In recent years, unmanned drones have been widely deployed and applied in civilian and military fields such as reconnaissance, surveillance, aerial photography, and disaster relief due to their low cost, high flexibility, and multi-mission adaptability. However, with the proliferation of unmanned drone applications, incidents of “black flights” and loss of control occur frequently, posing serious threats to airport takeoff and landing safety, protection of critical infrastructure, and public order, and even leading to major safety accidents. Therefore, the development of efficient and reliable identification technologies for “low, slow, and small” unmanned drone targets has become an urgent need for airspace regulation and security defense.
In existing research, radar detection, as an active sensing technology, has become a primary means for identifying “low, slow, and small” unmanned drones. People utilize radar’s capabilities, such as being unaffected by lighting and weather conditions and enabling all-weather continuous monitoring, to detect, track, and identify low-altitude small targets. However, due to the extremely small radar cross-section (RCS) of such targets, their echo signals are weak and easily淹没 in strong ground clutter and noise. Meanwhile, the lack of significant differentiating features among different unmanned drones makes traditional identification methods ineffective. Thus, there is a pressing need to develop a robust method to distinguish between different unmanned drone targets.
Since the rotors are relatively small components compared to the main body of the unmanned drone, the rotation of the rotors is referred to as micro-motion. The phenomenon of echo frequency shift caused by the motion of small components is also known as the micro-Doppler effect. Micro-Doppler characteristics reflect the periodic motion of the rotor blades. Different models of unmanned drones have varying rotor speeds and blade lengths due to functional and payload requirements, making it possible to estimate rotor parameters by analyzing micro-motion features and thereby identify different models of unmanned drones. Due to the insufficient resolution of low-frequency band radars, it is difficult to effectively and completely capture the micro-motion features of unmanned drone rotors in low signal-to-noise ratio environments. In contrast, millimeter-wave radar can not only effectively distinguish scattering centers such as unmanned drone rotors and fuselage due to its short wavelength characteristics, enabling high-resolution capture of rotor micro-motion features, but its inherent signal gain capability also significantly enhances echo signal strength. Together, these provide key technical support for stably detecting weak reflections of micro-motion features of unmanned drone rotors in low-altitude environments.
The process of identification based on micro-motion features mainly involves two aspects: feature extraction and classification algorithm research. The feature extraction stage involves time-frequency analysis methods, such as the commonly used Short Time Fourier Transform (STFT) method, Wavelet Transform (WT) method, etc. These methods convert noisy time-domain echo signals into the time-frequency domain for analysis. In the time-frequency diagram, the flashing frequency generated by the periodic rotation of the rotors and the maximum Doppler frequency generated by the tangential velocity of the blade tips can be extracted. For such methods, interpreting micro-motion features, such as Doppler shift and rotational frequency, are manually extracted, and the quality of extraction depends on the time-frequency resolution. However, the STFT method is limited by the Heisenberg uncertainty principle and cannot兼顾 time and frequency resolution under fixed window functions. Inappropriate window function selection can lead to spectral leakage. The WT method may suffer from cross-term interference when processing rotor echo signals with dense high-frequency components, leading to suboptimal analysis. To address such issues, time-frequency post-processing methods that refine time-frequency processing have been proposed, including Synchrosqueezing Transform (SST), Synchroextracting Transform (SET), Multisynchrosqueezing Transform (MSST), and cepstrum estimation, among others. These methods optimize the results of time-frequency analysis to compensate for the shortcomings of traditional methods. However, such time-frequency post-processing techniques are limited by the quality of time-frequency processing results. Under strong noise backgrounds, due to weak target signals and the presence of cross-term interference in multi-rotor echo signals, the energy redistribution is disturbed, affecting algorithm performance and even leading to misjudgments. Therefore, how to output high-quality time-frequency processing data and perform effective time-frequency post-processing is a critical issue that needs to be addressed.
