Spectral De-Aliasing Method of Micro-Motion Signals Based on a Complex-Valued U-Net Network

Abstract

Spectrum aliasing occurs in signal echoes when the sampling frequency does not comply with the Nyquist Sampling Theorem. In this scenario, the extraction of micro-motion parameters becomes challenging. This paper proposes a spectral de-aliasing method for micro-motion signals based on a complex-valued U-Net network. Zero interpolation is employed to insert zeros into the echo, effectively increasing the sampling frequency. After zero interpolation, the micro-motion signal contains both real micro-motion signal frequency components and new frequency components. Short-Time Fourier Transform (STFT) is then applied to transform the zero-interpolated echo from the time domain to the time–frequency domain. Furthermore, a complex-valued U-Net training model is utilized to eliminate redundant frequency components generated by zero interpolation, thereby achieving the frequency reconstruction of micro-motion signal echoes. Finally, the training models are employed to process the measured data. The theoretical analysis, simulations, and experimental results demonstrate that this method is robust and feasible, and is capable of addressing the problem of micro-motion signal echo spectrum aliasing in narrowband radar.

1. Introduction

Micro-motion refers to the vibrations, rotations, precessions, oscillations, and other minor movements of a target or its structural components, which represent the target’s specific motion state details [1]. Micro-motion exists in various target motion forms, such as tank track rolling [2], random ship swinging due to wave fluctuations [3], ballistic missile precession or nutation in mid-flight [4,5], helicopter rotor rotation [6], human posture recognition [7,8], and satellite and space debris spinning [9,10], among others. Consequently, radar target recognition technology based on micro-motion features is considered one of the most promising approaches in the field. In 2000, V. C. Chen introduced the concepts of micro-motion and micro-Doppler (M-D) to the radar domain [1]. Since then, the use of micro-motion features has attracted significant attention from academia and industry [9,10,11,12]. The micro-Doppler effect can reveal the unique micro-motion characteristics and structural features of a target or its components. Assuming that the target’s micro-motion information can be accurately extracted, rapid detection and recognition of the target can be achieved, providing critical military applications for accurately engaging targets.

The critical challenge in micro-motion signal processing is the processing of time-varying signals. Research on target micro-motion feature analysis and extraction primarily involves several aspects. The first aspect is the time–frequency transform method [13], which mainly includes Short-Time Fourier Transform (STFT), Wigner–Ville Distribution (WVD), Continuous Wavelet Transform (CWT), the S-Method (SM), Short-Time Fractional Fourier Transform (STFRFT) [14], and others. The second aspect focuses on signal separation techniques, which predominantly consist of Hilbert–Huang Transform (HHT), Empirical Mode Decomposition (EMD), and their improved methods, such as Complex Empirical Mode Decomposition (CEMD) [15]. In addition to these traditional methods, the statistical histogram approach in paper [16] outperforms the L-statistics in paper [17] and provides a method for separating the echoes of rigid bodies and rotation points. To address the issue of group target echo signal separation, paper [18] achieves group target signal separation using independent component analysis technology. The third aspect involves image-based micro-motion feature extraction techniques. Methods based on Hough Transform and extended Hough Transform are explored in paper [19]. Although this method yields improved results, the calculations involved present numerous challenges. Paper [20] utilizes image rotation correlation obtained from different coherent processing intervals (CPIs) to estimate speed. However, analyzing and extracting the micro-motion features of radar echoes with spectrum aliasing remains difficult due to frequency component aliasing. Orthogonal Frequency Division Multiplexing (OFDM) is proposed in paper [21] to solve the problem of spectrum aliasing. This method necessitates designing the waveform, which is challenging to implement. Translational compensation is performed in paper [22], but this method requires frequency estimation and struggles to effectively compensate targets with complex micro-motion characteristics in order to resolve the issue of spectrum aliasing.

In this paper, the U-Net network is employed to eliminate redundant frequency components caused by zero interpolation [23]. As a machine learning approach [24], deep learning classifies and recurses according to input data. U-Net [25] is a deep learning algorithm used in this paper. The U-Net network model was proposed in 2015 and was initially used for semantic segmentation, which combines lightweight and high performance. In essence, the U-Net network is also a full convolutional neural network model, and is composed of an encoder and a decoder. Its name comes from its architecture shape: the model presents a “U” shape as a whole. It was proposed to solve the problem of semantic segmentation in medical images, but its development in the following years also confirmed that it is a versatile tool for semantic segmentation tasks. Consequently, U-Net shows great potential in applications such as semantic segmentation, and target segmentation and detection, particularly in image segmentation. Currently, various U-Net networks have been developed, including H-Dense U-Net [26], RIC-U-Net [27], and RCNN-U-Net [28].

