A Micro-Doppler Frequency Ambiguity Resolution Method Based on Complex-Valued U-Net

In order to resolute the micro-Doppler frequency ambiguity caused by radar pulse repetition frequency not high enough (i.e., pulse dimension does not satisfy the requirement of Nyquist sampling theorem), this paper presents a micro-Doppler frequency ambiguity resolution method based on complex-valued U-net. The echo sequence is interpolated by zeros in the pulse dimension to increase the equivalent pulse repetition frequency, so that the echo sequence after zero interpolation contains the real micro-Doppler frequency; at the same time, some new frequency components are generated. The variation law of the echo sequence frequency after zero interpolation is analyzed. Then, the echo sequence in time domain after zero interpolation is transformed to the time-frequency domain by short-time Fourier transform (STFT). Finally, the time-frequency results can be segmented by the model, which is trained by complex-valued U-net to eliminate the redundant frequencies generated by zero interpolation; thus, the reconstruction of real micro-Doppler frequency is realized. Theoretical analysis and simulation results show that the proposed method can solve the problem of micro-Doppler frequency ambiguity. Compared with fully convolution network (FCN) and fully convolution residual network (FCRN), the proposed method has better performance and robustness.


Introduction
In modern warfare, helicopter has the advantages of vertical take-off and landing and strong transportation capabilities. It has become the preferred equipment for military operations such as ground attack, weapon delivery, and strategic support [1,2]. At the same time, helicopter has become one of the main targets of air defense operations in various countries. When this kind of target is hovering or flying at a slow speed, it is difficult to meet the requirements of detection and recognition by using traditional means, so it is particularly important to use micromotion characteristics for feature extraction and recognition [3][4][5]. However, for helicopter with fast rotation speed or long length, when the radar pulse repetition frequency cannot satisfy the Nyquist sampling theorem, the phenomenon of micro-Doppler frequency ambiguity will occur, which makes the feature extraction, parameter estimation, and recognition of rotor targets more difficult.
For modern radar, especially pulse-Doppler (PD) radar, it usually adopts different working modes [6,7] to deal with different tasks. e main working modes are low pulse repetition frequency (LPRF) mode [8], medium pulse repetition frequency (MPRF) mode [9][10][11], and high pulse repetition frequency (HPRF) mode [12,13]. However, range ambiguity or doppler ambiguity exists in different working modes. In view of this kind of problems, a lot of researches have been carried out on ambiguity resolution algorithms. At present, the methods of doppler ambiguity resolution are usually to form a group of pulse repetition frequency with multiple pulse repetition frequencies and then complete ambiguity resolution on this basis. e main algorithms include one-dimensional set algorithm [13], Chinese remainder theorem [14,15], and look-up table method [16,17]. Compared with the traditional doppler frequency ambiguity, the micro-Doppler frequency ambiguity is different. From the mechanism point of view, the micro-Doppler frequency is generated by the rotating parts of the target rather than the radial motion of the target. In addition, the relationship between traditional Doppler frequency and phase is generally linear, while the micro-Doppler frequency and phase satisfy the sinusoidal modulation relationship [18][19][20].
As a machine learning method [21][22][23], deep learning will classify and recurse according to the input data. Deep learning is mainly realized by neural network, which is an extensive, parallel, and interconnected network composed of adaptive simple units. Its structure can simulate the interaction of biological neural system to real world objects [24][25][26][27][28]. e deep learning algorithm used in this paper is U-net [29]. U-net is a symmetric coded-decoding network, which fuses the feature map generated by the coding layer with the feature map of the corresponding decoding layer to generate the final feature map of the decoding layer. U-net is mainly composed of convolution layers, pooling layers (downsampling layers), and upsampling layers and uses residual network for feature extraction with strong extraction performance. Because U-net uses convolution layers to replace the fully connected layer in convolutional neural network, the network carries out symmetrical coding and decoding, which causes the output vector of the network to be related to the input vector. erefore, U-net has good applications in target segmentation, semantic segmentation, and target detection, especially in the field of image segmentation [30][31][32].
In this paper, aimed at the problem of micro-Doppler frequency ambiguity, a method of micro-Doppler frequency ambiguity resolution based on complex-valued U-net and zero interpolation is proposed. eoretical analysis is focused on the reason of micro-Doppler frequency generation, and the change rule of micro-Doppler frequencies of echo sequence after zero interpolation. Because the pulse repetition frequency of the echo sequence is increased after zero interpolation, the echo sequence contains real micro-Doppler frequencies, and some redundant frequencies are also generated. en, the time-frequency results of the novel echo sequence are obtained by using STFT. e ambiguous time-frequency results corresponding to the micromotion sequence after zero interpolation are used as the input of the U-net network, and the unambiguous time-frequency results corresponding to the micromotion sequence after expanding the pulse repetition frequency are used as the labels of the U-net. e U-net is extended to the complex domain to train the model and generate the training model. Finally, compared with the fully convolution network (FCN) and fully convolution residual network (FCRN), the complex-valued U-net is used to eliminate the redundant frequency components to reconstruct the real micro-Doppler frequency. eoretical analysis and simulation results show that the proposed method is feasible and robust and can solve the problem of micro-Doppler frequency ambiguity.

