3-D Imaging of High-Speed Moving Space Target via Joint Parametric Sparse Representation

Transcription

1 Progress In Eletromagnetis Researh M, Vol. 8, , D Imaging of High-Speed Moving Spae Target via Joint Parametri Sparse Representation Yuxue Sun 1, *, Meng Jiang, Ying Luo 1, 2, 3,QunZhang 1, 2, 4, and Chunhui Chen 1 Abstrat The high-speed moving of spae targets introdues distortion and migration to range profile, whih will have a negative effet on three-dimensional 3-D) imaging of targets. In this paper, based on joint parametri sparse representation, a 3-D imaging method for high-speed moving spae target is proposed. First, the impat of high speed on range profile of target is analyzed. Then, based on an L-shaped three-antenna interferometri system, a dynami joint parametri sparse representation model of ehoes from three antennas is established. The ditionary matrix is refined by iterative estimation of veloity. Moreover, an improved orthogonal mathing pursuit OMP) algorithm is proposed to reover interferometri phase information. Finally, with the phase information, interferometri proessing is onduted to obtain the 3-D image of target satterers. The simulation results verify the effetiveness of the proposed method. 1. INTRODUCTION With the inreasing number of spae targets, suh as spae debris, satellites, and ballisti missiles, the spae environment is beoming more and more omplex. Thus it is of great signifiane to ondut traking, measuring, ategorization and reognition of spae targets to ensure spae seurity and stimulate the development of spae tehnology. Three-dimensional 3-D) imaging tehnique is apable of providing abundant features of target outline, size, miro-motional parameters, et. Thus the 3-D imaging tehnique for spae targets has been a hot researhing spot in radar imaging field [1]. The existing 3-D imaging methods of spae targets mainly inlude monostati-radar-based imaging methods [2, 3] and multistati-radar-based imaging methods [4, ]. Moreover, a novel interferometri 3-D imaging method of spae target based on an L-shaped three-antenna onfiguration has been proposed in the authors previous work [6]. On the ondition of multiple antennas, only miro-motional harateristi in the radial diretion of radar an be extrated from the eho of eah antenna. In addition, the extrated miro-motional harateristis from all antennas are approximately the same, sine there is not muh differene between their observation angles. However, the different loations of antennas produe deliate differenes among the ranges from target to eah antenna, whih will introdue orresponding phase differene i.e., interferometri phase) among the reeived ehoes of eah antenna. The interferometri phase in reverse reflets the position of target. Ref. [6] makes use of different miro-motional harateristis of multiple target satterers to separate the eho of eah satterer on range-slow-time plane. Then interferometri proessing is onduted to obtain interferometri phase, and a 3-D image of target is thus reonstruted through a geometrial transform between the interferometri phase and target position. Compared with Reeived 14 April 217, Aepted 24 June 217, Sheduled 1 July 217 * Corresponding author: Yuxue Sun sunyuxuejiayou@163.om). 1 Institute of Information and Navigation, Air Fore Engineering University, China. 2 National Laboratory of Radar Signal Proessing, Xidian University, China. 3 Collaborative Innovation Center of Information Sensing and Understanding, Xi an, China. 4 Key Laboratory for Information Siene of Eletromagneti Waves Ministry of Eduation), Fudan University, China. Army Engineering University, China.

