Focusing Translational Variant Bistatic Forward-Looking SAR Data Based on Two-Dimensional Non-Uniform FFT

Transcription

1 Progress In Eletromagnetis Researh M, Vol. 37, 1 10, 2014 Fousing Translational Variant Bistati Forward-Looking SAR Data Based on Two-Dimensional Non-Uniform FFT Chan Liu 1, Shunsheng Zhang 1, *, Chunyang Dai 2, and Ji Zhou 1 Abstrat Forward-looking imaging has extensive potential appliations, suh as self-navigation and self-landing. By hoosing proper geometry, bistati syntheti aperture radar (BiSAR) an break through the limitations of monostati SAR on forward-looking imaging and provide possibility of the forwardlooking imaging. In this speial bistati onfiguration, two problems involving large range ell migration (RCM) and large range-azimuth oupling are introdued by the forward-looking beam, whih make it diffiult to use traditional data fousing algorithms. To address these problems, a novel Omega-K algorithm based on two-dimensional non-uniform FFT (2-D NUFFT) for translational variant (TV) bistati forward-looking SAR (BFSAR) imaging is proposed in this paper. In this study, we derive an aurate spetrum expression based on two-dimensional priniple of stationary phase (2-D POSP). 2-D NUFFT is utilized to eliminate the range-variant term, whih an make full use of the data and improve the omputational effiieny as well. The experimental results, presented herein, demonstrate the effetiveness and advantages of the proposed algorithm. 1. INTRODUCTION Syntheti aperture radar is a radar imaging tehnology, whih is apable of produing high resolution images of an observed area under all-time and all-weather [1]. In some appliations, suh as military and airraft landing systems, a forward-looking imaging radar is highly desirable. However, traditional monostati SAR is not appliable in forward diretion due to azimuth ambiguities and poor Doppler resolution. To improve the Doppler resolution and avoid azimuth ambiguities simultaneously, BiSAR in priniple provides the possibility of the forward-looking image by hoosing proper geometry [2 4]. In reent years, some valuable studies on BiSAR image formation have been proposed, whih an be divided into two ategories: time domain and frequeny domain. Time-domain methods an fous bistati SAR data well, but these methods always have huge omputational osts. The proessing effiieny an be highly improved in the frequeny domain, the bistati point target referene spetrum (BPTRS) is an important basis [5 7]. However, the existene of the double square root (DSR) term in the bistati range makes it diffiult to get the exat solution for the two-dimensional (2-D) spetrum. Approximate solutions to get 2-D spetrum have been researhed, suh as the method of series reversion (MSR) [5], Loffeld s bistati formula (LBF) [8, 9], extended LBF (ELBF) [6, 12], and two-dimensional priniple of stationary phase (2-D POSP) [10, 11]. Among these methods, 2-D POSP an be utilized to ahieve an approximate one-to-one orrespondene between time and Doppler frequeny, thereby formulating an aurate BPTRS. The bistati forward-looking SAR (BFSAR) employs the spatially separated forward-looking reeiver and side-looking transmitter. Some theories for BFSAR have been developed in [13 16]. In [13], BFSAR is ategorized into three types: translational invariant (TI), translational variant (TV), and stationary transmitter ases. In the TV ases, the transmitter and reeiver platforms are moving along Reeived 5 April 2014, Aepted 9 June 2014, Sheduled 13 June 2014 * Corresponding author: Shunsheng Zhang (zhangss.bit@gmail.om). 1 Researh Institute of Eletroni Siene and Tehnology, University of Eletroni Siene and Tehnology of China, Chengdu, Sihuan , China. 2 Institute of Eletronis, Chinese Aademy of Sienes, Beijing , China.

