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Feb 26, 2016 - Hong-Cheng Zeng 1, Peng-Bo Wang 1,2, Jie Chen 1,2,*, Wei Liu 3, ... applications, where the transmitter could be a dedicated SAR [1–3] or ...
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A Novel General Imaging Formation Algorithm for GNSS-Based Bistatic SAR Hong-Cheng Zeng 1 , Peng-Bo Wang 1,2 , Jie Chen 1,2, *, Wei Liu 3 , LinLin Ge 4 and Wei Yang 1 1 2 3 4

*

School of Electronic and Information Engineering, Beihang University, Beijing 100191, China; [email protected] (H.-C.Z.); [email protected] (P.-B.W.); [email protected] (W.Y.) Collaborative Innovation Center of Geospatial Technology, Wuhan 430079, China Electronic and Electronic Engineering Department, University of Sheffield, Sheffield S1-3JD, UK; [email protected] School of Civil & Environment Engineering, The University of New South Wales (UNSW), Sydney NSW 20512, Australia; [email protected] Correspondence: [email protected]; Tel./Fax: +86-10-8233-7049

Academic Editor: Assefa M. Melesse Received: 14 December 2015; Accepted: 23 February 2016; Published: 26 February 2016

Abstract: Global Navigation Satellite System (GNSS)-based bistatic Synthetic Aperture Radar (SAR) recently plays a more and more significant role in remote sensing applications for its low-cost and real-time global coverage capability. In this paper, a general imaging formation algorithm was proposed for accurately and efficiently focusing GNSS-based bistatic SAR data, which avoids the interpolation processing in traditional back projection algorithms (BPAs). A two-dimensional point target spectrum model was firstly presented, and the bulk range cell migration correction (RCMC) was consequently derived for reducing range cell migration (RCM) and coarse focusing. As the bulk RCMC seriously changes the range history of the radar signal, a modified and much more efficient hybrid correlation operation was introduced for compensating residual phase errors. Simulation results were presented based on a general geometric topology with non-parallel trajectories and unequal velocities for both transmitter and receiver platforms, showing a satisfactory performance by the proposed method. Keywords: bistatic SAR; GNSS; bulk RCMC; modified hybrid correlation

1. Introduction Bistatic Synthetic Aperture Radar (SAR) plays an important role in various remote sensing applications, where the transmitter could be a dedicated SAR [1–3] or opportunistic illuminator systems, such as communication satellites or global navigation satellite systems (GNSS). Due to its safety, convenience and low cost, bistatic SAR using transmitters of opportunity has attracted more and more attention and developed very quickly, both theoretically and practically, over the past few years [4–8]. Our research focuses on the GNSS-based bistatic SAR, as the GNSS system with a designed permanent global coverage is an ideal choice for bistatic SAR compared with other opportunistic illuminators. Reflected GNSS signals can be utilized to image the interesting area as an innovative, all-time and all-weather microwave (L-band) remote sensing tool [9,10]. In addition, GNSS constellations consist of over 100 satellites (GPS, GLONASS, Galileo and BeiDou), and multi-angle observation from over 20 satellites in any given area is attainable [11], which could potentially be used for increasing the imaging formation space [12–15]. Furthermore, GNSS-based bistatic SAR can achieve easy synchronization aided by the GNSS time service [16,17]. Sensors 2016, 16, 294; doi:10.3390/s16030294

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For imaging, many modified algorithms for bistatic SAR with specific geometric configurations have been presented [18–22], such as the range Doppler algorithm (RDA), the wavenumber domain algorithm (WDA), the back-projection algorithm (BPA), the chirp scaling algorithm (CSA), etc. For GNSS-based bistatic SAR, however, none of the above-mentioned algorithms is applicable due to its unique topology and non-designed radar signal. Based on the assumption of parallel flight paths of transmitter and receiver, a modified RDA is proposed in [23], and a modified BPA is presented in [16], which has a high computational complexity for interpolation. Due to the unique range history, no frequency domain algorithm without any approximation has been proposed yet in the literature. On the other hand, the hybrid correlation algorithm is an accurate and flexible algorithm by performing two-dimensional (2-D) correlation between the echo signal and complex conjugate of the reference function for accurate focusing [24,25], but its efficiency will decrease dramatically with increased range cell migration (RCM) caused by long dwell time and therefore, it is not suitable for the GNSS-based bistatic SAR. In this paper, we consider the most complex geometry topology with non-parallel transmitter (GNSS satellites) and receiver (airplane) trajectories as well as unequal platform velocities, and propose a general imaging algorithm with bulk range cell migration correction (RCMC) and modified hybrid correlation processing. In the proposed algorithm, the bulk RCMC is employed to remove the phase resulted from RCM and range-azimuth coupling at the reference range; after this operation, all targets except for the reference one are coarsely focused; then, a modified hybrid correlation operation is performed with a short processing window in the range direction to compensate for the residual phase errors. In order to demonstrate the validity and feasibility of our proposed imaging formation algorithm, points targets simulations are carried out in this paper. The receiver is assumed to fly in a straight line, without any atmospheric turbulence during its flight. For motion error compensation, which is outside the scope of this paper, adequate processing algorithms about bistatic SAR motion compensation have been proposed [16,26,27] and they are relatively mature nowadays. Besides, other receiver and atmospheric errors (like clock slippage and local oscillator) compensation methods have been proposed in [16]. In the simulations, firstly, the comparison of simulation results of our proposed and traditional BPA are presented to prove the effectiveness of our methods. Furthermore, the results of computational time demonstrate that our proposed algorithm is more efficient. Then, the imaging results and qualities of point targets in different locations are carried out to prove the robustness of the proposed algorithm on full-scene images. The organization of this paper is as follows: Section 2 provides the geometric configuration and echo signal model of GNSS-based bistatic SAR, followed by detailed analysis of the 2-D point target spectrum and system resolution in Section 3. The general imaging algorithm is proposed in Section 4, supported by simulation results in Section 5. Conclusions are drawn in Section 6. 2. Signal Model of GNSS-Based Bistatic SAR Figure 1 shows the geometric configuration of GNSS-based bistatic SAR, where the origin O is set to be the scene center, the X-axis points to the East direction and the Y-axis points to the North. P(x p ,y p ) is an arbitrary point target in the imaging area (the targets are assumed to be located on the XOY plane). T and R are the transmitter (GNSS) and receiver (airplane), respectively. The transmitter is located at (xT (t),yT (t),zT (t)) with the velocity vector VT (VTx ,VTy ,VTz ), while the receiver at (xR (t),yR (t),zR (t)) has a velocity vector given by VR (VRx ,VRy ,VRz ). RR (t) and RT (t) denote receiver and transmitter slant range history, respectively. The instantaneous slant range RT (t) and RR (t) for target P can be expressed as R T ptq “ R R ptq “