Spectrum aliasing is common in radar signals [29,30]. The spectral de-aliasing techniques mainly rely on changes in sampling frequency to achieve spectrum de-aliasing. The frequency components are mainly fixed, that is, the frequency does not change over time, while micro-motion signals are generally time-modulated signals that change over time. Aimed at addressing the spectral aliasing problem in micro-motion signals, we propose a method that utilizes zero interpolation and complex-valued U-Net, which is used to resolve the micro-Doppler frequency ambiguity of a helicopter [23]. First, the influence of zero interpolation on the signal sequence is theoretically analyzed, and the modification of frequency components after zero interpolation in the signal sequence is examined. Since zero interpolation in the signal is equivalent to an increase in the sampling frequency, the signal contains real micro-Doppler frequency. However, this method also generates additional frequency components. Then, using the micro-motion model of the rotational signal as an example, the time–frequency results of the rotary micro-motion signal are obtained through STFT, which maintains consistent resolution in the entire time–frequency domain during signal analysis and exhibits strong integrity.

The U-Net is extended to the complex field for model training. The network input consists of the ambiguous time–frequency results of micro-motion signals after zero interpolation. The training model eliminates redundant frequency components in the time–frequency domain, compares the image segmentation effect with the fully convolutional network (FCN) [31] and the fully convolutional residual network (FCRN) [31] training models, and analyzes model performance using the test set. Finally, the experimental micro-motion signal generates time–frequency results after removing clutter through Moving Target Indicator (MTI) processing and STFT. The image segmentation employs a complex-valued U-Net with varying network parameters to remove redundant frequencies after zero interpolation. The performance of the networks is compared, and the influence of parameter changes on the network model’s performance is analyzed to determine the optimal network parameter configuration. Our theoretical analysis and simulation results demonstrate that this spectral de-aliasing method is robust and feasible, effectively addressing spectrum de-aliasing in micro-motion signals.

2. Related Work

2.1. Signal Model of Rotational Micro-Motion

Figure 1 is a diagram illustrating the radar receiving the echo of a rotating target. In Figure 1, a coordinate system with the radar as the origin is established. $R$ is the distance between the radar $O$ and the rotating target’s rotation center, $L$ is the blade length of rotating target, $d$ is the distance between the scattering points, $f_{rot}$ is the rotating target’s rotation frequency, and $\alpha$ and $\beta$ are the azimuth angle and the pitch angle between the radar $O$ and the rotating target’s rotation center, respectively. According to [14,32], when transmitting a continuous wave with a wavelength of $\lambda$, the maximum micro-Doppler frequency of the rotating target will originate from the tip of the scattering point of the blades. When electromagnetic waves are detected vertically on the blades, the max Doppler frequency of the rotating target is as follows:

$$f_{d\max }=\frac{4\pi f_{rot}L}{\lambda }\cos \beta$$

(1)

Equation (1) shows that the max Doppler frequency is related to the transmitting wavelength, blade length, rotation speed, and pitch angle. Simultaneously, to ensure that the signal is not aliased, the pulse repetition frequency (PRF) must meet the Nyquist Sampling Theorem:

$$PRF\ge 2f_{d\max }=\frac{8\pi f_{rot}L}{\lambda }\cos \beta$$

(2)

Therefore, in the analysis of rotating targets, it is crucial to select an appropriate pulse repetition frequency to avoid the adverse effects of spectrum aliasing.

2.2. Analysis of Zero Interpolation

To address the spectrum aliasing of micro-motion signals, zero interpolation of the sampling signal is equivalent to increasing its sampling frequency [33]. However, new frequency components will appear, enabling the processed signal to satisfy the Nyquist Sampling Theorem and reveal the true frequency in spectral analysis, thereby providing a novel method for resolving spectrum aliasing. The method involves interpolating zeros between the signal samples in order to increase the sampling frequency integer multiple. The signal will add $N$ new frequency components after interpolating $N$ multiples of zeros. Therefore, the sampling frequency is equivalent to $f_{sM}=(N+1)f_s$, and

Table 1. Frequency components after using zero interpolation.