2 126 Sun et al. monostati-radar-based imaging method, the interferometri imaging method provides a real target 3- D size for more reliable target reognition; meanwhile, it avoids omplex synhronization and joint proessing of ehoes from different radars ompared with multistati-radar-based imaging methods. Besides, the interferometri imaging method of spae target realizes a time-varying imaging during radar irradiation time with obtaining an illustration of target motion trajetory, thus providing a new diretion for 3-D imaging and harateristi extration of spae target. However, spae target is usually with a high-speed moving veloity, whih will introdue distortion and migration to range profile [7]. Interferometri 3-D imaging is onduted on range-slow-time plane, thereby the distortion and migration of range profile will bring many adverse effets to 3-D imaging, whih mainly inlude: i) the distortion of range profile will affet the extration of trajetory on range-slow-time profile, resulting in diffiulty of separating ehoes of multiple target satterers; ii) interferometri phase is hard to be extrated from distorted range profile; iii) impulse migration hanges the phase of target eho thus resulting in reonstruted target position obtained by interferometri proessing deviating from real value; iv) the migration of range profile affets the range position reonstrution of target satterer, sine the range position is diretly obtained by range-slow-time profile. Thus, ompensating for distortion and migration of range profile is an important step in the proedure of 3-D interferometri imaging of spae targets. This paper introdues parametri sparse representation into interferometri 3-D imaging of spae targets with high-speed motion. Aording to the harateristi of target eho, a sensing matrix is first onstruted, whih ontains unknown motion veloity of the target. And a joint sparse representation model of the ehoes from three antennas is aordingly set up. Then, the optimal veloity is obtained by iteratively refining the estimated veloity and sensing matrix. Based on the estimated optimal veloity, an improved orthogonal mathing pursuit OMP) algorithm is adopted to obtain the range position of target and phase information on range-slow-time profile. Finally, via interferometri proessing on the obtained phase information, the 3-D image of target is reonstruted. Simulation results verify that the proposed method is apable of ompensating the effet of high-speed motion while preserving the interferometri phase for 3-D imaging of spae targets. 2. SIGNAL MODEL The interferometri 3-D imaging system adopts three L-shaped antennas as shown in Fig. 1. Orthogonal antennas A, B and antennas A, C are omposed of interferometri antenna pairs on XOY plane and YOZ plane, respetively. The length of antenna baseline is L. Radar oordinate XYZ is originated at antenna A, and target oordinate xyz is originated at a point O on the target. The axes of target oordinate are parallel with those of radar oordinate, respetively. Antenna A transmits linear frequeny modulated LFM) signal. The reeived signal of antenna A is ) s A ˆt, K ) ˆt 2r Ak ˆt, t m )/ t m = σ k ret exp k=1 T p [j2πf ˆt 2r Akˆt, t m ) ) +jπμ ˆt 2r Akˆt, t m ) where ) { ˆt 1, Tp /2 ˆt T p /2 ret = 2) T p, others In Eq. 1), r Ak ˆt, t m ) is the range from the kth satterer to antenna A, the wave veloity, f the arrier frequeny, T p the pulse width, μ the modulation rate, ˆt = t t m the fast time, t m = mt, m =, 1, 2,...,M 1theslowtime,M the pulse number, T the pulse repetition duration, and σ k the sattering oeffiient of the kth satterer. The range from point O to antenna A is hosen as the referene range, whih is R ref t m )=R AO t m ). Sine the target moves with a high speed, it yields r Ak ˆt, t m )=r Ak t m )+v rˆt, wherer Ak t m ) is the range from the kth satterer to antenna A at eah slow-time sampling point, and v r is the radial veloity relative to radar. After dehirp, Eq. 1) beomes ) s Ad ˆt, K ) ˆt t m = σ Ak ret 2r Ak ˆt, t m )/ exp [ j φ 1 + φ 2ˆt + φ 3ˆt 2)] 3) T p k=1 ) 2 ] 1)

3 Progress In Eletromagnetis Researh M, Vol. 8, where σ Ak = σ k exp j 4π ) f R ΔAk t m ) φ 1 t m )= 4πμ r 2 2 Ak t m ) Rref 2 t m) ) φ 2 t m )= 4π μr ΔAk t m ) 4π f v r + 8πμ 2 r Ak t m ) v r ) φ 3 = 4π μv r + 4πμ 2 v2 r and R ΔAk t m ) = r Ak t m ) R ref t m ). σ Ak is the range profile of the kth satterer obtained by antenna A, of whih the phase term is the essene of interferometri imaging. Sine there is almost no differene between view angles of all antennas, the amplitudes of range profiles of them should remain a onstant [8, 9]. Thus the range profiles of antenna B and antenna C an be represented as σ Bk = σ k exp j 2π ) f R ΔAk t m )+R ΔBk t m )) 6) σ Ck = σ k exp j 2π ) f R ΔAk t m )+R ΔCk t m )) 7) where R ΔBk t m )=r Bk t m ) R ref t m )andr ΔCk t m )=r Ck t m ) R ref t m ). r Bk t m )andr Ck t m )are range from the kth satterer to antenna B and C, respetively. φ 1 is relatively small and generally an be ignored. In φ 2, the first term 4πμR ΔAk t m )/ is the initial range position of satterer and the next two terms will introdue range migration. φ 3 will introdue distortion to range profile. The migration and distortion of range profile will both affet the interferometri 3-D imaging in the later proessing, thus they must be ompensated before onduting interferometri proessing. It an be seen from Eq. ) that the migration and distortion terms are both deided by veloity of target, thus estimating and ompensating the veloity is the essene of eliminating migration and distortion of range profile. 4) O s R ref C L Figure 1. Geometry of 3-D imaging system and spae target. Z 3. PARAMETRIC SPARSE REPRESENTATION BASED 3-D IMAGING OF HIGH-SPEED SPACE MOVING TARGET 3.1. The Model of Joint Parametri Sparse Representation Interested spae target is sparse in the view of range diretion, that is, the target satterers over a ertain number of range bins. Disrete the interested area along the range diretion into N sub-bins

