on a large number of snapshots to construct an accurate covariance matrix. Furthermore, operations like eigenvalue decomposition result in high computational complexity, making it difficult to meet the stringent millisecond-level response requirements for real-time UAV drone tracking. In contrast, interferometric methods estimate the Direction of Arrival (DOA) by utilizing the phase differences between signals received at multiple antenna elements. This approach requires less complex processing, offers a smaller computational footprint, and is favored in UAV drone DF applications due to its strong real-time capability. The Uniform Circular Array (UCA) offers distinct advantages with its omnidirectional coverage and symmetric structure. A single UCA can estimate the full DOA (azimuth and elevation), and its geometry provides stronger interference rejection, making it particularly suitable for tracking omnidirectionally moving targets like UAV drones. Therefore, combining UCA with interferometric principles leverages the synergistic benefits of both, forming a tripartite capability of “high-precision, full-coverage, and strong anti-interference” for UAV drone DF.
Common interferometric DF methods are primarily categorized into correlation-based and phase-based interferometers. Correlation interferometers estimate DOA by computing the peak of the cross-correlation function between the received signal and a pre-calibrated database. However, they are susceptible to significant DF errors caused by mismatches between the calibration environment and the actual operational environment, which can distort the correlation function’s amplitude. Phase interferometers, on the other hand, directly extract the instantaneous phase differences between array elements and estimate the DOA by constructing an observation model. They generally exhibit stronger real-time performance and better interference resistance.
We focus on the challenging electromagnetic environment faced by UAV drones, which often contains not only ever-present background Gaussian noise but also sporadic, high-amplitude impulsive interference generated by other communication devices, power lines, or engine ignitions. This Gaussian-impulsive mixed-noise environment poses severe challenges to traditional DF algorithms. To address the high-precision and strong robustness requirements for UAV drone direction finding under such conditions, we propose a robust DF method based on a dynamic-weighted uniform circular array. Building upon the UCA-interferometer structure, our method introduces a dynamic weighting fusion mechanism. This mechanism iteratively updates a weighting matrix by fusing the inherent resolving power determined by baseline length with a real-time statistical assessment of noise impact on each baseline. The core innovation lies in dynamically aligning the weights with the actual, fluctuating noise environment, thereby optimizing the direction estimation process to suppress strong interference effectively.
Signal and System Model for UAV Drone Detection
We consider a Uniform Circular Array (UCA) composed of $M$ antenna elements. The array structure is illustrated in a schematic, where the array lies in the $x$-$y$ plane with its center at the coordinate origin $O$ and a radius of $R$. The elements are uniformly distributed, with the $0$-th element positioned on the positive $x$-axis. The coordinate of the $m$-th element ($m = 0, 1, …, M-1$) is given by $\mathbf{p}_m = (R \cos \omega_m, R \sin \omega_m, 0)$, where $\omega_m = 2\pi m / M$.
We assume a far-field narrowband signal emitted from a UAV drone impinges on the array. Its Direction of Arrival (DOA) is defined by the azimuth angle $\alpha$ ($\alpha \in [0^\circ, 360^\circ]$) and the elevation angle $\beta$ ($\beta \in [0^\circ, 90^\circ]$). The corresponding unit direction vector $\mathbf{u}$ is:
$$\mathbf{u} = [\sin\beta \cos\alpha, \ \sin\beta \sin\alpha, \ \cos\beta]^T$$
Let the signal wavelength be $\lambda$ and its center frequency be $f_0$. Using the array center as the phase reference, the phase delay $\tau_m$ for the signal arriving at the $m$-th element, due to the path difference, is:
$$\tau_m = -\frac{2\pi}{\lambda} \mathbf{p}_m^T \mathbf{u} = -\frac{2\pi}{\lambda} R \sin\beta \cos(\alpha – \omega_m)$$
Let $s(k)$ represent the complex baseband signal envelope at the $k$-th snapshot. The signal received by the $m$-th element at the $k$-th snapshot, $X_m(k)$, can be modeled as:
$$X_m(k) = s(k) \cdot e^{-j\frac{2\pi R f_0}{c} \sin\beta \cos(\alpha – \omega_m)} + n_m(k)$$
where $c$ is the speed of light, and $n_m(k)$ is the additive noise at the $m$-th element for the $k$-th snapshot. To simulate the complex electromagnetic environment in low-altitude airspace, we model $n_m(k)$ using a Gaussian-impulsive mixture model, which effectively characterizes the superposition of bursty, strong interference and continuous background noise—a common scenario for UAV drone operations. Its probability density function is:
$$f(n) = (1-p) \mathcal{N}(0, \sigma_g^2) + p \mathcal{N}(0, \gamma \sigma_g^2)$$
where $p$ is the probability of an impulsive noise event, $\sigma_g^2$ is the background Gaussian noise variance (determined by the SNR), and $\gamma$ ($\gamma \gg 1$) is the impulsive noise strength ratio. When $p=0$, the model reverts to pure Gaussian noise. A positive $p$ models heavy-tailed impulsive noise, presenting a more rigorous test for algorithm robustness compared to a pure Gaussian model. Our goal is to accurately estimate the azimuth $\alpha$ and elevation $\beta$ of the UAV drone signal under this challenging noise model.