Interpolating Zero Multiples $\mathit{N}$	Including Frequency Component
$N=0$	$f_1$
$N=1$	$f_1,f_1-f_s$
$⋮$	$⋮$
$N=2p$	$f_1,f_1-f_s,f_1+f_s,\cdots ,f_1-p{f}_s,f_1+p{f}_s$
$N=2p+1$	$f_1,f_1-f_s,f_1+f_s,\cdots ,f_1+p{f}_s,f_1-(p+1)f_s$

$p=1,\dots ,M,M=⌊\frac{N}{2}⌋$, $⌊\ast ⌋$ denotes the rounding down symbol.

displays the frequency components.

Figure 2 presents the time–frequency results of the rotating target. Figure 2a illustrates the time–frequency results of spectrum aliasing when the sampling frequency does not meet the Nyquist Sampling Theorem. The time–frequency results of the signal in Figure 2b–d correspond to interpolating one zero, two zeros, and three zeros, respectively. These results illustrate that the spectrum distribution becomes more uniform and the true frequency values appear with the ambiguous spectrum components through zero interpolation. The comparison reveals that as the number of interpolated zeros increases, the real frequency components appear, and the frequency components also multiply. In this paper, U-Net is employed for image segmentation to eliminate redundant frequency components.

2.3. Experimental Data Acquisition and Echo Data Processing

To verify the feasibility of the model, an experimental simulation is conducted. We use a specific narrow-band continuous-wave micro-radar. The radar emits sawtooth and triangular waves. Figure 3a displays the radar. Its carrier frequency is 24 GHz, its bandwidth is 20 MHz, its AD sampling rate is 400 kHz, and its pulse repetition frequency is 4000 Hz. Compared to rotation, according to Equation (1), the frequency range of the vibration generally does not produce spectral aliasing. An angular scatterer is rotated to simulate the target motion. The radar receives the echo signal and collects the echo data. Figure 3b illustrates the experimental data acquisition process. The rotation of the angular reflector simulates the target’s rotation, and the radar receives the micro-motion echo signal of the angular reflector.

STFT [17] is a classic time–frequency analysis method, and Equation (3) defines STFT. In Equation (3), $s\left(t\right)$ represents the receiving signal, and $w\left(t\right)$ represents the window function. In this study, STFT with an appropriate window length achieves suitable time–frequency resolution, while sampling reduces the size of the time–frequency transformation result of the complex-valued matrix.

$$STFT\left(t,f\right)={\displaystyle \int s\left(\tau \right)}w\left(\tau -t\right)\exp \left(-j2\pi f\tau \right)d\tau$$

(3)

Due to clutter in the data acquisition environment, echo signal processing is necessary. Moving Target Indicator (MTI) processing is used in this paper. Figure 4 shows MTI, an essential technology in the radar signal processing field.

$$Y(z)=X(z)-X(z-1)$$

(4)

Equation (4) displays the transfer function of MTI. $X(z)$ represents the result of z transformation of the receiving signal. It primarily uses the difference between the moving target echo and the clutter frequency domain to suppress clutter and retain the target echo by designing a filter with a specific notch. Figure 5 presents the time–frequency results after MTI processing. The time–frequency results indicate that MTI processing can suppress environmental clutter.

3. Method and Dataset

3.1. Design of the Complex-Valued U-Net

Based on the time-domain characteristics of micro-motion signals, this paper uses a U-Net in a complex field; the architecture is shown in Figure 6.

The network inputs are the time–frequency results in the complex field after zero interpolation. Every point in the instant frequency results has a real part and an imaginary part. For discrete echo sequences, STFT generates a 200 × 128 matrix time–frequency result, and the matrix is a complex-valued matrix.

To establish the experimental model, each value in the matrix in the time–frequency results is set as follows:

$$R_{ij}=x_{ij}+i{y}_{ij}$$

(5)

In Equation (5), $x_{ij}$ represents the real-part value of the complex values in the $i$ row and the $j$ column of the matrix, and $y_{ij}$ represents the imaginary-part value of the complex values in the $i$ row and the $j$ column, $i=1,2,3,\cdots ,200,j=1,2,3,\cdots ,128$.

