# Removal of Micro-Doppler Effects in ISAR Imaging Based on the Joint Processing of Singular Value Decomposition and Complex Variational Mode Extraction

Authors: Xu Wei; Jun Yang; Mingjiu Lv; Wenfeng Chen; Xiaoyan Ma; Saiqiang Xia; Ming Long
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

For inverse synthetic aperture radar (ISAR) imaging of targets with micromotion parts, the removal of micro-Doppler (m-D) effects is the key procedure. However, under the condition of a sparse aperture, the echo pulse is limited or incomplete, giving rise to the difficulty of eliminating m-D effects. Thus, a novel m-D effects removal algorithm is proposed, which can effectively eliminate m-D effects, as well as the interference introduced by noise and sparse aperture in the ISAR image of the main body. The proposed algorithm mainly includes two processing steps. First, combined with the cut-off point determined by the normalized singular value difference spectrum, the rough estimation of the main body is achieved by singular value decomposition (SVD). Then, the variational mode extraction (VME) is extended to complex variational mode extraction (CVME). The constrained variational problem constructed by bandwidth and spectral overlap constraints is solved by the alternating direction method of multipliers (ADMM), and the precise estimation of the main body is obtained. Experimental results based on both simulated and measured data demonstrate that the proposed algorithm can acquire the high-resolution ISAR image of the main body under noisy and sparse conditions.


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## Body

## 1. Introduction

As an advanced means of remote sensing, inverse synthetic aperture radar (ISAR) can realize two-dimensional high-resolution imaging of the noncooperative moving target, in which the range and azimuth resolutions depend on the bandwidth of radar transmitted signal and the rotation angle of the target relative to radar line of sight (RLOS), respectively [1]. With the maturity of ISAR imaging technology, ISAR plays a more and more important role in military and civil fields. Generally, in the process of ISAR imaging, after effective translational motion compensation (including range alignment [2] and phase adjustment [3]), the target imaging can be simply regarded as an ideal turntable model. Under the condition of full aperture, a focused ISAR image of the target can be obtained by adopting the traditional range-Doppler (RD) algorithm.In the real imaging scene, targets detected by the radar sometimes contain mechanical vibration or rotating structures, such as the rotor helicopter, the aircraft equipped with turboprops, and the ship with scanning antennas. These targets with micromotion parts produce not only the Doppler frequency of the main body but also additional time-varying Doppler frequency modulation, which is called micro-Doppler (m-D) effects [4]. When imaging the micromotion target, the m-D effects can be utilized as the internal feature of the target to improve the accuracy of automatic target recognition (ATR), but it will also interfere with the imaging of the main body and reduce the readability of the imaging result [5]. Therefore, in order to obtain a clear ISAR image of the main body, the removal of m-D effects has become an indispensable processing segment in the imaging process. In recent years, many m-D effects removal algorithms have been proposed to improve the imaging quality of the main body. According to the rotation radius of micromotion parts, m-D effects are mainly divided into the following two cases [6].For one thing, when the rotation radius is small (less than or equal to the range resolution cell), the micromotion parts show the same linear shape as the main body in the high-resolution range profile (HRRP) after range compression, so it is impossible to separate the two in HRRP directly. To solve the above problem, the adaptive linear frequency modulation representation is adopted to realize the separation of the main body and micromotion parts in the parameter space, but the construction of the linear frequency modulation basis function is difficult [7]. Stanković et al. combine short-time Fourier transform (STFT) with L-statistics to eliminate the m-D effects in the time-frequency domain [8], but the missing of the main body components may exist. In [9, 10], the missing main body signal can be reconstructed by the genetic algorithm (GA) and particle swarm optimization (PSO) algorithm, respectively. Then, a complete main body is obtained at the cost of time. In addition, the signal decomposition technologies can also acquire the effective separation of the main body and micromotion parts, such as complex empirical mode decomposition (CEMD) [11], complex local mean decomposition (CLMD) [12], and bivariate variational mode decomposition (BVMD) [13], but there are still some problems of difficult parameter setting and low calculation efficiency.For another, when the rotation radius is large (greater than the range resolution cell), the micromotion parts present a sinusoidal shape in HRRP, which is significantly different from the main body, so the m-D effects can be eliminated in HRRP directly. Thus, Zhang et al. propose the extended Hough transform (EHT) to extract the sinusoidal curve in HRRP to eliminate the micromotion signal component, but the amount of the parameter search is large [14]. In [15], based on the assumption that the echo amplitude of the main body is relatively stable, a spectrum cancellation technology is put forward for the main body imaging, but the phase information of the micromotion signal component cannot be completely eliminated, affecting the imaging quality of the main body. In view of the feature that the main body signal has row sparsity in HRRP, multiple sparse Bayesian learning (MSBL) [16] and Markov chain Monte Carlo (MCMC) sampling [17] are adopted to reconstruct the HRRP only containing the main body, and the ISAR image of the main body is obtained by fast Fourier transform (FFT) in azimuth, but the parameter adjustment process of the algorithms may be cumbersome. In [18], robust principal component analysis (RPCA) technology is performed to decompose each range cell into low-rank and sparse matrices, and the minimum entropy method is combined to obtain the high-resolution image of the main body, but the imaging efficiency needs to be further improved.However, during the actual imaging process, the time-sharing working mode of the radar, the noncooperation of the target, and strong noise interference will make the echo pulse limited or incomplete, namely, sparse aperture, leading to the high side and grating lobes in the RD imaging result [19], which may not only affect the imaging quality of the main body but also challenge the original m-D effects removal algorithms. In recent years, compressive sensing (CS) is widely applied to ISAR imaging and multiple-input multiple-output (MIMO) radar signal processing [20, 21]. Therefore, combined with its advantages in the high-precision reconstruction for undersampling data [22], the orthogonal matching pursuit (OMP) algorithm is utilized to reconstruct the ISAR image of the main body, but the ability of noise suppression is poor [23]. By improving the joint sparsity of the time-frequency domain, smoothed (SL2L0) algorithm is proposed in [24] to eliminate m-D effects, but the solution to the sparse reconstruction problem is limited by the time-frequency aggregation of STFT. In order to further improve the removal of m-D effects under the conditions of noise and sparse aperture, singular value decomposition [25] and complex variational mode extraction (SVD-CVME) are jointly processed to eliminate the m-D effects and the interference introduced by noise and sparse aperture in the ISAR image of the main body. In the proposed algorithm, combined with the signal model of the target with micromotion parts, SVD is used to achieve the rough estimation of the main body first. Then, the complex processing ability of variational mode extraction (VME) [26] is added to further eliminate the interference signal components affecting the main body imaging, obtain the precise estimation of the main body, and acquire the high-resolution ISAR image of the main body by FFT in azimuth. Finally, simulated and measured data are made use of to verify the effectiveness of the proposed algorithm for two kinds of m-D effects.The rest of this paper is organized as follows. Section2 constructs the signal model of the target with micromotion parts and compares the Doppler frequency differences between the main body and micromotion parts. Section 3 proposes the joint processing algorithm of SVD-CVME and sorts out the processing flow of the removal of m-D effects. Section 4 analyses the experimental results of the proposed algorithm under noisy and sparse conditions. The conclusions are briefly drawn in Section 5.