Analysis of Micro-Doppler Frequency
Characteristics of Helicopter 2.1. Echo Micro-Doppler Frequency Analysis. Figure 1 is the geometric relationship diagram between radar and rotor blades. According to [19,20], different blades have the same number of scattering points with the same interval and scattering coefficient. At this time, the echo of the rotor blades scattering point model can be expressed as where σ P i is the backscattering coefficient, R 0 is the distance between the radar and the rotor center O ′ , and f dP i,n (t) is the doppler frequency caused by the ith scattering point on the nth blade, as follows: where β is the pitch angle, f rot is the rotor rotation frequency, x i ( 0 ≤ x ≤ l, l is the blade length) is the distance between the scattering point P i and the rotor center, and θ 1 is the initial rotation angle.
According to equation (3), the peak Doppler frequency is closely related to radar wavelength, rotor rotation frequency, blade length, and pitch angle. At the same time, in order to ensure that the Doppler frequency is unambiguous, the pulse repetition frequency (sampling frequency on pulse dimension) needs to meet the requirements.
Equation (4) can be called the lower limit formula of PRF that the radar needs to meet when detecting the echo of the rotor blades. From the analysis of equation (4), the peak Doppler is proportional to the rotor rotation frequency, blade length, and pitch angle. e radar wavelength is determined by the radar carrier frequency. e higher the radar carrier frequency is, the greater the peak Doppler frequency is. Figure 2 shows the time-frequency results of the echo sequence when the PRF value does not meet (1000 Hz) and meets (2000 Hz) the PRF lower limit formula. rough comparison, it is found that when the PRF value does not meet the PRF lower limit formula, the spectrum aliasing will occur, that is, micro-Doppler frequency ambiguity. erefore, in the analysis of the rotor target, it is necessary to select the appropriate PRF value to avoid the adverse effects of frequency ambiguity.

Analysis of the Influence of Zero Interpolation on the Frequency Component of Signal.
Aiming at micro-Doppler frequency ambiguity, the method of sampling signal interpolation by zeros is equivalent to enlarging its sampling frequency, so that the processed signal satisfies Nyquist sampling theorem, and the real frequency appears in the spectral analysis, thus providing a new method for resolving spectrum aliasing. By this method, the echo sequence can be written as where t r is the sampling period, f s is the sampling frequency, f s ≥ 2 · max f 1 , f 2 , . . . , f K , the frequency range after sampling is [(− f s /2), (f s /2)], and the bandwidth is f s . Interpolating zeros to the echo sequence means that the sampling frequency changed. At the same time, it also changes the frequency information contained in the echo sequence and makes the sequence show new frequency components. After interpolating Mmultiples of zeros, the signal satisfies Nyquist sampling, and the signal can be expressed as e method of interpolating zeros is shown as above, that is, interpolating zeros with equal interval and equal number in the middle of the original sequence to increase the sampling frequency by an integral multiple. rough theoretical derivation, it can be found that the original sampling sequence is equivalent to adding M new frequency component after interpolating M multiples zeros, the energy of the signal is evenly distributed to each frequency component, and the amplitude of each frequency component becomes 1/M + 1 before zero interpolation. At this time, the sampling frequency is equivalent to f sM � (M + 1)f s , and the frequency component is shown in Table 1.
It can be seen from Table 1 that, for the undersampled echo sequence, we can make the sampling frequency meet the Nyquist sampling theorem by interpolating zeros in the signal processing, and then we can get the real frequency of the signal. In this paper, STFT is performed after interpolating zeros to the echo sequence of helicopter with three rotor blades, as shown in Figure 3. Compared with Figure 2, it can be found that, for the time-frequency results of rotor targets with frequency ambiguity, the real frequency value of the ambiguous spectrum component is shown in the timefrequency results through the zero interpolation, and the spectrum distribution is more uniform, which provides favorable conditions for the next step of image segmentation by using U-net.

Complex-Valued U-Net
Different from the conventional image data set commonly used in image segmentation, grey image and RGB threechannel image are used as input in training. In this paper, STFT is used to generate the time-frequency result of the signal after the zero interpolation. Any point on the timefrequency result is complex-valued, which is composed of real part and imaginary part. For radar echo, both the real part and the imaginary part contain useful information. In order to retain more useful information, U-net is used in the complex-valued, and the time-frequency result is divided into real channel and imaginary channel. en, the output data of the two channels are combined into a complex-value to realize image segmentation and remove the redundant part of the time-frequency result.