b“ b“

x T ptq ´ x p

‰2

x R ptq ´ x p

‰2

“ ‰2 ` y T ptq ´ y p ` z2T ptq “

` y R ptq ´ y p

‰2

` z2R ptq

(1)

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The range history R(t) is the sum of RT (t) and RR (t) R ptq “ R T ptq ` R R ptq

(2)

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Figure 1. Geometric configuration SatelliteSystem System (GNSS)-based bistatic Figure 1. Geometric configuration of of Global Global Navigation Navigation Satellite (GNSS)-based bistatic Synthetic Aperture Radar (SAR). Synthetic Radar (SAR). Figure 1. Aperture Geometric configuration of Global Navigation Satellite System (GNSS)-based bistatic Synthetic Aperture Radar (SAR).

As the SAR system signals,the thetrain train echo signals be divided As traditional the traditional SAR systemtransmits transmits pulse pulse signals, of of echo signals can can be divided a 2-D in memory the openingpulse and signals, closing of However, for for into ainto 2-D vector in memory bybythe opening and closing ofthe thereceiving-window. receiving-window. As the vector traditional SAR system transmits the train of echo signals can beHowever, divided GNSS-based bistatic SAR, transmitted signal frequency fixed, andthe thereceiving-window receiving-window is open GNSS-based SAR, the the transmitted signal frequency is is fixed, and is into a 2-Dbistatic vector in memory by the opening and closing of the receiving-window. However, for open all the time. Each echo of the transmitted GNSS code begins from the near-side target echo and GNSS-based bistatic SAR, the transmitted signal frequency is fixed, and the receiving-window is all the time. Each echo of the transmitted GNSS code begins from the near-side target echo and ends ends all bythe thetime. far side target echo, and there isGNSS an overlap between adjacent GNSS target code echoes as open the transmitted code begins from the near-side and by the far side target Each echo,echo andof there is an overlap between adjacent GNSS code echoes asecho indicated by indicated by the red parts shown in Figure 2a, which are shared by the far-side echo of the previous endsparts by the far side target echo, and there is an overlap code echoes as code the red shown in Figure 2a, which are shared by thebetween far-sideadjacent echo ofGNSS the previous GNSS GNSS code theparts near-side echo of the 2a, next GNSS Assuming that the receiver are by and the red shown in Figure which arecode. shared by the far-side echo of theclocks previous and indicated the near-side echothe ofGNSS the next GNSS code. Assuming that thethe receiver clocks are synchronized synchronized with satellites by a tracking algorithm [28], 2-D echo can be generated as GNSS code and the near-side echo of the next GNSS code. Assuming that the receiver clocks are with shown the GNSS satellites by a tracking algorithm [28], the part 2-D echo can be signal, generated as shown in in Figure 2b. If we just remove the overlapped of the echo the effective synchronized with the GNSS satellites by a tracking algorithm [28], the 2-D echo can be generated as Figure 2b. If wetime justofremove the overlapped part of the echo signal, the effective illuminated time illuminated target from different be uneven, which limit the shown in Figure 2b. If we just removerange the would overlapped part of the would echo signal, theapplication effective of target from different range would be uneven, which would limit the application of GNSS-based of GNSS-based bistatic SAR system. Generally speaking, as the length of aliasing part is significantly illuminated time of target from different range would be uneven, which would limit the application short enough and afterSAR imaging process with the unmatched parameters, power ofsignificantly ghost target bistatic SAR system. Generally speaking, as the length of aliasing isthe significantly short enough of GNSS-based bistatic system. Generally speaking, as the lengthpart of aliasing part is caused by the aliasing part is relatively small so that can be ignored. However, the valid range part and short after enough imagingand process with the unmatched parameters, the power of caused after imaging process with the unmatched parameters, theghost powertarget of ghost targetby the still part needs to be removed after the imaging caused byisthe aliasing part is so relatively small so that can be ignored. the valid range part to be aliasing relatively small that can beprocess. ignored. However, theHowever, valid range part still needs still needs beimaging removed process. after the imaging process. removed afterto the

(a)

(b)

(a) (b) Figure 2. Structures of the GNSS-based bistatic SAR echo signals: (a) echo signal structure; and (b) 2-D layout of the echoes. Figure 2. Structures of the GNSS-based bistatic SAR echo signals: (a) echo signal structure; and (b)

Figure 2. Structures of the GNSS-based bistatic SAR echo signals: (a) echo signal structure; and (b) 2-D 2-D layout of the echoes. layout of the echoes.

