CONSIDERING the high computational load of synthetic

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1 IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 5, NO. 4, OCTOBER Doppler Keystone Transform: An Approah Suitable for Parallel Implementation of SAR Moving Target Imaging Gang Li, Member, IEEE, Xiang-Gen Xia, Senior Member, IEEE, and Ying-Ning Peng, Senior Member, IEEE Abstrat In this letter, a syntheti aperture radar (SAR) data reformatting approah named Doppler Keystone transform (DKT) is proposed to orret the range migration of a moving target. By using the DKT, the SAR imaging program, i.e., the 2-D mathed filtering, an be transformed into separate 1-D operations along range or azimuth diretion, and therefore, the DKT is suitable for the parallel implementation of SAR imaging of the moving target. Our simulations show that by ombining the DKT and the Doppler phase ompensation methods, the moving target an be well imaged in high signal lutter-ratio ase. Index Terms Moving target imaging, parallel implementation, syntheti aperture radar (SAR). I. INTRODUCTION CONSIDERING the high omputational load of syntheti aperture radar (SAR) imaging, parallel proessing is adopted by many real-time SAR systems [13] [16]. In the appliation of SAR imaging of stationary sene, a well-known way of the parallel imaging is transforming the SAR 2-D mathed filtering into sequential 1-D operations along range or azimuth diretion and evenly partitioning the data and the operations on multiple parallel proessors [13], [14]. However, this way is not suitable for the parallel imaging of the moving target due to the fat that the unknown range migration of the moving target auses the oupling between the range and azimuth signals. The onventional algorithms of moving target imaging based on the motion parameter estimation [1] [3] or the Doppler parameter estimation [4] [7] are not designed for parallel proessing, and some elaborated partitions of the data and the omputations may be required to guarantee the effiieny of the implementation of these algorithms on multiple parallel proessors. In [8], a SAR data reformatting approah named Keystone transform (KT) was proposed to orret the range walk (the Manusript reeived September 21, 2007; revised November 30, 2007, January 13, 2008, Marh 10, 2008, and April 25, Current version published Otober 22, This work was supported by the Chuanxin Foundation from the Department of Eletroni Engineering, Tsinghua University. The work of X.-G. Xia was supported in part by the Air Fore Offie of Sientifi Researh (AFOSR) under Grant FA and in part by DEPSCoR under Grant W911NF through ARO. G. Li and Y.-N. Peng are with the Department of Eletroni Engineering, Tsinghua University, Beijing , China ( gangli@tsinghua.edu.n; ynpeng@tsinghua.edu.n). X.-G. Xia is with the Department of Eletrial and Computer Engineering, University of Delaware, Newark, DE USA ( xxia@ee.udel.edu). Digital Objet Identifier /LGRS linear omponent of the range migration), and it was ombined with the Doppler phase ompensation methods to image the moving target. The KT-based algorithm is inherently suitable for parallel implementation beause the SAR 2-D mathed filtering an be deomposed by the KT into separate 1-D operations along range or azimuth diretion without the knowledge of the target veloity, and, aordingly, the omputations an be parallelized to run on multiple proessors with the same (or very similar) program and at the same duration. However, the range migration orretion (RMC) by the KT is only approximate, and it leaves a residual term referred to as a range urvature (the quadrati omponent of the range migration). The range urvature effet beomes severe when the distane between the radar and the target is short or the SAR range resolution is high, whih may degenerate the imaging quality. To resolve this problem, in [9], the RMC is divided into two steps. The range urvature is first orreted by a seond-order KT (SKT), and then, the range walk is orreted by estimating the slope of the signature envelope in the 2-D time domain. The slope estimation of the 2-D signature envelope does not have an easy parallel arhiteture like the KT-based algorithm, and some elaborated segments and partitions of the data and the operations may be needed to implement the SKT-based algorithm in parallel. In this letter, a SAR data reformatting approah named Doppler KT (DKT) is proposed to orret the range migration of the moving target. It will be shown that both the range walk and the range urvature an be simultaneously orreted by the DKT and that by ombining the DKT and the Doppler phase ompensation methods, the moving target an be imaged in high signal lutter-ratio (SCR) ase. Moreover, the SAR 2-D mathed filtering is deomposed by the DKT into separate 1-D omputations along range or azimuth diretion. Therefore, the DKT is suitable for the parallel implementation of SAR imaging of the moving target. Another algorithm of moving target imaging based on extended hirp saling is also omposed of 1-D operations [20]. The differene between this letter and [20] is that, to orret the range migration of a moving target, data reformatting along the azimuth diretion is adopted in this letter, whereas a hirp saling vetor multipliation in the range diretion is used in [20]. The remainder of this letter is organized as follows. In Setion II, the KT and the SKT are briefly reviewed. In Setion III, the DKT-based algorithm is proposed, and its parallel implementation and omputational omplexity are disussed. In Setion IV, some simulation results X/$ IEEE Authorized liensed use limited to: Tsinghua University Library. Downloaded on November 6, 2008 at 00:43 from IEEE Xplore. Restritions apply.

