Micro‐motion parameter extraction of rotating target based on vortex electromagnetic wave radar

National Natural Science Foundation of China, Grant/Award Numbers: 61801516, 61971434 Abstract The vortex electromagnetic (EM) wave radar has the potential to obtain more accurate micro‐motion parameters for target recognition. However, with the existing algorithms of micro‐motion parameter extraction it is difficult to obtain the real rotation radius and tilt angle of a rotational target in the presence of multiple scattering points in the radar beam. A micro‐motion parameter extraction algorithm for rotating targets based on the vortex EM wave radar is proposed in this article. The angular Doppler is obtained from the dual‐ mode vortex EM echoes. The time interval between the maximum and minimum angular Doppler frequency is derived. The relationship between the time interval and micro‐ motion parameters is shown. By combining the linear Doppler and the angular Doppler, the micro‐motion parameters are roughly estimated. Then, fine micro‐motion parameters are obtained by using an iterative soft threshold algorithm. The proposed algorithm can extract the real rotation radius and tilt angle in the case of multiple scattering points. The performance and robustness of the algorithm are proved by simulations.


| INTRODUCTION
Recently, due to its unique properties, the vortex electromagnetic (EM) wave [1] has attracted the attention of radar and communication researchers [2,3]. Compared with the traditional EM wave [4], the vortex EM wave carries the orbital angular momentum (OAM), and the helical phase fronts structure [5] is generated in the beam. The vortex EM waves of different OAM modes α are orthogonal to each other [6]. With traditional non-vortex EM waves, the relative velocity between the radar and the target leads to the linear Doppler effect [7]. The linear Doppler has been used to extract micro-motion parameters of targets [8][9][10]. Because the linear Doppler is induced by the relative movement along the radar line-of-sight (LOS), only the components of micro-motions projected onto the LOS can be observed from the radar echoes. When the vortex EM waves is used to illuminate the target, the relative velocity causes linear Doppler and angular Doppler effects [11]. By combining the linear Doppler and angular Doppler effects, the real rotation radius and the tilt angle of the target can be extracted [12].
So far, the application of the vortex EM wave has been studied by many researchers, including two-dimensional highresolution imaging [13] and low signal-to-noise ratio (SNR) imaging [14]. However, the application of the vortex EM wave in extracting micro-motion parameters is still in the development stage. In [12], the Doppler effect and the micro-Doppler effect of a vortex EM wave are investigated and the angular Doppler frequency shift is deducted. The method of micro-motion parameter extraction is discussed under special circumstances in which the tilt angle is 0 or the rotation centre is on the Z-axis. This work can benefit the applications of vortex EM waves. However, the method of angular Doppler extraction and the angular Doppler frequency shift of the target composed of multiple scattering points are not discussed.
Due to the coupling relationship between the linear Doppler and the angular Doppler, the angular Doppler cannot be directly obtained. In [15], an effective linear and rotational Doppler separation method and a motion parameter estimation method are proposed. First, the time-frequency graph is obtained by performing a short-time Fourier transform (STFT) in the echo. Subsequently, the angular Doppler is extracted by performing Hough transform in the timefrequency graph. However, the presence of multiple scattering points in the radar beam is not considered. When the target contains multiple scattering points, the difference between the linear micro-Doppler of each scattering point makes it difficult to compensate them with a same compensation operation; therefore, the linear micro-Doppler cannot be removed and the angular micro-Doppler will be contaminated. As a result, it is difficult to estimate the micro-motion parameters accurately.
In this article, a micro-motion parameter extraction algorithm for a rotating target based on the vortex EM wave is studied to offer a solution to solve these problems. First, the extreme value of the angular Doppler frequency shift in which the target is located in an arbitrary position is derived. Subsequently, an extraction algorithm for the angular Doppler based on dual echoes is proposed. The range-slow-time profile can be obtained by performing fast Fourier transform (FFT) for the echo in the fast-time domain. The angular Doppler is extracted by conjugate multiplication for the dual-mode profiles, and the linear Doppler is extracted from the echo received at the centre of the antenna array. By combining the angular Doppler and the linear Doppler, the micro-motion parameters are estimated. Compared with the existing algorithms, the advantage of the proposed algorithm is that the real rotation radius and tilt angle can be estimated in the presence of multiple scattering points in the radar beam.
The article is organised as follows: In Section 2, the radar echo model and the position relationship between the target and the radar are displayed. The extreme value of the angular Doppler frequency shift is derived in Section 3. The method of extraction of micro-motion parameters is proposed in Section 4. In Section 5, the proposed algorithm is verified. Conclusions are given in Section 6.