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As a series of continuous wave signals are transmitted by GNSS-based bistatic SAR, its echo actually forms a one-dimensional vector. Assuming that encoding is used in each transmitted pulse g(t), g(t) can be expressed as g ptq “ a ptq cos p2π f 0 t ` θ0 q

(3)

where a(t), f 0 and θ 0 denote the primary GNSS ranging code envelope, signal carrier frequency and initial signal phase, respectively. Then, the transmitted signal p(t) can be expressed as 8 ÿ

p ptq “

g pt ´ nTq

(4)

n“´8

where T is the code period. Hence, the echo s(t) reflected from the target P can be modeled as „ „   8 ÿ R ptq R ptq s ptq “ σp t ´ σg t ´ nT ´ “ c c n“´8

(5)

where σ is target radar cross section (RCS), and c is the speed of light. Here, we assume that the transmitter antenna pattern gain is constant to all illuminated targets and the receiver antenna pattern is a rectangular window function. Then, the echo s(t) after quadrature demodulation can be written as „  " * 8 ÿ R ptq R ptq exp ´j2π (6) s ptq “ σa t ´ nT ´ c c n“´8 where the constant phase term has been ignored. Let R(nT) « R(t), and set η = nT + τ [24]. s(t) can be written in a 2-D form „  " * R ptq R pηq s pη, τq “ σa τ ´ exp ´j2π (7) c c where η and τ denote the slow-time and fast-time, respectively, and τ P [´T/2,T/2]. However, as shown in Figure 2b, the valid range part should be removed after the imaging process. Thus, in the final non-aliased image, τ falls within the following range „  R R (8) τ P ´ swath , swath 2c 2c where Rswath is the range swath. 3. GNSS-Based Bistatic SAR Signal Analysis In this section, formulation of the 2-D spectrum for a point target is first derived in detail and then system resolution of the GNSS-based bistatic SAR is obtained, followed by a suggested criterion for choosing opportunistic signal sources. 3.1. 2-D Point Target Spectrum Applying FFT with respect to τ, the echo signal s(η, τ) is transformed into the range frequency domain, yielding r Sr pη, f τ q “ s pη, τq exp t´j2π f τ τu dτ " * (9) p f 0 ` f τ q R pηq “ σA p f τ q exp ´j2π c where A(¨) denotes the spectral function of the range code signal, and fτ and f0 represent the range frequency and signal carrier frequency, respectively. Then, to obtain 2-D spectrum of the signal, the azimuth FFT is applied as ` ˘ r ( S2D f η , f τ “ Sr pη, f τ qexp ´j2π f η η dη " * (10) r p f 0 ` f τ q R pηq “ σA p f τ q Sr pη, f τ q exp ´j2π ´ j2π f η η dη c

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where fη is the azimuth frequency. By applying the principle of stationary phase (POSP) [29], we have 2π

p f 0 ` f τ q dR pηq ¨ ` 2π f η “ 0 c dη

(11)

It is difficult to obtain an explicit solution for Equation (11). However, the GNSS-based bistatic SAR signal exhibits a relatively small bandwidth compared to the traditional SAR system, and its resolution is very low. Therefore, by using the series reversion method in [18,30], the 2-D point target spectrum is approximately given by " * ` ˘ 2π p f 0 ` f τ q r S2D f η , f τ “ σA p f τ q exp ´j c # „ 2 + 1 fd ¨exp j2π ¨ ¨ q p fτ q ¨ ` fη 2 fr q p fτ q # (12) „ 3 + f3 fd 2 ` fη ¨exp j2π ¨ 3 ¨ q p f τ q ¨ q p fτ q 6 fr # „ 4 + 3 f 32 ´ f r f 4 3 fd ¨ q p fτ q ¨ ¨exp j2π ¨ ` fη q p fτ q 24 f r5 f0 , r is the target slant range at Doppler center time, and fd , fr , f3 and f4 denote the f0 ` fτ Doppler parameters, as given in the following Taylor series expansion ˆ ˙ 1 1 1 2 3 4 R pηq “ r ` λ f d η ` f r η ` f 3 η ` f 4 η (13) 2! 3! 4!

where q p f τ q “

All the coefficients can be evaluated at the aperture center, i.e., « ffˇ „ ˇ 1 dR T pηq dR R pηq ˇˇ 1 dR2T pηq dR2R pηq ˇˇ fd “ ` ` , fr “ ˇ ˇ ˇ λ dη dη λ dη 2 dη 2 η “0 η “0 « ffˇ « ffˇ 1 dR3T pηq dR3R pηq ˇˇ 1 dR4T pηq dR4R pηq ˇˇ f3 “ ` , f “ ` ˇ ˇ 4 ˇ ˇ λ λ dη 3 dη 3 dη 4 dη 4 η “0

(14)

η “0

Furthermore, the maximum values of the cubic and quartic phase terms in Equation (12) are ˇ ˇ ˇ ˇ ˇ π f 3 Ba3 ˇ ˇ π f 3 Ts3 ˇ ˇ ˇ ˇ ˇ φ3 « ˇ « 24 f r3 ˇ ˇ 24 ˇ ˇ ˇ ˇ ˇ ` ˇ π 3 f 2 ´ f f ˘ B 4 ˇ ˇ π `3 f 2 ´ f f ˘ T 4 ˇ r 4 r 4 ˇ ˇ aˇ s ˇ 3 3 φ4 « ˇ ˇ«ˇ ˇ ˇ ˇ ˇ ˇ 192 f r 192 f r5

(15)

where Ba = fr Ts is the Doppler bandwidth and Ts is the synthetic aperture time. Figure 3 shows the changing curve of ϕ3 , ϕ4 with respect to Ts , with the parameters listed in Section 5. As shown in Figure 3, both ϕ3 and ϕ4 are smaller than π/4 (black dash lines) when Ts is smaller than 24 s. For a general airplane receiver platform, such an upper bound for Ts is easily satisfied, and the fourth order range history is sufficient for accurate focusing. For some other long dwell time applications, like a fixed receiver [8], a higher order range history is necessary.

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satisfied, and the fourth order range history is sufficient for accurate focusing. For some other6 long Sensors 2016, 16, 294 of 15 dwell time applications, like a fixed receiver [8], a higher order range history is necessary.

Figure Figure 3. 3. Maximum Maximum phase phase values values of of the the cubic cubic and and quartic quartic phase phase terms terms in in the the 2-D 2-D spectrum. spectrum.