2 574 IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 5, NO. 4, OCTOBER 2008 where the linear and quadrati phase terms in t represent the oupling between f τ and t, and they are referred to as range walk and range urvature, respetively. If we let Fig. 1. Slant plane of SAR geometry. are given to demonstrate the performane of the DKT-based algorithm. II. REVIEW OF KT AND SKT In this setion, we briefly review the RMC by KT [8] and SKT [9]. The typial SAR geometry is shown in Fig. 1. The X-axis is the azimuth diretion, and the R-axis is the range diretion. The radar platform flies along the azimuth diretion with onstant veloity ν, and a target P moves with onstant azimuth speed ν x and range speed ν r. When the radar beam enter is passing by P, P is assumed to be loated at the range r 0, and this moment is defined as azimuth time (or slow time) t =0. The goal of this letter is to image P with the unknown (ν x,ν r ). By supposing that the radar transmits a linear frequenymodulated signal (or hirp signal), the baseband eho from P is [ s(t, τ) =A exp jπγ ( τ 2R(t) ) ] 2 [ exp ] j 4πf R(t) (1) where A is the omplex amplitude, γ is the hirp rate of the transmitted signal, τ is the range time (or fast time), is the speed of light, f is the arrier frequeny, R(t) is the instantaneous distane from the radar to the target, and it an be expressed as R(t) r 0 ν r t +((ν ν x ) 2 /2r 0 )t 2.In aordane with the stationary phase priniple [11], the Fourier transform of (1) in terms of τ yields [ s(t, f τ ) A exp jπ f τ 2 ] [ exp j 4π(f ] + f τ ) R(t) γ (2) where A is the onstant omplex amplitude, andf τ is the range frequeny. After the mathed filtering in range diretion, i.e., multiplying exp[jπf 2 τ /γ] to (2), (2) beomes s(t, f τ ) A exp j 4π(f + f τ ) r 0 + j 4π(f + f τ ) ν r t j 2π(f + f τ ) (ν ν x) 2 t 2 (3) r 0 then (3) beomes s (t,f τ )=A exp t = f f + f τ t (4) j 4π(f + f τ ) r 0 + j 4πf ν r t 2πf 2 j (f + f τ ) (ν ν x) 2 t 2 r 0 where the linear oupling between f τ and t has been removed, i.e., the range walk has been orreted. This is the basi idea of the KT [8]. By using 1/(f + f τ ) (f f τ )/f 2 in the last term of (5), one an see that the quadrati oupling between f τ and t is still left, i.e., the range urvature effet is still not orreted. When the distane between the radar and the target is short or the SAR range resolution is high, the range urvature effet will beome severe and ause the degeneration of the imaging quality. To resolve this problem, the SKT was proposed in [9], whih replaes (4) by ( ) 1/2 f t = t. (6) f + f τ Substituting (6) into (3) yields s (t,f τ ) A exp j 4π(f + f τ ) r 0 + j 4π ( f + 1 ) 2 f τ ν r t j 2πf (5) (ν ν x) 2 t 2 r 0. (7) It is lear that the quadrati oupling between f τ and t has been removed by the SKT, i.e., the range urvature has been orreted, but the linear oupling between f τ and t is still left, i.e., the range walk is still not orreted. If we perform the inverse Fourier transform in terms of f τ, the signature in the 2-D time domain will show a straight-line envelope, and then, the range walk an be orreted by estimating the slope of the 2-D signature envelope. In summary, the SKT orrets both the range walk and the range urvature, whereas the KT only orrets the range walk; thus, the SKT is better from the viewpoint of the imaging quality. However, the KT is better from the viewpoint of the parallel implementation. When the effet of f τ on the last term in (5) is ignored, i.e., when the range urvature is negleted, the SAR 2-D mathed filtering has been deomposed by the KT into separate 1-D operations along range or azimuth diretion. Then, in every step of the KT-based algorithm, one an easily segment the data and the operations aording to the range/azimuth index number and partition the omputations to run on multiple parallel proessors with the same (or very similar) program and at the same duration. The partitions of Authorized liensed use limited to: Tsinghua University Library. Downloaded on November 6, 2008 at 00:43 from IEEE Xplore. Restritions apply.