## 2. Signal Model

Without losing generality, considering the influence of micromotion parts in the imaging target, the ISAR imaging geometric model shown in Figure1 is constructed. In Figure 1, assuming that the translational motion compensation has been accurately completed, the target imaging can be simplified to an ideal turntable model. The imaging plane is composed of the coordinate system XOY. The origin O, Y−axis, and X−axis denote the target imaging center, the RLOS direction, and the direction perpendicular to RLOS, respectively. UO′V is the coordinate system where micromotion parts are located, and the origin O′ is the rotation center of micromotion parts. p and q represent an arbitrary scattering point on the main body and micromotion parts, respectively. The scattering point p rotates uniformly around the center O, with the radius rp, the angular velocity ωB , and the initial phase θp. The scattering point q rotates uniformly around the center O′, with the radius rq, the angular velocity ωR , and the initial phase θq. The rotation center O′ can be regarded as located on the main body and makes the same motion as the scattering point p with the radius rO′ and the initial phase θO′. R0 is the initial distance from the origin O to the radar.Figure 1
ISAR imaging geometric model of the target with micromotion parts.In order to obtain a longer detection range and higher range resolution simultaneously, a linear frequency modulation (LFM) signal is usually transmitted in the process of ISAR imaging, which can be expressed as [27](1)st^,tm=rectt^Tpexpj2πfct+12γt^2,where Tp, fc, γ, t, t^, and tm denote the pulse width, carrier frequency, frequency modulation slope, full time, fast time, and slow time, respectively. γ=B/Tp, and B is the bandwidth. t=t^+tm, tm=m⋅PRI, m=0,1,…,M−1, M is the number of pulses, and PRI is the pulse repetition interval rectu=1,u≤1/20,u>1/2b2−4ac.It is assumed that the target consists ofP scattering points from the main body and Q scattering points from micromotion parts. The echo signal received by the radar is represented as(2)srt^,tm=∑p=1Pσprectt^−2Rptm/cTpexpj2πfct−2Rptmc+12γt^−2Rptmc2+∑q=1Qσqrectt^−2Rqtm/cTpexpj2πfct−2Rqtmc+12γt^−2Rqtmc2,where c is the light velocity. σp and σq denote the scattering coefficients of the p−th main body scattering point and the q−th micromotion scattering point, respectively. To reduce the effective received bandwidth and realize broadband radar imaging efficiently, the target echo is processed by Dechirp first. Then, the residual video phase (RVP) and envelope skew terms are compensated [28], and the high-resolution range profile (HRRP) of the target after range compression can be obtained:(3)Sfr,tm=∑p=1PσpTpsincTpfr+2γcRΔptmexp−j4πλRΔptm+∑q=1QσqTpsincTpfr+2γcRΔqtmexp−j4πλRΔqtm,where fr is the range frequency, λ=c/fc is the wavelength, sincu=sinπu/πu, RΔptm=Rptm−Rref, and RΔqtm=Rqtm−Rref. Rptm and Rqtm denote the distance from the p−th main body scattering point and the q−th micromotion scattering point to the radar at tm time, respectively. Rref is the reference distance for Dechirp processing, which is usually selected as R0. Since ISAR imaging meets the far-field detection conditions, that is, rp,rq,rO′≪R0, after the translational motion compensation, RΔptm and RΔqtm are approximately expressed as(4)RΔptm≈rpsinωBtm+θp,RΔqtm≈rO′sinωBtm+θO′+rqsinωRtm+θq.In general, during the short coherent processing interval (CPI), the rotation angle of the main body is very small, while the rotation angle of micromotion parts is large [5]. Then, equation (4) can be further simplified as(5)RΔptm≈xpωBtm+yp,RΔqtm≈xO′ωBtm+yO′+rqsinωRtm+θq,where xp,yp denotes the initial coordinate of the p−th main body scattering point, xp=rpcosθp and yp=rpsinθp. xO′,yO′ denotes the initial coordinate of the rotation center O′ from micromotion parts, xO′=rO′cosθO′ and yO′=rO′sinθO′.Based on the assumption that migration through resolution cells (MTRCs) caused by the rotation is usually less than one range cell in short CPI [6], ignoring the influence of MTRC, and substituting equation (5) into equation (3), we can obtain(6)Sfr,tm=∑p=1PσpTpsincTpfr+2γcypexp−j4πλxpωBtm+yp+∑q=1QσqTpsincTpfr+2γcyO′+rqsinωRtm+θq⋅exp−j4πλxO′ωtm+yO′+rqsinωRtm+θq.According to the amplitude part of equation (6), the peaks of the p−th main body scattering point and the q−th micromotion scattering point are, respectively, located at(7)frp=−2γcyp,frq=−2γcyO′+rqsinωRtm+θq.When the rotation radiusrp of micromotion parts is less than or equal to the range resolution cell ρr=c/2B, the peaks of the main body and micromotion parts are in the shape of a straight line in HRRP, so the m-D effects cannot be eliminated directly in HRRP. When the relationship between rp and ρr meets: rq>ρr, the peaks of the main body and micromotion parts show the shape of a straight line and sinusoidal curve in HRRP, respectively, which are obviously different from each other.From the phase part of equation (6), the Doppler frequencies of the p−th main body scattering point and the q−th micromotion scattering point are, respectively,(8)fdp=−2λxpωB,fdq=−2λxO′ωB+rqωRcosωRtm+θq.It can be seen that regardless of the rotation radius of micromotion parts, the Doppler frequency of the main body is constant, and the Doppler frequency of micromotion parts changes with time; that is, the main body and micromotion parts show the shape of straight line and cosine curve in the time-frequency domain, respectively. Thus, on account of the Doppler frequency differences, the main body and micromotion parts can be separated, so as to eliminate the influence of m-D effects on the imaging of the main body.At last, in order to describe the matrix form of HRRP under the conditions of noise and sparse aperture, considering the additive complex Gaussian white noise, the HRRP of equation (6) can be written as Sn,m in discrete form, n=1,2,…,N, m=1,2,…,M, and N and M are the number of range cells and pulses, respectively. The full aperture signal sequence of the n−th range cell is defined as(9)Snf=Sn,1,Sn,2,…,Sn,M.Supposing that the sparse aperture is composed ofD subapertures, the d−th subaperture contains Ld pulses, whose corresponding sequence number is expressed as Md:Md+Ld−1, then the signal sequence of the d−th subaperture in the n−th range cell is described as(10)Snds=Sn,Md,Sn,Md+1,…,Sn,Md+Ld−1,where 1≤Md≤M−Ld+1, 1≤Ld≤M, and L1+L2+⋯+LD=M¯M¯≤M. Since each subaperture is arranged in sequence, and the d−1−th and d−th subapertures meet the following relationship: Md−1+Ld−1−1<Md, then the sparse aperture signal sequence of the n−th range cell is written as(11)Sns=Sn1s,Sn2s,…,SnDs.Keep the signals corresponding toSns in Snf and set the other signals to zeros. At this time, the signal sequence of the n−th range cell is recorded as Sn. The schematic diagram of signal sequences of Snf, Sns, and Sn is shown in Figure 2. Because of the fact that the sparse aperture sequence number of each range cell is the same, the matrix form of HRRP under the conditions of noise and sparse aperture can be expressed as S=S1T,S2T,…,SNTT.Figure 2
The schematic diagram of signal sequences.


## 3. The Proposed m-D Effects Removal Algorithm

Under the condition of full aperture, many effective m-D effects removal algorithms have appeared in recent years, but their applications are usually limited to one of the following two cases: (1) the rotation radius of micromotion parts is less than or equal to the range resolution cell; (2) the rotation radius of micromotion parts is larger than the range resolution cell. Concurrently, under the conditions of noise and sparse aperture, the performance of m-D effects removal of the existing algorithms becomes worse, and they cannot overcome the high side and grating lobes caused by the sparse aperture on the imaging result, seriously affecting the imaging quality of the main body. Thus, by the joint processing of SVD-CVME, the high-resolution ISAR image of the main body can be obtained under the conditions of noise and sparse aperture, which is suitable for the above two cases.


