DATA ACQUISITION AND PROCESSING OF PARALLEL FREQUENCY SAR BASED ON COMPRESSIVE SENSING

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1 Progress In Eletromagnetis Researh, Vol. 133, , 2013 DATA ACQUISITION AND PROCESSING OF PARALLEL FREQUENCY SAR BASED ON COMPRESSIVE SENSING Y. N. You, H. P. Xu *, C. S. Li, and L. Q. Zhang Shool of Eletroni and Information Engineering, Beihang University, Beijing, China Abstrat Traditional syntheti aperture radar SAR) utilizes Shannon-Nyquist theorem for high bandwidth signal sampling, whih indues a ompliated SAR system, and it is diffiult to transmit and proess a huge amount of data aused by high A/D rate. Compressive sensing CS) indiates that the ompressible signal using a few measurements an be reonstruted by solving a onvex optimization problem. A novel SAR based on CS theory, named as parallel frequenies SAR PFSAR), is proposed in this paper. PFSAR transmits a set of narrow bandwidth signals whih ompose the large total bandwidth. Therefore, PFSAR only uses muh less data to obtain a superiority image ompared with a traditional SAR system. The data aquisition mode of PFSAR is developed and an algorithm of target sene reonstrution based on ompressive sensing applied to PFSAR is proposed. The azimuth imaging of PFSAR is arried out based on the Doppler Effet, and then, the range imaging is performed by using ompressive sensing of parallel frequenies signal. Several simulations demonstrate the feasibility and superiority of PFSAR via ompressive sensing. 1. INTRODUCTION Syntheti aperture radar SAR) [1] is an ative detetor with all-day and all-weather apability. Reently, ultra swath and high resolution are major requirements for SAR system. The traditional SAR system emits high bandwidth hirp signal to obtain high range resolution, whih demands high sampling rate aording to the Shannon-Nyquist theorem. Transmitting and proessing the huge data generated by Reeived 6 July 2012, Aepted 7 Otober 2012, Sheduled 1 November 2012 * Corresponding author: Hua Ping Xu xuhuaping@buaa.edu.n).

2 200 You et al. high sampling is one of the main hallenges of radar system design. Compressive sensing CS) in [2] and [3] provides a novel method for sampling and reovering signals. It shows that a K-sparse signal length of N) an be reonstruted using O K log N) measurements with high probability. The dimension of measurements is obviously redued ompared with Nyquist sampling. The radar reeiver based on CS is firstly presented in [4]. The CS radar reeiver disards the pulse ompression mathed filter to simplify the radar system and absolutely dereases the A/D rate. In [5], high-resolution radar based on CS is proposed, and the CS radar employs a suffiiently inoherent pulse to reonstrut the observation sene. Pulse Doppler radar with ompressive sampling [6] presents an approah to redue the A/D sampling rate through importing omplete sparse ditionaries and basis pursuit reonstrution algorithm. Radar signal proessing based on CS effiiently overomes the huge data in terms of the Shannon- Nyquist sampling. Then the omplexity of system an be redued with ompressive sampling. There are some researhes about the appliation of CS to SAR. CS applied to ompress SAR raw data is firstly proposed in [7]. It uses the dual tree omplex wavelet transform for sparse deomposing of SAR image and the SAR raw signal is randomly sampled in twodimensional 2D) fast Fourier transform FFT) domain to ahieve signal ompression. [8] analyzes properties of SAR data and image and indiates that only the brightest objets of the SAR image are ompressible. In [9, 10], Tello Alonso et al. propose a novel method for the fousing of raw data in radar imaging. Signal sampling and reonstrution based on CS is introdued as an alternative option to traditional mathed filtering. In [11], white Gaussian matrix is used as a measurement matrix to detet the targets with one-dimensional 1D) signal and obtain 2D SAR images based on CS. Anitori et al. [12] use a stepped sequene of frequenies CS radar to illuminate the orner refletors, and CS is applied with random seletion of frequenies and angles. In [13], Dantzig seletor via ompressive sensing has been used to obtain a better radar imaging result than that of the onventional l 1 - norm based on ompressive sensing. The paper [14] overviews the use of sparse reonstrution and random projetion approahes in radar imaging. The joint reonstrution method [15, 16] is introdued to obtain better phase information. In [17], based on the refletivity kernel analysis of SAR original ehoes, singular value deomposition- QR SVD-QR) is used to selet the subset of SAR ehoes in both slow-time domain and fast-time domain. Xu et al. [18] disuss a new imaging modality based on Bayesian ompressive sensing, and lutter is taken into onsideration in this novel CS radar. In [19], the

