An Extended Wavenumber-Domain Algorithm Combined with Two-Step Motion Compensation for Bistatic Forward-Looking SAR

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1 Progress In Eletromagnetis Researh Letters, Vol. 63, 85 92, 2016 An Extended Wavenumber-Domain Algorithm Combined with Two-Step Motion Compensation for Bistati Forward-Looking SAR Yuebo Zha 1, * and Wei Pu 2 Abstrat With appropriate geometry onfigurations, bistati Syntheti Aperture Radar (SAR) an break through the limitations of monostati SAR on forward-looking imaging. Thanks to suh a apability, bistati forward-looking SAR (BFSAR) has extensive potential appliations. For the fousing problem of BFSAR, wavenumber-domain algorithm is aepted as the ideal solution. However, in pratial appliation, the proessing is limited beause of its inability to ombine the rangedependent motion ompensation (MoCo). To ope with suh a problem, an extended wavenumberdomain algorithm for BFSAR is derived in this paper. By modifying the referene funtion and mapping relationship in frequeny interpolation, the extended wavenumber-domain algorithm of BFSAR integrates a two-step motion ompensation. Simulation results verify the effetiveness of the proposed method. 1. INTRODUCTION In bistati syntheti aperture radar (SAR), as the transmitter and reeiver are mounted on separate platforms, it is possible to ahieve high-resolution image in the forward-looking area. Hene, bistati forward-looking SAR (BFSAR) has signifiant appliation value in the field of airplane navigation, landing, et. Some theories for BFSAR have been developed in [1], and some reonstrution methods, suh as range-doppler [3] and Chirp-Saling [4 6], are ongoing. In ontrast, wavenumber-domain algorithm [7] is viewed as an ideal solution to the BFSAR fousing problem as it solves the severe spatial dependene. Nevertheless, the wavenumber-domain algorithm annot satisfy the requirement of pratial appliation as it has no possibility of inluding a high preision, range-dependent motion ompensation (MoCo). Compared with the range-doppler (RD) and hirp saling (CS) algorithms with two-step MoCo, only the first order MoCo an be applied in the wavenumber-domain algorithm. The first order MoCo is the range-independent part of MoCo and is applied diretly before or after range ompression. In the ase of high-resolution systems, a preise ompensation of the range-dependent omponents of the motion error is essential. This proessing step, whih has to be implemented after range ell migration orretion (RCMC) and before azimuth ompression, is not possible. Reigber et al. propose an extended wavenumber-domain algorithm for monostati SAR [8]. By hanging the referene funtion multipliation and mapping relationship in frequeny interpolation, the algorithm an ope with range-dependent motion errors. However, it annot be diretly applied in bistati forward-looking mode as the different spetrum expression between monosati SAR and bistati SAR [9, 12]. Reeived 13 July 2016, Aepted 11 Otober 2016, Sheduled 31 Otober 2016 * Corresponding author: Yuebo Zha (zhayuebo@163.om). 1 China Eletronis Tehnology Group Corporation No. 38 Researh Institute, 199 Xiangzhang Avenue, Hi-Teh Zone, Hefei, Anhui 288, China. 2 Shool of Eletroni Engineering, University of Eletroni Siene and Tehnology of China, Chengdu, Sihuan , China.

2 86 Zha and Pu In the following, an extended wavenumber-domain algorithm with integrated two-step MoCo for BFSAR is proposed in this paper, whih is also an extension of the previous result [2]. The rest of this paper is organized as follows: An wavenumber-domain algorithm for BFSAR based on the point target referene spetrum (PTRS) alulated by MSR is presented in Setion 2. The extended wavenumber-domain algorithm is analytially derived in Setion 3. ompression is implemented independently, so that range-dependent MoCo an be integrated. Simulation results are arried out to verify the effetiveness of the proposed method in Setion 4. Finally, we draw our onlusions in Setion WAVENUMBER-DOMAIN ALGORITHM FOR BFSAR In this setion, the signal model of BFSAR is formulated, and the orresponding geometry is shown in Fig. 1. Transmitter and reeiver are assumed to move along two lines parallel to y axis with equal veloity V. The oordinate origin O(x ref, y ref, 0) is set to be the referene target and P (x, y, 0) is an arbitrary point target in the imaging area. The squint angles θ T P, θ R P and initial ranges R Ten P, R Ren P shown in Fig. 1 are measured at the omposite beam enter rossing time of target P. Figure 1. Imaging geometry onfiguration of BFSAR. Assume that Linear Frequeny Modulated (LFM) pulses are transmitted by the radar and the eho data from target P an be adequately desribed by [ ] τ 2rP (η)/ s(τ,η) =ret ret( η [ ) exp { jπk r τ 2r ] } P (η) 2 { exp j2π 2r } P (η) (1) T r T a λ where τ denotes the range time, and the azimuth time η is hosen to be zero at the omposite beam enter rossing time of the sene enter. λ is the wave-length, K r is the transmitted hirp rate, is the speed of propagation, T r is the timewidth of the LFM pulse and T a is the syntheti aperture time. In Eq. (1), r P (η) represents the instantaneous two-way range of the target P. r P (η) =r PT (η)+r PR (η) (2) where r PT (η) = RTen 2 P +(Vη y)2 2R Ten P (Vη y)osθ T P (3) r PR (η) = RRen 2 P +(Vη y)2 2R Ren P (Vη y)osθ R P (4) Here, method of series reversion (MSR) [10] is utilized to alulate the PTRS of BFSAR. Firstly, FFT in range diretion is performed for the raw data in Eq. (1). Then, the relationship between azimuth frequeny and time is derived by MSR, and therefore the stationary phase point of azimuth diretion an

