The proliferation of unmanned drone technology presents significant challenges for security and airspace management, particularly in complex urban environments. The passive detection of small, low-altitude, slow-moving unmanned drones remains a critical problem. Traditional radar-based systems struggle with the low Radar Cross-Section (RCS) of such targets, while active radio frequency (RF) detection methods depend on the target emitting signals and are susceptible to interference. This paper proposes a novel passive detection framework that leverages the inherent statistical non-Gaussianity and non-stationarity induced by a moving unmanned drone on the ambient electromagnetic channel. By analyzing higher-order statistical features of the channel impulse response, specifically the fourth-order cumulant, our method can detect the weak perturbation caused by a unmanned drone even under low signal-to-noise ratio (SNR) conditions.
The core premise is that a moving unmanned drone acts as a dynamic scatterer, subtly altering the multipath propagation structure of existing radio frequency signals (e.g., from cellular base stations or broadcast towers). These alterations are embedded within the received signal as weak, non-Gaussian components superimposed on a strong, quasi-stationary background consisting of the direct path and static multipath. To isolate this weak signature, we employ a two-stage processing chain. First, we construct a Hankel matrix from the slow-time sequence of received signal samples and apply Singular Value Decomposition (SVD). The dominant singular values, corresponding to the powerful static background, are truncated. The residual matrix is then reconstructed to obtain a signal where the background clutter is suppressed, and the dynamic components (including the unmanned drone reflection and noise) are relatively enhanced.
Second, to further discriminate the unmanned drone‘s signal from the remaining noise, we exploit the properties of higher-order statistics. For Gaussian processes, cumulants of order greater than two are theoretically zero. Since the ambient noise is often well-modeled as additive white Gaussian noise (AWGN), calculating the fourth-order cumulant of the reconstructed signal effectively suppresses the noise’s contribution while highlighting the non-Gaussian characteristics introduced by the unmanned drone. The temporal variance of this fourth-order cumulant serves as a sensitive detection statistic, exhibiting a systematic elevation and fluctuation when a unmanned drone is present compared to a normal, drone-free environment.
1. System and Signal Model
We consider a passive detection scenario with a single, non-cooperative transmitter (Tx) and a dedicated receiver (Rx). The transmitter emits a constant-power signal, $s(t)$, which can represent a generic communication signal like a Direct Sequence Spread Spectrum (DSSS) waveform. The receiver aims to detect the presence of an unmanned drone by monitoring changes in the propagation channel between the fixed Tx and Rx. The unmanned drone is modeled as a point scatterer moving through this environment.
The time-varying channel impulse response, $h(t, \tau)$, accounting for propagation paths up to the second-order reflection, is given by:
$$
h(t, \tau) = \sum_{i=1}^{L_1} \alpha_{1,i}(t) e^{j\phi_{1,i}(t)} \delta(\tau – \tau_{1,i}(t)) + \sum_{i=1}^{L_2} \alpha_{2,i}(t) e^{j\phi_{2,i}(t)} \delta(\tau – \tau_{2,i}(t))
$$
where:
- $L_1$ and $L_2$ are the number of first-order and second-order reflection paths, respectively.
- $\alpha_{k,i}(t)$, $\phi_{k,i}(t)$, and $\tau_{k,i}(t)$ are the time-varying amplitude, phase, and delay of the $i$-th path of order $k$ ($k=1,2$).
- Paths include reflections from static objects (buildings, ground) and the dynamic unmanned drone.
The received signal $y(t)$ is the convolution of the transmitted signal with the channel impulse response, plus additive white Gaussian noise $w(t)$:
$$
y(t) = s(t) * h(t) + w(t)
$$
For the $m$-th time frame (slow-time index), the discrete-time received signal at fast-time index $n$ can be decomposed as:
$$
y_m[n] = y_{b,m}[n] + y_{u,m}[n] + w_m[n]
$$
Here, $y_{b,m}[n]$ represents the composite background signal from static paths, $y_{u,m}[n]$ is the weak signal component reflected from the unmanned drone, and $w_m[n]$ is the AWGN. The primary challenge is to extract $y_{u,m}[n]$ from the dominant $y_{b,m}[n]$ and $w_m[n]$.