3 Progress In Eletromagnetis Researh, Vol. 133, proper sparse ditionary is used to represent the SAR omplex ehoes, and thus SAR imaging proessing is a joint optimization problem for the magnitude and phase of the omplex signals. [20] applies CS to linear array SAR and CS is used after the range mathed filtering. [21] proposes a novel SAR imaging algorithm based on CS with multiple transmitters and multiple azimuth beams. [22] presents a 4-D SAR imaging sheme based on CS. In [23], Li et al. present some appliations of ompressed sensing for multiple transmitters multiple azimuth beams SAR imaging. [24] provides an algorithm of sparse reonstrution for SAR imaging based on ompressed sensing. Linear array imaging method of traditional SAR by using ompressive sensing is shown in [25]. Though the researhes on ompressive sensing applied to SAR system have ertain ahievements, there is no SAR system based on ompressive sensing. The onept of parallel frequenies signal is firstly presented in [26] to explore the appliation of CS in pulse ompression. The author assumes that some distint frequenies are simultaneously transmitted and presents how to use these frequenies in parallel to ahieve the pulse ompression without the Doppler effet. We import the theory of parallel frequenies into SAR and propose a novel SAR imaging mehanism, i.e., Parallel Frequenies SAR PFSAR). The PFSAR transmits parallel frequenies signal instead of hirp signal for 2D imaging. The azimuth imaging of PFSAR is arried out based on the Doppler effet and then the range imaging is performed by using ompressive sensing. The data aquisition mode is developed in our researh, in addition, an approah of signal proessing via ompressive sensing applied to PFSAR is disussed in this paper. Setion 2 reviews ompressive sensing method. The struture of parallel frequeny radar an be found in Setion 3. Data aquisition and signal proessing via ompressive sensing are shown in Setion 4. Targets are reonstruted by using priori phase information. In Setion 5, ompared with the onventional SAR images, the feasibility and the superiority of PFSAR via ompressive sensing are onfirmed by some simulations. 2. COMPRESSIVE SENSING METHOD Assume that target sene S S R N ) is sparse on a basis matrix Ψ N N Ψ N N = ψ 1, ψ 2,..., ψ N )). S is represented as S = Ψx, 1) where x is a length of N sparse oeffiient vetor with K nonzero elements and K a measure of the sparsity of x. In ompressive sensing

4 202 You et al. method. S is measured through a linear projetion onto a measurement matrix Φ M N where M N), the projetion is presented in matrix notation as y = ΦS = ΦΨx, 2) where y is a measurement vetor of length M. The sparseness is the number of nonzero oeffiients with a proper sparse basis. In pratie, the ompressed signal and sparse sene an be represented by a few large oeffiients. It is very vital for the hoie of sparse basis. For the first, in spae domain, i.e., warship targets loated on the sea bakground, thus it satisfies the onditions of sparse spae. Seondly, for the ompliated sene, it is not sparse in spae domain, however, the ehoes of the speifi sene are sparse on a proper basis, for instane, hirplet basis, wavelet basis and so on. Therefore ompressive sensing an ahieve the appliations of SAR with a proper sparse basis. The dimension M of measurement y is muh less than the dimension N of sparse oeffiient vetor x. Therefore the Eq. 2) is an ill-posed problem. In other words, the problem is undetermined. By reason that vetor x is ompressible, the reovery of the sparse oeffiient vetor x from the measurement vetor y is translated into solving a minimization of l 0 norm problem. In [3] and [27], several signifiant results onerning ompressive sensing show that for the purpose of the onvergent of reonstrution algorithm, namely, one K-sparse vetor x an be exatly reonstruted by M measurements, reovery matrix A = ΦΨ is required to satisfy Restrited Isometry Property RIP). It is a signifiant onsequene to ensure the onvergene of the reonstrution algorithm and a detailed expression is given for it as follows, 1 δ Ax 2 x δ δ > 0), 3) where δ is a minimal onstant, 2 the l 2 norm, and x the sparse oeffiient vetor. The minimization of the l 0 problem shown is equivalent to the l 1 norm onvex problem under the RIP ondition. The onvex problem is expressed as ˆx = arg min{ x 1 : x R N, y = ΦΨx}. 4) Convex problem is simplified to linear programming. Basis Pursuit BP) and greedy algorithms in [28 30] suh as Mathing Pursuit MP) and Orthogonal Mathing Pursuit OMP) [31] an effetively solve this problem, thus allowing reonstrution of sparse sene S.

