RANGE DOPPLER ALGORITHM FOR BISTATIC SAR PROCESSING BASED ON THE IMPROVED LOFFELD S BISTATIC FORMULA

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1 Progress In Eletromagnetis Researh Letters, Vol. 27, , 2011 RANGE DOPPLER ALGORITHM FOR ISTATIC SAR PROCESSING ASED ON THE IMPROVED LOFFELD S ISTATIC FORMULA X. Wang 1, * and D. Y. Zhu 2 1 Nanjing University of Posts & Teleommuniations, Nanjing , China 2 Nanjing University of Aeronautis & Astronautis, Nanjing , China Abstrat This paper presents a new range Doppler algorithm RDA) for bistati syntheti aperture radar SAR) proessing in a general onfiguration based on a bistati point target referene spetrum: the improved extended Loffeld s bistati formula ILF). The ILF spetrum is proved to be omparably aurate with the spetrum derived using the method of series reversion MSR). ased on the expansion of the ILF spetrum, a new bistati RDA is developed to proess the azimuth invariant and variant bistati SAR data. Compared with existing bistati RDA, the new algorithm has a simpler formulation and is able to ope with moderate or high squint bistati SAR data. The simulated data in the azimuth invariant and variant bistati onfigurations are used to validate the new algorithm. 1. INTRODUCTION istati syntheti aperture radar SAR) 1 3] is a SAR system 4 6] whose transmitter and reeiver are plaed on separate platforms. The interest in bistati SAR systems has rapidly inreased in reent years. In the last deade, several approximate bistati point target spetrums 7 9] and bistati imaging methods 10 14] have been reported. Among these approximate spetrums, the Loffeld s bistati formula LF) 7], whih is developed earlier, has been used to derive bistati SAR imaging algorithm 10]. Nevertheless, LF Reeived 26 June 2011, Aepted 3 November 2011, Sheduled 18 November 2011 * Corresponding author: Xin Wang whxin2002@live.n).

2 162 Wang and Zhu beomes invalid when the Doppler hirp rates of the transmitter and reeiver are signifiantly different. Using the method of series reversion MSR), a more preise spetrum is derived in 8]. ased on MSR spetrum, several bistati imaging algorithms 11, 12] have been proposed. However, MSR spetrum is in the form of a power series, the auray of whih depends on the order and onverge speed of the series. As disussed in 13], LF beomes invalid in some extreme onfigurations i.e., the airborne-spaeborne bistati SAR). To extend LF for the general bistati SAR onfiguration, Wang et al. introdued a time-bandwidth produt TP) weighting operation in the derivation of LF and derived the extended Loffeld s bistati formula ELF) 9]. However, ELF spetrum is still not aurate enough, whih affets its appliation. Reently, a new spetrum, i.e., the improved LF ILF), is derived in 14] and proved to be more aurate than the original ELF. ased on the expansion of the ILF spetrum about range frequeny, a new bistati range Doppler algorithm RDA) is developed in this paper to proess the azimuth invariant and variant bistati SAR data. The new algorithm has a simple formulation and is able to ope with moderate or high squint SAR data. 2. THE ISTATIC POINT TARGET REFERENCE SPECTRUM In bistati SAR, assuming that radar transmits a linear frequeny modulated LFM) signal with hirp rate k and enter frequeny f, the eho from a single point target an be expressed as Sτ, t) = w r τ R 1t)+R 2 t) R 1 t) + R 2 t) exp j2πf ] w a t t )exp { jπk τ R 1t) + R 2 t) ] 2 } ], 1) where τ is the fast range) time and t the slow azimuth) time entered at t. R 1 t) and R 2 t) are the instantaneous slant ranges from the transmitter and the reeiver to the point target, respetively. In 1), w r τ) is the range envelope of the transmission signal and w a t t ) the azimuth envelope determined by the omposite antenna pattern. With the assumption of t = 0, the range history is written as Rt) = R 1 t) + R 2 t) = R v2 1 t2 2R 10 v 1 t sin θ 1 + R v2 2 t2 2R 20 v 2 t sin θ 2, 2)