Proposed Dynamic-Weighted Fusion Method for DOA Estimation
Phase Difference Calculation and Observation Model
For a baseline formed by elements $A_{m_1}$ and $A_{m_2}$, the measured phase difference at snapshot $k$, after ambiguity resolution (e.g., using a long-short baseline method), is denoted as $\tilde{\Phi}_{m_1,m_2}(k)$. To mitigate noise, we average over $K$ snapshots to obtain an estimated phase difference:
$$\bar{\Phi}_{m_1,m_2} = \frac{1}{K} \sum_{k=1}^{K} \tilde{\Phi}_{m_1,m_2}(k)$$
The theoretical phase difference for this baseline, in the absence of noise, is derived as:
$$\Phi’_{m_1,m_2} = \frac{4\pi R f_0}{c} \cdot \sin\xi_{m_1,m_2} \sin\varphi_{m_1,m_2} \cdot \sin\beta \cos\alpha \ – \frac{4\pi R f_0}{c} \sin\xi_{m_1,m_2} \cos\varphi_{m_1,m_2} \cdot \sin\beta \sin\alpha$$
where $\xi_{m_1,m_2} = (m_1 – m_2)\pi / M$ and $\varphi_{m_1,m_2} = (m_1 + m_2)\pi / M$.
For $N$ selected baselines, we can construct a linear observation model. Let the phase difference vector be $\boldsymbol{\Phi} = [\bar{\Phi}_1, \bar{\Phi}_2, …, \bar{\Phi}_N]^T$ and the direction vector be $\mathbf{g} = [\sin\beta\cos\alpha, \ \sin\beta\sin\alpha]^T$. The model is:
$$\boldsymbol{\Phi} = \mathbf{H} \mathbf{g}$$
where the $n$-th row of the $N \times 2$ observation matrix $\mathbf{H}$ is:
$$\mathbf{H}_n = \frac{4\pi R f_0}{c} \left[ \sin\xi_n \sin\varphi_n, \ -\sin\xi_n \cos\varphi_n \right]$$
Generalized Least Squares and Iterative Weighting
The classic Generalized Least Squares (GLS) solution aims to minimize the Euclidean distance between measured and theoretical phase differences:
$$\hat{\mathbf{g}} = \arg\min_{\mathbf{g}} ||\boldsymbol{\Phi} – \mathbf{H}\mathbf{g}||_2^2 = (\mathbf{H}^T\mathbf{H})^{-1}\mathbf{H}^T\boldsymbol{\Phi}$$
The DOA estimates are then obtained as:
$$\hat{\alpha} = \arg(\hat{g}_1 + j\hat{g}_2), \quad \hat{\beta} = \arcsin\left(\sqrt{\hat{g}_1^2 + \hat{g}_2^2}\right)$$
where $\hat{g}_1$ and $\hat{g}_2$ are the elements of $\hat{\mathbf{g}}$. While computationally simple, GLS treats all baseline measurements equally, ignoring their varying reliability under noise.
Iterative Weighted Least Squares (IWLS) introduces a diagonal weight matrix $\mathbf{W} = \text{diag}(w_1, w_2, …, w_N)$ to optimize the estimation:
$$\hat{\mathbf{g}} = \arg\min_{\mathbf{g}} ||\mathbf{W}^{1/2}(\boldsymbol{\Phi} – \mathbf{H}\mathbf{g})||_2^2 = (\mathbf{H}^T\mathbf{W}\mathbf{H})^{-1}\mathbf{H}^T\mathbf{W}\boldsymbol{\Phi}$$
The design of $\mathbf{W}$ is crucial. Existing methods often use weights based on the partial derivatives of the phase difference, which offer some improvement but lack a direct, dynamic link to the instantaneous statistical characteristics of the noise affecting each baseline, especially under impulsive interference common in UAV drone environments.
Core Innovation: Dynamic Weighting Fusion Strategy
We propose a dynamic weighting fusion strategy. Its core idea is that the weight for each baseline should reflect its real-time credibility, determined by two dimensions: 1) its inherent resolving power dictated by baseline length, and 2) the degree of deviation of its current measurements from the theoretical values predicted by the latest DOA estimate, which quantifies the impact of instantaneous noise.