### 3.1. SVD

As an effective signal processing method, SVD is widely used in signal noise reduction, weak signal extraction, and mechanical fault diagnosis [29]. Different from traditional filtering methods, SVD uses singular values to reflect the different component information of the original signal and separates the signal and noise through the selection of the number of singular values, which is suitable for the case that the frequency components of the signal overlap each other. According to the previous analysis of the signal model of the target with micromotion parts, the Doppler frequencies of the main body and micromotion parts are obviously different, and there is a partial overlap in the frequency or time-frequency domains. In this paper, SVD is used to separate the main body and micromotion parts to eliminate the m-D effects and retain the component of the main body.Before using SVD for signal processing, it is necessary to construct a data matrix related to the original signal. In [30], it points out that constructing a one-dimensional signal sequence into a Hankel matrix is the most commonly used and effective method. Take the signal sequence Sn of the n−th range cell as an example for analysis. For ease of presentation, let G=Sn, G=g1,g2,…,gM, and convert G to multidimensional sequences G1,G2,…,GI, in which Gi is Gi=gi,gi+1,…,gi+J−1. Then, the Hankel matrix H is constructed as(12)H=G1G2⋮GI=g1g2⋯gJg2g3⋯gJ+1⋮⋮⋱⋮gIgI+1⋯gM,where H∈ℂI×J, 1<J<M, and M=I+J−1.For the Hankel matrixH composed of the signal sequence, its SVD is defined as(13)H=UΣVH,where ∑∈ℝI×J is a diagonal matrix, ∑=DI<J or ∑=DTI≥J, D=diagξ1,ξ2,…,ξL,0, L=minI,J, ξ1≥ξ2≥⋯≥ξL≥0, and ξl is the l−th singular value of H, l=1,2,⋯,L. U∈ℂI×I and V∈ℂJ×J are orthogonal matrices, representing the left singular matrix and the right singular matrix of ∑, respectively. For the essence of SVD, each singular value corresponds to a signal component, so the SVD of H can also be written as(14)H=ξ1u1v1H+ξ2u2v2H+⋯+ξLuLvLH=H1+H2+⋯+HL,where ul∈ℂI×1 and vl∈ℂJ×1 are the l−th column vector of U and V, respectively. Hl=ξlulvlH is the Hankel matrix formed by the l−th signal component.It can be seen from [29] that when SVD is used for signal noise reduction, the signal components usually focus on a few larger singular values, and other smaller singular values represent the noise components. Similarly, the Doppler frequency of the main body is constant, and the energy is relatively concentrated in the frequency or time-frequency domains, while the Doppler frequency of micromotion parts changes with time, and the energy is relatively dispersed in the frequency or time-frequency domains. Therefore, after SVD of the signal sequence of a certain range cell, the singular value of the main body is much larger than that of micromotion parts. In order to effectively eliminate the m-D effects to obtain the focused ISAR image of the main body, it is necessary to find the cut-off point between the singular values of the main body and micromotion parts and retain all the singular values of the main body as much as possible.At present, there is no unified method for the selection of a singular value cut-off point. In [31], the skewness coefficient and Spearman correlation coefficient of each singular value distribution are calculated through multiple simulation experiments to judge the cut-off point of singular values, but the demand for data is large and cannot meet the requirements of real time. Another simple and widely used method is to determine the cut-off point through a singular value difference spectrum. The effectiveness of the singular value difference spectrum is theoretically analyzed in [29]. According to the singular value obtained by SVD, the singular value difference spectrum w is defined as(15)w=w1,w2,…,wL−1,where wl=ξl−ξl+1, l=1,2,…,L−1.When the signal component is simple and the noise is less, the singular value cut-off point determined by the peak ofw can realize the effective separation of signal and noise. However, under the conditions of noise and sparse aperture, if the peak of w is directly used to determine the cut-off point between the singular values of the main body and micromotion parts, some parts of the main body may be missing. Thus, the method proposed in [25] is adopted to determine the cut-off point. Firstly, the maximum value of w is normalized as w¯=w/maxw, and then a small threshold μ is given. Finally, w¯ is judged one by one. When the k+1−th normalized difference spectrum w¯k+1k=1,2,…,L−2 is less than μ for the first time, the cut-off point of singular values is determined as L′=k. In combination with equation (14), the estimated value of the main body signal sequence is(16)GB=H′1,1,…,H′1,J,H′2,J,…,H′I,J,where H′=H1+H2+⋯+HL′, and H′i,j is the element of the i−th row and j−th column in H′.


### 3.2. CVME

In view of the influence of noise and sparse aperture, as well as the inaccurate selection of threshold when determining the cut-off point of singular values, after using SVD to separate the main body and micromotion parts, there will inevitably be residual m-D effects and the interference introduced by noise and sparse aperture. Therefore, after SVD processing, the residual interference signal components need to be eliminated to obtain the focused high-resolution ISAR image.To further eliminate the interference signal components affecting the main body imaging, a complex signal decomposition method is proposed, namely, CVME. As a simplified method of variational mode decomposition (VMD) [32], variational mode extraction (VME) [26] can efficiently extract the specific or expected mode in the original signal without presetting the exact number of modes (decomposition layers) like VMD. At the same time, it has a strict mathematical theoretical framework and is robust to the suppression of noise and sampling error. However, VME can only process real signals and cannot be directly used to eliminate the m-D effects. It is necessary to increase the capacity of complex processing. The common methods are to project the complex signals in different directions or separate the positive and negative frequency components [33, 34], but these methods have the problems of loss of signal components and a large amount of calculation. Therefore, the idea of [35] is adopted in this paper. During the construction of the bandwidth constraint criterion, the Hilbert transform is ignored and complex signals are directly used to represent this criterion. Then, VME is extended to CVME to realize the decomposition of complex signals.Next, take the signal sequenceSn of the n−th range cell as an example for analysis. According to Section 3.1, the estimated value of the main body signal sequence after SVD processing is GB. For ease of representation, let ytm=GB, and assume that ytm after SVD processing can be expressed as(17)ytm=ybtm+yrtm,where ybtm is the desired main body signal component. yrtm is the residual signal components, which are independent of the main body and need to be eliminated before imaging.To successfully extract the desired main body signal components for CVME, the following two constraints need to be met [26]:(1)
Bandwidth Constraint. The desired main body signal component ybtm should be compact around its center frequency ωb; that is, the criteria of bandwidth constraint are established as follows:(18)J1=∂tmybtmexp−jωbtm22,where∂tm⋅ represents the partial derivative with respect to tm, and ⋅2 stands for l2 norm.(2)
Spectral Overlap Constraint. Firstly, a proper filter β^ω is designed. The spectrum of the filter has an infinite gain at ω=ωband behaves like a Wiener filter when far away from ωb. Then, to ensure the integrity of the desired signal extraction, the spectral overlap between the residual signal components yrtm and ybtm should be minimized, which is equivalent to the minimum energy of yrtm around ωb; that is, the criteria of spectral overlap constraint are established as follows:(19)J2=βtm∗yrtm22,where “∗” denotes the convolution operation. βtm is the impulse response of the designed filter β^ω, β^ω=1/αω−ωb2, and α is the bandwidth control parameter of the filter.Considering the bandwidth and spectral overlap constraints above, the extraction of the desired main body signal component can be transformed into the following constrained variational problem:(20)minyb,ωb,yrαJ1+J2s.t.ytm=ybtm+yrtm.In order to make equation (20) have better convergence and accurate reconstruction performance, we introduce a combination of quadratic penalty term and Lagrange multiplier into equation (20) to establish the augmented Lagrangian function, as follows:(21)Lyb,ωb,yr,η=α∂tmybtmexp−jωbtm22+βtm∗yrtm22+ytm−ybtm+yrtm22+ηtm,ytm−ybtm+yrtm,where ηtm is the Lagrangian function. ⋅ denotes the inner product in the complex domain. Making use of Parseval’s equality [26]: xtm22=x^ω22, where x^ω is the Fourier transform of xtm, the time domain form of equation (21) can be transformed into the frequency domain form, and then by changing variables in the first item, equation (21) can be rewritten as(22)Lyb,ωb,yr,η=αjω−ωby^bω22+β^ωy^rω22+y^ω−y^bω+y^rω22+η^ω,y^ω−y^bω+y^rω≈αjω−ωby^bω22+β^ωy^rω22+y^ω−y^bω−y^rω+η^ω222.According to [32], the saddle point of the above multivariable problem is the frequency domain solution of the constrained variational problem shown in equation (20). To solve this problem efficiently, an effective solution method is given in VMD [36], namely, the alternating direction method of multipliers (ADMM). In this method, equation (22) is transformed into a series of suboptimization problems, and different variables yb,ωb,yr,η are updated alternately in the iterative process. In the k+1−th iteration, the first step of CVME is to update the desired main body signal component y^bk+1ω, and its suboptimization problem is(23)y^bk+1ω=argminy^bα∫−∞∞ω−ωbk2y^bω2dω+∫−∞∞y^ω−y^bω−y^rkω+η^kω22dω,where k is the number of iterations, k=0,1,2,…. The first derivative with respect to y^bω is calculated, and we can get the updated form of y^bk+1ω, as follows:(24)y^bk+1ω=y^ω−y^rkω+η^kω/21+αω−ωbk2.In the second step, the suboptimization problem of the center frequencyωbk+1 is(25)ωbk+1=argminωbα∫−∞∞ω−ωb2y^bk+1ω2dω+1α2∫−∞∞y^rkω2ω−ωb4dω,where the first derivative with respect to ωb is calculated, and the updated form of ωbk+1 can be obtained as(26)ωbk+1=α∫−∞∞ωy^bk+1ω2dω−2/α2∫−∞∞y^rkω2/ω−ωbk5dωα∫−∞∞y^bk+1ω2dω.In the third step, the suboptimization problem of residual signal componentsy^rk+1ω is(27)y^rk+1ω=argminy^r1α2∫−∞∞y^rω2ω−ωbk+14dω+∫−∞∞y^ω−y^bk+1ω−y^rω+η^kω22dω,where the first derivative with respect to y^rω is calculated, and the updated form of ωbk+1 can be expressed as(28)y^rk+1ω=α2ω−ωbk+14y^ω−y^bk+1ω+η^kω/21+α2ω−ωbk+14.The last step is to update the Lagrangian multiplierη^k+1ω by adopting the dual ascent method [32], and the updated form of which is(29)η^k+1ω=η^kω+τy^ω−y^bk+1ω−y^rk+1ω,where τ is the update parameter of the Lagrangian multiplier, which can be set by the method proposed in [32].So far, one iteration of CVME can be realized according to equations (24), (26), (28), and (29). When the desired main body signal component y^bk+1ω meets the following convergence condition, the iterative process is ended in advance, the extracted main body component y^bω at this time is output, and then the inverse Fourier transform is performed to obtain the estimated time-domain signal y^btm of the main body.(30)re=y^bk+1ω−y^bkω22/y^bkω22<ε,where ε is the tolerance of convergence condition.