The received power for different path types follows standard propagation models, as summarized below:
| Path Type | Power Model | Parameters |
|---|---|---|
| Line-of-Sight (LOS) | $P_{los} = \dfrac{P_t G_t G_r \lambda^2}{(4\pi R)^2}$ | $P_t$: Tx power, $G_t/G_r$: Antenna gains, $\lambda$: Wavelength, $R$: Tx-Rx distance. |
| First-Order Reflection | $P_{ref} = \dfrac{P_t G_t G_r \lambda^2 \sigma}{(4\pi)^3 D_1^2 D_2^2}$ | $\sigma$: RCS of scatterer, $D_1$, $D_2$: Tx-scatterer and scatterer-Rx distances. |
| Second-Order Reflection | $P_{double} = \dfrac{P_t G_t G_r \lambda^2 \sigma_1 \sigma_2}{(4\pi)^3 D_1^2 D_2^2 D_3^2}$ | $\sigma_1$, $\sigma_2$: RCS of two scatterers, $D_1$, $D_2$, $D_3$: Segmented distances. |
The signal from the unmanned drone, especially as a second-order reflector, is typically very weak, i.e., $P_{double}^{drone} \ll P_{los}, P_{ref}^{static}$, making direct detection infeasible without sophisticated processing.
2. Background Signal Suppression via Hankel-SVD
The received signal is organized into a two-dimensional matrix $\mathbf{Y} \in \mathbb{C}^{N \times M}$, where $N$ is the number of fast-time samples per frame and $M$ is the number of slow-time frames (snapshots). The $m$-th column is the $m$-th frame’s signal.
For a specific fast-time sample index $n$, we extract the slow-time sequence across all $M$ frames: $\mathbf{y}_n = [y_1[n], y_2[n], \dots, y_M[n]]^T$. This sequence contains the temporal evolution of the signal at that specific delay, which is influenced by the motion of the unmanned drone. A Hankel matrix $\mathbf{Y}_n$ is constructed from $\mathbf{y}_n$:
$$
\mathbf{Y}_n = \begin{bmatrix}
y_1[n] & y_2[n] & \cdots & y_P[n] \\
y_2[n] & y_3[n] & \cdots & y_{P+1}[n] \\
\vdots & \vdots & \ddots & \vdots \\
y_L[n] & y_{L+1}[n] & \cdots & y_M[n]
\end{bmatrix}
$$
where $L \approx M/2$ and $P = M – L + 1$. This matrix embeds the temporal structure of the signal. Performing Singular Value Decomposition (SVD) on $\mathbf{Y}_n$ yields:
$$
\mathbf{Y}_n = \mathbf{U} \mathbf{\Sigma} \mathbf{V}^T = \sum_{i=1}^{r} \sigma_i \mathbf{u}_i \mathbf{v}_i^T
$$
where $\sigma_1 \ge \sigma_2 \ge \dots \ge \sigma_r > 0$ are the singular values, and $\mathbf{u}_i$, $\mathbf{v}_i$ are the corresponding left and right singular vectors. The key observation is that the large singular values ($i=1, \dots, k$) predominantly correspond to the highly correlated, strong static background signal $y_b$. The smaller singular values correspond to weaker dynamic components (including the unmanned drone signal $y_u$) and noise.
To suppress the background, we compute a residual matrix by reconstructing $\mathbf{Y}_n$ using only the singular values beyond an adaptively determined index $k$:
$$
\hat{\mathbf{Y}}_n = \sum_{i=k+1}^{r} \sigma_i \mathbf{u}_i \mathbf{v}_i^T
$$
The threshold index $k$ is selected based on the Singular Value Difference Spectrum (SVDS). Let $d_i = \sigma_i – \sigma_{i+1}$. The first significant local peak in $\{d_i\}$ indicates the transition from dominant background components to weaker dynamic/noise components. We define $k$ as:
$$
k = \min \left\{ i \, \middle| \, d_i > d_{i-1}, \, d_i \ge d_{i+1}, \, d_i \ge \mu_d + \eta \sigma_d \right\}
$$
where $\mu_d$ and $\sigma_d$ are the mean and standard deviation of the SVDS, and $\eta$ is a significance factor (e.g., $\eta=1$). Finally, the residual matrix $\hat{\mathbf{Y}}_n$ is transformed back into a one-dimensional slow-time sequence $\hat{y}_n[m]$ using diagonal averaging. Repeating this process for all $n$ and recombining yields the background-suppressed signal matrix $\hat{\mathbf{Y}}$.