2 2 Liu et al. different trajetories with different veloities. Compared with the other two ases, TV-BFSAR is more flexible and an be used for reonnaissane, guidane and so on. However, in TV-BFSAR, the large range-azimuth oupling and large range ell migration (RCM) are the major problems. Moreover, the spatial variations of TV-BFSAR raw data have 2-D harateristis. Hene, many algorithms used in BiSAR annot fous TV-BFSAR data well, and hoosing proper imaging method is a key issue. In [10], range-doppler algorithm (RDA) and hirp saling algorithm (CSA) based on 2-D POSP for BiSAR is proposed. But the spatial variations of the Doppler parameters in the azimuth diretion are ignored, whih auses signifiant performane degradation of RCM orretion and azimuth ompression. Zhang and Li [14] introdue a novel RDA for BFSAR onfiguration. However, owing to the approximation of the phase history, the fousing auray is severely affeted. Additionally, this method is only able to get narrow-swath BFSAR image. An Omega-K imaging algorithm for TI-BiSAR has been proposed in [17]. But the one-dimensional (1-D) Stolt interpolation used in [17] an only eliminate the spatial variations in the range diretion. Due to the 2-D spatial variation harateristis, this algorithm is no longer appliable for TV-BFSAR imaging. Using ELBF, an Omega-K algorithm for BiSAR is presented in [12]. For the range-variant term, it adopts 2-D Stolt interpolation to orret the nonlinear dependene on the range and azimuth frequenies. 2-D Stolt interpolation transforms the non-uniform frequeny domain into a regular spaed retangle [18, 19]. For TV mode, the imaging blok size beomes very small, leading low utilization of the olleted data in the spatial frequeny domain. Hene, this method based on Stolt interpolation is not appliable for TV-BFSAR any more. Non-uniform FFT (NUFFT) is an aurate and effiient tehnique to evaluate the non-uniform disrete Fourier transform (NUDFT) [19 23]. Reently, NUFFT tehnique was introdued to the researh field of SAR data proessing [19, 21, 22]. In this paper, a novel Omega-K algorithm based on 2-D NUFFT for TV-BFSAR imaging is proposed. Based on the aurate spetrum expression derived by using 2-D POSP, 2-D NUFFT is utilized to ompute the optimum reonstrution image, and omes at low omputational ost saled with omplexity O(MNlog 2 MN). Compared with the Omega-K method presented in [12], the proposed algorithm an make full effiient use of the olleted data in the spatial frequeny domain. Performane omparisons of the proposed algorithm with the algorithms in [10, 12] are onduted to demonstrate the advanes of the Omega-K algorithm presented in this paper. The rest of this paper is organized as follows. The signal model of BFSAR is established in Setion 2. In Setion 3, the spetrum derived using 2-D POSP is rewritten and a novel Omega-K algorithm based on 2-D NUFFT is proposed. Setion 4 onduts the performane omparisons of the proposed algorithm with onventional algorithms. The onlusions of the paper are drawn in Setion SIGNAL MODEL Figure 1 shows the imaging geometry of the bistati forward-looking SAR, inluding parallel traks and unequal veloities. The reeiving antenna is forward looking, while the transmitter antenna is squint looking. The sene enter is set to be the origin of oordinates and the height of the sene is assumed to be zero. After demodulation to baseband, the reeived signal an be expressed by the range time τ and azimuth time η ( s(τ, η) = p τ R (η) ) ( exp j2π R (η) ) (1) λ represents the speed of light and λ the arrier wavelength. p( ) is defined as the transmitted signal, and the instantaneous range history R(η) is given by R(η) = R T (η) + R R (η) (2) R T (η) = R 2 0T + (η η 0T ) 2 v 2 T (3) R R (η) = R 2 0R + (η η 0R ) 2 v 2 R here R 0T is the minimum slant range from transmitter to sene enter and R 0R the height of reeiver. η 0T and η 0R stand for initial offset time of transmitter and reeiver, respetively. The moving veloities of transmitter and reeiver are respetively denoted by and v R.