### 3.3. Flow Chart of the Proposed Algorithm

In order to clearly describe the proposed m-D effects removal algorithm, the algorithm flow chart shown in Figure3 is constructed. In Figure 3, the proposed algorithm mainly involves two processing steps of SVD and CVME to obtain the rough estimation and precise estimation of the main body, respectively. The imaging quality of the main body is improved by the joint processing of SVD-CVME.Figure 3
Flow chart of the proposed m-D effects removal algorithm.In terms of parameter initialization of SVD, in [29], it points out that when I and J are close or equal, the Hankel matrix constructed by the input signal sequence is conducive to improve the performance of SVD. Thus, we let I=M/2+1 (M is even) or I=M+1/2 (M is odd) in this paper. In addition, for the selection of the singular value cut-off point between the main body and micromotion parts, since the input signal sequence is affected by noise and sparse aperture, it is impossible to accurately give the cut-off point to completely separate the signal components of the main body and micromotion parts. Here, the threshold μ is set to 0.01 to retain the signal components of the main body as much as possible.In CVME, the bandwidth control parameterα is an important parameter. A smaller α makes the filter bandwidth larger and affects the elimination effect of residual signal components. A larger α makes the filter bandwidth smaller and the expected extracted signal components may suffer from partial loss. Thus, a simple heuristic method [37] is adopted to change α. In the iterative process, α is multiplied from αmin to αmax, so as to retain the desired signal components and eliminate the remaining signal components, in which αmin and multiple γ are simply set to 1 and 2, and αmax is defined in the simulation experiment. At the same time, set the iterative convergence condition as shown in equation (30) and give the maximum number of iterations to reduce the iterative time of CVME. Generally, when ε is set to 10−6 and Imax is set to 100, good decomposition results can be obtained, and the amount of computation of cyclic iteration is relatively small.To further reduce the joint processing time of SVD-CVME, FFT in azimuth is performed on the initial HRRP to obtain the initial ISAR imageX0∈ℂN×M. X0 is transformed into X¯0=X0/maxn,mX0 via the operations of modulus and maximum normalization, and the maximum value of each range cell is calculated as C=maxmX¯0, C=c1,c2,…,cN. Then, the threshold is determined as [38](31)th=Ec+1N−1∑n=1Ncn−Ec21/2,where Ec=1/N∑n=1Ncn, and according to the threshold th, we can estimate the number of range cells NS where the imaging target is located, and carry out the subsequent processing of SVD and CVME, respectively.In the end, we further analyze the computational complexity of the proposed algorithm, which consists of SVD and CVME. The main computational burden of SVD comes from SVD processing of the Hankel matrix, and its computational complexity is approximatelyOIJ2. CVME mainly involves matrix multiplication. Assuming that the number of iterations of CVME is N1, its computational complexity is approximately ON1M. Thus, combined with the number of range cells NS, the total computational complexity of the SVD-CVME algorithm is approximately ONSIJ2+N1M.


## 3.1. SVD

As an effective signal processing method, SVD is widely used in signal noise reduction, weak signal extraction, and mechanical fault diagnosis [29]. Different from traditional filtering methods, SVD uses singular values to reflect the different component information of the original signal and separates the signal and noise through the selection of the number of singular values, which is suitable for the case that the frequency components of the signal overlap each other. According to the previous analysis of the signal model of the target with micromotion parts, the Doppler frequencies of the main body and micromotion parts are obviously different, and there is a partial overlap in the frequency or time-frequency domains. In this paper, SVD is used to separate the main body and micromotion parts to eliminate the m-D effects and retain the component of the main body.Before using SVD for signal processing, it is necessary to construct a data matrix related to the original signal. In [30], it points out that constructing a one-dimensional signal sequence into a Hankel matrix is the most commonly used and effective method. Take the signal sequence Sn of the n−th range cell as an example for analysis. For ease of presentation, let G=Sn, G=g1,g2,…,gM, and convert G to multidimensional sequences G1,G2,…,GI, in which Gi is Gi=gi,gi+1,…,gi+J−1. Then, the Hankel matrix H is constructed as(12)H=G1G2⋮GI=g1g2⋯gJg2g3⋯gJ+1⋮⋮⋱⋮gIgI+1⋯gM,where H∈ℂI×J, 1<J<M, and M=I+J−1.For the Hankel matrixH composed of the signal sequence, its SVD is defined as(13)H=UΣVH,where ∑∈ℝI×J is a diagonal matrix, ∑=DI<J or ∑=DTI≥J, D=diagξ1,ξ2,…,ξL,0, L=minI,J, ξ1≥ξ2≥⋯≥ξL≥0, and ξl is the l−th singular value of H, l=1,2,⋯,L. U∈ℂI×I and V∈ℂJ×J are orthogonal matrices, representing the left singular matrix and the right singular matrix of ∑, respectively. For the essence of SVD, each singular value corresponds to a signal component, so the SVD of H can also be written as(14)H=ξ1u1v1H+ξ2u2v2H+⋯+ξLuLvLH=H1+H2+⋯+HL,where ul∈ℂI×1 and vl∈ℂJ×1 are the l−th column vector of U and V, respectively. Hl=ξlulvlH is the Hankel matrix formed by the l−th signal component.It can be seen from [29] that when SVD is used for signal noise reduction, the signal components usually focus on a few larger singular values, and other smaller singular values represent the noise components. Similarly, the Doppler frequency of the main body is constant, and the energy is relatively concentrated in the frequency or time-frequency domains, while the Doppler frequency of micromotion parts changes with time, and the energy is relatively dispersed in the frequency or time-frequency domains. Therefore, after SVD of the signal sequence of a certain range cell, the singular value of the main body is much larger than that of micromotion parts. In order to effectively eliminate the m-D effects to obtain the focused ISAR image of the main body, it is necessary to find the cut-off point between the singular values of the main body and micromotion parts and retain all the singular values of the main body as much as possible.At present, there is no unified method for the selection of a singular value cut-off point. In [31], the skewness coefficient and Spearman correlation coefficient of each singular value distribution are calculated through multiple simulation experiments to judge the cut-off point of singular values, but the demand for data is large and cannot meet the requirements of real time. Another simple and widely used method is to determine the cut-off point through a singular value difference spectrum. The effectiveness of the singular value difference spectrum is theoretically analyzed in [29]. According to the singular value obtained by SVD, the singular value difference spectrum w is defined as(15)w=w1,w2,…,wL−1,where wl=ξl−ξl+1, l=1,2,…,L−1.When the signal component is simple and the noise is less, the singular value cut-off point determined by the peak ofw can realize the effective separation of signal and noise. However, under the conditions of noise and sparse aperture, if the peak of w is directly used to determine the cut-off point between the singular values of the main body and micromotion parts, some parts of the main body may be missing. Thus, the method proposed in [25] is adopted to determine the cut-off point. Firstly, the maximum value of w is normalized as w¯=w/maxw, and then a small threshold μ is given. Finally, w¯ is judged one by one. When the k+1−th normalized difference spectrum w¯k+1k=1,2,…,L−2 is less than μ for the first time, the cut-off point of singular values is determined as L′=k. In combination with equation (14), the estimated value of the main body signal sequence is(16)GB=H′1,1,…,H′1,J,H′2,J,…,H′I,J,where H′=H1+H2+⋯+HL′, and H′i,j is the element of the i−th row and j−th column in H′.