3 Progress In Eletromagnetis Researh Letters, Vol. 27, O z A R 20 y H R 10 aseline Transmitter flat path x Transmitter aperture enter R 20 O z y A H aseline R 10 Transmitter flat path x Transmitter aperture enter Reeiver flat path Reeiver aperture enter a) Reeiver flat path Reeiver aperture enter Figure 1. Geometries of the bistati azimuth a) invariant and b) variant SAR in the simulation. b) where R 10 = R 1 t) t=0, R 20 = R 2 t) t=0, v 1 and v 2 are the veloities of the transmitter and the reeiver, respetively. The angles θ 1 and θ 2 in 2) refer to the instantaneous squint angles of the transmitter and the reeiver at the omposite beam enter ross time, respetively. Performing 2-D Fourier transform on 1), the ILF 12] an be obtained as ) fr Sf r, f a ) = w r w a t k ) exp jϕf r, f a )], 3) where f r and f a in 3) denote the range and azimuth frequeny, respetively, the bandwidth of LFM signal, and t k the stationary phase point. The phase term in 3) is where ϕf r, f a ) = 2π R 10 os θ 1 F 1 = 2π f r + f ) 2 2 fat 2, F 1 2π R 20 os θ 2 F 2 f at R 10 sin θ 1 v 1 + f ar R 20 sin θ 2 v 2 F 2 = f r + f ) 2 2 far 2 v2 2, f at = k t f a f ] r + f v 1 sin θ 1 + v 2 sin θ 2 ) f ar = k r f a f ] r + f v 1 sin θ 1 + v 2 sin θ 2 ) v 2 1 ) π f 2 r k, 4) + f r + f v 1 sin θ 1, + f r + f v 2 sin θ 2,

4 164 Wang and Zhu k t = and k r = v1 2 os θ2 1 R 20 v1 2 os θ2 1 R 20 + v2 2 os θ2 2 R, 10 v2 2 os θ2 2 R 10 v1 2 os θ2 1 R 20 + v2 2 os θ2 2 R COMPARALE PRECISION OF THE ILF SPECTRUM AND THE MSR SPECTRUM Among the existing bistati point target spetrums, MSR spetrum is the most preise, espeially in mathematial analysis. To verify the preision of ILF spetrum, we will expand the phase term of ILF and ompare the expansion with the MSR spetrum. Expanding the phase term in 4) about the Doppler entriod frequeny f a = fr+f v 1 sin θ 1 + v 2 sin θ 2 ), we an obtain { ϕf r, f a ) π f r 2 k 2π f r+f R 10 +R 20 )+2π f ) r+f 1 fa 2 +k 1 4k 2 f r +f + k ) 3 fa 3 ) A 2 2k 2 ) 3 +k f r +f A k2 4 + k ] )} 4 fa 4 2k 2 ) 4 +k 1, 5) f r +f where A i = di R 1 t) dt i, t=t k i = di Rt) dt i, t=t i = di R 2 t) dt i, i = 2, 3). t=t Equation 5) is almost the same as MSR spetrum s phase term that retains the terms in the series up to quarti term. As the fourth order term is usually very small, the little differene will not affet the auray of the ILF spetrum. Hene, the onlusion that the two spetrums are omparably aurate an be obtained. As the MSR spetrum is expressed as a series, the auray of whih is affeted by the expansion order and onverge speed of the series used in its derivation. The appliation of ILF is more onvenient than MSR spetrum. 4. THE NEW ISTATIC RDA Expanding the phase term in 4) with respet to the range frequeny f r, an expliit form of the ILF spetrum s phase an be formulated

5 Progress In Eletromagnetis Researh Letters, Vol. 27, as ϕf r, f a ) = ϕ ar f r, f a ) + ϕ sr f r, f a ) + ϕ rm f r, f a ) + ϕ az f r, f a ) 6) where where ϕ ar = π f r 2, k r ϕ sr = πfr 2 kt 2 fa 2 k R 10 os θ 1 v1 2f 4 H1 4 + R 20 os θ rf 2 2 ] a 2 v2 2f 4 H2 4 { πfr 3 kt 2 f 2 a R 10 os θ 1 v1 2f 5 H1 5 f d )] 1 f v2 2 k r f a + d 2 k + R 20 os θ rf 2 2 a 2 v2 2f 5 H2 5 f d 1 f v2 2 k r f a + d 2 R10 os θ 1 ϕ rm = 2πf r H 1 1 d2 2 k r f a v2 2 d 2 v2 2f R 10 os θ 1 ϕ az = 2π f H 1 +k r R 20 sin θ 2 v 2 1 d2 1 ) + 7) )]}, 8) ) k t f a v1 2 d 1 v1 2f + R 20 os θ 2 H 2 d1 R 10 sin θ 1 + d 2R 20 sin θ 2 v 1 v 2 R 20 os θ 2 H 2 + k t R 10 sin θ 1 v 1 )], 9) + f ) ] f a, 10) d 1 = k r v 1 sin θ 1 k t v 2 sin θ 2, d 2 = k t v 2 sin θ 2 k r v 1 sin θ 1, ) 2 2 f d 1 + k t f a H 1 = 1 f 2 v1 2, ) 2 2 f d 2 + k r f a H 2 = 1 f 2 v2 2. With the typial proess flow of RDA, the new algorithm an be developed. After range ompressed using the phase term in 7), the data is transformed into the 2D domain by an azimuth FFT and range ompressed again. The range ompressed data is then transformed into the RD domain via a range IFFT and RCM orreted via interpolation aording to Equation 9), where a sin interpolator is used in our simulation in Setion 5. Finally, an azimuth mathed filter with the