Regarding classification algorithm research, early studies typically used common machine learning algorithms such as random forests and support vector machines. In deep learning, commonly used network architectures include AlexNet and its variants, VGG16, and ResNet18, among others. More recent methods go further by fusing multiple features extracted from K-band radar of unmanned drones and inputting them into neural networks to achieve higher recognition accuracy. Additionally, some studies input multi-band, multi-angle time-frequency spectrograms into Long Short-Term Memory (LSTM) networks with added attention mechanisms, which also significantly improves target recognition accuracy. Some research also focuses on using deep learning networks to process raw echo signals, performing deeper feature extraction end-to-end. Although these methods significantly improve recognition accuracy, the models themselves rely on large amounts of data for training and have high requirements for input data quality, making them unsuitable for real-time processing needs.
In addition to using micro-motion features of unmanned drones for identification, there are currently methods for unmanned drone identification based on features such as RCS, polarization, high-resolution range profiles, and motion trajectories. However, due to their generally insufficient robustness in low-altitude complex environments, their application effectiveness is limited.
From the previous discussion, extensive research has verified the effectiveness of using micro-motion features of unmanned drone rotors to identify different models of unmanned drones. This paper, based on the motion model of unmanned drones, constructs an echo signal model of unmanned drone rotors based on millimeter-wave radar, utilizes the high-resolution characteristics of millimeter-wave radar to capture micro-motion features of unmanned drone rotors at low altitudes, and establishes inverse equations for rotor physical parameters to analyze their speed, blade length, etc. It should be noted that the extraction accuracy of micro-Doppler feature values in the inverse equations directly affects the estimation of blade length.
Aiming at the problems of severe spectral leakage and significant cross-term interference in traditional time-frequency analysis, which make it difficult to extract micro-motion features of rotors, this paper proposes a high-precision feature extraction method based on TELSET. This method introduces a pre-energy threshold filter, sets a threshold based on the global background energy distribution in the time-frequency domain, filters out low-energy noise and artifacts, and achieves effective “purification” of the time-frequency diagram. Then, based on the differences in echo energy of multiple rotors due to different spatial positions and radar cross-sections, it innovatively proposes local maximum synchronous extraction along the time axis. On the filtered time-frequency diagram, Gaussian smoothing and local extremum detection are performed for each frequency slice to accurately capture the flashing frequency trajectory of the rotors, constructing a highly sparse TELSET time-frequency representation, effectively suppressing cross-term interference while sharpening micro-Doppler ridges. Based on the extracted high-quality time-frequency ridges, high-precision extraction of micro-Doppler feature values can be achieved, and combined with inverse equations, blade length can be estimated, ultimately supporting effective identification of different models of unmanned drones.
Rotor Echo Signal Model
Currently, most small unmanned drones are similar in overall configuration, generally consisting of a main fuselage and rotors, among other components. However, different models have differences in specific parameters such as rotor speed, number of blades, and blade length. Different kinematic characteristics exhibit unique modulation characteristics in the echoes. Based on the specificity of micro-motion features of different unmanned drone rotors, this paper constructs a highly robust rotor echo model, providing a theoretical basis for subsequent signal analysis and micro-motion feature extraction.
In the coordinate system, the radar is located at the origin $O$, the center coordinate of the unmanned drone is $O_u$, and the rotation axis center of its $n$-th rotor is denoted as $O_n$. The azimuth and elevation angles of the unmanned drone relative to the radar are $\alpha$ and $\beta$, respectively, and the distance between the radar and the center of the unmanned drone is $R$. The main structure of the unmanned drone connects the centers of the four rotors, and the connection length is approximately $d$. The initial phase of the $n$-th rotor axis relative to the unmanned drone body is $\varphi_n$; the initial phase of the $m$-th blade on the $n$-th rotor is $\varphi_{nm}$. The projections of $O_u$ and $O_n$ on the $xoy$ plane are $O’_u$ and $O’_n$, respectively.
Assuming the unmanned drone is always in a hovering state, the distance between the radar and the center of each rotor axis can be calculated as:
$$R_n = \sqrt{R^2 + d^2 – 2Rd \sin \beta \cos(\alpha – \varphi_n)}$$
Let the rotation frequency of the $n$-th rotor of the unmanned drone be $f_n$, the length of each blade be $L$, the radar carrier frequency be $f_c$, the wavelength be $\lambda$, and the speed of light be $c$. Let $P$ be a scattering point on the blade of the $n$-th rotor, and the distance from this scattering point to the rotor center be $l_P$. Then the instantaneous distance $R_P(t)$ from point $P$ to the radar can be expressed as:
$$R_P(t) = R_n + l_P p(t)$$
where $p(t) = \sin \beta \cos(2\pi f_n t + \varphi_{nm} + \alpha)$.