## 3.2. CVME

In view of the influence of noise and sparse aperture, as well as the inaccurate selection of threshold when determining the cut-off point of singular values, after using SVD to separate the main body and micromotion parts, there will inevitably be residual m-D effects and the interference introduced by noise and sparse aperture. Therefore, after SVD processing, the residual interference signal components need to be eliminated to obtain the focused high-resolution ISAR image.To further eliminate the interference signal components affecting the main body imaging, a complex signal decomposition method is proposed, namely, CVME. As a simplified method of variational mode decomposition (VMD) [32], variational mode extraction (VME) [26] can efficiently extract the specific or expected mode in the original signal without presetting the exact number of modes (decomposition layers) like VMD. At the same time, it has a strict mathematical theoretical framework and is robust to the suppression of noise and sampling error. However, VME can only process real signals and cannot be directly used to eliminate the m-D effects. It is necessary to increase the capacity of complex processing. The common methods are to project the complex signals in different directions or separate the positive and negative frequency components [33, 34], but these methods have the problems of loss of signal components and a large amount of calculation. Therefore, the idea of [35] is adopted in this paper. During the construction of the bandwidth constraint criterion, the Hilbert transform is ignored and complex signals are directly used to represent this criterion. Then, VME is extended to CVME to realize the decomposition of complex signals.Next, take the signal sequenceSn of the n−th range cell as an example for analysis. According to Section 3.1, the estimated value of the main body signal sequence after SVD processing is GB. For ease of representation, let ytm=GB, and assume that ytm after SVD processing can be expressed as(17)ytm=ybtm+yrtm,where ybtm is the desired main body signal component. yrtm is the residual signal components, which are independent of the main body and need to be eliminated before imaging.To successfully extract the desired main body signal components for CVME, the following two constraints need to be met [26]:(1)
Bandwidth Constraint. The desired main body signal component ybtm should be compact around its center frequency ωb; that is, the criteria of bandwidth constraint are established as follows:(18)J1=∂tmybtmexp−jωbtm22,where∂tm⋅ represents the partial derivative with respect to tm, and ⋅2 stands for l2 norm.(2)
Spectral Overlap Constraint. Firstly, a proper filter β^ω is designed. The spectrum of the filter has an infinite gain at ω=ωband behaves like a Wiener filter when far away from ωb. Then, to ensure the integrity of the desired signal extraction, the spectral overlap between the residual signal components yrtm and ybtm should be minimized, which is equivalent to the minimum energy of yrtm around ωb; that is, the criteria of spectral overlap constraint are established as follows:(19)J2=βtm∗yrtm22,where “∗” denotes the convolution operation. βtm is the impulse response of the designed filter β^ω, β^ω=1/αω−ωb2, and α is the bandwidth control parameter of the filter.Considering the bandwidth and spectral overlap constraints above, the extraction of the desired main body signal component can be transformed into the following constrained variational problem:(20)minyb,ωb,yrαJ1+J2s.t.ytm=ybtm+yrtm.In order to make equation (20) have better convergence and accurate reconstruction performance, we introduce a combination of quadratic penalty term and Lagrange multiplier into equation (20) to establish the augmented Lagrangian function, as follows:(21)Lyb,ωb,yr,η=α∂tmybtmexp−jωbtm22+βtm∗yrtm22+ytm−ybtm+yrtm22+ηtm,ytm−ybtm+yrtm,where ηtm is the Lagrangian function. ⋅ denotes the inner product in the complex domain. Making use of Parseval’s equality [26]: xtm22=x^ω22, where x^ω is the Fourier transform of xtm, the time domain form of equation (21) can be transformed into the frequency domain form, and then by changing variables in the first item, equation (21) can be rewritten as(22)Lyb,ωb,yr,η=αjω−ωby^bω22+β^ωy^rω22+y^ω−y^bω+y^rω22+η^ω,y^ω−y^bω+y^rω≈αjω−ωby^bω22+β^ωy^rω22+y^ω−y^bω−y^rω+η^ω222.According to [32], the saddle point of the above multivariable problem is the frequency domain solution of the constrained variational problem shown in equation (20). To solve this problem efficiently, an effective solution method is given in VMD [36], namely, the alternating direction method of multipliers (ADMM). In this method, equation (22) is transformed into a series of suboptimization problems, and different variables yb,ωb,yr,η are updated alternately in the iterative process. In the k+1−th iteration, the first step of CVME is to update the desired main body signal component y^bk+1ω, and its suboptimization problem is(23)y^bk+1ω=argminy^bα∫−∞∞ω−ωbk2y^bω2dω+∫−∞∞y^ω−y^bω−y^rkω+η^kω22dω,where k is the number of iterations, k=0,1,2,…. The first derivative with respect to y^bω is calculated, and we can get the updated form of y^bk+1ω, as follows:(24)y^bk+1ω=y^ω−y^rkω+η^kω/21+αω−ωbk2.In the second step, the suboptimization problem of the center frequencyωbk+1 is(25)ωbk+1=argminωbα∫−∞∞ω−ωb2y^bk+1ω2dω+1α2∫−∞∞y^rkω2ω−ωb4dω,where the first derivative with respect to ωb is calculated, and the updated form of ωbk+1 can be obtained as(26)ωbk+1=α∫−∞∞ωy^bk+1ω2dω−2/α2∫−∞∞y^rkω2/ω−ωbk5dωα∫−∞∞y^bk+1ω2dω.In the third step, the suboptimization problem of residual signal componentsy^rk+1ω is(27)y^rk+1ω=argminy^r1α2∫−∞∞y^rω2ω−ωbk+14dω+∫−∞∞y^ω−y^bk+1ω−y^rω+η^kω22dω,where the first derivative with respect to y^rω is calculated, and the updated form of ωbk+1 can be expressed as(28)y^rk+1ω=α2ω−ωbk+14y^ω−y^bk+1ω+η^kω/21+α2ω−ωbk+14.The last step is to update the Lagrangian multiplierη^k+1ω by adopting the dual ascent method [32], and the updated form of which is(29)η^k+1ω=η^kω+τy^ω−y^bk+1ω−y^rk+1ω,where τ is the update parameter of the Lagrangian multiplier, which can be set by the method proposed in [32].So far, one iteration of CVME can be realized according to equations (24), (26), (28), and (29). When the desired main body signal component y^bk+1ω meets the following convergence condition, the iterative process is ended in advance, the extracted main body component y^bω at this time is output, and then the inverse Fourier transform is performed to obtain the estimated time-domain signal y^btm of the main body.(30)re=y^bk+1ω−y^bkω22/y^bkω22<ε,where ε is the tolerance of convergence condition.