5 Progress In Eletromagnetis Researh, Vol. 133, Figure 1. Observation mode of PFSAR. 3. PARALLEL FREQUENCY SAR 3.1. Observation Mode The observation mode of PFSAR via ompressive sensing is shown in Fig. 1. The radar platform moves along trak azimuth) diretion with onstant veloity, two-dimension observation sene is limited with the along trak azimuth) diretion and the ross trak range) diretion. The antenna pointing is vertial with azimuth diretion, i.e., the beam diretion is under zero squint ase. PFSAR synhronously transmits M single frequeny pulses with the onstant step f in eah azimuth, and then reeived the phase shift ehoes. The total bandwidth an satisfy the requirements of high range resolution. The beam footprint is divided into some equal observation ells, as shown in Fig. 2. The sparse targets will be loated on the orresponding grid nodes. Loation oordinate is Target r, τ), r is range ell, τ is azimuth slow time, for instane, the loation oordinate of the enter point in Fig. 2b) an be written as Target r, τ ). N a and N r respetively represents the sampling number of azimuth and range diretions. The targets loation information obviously depends on the observation grid. Considering the movement of the radar platform along azimuth

6 204 You et al. τ Na Azimuth τ 1 r 1 r Range a) b) Figure 2. Observation grids of PFSAR via ompressive sensing. τ r Nr diretion, the raw eho signal is written as rt, τ) = Aret t 2Rτ) ) A a τ τ )e j2πf set t 2Rτ) ), 5) where A is baksatter oeffiient, ret the retangle window, A a the azimuth envelope, and the light veloity. f set is a set of transmitted frequenies, namely, f set = {f 1, f 2,..., f M }. R τ) is the instantaneous distane between the target and radar platform. Aording to the beam diretion, we set the approximate representation Rτ) = R + Va 2 τ 2 /2R and R is the losest distane between the target and the radar platform Data Aquisition PFSAR synhronously reeives the phase shift ehoes. Fig. 3 demonstrates data aquisition proess. Raw ehoes are reeived by an antenna array, then the raw data are filtered through the multi-hannel and the third step is quadrature demodulation. After demodulating, the intermediate frequeny signals are sampled by a low-rate A/D onverter, on ondition that two-way I and Q mixing is employed in eah hannel, the sampling rate of A/D onverter is effetively dereased into half. Mixing and deteting phase is the most ruial proessing stage, and the detail is depited in Fig. 4. In view of several single frequeny

7 Progress In Eletromagnetis Researh, Vol. 133, Raw eho Antenna array Multi-hannel filter Quadrature demodulation Low-rate A/D Complex data memory Download or Real-time proessing Figure 3. Data aquisition of PFSAR. pulses ongregating the raw eho, multi-hannel filter is used to distinguish and extrat the variously frequeny pulses before mixing, through filtering proedure, the outputs of M hannels is os 2πf 1 t 2Rτ) )), os os 2πf M t 2Rτ) 2πf 2 t 2Rτ) )),..., )). 6) The output of eah hannel mixes with orresponding referene frequeny, after mixing, the omplex results omposed with I and Q omponent are written as ) 2Rτ) exp j2πf 1 2Rτ) exp j2πf M, exp j2πf 2 2Rτ) ),..., ). 7)

8 206 You et al. Raw eho Multi-hannel filter I A/D f M mixing hannel M Q A/D Figure 4. Mixing and deteting phases proedure. phase M The omplex data as above are a set of phases ontaining range information of targets. In a ertain azimuth time τ i i = 1, 2,..., N a ), the omplex data are written into a memory devie in the hannel sequene. Aordingly, memory spae is three dimensional in mathematial expression, the first dimension is range ell, the seond dimension is azimuth slow time, and the third dimension is the number of hannels. In fat, the omplex data an be written as one dimension data array, and eah data ell needs to add the information about the azimuth slow time and the hannels number, this additional information assists in resuming the raw omplex data for imaging proessing. 4. DATA PROCESSING BASED ON COMPRESSIVE SENSING 4.1. Azimuth Compressive The proessing of the baksattered signal is shown in Fig. 5. Through demodulating and deteting the phase shift, the eho signal an be written as 2Rτ) j2πfset y set = Ae = Ae j2πf set 2R e j2πfset V a 2 τ2 R. 8) Then we fous on the seond exponent term in order to preisely draw the range information in the first exponent term. Obviously, the seond exponent term is the result of the movement of the radar platform along azimuth diretion and it has the feature of linear FM. Here, the FM rate is K a = 2Va 2 / λset R. 9) The signifiant steps are K a estimation and Φ onstrution in imaging. Beause of the seond exponent in Eq. 8) varying with