3.2. System System Resolution of GNSS-Based Bistatic SAR Azimuth resolution resolutionand andrange rangeresolution resolution important parameters a GNSS-based bistatic areare important parameters of aof GNSS-based bistatic SAR SAR system. The general analysis approaches for resolution spatial resolution ofSAR bistatic been system. The general analysis approaches for spatial of bistatic haveSAR beenhave presented presented in the [31,32], and the generalized ambiguity functionfunction and point-spread function of in [31,32], and generalized ambiguity function and point-spread of GNSS-based multistatic GNSS-based multistatic been given in [12]. Based on this literature,ofthe system resolution SAR have been given in SAR [12]. have Based on this literature, the system resolution GNSS-based bistatic of GNSS-based bistatic SAR with aisgeneral configuration is analyzed in this section. SAR with a general configuration analyzed in this section. According to the principle of gradient gradient method method [31,33], the the directions directions of range and azimuth resolutions are ı ” ı ” px,y  x , yq p0q   x ,pyx,y  q p0q grad f grad R    grad df d  0  grad R  0  ıˇ  ,, u ” ” ıˇ uau“ ˇˇ ur “ ˇˇ (16) (16) ˇ px,y q ˇ a r p x,y  x , y  p0q ˇ  x , y  q p0q R f ˇgrad ˇ ˇgrad    grad df grad R 0  0

 d    where fd is the Doppler frequency and it can be found by differentiation of the range history with where fd is the Doppler frequency and it can be found by differentiation of the range history with respect to η. grad(¨) is the gradient function. The superscript (x,y) represents the target located at respect to η. grad(·) is the gradient function. The superscript (x,y) represents the target located at (x,y). Unlike the traditional monostatic SAR system, the range and azimuth resolution vectors of (x,y). Unlike the traditional monostatic SAR system, the range and azimuth resolution vectors of GNSS-based bistatic SAR are not orthogonal, and even almost in parallel with each other in some cases. GNSS-based bistatic SAR are not orthogonal, and even almost in parallel with each other in some The angle θ between the two vectors can be calculated by cases. The angle θ between the two vectors can be calculated by (17) θ “ a cos pua ¨ ur q   a cos  u a  u r  (17) Figure 4 shows the iso-range, iso-Doppler lines and the angle θ for a given area, with the parameters in Section 5. For iso-Doppler the GPS SVNlines 12 satellite, shown in aFigures and 5a, the the Figure listed 4 shows the iso-range, and the as angle θ for given 4a area, with gradients of listed iso-range and iso-Doppler nearly in parallel, have a negative impact parameters in Section 5. For thelines GPSare SVN 12 satellite, as which shownwill in Figures 4a and 5a, the on 2-D target detectionand after imaging. However, as demonstrated Figures anda 5b, the direction gradients of iso-range iso-Doppler lines are nearly in parallel, by which will 4b have negative impact of range resolution is approximately to the direction of azimuth resolution with the on 2-D target detection after imaging. perpendicular However, as demonstrated by Figures 4b and 5b, the direction geometry topology of 2. of range resolution is SVN approximately perpendicular to the direction of azimuth resolution with the

geometry topology of SVN 2.

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(a(a) )

(b(b) )

Figure The Iso-range SVN 12; and (b) SVN 2.2. Figure TheIso-range Iso-rangeand andIso-Doppler Iso-Dopplerlines linesof: of: (a) SVN 12; and (b) SVN Figure 4.4.4.The and Iso-Doppler lines of:(a) (a) SVN 12; and (b) SVN 2.

(a(a) )

(b(b) )

Figure The ofofof Range and Azimuth resolutions: (a) and Figure TheAngle Anglebetween betweenthe the directions Range and Azimuth resolutions: (a)SVN12; SVN12; and(b) (b) Figure 5.5.5.The Angle between thedirections directions Range and Azimuth resolutions: (a) SVN12; and SVN 2. SVN 2. (b) SVN 2.

Therefore, toto obtain better 2-D signal source should be chosen Therefore, obtainbetter better2-D 2-Dresolutions, resolutions,the thebest besttransmitter transmitter signal source should bebe chosen Therefore, to obtain resolutions, the transmitter signal source should chosen firstly among all visible GNSS satellites. Furthermore, according to the gradient of range history and firstly among visibleGNSS GNSSsatellites. satellites.Furthermore, Furthermore, according history and firstly among allall visible accordingto tothe thegradient gradientofofrange range history and Doppler frequency, the range and azimuth Doppler frequency, the rangeand andazimuth azimuthresolution resolutioncan canbe beobtained obtainedby by [31] Doppler frequency, the range resolution can be obtained by[31] [31]

1 11 1 c 1 11   11, ,     c¨ c ρ a “aˇˇa ” px,y  xq,xy,y  ıˇˇ ¨ TT , ρrrr “ ˇˇ  ” x,xy,y   Bıˇ wˇ w Bw  RRpx,y  B grad grad  0 0ˇ  Tsss  0 0qp0q grad grad R fdfdfd p0q ˇgrad ˇgrad   ˇ

(18) (18)(18)

where Bw the To understand thespatial spatialvariant variant 2-D resolution clearly, Figure 6a,b the signal bandwidth. 2-D resolution clearly, Figure 6a,b where BBwis wisis thesignal signalbandwidth. bandwidth.To Tounderstand understandthe the spatial variant 2-D resolution clearly, Figure 6a,b where shows the 2-D resolution distribution with configurations ofofGPS SVN12 2.2.shown As shows the 2-D resolution distribution with thethe configurations of GPS SVN12 andand GSPGSP SVNSVN 2. As shows the 2-D resolution distribution with the configurations GPS SVN12 and GSP SVN As 6,6,unlike SAR, azimuth range are spatially inshown Figure 6, unlike monostatic SAR, the azimuth rangeand resolutions are spatially and shownin inFigure Figurethe unlikethe themonostatic monostatic SAR,the theand azimuth and rangeresolutions resolutions arevariant spatially variant and both of them depend on the specific system bistatic geometry [31]. variant anddepend both of them the specific system bistatic geometry [31]. both of them on thedepend specificonsystem bistatic geometry [31].