4 128 Sun et al. whih an be represented as R Δ =[r Δ1,...,r Δi,...,r ΔN ] i =1,...,N 8) where r Δi = r i R ref and r i is the range from the ith range bin to radar. During the imaging time, the mth pulse reeived by antenna A is s Adm ˆt ) K = σ Amk ϕ 9) where T p k=1 ) ˆt 2r Ak / ϕ =ret exp j φ 1 + φ 2ˆt + φ 3ˆt 2)) 1) And ϕ represents the kernel funtion. After being disretized, Eq. 9) an be rewritten as s Adm = Φ v r ) σ Am + E m 11) where Φ v r ) = [ϕ 1,...,ϕ i,...,ϕ N ] 12) expj φ 1 r Δi )+φ 2 r Δi ) ˆt 1 + φ 3 r Δi ) ˆt 2 1 )). ϕ i = expj φ 1 r Δi )+φ 2 r Δi ) ˆt j + φ 3 r Δi ) ˆt 2 j )) 13). expj φ 1 r Δi )+φ 2 r Δi ) ˆt N + φ 3 r Δi ) ˆt 2 N )) φ 1 r Δi )= 4πμr Δi 2 r Δi +2R ref ) φ 2 r Δi )= 4π v r f 2μr ) ) Δi + R ref ) + μr Δi 14) φ 3 r Δi )= 4π ) μv r μv2 r σ Am [σ = Am1 exp j 4π ) f r Δ1,...,σ Ami exp j 4π ) f r Δi,...,σ AmN exp j 4π )] T f r ΔN 1) σ Ami is the sattering oeffiient of the ith range bin in interested area, T =[ˆt 1,...,ˆt j,...,ˆt N ]the sampling time sequene, N the number of sampling data, and E m the noise vetor. Through solving Eq. 11), the range profile σ Am an be obtained. The sensing matrix Φv r ) is indeterminate sine the veloity is unknown. Different veloities orrespond to different sensing matrixes. Thus v r should be estimated at first. With a well estimated v r, the sensing matrix will math with eho signal. Thereby a foused range profile ould be obtained. In this paper, an iterative optimization algorithm is applied to estimate v r. The veloity and sensing matrix are updated iteratively until the optimal solution is obtained. Then the range profile σ Am anbesolvedbythefollowing optimizing model: min σ Am s.t. sadm Φ v r ) σ Am 2 ε 16) where ε is noise level. In an interferometri imaging system, the phase information of eho should be as aurate as possible to ensure the imaging preision [1]. Thus a joint sparse representation model is established for ensuring the oherene of the reonstruted phase information from three antennas. Eq. 16) an be rewritten as min σ 2,1 m s.t. sm Φ v r ) σ m 2 ε 17) where σ m 2,1 = σ Am 2 + σ Bm 2 + σ Cm 2 is mixed norm, s m = [s Adm s Bdm s Cdm ] and σ m = [σ Am σ Bm σ Cm ]. Hereto, the problem of 3-D imaging is transformed into a problem of solving Eq. 17).

