THE INTERFEROMETRY PHASE OF INSAR COHER- ENT JAMMING WITH ARBITRARY WAVEFORM MOD- ULATION

Transcription

1 Progress In Eletromagnetis Researh, Vol. 14, , 01 THE INTERFEROMETRY PHASE OF INSAR COHER- ENT JAMMING WITH ARBITRARY WAVEFORM MOD- ULATION Q. Liu *, S. Xing, X. Wang, J. Dong, D. Dai, and Y. Li Shool of Eletroni Siene and Engineering, National University of Defense Tehnology, Changsha , China Abstrat This paper fouses on the interferometry phase of an ative oherent jamming in InSAR Interferometry Syntheti Aperture Radar images. Based on the signal models of oherent jammer, the jamming s imaging results are derived by employing the Omega-K algorithm. By omparing the imaging results of InSAR s two hannels, the jamming s interferometry phases for both working modes, the single-pass and repeat-pass modes, are proved to be onstants. And the values of the interferometry phases are determined by the jammer s geometry position relative to InSAR baseline, but independent of the jamming s waveform modulation and its bakground terrain. 1. INTRODUCTION The emerging of Syntheti Aperture Radar SAR in the 60 s of last entury is a milestone for two-dimensional radar imaging. Hitherto, there have still been many SAR tehnologies needed to be refined to ahieve better system performanes and image proessing, e.g., two-dimensional sidelobe redution 1], Automati Target Reognition ATR, 3], sparse reonstrution with limited information available 4], et. Also, its basi funtion of twodimensional imaging has been extended to inlude many branhes of SAR family, e.g., Polarimetri SAR PolSAR 5], multiband SAR 6], bistati SAR 7, 8], Multi-Input Multi-Output SAR MIMO SAR 9], and Interferometry SAR InSAR. Although InSAR has been developed sine long time ago and widely used in many areas 10 1], its performane in a hostile eletromagneti environment has not Reeived 16 November 011, Aepted 6 January 01, Sheduled 1 January 01 * Corresponding author: Qingfu Liu liu @163.om.

2 10 Liu et al. yet been thoroughly studied. In real world, there exist many kinds of intentional and/or not jammings or interferenes 13, 14], e.g., a ivi TV transmitter, a radio station, or even a military jammer. Among all of the jammings, the military intentional jamming may prominently worsen the interferometry performane of InSAR. During the past two deades, many jamming-suppression methods have been developed to overome the radio frequeny interferene RFI 15], and most of them are espeially effetive in the suppression of narrow band interferene NBI 16]. RFI is usually produed unintentionally by the ivil transmitters and an be suppressed in a relatively easier way. However, the jammings produed intentionally by a military jammer are usually muh more diffiult to be suppressed, sine most of them share the same bandwidth with radar and their waveform modulations diffiult to be known previously 17]. Aording to the oherene between jamming and SAR transmit signal, the jamming an be approximately divided into three types 17]. The first type is the inoherent jamming, whih shares the same bandwidth but is inoherent to the SAR transmit signal in the fast time 18, 19]. The seond type is known as the partially oherent jamming, whih is oherent in the fast time but inoherent in the slow time 0, 1]. The third type is alled the oherent jamming, whih is oherent in both the fast and slow times 17, ]. On the ontrary, if the jamming an be deteted by SAR, it an be aneled through a number of Eletroni Counter-Counter Measures ECCM tehnologies, e.g., the wideband sidelobe-anellation 3], the spatial filtering 4] and the adaptive beamforming 5]. The main purpose of this paper is to analyze the interferometry phase of a oherent jamming whih is now widely employed in modern Digital Radio Frequeny Memory DRFM based jammer. And the differenes of the interferometry phases between the hostile jamming and the real targets may provide a new approah to distinguish them in the Eletroni Counter Measures ECM environment. By the way, a SAR transponder 6 8] also works in a way very like the oherent jamming exept that it is used for the alibration purpose and its signal parameters an be pre-known. In this point of view, both the oherent jammer and the transponder should have the same kind of interferometry phase if they are all with single transmit antenna. Thus, the results presented in the paper also hold true for an ative transponder.