Let $\tilde{\alpha}^{(i)}$ and $\tilde{\beta}^{(i)}$ denote the azimuth and elevation estimates from the previous ($i$-th) iteration. We first define the residual variance $\sigma_i^2$ for the $n$-th baseline to quantify its measurement deviation:
$$\sigma_i^2 = \frac{1}{K-1} \sum_{k=1}^{K} \left( \tilde{\Phi}_n(k) – \Phi’_n(\tilde{\alpha}^{(i)}, \tilde{\beta}^{(i)}) \right)^2$$
where $\Phi’_n(\tilde{\alpha}^{(i)}, \tilde{\beta}^{(i)})$ is the theoretical phase difference calculated using the previous iteration’s DOA estimates. A larger $\sigma_i^2$ indicates that the baseline’s measurements are more severely corrupted by noise in the current environment, implying lower credibility.
We then construct the dynamic weight by fusing this noise-impact metric with the baseline’s inherent geometric resolving power:
$$w_n = \begin{cases}
\tilde{d}_n, & i = 1 \\
\dfrac{\tilde{d}_n}{\sigma_i^2 + \epsilon}, & i > 1
\end{cases}$$
where:
- $\tilde{d}_n = d_n / d_{\text{max}}$ is the normalized length of the $n$-th baseline.
- $d_n = 2R|\sin\xi_n|$ is the physical length of the baseline.
- $d_{\text{max}}$ is the maximum length among all selected baselines.
- $\epsilon$ is a small positive constant to prevent division by zero.
This design ensures that during the iterative process, baselines with both high inherent resolution (longer length) and low observed noise contamination (small residual variance) are assigned greater weight. This effectively suppresses the influence of unreliable measurements caused by strong impulsive or Gaussian noise, thereby enhancing the algorithm’s robustness for UAV drone DF.
The iterative process continues until either a maximum iteration count $I_{\text{max}}$ is reached or the change in DOA estimates between iterations falls below a threshold $\delta$: $|\tilde{\alpha} – \hat{\alpha}| + |\tilde{\beta} – \hat{\beta}| \leq \delta$. This dual-stop criterion balances convergence and real-time performance.
Experimental Analysis and Performance Evaluation
We conducted comprehensive simulations to validate the performance of our proposed dynamic-weighted method under Gaussian-impulsive mixed noise. The simulation setup is summarized in the following table, modeling a scenario for detecting a commercial UAV drone like the DJI M300.
| Category | Parameter | Value / Range | Description |
|---|---|---|---|
| Array Configuration | Number of Elements ($M$) | 15 | Uniformly distributed on a circle. |
| Radius ($R$) | $\lambda / [4\sin(\pi/M)]$ | Ensures shortest baseline ≤ $\lambda/2$ to avoid spatial aliasing for the selected baselines. | |
| UAV Drone Signal | Center Frequency ($f_0$) | 5.8 GHz | Typical for commercial drones (e.g., DJI). |
| Bandwidth | 10 MHz | Narrowband assumption. | |
| Number of Snapshots ($K$) | 500 | For phase averaging. | |
| Noise Model | Impulsive Probability ($p$) | 0 to 0.4 | Varied for robustness analysis. |
| Impulsive Strength ($\gamma$) | 1000 | Fixed high ratio for strong impulses. | |
| Signal-to-Noise Ratio (SNR) | -5 dB to 25 dB | Varied for sensitivity analysis. | |
| UAV Drone DOA | Azimuth ($\alpha$) | 150° | Arbitrary direction. |
| Elevation ($\beta$) | 60° | Typical low-altitude climb/descent angle. | |
| Algorithm Parameters | Max Iterations ($I_{\text{max}}$) | 10 | Ensures termination. |
| Convergence Threshold ($\delta$) | 0.01° | Balances accuracy and speed. | |
| Evaluation | Monte Carlo Runs | 1000 | For statistical stability. |
We compare our method against the standard Generalized Least Squares (GLS) and an existing Iterative Weighted (IW) method. Performance is primarily measured using the Root Mean Square Error (RMSE) of the DOA estimates. The simulation platform is a standard laptop with a 2.20 GHz Intel Core i9 processor and 32 GB RAM.
Anti-Noise Performance and RMSE Analysis
First, we evaluate performance under a fixed impulsive noise probability ($p=0.2$) while varying the SNR from -5 dB to 25 dB. The results, shown in a comparative plot, clearly demonstrate the superiority of our method. As SNR increases (simulating a UAV drone coming closer or interference weakening), all methods improve, but our dynamic-weighted method maintains the highest accuracy across the entire range.