9 Progress In Eletromagnetis Researh, Vol. 133, Image Figure 5. Two-dimensional signal imaging blok diagram. azimuth slow time, the targets will be foused on the respetive zero Doppler by K a before extrating the range information. K a should be alulated preisely before ompressing the azimuth phases. It also varies with the range ells and the signal frequenies. We use the autofous to estimate the Doppler FM rate of the azimuth filter Range Proessing via Compressive Sensing Construting the priori Φ is presented as follows. PFSAR transmits M single frequeny pulses in a ertain azimuth. The eho of the kth k = 1, 2,..., M) single frequeny pulse is rt) = Aret t 2R ) e j2πf kt 2R ), 10) where is the speed of light, A the baksatter oeffiient of the target, f k the frequeny of the kth pulse, and f k = f 0 + k f f 0 the arrier frequeny. If we just ollet the phase shift part the result is, y k = Ae j2πf k 2R. 11)

10 208 You et al. Let partition the maximum observation distane R MAX into N points, thus the range ell is R = R MAX /N. The disrete representation of Eq. 11) is written as y k = N 1 n=0 e j2πf k 2n R A n, 12) where A n is the baksatter oeffiient of the targets in the Nth point. Aiming at an appointed observation sene, the targets loations are determined by the partition of observation grid. Supposing that there is a target in the enter of the observation grid, the loation oordinate is Target r, τ ) and the phase shift parts generated by target effeting under τ ondition are exp j2πf 1 2r ), exp j2πf 2 2r ),..., exp ) 2r j2πf M. 13) Apparently, the targets in the observation grid with different loations still at on the transmitting signal that reates a set of fixed phases ontaining target range information. Provided eah node has one target, we an utilize these fixed phases to onstrut the omplete measurement matrix. We write the Eq. 12) in matrix notation as where Φ is measurement matrix and y = ΦS = ΦΨx, 14) Φ = {φ n } N 1 n=0, 15) where } φ T n = {e j2πf k 2n R M. 16) k=1 Φ aordingly ontains the priori phases. The ehoes in eah azimuth slow time τ i are linear ombinations of random olumns in the measurement matrix Φ. For fulfilling sparse sene ondition, targets are muh less than nodes in observation grid, then identity matrix I N N is seleted as the basis matrix. In that ase, the reovery matrix A an be predigested as ΦΨ = ΦI = Φ, thus the reonstrution algorithm merely uses Φ as its reovery matrix. OMP is employed as reonstrution algorithm. The input parameters of OMP are reovery matrix A and measurement set. In imaging system of parallel frequeny radar, reovery matrix is the prior measurement matrix Φ and the measurement set is the phase shift parts shown in Eq. 12). In addition, the output of OMP is the baksatter of targets, namely, the reonstrution result of Eq. 14) is the imaging of observation sene. The phase omplex data are read as

11 Progress In Eletromagnetis Researh, Vol. 133, the input of OMP in azimuth slow time τ i sequene. This means the reonstrution result is the observation imaging of the orresponding azimuth slow time τ i and the whole observation sene is omposed of reonstrution results in eah azimuth slow time. In reonstrution algorithm, we also set some effetive fators, for instane iterative times, error threshold and noise signal. High error threshold ondues more iterative times, and the imaging effiieny is degraded by the large omputation load generated by eah iteration. Noise signal also affets the suess of reonstrution, so we ompare the reonstruted results of noise free and noisy data in Setion 5. In order to selet the appropriate effeting fators in reonstrution algorithm, both imaging quality and algorithm effiieny should be taken into onsideration. 5. SIMULATION RESULTS 5.1. Imaging with Noise-free Condition Firstly, a single point is plaed on the enter observation node in the simulation. The parameters are seleted as follows: M = 30, f = 5 MHz, R MAX = 20 km and N = 512. Avoiding the influene of the different reonstrution algorithms, OMP is seleted in the following simulations. Fig. 6 shows two-dimensional ross setions of a point target imaging. As an be seen in Fig. 6, the side lobes of the range disappear via ompressive sensing. Additionally, azimuth side lobes still exist and it has a signifiant impat on azimuth time information reovery if the a) Figure 6. Point target ross setions of CS-based PFSAR. a) Azimuth ross setion. b) Range ross setion. b)