3 Progress In Eletromagnetis Researh M, Vol. 37, Transmitter ν T Reeiver ν R R (η) R R (η) T Z Y Footprint X Figure 1. Imaging geometry of bistati forward-looking SAR. 3. IMAGE FORMATION ALGORITHM 3.1. Rewriting The Signal Spetrum 2-D FFT is made toward (1) using 2-D POSP [10], then ignore the amplitude, the spetrum expression of bistati forward-looking eho signal an be desribed as S (f τ, f η ) = p (f τ ) exp [ jϕ (f τ, f η )] (4) [ R0T ϕ (f τ, f η ) = 2π (f ηt η 0T + f ηr η 0R ) + 2π F T W + R ] 0R F RW ( ) 2 F T W = (f τ + f 0 ) 2 fηt F RW = (f τ + f 0 ) 2 ( ) 2 fηr v R (5) f ηt = k T (f η f DR f DT ) + f DT f ηr = k R (f η f DR f DT ) + f DR f DT = sin θ ST f DR = v R sin θ SR / (f τ + f 0 ) / (f τ + f 0 ) k ηt k ηr k T = k R = k ηt + k ηr k ηt + k ηr (6) k ηt = v2 T os3 θ ST λr 0T k ηr = v2 R os3 θ SR λr 0R f τ and f η stand for the range and azimuth frequeny, respetively. f 0 is the arrier frequeny of the transmitted signal. The squint angles of transmitter and reeiver at the omposite beam enter rossing time are respetively denoted by θ ST and θ SR. To failitate the development of proessing algorithms, we bilinearly express η 0T, R 0T, and R 0R in terms of (r, η 0R ) [9]. η 0T = p 10 + p 11 r + p 12 η 0R R 0T = p 20 + p 21 r + p 22 η 0R R 0R = r + R m (7) R m is the losest range from the sene enter to the ideal trajetory of reeiver and r = R 0R R m. Substituting (7) into (5), the phase history an be expressed as ϕ (f τ, f η ) 2π [p 10 + p 11 (r + R m )] k T f η + 2π (p 12 k T + k R ) η 0R f η + 2π (r + R m) F RW + 2π (p 20+p 21 r+p 22 η 0R ) F T W + 2π (k T v R sin θ SR k R sin θ ST ) (f τ +f 0 ) [(1 p 12 )η 0R p 10 p 11 r] (8)

4 4 Liu et al. Similar to [8], (8) an be deomposed into two omponents: QM phase term and BD phase term. ϕ (f τ, f η ) = ϕ QMW (f τ, f η ) + ϕ BDW (f τ, f η ) (9) ϕ QMW (f τ, f η ) = 2πp 10 k T f η + 2πp 11 rk T f η + 2π (p 12 k T + k R ) η 0R f η + 2π p 22η 0R F T W + 2π (r + R m) F RW + 2π (p 20 + p 21 r) F T W (10) ϕ BDW (f τ, f η ) = 2π (k T v R sin θ SR k R sin θ ST ) (f τ + f 0 ) (η 0R η 0T ) + 2πp 11 R m k T f η (11) From (10), it an be readily shown that 2πp 22 η 0R F T W / ontains an azimuth-dependent range ell migration (RCM) term whih is introdued by the azimuth-variant baseline between transmitter and reeiver. For larity, we expand F T W with respet to f τ and f η as ( 2π p 22η 0R F T W 2π ( ) 2 p 22η 0R f0 2 fd + f d ) 2 f 2 0 ( fd ) 2 + f τ ( ) λfd 2 1 ( ) 2fd f 2 0 ( fd ) 2 f η (12) f d = v R sin θ SR /λ + sin θ ST /λ. In (12), the first two terms denote the residual phase terms and are negligible; the third term is the azimuth-dependent RCM and the last term represents the azimuth-saling term [9]. Therefore, (10) and (11) an be rewritten as ϕ QMW (f τ, f η ) = 2πp 10 k T f η +2πp 11 rk T f η + 2π (r+r m) F RW + 2π (p 20+p 21 r) F T W + 2πβ η η 0R f η (13) ϕ BDW (f τ, f η ) = ϕ BDW (f τ, f η ) + 2πp 22η 0R f τ ( ) (14) 2 1 λfd β η = p 12 k T + k R 3.2. Omega-K Algorithm Based on 2-D NUFFT v 2 T λp 22 f d ( ) (15) 2 1 λfd To deal with ϕ BDW, we perform a preproessing operation in the range-azimuth subsetions to remove it. The preproessing funtion is given as H pre (f τ, f η ) = exp [j ϕ BDW (f τ, f η )] (16) For further larity, ϕ QMW is deomposed into a range-invariant term, range-variant term and azimuth-saling term [12]. Hene, (13) an be expressed as