5 Progress In Eletromagnetis Researh M, Vol. 8, D Imaging Based on Improved Orthogonal Mathing Pursuit Algorithm The amplitudes of range profiles from the three antennas keep onsistent, as mentioned before, thus the joint range profiles an be represented as σ m = Amp ψ 18) where Amp = diagσ), σ = [σ Am1,...,σ Ami,...,σ AmN ] and ψ=[expjψ A ), expjψ B ), expjψ C )] is the phase vetor of R Δ. ψ A = [ψ A1,...ψ Ai,...ψ AN ] T, ψ B = [ψ B1,...ψ Bi,...ψ BN ] T, ψ C = [ψ C1,...ψ Ci,...ψ CN ] T. An improved OMP algorithm is developed to solve Eq. 17). The detail is as follows: Step 1 Initializing: The residual is r = s m. Iterative times are n =1andp =1. Theposition vetor is Pos =. Mathing vetor set is At =. Sparseness is Sp, and initial veloity is v rn. Step 2 Seek for the mathing vetor and position: Searh for the position of max Φ H n v rn)r )and denote its row number and olumn number as row and ol, respetively. Update Pos =[Pos,row], At =[At,ϕ row ]. Step 3 Estimate the amplitude: amp = At H At ) 1 At H s ol 19) where s ol is the olth olumn of s m.denoteval =diagamp). Step 4 Estimate the phase: pha = Val H At H At Val ) 1 ValH At H s ol 2) Step Update the residual: r = s m At Val pha 21) Step 6 Iterate to get the solution: p = p +1andΦ n ϕ row )=. Repeat Step 2 to Step until p = Sp. Step 7 Estimate the veloity: n = n + 1 and set the termination threshold η. Construt σ mn 1) by obtained amp and pha, then expand the sensing matrix using Taylor series expansion, that is ) ) Φ n 1 v r ) Φ n 1 vrn 1) Φn 1 vrn 1) + v r Δv r 22) vr=vrn 1) where Δv r is the inrement of veloity. Substitute Eq. 22) into the objet funtion v r = arg min s m Φ n 1 v rn 1) ) σ mn 1) 2, then it yields minδv r s m Φ n 1 v rn 1) ) σ mn 1) T n 1 Δv r 2,where T n 1 = Φ n 1v r ) v r σ mn 1) 23) vr=vrn 1) Obtain the solution as Δv r = T H T ) 1 T H s m Φ n 1 vrn 1) ) σ mn 1) ) 24) The veloity is omputed as v rn = v rn 1) +Δv r. Substitute the new v rn into 22) and repeat the above proessing until Δv r η. Step 8 Obtain the range profile: onstrut Φ n v rn )usingv rn and repeat Step 2 to Step 6. A foused range profile is finally obtained. The obtained position vetor Pos ontains the positions of satterers in the range diretion. Correspond these positions to R Δ and thus the y-axis positions of satterers are obtained. pha ontains the phase information for interferometri proessing and pha =[phaa, phab, phac] T,wherephaA, phab and phac are the phases of range profiles from antenna A, B and C, respetively. Use the phase information, ombining with Eqs. 4), 6) and 7), to ondut interferometri proessing. The interferometri phases are ψ ABk = anglephaa k phab k)=2πr ΔAk R ΔBk )/λ 2) ψ ACk = anglephaa k phac k)=2πr ΔAk R ΔCk )/λ 26)

6 13 Sun et al. where represents onjugating operation; phaa k, phab k and phac k are the kth elements in orresponding phase vetors; λ is wave length. Aording to the geometrial relationship in Fig. 1, it yields R ΔAk R ΔBk =r Ak R ref ) r Bk R ref )= 2LX + x k ) L 2 r Ak + r Bk 27) In far field, it yields r Ak + r Bk r Ak + r Ck 2 y k + Y ), where y k is the reonstruted y-axis position of the kth satterer. Thus, reonstruted x-axis and z-axis position of the kth satterer are x k = ψ ABλ y k + Y ) + L 2πL 2 X 28) z k = ψ ACλ y k + Y ) + L 2πL 2 Z 29) On the basis of the reonstruted 3-D positions [x k,y k,z k ],k =1, 2,...,K of target satterers, the 3-D image is thus obtained. During imaging time, repeat the 3-D imaging proessing for eah eho pulse, time-varying 3-D images are reonstruted. 4. SIMULATIONS AND ANALYSIS 4.1. Simulation Results The antenna onfiguration in Fig. 1 is adopted. Suppose that the arrier frequeny of transmitted signal is 2 GHz, bandwidth 7 MHz, pulse width 2 µs, pulse repetition frequeny 8 Hz, imaging time 1 s, range resolution.2 m and length of antenna baseline L = 1 m. The veloity of target is 4,, 4) m/s, and the radial veloity is v r = 66.9 m/s. The target enter is at,, ) km at initial time. Suppose that there are three satterers. Considering the omplex miro-motion of spae targets, rotation and preession are simulated here. One of the three satterers is rotating with rotation veloity ω =.4,, 3.94) rad/s. The other two are undergoing preessional motion, whih ontains the same rotation with the first satter and oning of veloity ω s =,, 18.8) rad/s. The rotation period is.2 s and the oning period is the.33 s. Under Nyquist sampling, range profile at eah time slie will be affeted by high speed motion. For an instane, the range profile at t m =.6sisshowninFig.2a). Fig. 2b) shows the range profile when the veloity of target is m/s at t m =.6s. It an be seen that range profile under high speed is seriously migrated and distorted. The range-slow-time profile is shown in Fig. 2). To better illustrate the figures, time span [,.4 s] in figures is taken out. The migrated and distorted range-slow-time profile will affet both the abstrating the y-axis position of satterer and interferometri phase. Fig. 2d) shows y-axis positions abstrated by range-slow-time profile in Fig. 2). Fig. 2e) shows the theoretial y-axis positions. As shown, abstrated y-axis positions are also migrated and distorted with many redundant points. The initial veloity of target is set as m/s, and the estimated veloity is 61.2 m/s. The ompressing ratio is.8. In order to illustrate the performane of the veloity estimation in the proposed method, Frational Fourier transform FrFT) based veloity estimation algorithm in [11] is ompared with the proposed algorithm. The FrFT based veloity estimation algorithm makes use of a priori information that the eho of high-speed moving target is multiomponent hirp signal and searhes for the optimal order p to obtain veloity. While hoosing different order-searhing auray lasses δ, different veloity auraies are obtained. Table 1 shows the simulation results of the proposed algorithm and FrFT based algorithm with different order-searhing auray lasses. Eah result is a mean of Table 1. Comparison of two veloity-estimated algorithms. FrFT Proposed δ = e 3 δ = e 4 algorithm Estimated veloity/m/s) Computing time/s