3 Progress In Eletromagnetis Researh Letters, Vol. 63, be dedued. Using the stationary phase point, analytial expression of PTRS of BFSAR with respet to target P is alulated. S(f τ,f η )=W r (f τ )W az (f η )exp{jφ P (f τ,f η )} { = W r (f τ )W az (f η )exp j2πr } en P y F ref (f η,f τ ) j2πf η v jπf2 τ (5) K r In Eq. (5), φ P (f τ, f η ) is the phase of PTRS, and F ref (f η, f τ ) is the frequeny term. 2 ( F ref (f η,f τ ) = 4α ref (f + f τ ) f η +(f + f τ ) k ) 2 ref1 3 ( β ref 8α 3 ref (f + f τ ) 2 f η +(f + f τ ) k ) 3 ref1 4 (9βref 2 /α5 ref 4γ ref/α 4 ref ) ( 64(f + f τ ) 3 f η +(f + f τ ) k ) 4 ref1 +(f + f τ ) (6) R en P is the two-way range of target P measured at the omposite beam enter rossing time; f orresponds to the enter frequeny; f η and f τ denote the azimuth and range frequenies, respetively. W r ( ) represents the spetral shape of the transmitted pulse, and W az (η) denotes the shape of Doppler spetrum. α ref, β ref and γ ref are the rewritten results of k ref2, k ref3 and k ref4 by performing some algebrai manipulations, where k ref1, k ref2, k ref3 and k ref4 represent the first, seond, third and fourth order derivatives of the instantaneous two-way range history of the referene target at η = 0, respetively (seen in [11]). Based on the PTRS in Eq. (5), an wavenumber-domain algorithm an be derived. Firstly, a referene funtion multipliation (RFM) is performed to aomplish the 2-dimensional bulk fous. { S RF M(f τ,f η )=exp j 2πR } en ref F ref (f η,f τ )+j πf2 τ y ref + j2πf η (7) K r v Then, a Stolt interpolation operation an be utilized to handle the range-dependent part of RCM, azimuth ompression, and seond range ompression (SRC) orretly. f τ + f = F ref (f η,f τ ) (8) Nevertheless, as RCMC and azimuth ompression are performed simultaneously, it annot integrate the range-dependent MoCo. 3. EXTENDED WAVENUMBER-DOMAIN FOR BISTATIC FORWARD-LOOKING SAR Assoiated with PTRS in Setion 2, an extended wavenumber-domain algorithm for BFSAR is proposed in this setion. The proposed algorithm uses modified RFM and modified Stolt interpolation in order to separate azimuth ompression. Therefore, range-dependent MoCo an be applied before azimuth ompression. The detail of the proposed method is presented Modified RFM Firstly, range ompression is performed, and the first order MoCo is aomplished. Then, the data after first order MoCo is transformed into 2-D frequeny domain. In Eq. (7), the RFM fouses the referene target ompletely. An alternative way is to transform the frequeny fator in Eq. (7) into a different Fourier integral, whih separates the RCM orretion and SRC from the atual azimuth ompression. The frequeny term in Eq. (6) is expanded into the series of range frequeny f. F ref (f τ,f η ) F az (f η )+F rm (f η )f τ + F sr (f η )fτ 2 (9) where F az (f η ) denotes the azimuth ompression term, F az (f η )=F ref (f η,f τ ) fτ =0 (10)