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(a) (a)

(b) (b)

Figure 6. The 2-D resolution distribution of GNSS-based bistatic SAR: (a) SVN12; and (b) SVN 2. Figure Figure 6. 6. The The 2-D 2-D resolution resolution distribution distribution of of GNSS-based GNSS-based bistatic bistatic SAR: SAR: (a) (a) SVN12; SVN12; and and (b) (b) SVN SVN 2. 2.

4. A General Imaging Algorithm 4. AAGeneral GeneralImaging ImagingAlgorithm Algorithm 4. Based on the analysis in Section 3, a novel general imaging algorithm is proposed in this Based on inin Section 3, a3,novel general imaging algorithm is proposed in this section. There areanalysis two major challenges GNSS-based bistatic SAR imaging. The firstsection. the Based on the the analysis Section afornovel general imaging algorithm is proposed inis this There are two history major for bistatic SAR imaging. The is the intricate intricate range caused the GNSS-based specific geometry. In general, the first transmitter and section. There are twochallenges major by challenges for system GNSS-based bistatic SAR imaging. The first is the range history caused the specific geometry. In general, the transmitter and the receiver’s receiver’s movements are independent, and their trajectories andIn speeds are essentially different. As intricate range historyby caused by thesystem specific system geometry. general, the transmitter and the are their and trajectories and speeds are essentially different. As a result, amovements result, the 2-Dindependent, spectrum ofand GNSS-based bistatic SAR isand extremely complicated, as shown in receiver’s movements are independent, their trajectories speeds are essentially different. As the 2-D spectrum GNSS-based SARbistatic is extremely as shown in Equation (12),2-D andof the solutions tobistatic range cell migration correction are therefore more complex. The aEquation result, the spectrum of GNSS-based SAR iscomplicated, extremely complicated, as shown(12), in and theone solutions to range cell migration correction are correction therefore more complex. The second one is second is the dwell time. GNSS-based bistatic SAR system, long dwell timecomplex. is necessary Equation (12), andlong the solutions to In range cell migration are therefore more The theachieve longone dwell time. GNSS-based SAR system, long system, dwell isdwell necessary to necessary achieve to aisreasonable azimuth and sufficient image SNR time for radar applications, whicha second the longIndwell time.resolution In bistatic GNSS-based bistatic SAR long time is reasonable resolution and sufficientand image SNR for radarSNR applications, leads towhich large leads to large range migration and extremely bigsufficient data size. to achieve aazimuth reasonable azimuth resolution image for radar which applications, onrange the aforementioned discussion, a novel general imaging algorithm based on hybrid rangeBased migration and extremelyand big extremely data size. big leads to large migration data size. correlation is derived. The block diagram of the proposed algorithm is shown Figure 7. Based processing on the aforementioned aforementioned discussion, novel general imaging algorithm basedinon on Based on discussion, aa novel general imaging algorithm based hybrid There are three parts: the first part is range compression, which is carried out to implement matched correlation processing is derived. The block diagram of the proposed algorithm is shown in Figure correlation The block diagram of the proposed algorithm is shown in Figure 7.7. filtering the range the is second is bulk compression, whichout is used to decrease range matched There areinthree parts:direction; the first part rangepart compression, which is carried to implement matched migration focus direction; the targets reference last part is which performing modified filtering in and the range bulkthe useda to to decreasehybrid filtering the theatsecond partrange; is bulk compression, is used decrease range correlation processing to acquire accurate focused image. migration and and focus the the targets an at reference range; the the last part is performing performing a modified hybrid migration focus targets at last correlation processing to acquire an accurate focused image. correlation

Figure 7. Block diagram of the proposed imaging algorithm. Figure 7. Block diagram of the proposed imaging algorithm.

Figure 7. Block diagram of the proposed imaging algorithm.

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4.1. Range Compression To maximize signal-to-noise ratio and obtain fine range resolution of the sensed objects, range compression processing is performed. As in a single GNSS constellation system there are over 20 GNSS satellites and at least six to eight illuminating the same scene, the receiver of GNSS-based bistatic SAR actually records all the visible satellite signals. Therefore, the echo of GNSS-based bistatic SAR is the superposition of different reflected signal echoes. The GNSS network uses a high-rate pseudo random noise (PRN) sequence for each satellite. As a result, the PRN codes have particularly excellent auto-correlation and cross-correlation properties, and these cross-correlation values are so small that they usually can be ignored [34]. Therefore, the corresponding PRN code could be employed as the matched filter. Furthermore, after range compression, the reflected signals from different visible GNSS satellites are separated. Range compression can be realized in the azimuth time range frequency domain via a range FFT on raw data and reference signal, matched filter multiplication and a range IFFT. The reference signal is determined by pseudo PRN sequence of the chosen satellite, and it can be conducted by the signal synchronization algorithm [16]. After matched filtering, the signal can be modeled as follows: " * p f 0 ` f τ q R pηq Src pη, f τ q “ σFf p f τ q exp ´j2π (19) c where F f (¨) represents the spectrum of the auto-correlation function Fτ (¨) between the echo data and reference signal. It is also highlighted that, if the aggregate E5 signal of Galileo system is used, range signal suppression technology should be adopted after range compression [35]. 4.2. Bulk RCMC For a GNSS-based bistatic SAR system, the imaging operation has to deal with a huge amount of raw data due to the large RCM. As a result, it will lead to a very high computational load and low efficiency for traditional imaging algorithms. The total RCM can be divided into two parts: bulk RCM and residual RCM. Bulk RCM represents the RCM of a reference target, while the remaining component of RCM is the residual RCM. Therefore, bulk RCMC is performed in the proposed algorithm to decrease the range migration and enhance the processing efficiency. It starts with an azimuth FFT to transform the signal into the 2-D frequency domain. Then, multiplication with the reference function is performed to remove the bulk RCM, the frequency modulation in azimuth and range-azimuth coupling at the reference slant range. According to the 2-D spectrum in Equation (12), the reference function is given by "