Firstly, the maximum value is determined using the following modulus:

$$R_{\max }=\max \left\{\sqrt{x_{ij}{}^2+y_{ij}^2}\right\}$$

(6)

Then, we normalize the matrix. The real part and the imaginary part are separated, resulting in the following real-part matrix:

$$R_{ij}^{{}_{real}}=\frac{x_{ij}}{R_{\max }}=\left[\begin{array}{cccc}\frac{x_{1,1}}{R_{\max }}& \cdots & \cdots & \frac{x_{1,128}}{R_{\max }}\\ ⋮& \ddots & \ddots & ⋮\\ ⋮& \ddots & \ddots & ⋮\\ \frac{x_{200,1}}{R_{\max }}& \cdots & \cdots & \frac{x_{200,128}}{R_{\max }}\end{array}\right]$$

(7)

The imaginary-part matrix is as follows:

$$R_{ij}^{inag}=\frac{y_{ij}}{R_{\max }}=\left[\begin{array}{cccc}\frac{y_{1,1}}{R_{\max }}& \cdots & \cdots & \frac{y_{1,128}}{R_{\max }}\\ ⋮& \ddots & \ddots & ⋮\\ ⋮& \ddots & \ddots & ⋮\\ \frac{y_{200,1}}{R_{\max }}& \cdots & \cdots & \frac{y_{200,128}}{R_{\max }}\end{array}\right]$$

(8)

Both matrices are substituted into the fully convolutional network for encoding and decoding operations. A superior network model is identified by adjusting the number of network layers and convolution kernel size. According to engineering practice, the convolutional kernels are set to $3\times 3$, $5\times 5$, and $7\times 7$, and the number of layers is set to 5, 7, and 9.

Table 2. U-Net parameter configuration.