ϕ QMW (f τ, f η ) = φ 1 (f τ, f η ) + φ 2 (f τ, f η ) + φ AS (f η ) (17) φ 1 (f τ, f η ) = 2πp 10 k T f η + 2π R mf RW + 2π p 20F T W (18) φ 2 (f τ, f η ) = 2π (1 + p 21) r (p 11 k T f η + F RW + p 21 F T W ) (1 + p 21 ) (19) φ AS (f η ) = 2πβ η η 0R f η (20) φ 1 (f τ, f η ) represents the range-invariant omponent. It an be removed with referene funtion multipliation (RFM) in the 2-D frequeny domain. φ AS (f η ) is the azimuth-saling term. In this paper, the orretion fator for φ AS (f η ) is substituted by averaged values over slant range and azimuth time. Thus, the azimuth-saling term an also be orreted with RFM in the 2-D frequeny domain. The RFM filter is obtained as following ( H RF M (f τ, f η ) = exp {j [φ 1 (f τ, f η ) + φ AS (f η )]} p (f τ ) exp j2π R ) m + p 20 f τ (21)

5 Progress In Eletromagnetis Researh M, Vol. 37, The last exponential term of (21) denotes a phase orretion to establish the phase referene to the sene enter. φ 2 (f τ, f η ) stands for the range-variant omponent, whih ontains the range-variant RCM, rangeazimuth oupling and azimuth modulation. Obviously, it is signifiant to orret φ 2 (f τ, f η ) in a subsequent operation to fous preisely over the whole sene. The oupling between f τ and f η an be removed if we resale f τ by the frequeny mapping f τ = (p 11k T f η + F RW + p 21 F T W ) (1 + p 21 ) f 0 (22) After mapping operation, the data from (f τ, f η ) spae to (f τ, f η ) spae, (19) an be onverted to φ 2 (f τ, f η ) = 2π (1 + p 21) r (f τ + f 0 ) (23) It an be readily shown from (22) that the range-azimuth oupling an be eliminated and RCM an be suffiiently orreted. However, due to non-uniform distribution of the spatial frequenies f τ and f η, the foused image annot be obtained only by 2-D inverse FFT (IFFT). Stolt interpolation is in general utilized to solve this problem, whih transforms the 2-D non-uniform frequeny domain into a regular spaed retangle. Then 2-D IFFT is employed to generate an image [17]. This method inevitably makes the supported domain small and dereases the data utilization. Using the simulation parameters defined in Setion 4, the non-uniform distribution of the spatial frequenies f τ and f η is showed in Fig. 2. In this kind of TV-BFSAR ase, the method based on Stolt interpolation an only utilize a small part (the red retangle in Fig. 2) in the skew 2-D support. Obviously, the data utilization is quite low, whih affets the image performane severely. Mapped range frequeny (MHz) Azimuth frequeny (Hz) Figure 2. Frequeny samples point in (f τ, f η ) spae. NUFFT is an aurate and effiient tehnique to evaluate the non-uniform disrete Fourier transform (NUDFT) [19 23]. Aording to [20], the reonstrution of target sattering funtion an be represented by I k,l = M 1 N 1 m=0 n=0 [ S m,n exp ( j 2π M k α m,n + j 2π )] N l β m,n M/2 k < M/2, N/2 l < N/2 (24) S m,n = S (m f τ, n f η ) (25) α m,n = M x 2π [α (m f τ, n f η ) α 0 ] (26) β m,n = N y 2π [β (m f τ, n f η ) β 0 ] (27) x, y, f τ and f η denote the sampling interval in X-axis, Y -axis, range frequeny and azimuth frequeny, respetively. M and N represent the number of sampling points in range frequeny and azimuth frequeny. After RFM, the signal in the 2-D frequeny domain is denoted by S. α and β is the spatial frequenies f τ and f η. (α 0, β 0 ) is the midpoint of the irregular spatial frequenies region.