The baseband echo signal $s_P(t)$ reflected from point $P$ to the radar can be obtained as:
$$s_P(t) = e^{-j4\pi f_c R_P(t)/c}$$
Substituting equation (2) into equation (3) gives:
$$s_P(t) = e^{-j4\pi [R_n + l_P p(t)]/\lambda}$$
To solve for the overall baseband echo of the blade, it is necessary to integrate the echo signal of scattering point $P$ over its entire length $L$, i.e.:
$$s_{nm}(t) = \int_0^L s_P(t) \, dl_P$$
Through integration, the radar echo signal of the $m$-th blade of the $n$-th rotor of the unmanned drone can be obtained as:
$$s_{nm}(t) = L \cdot \text{sinc}\left[2\pi L p(t)/\lambda\right] \cdot e^{-j4\pi [R_n + 0.5L p(t)]/\lambda}$$
Extending the echo model of a single blade to the entire unmanned drone, the total echo signal for an unmanned drone with $N$ rotors and $M$ blades per rotor is:
$$s(t) = \sum_{n=1}^{N} \sum_{m=1}^{M} \left\{ L \cdot \text{sinc}\left[2\pi L p(t)/\lambda\right] \cdot e^{-j4\pi [R_n + 0.5L p(t)]/\lambda} \right\}$$
From equation (7), it can be seen that the rotor echo model mathematically appears as a complex exponential term amplitude-modulated by a sinc function, indicating that it is not a single frequency component but a modulated signal with a certain bandwidth. This model effectively establishes a quantitative relationship between the geometric configuration, motion parameters of the unmanned drone, and radar echo characteristics, providing a theoretical basis for analyzing the dynamic scattering mechanism of the rotors. Based on this, the relationship between the flashing time width and the maximum Doppler frequency in the rotor echo signal will be further discussed below to clarify how to invert the rotor blade length from the echo.
Rotor Echo Signal Parameter Estimation Method
Constructing the Inverse Equation for Unmanned Drone Rotor Blade Length
Converting the echo signal containing rotor information into the time-frequency domain, the time-frequency spectrogram will show wing-like flashing phenomena, which are the result of periodic rotation of the blades. Each flash occurs when the rotor blade cuts into the radar beam once, so the time interval between adjacent flashes is called the flashing period $T_P$:
$$T_P = \frac{T_{\text{rot}}}{k}$$
where $T_{\text{rot}}$ is the rotor rotation period, and $k$ is the number of flashes per period. When the number of blades $M$ is even, $k = M$; when $M$ is odd, $k = 2M$.
From equation (7), it can be seen that the rotor echo appears as a periodic waveform in the form of a sinc function in the time domain, different from the flashing period $T_P$ in the time-frequency domain. Its flashing time width $\Delta T$ is defined as the time interval between the first zero points on both sides of the main lobe in the time domain. By solving for the zero points of the sinc function, the calculation formula for $\Delta T$ can be derived as:
$$\Delta T = \frac{1}{B_{\text{MDS}}} – \left(-\frac{1}{B_{\text{MDS}}}\right) = \frac{2}{B_{\text{MDS}}}$$
where $B_{\text{MDS}}$ is the bandwidth of the micro-Doppler signal.
The bandwidth of the micro-Doppler signal $B_{\text{MDS}}$ is not arbitrary; it is determined by the physical parameters of the rotor. For a rotor model that produces sinusoidal frequency modulation, its instantaneous frequency varies between $+f_d^{\max}$ and $-f_d^{\max}$. Therefore, the total bandwidth generated is:
$$B_{\text{MDS}} = 2 f_d^{\max}$$
Substituting equation (10) into equation (9) gives the relationship between the time domain and frequency domain of the rotor echo signal:
$$\Delta T = \frac{1}{f_d^{\max}}$$
That is, the flashing time width $\Delta T$ is inversely proportional to the maximum Doppler frequency $f_d^{\max}$.