## 3.3. Flow Chart of the Proposed Algorithm

In order to clearly describe the proposed m-D effects removal algorithm, the algorithm flow chart shown in Figure3 is constructed. In Figure 3, the proposed algorithm mainly involves two processing steps of SVD and CVME to obtain the rough estimation and precise estimation of the main body, respectively. The imaging quality of the main body is improved by the joint processing of SVD-CVME.Figure 3
Flow chart of the proposed m-D effects removal algorithm.In terms of parameter initialization of SVD, in [29], it points out that when I and J are close or equal, the Hankel matrix constructed by the input signal sequence is conducive to improve the performance of SVD. Thus, we let I=M/2+1 (M is even) or I=M+1/2 (M is odd) in this paper. In addition, for the selection of the singular value cut-off point between the main body and micromotion parts, since the input signal sequence is affected by noise and sparse aperture, it is impossible to accurately give the cut-off point to completely separate the signal components of the main body and micromotion parts. Here, the threshold μ is set to 0.01 to retain the signal components of the main body as much as possible.In CVME, the bandwidth control parameterα is an important parameter. A smaller α makes the filter bandwidth larger and affects the elimination effect of residual signal components. A larger α makes the filter bandwidth smaller and the expected extracted signal components may suffer from partial loss. Thus, a simple heuristic method [37] is adopted to change α. In the iterative process, α is multiplied from αmin to αmax, so as to retain the desired signal components and eliminate the remaining signal components, in which αmin and multiple γ are simply set to 1 and 2, and αmax is defined in the simulation experiment. At the same time, set the iterative convergence condition as shown in equation (30) and give the maximum number of iterations to reduce the iterative time of CVME. Generally, when ε is set to 10−6 and Imax is set to 100, good decomposition results can be obtained, and the amount of computation of cyclic iteration is relatively small.To further reduce the joint processing time of SVD-CVME, FFT in azimuth is performed on the initial HRRP to obtain the initial ISAR imageX0∈ℂN×M. X0 is transformed into X¯0=X0/maxn,mX0 via the operations of modulus and maximum normalization, and the maximum value of each range cell is calculated as C=maxmX¯0, C=c1,c2,…,cN. Then, the threshold is determined as [38](31)th=Ec+1N−1∑n=1Ncn−Ec21/2,where Ec=1/N∑n=1Ncn, and according to the threshold th, we can estimate the number of range cells NS where the imaging target is located, and carry out the subsequent processing of SVD and CVME, respectively.In the end, we further analyze the computational complexity of the proposed algorithm, which consists of SVD and CVME. The main computational burden of SVD comes from SVD processing of the Hankel matrix, and its computational complexity is approximatelyOIJ2. CVME mainly involves matrix multiplication. Assuming that the number of iterations of CVME is N1, its computational complexity is approximately ON1M. Thus, combined with the number of range cells NS, the total computational complexity of the SVD-CVME algorithm is approximately ONSIJ2+N1M.


## 4. Experiments and Analysis

In this section, the effectiveness of the proposed m-D effects removal algorithm is verified. Under the conditions of noise and sparse aperture, based on simulated and measured data, the imaging performance of the proposed SVD-CVME algorithm is compared with CEMD [11], BVMD [13], and OMP [23].


### 4.1. Experimental Results of Simulated Data

A scattering point model of the target with micromotion parts is constructed, as shown in Figure4. The model consists of 31 scattering points, of which 27 and 4 scattering points represent the main body and micromotion parts, respectively. To analyze the two cases mentioned above in the removal of m-D effects, two sets of simulated data are constructed, and noted as data1 and data2, respectively. Some parameters of their micromotion parts are different, such as rR=0.2m and ωR=4πrad/s for data1 and rR=3m and ωR=12πrad/s for data2. Other parameters of the two sets of simulated data are the same, as follows: radar carrier frequency 10 GHz, bandwidth 600 MHz, pulse repetition frequency 300 Hz, and pulse width 150 μs. The angular velocity ωB of the main body is 0.02πrad/s, and the initial phases of the four scattering points of micromotion parts are set as 0,π,π/2,3π/2, respectively. The scattering intensity of the main body and micromotion parts is the same. In the process of ISAR imaging, the complete radar echo contains 256 pulses, and each pulse contains 256 samples.Figure 4
A scattering point model of the target with micromotion parts.Under the conditions of no noise and full aperture, HRRPs and ISAR images of data1 and data2 are shown in Figure5. It can be seen that the main body and micromotion components cannot be distinguished in HRRP of data1, while micromotion signal components can be seen in HRRP of data2. The imaging results of the main body for two sets of simulated data are affected by the m-D effects. Next, under the conditions of noise and sparse aperture, the imaging performance of the main body after the removal of m-D effects is analyzed. Moreover, the imaging performance of different algorithms is measured by three indexes [19], such as image entropy (IE), image contrast (IC), and running time (Time). Among them, Time is described by the running time of the computer. The definitions of IE and IC can be expressed as follows according to the imaging result X=xnmN×M.(32)IE=−∑n=1N∑m=1MxnmsumXlnxnmsumX,IC=NMsumX2∑n=1N∑m=1Mxnm2−11/2,where sumX=∑n=1N∑m=1Mxnm.Figure 5
HRRPs and ISAR images of data1 and data2. (a) HRRP of data1. (b) ISAR image of data1. (c) HRRP of data2. (d) ISAR image of data2.First of all, combined with data1 and data2, the imaging performance of different algorithms under noisy conditions is compared. In the simulation process, the upper limit of the bandwidth control parameter for SVD-CVME is set asαmax=1000. Similar to SVD-CVME, CEMD and BVMD only process the range cells where the imaging target is located. In our all experiments, signal-to-noise ratio (SNR) is defined as the ratio of echo signal energy after pulse compression to noise energy, and azimuth sparsity rate (ASR) is defined as the ratio of the number of pulses taken out to the total number of pulses. When ASR is set to 0.5, for data1 and data2, the imaging results of CEMD, BVMD, OMP, and SVD-CVME with SNRs of 5 dB and 0 dB are shown in Figures 6 and 7, respectively. The IE, IC, and Time of imaging results of different algorithms are presented in Table 1.Figure 6
Imaging results of different algorithms under different SNRs for data1. (a) CEMD. (b) BVMD. (c) OMP. (d) SVD-CVME.Figure 7
Imaging results of different algorithms under different SNRs for data2. (a) CEMD. (b) BVMD. (c) OMP. (d) SVD-CVME.Table 1
IE, IC, and time of imaging results in Figures6 and 7.
DataIndexSNR = 5 dBSNR = 0 dBCEMDBVMDOMPSVD-CVMECEMDBVMDOMPSVD-CVME1IE7.71167.65896.78055.90007.76327.67196.98205.9260IC10.429410.810010.868718.12229.846210.37038.863417.8577Time (s)4.59071.29930.21780.29144.72541.30050.20620.28232IE7.68377.63626.80515.82897.72817.70766.98795.9245IC10.291710.711210.627218.88309.845210.10738.808417.8222Time (s)4.45711.18500.17510.29444.50601.23250.20500.2833As is seen from Figures6 and 7, for CEMD and BVMD, there are some side lobe interferences around the scattering points of the main body, and its imaging resolution is not high. For OMP, although the scattering points of the main body can be reconstructed quickly, the ability of noise suppression is poor, and there are some residual m-D effects. Relatively speaking, under different SNRs, the imaging quality of the main body for SVD-CVME is the best, and the imaging results in the two sets of simulated data have few differences, indicating that the proposed algorithm has good noise suppression and the ability of m-D effects removal and is not sensitive to the rotation radius of micromotion parts. The above conclusions are further illustrated in the numerical results shown in Table 1. Compared with other algorithms, SVD-CVME has the smallest IE and the largest IC, but the imaging time is slightly longer than OMP.Then, under the condition of sparse aperture, the imaging performance of different algorithms for the two sets of simulated data is compared. The parameter settings of all algorithms are the same as those in the discussion of noise. Under the condition of SNR = 10 dB, for data1 and data2, the imaging results of four algorithms with ASRs of 0.4 and 0.2 are shown in Figures8 and 9, respectively. The IE, IC, and Time of imaging results of different algorithms are presented in Table 2.Figure 8
Imaging results of different algorithms under different ASRs for data1. (a) CEMD. (b) BVMD. (c) OMP. (d) SVD-CVME.Figure 9
Imaging results of different algorithms under different ASRs for data2. (a) CEMD. (b) BVMD. (c) OMP. (d) SVD-CVME.Table 2
IE, IC, and time of imaging results in Figures8 and 9.
DataIndexASR = 0.4ASR = 0.2CEMDBVMDOMPSVD-CVMECEMDBVMDOMPSVD-CVME1IE7.52547.58686.58815.87377.72967.64356.71985.9113IC10.990610.997612.524518.23448.60619.887911.375717.8645Time (s)4.59721.19040.19950.29055.14041.23300.15890.28672IE7.75637.87206.62635.88647.62037.91086.76035.9090IC10.06839.684312.111618.34668.75879.047210.976517.7186Time (s)4.51471.23290.20140.29634.97621.21330.16750.2806According to Figures8 and 9 and Table 2, for the two sets of simulated data, SVD-CVME has the best removal of m-D effects and imaging quality of the main body under the condition of sparse aperture. Moreover, IE and IC of SVD-CVME are the smallest and largest, respectively, further indicating that it has better imaging performance than other algorithms. On the contrary, with the decrease of ASR, the imaging quality of the main body for CEMD and BVMD becomes worse. For OMP, some m-D effects are not eliminated, which seriously affects the imaging result of the algorithm on the main body.