A	B	C	D	E	F	G	H	I
5 Weight Layers	5 Weight Layers	5 Weight Layers	7 Weight Layers	7 Weight Layers	7 Weight Layers	9 Weight Layers	9 Weight Layers	9 Weight Layers
3Conv $3\times 3$, ReLU	3Conv $5\times 5$, ReLU	3Conv $7\times 7$, ReLU	3Conv $3\times 3$, ReLU	3Conv $5\times 5$, ReLU	3Conv $7\times 7$, ReLU	3Conv $3\times 3$, ReLU	3Conv $5\times 5$, ReLU	3Conv $7\times 7$, ReLU
Max pool $2\times 2$ 2Conv $3\times 3$, ReLU	Max pool $2\times 2$ 2Conv $5\times 5$, ReLU	Max pool $2\times 2$ 2Conv $7\times 7$, ReLU	Max pool $2\times 2$ 2Conv $3\times 3$, ReLU	Max pool $2\times 2$ 2Conv $5\times 5$, ReLU	Max pool $2\times 2$ 2Conv $7\times 7$, ReLU	Max pool $2\times 2$ 2Conv $3\times 3$, ReLU	Max pool $2\times 2$ 2Conv $5\times 5$, ReLU	Max pool $2\times 2$ 2Conv $7\times 7$, ReLU
Max pool $2\times 2$ 2Conv $3\times 3$, ReLU	Max pool $2\times 2$ 2Conv $5\times 5$, ReLU	Max pool $2\times 2$ 2Conv $7\times 7$, ReLU	Max pool $2\times 2$ 2Conv $3\times 3$, ReLU	Max pool $2\times 2$ 2Conv $5\times 5$, ReLU	Max pool $2\times 2$ 2Conv $7\times 7$, ReLU	Max pool $2\times 2$ 2Conv $3\times 3$, ReLU	Max pool $2\times 2$ 2Conv $5\times 5$, ReLU	Max pool $2\times 2$ 2Conv $7\times 7$, ReLU
Up-conv $2\times 2$ Conv $3\times 3$, ReLU Copy and Corp 2Conv $3\times 3$, ReLU	Up-conv $2\times 2$ Conv $5\times 5$, ReLU Copy and Corp 2Conv $5\times 5$, ReLU	Up-conv $2\times 2$ Conv $7\times 7$, ReLU Copy and Corp 2Conv $7\times 7$, ReLU	Max pool $2\times 2$ 2Conv $3\times 3$, ReLU	Max pool $2\times 2$ 2Conv $5\times 5$, ReLU	Max pool $2\times 2$ 2Conv $7\times 7$, ReLU	Max pool $2\times 2$ 2Conv $3\times 3$, ReLU	Max pool $2\times 2$ 2Conv $5\times 5$, ReLU	Max pool $2\times 2$ 2Conv $7\times 7$, ReLU
Up-conv $2\times 2$ Conv $3\times 3$, ReLU Copy and Corp 2Conv $3\times 3$, ReLU Conv $1\times 1$, ReLU Copy and Corp	Up-conv $2\times 2$ Conv $5\times 5$, ReLU Copy and Corp 2Conv $5\times 5$, ReLU Conv $1\times 1$, ReLU Copy and Corp	Up-conv $2\times 2$ Conv $7\times 7$, ReLU Copy and Corp 2Conv $7\times 7$, ReLU Conv $1\times 1$, ReLU Copy and Corp	Up-conv $2\times 2$ Conv $3\times 3$, ReLU Copy and Corp 2Conv $3\times 3$, ReLU	Up-conv $2\times 2$ Conv $5\times 5$, ReLU Copy and Corp 2Conv $5\times 5$, ReLU	Up-conv $2\times 2$ Conv $7\times 7$, ReLU Copy and Corp 2Conv $7\times 7$, ReLU	Max pool $2\times 2$ 2Conv $3\times 3$, ReLU	Max pool $2\times 2$ 2Conv $5\times 5$, ReLU	Max pool $2\times 2$ 2Conv $7\times 7$, ReLU
			Up-conv $2\times 2$ Conv $3\times 3$, ReLU Copy and Corp 2Conv $3\times 3$, ReLU	Up-conv $2\times 2$ Conv $5\times 5$, ReLU Copy and Corp 2Conv $5\times 5$, ReLU	Up-conv $2\times 2$ Conv $7\times 7$, ReLU Copy and Corp 2Conv $7\times 7$, ReLU	Up-conv $2\times 2$ Conv $3\times 3$, ReLU Copy and Corp 2Conv $3\times 3$, ReLU	Up-conv $2\times 2$ Conv $5\times 5$, ReLU Copy and Corp 2Conv $5\times 5$, ReLU	Up-conv $2\times 2$ Conv $7\times 7$, ReLU Copy and Corp 2Conv $7\times 7$, ReLU
			Up-conv $2\times 2$ Conv $3\times 3$, ReLU Copy and Corp 2Conv $3\times 3$, ReLU Conv $1\times 1$, ReLU Copy and Corp	Up-conv $2\times 2$ Conv $5\times 5$, ReLU Copy and Corp 2Conv $5\times 5$, ReLU Conv $1\times 1$, ReLU Copy and Corp	Up-conv $2\times 2$ Conv $7\times 7$, ReLU Copy and Corp 2Conv $7\times 7$, ReLU Conv $1\times 1$, ReLU Copy and Corp	Up-conv $2\times 2$ Conv $3\times 3$, ReLU Copy and Corp 2Conv $3\times 3$, ReLU	Up-conv $2\times 2$ Conv $3\times 3$, ReLU Copy and Corp 2Conv $5\times 5$, ReLU	Up-conv $2\times 2$ Conv $3\times 3$, ReLU Copy and Corp 2Conv $7\times 7$, ReLU
						Up-conv $2\times 2$ Conv $3\times 3$, ReLU Copy and Corp 2Conv $3\times 3$, ReLU	Up-conv $2\times 2$ Conv $5\times 5$, ReLU Copy and Corp 2Conv $5\times 5$, ReLU	Up-conv $2\times 2$ Conv $7\times 7$, ReLU Copy and Corp 2Conv $7\times 7$, ReLU
						Up-conv $2\times 2$ Conv $3\times 3$, ReLU Copy and Corp 2Conv $3\times 3$, ReLU Conv $1\times 1$, ReLU Copy and Corp	Up-conv $2\times 2$ Conv $5\times 5$, ReLU Copy and Corp 2Conv $5\times 5$, ReLU Conv $1\times 1$, ReLU Copy and Corp	Up-conv $2\times 2$ Conv $7\times 7$ ReLU Copy and Corp 2Conv $7\times 7$, ReLU Conv $1\times 1$, ReLU Copy and Corp

displays the network structure of the U-Net designed in this paper.