Since the maximum Doppler frequency $f_d^{\max}$ is generated by the maximum linear velocity of the rotor blade tip $v_{\text{tip}}$:
$$f_d^{\max} = \frac{2 v_{\text{tip}} \cos \beta}{\lambda}$$
where the blade tip linear velocity $v_{\text{tip}} = 2\pi L f_{\text{rot}}$, $\beta$ is the radar elevation angle, and $\cos \beta$ is 1 when the direction of the blade tip’s rotational velocity is parallel to the radar line-of-sight direction. $L$ is the blade length, and $f_{\text{rot}}$ is the reciprocal of the rotation period $T_{\text{rot}}$, i.e., the rotation frequency.
Substituting $v_{\text{tip}}$ and $f_{\text{rot}}$ into equation (12) gives:
$$f_d^{\max} = \frac{2 (2\pi L f_{\text{rot}}) \cos \beta}{\lambda} = \frac{4\pi L f_{\text{rot}} \cos \beta}{\lambda}$$
From equation (13), the inverse equation for the rotor blade length $L$ can be obtained:
$$L = \frac{\lambda f_d^{\max}}{4\pi f_{\text{rot}} \cos \beta}$$
This section, based on the flashing characteristics of rotor echoes and the micro-Doppler effect, establishes an inverse equation for rotor blade length. The core of the inversion lies in using the maximum Doppler frequency $f_d^{\max}$ generated by the blade tip during rotation and the rotation frequency $f_{\text{rot}}$ to establish a quantitative relationship with the blade length $L$. This equation shows that the accuracy of blade length inversion directly depends on the accurate extraction of micro-motion feature parameters. The following will focus on analyzing how to extract micro-Doppler feature values with high precision to provide reliable input for the inverse equation.
High-Resolution Feature Extraction Method Based on TELSET
Due to the inherent time-frequency resolution矛盾 of traditional time-frequency analysis methods, it is difficult to clearly capture the high-frequency micro-motion features of low, slow, and small unmanned drone rotors. In low-altitude scenarios, the signal-to-noise ratio of rotor echo signals is extremely low, and the randomness of blade initial phases introduces severe cross-term interference, causing micro-Doppler time-frequency ridges to be highly aliased in the time-frequency domain, making feature separation difficult. Original time-frequency post-processing algorithms such as SET and LSET are based on frequency extraction operators of phase derivatives. Under ideal conditions, ridge localization is accurate, but under low signal-to-noise ratio conditions, they are sensitive to noise and susceptible to phase ambiguity, making it difficult to stably extract ridges in actual complex environments. To address this, this paper proposes a high-resolution micro-motion feature extraction method based on TELSET. This method, based on the differences in echo energy exhibited by different rotors due to spatial positions and radar cross-sections under the same radar acquisition, performs energy threshold discrimination on the time-frequency matrix after STFT, effectively filtering out energy diffusion and artifacts below the threshold. On the “purified” time-frequency diagram, local maximum synchronous extraction transform is performed along the time dimension to further optimize time-frequency ridge localization, thereby obtaining a micro-motion feature representation with higher concentration and reliability. To systematically explain the TELSET method, its processing steps will be described in detail below. The specific steps of the TELSET method are as follows:
Step 1: First, preprocess the original echo signal by suppressing static clutter through moving target detection to initially improve the target signal-to-noise ratio. Then, locate the range cell where the target is located based on the accumulated signal amplitude. Finally, extract the echo signal within this range cell for time-frequency analysis.
Step 2: Let the echo signal received by the radar be a multi-component signal. The general formula can be expressed as:
$$s(t) = \sum_{i=1}^{I} A_i(t) e^{j\phi_i(t)} + n(t)$$
where $I$ is the number of signal components, $A_i(t)$ is the instantaneous amplitude of the $i$-th component, $\phi_i(t)$ is the instantaneous phase of the $i$-th component, and $n(t)$ is additive noise.