`

Hre f f η , f τ

˘

* 2π p f 0 ` f τ q r “ exp j c # „ 2 + 1 fd ¨exp ´j2π ¨ ¨ q p fτ q ¨ ` fη 2 fr q p fτ q # „ 3 + f3 f d ¨exp ´j2π ¨ 3 ¨ q2 p f τ q ¨ ` fη q p fτ q 6 fr # „ 4 + 3 f 32 ´ f r f 4 3 fd ¨exp ´j2π ¨ ¨ q p fτ q ¨ ` fη q p fτ q 24 f r5

(20)

where Rre f is reference range, and fd_re f , fr_re f , f3_re f , and f4_re f are the corresponding Doppler parameters. After the bulk RCM component is removed and the target phase at the reference range is compensated, a residual phase still exists for targets at other range bins and differential RCM is still present. The residual RCM is range dependent and is much smaller than the bulk RCM.

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The bulk RCMC is introduced as a coarse focusing stage in Section 4.2, and the RCM is reduced accordingly. However, the form of target range history is changed after performing the bulk RCMC. 4.3. Modified Hybrid Correlation Processing Therefore, the traditional hybrid correlation processing is not suitable for differential focusing any more.The In this a modified hybrid processing is proposed due tothe theRCM different range bulkstage, RCMC is introduced as acorrelation coarse focusing stage in Section 4.2, and is reduced history. To correct the residual RCM for all targets, a processing window in range direction is accordingly. However, the form of target range history is changed after performing the bulk RCMC. applied. The offset of the correlation window in time domain is Therefore, the traditional hybrid correlation processing is not suitable for differential focusing any more. In this stage, a modified hybrid is proposed due to the different range   R correlation    Rref  processing  j 2  R F  for all targets, a exp  ,   RCM    Rrefinrange  direction is applied. (21) history. To correct the hresidual processing window c   The offset of the correlation window in time domain is „ of reference target.  processing where Rref(η) is the range history ı) R pηq ´ Rre f pηq The!signal”after hybrid correlation h pη, τq “ Fτ τ ´ exp ´j2π R pηq ´ Rre f pηq (21) can be expressed as c





    R The m f  signal Rref f after hybrid mtarget. where Rre f (η) is the range history of reference correlation processing      S f ,   S ' f ,  i   fs   H a* f , (22)     can be expressed as hc  2 c   i 0       « ˜ [ ` ˘ ` ˘ _¸ff m ` ˘ ÿ ` ˘ R f η ´ Rre f f η m Shc f η , τ the “ system S1 f η ,signal τ ` iafter ´ ` ¨ Ha˚ hybrid f η , τ correlation (22) where S’[·] represents the range IFFT in thef s modified 2 c i “ 0 processing stage, m is the length of correlation window in the range direction, fs is the range is the azimuth FFTafter of correlation the modified superscript * represents the sampling rate, Ha(fη, τ) the where S’[¨] represents system signal the range window, IFFT in the hybrid correlation





 

 





η, τ), and the function. As the RCM has been complex conjugate Ha(flength  represents processing stage, m of is the of correlation window infloor the range direction, fs isbulk the range sampling rate, Ha (fηat, τ) the azimuth FFT of correlation window, the superscript complex removed theisbulk RCMC stage, a short correlation window in the range* represents direction isthe used in the t¨u conjugate of H (f , τ), and represents the floor function. As the bulk RCM has been removed at the a η correlation processing, which means the efficiency of the proposed imaging modified hybrid bulk RCMC stage, a short correlation window in the range direction is used in the modified hybrid algorithm is improved. correlation processing, which means the efficiency of the proposed imaging algorithm is improved. After hybrid correlation processing, the residual RCM and the residual phase error are After hybrid correlation processing, the residual RCM and the residual phase error are completely completely corrected. Then, an azimuth IFFT is performed, leading to an accurately focused image. corrected. Then, an azimuth IFFT is performed, leading to an accurately focused image. 5. Simulation and Discussions 5. Simulation and Discussions In this section, the C/A code of GPS is adopted as the opportunistic signal for bistatic SAR In this As section, code of GPS adopted as the opportunistic signal SAR simulation. shownthe in C/A Figures 4 and 5, theisgeometric topology with satellite SVN for 2 is bistatic favorable to simulation. shown in Figures 4 and 5atthe topology withthe satellite SVNand 2 ismotion favorable to achieve 2-D As (range-azimuth) resolution thegeometric same time. Therefore, C/A code state achieve 2-D (range-azimuth) resolution at the same time. Therefore, the C/A code and motion state of of the SVN 2 are used in the subsequent simulations to demonstrate the performance of the proposed the SVNimaging 2 are used in the subsequent simulations to demonstrate performance of the 8proposed general algorithm. The simulation scene with point targetsthe is illustrated in Figure and the general imaging algorithm. The simulation scene with point targets is illustrated in Figure 8 and and specific parameters are listed in Table 1, where we can see that the trajectories of transmitter the specific parameters are listed in Table 1, where we can see that the trajectories of transmitter and receiver are non-parallel, and their velocities are non-parallel and different. receiver are non-parallel, and their velocities are non-parallel and different.

Figure 8. 8. Distribution Distribution of of the the simulation simulation scene. scene. Figure

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Table 1. Main simulation parameters. Table 1. Main simulation parameters.