3 Progress In Eletromagnetis Researh, Vol. 14, Table 1. symbols. The list of parameter names and their orresponding Parameter Name Symbol Parameter Name Symbol The veloity of light The Doppler frequeny f a The fast time t r of SAR transmitted The arrier frequeny signal The slow time t a The veloity of SAR platform The minimum slant range between the slave antenna phase enter and the transponder s transmit antenna phase enter R s The minimum slant range between the master antenna s phase enter and the transponder s transmit antenna phase enter The fast time frequeny f r The referene slant range f v a R m R ref. THE JAMMING S INTERFEROMETRY PHASE IN REPEAT-PASS MODE Firstly, we present the signal model of the jamming reeived by both master and slave hannels of InSAR. The analyti expression of the jamming s two dimensional spetrum is derived by applying the priniple of stationary phase POSP 9]. Then the Omega-K 9] algorithm is employed to derive the imaging result of the jamming. By omparing the imaging results of the master and the slave hannels, the jamming s interferometry phase an be derived. The symbols that will be used in this paper are listed in Table The Jamming s Two-dimensional Spetrum The stop-and-go approximation is widely used for SAR imaging with pulse transmit signal. For an ideal point-like target on the ground, its eho reeived by SAR is always delayed in aordane with the distane between the jammer and SAR system. And this will introdue the demodulated eho with a two-way delay in envelope and a quadrati modulation in phase. Interestingly, the jamming produed by a oherent jammer is similar to the point-like target s eho exept that the baseband waveform an be arbitrarily modulated for the former. Therefore, the oherent jamming has a similar signal

4 104 Liu et al. model to that of point-like target s eho, if the jammer has only single transmit antenna. And its transmit signal reeived by InSAR master hannel an be written as g m t r, t a = g t r R m t a 1, t a exp j4πf R m t a 1} 1 where R m t a is the slant range between the antenna phase enters of InSAR master hannel and the transponder with respet to different slow time t a, namely, R m t a = R m + v at a g t r, t a represents the base band omponent or the arbitrary waveform modulation of the transponder s transmitted signal. The seond term in 1 is aused by the arrier frequeny modulation. If the Fourier transform of g t r, t a with respet to t r is denoted as G f r, t a ft g t r, t a } 3 Then the Fourier transform of g t r 1 R m t a, t a ] an be expressed as ft g t r 1 R m t a, t a ]} =exp j4πfr 1 R m t a } G f r, t a 4 And the Fourier transform of g m t r, t a in 1 an be written as G m f r,t a =ft g t r R m t a 1, t a exp j4πf R m t a 1}} where =G f r, t a exp jθ m 1 f r, t a } 5 θ m 1 f r, t a = 4π f + f r R m t a 1 6 There are two terms in 5, the first one is a slow-hanging part with respet to t a, and the seond is a fast-hanging part. Also, the Fourier transform of G m f r, t a in 5 with respet to t a an be obtained by applying the POSP diretly to 5 as follows G m f r,f a = Gf r, t a exp j4πf +f r R m t a 1} exp jπf a t a }dt a = Gf r, t a expjθ m f r, t a }dt a 7 where θ m f r, t a = π f a t a f + f r R m t a 1] = π f a t a f + f r 1 ] Rm + vat a The partial derivative of θ m f r, t a with respet to t a an be expressed as ] θ m f r, t a = π f a f + f r vat a t a 9 Rm + vat a 8

5 Progress In Eletromagnetis Researh, Vol. 14, as Then the stationary point of the integrand in 7 an be derived f a R m t a = 10 v a f + f r v a ] f a By substituting 10 into 7, the integral result an be gotten as G m f r, f a = G 1 f r, f a exp jθ m 3 f r, f a } 11 where f a R m G 1 f r, f a = G f r, 1 v a f + f r v a ] f a and f a R m θ m 3 f r, f a = θ m f r, v a f + f r v a ] f a = 4πR m f + f r fa 13 Now the jamming reeived by the master hannel has been expressed in two-dimensional frequeny domain. Obviously, this expression does not depend on the SAR imaging algorithm itself, but depend on the parameters of the SAR transmit signals, the modulation of the jammer, and the relative geometri positions between SAR and the jammer, et.... The Jamming s Imaging Results and Its Interferometry Phase In this part of the paper, it is the imaging result of the jamming that makes the main onern. Without loss generality, the Omega-K algorithm is used in the following imaging proess, where the referene phase term in the Omega-K algorithm an be written as 9] θ ref f r, f a = πf r K r + 4πR ref f + f r fa 4Va Multiply 11 with the referene phase term in 14 to generate = G m f r, f a exp jθ ref f r, f a } exp j 4πR m f + f r f a 4v a } G 1 f r, f a } 14 exp jθ ref f r, f a } G m f r, f a 15