6 6 Liu et al. Aording to [20], 2-D NUFFT an substitute for Stolt interpolation and 2-D IFFT to get the foused image, but also an effetively utilize the eho information. Additionally, the diret omputing of (24), a two-dimensional disrete Fourier transform proess, takes O(M 2 N 2 ) arithmetial operations, while 2-D NUFFT based on Least-Square an ompute the same result in only O(MN log 2 MN) operations whih is the same to the omplexity for the algorithm based on 2-D POSP in [10] and the omega-k algorithm in [12]. Hene, 2-D NUFFT for BFSAR image formation is a good balane between omputational load and fousing performane. The proessing of the proposed algorithm onsists of the following steps. (1) Using 2-D POSP, 2-D spetrum expression is generated. (2) Based on bilinear regression and Taylor expansion, the phase history is rewritten, and it an be expressed as QM term and BD term. (3) Preproessing operation to deal with the BD term. (4) RFM to remove the range-invariant omponent and the azimuth-saling term. (5) Mapping operation to eliminate the range-azimuth oupling and orret the RCM. (6) 2-D NUFFT operation to linearize both 2-D frequeny and obtain the foused image. 4. SIMULATION RESULTS The purpose of the simulation is to verify the effetiveness of the proposed imaging algorithm. The simulation parameters are listed in Table 1. Firstly, point satterers simulation is onduted to validate the proposed algorithm in this paper. Then, we apply the traditional methods in [10, 12] to BFSAR onfiguration. Performane omparisons of the proposed algorithm with the traditional methods demonstrate that the former an fous TV-BFSAR better. Table 1. Simulation radar parameters. Transmitter initial loation ( 3, 0, 8) km Carrier frequeny 9.65 GHz Reeiver initial loation (0, 2, 4) km Range bandwidth 100 MHz Veloity of transmitter 200 m/s PRF 1000 Hz Veloity of reeiver 100 m/s Pulse width 10 µs Y 500 m 500 m P m 500 m O 500 m 500 m X P m 500 m Figure 3. Loation of point targets for simulation. Figure 4. Two-dimensional spatial-domain result after the 2-D NUFFT operation.

7 Progress In Eletromagnetis Researh M, Vol. 37, Point Satterers Simulation The simulated senario onsists of nine point targets in the ground plane z = 0, whih is shown in Fig. 3. The referene point O is loated at the enter of the sene, and the other eight points are all loated in a 1100 m 1100 m square. Among the nine point targets, P 1 is the furthest-range point target, and P 2 is the shortest-range point target. Using the parameters of Table 1, the nine point targets are reonstruted via the proposed algorithm. The result is shown in Fig. 4. Obviously, 2-D NUFFT aomplishes range-azimuth oupling orretion and RCM orretion Performane Comparisons A smaller senario (300 m 300 m) is hosen to test the algorithms in order to ompare the foused performane of the algorithms in more detail. In this senario, points A(100, 100), B(0, 0), C( 100, 100) are analyzed to quantify the imaging preision of the proposed algorithm. In order to show the details, the results are interpolated by a fator of 16. Figs. 5(a) () give the ontours of points A, B, and C proessed by the proposed Omega-K algorithm. Comparatively, Figs. 5(d) (f) give the ontours of points A, B, and C proessed by the algorithm based on ELBF in [12], and Figs. 5(g) (i) give those ontours proessed by the RD algorithm based on 2-D POSP in [10]. In Fig. 5, the horizontal axis and the vertial axis are range samples and azimuth samples, respetively. By the simulation Point Target A Point Target B Point Target C (a) (b) () (d) (e) (f) (g) (h) (i) Figure 5. Contours of points A, B, and C in the 300 m 300 m square. (a) () Proessed by the proposed method. (d) (f) Proessed by the method in [12]. (g) (i) Proessed by the method in [10].