6 166 Wang and Zhu phase term being the negative of 10) is used to azimuth ompress the signal and implement the algorithm. The new bistati RDA an be used to proess the azimuth invariant and variant bistati SAR data diretly. When the transmitter and the reeiver platforms move along unparallel paths with different veloities, the algorithm an be used to proess the reeived signal diretly. However, the loations of the foused point targets in the image in the azimuth variant onfiguration will be different from that in the azimuth invariant ase. 5. SIMULATION Two bistati onfigurations i.e., azimuth invariant and variant ases) will be simulated to validate the new bistati RDA. Table 1 lists the parameters. Seven point targets with interval of 200 m along the bistati bisetor line are used in the simulation, where the geometries in the azimuth invariant and variant ases are depited in Fig. 1a) and Fig. 1b), respetively. Point targets in the simulation are depited along the bistati bisetor line OH in the figures, where the point target A lies at the sene enter, and the point target lies 600 m away from the sene enter Ground range m) Ground range m) Azimuth pixel Range pixel A a) Azimuth pixel Range pixel Figure 2. Images in azimuth a) invariant and b) variant ase. Table 1. Simulation parameters. A b) Azimuth invariant Azimuth variant Transmitter Reeiver Transmitter Reeiver Veloity in x diretion 190 m/s 190 m/s 190 m/s m/s Veloity in y diretion 0 m/s 0 m/s 0 m/s 60.0 m/s Squint angle istati entriod range m m Range bandwidth 80 MHz 80 MHz aseline m m

7 Progress In Eletromagnetis Researh Letters, Vol. 27, The foused results in the azimuth invariant and variant ases are shown in Fig. 2 a) and Fig. 2b), respetively, where the orresponding ground range of the foused point targets is denoted on the top of the Fig. 2. The impulse responses analyses of the point targets A and in azimuth invariant and variant ases are shown in Fig. 3 and Fig. 4, respetively. It ould be seen from Figs. 3 4 that PT A loated at the a) b) ) Figure 3. Point target analysis in azimuth invariant ase. a) Range profile and b) azimuth profile of the foused point target A. ) Range profile and d) azimuth profile of the foused target. 1 range pixel = m, 1 azimuth pixel = m). d) a) b)

8 168 Wang and Zhu a) Figure 4. Point target analysis in azimuth variant ase. a) Range profile and b) azimuth profile of the foused point target A. ) Range profile and d) azimuth profile of the foused target. 1 range sample = m, 1 azimuth pixel = m). sene enter is well foused while PT is notieably degraded. The degraded image quality of PT is due to the appliation of the rangeinvariant SRC filter. If we restrit the quadrati phase error QPE) in SRC within ± π 2, the algorithm an handle a range invariane region of m in the azimuth invariant ase and m in azimuth variant ase. 6. CONCLUSION A new RDA is developed in this paper based on a bistati point target spetrum, i.e., the improved LF. The spetrum has an analogous analytial expression with the monostati point target 2-d spetrum and is more aurate than original ELF. The ILF spetrum is expanded to the third order to implement the new RDA. The new algorithm has a simpler formulation than existing bistati RDA and is able to ope with moderate high squint data. oth the azimuth invariant and variant bistati SAR modes are simulated. Simulation results have validated our approah. REFERENCES 1. Sun, J., S. Mao, G. Wang, and W. Hong, Extended exat transfer funtion algorithm for bistati SAR of translational invariant ase, Progress In Eletromagnetis Researh, Vol. 99, , Wu, J., J. Yang, Y. Huang, Z. Liu, and H. Yang, A new look at the point target referene spetrum for bistati SAR, Progress In Eletromagnetis Researh, Vol. 119, , b)

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