In Table 2, 3conv $3\times 3$ ReLU represents three convolution operations, with a convolution kernel size of $3\times 3$ and an activation function of ReLU; Max pool $2\times 2$ denotes maximum pooling, with a step of $2\times 2$; and Up-conv $3\times 3$ signifies up-convolution, with a convolution kernel size of $3\times 3$ Figure 7 illustrates the structure and operation flow.

3.2. The Training Model

In this paper, the time–frequency results of the micro-motion signal echo after zero interpolation serve as the dataset inputs, while the unambiguous time–frequency results of the micro-motion signal obtained by expanding the sampling frequency are the dataset labels. The U-Net in the complex field is trained to generate the model, which is evaluated and compared using the convolutional network FCN and FCRN models. A variety of performance tests are conducted on the designed U-Net to analyze the network performance. Subsequently, the number of layers and convolution kernel sizes of the complex-valued U-Net are changed to generate the training model. After comparing performance, the time–frequency results of the experimental data are tested, and we analyze the influence of the number of layers and the convolution kernel model to solve the problem of spectrum aliasing, and select the best one. Figure 8 demonstrates the training model process and

Table 3. Configuration of deep learning platform in training.

Configuration	Content
CPU	AMD Ryzen 7 5700U
GPU	RTX 3050
Framework of deep learning	TensorFlow 2.6.0
System	Windows 10

shows the training environment.

3.3. Dataset

As can be seen from Equation (2), many factors affect the rotating target echo signal, such as the number of rotating target blades, rotating speed, signal-to-noise ratio, etc. In this paper, the blade tip of the rotor target is equivalent to the scattering point to simulate the rotating target. Through the control variate method, the parameters are set to generate the dataset for training. The initial blade phase of the micro-motion signal is random. The range of the initial phase satisfies the phase $\mathsf{\Phi }\subset [0,2\pi ]$. The wavelength is $\lambda =0.3 \mathrm{m}$. We set the speed and blade length according to the highest frequency range. For the generation of a simulation dataset, to match with the experimental simulation data, this paper uses a single rotor and narrow-band radar echo model modeling; the rotation speed is set to $4–5 \mathrm{rad}/\mathrm{s}$, the blade length is set to $0.36\sim 0.46 \mathrm{m}$, the sampling frequency is $4000 \mathrm{Hz}$, and the time is 1 s. The STFT is carried out for the echo signal generated by the echo model after interpolating the zeros once. The time–frequency result of the first 0.32 s is taken as the input of the training set. The sampling signal is increased to $8000 \mathrm{Hz}$, with the other parameters remaining the same, and the time–frequency results generated by the Short-Time Fourier transform of the echo are taken as the labels of the training set. According to the task, 2000 sets of data are generated as training sets. The number of test sets is 10% of the training set. The parameter design of the test set is the same as that of the training set.

3.3.1. Dataset of Zero Interpolation

The dataset does not consider noise, comparing network performance in an ideal case. The dataset input consists of the time–frequency results after interpolating zero once, and Figure 9 shows the training labels.

3.3.2. Dataset under Different SNRs

For the actual micro-motion signal, due to the presence of noise, its spectra are not as clean after STFT. Therefore, a dataset is generated for the initial signals under different SNRs, with the SNR set to $0\sim 10 \mathrm{dB}$. The dataset input consists of the time–frequency results with $\mathrm{SNR}=5.584 \mathrm{dB}$, and Figure 10 shows the training label.

3.3.3. Dataset of Experimental Data

After clutter suppression via MTI processing, the echo of the experimental data is analyzed via STFT to generate the time–frequency results, as shown in Figure 11a. To generate the model for effective image segmentation, the same MTI processing is performed on the input dataset. The time–frequency results of the input are shown in Figure 11b. Compared with the time–frequency results of the micro-motion signal sequences, the intensity of the sinusoidal curves is not uniform, and there are sporadic flicker residues beside the sinusoidal curves.

4. Performance Analysis

4.1. Network Performance Analysis

This paper used the complex-valued U-Net compared with the FCN and the FCRN. Figure 12 shows the training loss curves of these networks. For the FCN and the FCRN, the loss is far greater than that of the complex-valued U-Net. With an increasing number of training epochs, the loss is not stable. The FCRN even shows an overfitting phenomenon as its loss suddenly increases rapidly, and then, after a few epochs, the loss is reduced. However, with increased training epochs, the loss of the complex-valued U-Net remains low, demonstrating excellent performance.