4 88 Zha and Pu F rm (f η ) denotes the range ell migration (RCM) fator, F rm (f η,f τ )=F ref (f η,f τ ) fτ =0 (11) and F sr (f η,f τ ) is the seond range ompression part. F sr (f η,f τ )= 1 2 F ref (f η,f τ ) fτ =0 (12) In order to separate azimuth ompression, F az (f η ) is subtrated from the referene funtion, and Eq. (7) is hanged into { S RF M (f τ,f η )=exp j 2πR } en ref y ref [F rm (f η )+F sr (f η )] + j2πf η (13) v where R en ref is two-way range of the referene target measured at the omposite beam enter rossing time. Here, only bulk RCM and bulk SRC are aomplished in RFM, while bulk azimuth ompression is separated. Residual phase of PTRS in Eq. (5) onverts into φ P 1 (f τ,f η )= j 2πR en ref y y ref F az (f η ) j2πf η j 2πΔR Pen F ref (f η,f τ ). (14) v where ΔR Pen represents the range oordinate of target P. ΔR en P = R en P R en ref (15) 3.2. Modified Stolt Interpolation and Compression In order to separate azimuth ompression, a hange of variables, different from the Stolt mapping in Eq. (8), is applied. f τ new + f = F rm (f η )f τ + F sr (f η )fτ 2 (16) The modified Stolt mapping additionally introdues a frequeny shift and performs residual RCMC and residual SCR. F az (f η ) denotes residual azimuth ompression in Stolt interpolation. Hene, residual azimuth ompression is eliminated in Eq. (16). The residual phase of PTRS after modified Stolt interpolation hanges into φ P 2 (f τ,f η )= j 2πΔR Pen { } y y ref fτnew + f 2πfη j 2πΔR Pen F 1 ref (f η ) v j 2πR en ref F 1 ref (f η ) (17) At this point, the range-dependent motion ompensation an be applied beause RCM orretion has been done ompletely, while azimuth ompression has not been implemented yet. Here, phase orretion of the form { S amf (f η )=exp j 2πR Pen F az (f η ) performs the final azimuth fousing. After multiplied by the azimuth ompression phase, the residual phase of PTRS is φ P 3 (f τ new,f η )=j 2πΔR Pen { f + f y y ref τ new} j2πfη (19) v The data in Eq. (19) is transformed into 2-D spatial domain by using 2-D IFFT to get a well foused image. To make it lear, flowhart of the proposed extended wavenumber-domain algorithm for BFSAR is presented in Fig. 2. } (18)

5 Progress In Eletromagnetis Researh Letters, Vol. 63, Raw data Compression First order MoCo 2-D FFT S (, ) RFM f f τ η Modified Stolt interpolation -dependent MoCo Samf ( ) f η 2-D IFFT image Figure 2. Flowhart of the extended wavenumber-domain algorithm for BFSAR. Table 1. Simulation parameters. Parameter Carrier frequeny Band width Syntheti aperture time Nominal Radar platform veloity Pulse repetition frequeny Coordinates of transmitter Coordinates of reeiver Value 9.6GHz MHz 5 s m/s 0 Hz (6, 3, 6) km (0, 6, 4) km 4. SIMULATION RESULTS To evaluate the performane of the proposed extended wavenumber-domain algorithm, simulations for BFSAR are arried out in this setion. The simulation parameters are shown in Table 1. 9 targets are set in the sene area as shown in Fig. 3. The distane between two adjaent targets along azimuth diretion is m while the adjaent distane along range axis is m. To better evaluate the performanes of the proposed method, the raw data are added by the white Gauss noise with SNR = 5 db. Motion errors are added to the raw data of BFSAR. Let Δx T and Δx R denote the deviations in x diretion for the transmitter and reeiver, respetively, while Δz T and Δz R denote the height deviations of the transmitter and reeiver, respetively. Funtions of these deviations with respet to the azimuth time t are { ΔxT =2os(2π0.8t), Δx R =2os(2π0.8t) (20) Δz T =2.5os(2π1.5t), Δz R =2.5os(2π1.5t)

6 90 Zha and Pu m m Figure 3. Target area used in the simulation (Samples) (Samples) (Samples) (a) (Samples) (b) Figure 4. (a) Wavenumber-domain without range-dependent MoCo. (b) Extended wavenumberdomain with two-step MoCo. Figure 4(a) indiates the imaging result using the Omega-K algorithm in [11] with first order MoCo. In the presene of range dependant motion errors, the targets with the referene range sum are well foused, while the fous quality of the targets apart from the swath enter deteriorates seriously. Fig. 4(b) shows the imaging result by the proposed extended wavenumber-domain algorithm with twostep MoCo. With both range independent and dependent MoCo, all the targets are foused well. To quantify the preision of the proposed algorithm, targets O, A and B are analyzed in more detail. Target O is assumed to be loated in the sene enter, and the oordinates of target A and B are (,, 0) m and (0,, 0) m, respetively. In order to highlight the performane of this approah further, points A, O, and B are analyzed in more detail. Fig. 5 shows the ontours of point A, O, andb proessed by the extended wavenumber-domain with two-step MoCo. To show the details, the results are interpolated eight times. Fig. 6 gives the ontours of points A, O, andb proessed by the wavenumber-domain without range-dependent MoCo. As an be seen from Fig. 5 and Fig. 6, only the targets in the referene azimuth bin an be foused well by the two-step MoCo. The targets at the orners of the senario are partially unfoused. However, we an find that the proposed method an fous these targets simultaneously. The imaging results display good shape of a 2-D sin funtion whih is the point spread funtion of a SAR system. The quality measurements of these results are listed in Table 2. The peak sidelobe ratio (PSLR), integrated sidelobe ratio (ISLR) and impulse-response width (IRW) are used as quality riteria. For the