| OBSERVATION MODEL
So far, many methods of generating vortex electromagnetic waves have been proposed [16][17][18]. The single-in-multiple-out model of generating vortex EM wave is used in this article. As shown in Figure 1, the linear frequency modulation (LFM) signal s trans ðtÞ emitted by an antenna located at the origin of the coordinate can be given as where rectðt=T p Þ is a rectangular window, T p is the pulse duration, f c is the carrier frequency, γ is the frequency modulation rate. The echo is received by the uniform circular array (UCA) with radius a. The UCA is composed of N antennas, which are placed equidistant in a counter clockwise direction, and each antenna is multiplied by a phase term expðiαð2π=NÞnÞ; n ¼ 0; 1; …; N − 1, where α is the OAM modes. Assuming that the targets are the ideal scattering point, the multi-target echo sðt; αÞ received by the UCA can be expressed as shown in [16].
The geometric relationship between the target and the radar array is shown in Figure 1. O is the origin of the Cartesian coordinate system OXY Z, located at (0,0,0). O 0 is the origin of the reference coordinate system O 0 X 0 Y 0 Z 0 , located at ðx c ; y c ; z c Þ. The reference coordinate system is parallel to the Cartesian coordinate system. The scattering point P is located in the target-local coordinate system O 0 xyz, rotating around the origin O 0 . The location of P in the spherical coordinates can be written as ðr p ; θ p ; φ p Þ. The rotation frequency is f and the rotation radius is r a . Therefore, the coordinates of P in the target-local coordinate system are � where F I G U R E 1 Geometry of uniform circular array (UCA) radar and target R init ¼ ϕ e and φ e are the degree of rotation of the target about the Z 0 -axis, and only the initial phase will be affected by them.
Since the micro-motion parameter extraction is independent of the initial phase of the target, the rotation matrix R init can also be expressed as a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 It is assumed that the target is relatively stationary on the radar during the pulse duration. The slow time t m is the pulse interval time sequence, and the fast time t is expressed as the pulse duration time sequence. sðt; αÞ can be rewritten as Finally, the echo s o ðt; t m ; αÞ received by the antenna at the centre of the array can be expressed as

| ANGULAR DOPPLER EFFECT
In this section, the extreme value and the extreme value point of the angular Doppler frequency shift are derived. When the scattering point rotates in space, the azimuth angle φ p ðt m Þ can be expressed as shown in [12].
The angular Doppler frequency shift f A ðt m Þ is ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi where Since θ e ≠ 0 and the rotation centre is not on the Z-axis, the frequency shift of the angular Doppler is complicated. It can be found that the cosine functions with different initial phases exist in the numerator and denominator of Equation (8); as a result, the extreme value and the extreme value point of the angular Doppler frequency shift are difficult to obtain by derivation. The denominator of f A ðt m Þ can be rewritten as ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi where r c ¼ ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi and Aðt min Þ is the minimum of Aðt m Þ. When r c ≤ r a , the minimum of Aðt m Þ tends to zero. The interval of extremum of the denominator is related to the interval of extremum of f A ðt m Þ. The interval of extremum points of Aðt m Þ can be roughly estimated by the interval of extremum points of f A ðt m Þ. On ignoring the constant term in Equation (11), dðt m Þ can be expressed as dðt m Þ ¼ 0:5r 2 a sin 2 θ e cos 4 πf t m þ 2r a ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi Due to the influence of the initial phase in the cosine function, the two cases y c ¼ 0 and y c ≠ 0 need to be discussed separately.
The cosine function is expressed as a Maclaurin series and dðt m Þ can be rewritten as In Equation (10), d 2 ðt m Þ changes with t m and it can be expressed as It is assumed that the superscript "0" is the derivation operation. d 0 2 ðt m Þ can be obtained as The extreme value points t 1 and t 2 are presented as follows: ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi By substituting t 1 and t 2 into (10), the extreme values f 1 and f 2 can be given by It can be seen that f 1 and f 2 will be affected by the rotation frequency f , the tilt angle θ e , the rotation radius r a , and the rotation centre ðx c ; y c ; z c Þ. In this section, the time interval Δt between the maximum value point and the minimum value point is used to extract the micro-motion parameters. Δt is Δt ¼ jt 1 − t 2 j ¼ ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi À x c þ r a sin 2 θ e � 2 þ 2y 2 c q y c πf ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi � If the target rotation centre ðx c ; y c ; z c Þ is known, then r a sin 2 θ e can be obtained. r a sin 2 θ e ¼ y c ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ðπf when y c ¼ 0, the initial phase θ c in Equation (10) is eliminated. dðt m Þ is rewritten as It can be found that the number of extreme points is affected by x c and r a sin 2 θ e when y c ¼ 0. So, the two cases, x c ≥ r a sin 2 θ e and x c < r a sin 2 θ e , are discussed separately.
In this case, the extreme points t 1 and t 2 of Equation (22) can be expressed as By substituting t 1 and t 2 into (8), f 1 and f 2 can be obtained The ratio y 1 is when the value of x c is known, the radius of rotation r a can be obtained.
In this case, the extreme points t 1 ; t 2 ; t 3 of Equation (22) can be expressed as By substituting t 1 ; t 2 ; t 3 into Equation (10), the extreme values f 1 ; f 2 ; f 3 can be displayed as follows: The micro-motion information can be obtained by the time interval Δt of extreme points and Δt is