8 8 Liu et al. results, it is lear that the proposed algorithm and the RD algorithm based on 2-D POSP in [10] an fully fous all the targets at the same time, as simulation results of the method in [12] is not so well. On the other hand, points P 1, O, and P 2 in the 1100 m 1100 m square are also analyzed in detail. Similarly, Fig. 6 respetively give the ontours of points P 1, O, and P 2 proessed by the proposed Omega-K algorithm, and the methods in [10, 12]. In Fig. 6, the horizontal axis and the vertial axis are range samples and azimuth samples, respetively. The simulation results show that the algorithm proposed in this paper an fully fous all the targets at the same time. By ontrast, the methods in [10, 12] an fous the target at the referene position well, as targets at the orners of senario are only partially foused. In order to evaluate the reonstruted point target at the orner of senario, the Peak Side-Lobe Ratio (PSLR) and the Integrated Side-Lobe Ratio (ISLR) are utilized as measurement of performane, and the evaluation results of target P 1 are tabulated in Table 2. In addition, the azimuth profiles with respet to target P 1 are shown in Fig. 7, the red solid line, green solid line, and blue dotted line represent the results using the proposed method in this paper and the methods in the literatures, respetively [10, 12]. It an be seen that the proposed algorithm in this paper has good fousing performane, as the fousing performane of the imaging algorithm in [12] and the method in [10] Point Target P Point Target O Point Target P 1 2 (a) (b) () (d) (e) (f) (g) (h) (i) Figure 6. Contours of points P 1, O, and P 2 in the 1100 m 1100 m square. (a) () Proessed by the proposed method. (d) (f) Proessed by the method in [12]. (g) (i) Proessed by the method in [10].

9 Progress In Eletromagnetis Researh M, Vol. 37, Table 2. Imaging performane of P 1. Range Azimuth PSLR (db) ISLR (db) PSLR (db) ISLR (db) The proposed method The method in [12] The method in [10] The proposed method The method in [12] The method in [10] Magnitude (db) Figure 7. Azimuth profiles of targets P Azimuth (samples) is not so well. The simulation results for different senarios indiate that the proposed algorithm in this paper has nearly ideal fousing performane, even in a large senario. However, the imaging algorithm of Wang [12] has bad imaging results mainly beause the approximate spetrum based on ELBF and the utilization of Stolt interpolation make the algorithm unsuitable for TV-BFSAR. Due to ignoring the spatial variations of the Doppler parameters in the azimuth diretion, the RD algorithm based on 2-D POSP in [10] an foused well in a small sene, as it an only partially fous targets at the orners of a large senario. Thus, if the method in [10] is applied to BFSAR onfiguration, it is limited to small senarios. 5. CONCLUSIONS This paper presents a novel Omega-K imaging algorithm for translational variant bistati forwardlooking SAR imaging. The ore of the Omega-K algorithm is to obtain the preise spetrum based on 2-D POSP, and eliminate the range-azimuth oupling and orret the RCM using 2-D NUFFT. NUFFT alulates omplex kernels based on the given sampling pattern to minimize the interpolation error, and sales with omplexity O(MNlog 2 MN). The simulation results validate that the proposed algorithm has the ability to fous TV-BFSAR data within a 1100 m 1100 m region. Moreover, performane omparisons prove that the imaging results of 2-D NUFFT proessing have nearly ideal fousing performane with low omputational ost. 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, Balke, J., Field test of bistati forward-looking syntheti aperture radar, Pro. Reord of the IEEE 2005 Int. Radar Conf., , Waltersheid, I., T. Espeter, J. Klare, and A. Brenner, Bistati spaeborne-airborne forwardlooking SAR, Pro. of EUSAR, , 2010.