Inputting an image from the test set into the models from the networks, this paper compares the output results. The results show that the FCN and the FCRN exhibit poor micro-motion spectral de-aliasing performance, not only failing to remove the redundant flicker components but also causing the flicker in the initial time–frequency results to become blurred, making the extraction of useful information more challenging. The complex-valued U-Net retains the required flicker and envelope information and effectively removes redundant flicker. Figure 13 displays the outputs of the three network training models on the same image from the test set.

The complex-valued U-Net demonstrates strong image segmentation ability [34]. This paper compares two well-known target image quality indicators, the Peak Signal–Noise Ratio (PSNR) and Structural Similarity Measure (SSIM), to further analyze the model’s performance. The value range of SSIM is between 0 and 1. When two images are perfectly similar, the SSIM value is 1. The closer the SSIM value is to 1, the more similar the two images are. The PSNR value is typically expressed in dB. When the PSNR value is higher than 40 dB, the processed image is virtually indistinguishable from the original image. If it falls between 20 dB and 40 dB, it can be considered that, although the image exhibits distortion, it aligns with the processing objective. If it is below 20 dB, the image processing effect is deemed insufficient to meet the requirements.

Taking 10 samples from the test set, this paper compares the PSNR and SSIM of the three network models.

Table 4. SSIM of the three networks.

	FCN	FRCN	U-Net
1	0.716	0.659	0.947
2	0.778	0.737	0.944
3	0.691	0.636	0.957
4	0.749	0.704	0.945
5	0.716	0.659	0.956
6	0.743	0.691	0.941
7	0.770	0.728	0.956
8	0.788	0.759	0.940
9	0.760	0.712	0.930
10	0.779	0.740	0.952
avg	0.749	0.7025	0.9468

and

Table 5. PSNR of the three networks.

	FCN	FRCN	U-Net
1	20.68	18.95	43.33
2	20.75	19.26	42.71
3	20.64	19.00	43.51
4	20.65	19.09	43.16
5	20.71	19.02	44.69
6	20.69	19.09	42.53
7	20.70	19.20	44.97
8	20.73	19.55	44.51
9	20.68	19.18	41.34
10	20.73	19.23	44.46
avg	20.70	19.16	43.52

display the SSIM and the PSNR for the three networks, respectively. The comparison reveals that the average value of SSIM for the complex-valued U-Net is 0.9468, close to 1, and the average value of PSNR is 43.52 dB, greater than 40 dB, indicating that this model exhibits excellent image segmentation performance. In contrast, the other two networks perform poorly and are unable to complete the task of image segmentation.

4.2. Performance Analysis of the Complex-Valued U-Net

The performance of the complex-valued U-Net is analyzed in this paper using the micro-motion signal after interpolating zero once. The analysis includes two parts: analysis under different SNRs and analysis under different network parameters.

4.2.1. Performance Analysis under Different SNRs

For the complex-valued U-Net used in this paper, the training model is used to test and validate the test set while analyzing its performance under different SNRs. According to the SNR range of the training set, the test set employed in this analysis is separated by 1 dB, and time–frequency results with SNRs from 0 dB to 10 dB are selected for testing. Figure 14 displays input data with 0 dB, 5 dB, and 10 dB from the test set, and Figure 15 shows the output. The results demonstrate that with the increase in the signal-to-noise ratio, the flicker does not change significantly, but the envelope weakens as the signal-to-noise ratio decreases. Compared to the test set input, its envelope is still more pronounced; compared to the test set label, it exhibits some loss. The PSNR and SSIM under different SNR reduction results are provided in

Table 6. SSIM and PSNR of complex-valued U-Net.

	0 dB	1 dB	2 dB	3 dB	4 dB	5 dB	6 dB	7 dB	8 dB	9 dB	10 dB
SSIM	0.322	0.359	0.302	0.448	0.455	0.424	0.443	0.446	0.415	0.480	0.392
PSNR (dB)	20.48	21.08	17.95	22.85	24.02	24.49	25.23	26.31	26.48	27.59	28.19

. Through comparison, it is discovered that PSNR and SSIM are significantly reduced in the presence of noise. The randomness of the noise leads to a difference between the input noise and the output noise when generating the dataset. Valuable information primarily resides in the sinusoidal signal. According to the model’s time–frequency results, the model can better retain the sinusoidal signal, and low image index parameters minimally impact subsequent parameter extraction. The performance of the complex-valued U-Net under different SNRs is superior, effectively addressing spectrum aliasing.