### 4.2. Experimental Results of Measured Data

To further explore the effectiveness of the SVD-CVME algorithm, the measured data An-26 is adopted for verification. Its structural diagram and optical image are shown in Figures10(a) and 10(b). The main parameters of the measured data are set as follows: carrier frequency 5.52 GHz, bandwidth 400 MHz, pulse repetition frequency 400 Hz, and pulse width 25.6 μs. During ISAR imaging, the complete radar echo contains 1024 pulses, and each pulse contains 256 samples. Under the conditions of no noise and full aperture, HRRP and ISAR images of the measured data are shown in Figures 10(c) and 10(d). From the imaging result, the m-D effects produced by the turboprops on both sides of the fuselage affect the imaging quality of the main body of the target, which should be effectively eliminated before imaging. Next, based on the measured data, the imaging performance of the four algorithms mentioned above under the conditions of noise and sparse aperture is analyzed.Figure 10
The measured data An-26. (a) Structural diagram. (b) Optical image. (c) HRRP. (d) ISAR image.Above all, the imaging performance of different algorithms under noisy conditions is compared. In the simulation process, the upper limit of the bandwidth control parameter for SVD-CVME is set asαmax=200. Under the condition of ASR = 0.5, the imaging results of four algorithms with SNRs of 5 dB and 0 dB are exhibited in Figure 11, and the respective numerical results are calculated in Table 3.Figure 11
Imaging results of different algorithms under different SNRs for An-26. (a) CEMD. (b) BVMD. (c) OMP. (d) SVD-CVME.Table 3
IE, IC, and time of imaging results in Figure11.
IndexSNR = 5 dBSNR = 0 dBCEMDBVMDOMPSVD-CVMECEMDBVMDOMPSVD-CVMEIE9.41129.38288.81167.76629.49679.54828.89737.8502IC6.72917.40247.094315.87585.91966.32776.210515.7141Time (s)11.48499.43975.00266.223711.52379.87225.08286.1768It can be seen that the ISAR images obtained by CEMD and BVMD are interfered with by a large number of side lobes, affecting the distinguishability of the main body. The imaging time of OMP is short, but the ability of noise suppression is poor. Under noisy conditions, SVD-CVME has the best removal of m-D effects, the ability of noise suppression, and the imaging quality of the main body. The imaging result of SVD-CVME has the smallest IE and the largest IC. However, the cell by cell processing mode of SVD-CVME leads to a long imaging time and the imaging resolution also needs to be further improved.Then, under the condition of sparse aperture, the imaging performance of different algorithms is compared. The parameter settings of all algorithms are the same as those in the discussion of noise. Under the condition of SNR = 10 dB, the imaging results of four algorithms with ASRs of 0.4 and 0.2 are exhibited in Figure12, and the respective numerical results are calculated in Table 4.Figure 12
Imaging results of different algorithms under different ASRs for An-26. (a) CEMD. (b) BVMD. (c) OMP. (d) SVD-CVME.Table 4
IE, IC, and time of imaging results in Figure12.
IndexASR = 0.4ASR = 0.2CEMDBVMDOMPSVD-CVMECEMDBVMDOMPSVD-CVMEIE9.38299.39598.68637.77889.96629.70618.76917.8605IC6.95347.52938.297015.98204.28515.94697.377515.2223Time (s)11.34829.79774.25826.161612.55239.68732.62556.2577In Figure12, although the imaging resolution of SVD-CVME is not high under the condition of sparse aperture, the imaging results of the main body are relatively complete and m-D effects are basically eliminated. However, the imaging results of the main body obtained by other algorithms are poor and cannot meet the imaging requirements. The numerical results calculated in Table 4 also validate the effectiveness of SVD-CVME, which has the smallest IE and the largest IC, and the imaging efficiency which is second only to OMP.