6 106 Liu et al. By applying the Stolt mapping 9] 15 an be transformed to f + f r f a 4v a G m f r, f a = G1 f + f r + fa where 4v a = f + f r 16 f, f a ] exp j π f + f K r + fa r f exp j4π R m R ref 1 f + f r } G f r, f a exp j4π Rm R ref 1 f +f r} 17 G f r, f a = G1 exp j π K r f + f r + fa f, f a f + f r + f a 4v a ] f 18 The jamming s imaging result by using the Omega-K algorithm an be obtained through the two-dimensional inverse Fourier transform of 17 g m t r, t a =g tr R m R ref 1, t a exp j4πrm R ref 1 f } 19 where g t r, t a is the two dimensional inverse Fourier transform of G f r, f a. Being similar to 1, the jamming reeived by the slave hannel is is g s t r, t a = g t r R m t a 1, t a exp j4πf R t a 1} 0 After the imaging proess, the imaging output of the slave hannel g s t r, t a =g tr R s R ref 1, t a exp j4πrs R ref 1 f } 1 The differene between t r 1 R m R ref and t r 1 R s R ref in both 0 and 1 will indue a envelope shift between g m t r, t a and g s t r, t a i.e., the -D imaging results of the two hannels of InSAR. This effet an be ompensated by the image

7 Progress In Eletromagnetis Researh, Vol. 14, registration proessing of InSAR. Here we suppose that the slave image is adjusted in aordane with the master image, then g s t r, t a should be rearranged as g s t r, t a =g tr R m R ref 1, t a exp j4πrs R ref 1 f } By omparing the seond terms in both 19 and, the interferometry phase of jamming an be obtained as θ IF = 4π R m R s 1 f 3 Note that the interferometry phase given in 3 is a onstant, whih is irrelevant to the fast time t r and slow time t a, and independent of the jamming s waveform modulations. Indeed, it is equal to the interferometry phase of the terrain where the jammer lies in. In other words, the interferometry phase of the jamming is determined by the jammer s geometry position relative to InSAR s two hannels. 3. THE JAMMING S INTERFEROMETRY PHASE IN SINGLE-PASS MODE In single-pass mode, the jamming reeived by the master hannel is the same as the one in repeat-pass mode whih is given in 1. Therefore, its imaging result is also the same as that expressed in 19, and will not be disussed anymore in the following part. However, the jamming reeived by the slave hannels are different from eah other for both repeat-pass and single-pass modes, where the latter an be expressed as g s t r, t a = g t r R m t a + R s t a ] 1, t a exp jπf 1 R m t a + R s t a ] } 4 Compared with 1, it should be noted that the two-way time delay in 1 is now replaed with the summation of two separate delays. In order to apply the Omega-K algorithm to 4, the jamming s two dimensional spetrum should be derived as follows. Aording to the expression of 4, the Fourier transform of g s t r, t a with respet to t r an be written as G s f r, t a =G f r, t a exp jπ f +f r 1 R m t a +R s t a ] } 5 where the first and seond terms are both the slow hanging and fast hanging parts with respet to the slow time t a. Therefore, the POSP an also be applied to 5 to derive the two-dimensional spetrum.