4.2.2. Performance Analysis under Different Network Parameters

In this paper, we altered the convolution kernel size and network layers, constructed nine networks, trained the network, generated the training model, used the test set to compare different network performances, and identified the network with the best performance.

The convolution kernel size and network layers directly influence the training results. Figure 16 depicts the loss curves of network training.

Taking the loss curves of networks A, D, and G as examples, the convolution kernel size for the three networks is 3 × 3, but the number of network layers varies, with 5, 7, and 9 layers, respectively. Examining the loss curves of networks A, B, and C, the number of layers for these three networks is the same, and the convolution kernel sizes are 3 × 3, 5 × 5, and 7 × 7, respectively. After comparing, it is found that the networks exhibit lower loss values when the number of network layers is 5, and network A has the largest loss value. This demonstrates the importance of selecting suitable parameters.

Fifty time–frequency results output from the training model are calculated. Figure 17 and Figure 18 show the SSIM and PSNR of the 50 time–frequency results from the test set, respectively. After comparison, it is evident that networks A–I perform differently. Except for network A, the other networks demonstrate good performance.

4.3. Experimental Results

MTI processing is applied to process the experimental data, and the results after STFT are used as inputs to compare the performance of the complex-valued U-Net under different network parameters. Figure 19 presents the output of each training model.

Figure 19 (raw) shows the time–frequency results without processing. Figure 19 (input) shows the time–frequency results of the input network models, and Figure 19a–i correspond to the output results of the models generated by networks A–I. Figure 20 presents the results of each training model after IRadon transform. Figure 20 (raw) shows the results without processing after IRadon transform. Figure 20 (input) shows the results of the input after IRadon transform and Figure 20a–i correspond to the results of networks A–I after IRadon transform.

We use IRadon transform and Hough transform to extract the speed features from the input and output results, and the speed extraction results are shown in in

Table 7. The Speed ($\mathrm{rad}/\mathrm{s}$) extraction results.

Algorithm	Raw	Input	A	B	C	D
IRadon	3.7	2.81	2.86	2.85	2.86	2.83
Hough	3.96	2.19	2.47	2.48	2.47	3.96
Algorithm	E	F	G	H	I
IRadon	2.84	2.83	2.86	2.87	2.86
Hough	2.19	2.48	2.47	2.19	2.19

.

As shown in Figure 19, as the number of network layers and the size of the convolutional kernel increase, the model has a more significant effect on removing excess frequency components from the input time–frequency results. In the transformation results of Figure 20, the focus is also higher. The estimation results in Table 7 also indicate that the proposed method can solve the problem of spectrum aliasing in micro-motion signals and improve the accuracy of micro-motion feature extraction. Upon comparison, the set of network parameters significantly influences the experimental data in addressing the problem of spectrum aliasing. Due to the numerous errors in acquiring experimental data, the obtained data differ considerably from the theoretical analysis, directly affecting the model’s performance. For the measured data, as the number of network layers increases, the effect of removing redundant frequency components improves. This means that the SSIM and PSNR derived from the test set can only serve as references. Selecting the most suitable training parameters through the measured data, generating the training model, and removing redundant frequency components remain necessary.

5. Conclusions

Aiming to address the problem of spectrum aliasing caused by frequency components being higher than the sampling frequency in micro-motion signal echo signals, a complex-valued U-Net is employed to eliminate redundant frequency components generated via zero interpolation. Moreover, the performance of the complex-valued U-Net is compared with that of FCN and FCRN. Simultaneously, different SNR test sets and network parameters are utilized to investigate the performance of the complex-valued U-Net in segmenting micro-motion signals. The measured data are obtained through experiments, and the optimal network parameters are acquired by employing the models under different network parameters, proving that the image segmentation model can be applied in spectral de-aliasing. The spectrum aliasing of micro-motion signals can be effectively addressed using zero interpolation and the complex-valued U-Net. However, due to limitations in the experimental hardware, the amount of experimental data is relatively small, and the model’s performance could be further optimized. In the future, more types of micro-motion signals will be used to test the model’s performance. For the network, the next steps could be to further optimize the network structure, improve the model training dataset, and improve the model’s robustness and usage range.