Design of Complex-Valued U-Net.
is paper uses a complex domain U-net, and the architecture is shown in Figure 4. e input of the network is a two-dimensional complex time-frequency result after zero interpolation. Any point on the instant frequency result is a complex number. For discrete echo sequences, STFT is used here to generate a 400 × 400 matrix time-frequency result, with each point in the time-frequency result set to In the equation, c � 1, 2, 3, . . . , 400, z � 1, 2, 3, . . . , 400. Firstly, we take the modulus to find the maximum value as en, we separate the real part and the imaginary part, and the real part matrix is e imaginary part matrix is ese two matrices are used as two channels input data for encoding and decoding operations in a complex U-net. e U-net in this paper has seven layers, and the input and output time-frequency result sizes are 400 × 400. e network training steps are as follows: (1) e time-frequency result of the input data set is processed. e time-frequency result is divided into real part matrix and imaginary part matrix. e matrix is processed according to equation (9) and equation (10) as the input of the network.  (2) e first four layers are convolution and pooling, which compresses the data and extracts some simple features. e last three layers are upsampling and convolution. Some deeper features can be obtained through upsampling convolution. rough the jump structure, the data of the first three layers can be combined with the data of the last three layers and get better image segmentation performance by combining with the data of the deep layer. At last, the image segmentation is realized by the 1 × 1 convolution kernel of the seventh layer. e structure and operation flow of complex-valued U-net are shown in Figure 5.
e network structure is as follows: In this network structure, Conv is convolution, BN is batch normalization, upconv is upconvolution for upsampling, and the activation function is ReLU, which is called rectified linear unit shown in the following equation: Mathematical Problems in Engineering the performance of the designed U-net is tested. e training model flow is shown in Figure 6. e training environment configuration and hardware are shown in Table 2.

Dataset
According to equation (2), there are many factors that affect the time-frequency results of the rotor targets' echo sequence, such as the number of rotor blades, rotational speed, initial phase, and signal-to-noise ratio (SNR). In this paper, the method of controlling variables, setting parameters, and generating data sets are used for training. e initial phase of the echo is random. According to the number of blades, the range of the initial phase satisfies phase Φ ⊂ [0, 2π/N] and N is the number of blades. e carrier frequency f c � 1GHzand the wavelengthλ � 0.3m. e rotational speed and blade length are set according to the value range of the highest frequency. e performance of the network model is analyzed under different interpolating zero multiples, different number of blades, and different SNR.

Different Interpolating Zero Multiples.
For zero interpolation, the number of interpolating zeros directly affects the number of frequency components and the magnitude of the spectrum. Based on the target echo of three blades rotor targets, the highest frequency in the echo is set to 1000Hz ∼ 2000Hz, 2000Hz ∼ 3000Hz, 3000Hz ∼ 4000Hz, and the sampling frequency is set to 2000Hz. e original echo sequence is interpolated by the zero interpolation, and the time-frequency result is generated by using STFT as the input data. Increase the sampling frequency to 4000Hz,6000Hz, 8000Hzand the time-frequency results after STFT as output training labels for training. e specific   Table 3. e input timefrequency results and the output training label time-frequency results are shown in Figures 7 and 8.

Different Number of Blades.
For rotor targets, when the number of rotor blades is different, the characteristics of time-frequency results are different. Parameters are set to make the rotational speed of the blade 5 ∼ 7.5rad/sand the length of the rotator 5 − 7.5m. e highest frequency in the echo sequence is 1000Hz ∼ 2000Hz, and the sampling frequency is 2000Hz.By using the zero interpolation, the sampling frequency is expanded to 4000Hz, and the timefrequency result is generated by using STFT as the input data of the U-net. e time-frequency result generated by STFT after increasing the sampling frequency to 4000Hzis as an output training label. e input time-frequency results and the output training label time-frequency results are shown in Figure 9 and Figure 10.

Different Signal-to-Noise Ratios.
For radar echo, the time-frequency result after STFT is not so clear, due to the presence of noise. e initial parameters under different SNR are used to generate dataset by using parameters in Section 4.2. An input time-frequency result and an output training label are shown in Figure 11.

Composition of Dataset.
e dataset consists of the training set and the test set. e training set is composed of three parts, the data set of different interpolating zero multiples, the data set of different number of rotor blades, and the data set of different SNRs. e composition of the test set is the same as the data set. e composition of training set is shown in Table 4, and the composition of test set is shown in Table 5.