8 108 Liu et al. The Fourier transform of the seond term in 5 an be expressed as ft exp jπ f + f r R m t a + R s t a ] 1} = exp jπ f +f r R m t a +R s t a ] 1} exp jπf a t a } dt a exp jθ f r, t a } dt a 6 where θ f r, t a =π f a t a f + f r R m t a + R s t a ] 1] =π f a t a 1 f +f r R m +vat a+ ] Rs +vat a 7 It is a diffiult task to resolve the integral in 6. Therefore, an approximation is made in the following to resolve this problem. If the differene between the two slant ranges R m and R s is expressed as δ R = R m R s 8 Then the first square root in 7 an be approximated as R m + vat a = R s + δ R + vat a = Rs + R s δ R + δr + v at a 9 Considering δ R R s and δ R v a t a, it has δ R R s and δ R v at a. So 9 an be approximated as R m + v at a R s + R s δ R + v at a 30 It is useful to make the following paraboli approximation R m + vat a Rs + vat R s a + δ R 31 R s + vat a The paraboli approximation an also be applied to the square root term Rm + vat a in 7 to yield θf r,t a π f a t a f +f r R s+v at R s a + δ R + ] R R s+v at s +vat a a =π f a t a f + f r ] Rs +vat R s a+ δ R 3 R s + vat a Then the first derivative of θ f r, t a with respet to t a an be written as ] θ f r, t a =π f a f +f r vat a R s v at a R s +vat a Rs + vat a 3/ δ R 33

9 Progress In Eletromagnetis Researh, Vol. 14, It has the following inequality R s vat a Rs +vat a 3/ δ R v at a Rs +vat δ R = a For Namely, δ R R s+v at a 1, then v at a R s + v at a δ R R s + v at a v at a R s +v at a v at a R s + v at a R s v at a R s + v at a 3/ δ R v at a R s + v at a δ R R s +v at a Aording to the inequality given by 36, the seond term in the braket of 33 an be negleted, and θ f r, t a an be approximated as ] θ f r, t a π f a f + f r vat a 37 R s + vat a By setting the value of θ f r, t a equal to zero, the stationary point of the integrand in 6 an be written as t a = v a f a R s f + f r fa 38 By substituting 31 and 38 into 7, the stationary point of the phase an be expressed as shown in the appendix θ f r, f a R s va f + f r fa 4πR s f + f r f a 4v a π f + f r fa δ R 39 By applying POSP to 5, its Fourier transform of G s f r, t a with respet to t a an be written as G s f r, f a = exp π j 4πR s f + f r f a 4v a ]} f + f r fa δ R G 1 f r, f a 40

10 110 Liu et al. Now we have derived the jamming s two-dimensional spetrum G s f r, f a, whih is needed for the Omega-K algorithm. Multiplying 40 with the referene funtion shown in 14, it has where G s f r, f a exp jθ ref f r, f a } = G 1 f r, f a exp jθ ref 1 f r, f a } G s f r, f a 41 θ ref 1 f r, f a = 4π R s R ref f + f r fa πδ R f + f r f a 4v a πf r K r 4 By applying the Stolt mapping presented in 16, 41 an be transformed to G s f r,f a = G1 f + f r + fa f, f a 4v a ] exp j π f + f K r + fa r f exp j 4π R s R ref + 0.5δ R f + f } r = G f r, f a exp j 4π R s R ref +0.5δ R f +f } r 43 Then the two-dimensional inverse Fourier transform of 41 an be expressed as g s t r, t a = g tr 1 R s R ref + 0.5δ R, t a exp j4π R s R ref + 0.5δ R 1 f } 44 After image registration proessing of InSAR, the slave image g m t r, t a should be rewritten as g m t r, t a = g tr R s R ref 1, t a exp j4π R s R ref + 0.5δ R 1 f } 45 As mentioned above, the imaging results of the master hannel an also be expressed as 19. Then the interferometry phase of

11 Progress In Eletromagnetis Researh, Vol. 14, the jamming in the single pass mode an be derived by omparing the imaging results of the master and slave hannels, i.e., the phase differene between 19 and 45. And it an be written as θ IF = πδ R 1 f = π R m R s 1 f 46 By omparing 3 and 46, it an be seen that the values of the jamming s interferometry phase for both repeat-pass and singlepass modes are onstants. Both of them have the same interferometry value whih is determined by the geometri position of the jammer, and is independent of the jammer s waveform modulations. a b Figure 1. The InSAR images of the jamming with periodi modulation. a The -D image of the master hannel. b The interferometry phase. The interferometry phase after removing the flat earth phase and phase filtering.