3. Drone Detection via Fourth-Order Cumulant Variance
The signal $\hat{\mathbf{Y}}$ now contains the suppressed dynamic components. However, the residual noise can still mask the very weak unmanned drone signal. To enhance the signal-to-noise ratio for detection, we leverage the properties of higher-order cumulants. The $k$-th order cumulant of a zero-mean random process is zero for $k>2$ if the process is Gaussian. Therefore, the fourth-order cumulant is an excellent tool for suppressing Gaussian noise while preserving the non-Gaussian characteristics of the unmanned drone‘s reflected signal.
For a discrete, zero-mean signal $x[n]$, the fourth-order cumulant is defined as:
$$
\begin{aligned}
C_{4x}(\tau_1, \tau_2, \tau_3) &= \text{cum}[x[n], x[n+\tau_1], x[n+\tau_2], x[n+\tau_3]] \\
&= \mathbb{E}[x[n]x[n+\tau_1]x[n+\tau_2]x[n+\tau_3]] \\
&\quad – \mathbb{E}[x[n]x[n+\tau_1]]\mathbb{E}[x[n+\tau_2]x[n+\tau_3]] \\
&\quad – \mathbb{E}[x[n]x[n+\tau_2]]\mathbb{E}[x[n+\tau_1]x[n+\tau_3]] \\
&\quad – \mathbb{E}[x[n]x[n+\tau_3]]\mathbb{E}[x[n+\tau_1]x[n+\tau_2]]
\end{aligned}
$$
Computing the full 3D cumulant $C_{4x}(\tau_1, \tau_2, \tau_3)$ is computationally intensive ($\mathcal{O}(NL^3)$). For efficient real-time detection, we compute a 2D slice by fixing one lag, e.g., $\tau_2 = 0$:
$$
\begin{aligned}
C_{4x}(\tau_1, 0, \tau_3) &= \mathbb{E}[x[n]x[n+\tau_1]x[n]x[n+\tau_3]] \\
&\quad – \mathbb{E}[x[n]x[n+\tau_1]]\mathbb{E}[x[n]x[n+\tau_3]] \\
&\quad – \mathbb{E}[x[n]^2]\mathbb{E}[x[n+\tau_1]x[n+\tau_3]] \\
&\quad – \mathbb{E}[x[n]x[n+\tau_3]]\mathbb{E}[x[n+\tau_1]x[n]]
\end{aligned}
$$
This reduces complexity to $\mathcal{O}(NL^2)$. For the $m$-th time frame’s background-suppressed signal $\hat{y}_m[n]$ of length $N$, we estimate $C_{4x}^{(m)}(\tau_1, 0, \tau_3)$ for $\tau_1, \tau_3 \in \{-L, \dots, 0, \dots, L\}$. We arrange these values into a matrix $\mathbf{S}_m \in \mathbb{R}^{(2L+1) \times (2L+1)}$:
$$
\mathbf{S}_m(i, j) = C_{4x}^{(m)}(i-L-1, 0, j-L-1)
$$
The presence of a unmanned drone introduces a time-varying, non-Gaussian component, causing the elements of $\mathbf{S}_m$ to fluctuate more significantly across frames $m$ compared to a drone-free scenario. To capture this temporal non-stationarity, we compute the variance of the elements within each matrix $\mathbf{S}_m$ as our detection statistic:
$$
\sigma_m^2 = \text{Var}(\mathbf{S}_m) = \frac{1}{(2L+1)^2} \sum_{i=1}^{2L+1} \sum_{j=1}^{2L+1} \left( S_m(i, j) – \bar{S}_m \right)^2
$$
where $\bar{S}_m$ is the mean of all elements in $\mathbf{S}_m$. In a normal environment, $\sigma_m^2$ remains at a low baseline level determined by estimation variance. When an unmanned drone is present, $\sigma_m^2$ exhibits a systematic elevation and distinct temporal fluctuations. A detection decision is made by comparing a moving average or specific features of the $\sigma_m^2$ sequence against a threshold derived from background statistics.
4. Simulation Analysis and Performance Evaluation
A comprehensive simulation was conducted in MATLAB to validate the proposed method. A 500m x 500m urban area was modeled with multiple static scatterers. A transmitter and receiver were placed 500m apart at 60m height. An unmanned drone starts at (200m, 100m, 60m) and flies along the x-axis at 5 m/s. Key simulation parameters are listed below:
| Parameter | Value |
|---|---|
| Simulation Area | 500 m × 500 m |
| Tx/Rx Height | 60 m |
| Tx-Rx Distance | 500 m |
| Drone Initial Position | (200, 100, 60) m |
| Drone Speed | 5 m/s |
| Total Observation Time ($T_{total}$) | 15 s |
| Slow-Time Interval ($\Delta t$) | 0.01 s |
| Signal Bandwidth ($B$) | 100 MHz |
| Time Resolution ($T_c = 1/B$) | 10 ns |
The received signal for both normal (drone-absent) and intruded (drone-present) scenarios was generated. The Hankel-SVD processing successfully suppressed the strong direct path and dominant static reflections, as evidenced by the time-delay power profiles before and after processing. The weak returns from the unmanned drone became more discernible in the residual signal.