Network Performance Comparison.
ree full convolution neural networks are used to compare the performance including FCN, FCRN, and U-net. Figure 12 shows the loss curves of the three networks during training. By comparing the loss curves of three networks, we can find that the loss of FCN and FCRN is much greater than the loss of U-net. With the increase of training cycles, the loss of U-net tends to be stable and less. For the other two networks, the loss decreases with the training cycles, but the fluctuation of the network is violent, which indicates that the training effect and robustness of the U-net model are the best.
Enter a picture of the test set and input three network training models to compare the output time-frequency results. From the results, we can find that the FCN and FCRN of image segmentation performances are extremely poor; not only do they not remove the extra flicker components, but also they blur the original time-frequency results, making it more difficult to extract useful information. e U-net can effectively remove redundant flickers while retaining the required flicker and envelope information. Figure 13 shows the results of the same input picture test with three network training models.

Impact Analysis of Zero Interpolation on U-Net
Performance. Figure 14 is training loss curves of the relationship between training rounds with different interpolating zeros multiples. In Figure 14, fs2/fs3/fs4 means interpolating once/double/treble of zeros. By comparing the loss curves of different zero interpolating multiples, we can see that, with the increasing of training cycles, the loss rate decreases and finally tends to be stable. However, with the increasing of interpolation zeros, the parameter quantity of time-frequency results increases, the parameter quantity of network model increases, the range value of loss tends to increase stably, and loss tends to fluctuate more easily.
As the number of interpolating zero multiples increases, there is a greater difference in the details of the output results, the number of interpolation zeros increases, the envelope part of the flicker is fragile, and the envelope part of the package flicker is ambiguous. Figure 15 is the outputs after the part of test set using the training model. e simulation results show that the zero interpolation and the complex-valued U-net model can remove the problem of micro-Doppler ambiguity due to sampling frequency limitation of the target echo, which creates the possibility for the next step in parameter estimation and feature extraction.

Performance Analysis of Complex-Valued U-Net.
Due to the more zeros interpolated by the zero interpolation, the amount of data became larger, and the network model image segmentation effect will be affected. In this paper, the U-net model is analyzed by using the echo sequences with zero interpolation of once of zeros. Performance analysis is divided into two parts: one is performance analysis under different SNRs, and the other is performance analysis under different number of rotor blades.

Different Interpolating Performance Analysis under Different SNR.
For the complex-valued U-net, the test set is tested and validated with the generated training model when it performs performance analysis under different SNRs. e input parts of the test set are shown in Figure 16, and the output results are shown in Figure 17. Due to the SNR range, the training set and the test set in this study select 0 dB, 3 dB, 10 dB for testing. e result shows that the flicker does not change much with the increase of SNR, but the envelope becomes weaker with the decrease of SNR. e envelope is more obvious than the input of test set, and there is still some loss compared to the label of test set. rough comparison, it is found that the complex-valued U-net has better performance under different SNRs and can effectively solve the problem of frequency ambiguity in the case of noise.

Performance Analysis under Different Number of Rotor
Blades. For rotor targets, the number of rotor blades is different, but the time-frequency characteristics are very     similar. e performance of complex-valued U-net in the case of different number of blades is tested in this paper, and the training model is generated and tested by using the dataset of different number of blades. e results of timefrequency results of different number of blades are shown in Figure 18. By comparing the input and output images of 1-6 blades in the test set, it can be found that the training model has a better capability in removing redundant flicker, and a large amount of useful information is retained in envelope preservation.   Mathematical Problems in Engineering

Conclusions
In order to resolute the problem of micro-Doppler frequency ambiguity, that is, the highest frequency in echo, which is higher than the sampling frequency, zero interpolation is proposed to increase the sampling frequency, and the complex-valued U-net is proposed to remove the redundant frequency components generated by zero interpolation after the real frequency is restored by zero interpolation. is method achieves the purpose of reconstructing the real micro-Doppler frequency. rough the experimental analysis, the zero interpolation and complex-valued U-net can effectively solve the problem of micro-Doppler ambiguity in helicopter rotor echo. Compared with FCN and FCRN, the complex-valued U-net can effectively remove redundant flickers while retaining the required flicker and envelope information and has excellent performance to segment the time-frequency image of rotor targets as follows: (1) It can remove the redundant frequency components generated by zero interpolation in different zero interpolating multiples. With the number of interpolation zeros increasing, it still has good performance.
(2) In a large signal-to-noise ratio range, it can effectively reduce the influence of noise on time-frequency results and remove the redundant frequency components generated by zero interpolation. (3) For rotor targets, it can generate a training model to remove redundant frequency components generated by zero interpolation effectively for different number of rotor blades. e training model can save real flicker, and a large amount of useful information is retained in envelope preservation for feature extraction, parameter estimation, and recognition.
However, due to the limitation of experimental hardware and the amount of experimental data, the performance of the model is not optimal. e performance of the model will be tested by using the measured data for further research.

Data Availability
e numerical data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare no conflicts of interest.