12 11 Liu et al. 4. SIMULATION AND ANALYSIS To further analyze the interferometry phase of the jamming, a simulation of airborne InSAR is presented in this part. Sine the simulation of SAR ehoes with large sene is rather omputational expensive 30], the images with omplex Gaussian lutter are employed here without resorting to ehoes simulation. The InSAR parameters are listed as follows. The baseline length is m, the flight height is 8 km, the baseline depression angle is 60 deg. Both the first and the seond simulation senarios share the same terrain, whih is a flat plane with an area of 500 m 500 m in both the ground range and vertial diretions. The jammer is in the enter of the terrain. The jammingto-lutter power ratio JCR is 6 db. The -D images for both the master and slave hannels are similar to eah other in magnitude, so only the master image is presented as shown in Figure 1a. a b Figure. The InSAR images of the jamming with periodi and multi-delay modulations. a The -D imaging result of the master hannel. b The interferometry phase. The interferometry phase after removing the flat earth phase and phase filtering.

13 Progress In Eletromagnetis Researh, Vol. 14, The periodi modulation jamming 17] is utilized in the first simulation senario, and this kind of jamming an produe multiple disrete lines in the aross range diretion as shown in Figure 1a. The interferometry phase of the jamming does not show muh differene from that of the terrain, beause the jamming is distributed around the jammer and beause the flat ground phase does not hange drastially within a limited ground range shown in Figure 1b. After the removal of the flat ground phase, the interferometry phase of the terrain beome null, whereas a phase opposite to the flat ground phase is added to the jamming as shown in Figure 1. Therefore, the value of the jamming s interferometry phase should be a onstant in the original InSAR interferometry image given in Figure 1b. The multi-delay modulation is employed along with the periodi modulation by the jammer in the seond simulation senario. This kind of modulation an produe multi jamming strip along the ground range diretion as shown in Figure a. There are three jamming strips in Figure a, where the first and the third strips are before and after the jammer in the ground range diretion, respetively; the seond strip is around the jammer. As shown in Figure b, all of the a b d

14 114 Liu et al. e Figure 3. The InSAR images of the jamming with periodi and multidelay modulations. a The -D imaging result of the master hannel jamming-free. b The -D imaging result of the master hannel being jammed. The interferometry phase jamming-free. d The interferometry phase being jammed. e The interferometry phase after removing the flat earth phase and phase filtering jammingfree. f The interferometry phase after removing the flat earth phase and phase filtering being jammed. three jammings share the same interferometry phase, and are equal to the interferometry phase of the terrain where the jammer lies in. The first and the third jamming strips an be easily distinguished from the bakground interferometry phase, sine they are away from the jammer in the ground range diretion. After the removal of the flat ground phase, all of the three jamming strips are added with an opposite flat ground phase as shown in Figure. A terrain with height variation is used in the third simulation senario. Some dark areas an be seen in Figures 3a and b. They are indeed the shadows of the mountains. The same jamming is used in both the seond and third simulation senarios. As shown in Figure 3, the jamming in the third simulation senario have the same interferometry phase as that in the seond one. Therefore, the interferometry phase of the jamming in the InSAR images are independent from its bakground terrain. 5. CONCLUSION For both the repeat-pass and single-pass modes, the interferometry phases of the jamming are onstants. They are equal to the interferometry phases of terrain where the jammer lies in and independent of the jamming s waveform modulations. Although only f

15 Progress In Eletromagnetis Researh, Vol. 14, the ase of the oherent jammer is disussed throughout this paper, the interferometry phase of any other oherent ative transmitter with single transmit antenna should be a onstant. This will also hold true for a SAR transponder when it is used for alibration purpose. Therefore, the work of this paper an be developed to reognize the ative oherent transmitter in InSAR images. APPENDIX A. THE JAMMING S STATIONARY POINT PHASE IN INSAR SLAVE CHANNEL Substituting 31 into 7 yields: θ f r,t a π f a t a 1 f +f r ] Rs +vat R s a+ δ R R s +v at a A1 Substituting 38 into R s + v at a yields: R s + v at a ta= v a fars f+fr f a 4v a = Rs + 4v a = 4v a f + f r Rs f ] + f r R = s f + f r fa f + f r fa Therefore, 39 an be derived as f a R s θ f r, v f + f r fa f a R s f + f r f a 4v a π f a R s v f + f r fa f + f r f + f r R s f + f r fa + f + f r fa f +f r = π f a R s f +f r R s 1 f +f r fa v f +f r fa 4v a = 4πR s f + f r f a 4v a π δ R δ R f + f r fa δ R ] A A3

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