| MICRO-MOTION PARAMETER EXTRACTION
In this section, the extraction algorithm of micro-motion parameters is proposed. First, an extraction method of the angular Doppler based on dual echoes is proposed. Subsequently, rough micro-motion parameters are extracted by combining the linear Doppler and the angular Doppler. Lastly, the iterative soft thresholding (IST) algorithm [19] is used to estimate the fine micro-motion parameters. First, an extraction method of the angular Doppler based on dual echoes is proposed. It is assumed that the reference signal is s ref ðtÞ ¼ rectðt=T p Þe -i2πðf c tþ0:5γt 2 Þ . The echo sðt; t m ; αÞ is multiplied by the reference signal s ref ðtÞ, and then it leads to It is assumed that FFTf g is the fast Fourier transform. Sðf r ; t m ; αÞ ¼ FFTfs 1 ðt; t m ; αÞg can be expressed as and The amplitude spectrum of Hðf r Þ is similar to that of the sinc function. After removing the RVP term, the dual-mode signals can be expressed as In Equation (30), the phase terms are consistent except for e −iαφ p ðt m Þ . The two signals of Equation (30) are conjugate multiplied where P j;k ðf r ; t m Þ ¼ e −i2πðf r þγτ j ðt m ÞÞτ j ðt m Þ e i2πðf r þγτ k ðt m ÞÞτ k ðt m Þ , and the superscript " * " represents the conjugate operation.
Assuming that the rotating target is located in a range resolution unitf r ¼ −γτ j ðt m Þ, the peak S peak ðt m ; αÞ of Equation (33) can be expressed as The linear Doppler of Equation (34) is suppressed and the angular Doppler is extracted. Assuming that there is only a single scattering point in the range resolution unit, S 2 ðf r ; t m ; αÞ can be rewritten as The linear Doppler of Equation (35) is eliminated and the angular Doppler is extracted. In both cases, the angular Doppler can be extracted.
Subsequently, the rough micro-motion parameters are extracted by combining the linear Doppler and the angular Doppler. Since s o ðt; t m ; αÞ contains only the linear Doppler, it can be used to extract the radial length and rotation frequency. The range-slow-time profile is obtained by performing the FFT for s o ðt; t m ; αÞ in the fast-time domain. Then, the time-frequency graph is obtained by performing the STFT [20] for the peak of the range-slow-time profile.
The rotation frequency f can be estimated by using the autocorrelation method on the time-frequency diagram. The maximum frequency shift f max in the time-frequency diagram is and the radial length r 0 is By combining r a sin 2 θ e and r 0 , the rotation radius and tilt angle can be roughly estimated.
Lastly, the IST algorithm is used to estimate the fine micromotion parameters. The optimisation goal of the IST algorithm can be expressed as where D is the dictionary, ‖‖ 2 is 2-norm, ‖‖ 1 is 1-norm, and μ is the regularisation coefficient. X is a column vector. To construct the dictionary of the IST algorithm, the parameters σ j ; n; θ e ; r a ; f in S peak ðt m ; αÞ need to be obtained. The number of scattering points n can be obtained by the time-frequency diagram of the linear Doppler. According to the peak ratio of different scattering points, the relative value σ j of the normalised scattering coefficient can be obtained. Since the centre of rotation and r 0 are known, when θ e is determined, r a is also determined. Ignoring the constant t m and α in the dictionary, S peak ðt m ; αÞ under different tilt angles can be expressed as S p ðθÞ.
The target tilt angle can be obtained by the optimisation result of the IST algorithm. The maximum value in X is selected, and the corresponding θ k is the estimated value of the tilt angle.
The flow of the micro-motion parameter extraction method is shown in Figure 2.