The efficacy of the fourth-order cumulant variance $\sigma_m^2$ as a detection statistic is demonstrated in the following figure. The temporal trace of $\sigma_m^2$ shows a clear and sustained elevation when the unmanned drone is present, compared to the low, stable baseline in the normal scenario. The increased fluctuations correspond to the dynamic perturbation of the channel by the moving unmanned drone.
The performance was quantified using detection probability ($P_d$) versus SNR and drone altitude via Monte Carlo simulations (300 trials). The detection threshold was set as $\mu_{bg} + 3\sigma_{bg}$, where $\mu_{bg}$ and $\sigma_{bg}$ are the mean and standard deviation of $\sigma_m^2$ in the normal scenario. A detection was declared if the $\sigma_m^2$ sequence exceeded this threshold for a significant portion of the observation window.
| Drone Altitude (m) | $P_d$ @ SNR=5 dB | $P_d$ @ SNR=10 dB | $P_d$ @ SNR=15 dB | $P_d$ @ SNR=20 dB |
|---|---|---|---|---|
| 45 | 0.52 | 0.61 | 0.68 | 0.73 |
| 50 | 0.58 | 0.66 | 0.72 | 0.78 |
| 55 | 0.63 | 0.72 | 0.79 | 0.83 |
| 60 | 0.71 | 0.78 | 0.82 | 0.85 |
| 65 | 0.70 | 0.77 | 0.81 | 0.84 |
| 70 | 0.69 | 0.76 | 0.80 | 0.83 |
| 75 | 0.68 | 0.75 | 0.79 | 0.82 |
The results show robust performance across SNR levels, with $P_d$ reaching 0.85 at 20 dB SNR when the unmanned drone flies at 60m (equal to Tx/Rx height). Performance degrades slightly at lower altitudes due to increased ground clutter and shadowing but remains effective. The method’s robustness to noise is attributed to the Gaussian-suppressing property of the fourth-order cumulant.
We also compared our method (Hankel-SVD + Fourth-Order Cumulant Variance) against two baseline methods applied to the same background-suppressed signal: Energy Detection and Second-Order Variance Detection. The comparison was conducted under varying SNR and with an increasing number of multipath scatterers.
| Method | Avg. $P_d$ (SNR=5-20 dB) | Robustness to Dense Multipath | Computational Complexity |
|---|---|---|---|
| Proposed Method | 0.74 | High (Cumulant suppresses noise) | Moderate ($\mathcal{O}(MNL^2)$) |
| Energy Detection | 0.58 | Low (Sensitive to residual clutter energy) | Low ($\mathcal{O}(MN)$) |
| Variance Detection | 0.62 | Medium | Low ($\mathcal{O}(MN)$) |
The proposed method consistently outperforms the baselines, especially in low-SNR and dense multipath conditions, confirming the advantage of using higher-order statistics for detecting the weak, non-Gaussian signature of an unmanned drone.
5. Conclusion
This paper presented a novel passive detection method for low-altitude, slow-moving unmanned drones in complex urban environments. The method capitalizes on the subtle perturbations a moving unmanned drone induces in the statistical properties of an ambient wireless channel. By first applying a Hankel matrix-based SVD to suppress the powerful static background clutter and then employing the variance of the fourth-order cumulant as a detection statistic, the method effectively isolates and amplifies the weak, non-Gaussian signature of the unmanned drone while inherently suppressing Gaussian noise.
Simulation results validated the method’s effectiveness, showing a clear discrimination between normal and drone-present scenarios through a significant elevation in the cumulant variance metric. The method demonstrated robust detection probability across various SNRs and drone altitudes, outperforming conventional energy-based or second-order statistical detectors. This framework provides a promising, transmitter-agnostic approach for the passive monitoring of airspace against unauthorized unmanned drone incursions, addressing a critical need in modern urban security. Future work will focus on real-time implementation, multi-drone discrimination, and testing with real-world RF signals.