After phase unwrapping of the slow-time signal, quadratic polynomial fitting is used to estimate and compensate for polynomial phase errors in the signal, thereby obtaining a phase-corrected signal. Performing STFT on this signal yields:
$$\text{STFT}(t, f) = \int_{-\infty}^{\infty} s(\tau) w(\tau – t) e^{-j2\pi f \tau} \, d\tau$$
Discrete implementation form:
$$\text{STFT}[n, k] = \sum_{m=0}^{L_w-1} s[n + m] w[m] e^{-j2\pi km / K}$$
where $w[m]$ is the Chebyshev window function, $L_w$ is the window length, and $K$ represents the number of frequency points.
Step 3: Aiming at the problem that the original LSET is susceptible to noise interference in low-altitude environments for ridge extraction, this paper率先 introduces a pre-energy threshold filtering环节 to improve the accuracy of subsequent time-frequency ridge extraction by filtering out low-energy noise points. Among them, to stably estimate the threshold, the global background energy $E_{\text{global}}$ is calculated using the mean that均衡 reflects the overall energy level. The calculation formula is as follows:
$$E_{\text{global}} = \text{mean}(P(t, f))$$
where the energy of each time-frequency point $(t, f)$ is $P(t, f) = |S(t, f)|^2$.
The final filtering threshold is determined by the global background energy and the threshold factor:
$$T = \eta \cdot E_{\text{global}}$$
where the threshold factor $\eta$ needs to be set through multiple对照 experiments, optimized based on indicators such as time-frequency concentration and rotor parameter estimation error, ensuring optimal balance between feature retention and noise suppression. Results show that it dynamically adapts to the actual signal-to-noise ratio of the echo signal, with a negative correlation between the two: high signal-to-noise ratio scenarios correspond to 0.3 to 0.8, where low-intensity filtering can retain key features; low signal-to-noise ratio scenarios require increasing to 1.0 to 1.5, strengthening noise and cross-term suppression, while微调 according to range cells based on distance attenuation characteristics. Finally, energy threshold discrimination is performed on the time-frequency matrix, retaining the target energy region:
$$\text{STFT}_{\text{filtered}}(t, f) =
\begin{cases}
\text{STFT}(t, f) & \text{if } P(t, f) > T \\
0 & \text{otherwise}
\end{cases}$$
Step 4: For slow-varying signals such as micro-Doppler signals, assume that at any time $\tau$, when $|\tau – t|$ is sufficiently small, the amplitude and phase of its components can be approximated as linear changes. According to Taylor’s formula, the instantaneous amplitude $A_i(\tau)$ and phase $\phi_i(\tau)$ of the $i$-th component can be approximated to first-order terms at time point $t$:
$$A_i(\tau) \approx A_i(t)$$
$$\phi_i(\tau) \approx \phi_i(t) + \omega_i(t)(\tau – t)$$
where $\omega_i(t) = \phi_i'(t)$ is the instantaneous frequency. This conclusion can be understood as the micro-Doppler component approximating a complex exponential signal within a short-time window. Its energy is concentrated near the instantaneous frequency in the time-frequency plane, forming time-frequency ridges. However, extremum search based on instantaneous frequency extraction operators can accurately provide ideal extraction results only when the frequencies of each mode are sufficiently separated. But in actual complex scenarios, the instantaneous frequency of rotor signal components changes rapidly. When there is strong noise or cross-term interference, this extraction method may be sensitive to noise, producing false frequency estimates, seriously affecting subsequent ridge estimation accuracy. To overcome this limitation, this paper proposes local maximum synchronous extraction along the time axis. Based on the filtered time-frequency matrix $\text{STFT}_{\text{filtered}}(t, f)$, for each frequency slice $M_f(t) = |\text{STFT}_{\text{filtered}}(t, f)|$, apply Gaussian sliding average:
$$\tilde{M}_f(t) = (g_\sigma * M_f)(t) = \int_{-\infty}^{\infty} g_\sigma(\tau) M_f(t – \tau) \, d\tau$$
where $g_\sigma(t) = \frac{1}{\sqrt{2\pi}\sigma} e^{-t^2/(2\sigma^2)}$ is the Gaussian kernel function.