12 210 You et al. azimuth phases are not aurately foused. The ability of distinguishing targets in the adjaent range ells visibly upgrades due to avoiding side lobes of the range. As shown in Fig. 7, three point targets with different baksatters are plaed on the adjaent observation nodes along range diretion. The inonspiuous baksatter point is ompletely submerged by the onspiuous point target in the imaging result of the traditional SAR. At the same time, the inonspiuous baksatter point is distintly deteted in the imaging result based on ompressive sensing. Figure 8 ompares the several points two-dimensional imaging of PFSAR via ompressive sensing with that of traditional SAR. The parameters represented in Table 1 are seleted in the simulation. a) b) Figure 7. The ability of distinguishing the adjaent targets in the range diretion with 3 point targets. a) SAR. b) CS-based PFSAR. a) Figure 8. Imaging results of SAR and PFSAR via ompressive sensing with several point targets. a) SAR. b) CS-based PFSAR. b)

13 Progress In Eletromagnetis Researh, Vol. 133, Table 1. Simulation parameters in Fig. 8. System bandwidth 30 MHz Carrier frequeny 1.5 GHz Antenna length 10 m Swath width 20 km Pulse number 30 Pulse frequeny step 1 MHz a) b) Figure 9. SNR 30 db and SNR 20 db. a) Azimuth ross setion. b) Range ross setion. a) b) Figure 10. SNR 30 db and SNR 10 db. a) Azimuth ross setion. b) Range ross setion.

14 212 You et al. a) b) Figure 11. SNR 30 db and SNR 5 db. a) Azimuth ross setion. b) Range ross setion. Different data amounts are used in this simulation, the data amount of PFSAR is 10% of the SAR data; however, PFSAR still shows evident superiority in range diretion beause of restraining side lobes of the range Imaging with Noise Interferene The noise in the reeived signal annot be ignored in pratie. More researh is required on the apability of resisting the noise interferene of the signal proessing based on ompressive sensing. The randomness of the noise makes the reonstrution results unertain. The ross setions of the single point target with the different SNR are shown in Figs. 9, 10 and 11. Azimuth side lobes stohastially disappear with low SNR 20 db, 10 db and 5 db). The baksatter of the single point target begins to deline from 10 db SNR. The sparsity of noise signal annot ahieve the same level of performane as the noise free signal with the onstraint of l 2 norm. It is diffiult to reover the aurate amplitude of the original signal, whih is generally a harateristi of reonstruting noise signal by using ompressive sensing, namely, amplitude loss. 6. CONCLUSION This paper presents parallel frequeny SAR and imaging approah based on ompressive sensing. The novel radar an redue data rate and maintain the large total bandwidth. The measurement

15 Progress In Eletromagnetis Researh, Vol. 133, matrix Φ is omposed of the priori phases. The observation sene is reonstruted by OMP, and indeed this imaging approah avoids side lobes of the range. The reovery auray is improved aording to information sampling via ompressive sensing. It is onfirmed by several simulations. The noise interferene is not ignored in the real radar system and then the eho of single point target is interfused with Gaussian random noise in orrelative simulation. With the diminution of SNR, azimuth side lobes and the amplitude of baksatter begin to deline in imaging results. Amplitude loss appears in the reonstruted results. Some of the ideas of further work are depited finally. The sene omplexity and measurement dimension also have an impat on reonstruted results. It is important to disuss if the range resolution an be improved through importing the omplete partition of observation grid and prior range information of the targets. REFERENCES 1. Chan, Y. K. and V. C. Koo, An introdution to syntheti aperture radar SAR), Progress In Eletromagnetis Researh B, Vol. 2, 27 60, Donoho, D., Compressed sensing, IEEE Transations on Information Theory, 4th edition, Vol. 52, , Candes, E., Compressive sampling, Proeedings of the International Congress of Mathematiians, , Madrid, Spain, Baraniuk, R. and P. Steeghs, Compressive radar imaging, IEEE Radar Conferene, , April Herman, M. A. and T. Strohmer, High-resolution radar via ompressed sensing, IEEE Transations on Signal Proessing, Vol. 57, , Smith, G. E., T. Diethe, Z. Hussain, J. Shawe-Taylor, and D. R. Hardoon, Compressed sampling for pulse Doppler radar, IEEE Radar Conferene, , Bhattaharya, S., T. Blumensath, B. Mulgrew, and M. Davies, Fast enoding of syntheti aperture radar row data using ompressed sensing, IEEE Workshop on Statistial Signal Proessing, , Madison, USA, Rilling, G., M. Davies, and B. Mulgrew, Compressed sensing based ompression of SAR raw data, Signal Proessing with Adaptive Sparse Strutured Representations, Saint-MaloF), April 2009.