## 4.1. Experimental Results of Simulated Data

A scattering point model of the target with micromotion parts is constructed, as shown in Figure4. The model consists of 31 scattering points, of which 27 and 4 scattering points represent the main body and micromotion parts, respectively. To analyze the two cases mentioned above in the removal of m-D effects, two sets of simulated data are constructed, and noted as data1 and data2, respectively. Some parameters of their micromotion parts are different, such as rR=0.2m and ωR=4πrad/s for data1 and rR=3m and ωR=12πrad/s for data2. Other parameters of the two sets of simulated data are the same, as follows: radar carrier frequency 10 GHz, bandwidth 600 MHz, pulse repetition frequency 300 Hz, and pulse width 150 μs. The angular velocity ωB of the main body is 0.02πrad/s, and the initial phases of the four scattering points of micromotion parts are set as 0,π,π/2,3π/2, respectively. The scattering intensity of the main body and micromotion parts is the same. In the process of ISAR imaging, the complete radar echo contains 256 pulses, and each pulse contains 256 samples.Figure 4
A scattering point model of the target with micromotion parts.Under the conditions of no noise and full aperture, HRRPs and ISAR images of data1 and data2 are shown in Figure5. It can be seen that the main body and micromotion components cannot be distinguished in HRRP of data1, while micromotion signal components can be seen in HRRP of data2. The imaging results of the main body for two sets of simulated data are affected by the m-D effects. Next, under the conditions of noise and sparse aperture, the imaging performance of the main body after the removal of m-D effects is analyzed. Moreover, the imaging performance of different algorithms is measured by three indexes [19], such as image entropy (IE), image contrast (IC), and running time (Time). Among them, Time is described by the running time of the computer. The definitions of IE and IC can be expressed as follows according to the imaging result X=xnmN×M.(32)IE=−∑n=1N∑m=1MxnmsumXlnxnmsumX,IC=NMsumX2∑n=1N∑m=1Mxnm2−11/2,where sumX=∑n=1N∑m=1Mxnm.Figure 5
HRRPs and ISAR images of data1 and data2. (a) HRRP of data1. (b) ISAR image of data1. (c) HRRP of data2. (d) ISAR image of data2.First of all, combined with data1 and data2, the imaging performance of different algorithms under noisy conditions is compared. In the simulation process, the upper limit of the bandwidth control parameter for SVD-CVME is set asαmax=1000. Similar to SVD-CVME, CEMD and BVMD only process the range cells where the imaging target is located. In our all experiments, signal-to-noise ratio (SNR) is defined as the ratio of echo signal energy after pulse compression to noise energy, and azimuth sparsity rate (ASR) is defined as the ratio of the number of pulses taken out to the total number of pulses. When ASR is set to 0.5, for data1 and data2, the imaging results of CEMD, BVMD, OMP, and SVD-CVME with SNRs of 5 dB and 0 dB are shown in Figures 6 and 7, respectively. The IE, IC, and Time of imaging results of different algorithms are presented in Table 1.Figure 6
Imaging results of different algorithms under different SNRs for data1. (a) CEMD. (b) BVMD. (c) OMP. (d) SVD-CVME.Figure 7
Imaging results of different algorithms under different SNRs for data2. (a) CEMD. (b) BVMD. (c) OMP. (d) SVD-CVME.Table 1
IE, IC, and time of imaging results in Figures6 and 7.
DataIndexSNR = 5 dBSNR = 0 dBCEMDBVMDOMPSVD-CVMECEMDBVMDOMPSVD-CVME1IE7.71167.65896.78055.90007.76327.67196.98205.9260IC10.429410.810010.868718.12229.846210.37038.863417.8577Time (s)4.59071.29930.21780.29144.72541.30050.20620.28232IE7.68377.63626.80515.82897.72817.70766.98795.9245IC10.291710.711210.627218.88309.845210.10738.808417.8222Time (s)4.45711.18500.17510.29444.50601.23250.20500.2833As is seen from Figures6 and 7, for CEMD and BVMD, there are some side lobe interferences around the scattering points of the main body, and its imaging resolution is not high. For OMP, although the scattering points of the main body can be reconstructed quickly, the ability of noise suppression is poor, and there are some residual m-D effects. Relatively speaking, under different SNRs, the imaging quality of the main body for SVD-CVME is the best, and the imaging results in the two sets of simulated data have few differences, indicating that the proposed algorithm has good noise suppression and the ability of m-D effects removal and is not sensitive to the rotation radius of micromotion parts. The above conclusions are further illustrated in the numerical results shown in Table 1. Compared with other algorithms, SVD-CVME has the smallest IE and the largest IC, but the imaging time is slightly longer than OMP.Then, under the condition of sparse aperture, the imaging performance of different algorithms for the two sets of simulated data is compared. The parameter settings of all algorithms are the same as those in the discussion of noise. Under the condition of SNR = 10 dB, for data1 and data2, the imaging results of four algorithms with ASRs of 0.4 and 0.2 are shown in Figures8 and 9, respectively. The IE, IC, and Time of imaging results of different algorithms are presented in Table 2.Figure 8
Imaging results of different algorithms under different ASRs for data1. (a) CEMD. (b) BVMD. (c) OMP. (d) SVD-CVME.Figure 9
Imaging results of different algorithms under different ASRs for data2. (a) CEMD. (b) BVMD. (c) OMP. (d) SVD-CVME.Table 2
IE, IC, and time of imaging results in Figures8 and 9.
DataIndexASR = 0.4ASR = 0.2CEMDBVMDOMPSVD-CVMECEMDBVMDOMPSVD-CVME1IE7.52547.58686.58815.87377.72967.64356.71985.9113IC10.990610.997612.524518.23448.60619.887911.375717.8645Time (s)4.59721.19040.19950.29055.14041.23300.15890.28672IE7.75637.87206.62635.88647.62037.91086.76035.9090IC10.06839.684312.111618.34668.75879.047210.976517.7186Time (s)4.51471.23290.20140.29634.97621.21330.16750.2806According to Figures8 and 9 and Table 2, for the two sets of simulated data, SVD-CVME has the best removal of m-D effects and imaging quality of the main body under the condition of sparse aperture. Moreover, IE and IC of SVD-CVME are the smallest and largest, respectively, further indicating that it has better imaging performance than other algorithms. On the contrary, with the decrease of ASR, the imaging quality of the main body for CEMD and BVMD becomes worse. For OMP, some m-D effects are not eliminated, which seriously affects the imaging result of the algorithm on the main body.


## 4.2. Experimental Results of Measured Data

To further explore the effectiveness of the SVD-CVME algorithm, the measured data An-26 is adopted for verification. Its structural diagram and optical image are shown in Figures10(a) and 10(b). The main parameters of the measured data are set as follows: carrier frequency 5.52 GHz, bandwidth 400 MHz, pulse repetition frequency 400 Hz, and pulse width 25.6 μs. During ISAR imaging, the complete radar echo contains 1024 pulses, and each pulse contains 256 samples. Under the conditions of no noise and full aperture, HRRP and ISAR images of the measured data are shown in Figures 10(c) and 10(d). From the imaging result, the m-D effects produced by the turboprops on both sides of the fuselage affect the imaging quality of the main body of the target, which should be effectively eliminated before imaging. Next, based on the measured data, the imaging performance of the four algorithms mentioned above under the conditions of noise and sparse aperture is analyzed.Figure 10
The measured data An-26. (a) Structural diagram. (b) Optical image. (c) HRRP. (d) ISAR image.Above all, the imaging performance of different algorithms under noisy conditions is compared. In the simulation process, the upper limit of the bandwidth control parameter for SVD-CVME is set asαmax=200. Under the condition of ASR = 0.5, the imaging results of four algorithms with SNRs of 5 dB and 0 dB are exhibited in Figure 11, and the respective numerical results are calculated in Table 3.Figure 11
Imaging results of different algorithms under different SNRs for An-26. (a) CEMD. (b) BVMD. (c) OMP. (d) SVD-CVME.Table 3
IE, IC, and time of imaging results in Figure11.
IndexSNR = 5 dBSNR = 0 dBCEMDBVMDOMPSVD-CVMECEMDBVMDOMPSVD-CVMEIE9.41129.38288.81167.76629.49679.54828.89737.8502IC6.72917.40247.094315.87585.91966.32776.210515.7141Time (s)11.48499.43975.00266.223711.52379.87225.08286.1768It can be seen that the ISAR images obtained by CEMD and BVMD are interfered with by a large number of side lobes, affecting the distinguishability of the main body. The imaging time of OMP is short, but the ability of noise suppression is poor. Under noisy conditions, SVD-CVME has the best removal of m-D effects, the ability of noise suppression, and the imaging quality of the main body. The imaging result of SVD-CVME has the smallest IE and the largest IC. However, the cell by cell processing mode of SVD-CVME leads to a long imaging time and the imaging resolution also needs to be further improved.Then, under the condition of sparse aperture, the imaging performance of different algorithms is compared. The parameter settings of all algorithms are the same as those in the discussion of noise. Under the condition of SNR = 10 dB, the imaging results of four algorithms with ASRs of 0.4 and 0.2 are exhibited in Figure12, and the respective numerical results are calculated in Table 4.Figure 12
Imaging results of different algorithms under different ASRs for An-26. (a) CEMD. (b) BVMD. (c) OMP. (d) SVD-CVME.Table 4
IE, IC, and time of imaging results in Figure12.
IndexASR = 0.4ASR = 0.2CEMDBVMDOMPSVD-CVMECEMDBVMDOMPSVD-CVMEIE9.38299.39598.68637.77889.96629.70618.76917.8605IC6.95347.52938.297015.98204.28515.94697.377515.2223Time (s)11.34829.79774.25826.161612.55239.68732.62556.2577In Figure12, although the imaging resolution of SVD-CVME is not high under the condition of sparse aperture, the imaging results of the main body are relatively complete and m-D effects are basically eliminated. However, the imaging results of the main body obtained by other algorithms are poor and cannot meet the imaging requirements. The numerical results calculated in Table 4 also validate the effectiveness of SVD-CVME, which has the smallest IE and the largest IC, and the imaging efficiency which is second only to OMP.


## 5. Conclusions

Under the conditions of noise and sparse aperture, a new m-D effects removal algorithm is proposed in this paper, which is suitable for the cases of small and large rotation radius of micromotion parts. In order to effectively eliminate the m-D effects and the interference introduced by noise and sparse aperture, SVD is used to retain the main signal components of the target as much as possible. Then, VME is extended to CVME through the complex signal decomposition technology, further extracting the main body component of the target and eliminating the residual interference signal components. Through the joint processing of SVD-CVME, the imaging quality of the main body is improved. Based on extensive experimental results of both simulated and measured data, we can conclude that the proposed algorithm obtains better imaging results of the main body under different SNRs and ASRs. However, the proposed algorithm also has some limitations. The cell by cell processing mode and FFT in azimuth may lead to long imaging time and low imaging resolution, which will be the problems to be solved in the next step of this paper. In addition, the removal of m-D effects under the condition of complex motion is also a difficult problem worthy of study.



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*Source: 6141278-2022-04-27.xml*