Then, set a minimum significance on the smoothed sequence to detect local maximum points:
$$\mathcal{P}_f = \left\{ t_0 \mid \tilde{M}_f(t_0) = \max_{t \in [t_0 – \Delta, t_0 + \Delta]} \tilde{M}_f(t), \tilde{M}_f(t_0) > \epsilon \right\}$$
where $\Delta$ is the local neighborhood width, and $\epsilon$ is the minimum significance threshold.
Step 5: Deeply integrate energy threshold filtering and local maximum synchronous extraction transform to form a sparse TELSET time-frequency representation:
$$\text{TELSET}(t, f) =
\begin{cases}
\text{STFT}_{\text{filtered}}(t, f) & \text{if } t \in \mathcal{P}_f \\
0 & \text{otherwise}
\end{cases}$$
This representation retains energy information only on the time-frequency trajectories that pass the screening, thereby generating a highly sparse time-frequency distribution with energy focused on real signal components.
Its discrete implementation form is:
$$\text{TELSET}[n, k] =
\begin{cases}
\text{STFT}_{\text{filtered}}[n, k] & \text{if } n \in \mathcal{P}_k \\
0 & \text{otherwise}
\end{cases}$$
By setting a global energy-based threshold, non-target noise points in the time-frequency domain are discriminated. Utilizing the periodic rhythm of rotor rotation, local maximum search is performed along the time axis, focusing more on the flashing trajectory of the rotors, effectively marking the precise moments when different signal components pass through each frequency point, thereby accurately estimating rotor speed. Additionally, the TELSET method can剔除无效能量区域 near the extreme points of Doppler frequency, suppressing Doppler frequency ambiguity and energy diffusion. This processing retains the maximum Doppler frequency of the rotors while significantly reducing the computational complexity of time-frequency analysis.
Experimental Verification and Analysis
Data Analysis
To verify the robustness of the proposed method, this paper analyzes millimeter-wave radar measured data and a public dataset (LSS-FMCWR-1.0: Multi-band FMCW Radar Low Slow Small Detection Dataset) respectively. The specific process is as follows. The experimental data were collected using a millimeter-wave radar system composed of an IWR1443 BOOST evaluation board and a DCA1000 EVM data acquisition board. One-third of the transmit (TX) antennas and four out of four receive (RX) antennas were enabled. The laptop used to control the millimeter-wave radar system is equipped with an Intel i7-9750H CPU and 16GB of memory. Measured data analysis was performed using three devices with differences in blade length and speed: DJI Mavic3E, DJI Mini3, and a single-rotor device.
With the辅助 verification of an infrared laser tachometer, the rotor parameters of the unmanned drones related to the experiment are shown in Table 1. Regarding data processing, all radar signal processing algorithms were implemented on MATLAB 2023b, yielding可观 results.
| Unmanned Drone Type | Number of Rotors | Rotation Speed (r/min) | Blade Length (cm) |
|---|---|---|---|
| DJI Mini3 | 4 | 6750 | 7.62 |
| DJI Mavic3E | 4 | 7950 | 11.95 |
| Single-rotor | 1 | 1940 | 9 |
The main technical parameters involved in the experiment for the millimeter-wave radar measured dataset are shown in Table 2.
| Parameter | Parameter Value |
|---|---|
| Carrier Frequency | 77 GHz |
| Signal Bandwidth | 2943 MHz |
| Pulse Repetition Frequency | 64 kHz |
| Sampling Rate | 18 MHz |
| Range Resolution | 0.05 m |
| Elevation Angle | 20°-80° |
| Collection Distance | 1 m, 2 m, 4 m |
To accurately capture the micro-motion features of the rotors, this study adopted a high pulse repetition frequency modulation strategy. Although this choice sacrifices some maximum unambiguous range, the experiment successfully extracted micro-Doppler feature values of short-rotor unmanned drones with speeds of 2000 RPM to 9000 RPM with high precision, enabling accurate identification of low-altitude unmanned drones. Below, micro-motion feature extraction will be performed for three devices with differences in blade length and speed: DJI Mini3, DJI Mavic3E, and a single-rotor device.