Parameters Parameters Receiver center position Receiver center position Receiver velocity Receiver velocity Transmitter center position (SVN Transmitter center position (SVN 2) 2) Transmitter center position (SVN 12) Transmitter center position (SVN 12) Transmitter center velocity (SVN 2) Transmitter center velocity (SVN 2) Transmitter center velocity (SVN 12) TransmitterSynthetic center velocity (SVN aperture time12) Synthetic aperture time Signal bandwidth (GPS C/A) Signal bandwidth (GPS C/A) Signal wavelength Signal Sampling wavelength rate SamplingPRF rate PRF

Value Value (6, ´25,5) km km (6, −25,5) (´30,(−30, 60, 0) 60,m/s 0) m/s (1.0235, ´1.5541, 1.2402) ˆ 10×4 10 km 4 km (1.0235, −1.5541, 1.2402) (´1.8028, 1.6145, 0.4677) ˆ 104 km (−1.8028, 1.6145, 0.4677) × 104 km (185.6, ´2113.7, ´1800.0) m/s (185.6, −2113.7, −1800.0) m/s (´2096.1, ´730.7, ´2321.8) m/s (−2096.1,10.0 −730.7, s −2321.8) m/s 10.0 s 2.046 MHz 2.046 0.19 m MHz 0.19 m 5.0 MHz 1005.0 HzMHz 100 Hz

5.1. Comparative Experiments and Analysis

5.1. Comparative Experiments and Analysis

To show the performance of the proposed algorithm, the imaging results of scene center C with To show the performance of the proposed algorithm, the imaging results of scene center C the traditional BPA [16] and algorithmalgorithm are shown Figurein9. Figure In order with the traditional BPA our [16]proposed and our proposed areinshown 9. to In compare order to their imaging results, imaging results the proposed havealgorithm projectedhave ontoprojected the ground compare theirthe imaging results, the of imaging results ofalgorithm the proposed ontoplane (not the the ground slant range the final results of BPA are acquired ground plane directly. planeplane), (not theasslant rangeimaging plane), as the final imaging results of in BPA are acquired in Comparing these directly. two imaging results, wetwo would not be able to a clear between ground plane Comparing these imaging results, wesee would not difference be able to see a clear them. difference the between Furthermore, theresolution spatial resolution (azimuth resolution Furthermore, spatialthem. resolution (azimuth and range resolution), peakand siderange lobe ratio resolution), peak side lobe ratio (PSLR) and integrated side lobe ratio (ISLR) for target C are listed inresults (PSLR) and integrated side lobe ratio (ISLR) for target C are listed in Table 2. Both the imaging 2. Both the imaging results qualities indicate our proposed algorithm has same and Table imaging qualities indicate that and our imaging proposed algorithm hasthat been adequately and equally been adequately and equally same effective with the traditional one. On the other hand, the effective with the traditional one. On the other hand, the computational time of traditional BPA is computational time of traditional BPA is 172.9 s, with the parameter listed in Table 1. However, for 172.9 s, with the parameter listed in Table 1. However, for our proposed algorithm, less CPU time is our proposed algorithm, less CPU time is required (68.2 s), which means the proposed algorithm is required s), which means the proposed algorithm are is more efficient. bewith noted that the more (68.2 efficient. It should be noted that the simulations carried out on It a should computer Xeon simulations are carried out on a computer with Xeon E5649 2.53 GHz processors and 32 GB RAM, E5649 2.53 GHz processors and 32 GB RAM, and all programs adopt the parallel codes (quad-core) and all programs adopt the parallel codes (quad-core) in MATLAB. in MATLAB.

(a)

(b)

Figure 9. Contour plots of Target C: imaging (a) imaging results of traditional Back ProjectionAlgorithm Algorithm(BPA); Figure 9. Contour plots of Target C: (a) results of traditional Back Projection (BPA); and (b) imaging results of our proposed algorithm. and (b) imaging results of our proposed algorithm. Table 2. Imaging quality analysis of Comparative Experiments.

Table 2. Imaging quality analysis of Comparative Experiments.

Azimuth Resolution (m) PSLR (dB) Traditional BPA 30.08 Azimuth −13.30 Proposed Algorithm −13.30 Resolution30.08 (m) PSLR (dB)

Traditional BPA Proposed Algorithm

30.08 30.08

´13.30 ´13.30

Range ISLR (dB) Resolution (m) PSLR (dB) ISLR (dB) Range −19.43 −10.22 82.05 −27.34 −10.23 82.05 −19.42 ISLR (dB) ISLR (dB) Resolution (m)−27.34 PSLR (dB)

´10.22 ´10.23

82.05 82.05

´27.34 ´27.34

´19.43 ´19.42

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5.2. Imaging Simulation Experiments and Analysis

To demonstrate demonstratethe thevalidity validity and feasibility of proposed our proposed imaging formation algorithm, To and feasibility of our imaging formation algorithm, points points targets simulation experiments areout carried in this section. Figurethe 10 processing shows the targets imagingimaging simulation experiments are carried in thisout section. Figure 10 shows processing results at different imaging After the range compression stage, matched filtering results at different imaging stages. Afterstages. the range compression stage, matched filtering in the range in the range is completed and the can been seen clearly Figure In this dimension is dimension completed and the large RCM canlarge been RCM seen clearly in Figure 10a. Inin this case, 10a. the RCM is case, the is more thanwhich 100 range gates, the which will reduce efficiency of the process, more thanRCM 100 range gates, will reduce efficiency of thethe imaging process, as aimaging long correlation as a long(means correlation function m adopted = 128 atinleast) has tohybrid be adopted in traditional function m = 128 at least)(means has to be traditional correlation algorithms.hybrid Then, correlation algorithms. Then, after the bulk asmigration shown inisFigure 10b, the range after performing the bulk RCMC, as performing shown in Figure 10b, RCMC, the range reduced significantly migration is reduced significantly onlya residual RCM exists. with As a sixteen result, apoints correlation window and only residual RCM exists. As aand result, correlation window (m = 16) in the with sixteen points (m = 16) in for thethe range dimension is suggested forathe general case,inwhich indicates range dimension is suggested general case, which indicates notable saving computational a notable saving computational complexity the proposed modified correlation operation. complexity by the in proposed modified correlationby operation. Even more savings can be achieved in Even more savings can be achieved in long dwell time situations, such as a fixed receiver. The final long dwell time situations, such as a fixed receiver. The final imaging result is shown in Figure 10c, imaging is shown 10c, where the topology distortionand caused by the specific geometric where theresult distortion causedinbyFigure the specific geometric mismatching 2-D resolution can topology and mismatching 2-D resolution can be observed. be observed.