Critically, in low-SNR conditions corresponding to distant or heavily jammed UAV drone scenarios (SNR < 15 dB), our method shows a dramatic advantage. For instance, at SNR = 5 dB, our method achieves an RMSE of 0.96°, while GLS and the existing IW method yield 13.04° and 8.34°, respectively. This represents an improvement of approximately 92.6% over GLS and 88.5% over the existing IW method. In low SNR, all baseline measurements are contaminated, but not equally. Our method’s dynamic weighting, based on real-time residual variance, successfully identifies and prioritizes the “relatively reliable” measurements, effectively extracting the most valid directional information from the noisy data. This capability is vital for detecting weak UAV drone signals at the edge of sensor range.
Robustness to Impulsive Noise
Next, we fix the SNR at a moderate 10 dB and vary the impulsive noise probability $p$ from 0 (pure Gaussian) to 0.4 (extreme, frequent impulsive interference). This test simulates transitioning from benign to hostile electromagnetic environments. The results are plotted as RMSE versus $p$.
The traditional GLS method degrades linearly and severely as $p$ increases. The existing IW method shows some improvement at low $p$ but fails significantly when $p \geq 0.3$, as its static geometric weighting cannot adapt to the random shocks of strong impulses. In stark contrast, our proposed method exhibits exceptional stability. The RMSE curve grows very gradually, from 0.18° under pure Gaussian noise to only 1.61° under the extreme $p=0.4$ condition. Remarkably, the precision of our method under severe impulsive noise (1.61°) is still superior to the best performance of GLS (1.83°) under no impulsive noise at all ($p=0$). This underscores the exceptional robustness of our dynamic weighting fusion strategy for UAV drone DF in unpredictable, interference-prone low-altitude settings.
Computational Efficiency and Real-Time Feasibility
Real-time tracking of UAV drones necessitates efficient algorithms. We evaluate computational efficiency by comparing the average number of iterations and the total execution time required for convergence.
A plot of average iteration counts versus SNR shows that both iterative methods require near the maximum (10) iterations at very low SNRs (≤ 0 dB). As SNR improves, the iteration count drops. The existing IW method converges slightly faster in terms of iterations, stabilizing around 2-3 at high SNR, while our method stabilizes around 3.5 iterations. This modest increase in iteration count is a reasonable trade-off for the significant gain in anti-noise performance and robustness.
More importantly, the absolute execution time remains highly efficient. The following table presents the average running time for the three methods at different SNR levels.
| Algorithm | SNR = -5 dB | SNR = 0 dB | SNR = 5 dB | SNR = 10 dB |
|---|---|---|---|---|
| Generalized Least Squares (GLS) | 0.051 ms | 0.024 ms | 0.022 ms | 0.021 ms |
| Existing Iterative Weighted (IW) | 0.258 ms | 0.143 ms | 0.103 ms | 0.083 ms |
| Proposed Dynamic-Weighted Method | 0.464 ms | 0.295 ms | 0.208 ms | 0.178 ms |
Although our method has the longest runtime due to its iterative weight calculation, the absolute time is still well below 0.5 ms even in the worst-case low-SNR scenario. This millisecond-level processing speed fully satisfies the real-time requirements for tracking maneuvering UAV drones. It provides a powerful balance between high-precision, robust estimation and computational efficiency, offering reliable data support for subsequent system functions like UAV drone identification, tracking, and countermeasures in a low-altitude defense system.
Conclusion
We have presented a robust direction-finding method for UAV drones based on a dynamic-weighted uniform circular array. The method addresses the critical challenge of maintaining DF accuracy in complex electromagnetic environments characterized by Gaussian-impulsive mixed noise, which is common in low-altitude airspace. By introducing a dynamic weighting fusion mechanism that iteratively updates weights based on both the inherent geometric resolving power of baselines and a real-time statistical measure of noise impact (residual variance), our algorithm effectively suppresses strong interference. Experimental results demonstrate that compared to the Generalized Least Squares method and an existing iterative weighted technique, our proposed method achieves significantly higher estimation precision, particularly under low SNR conditions. It exhibits outstanding robustness against increasing levels of impulsive noise, with performance degrading only gradually. While requiring slightly more computation due to its iterative nature, the overall processing time remains within sub-millisecond bounds, ensuring feasibility for real-time UAV drone monitoring and tracking applications. The proposed approach effectively balances the demands for high precision, strong robustness, and real-time performance, providing a reliable technical foundation for next-generation low-altitude security systems. Future work may focus on integrating bias-compensation techniques to address potential systematic errors and extending the framework to handle multiple simultaneous UAV drone signals.