3 LI et al.: DKT: AN APPROACH SUITABLE FOR PARALLEL IMPLEMENTATION OF SAR MOVING TARGET IMAGING 575 the data and the omputations are simple but do guarantee the high effiieny of the parallel implementation, and thus, we an say that the KT-based algorithm is inherently suitable for parallel implementation. In ontrast, the slope estimation of the 2-D signature envelope is required for the SKT-based algorithm, whih may be diffiult to be implemented in parallel with high effiieny (at least more diffiult than the KT-based algorithm) unless some elaborated partitions of the data and the omputations are speifially designed. We next propose an alternative method. III. DKT-BASED ALGORITHM In this setion, first, the DKT-based algorithm is desribed, then its parallel implementation is introdued, and finally, its omputational omplexity is disussed. A. Algorithm Formulation In aordane with the stationary phase priniple [11], we perform the Fourier transform on (3) in terms of t and then have s(f t,f τ ) A (f + f τ ) exp j4πr 0 r 0 +jπ 2(ν ν x ) 2 (f + f τ ) (f t f d,τ ) 2 (8) where A is the onstant omplex amplitude, f t is the Doppler frequeny, f d,τ is the Doppler entroid for various f τ, and it an be expressed as f d,τ =2ν r (f + f τ )/. The seond term of (8) inludes the oupling between f τ and f t, and it implies the range migration in the 2-D frequeny domain. Now, we resale the f t axis by the following transform: (f t f d,τ )= f + f τ f (f t f d ) (9) where f d Δ = fd,τ fτ =0 =2ν r f /, i.e., we substitute (9) into (8), and then, we have s (f t,f τ ) A (f + f τ ) exp j4πr 0 r 0 +jπ 2(ν ν x ) 2 (f t f d ) 2 f. (10) One an see that the oupling between f τ and f t has been ompletely removed in (10), i.e., the range migration has been orreted. The transform in (9) is named DKT, whih an be implemented by interpolations as done in [8] or by a omputationally effiient saling priniple as done in [10]. It is noted that the DKT in (9) requires the estimates of f d,τ for every f τ. This is easily ahieved by applying the lassial algorithms of the Doppler entroid estimation, suh as the energy balaning or the optimal estimator, et. [12], on the Doppler spetra for Fig. 2. Flowhart of the DKT-based algorithm. every f τ. After the RMC by the DKT, we take the inverse Fourier transform in terms of f τ in (10) and have ( s (f t,τ) A δ τ 2r ) 0 r 0 exp jπ 2(ν ν x ) 2 (f t f d ) 2 (11) f where A is the onstant omplex amplitude. Then, the Doppler phase ompensation approahes, suh as the time frequeny analysis [4], [6], an be used for azimuth mathed filtering, and as a result, the fine SAR imaging results an be obtained. B. Parallel Implementation The aforementioned desriptions imply that the SAR 2-D mathed filtering an be deomposed by the DKT into separate 1-D operations along range or azimuth diretion. Therefore, similar to the KT-based algorithm, the DKT-based algorithm is also inherently suitable for parallel implementation. The flowhart of the parallel implementation of the DKT-based algorithm is shown in Fig. 2, where SAR imaging of the moving target is aomplished by sequential steps and eah step onsists of 1-D operations along range or azimuth diretion. The orner turn in Fig. 2 is the data transpose proessing whih is helpful to quikly aess the data matrix and effiiently perform the vetor omputation [13], [14]. Similar to the parallel implementation of the SAR imaging of stationary sene [13], [14], in eah step in Fig. 2, the data an be evenly segmented into several subbloks, and these subbloks and orresponding omputations an be parallelized to run on multiple proessors with the same (or very similar) program and at the same duration. By using the interpolation-free approah proposed in [10] to Authorized liensed use limited to: Tsinghua University Library. Downloaded on November 6, 2008 at 00:43 from IEEE Xplore. Restritions apply.