The parameter estimation process takes DJI Mini3 as an example. Based on the reasonable selection of target range cells from the MTI range map, after short-time Fourier transform, TELSET algorithm processing is performed to concentrate energy on the time-frequency trajectory, obtaining comb-like flashes of the rotor. Regarding the selection of peak points in the TELSET diagram, to counteract the relative frequency difference caused by changes in the relative angle between the blade rotation and the radar, along the positive and negative directions of the frequency axis, find the continuous peaks with the most concentrated energy, take the absolute values of their frequency values, and average them. The maximum Doppler frequency value corresponding to the blade tip can be estimated to be approximately 24125 Hz. The time profile at the selected frequency slice is plotted. The flashing frequency is estimated using the reciprocal of the time interval corresponding to adjacent peaks. Through calculation, the flashing frequency value is approximately 226.95 Hz. Combined with prior information of the general configuration of low, slow, and small unmanned drones, i.e., each rotor of a small multi-rotor unmanned drone is equipped with two blades, the blade rotation speed estimate is approximately 113.48 Hz. Then, combined with the radar elevation angle during collection of approximately 25°, substituting into equation (14), the blade length estimate can be derived as approximately 7.28 cm. To address extraction deviations that may be caused by fuselage leveling jitter, additional maximum Doppler peak points and rotation frequencies are estimated, and all results are averaged as the final estimate.
For the public dataset (LSS-FMCWR-1.0: Multi-band FMCW Radar Low Slow Small Detection Dataset) time-frequency analysis, the following are the TELSET processed time-frequency diagram and the time profile at the selected frequency for the echo signal of DJI Inspire2 unmanned drone collected in the L-band. The specific experimental parameters for this selected data segment are as follows: radar operating center frequency is approximately 1.45 GHz, modulation bandwidth is 100 MHz, sampling frequency is 500 kHz, modulation period is 0.3 ms, radar observation elevation angle is horizontal, the horizontal distance between the unmanned drone and the radar is approximately 11 m, and the rotor blade length is 19 cm. Due to the low range resolution of the L-band, multiple rotor echo signals may alias in the selected range cell, causing energy diffusion and artifacts in the time-frequency diagram after short-time Fourier transform. By using energy threshold filtering to eliminate these interferences, local maximum search on frequency slices on the purified time-frequency diagram can effectively locate time-frequency trajectories and extract flashing frequencies. Based on this, the blade length parameter estimation is also closer to the true value.经检验, the TELSET method estimates the blade length of Inspire2 to be approximately 17.96 cm, which is more accurate than the original LSET method, reducing the blade length estimation error from 7% to 5.46%, further verifying the robustness of the method.
Performance Comparison
This study uses relative error as an evaluation metric. The calculation formula is:
$$\text{Relative Error} = \frac{|\text{Estimated Value} – \text{True Value}|}{|\text{True Value}|} \times 100\%$$
Taking the measured data of the DJI Mini3 rotor unmanned drone by millimeter-wave radar as an example for analysis, the parameter estimation results of the TELSET method compared with other time-frequency analysis methods are detailed in Table 3. Statistical results based on 50 Monte Carlo experiments show that the proposed TELSET method has relative errors of 0.87% and 4.46% when estimating rotor speed and blade length, respectively. The blade length estimation accuracy is improved by an average of 6.17% compared to the STFT method, verifying the effectiveness of the TELSET method in rotor parameter estimation.
| Time-Frequency Analysis Method | Rotation Speed (r/s) | Blade Length (cm) | ||||
|---|---|---|---|---|---|---|
| Theoretical Value | Estimated Value | Relative Error | Theoretical Value | Estimated Value | Relative Error | |
| STFT | 112.5 | 100.67 | 10.52% | 7.62 | 8.43 | 10.63% |
| SET | 112.5 | 115.94 | 3.06% | 7.62 | 7.01 | 8.01% |
| LSET | 112.5 | 111.11 | 1.24% | 7.62 | 7.18 | 7.83% |
| TELSET | 112.5 | 113.48 | 0.87% | 7.62 | 7.28 | 4.46% |
The rotor parameter estimation results of the TELSET method for all models in this experiment are shown in Table 4.