| SIMULATION RESULTS AND ANALYSIS
In this section, the extreme value of the angular Doppler frequency in which the target is located in an arbitrary position and the extraction methods of the micro-motion parameters under different conditions are verified. In Section 1, the singlepoint target model is adopted so that the micro-Doppler shift can be calculated precisely from the phase derivation of the echo. This will be helpful to better verify the analyses of Section III and IV. In Section 2, a target model with 2 scattering points is used in the simulation to investigate the performance of the method under the condition of multiple scattering points.

| Micro-Doppler effect
The parameters of the radar and target are listed Table 1. The rotation radius is 0.2 m. It is assumed that radial length r 0 of 0.0684 m and rotation centre are known.
(1) y c ≠ 0 When the rotation centre is ð0:1; 0:1; 50Þm, the angular Doppler shift is as shown in Figure 3. The solid line is the theoretical value of the angular Doppler frequency, and the dashed line is the approximate value of the angular Doppler frequency after Maclaurin expansion. The maximum is 149.2 Hz at 0.0818s, and the minimum is 22.52 Hz at 0.0508s. It is found that the theoretical curve can be expressed accurately by the approximate curve.
According to Equation (17), the extreme values are 22.9 and 104.4 Hz, respectively. The minimum can be estimated accurately. Because the curve changes sharply at the maximum, it leads to high error. According to Equation (19), the time interval between the extreme points is Δt ¼ 0:03s. The time interval can be estimated accurately. The target tilt angle can be obtained by combining Equations (20) and (37). The tilt angle of the target is estimated as 0.3585 rad, and the radius of rotation is estimated as 0.1949 m. The micro-motion parameter can be reconstructed with high precision.
(2) y c ¼ 0; x c ≥ r a sin 2 θ e When the centre of rotation is (0.1,0,50) m, the angular Doppler frequency is as shown in Figure 4. The maximum is 75.18 Hz at 0.025s, and the minimum is 25.06 Hz at 0.05s. According to Equation (24), the extreme values are 25.05 and 75.17 Hz, respectively. The maximum and minimum can be estimated accurately. According to Equation (25), the rotation radius can be estimated as 0.2 m. Then the tilt angle can be estimated as 0.3484 rad. The micro-motion parameters can be reconstructed with high precision.  (20) and (37). The rotation radius is estimated as 0.1794 m. Δt cannot be accurately estimated by the interval of the extremum points; therefore, the micro-motion parameters cannot be accurately estimated.
The extreme value of the angular Doppler frequency is verified by simulation. If the linear Doppler and angular Doppler can be obtained, the micro-motion parameters can be estimated accurately. Compared with the method in [12], the tilt angle and rotation radius can be extracted, in which the target is located in an arbitrary position.