4 576 IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 5, NO. 4, OCTOBER 2008 implement (9), most omputations of the DKT-based algorithm in Fig. 2 are fast Fourier transforms (FFTs), inverse FFTs (IFFTs), omplex vetor multipliations, and salar operations, all of whih are ommon operations in popular parallel systems [14] [16]. In this way, the high implementation effiieny of the DKT-based algorithm an be ahieved in popular parallel systems. C. Computational Complexity The DKT-based algorithm is ompared with the SKT-based algorithm in terms of the omputational omplexity as follows. From Fig. 2, one an see that, exept Steps 4 and 5, other steps are also required for the SKT-based algorithm [9]. Moreover, both the SKT and the DKT an be implemented by the same way, i.e., by the interpolation [8] or the hirp saling method [10]. Therefore, with the given size of the raw data blok to be proessed, we only need to ompare the omputational amounts between the Step 4 in Fig. 2 and the range walk orretion in [9]. In what follows, we assume that the data blok onsists of N N samples. In [9], the range walk should be orreted sequentially with the following two steps: 1) The slope of the 2-D signature envelope should be estimated. Sine in [9] no speifi slope estimation is mentioned, let us assume that Radon transform [17] is used for the estimation sine it is a popular and effiient method. The omputational omplexity of the Radon transform is about O(N 3 ), and it may be redued to O(N 2 log 2 N) by some fast ways, whih omes at the ost of inreasing errors [18], and 2) after the slope estimation, the interpolation along the range diretion should be operated sine the range walk value might not be multiple times of the range sampling unit. In Step 4 of the DKT-based algorithm, for eah f τ, f d,τ an be estimated by the orrelation operation (see [12, eqs. 14 and 15]), whih an be implemented by [19] Fig. 3. Resolutions versus various SCRs. (a) Range resolution. (b) Azimuth resolution. f d,τ =arg ft U(f t,f τ )=0 U(f t,f τ )=IFFT ft FFT ft [W (f t,f τ )] onj (FFT ft [G(f t,f τ )]) (12) where W (f t,f τ ) is the azimuth power spetrum, G(f t,f τ ) is the weighting funtion of the optimal estimator of the Doppler entroid [12], and onj( ) denotes the onjugate operation. From (12), one an alulate that the omputational omplexity of Step 4 in Fig. 2 is about O(N 2 log 2 N), and note that it does not need the additional interpolation and does not inrease the error. Therefore, the DKT-based algorithm has lower omputation omplexity than the one in [9]. Moreover, as mentioned in the previous setion, the DKT-based algorithm is inherently suitable for parallel implementation, whereas to perform the SKT-based algorithm in [9] in parallel, some elaborated data partitions may be needed for the Radon transform and the interpolation in range diretion. IV. SIMULATION RESULTS To demonstrate the performane of the DKT-based algorithm, some simulations are provided in this setion. The radar Fig. 4. Imaging results of the DKT-based algorithm. (a) Two-dimensional IRF. (b) Range profile of the IRF. () Azimuth profile of the IRF. parameters are as follows: pulse repetition frequeny PRF = 1 khz, arrier frequeny f =10GHz, slant range r 0 =10km, platform speed ν = 100 m/s, pulse bandwidth is 200 MHz, and syntheti aperture time is 4 s. A simulated point target is moving with the onstant speed vetor (ν x,ν r )=(3, 4) m/s. From these parameters, one an alulate that the theoretial range and azimuth resolutions are about 0.75 and 0.39 m, respetively. Authorized liensed use limited to: Tsinghua University Library. Downloaded on November 6, 2008 at 00:43 from IEEE Xplore. Restritions apply.

5 LI et al.: DKT: AN APPROACH SUITABLE FOR PARALLEL IMPLEMENTATION OF SAR MOVING TARGET IMAGING 577 It is assumed that the distribution of the lutter is Gaussian, and the optimal estimator of the Doppler entroid [12] is used to estimate f d,τ in the DKT [see (9)]. Fig. 3 shows the range and azimuth resolutions for various SCRs, where 100 Monte Carlo experiments are run for eah simulation. In Fig. 3(a), the range resolution varies nonmonotonially versus various SCRs. The reason is that the range resolution is determined by the pulse bandwidth and that the range-diretion mathed filtering has been done before the DKT. In Fig. 3(b), the azimuth resolution beomes oarse in the lower SCR ase. The reason is that, in the lower SCR ase, the DKT produes the inomplete RMC due to the larger estimation errors of all f d,τ. This implies that higher SCR is required for the DKT-based algorithm. For example, if the desired azimuth resolution is smaller than 1 m, the SCR higher than 12 db is required. When SCR =20dB, the imaging results of the DKT-based algorithm are shown in Fig. 4, where the 2-D impulse response funtion is shown in Fig. 4(a) and its range and azimuth profiles are shown in Fig. 4(b) and (), respetively. This proves that the DFT-based algorithm works well in high SCR ase. V. C ONCLUSION In this letter, a SAR data reformatting approah in the 2-D frequeny domain, whih is named DKT, is proposed to orret the range migration of the moving target. Our simulation results show that by ombining the DKT with the Doppler phase ompensation approahes, the DKT-based algorithm an well image the moving target in high SCR ase. Moreover, by using the DKT, the SAR 2-D mathed filtering an be deomposed into separate 1-D omputations along the range or azimuth diretion; therefore, the DKT-based algorithm is suitable for the parallel implementation of SAR imaging of the moving target. REFERENCES [1] S. Werness, W. Carrara, L. Joye, and D. 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