10 10 Liu et al. 4. Espeter, T., I. Waltersheid, J. Klare, A. R. Brenner, and J. H. G. Ender, Bistati forward-looking SAR: Results of a spaeborne-airborne experiment, IEEE Geosiene and Remote Sensing Letters, Vol. 8, No. 4, , Neo, Y. L., F. Wong, and I. G. Cumming, A two-dimensional spetrum for bistati SAR proessing using series reversion, IEEE Geosiene and Remote Sensing Letters, Vol. 4, No. 1, 93 96, Wang, R., O. Loffeld, Q. Ul-Ann, et al., A bistati point target referene spetrum for general bistati SAR proessing, IEEE Geosiene and Remote Sensing Letters, Vol. 5, No. 3, , Wu, J. J., J. Y. Yang, Y. L. Huang, Z. Liu, and H. G. Yang, A new look at the point target referene spetrum for bistati SAR, Progress In Eletromagnetis Researh, Vol. 119, , Loffeld, O., H Nies, V. Peters, and S. Knedlik, Models and useful relations for bistati SAR proessing, IEEE Transations on Geosiene and Remote Sensing, Vol. 42, No. 10, , Natroshvili, K., O. Loffeld, H. Nies, A. Medrano-Ortiz, and S. Knedlik, Fousing of general bistati SAR onfiguration data with 2-D inverse saled FFT, IEEE Transations on Geosiene and Remote Sensing, Vol. 44, No. 10, , Wang, R., Y. K. Deng, O. Loffeld, et al., Proessing the azimuth-variant bistati SAR data by using monostati imaging algorithms based on two-dimensional priniple of stationary phase, IEEE Transations on Geosiene and Remote Sensing, Vol. 49, No. 10, , Li, J., S. S. Zhang, and J. F. Chang, Bistati forward-looking SAR imaging based on twodimensional priniple of stationary phase, Pro. of MMWCST, , Wang, R., O. Loffeld, H. Nies, and J. H. G. Ender, Fousing spaeborne/airborne hybrid bistati SAR data using wavenumber-domain algorithm, IEEE Transations on Geosiene and Remote Sensing, Vol. 47, No. 7, , Wu, J. J., J. Y. Yang, Y. L. Huang, H. G. Yang, and H. C. Wang, Bistati forward-looking SAR: Theory and hallenges, Pro. Reord of the IEEE 2009 Int. Radar Conf., 1 4, Zhang, S. S. and J. Li, Forward-looking bistati SAR imaging based on high order range equation and high order phase ompensation, Journal of Eletromagneti Waves and Appliations, Vol. 26, Nos , , Qiu, X. L., D. H. Hu, and C. B. Ding, Some refletions on bistati SAR of forward-looking onfiguration, IEEE Geosiene and Remote Sensing Letters, Vol. 5, No. 4, , Wu, J. J., Z. Y. Li, Y. L. Huang, J. Y. Yang, H. G. Yang, and Q. H. Liu, Fousing bistati forwardlooking SAR with stationary transmitter based on Keystone transform and nonlinear hirp saling, IEEE Geosiene and Remote Sensing Letters, Vol. 11, No. 1, , Liu, B., T. Wang, Q. Wu, and Z. Bao, Bistati SAR data fousing using an omega-k algorithm based on method of series reversion, IEEE Transations on Geosiene and Remote Sensing, Vol. 47, No. 8, , Xiong, T., M. D. Xing, X. G. Xia, and Z. Bao, New appliations of Omaga-K algorithm for SAR data proessing using effetive wavelength at high squint, IEEE Transations on Geosiene and Remote Sensing, Vol. 51, No. 5, , Han, K., C. C. Chen, D. D. Shen, F. Pan, et al., An aurate 2-D nonuniform fast Fourier transform method applied to high resolution SAR image reonstrution, Pro. of Metamaterials, 1 4, Dai, C. Y. and X. L. Zhang, Bistati polar format algorithm based on NUFFT method, Journal of Eletromagneti Waves and Appliations, Vol. 25, Nos , , Andersson, F., R. Moses, and F. Natterer, Fast Fourier methods for syntheti aperture radar imaging, IEEE Transations on Aerospae and Eletroni Systems, Vol. 48, No. 1, , Liu, Z., J. Y. Yang, X. L. Zhang, H. Huang, and W. C. Li, Imaging algorithm based on leastsquare NUFFT method for spaeborne/airborne squint mode bistati SAR, Pro. of IGARSS, , Liu, Q. H. and N. Nguyen, An aurate algorithm for nonuniform fast Fourier transforms (NUFFT s), IEEE Mirowave and Guided Wave Letters, Vol. 8, No. 1, 18 20, 1998.