7 Progress In Eletromagnetis Researh M, Vol. 8, normalized amplitude normalized amplitude range/m range/m -1 1 range/m a) b) ) y-value/m d) range/m e) Figure 2. Initial proessing result of eho. a) Range profile when v r = 66.9 m/s. b) Range profile when v r = m/s. ) Range-slow-time profile. d) Abstrated y-axis position. e) Theoretial y-axis position. Monte Carlo experiments. It an be seen that the FrFT based algorithm spends muh longer time when order-searhing auray lass is inreased, while the proposed algorithm an obtain a relatively aurate veloity in a short period of time. The sensing matrix is then onstruted with the estimated veloity, after the y-axis position and phase information are reonstruted using the improved OMP algorithm. Reonstruted y-axis position is shown in Fig. 3a). It an be seen that migration and distortion are well ompensated. Finally, interferometri proessing is onduted to obtain x-axis and z-axis positions. The results are shown in Fig. 3b) and Fig. 3), respetively. With the reonstruted 3-D positions, a 3-D time-varying image of target is obtained as Fig. 3d). The results show that the proposed method an effetively ompensate the distortion and migration introdued by high-speed motion and an reover the phase information to ondut 3-D imaging of spae target. 1 1 y-value/m x-value/m a) b)

8 132 Sun et al. 1 theoretial trajetory reonstruted trajetory z-value/m ) z-axis/m x-axis/m d) -1 y-axis/m Figure 3. The reonstruted positions and 3-D image. a) Reonstruted y-axis position. b) Reonstruted x-axis position. ) Reonstruted z-axis position. d) Reonstruted 3-D trajetory Robustness Analysis In order to analyze the robustness of the proposed method, Gaussian white noise is added to the signal of 4.1. When signal to noise ratio SNR) is 1dB, the reonstruted y-axis, x-axis and z-axis positions are shown in Figs. 4a) 4), respetively. It an be seen that reonstruted y-axis position is y-value/m 1-1 theoretial result reonstruted result x-value/m theoretial result reonstruted result z-value/m 1-1 theoretial result reonstruted result a) b) ) Figure 4. The reonstruted positions. a) Reonstruted y-axis position. b) Reonstruted x-axis position. ) Reonstruted z-axis position. MSE/m x-axis z-axis y-axis MSE/m SNR=-6dB SNR=dB SNR=dB SNR=dB SNR=1dB SNR/dB ompressing ratio Figure. MSEs under different SNRs. Figure 6. The relationship of ompressing ratio and MSE.