| Unmanned Drone Type | Rotation Speed (r/s) | Blade Length (cm) | ||||
|---|---|---|---|---|---|---|
| Theoretical Value | Estimated Value | Relative Error | Theoretical Value | Estimated Value | Relative Error | |
| DJI Mini3 | 112.5 | 111.52 | 0.86% | 7.62 | 7.28 | 4.46% |
| DJI Mavic3E | 132.5 | 130.08 | 1.83% | 11.95 | 12.93 | 8.20% |
| Single-rotor | 32.33 | 31.37 | 2.97% | 9 | 8.82 | 2.03% |
To quantify the advantage of the TELSET method in time-frequency concentration performance, this study uses third-order Renyi entropy as a core metric. This metric is highly sensitive to dominant energy components in the signal. The lower the entropy value, the higher the concentration of signal energy in the time-frequency domain. Under the condition of ensuring the same experimental parameters, the entropy estimates of the time-frequency distributions of three unmanned drones are shown in Figure 1. In the rotor parameter estimation of DJI Mini3, the third-order Renyi entropy of the TELSET method is significantly reduced by 2.86 dB compared to the traditional STFT method. This substantial reduction confirms the superiority of the TELSET method in concentration performance.
To quantitatively evaluate the computational efficiency of the TELSET method, the computation time and algorithm time complexity trends of TELSET were compared with various time-frequency analysis methods. The comparison results of computation time are shown in Table 5.
| Time-Frequency Analysis Method | Computation Time (s) |
|---|---|
| STFT | 0.0983 |
| SET | 1.1476 |
| LSET | 1.0464 |
| TELSET | 0.9873 |
From Table 5, it can be seen that since the SET method is based on phase derivative calculation and energy redistribution, its computation time is higher than other time-frequency processing methods. TELSET does not require过多假设 of prior knowledge of the signal but filters based on the energy amplitude of time-frequency coefficients. This method acts directly in the time-frequency domain. Although energy threshold processing is added, the computation time remains basically unchanged, meeting real-time processing needs. To cover the complete rotation cycle of unmanned drone rotors with different speeds, match the commonly used data volume range in engineering, and comply with the power-of-two characteristic of FFT operations, three groups of slow-time sequence lengths of 256, 512, and 1024 were selected. The computation time of four time-frequency processing methods—STFT, SET, LSET, and TELSET—was tested 5 times repeatedly and averaged to analyze their time complexity characteristics. The comparison results are shown in Figure 2.
Conclusion
Aiming at the limitations of existing time-frequency analysis methods in extracting micro-motion features of “low, slow, and small” unmanned drones, such as spectral leakage and cross-term interference, this study proposes a new feature extraction method based on TELSET. This method constructs an energy threshold to effectively suppress energy diffusion and cross-term interference of multi-rotor echo signals in the time-frequency plane. Using local maximum extraction on frequency slices, it can effectively sharpen time-frequency ridges and extract flashing frequencies with high precision, improving time-frequency energy concentration while meeting real-time processing needs. The research covers millimeter-wave radar measured data of various models such as DJI Mini3, DJI Mavic3E, and a single-rotor device, and robustness was verified on the LSS-FMCWR-1.0 public dataset. Through experimental analysis, the proposed TELSET method can generate clearer and more focused time-frequency spectrograms compared to the STFT method, with its third-order Renyi entropy reduced by 2.86 dB. In the parameter estimation of rotor speed and blade length for DJI Mini3, the relative errors of the TELSET method are reduced to 0.87% and 4.46%, respectively, improving blade length estimation accuracy by an average of 6.17% compared to the STFT method. It is worth noting that this research, as a proof of principle, mainly focuses on the specific stage of close-range high-precision detection. The attenuation of high-frequency signals in millimeter-wave radar for long-range detection is a challenge that needs to be addressed jointly for engineering applications. Future research will致力于攻克 remote detection technical bottlenecks, exploring the integration of advanced signal processing algorithms and deep neural network models to expand the application boundaries of this method.