| Micro-motion parameter extraction
In this section, the effectiveness of the micro-motion parameter extraction method is verified. In practical F I G U R E 3 Theoretical and approximate curves YUAN ET AL.
applications, the centre of rotation deviates from the Z-axis. The proposed algorithm can be applied, as estimated, in the presence of multiple scattering points in the radar beam. In this case, the performance of the algorithm is discussed when there are single or multiple scattering points in the range resolution unit. The radar and target parameters are shown in Table 1. The target consists of two scattering points symmetrical about the centre of rotation, and the centre of rotation is (0.1,0.15,50) m. The scattering coefficient of point 1 is 1 and that of point 2 is 0.8.
When the radius of rotation is 1 m, the range-slow-time image can be obtained by performing FFT for s o ðt; t m ; αÞ, and the image is shown in Figure 6 (a). There is only one scattering point in a range resolution unit, and the radial length can be estimated as 0.45 m. Compared with the theoretical value 0.3387 m, the radial length is not accurately estimated. The time-frequency graph can be obtained by performing STFT for the echo of the red box in Figure 6a. The time-frequency graph of a single range unit is shown in Figure 6b. The maximum frequency is 2875 Hz. According to Equation (37), the radial length r 0 can be estimated as 0.3432 m. Since the amplitude ratio of the curve peaks of the two scattering points is approximately 0.8, the normalised scattering coefficients of the two scattering points are estimated as 1 and 0.8. To separate the echoes of different scattering points, the target trajectory in the range-slow-time domain needs to be obtained. The trajectories of the scattering points are obtained and extracted by the skeleton. The range units on the trajectory are extracted, and the scattering points are separated. The time-frequency graph of scattering point 1 is shown in Figure 6c.  Figure 7b. By selecting the maximum frequency and the minimum frequency, the time interval Δt ¼ 0:036sis obtained. According to Equations (20) and (37), the rough micro-motion parameters, including the rotation radiusr a ¼ 0:9387m and tilt angleθ e ¼ 0:3774rad, are obtained.
The search range is set near the rough value. So, according to the obtained rough value, the parameters of the IST algorithm can be set as θ 0 ¼ 0:2264rad, θ k ¼ 0:5241rad, and θ k − θ k−1 ¼ 0:006. The rotation radiusr a ¼ 1:041m and tilt angleθ e ¼ 0:3394rad are precisely estimated by the IST algorithm.
When the radius of rotation is 0.2 m, the time-frequency graphs of the linear Doppler and angular Doppler are shown in Figure 8. In this case, the target motion is a line in the rangeslow-time domain. The linear Doppler time-frequency diagram is shown in Figure 8a. According to Equation (37), the target radial length is calculated as r 0 ¼ 0:0671m. The normalised scattering coefficient of the two scattering points are estimated as 1 and 0.8, respectively. Subsequently, the time-frequency diagram of the angular Doppler is shown in Figure 8b. The time interval between the maximum frequency location and the minimum frequency location is 0.029s. Compared with the ideal time  interval of 0.0266s, the time interval cannot be accurately estimated. According to Equations (20) and (37), the tilt angle is estimated as 0.8563 rad and the rotation radius is estimated as 0.0899 m. To obtain the micro-motion parameters, the IST algorithm is applied. The parameters of the IST algorithm can be set as θ 0 ¼ 0:297rad, θ k ¼ 1:412rad, and θ k − θ k−1 ¼ 0:022rad. The rotation radius is estimated asr a ¼ 0:2022m and the tilt angle is estimated asθ e ¼ 0:3451rad. The target micro-motion parameters can be accurately estimated. Compared with the method in [15], the tilt angle and rotation radius can be extracted in the presence of multiple scattering points.
It is found that the proposed algorithm mainly consists of two parts: the micro-motion parameters, which are roughly estimated from the time-frequency diagrams of angular Doppler and the fine estimated value, which is further obtained by the IST algorithm. The first part does not require iteration and search operations, and so it takes a very short computation time, which is consistent with the traditional method. The second part contains iteration and search operations; however, because the micro-motion parameters have been roughly estimated in the first part, the computation time will not be very long. On our personal computer, with 16 GB memory and GeForce MX350 graphics card, it takes about 4s to run the programme.

| Robustness analysis
Since radar works in a noisy environment, the robustness of the algorithm to noise is demonstrated. In this section, the complex Gaussian white noise is added to the echo. The radar First, the performance of the proposed algorithm is demonstrated when the rotation radius is 1 m. When the signal-to-noise ratio (SNR) is 0dB, the time-frequency diagram of the linear Doppler and the angular Doppler are shown in Figure 9 (a) and (b), respectively. According to Figure 9b, the time interval between the maximum frequency and the minimum frequency is 0.035s. The radial length r 0 ¼ 0:3432m is obtained by Figure 9a. The rotation radiusr a ¼ 1:0811m and tilt angleθ e ¼ 0:3266rad are obtained. Even if the SNR is 0 dB, the micro-motion parameters can be accurately extracted.
To verify the robustness of the algorithm, the error curve of the tilt angle at different SNR is exhibited in Figure 10a, and the error curve of the rotation radius at different SNR is exhibited in Figure 10b. Because scattering points need to be separated, the operation is greatly affected by noise. With the decrease of SNR, the estimation accuracy of micro-motion parameters also decreases.
Finally, the performance of the proposed algorithm is demonstrated when the rotation radius is equal to 0.2 m. When the SNR is 0dB, the time-frequency diagram of the linear Doppler is shown in Figure 11. The radial length r 0 ¼ 0:0671m can be estimated by Figure 11. The IST algorithm is applied, and the rotation radiusr a ¼ 0:1996m and the tilt angle ŝ θ e ¼ 0:3459rad are estimated.
The error curve of the tilt angle at different SNR is exhibited in Figure 12a, and the error curve of the rotation radius at different SNR is exhibited in Figure 12b. When the rotation radius is equal to 0.2 m, there is no need to separate the scattering points, and the algorithm performance is less affected by noise. Compared with the case of 1 m rotation radius, the micro-motion parameters can be estimated more accurately in low SNR.

| CONCLUSION
Vortex electromagnetic waves have attracted much attention in radar applications. In this article, the extraction algorithm of micro-motion parameters for rotating targets is studied. Compared with the existing algorithms, the advantage of the proposed algorithm is that it can be applied in the presence of