(a)

(b)

(c)

Figure 10. 10. Processing stages: (a) range frequency frequency Figure Processing results results at at different different stages: (a) range range compression compression result result in in range azimuth time bulk RCMC RCMC result result in in range range frequency frequency azimuth azimuth time domain; and final azimuth time domain; domain; (b) (b) bulk time domain; and (c) (c) final imaging result after azimuth compression. imaging result after azimuth compression.

11shows showsthe theinterpolated interpolated imaging results targets N/C/F in slant range plane, and Figure 11 imaging results for for targets N/C/F in slant range plane, and their theirprofiles 2-D profiles (fastand time andtime) sloware time) are demonstrated in 12. Figure 12. As in shown Figure 12a, 2-D (fast time slow demonstrated in Figure As shown Figurein12a, the slow the slow profile is still alike sinctraditional envelopeSAR likeresults. traditional SARasresults. However,transmit as the time profiletime is still a sinc envelope However, the non-designed non-designed radarbistatic signalSAR, in GNSS-based fastfunction time profile is not a sinc radar signal intransmit GNSS-based the fast timebistatic profileSAR, is notthe a sinc anymore and it is function anymore it is the envelope of the auto-correlation function of C/A as shown in the envelope of theand auto-correlation function of C/A code, as shown in Figure 12b.code, As demonstrated Figure As 11 demonstrated both Figures 11 dimension and 12, low side-lobe without in rangeany dimension is in both 12b. Figures and 12 low in side-lobe in range is achieved weighting achieved without any weighting the spatial (azimuthwiden resolution operation. Furthermore, the spatialoperation. resolutionFurthermore, (azimuth resolution andresolution range resolution), ratio, and range resolution), widen ratio,are PSLR and ISLR for N/C/F geometric are listed in Table 3. In this PSLR and ISLR for targets N/C/F listed in Table 3. Intargets this specific configuration, the specific geometric the azimuth resolution in rapidly different spatial azimuth resolution configuration, changes significantly in different spatial changes locationssignificantly because of the changing locations because of theboth rapidly changing rangeresults history.and Finally, bothqualities the sceneofimaging resultshave and range history. Finally, the scene imaging imaging point targets imaging qualities of point targets have demonstrated excellent focusing performance of the demonstrated excellent focusing performance of the proposed algorithm. proposed algorithm. Table 3. Imaging quality analysis of point targets. Table 3. Imaging quality analysis of point targets. Azimuth

N N C C F F

Azimuth ρ a,m (m) Widen Ratio PSLR (dB) ρa,m (m) Widen Ratio PSLR (dB) 48.16 1.032 ´13.26 48.16 1.032 −13.26 30.08 1.029 ´13.30 30.08 1.029 −13.30 23.61 1.031 ´13.24 23.61 1.031 −13.24

Range

ISLR (dB)

ISLR (dB) ´10.20 −10.20 ´10.23 −10.23 ´10.19 −10.19

Range ρr,m (m) Widen Ratio PSLR (dB) ρr,m (m) Widen Ratio PSLR (dB) 81.54 1.012 ´27.09 81.54 1.012 −27.09 82.05 1.011 ´27.34 82.05 1.011 −27.34 82.68 1.014 ´26.99 82.68 1.014 −26.99

ISLR (dB)

ISLR (dB) ´19.40 −19.40 ´19.42 −19.42 ´19.37 −19.37

*ρ a,m and ρr,m denote the measured azimuth resolution and range resolution in XOY plane, and widen ratio

*ρrepresents a,m and ρr,mthe denote the measured azimuth resolution and range in XOY and widen ratio between the measured and ideal resolution. Theresolution ideal resolution canplane, be calculated by Equation (18) or the method in reference [31,32]. ratio represents the ratio between the measured and ideal resolution. The ideal resolution can be calculated by Equation (18) or the method in reference [31,32].

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((aa))

b)) ((b

((cc)) Figure 11. Interpolation imaging results of: (a) target N; (b) target F; and (c) target C.

Figure 11. Interpolation imagingresults results of: of: (a) (b)(b) target F; and (c) target C. Figure 11. Interpolation imaging (a)target targetN;N; target F; and (c) target C.

((aa))

b)) ((b

Figure 12. 12. Interpolation Interpolation imaging imaging results results of: of: (a) (a) slow slow time time profile; profile; and and (b) (b) fast fast time time profile. profile. Figure

Figure 12. Interpolation imaging results of: (a) slow time profile; and (b) fast time profile.

6. Conclusions In this paper, a novel bistatic SAR system based on GNSS signals has been introduced, where the mechanism for its 2-D echo signal acquisition is analyzed and its 2-D point target spectrum

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model provided. To focus the echo data more accurately and efficiently, a general imaging algorithm without any interpolation processing was proposed. Due to the complex range history and large RCM in GNSS-based bistatic SAR, a bulk RCMC is applied first to remove the RCM of the reference target, avoiding a long and time-consuming range correlation window processing in the focusing stage; then, a modified hybrid correlation operation is carried out for accurate focusing of the echo data, with a much shorter correlation processing window (in most cases, sixteen points would be enough). As demonstrated by simulations results, the proposed GNSS-based bistatic SAR system works effectively and the imaging algorithm performs well. Acknowledgments: This work was supported by National Natural Science Foundation of China (NSFC) under Grant No.61132006 and National Natural Science Foundation of China (NSFC) under Grant No. 61171123. Author Contributions: Hong-Cheng Zeng and Peng-Bo Wang conceived and designed the experiments and implemented the methods; Hong-Cheng Zeng and Wei Yang analyzed the data; Jie Chen, Wei Liu and Lin Lin Ge supervised the research, including the experiments and development; and Hong-Cheng Zeng wrote the paper. Conflicts of Interest: The authors declare no conflict of interest.

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