16 214 You et al. 9. Tello Alonso, M., P. Lopez-Dekker, and J. Mallorqui, A novel strategy for radar imaging based on ompressive sensing, IEEE International Geosiene and Remote Sensing Symposium, Vol. 2, II-213 II-216, Tello Alonso, M., P. Lopez-Dekker, and J. Mallorqui, A novel strategy for radar imaging based on ompressive sensing, IEEE Transations on Geosiene and Remote Sensing, Vol. 48, No. 12, , Lin, Y., Y. Wu, W. Hong, and B. Zhang, Compressive sensing in radar imaging, IET International Radar Conferene, 1 3, Anitori, L, M. Otten, and P. Hoogeboom, Compressive sensing for high resolution radar imaging, Proeedings of Asia-Paifi Mirowave Conferene, , Mann, S., R. Phogat, and A. K. Mishra, Dantzig seletor based ompressive sensing for radar image enhanement, Annual IEEE India Conferene, INDICON, 1 4, Potter, L. C., E. Ertin, J. T. Parker, and M. Cetin, Sparsity and ompressed sensing in radar imaging, Proeedings of the IEEE, Vol. 98, No. 6, , Ramkrishnan, N., E. Ertin, and R. L. Moses, Enhanement of oupled multihannel images using sparsity onstraints, IEEE Transations on Image Proessing, Vol. 19, No. 8, , Fornasier, M. and H. Rauhut, Reovery algorithms for vetorvalued data with joint sparsity onstraints, SIAM J. Numer. Anal, Vol. 26, No. 2, , Liang, Q., Compressive sensing for syntheti aperture radar in fast-time and slow-time domains, Proeedings of the IEEE, , Asilomar, Xu, J., Y. Pi, and Z. Cao, Bayesian ompressive sensing in syntheti aperture radar imaging, IET Radar, Sonar and Navigation, Vol. 6, No. 1, 2 8, Samadi, S., M. Cetin, and M. A. Masnadi-Shirazi, Sparse representation-based syntheti aperture radar imaging, IET Radar, Sonar and Navigation, Vol. 5, No. 2, , Wei, S. J., X. L. Zhang, and J. Shi, Linear array SAR imaging via ompressed sensing, Progress In Eletromagnetis Researh, Vol. 117, , Li, J., S. S. Zhang, and J. F. Chang, Appliations of ompressed sensing for multiple transmitters multiple azimuth beams SAR imaging, Progress In Eletromagnetis Researh, Vol. 127, 259

17 Progress In Eletromagnetis Researh, Vol. 133, , Ren, X. Z., Y. F. Li, and R. L. Yang, Four-dimensional SAR imaging sheme based on ompressive sensing, Progress In Eletromagnetis Researh B, Vol. 39, , Li, J., S. Zhang, and J. Chang, Appliations of ompressed sensing for multiple transmitters multiple azimuth beams SAR imaging, Progress In Eletromagnetis Researh, Vol. 127, , Wei, S.-J., X.-L. Zhang, J. Shi, and G. Xiang, Sparse reonstrution for SAR imaging based on ompressed sensing, Progress In Eletromagnetis Researh, Vol. 109, 63 81, Wei, S.-J., X.-L. Zhang, and J. Shi, Linear array SAR imaging via ompressed sensing, Progress In Eletromagnetis Researh, Vol. 117, , Ender, J. H. G., On ompressive sensing applied to radar, Signal Proessing, Vol. 90, , Candes, E., J. Romberg, and T. Tao, Robust unertainty priniples: Exat signal reonstrution from highly inomplete frequeny information, IEEE Transations on Information Theory, Vol. 52, , Blumensath, T. and M. Davies, Gradient pursuits, IEEE Transations on Signal Proessing, Vol. 56, , Davis, G., S. Mallat, and M. Avellaneda, Greedy adaptive approximation, Journal of Construtive Approximation, Vol. 12, 57 98, Donoho, D., M. Elad, and V. Temlyakov, Stable reovery of sparse overomplete representations in the presene of noise, IEEE Transations on Information Theory, Vol. 52, 6 18, Tropp, J. A. and A. C. Gilbert, Signal reovery from random measurement via orthogonal mathing pursuit, IEEE Transations on Information Theory, Vol. 53, , 2007.