7 Progress In Eletromagnetis Researh Letters, Vol. 63, three targets foused by the proposed extended wavenumber-domain algorithm, the PSLR and ISLR both agree well with the theoretial values of db and db, respetively. Moreover, in Table 2, for the image results foused by the wavenumber-domain algorithm without range-dependent MoCo, targets A and B are defoused by the range-dependent motion errors. The quality measurements valid the effetiveness of the proposed algorithm further. The Omega-K algorithm in Table 2 means the imaging algorithm in [11] (a) (b) () Figure 5. Extended wavenumber-domain with two-step MoCo. (a) Target A. (b)targeto. ()Target B (a) (b) () Figure 6. Wavenumber-domain without range-dependent MoCo. (a) Target A. (b) Target O. () Target B. Table 2. Imaging quality parameters. Omega-K without range-dependent MoCo Extended Omega-K with two-step MoCo PSLR (db) ISLR (db) IRW (m) PSLR (db) ISLR (db) IRW (m) Target A Target O Target B Target A Target O Target B

8 92 Zha and Pu 5. CONCLUSION An extension of wavenumber-domain sheme of BFSAR has been proposed in this paper, whih diretly integrates a two-step MoCo. This algorithm resolves the issue of inability of the onventional wavenumber-domain approah to perform a range-dependant MoCo, whih is neessary when proessing airborne BFSAR data. This method has the advantages of being able to integrate two-step MoCo as well as maintaining fousing preision. Performane of the proposed algorithm has been demonstrated by the simulations. REFERENCES 1. Wu, J., J. Yang, Y. Huang, H. Yang, and H. Wang, Bistati forward-looking SAR: Theory and hallenges, 9 IEEE Radar Conferene, 1 4, IEEE, Pu, P., J. Yang, Y. Huang, H. Yang, and W. Li, A residual range ell migration orretion algorithm for SAR based on low-frequeny fitting, 2015 IEEE Radar Conferene, , IEEE, Qiu, X., D. Hu, and C. Ding, Some refletions on bistati SAR of forward-looking onfiguration, IEEE Geosiene and Remote Sensing Letters, Vol. 5, No. 4, , Wu, J., J. Yang, Y. Huang, and H. Yang, Fousing bistati forward-looking SAR using hirp saling algorithm, 2011 IEEE Radar Conferene (RADAR), , IEEE, Li, Z., J. Wu, W. Li, Y. Huang, and J. Yang, One-stationary bistati side-looking sar imaging algorithm based on extended keystone transforms and nonlinear hirp saling, IEEE Geosiene and Remote Sensing Letters, Vol. 10, No. 2, , Wu, J., Z. Li, Y. Huang, J. Yang, H. Yang, and Q. H. Liu, Fousing bistati forward-looking SAR with stationary transmitter based on keystone transform and nonlinear hirp saling, IEEE Geosiene and Remote Sensing Letters, Vol. 11, No. 1, , Jan Shin, H.-S. and J.-T. Lim, Omega-k algorithm for airborne forward-looking bistati spotlight SAR imaging, IEEE Geosiene and Remote Sensing Letters, Vol. 6, No. 2, , Reigber, A., E. Alivizatos, A. Potsis, and A. Moreira, Extended wavenumber-domain syntheti aperture radar fousing with integrated motion ompensation, IEE Proeedings Radar, Sonar and Navigation, Vol. 153, No. 3, , Jun Loffeld, O., H. Nies, V. Peters, and S. Knedlik, Models and useful relations for bistati SAR proessing, GIEEE Transations on eosiene and Remote Sensing, Vol. 42, No. 10, , Ot 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, Liu, H., 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, , Yang, J., Y. Huang, H. Yang, J. Wu, W. Li, Z. Li, and X. Yang, A first experiment of airborne bistati forward-looking SAR Preliminary results, Pro. IGARSS, , Melbourne, VIC, Jul