9 Progress In Eletromagnetis Researh M, Vol. 8, little affeted by noise, while reonstruted x-axis and z-axis positions are distributed randomly around theoretial values. In order to refine the reonstruted results, urve fitting operation an be applied to the reonstruted positions of eah satterer. After urve fitting operation, the reonstruted position of eah satterer will be a smooth urve. Through independent trials under different SNR values, the mean square errors MSEs) of reonstruted positions after urve fitting operation ompared with theoretial values are shown in Fig.. It an be seen that reonstruted x-axis and z-axis positions obtained by interferometri proessing are more easily affeted by noise than reonstruted y-axis position. When SNR is below 3 db, the reonstruted positions seriously deviate from theoretial values. Fig. 6 illustrates the relationship of data ompressing ratio and MSE of 3-D position with different SNR values. It an be seen that MSE inreases with the inreasing of ompressing ratio. And when ompressing ratio is below.8, the reonstruted result is relatively aurate.. CONCLUSIONS This paper proposes a 3-D imaging method for high-speed moving spae target. Through solving a joint sparsity model with veloity to be estimated, the estimation of veloity is aomplished, and the omplex range profile is obtained. No extra veloity ompensation is needed. Compared with FrFT based veloity estimation algorithm, the proposed algorithm adopts a sparse sampling mode, thus taking up fewer resoures. Meanwhile, the proposed algorithm has a higher auray and faster alulation speed. Moreover, the onstrution of the joint sparse parametri model guarantees the phase oherene of three antennas, whih then makes sure that the interferometri 3-D imaging ahieve a good result. In general, the proposed 3-D imaging method an ompensate the effet of high-speed motion and realize time-varying 3-D imaging of spae target, whih provides a support for measurement, ategorization and reognition of spae targets. ACKNOWLEDGMENT This work was supported by the National Natural Siene Foundation of China under Grant and , the Youth Siene and Tehnology New Star Projet of Shaanxi Provine under Grant 216KJXX-49 and the Natural Siene Foundation Researh Program of Shaanxi Provine under Grant 21JM636. REFERENCES 1. Zhang, L., M. D. Xing, C. W. Qiu, and Z. Bao, Two-dimensional spetrum mathed filter banks for high-speed spinning-target three-dimensional IASR imaging, IEEE Geosi. Remote Sens. Lett., Vol. 6, No. 3, , Wang, Q., M. D. Xing, L. G. Yue, and Z. Bao, High-resolution three-dimensional radar imaging for rapidly spinning targets, IEEE Trans. Geosi. Remote Sens., Vol. 46, No. 1, 22 3, Bai, X. R., M. D. Xing, F. Zhou, and Z. Bao, High-resolution three-dimensional imaging of spinning spae debris, IEEE Trans. on Geosi. Remote Sens., Vol. 47, No. 4, , Ai, X. F., Y. Huang, F. Zhao, J. H. Yang, Y. Z. Li, and S. P. Xiao, Imaging of spinning targets via narrow-band T/R-R bistati radars, IEEE Geosi. Remote Sens. Lett., Vol. 1, No. 2, , Luo, Y., Q. Zhang, N. Yuan, F. Zhu, and F. F. Gu, Three-dimensional preession feature extration of spae targets, IEEE Trans. Aerosp. Eletron. Syst., Vol., No. 2, , Sun, Y. X., C. Z. Ma, Y. Luo, Y. A. Chen, and Q. Zhang, An interferometri-proessing based three-dimensional imaging method for spae rotating targets, IET 216 International Radar Conferene, , Guangzhou, China, Otober Xing, M. D., Q. Wang, G. Y. Wang, and Z. Bao, A mathed-filter-bank-based 3-D imaging algorithm for rapidly spinning targets, IEEE Trans. Geosi. Remote Sens., Vol. 47, No. 7, , 29.

10 134 Sun et al. 8. Pang, C. S., T. Shan, T. Ran, and N. Zhang, Detetion of high-speed and aelerated target based on the linear frequeny modulation radar, IET Radar Sonar Navig., Vol. 8, No. 1, 37 47, Chen, Q. Q., G. Xu, L. Zhang, M. D. Xing, and Z. Bao, Three-dimensional interferometri inverse syntheti aperture radar imaging with limited pulses by exploiting joint sparsity, IET Radar Sonar Navig., Vol. 9, No. 6, , Liu, Y. B., N. Li, R. Wang, and Y. K. Deng, Ahieving high-quality three-dimensional InISAR imageries of maneuvering target via super-resolution ISAR imaging by exploiting sparseness, IEEE Geosi. Remote Sens. Lett., Vol. 11, No. 4, , Liao, K., H. Chen, and D. Y. Zhou, High speed motion ompensation based on the Frational Fourier transform, 213 IEEE 4th International Conferene on Eletronis Information and Emergeny Communiation, , Beijing, China, November 213.