Bistatic Fixed-receiver Parasitic SAR Processor Based ... - IEEE Xplore
Bistatic Fixed-receiver Parasitic SAR Processor Based on the Back-propagation Algorithm Jesus Sanz-Marcos, Pau Prats, Jordi J. Mallorqui Signal Theory and Communications Dept. Universitat Politecnica de Catalunya (UPC) Barcelona, Spain {jsanz,pprats,mallorqui}@tsc.upc.edu Abstract— In bistatic SAR systems the transmitter and the receiver are not in the same platform and therefore can follow different trajectories. Parasitic receivers record the backscattered echo of the signal transmitted from existing monostatic SARs, such as satellites or airplanes, and represent a challenging opportunity to start to understand the complexities of bistatic systems before constellations of SAR satellites are deployed on space. The case under study is based on a transmitting antenna following a rectilinear trajectory, while the receiver remains on the ground in the top of a high tower looking down to the illuminated scene. In this case, efficient SAR monostatic algorithms (Chirp scaling, Range Doppler, Omega-K, etc) are not suitable to accurately focus bistatic SAR data with fixed receiver due to the topography dependence of the impulse response. In this paper we will demonstrate the problems that arise when using already mentioned standard processing algorithms and we will propose a way to use the backpropagation technique adapted to the bistatic fixed-receiver case. The major problem regarding focusing raw data from these systems is that the target impulse response not only depends on the distance from the target to the transmitter but also on the distance to the receiver which is constant and adds a delay which depends on the target complete position (azimuth, range and height). First the algorithm will be suited to a flat ground model and later to a real case where the Digital Elevation Model is known. The implemented algorithms will be evaluated and verified with simulated scenes formed by ideal targets. Keywords-component; bistatic; propagation; fixed receiver
processor;
SAR;
back-
There are some limitations of course, for example the reduced size of the image, synchronization issues, etc.
Figure 1. Bistatic system geometry with static reveicer (left) and bistatic ambiguity for two targets at the same bistatic distance (right)
Consider the geometry depicted in Fig. 1a, where instead of the usual cylindrical coordinate system, a Cartesian coordinate system has been used to define transmitter, receiver and target positions. The starting expression for the impulse response of a given target as a function of its geometry is: target : (R1 , x, h ) → g (t az , t rg ) = 1 2π = p t rg − R (t az ) exp − j R (t az ) wT (x − vt az , r1 )wR c λ R (t az ) = R12 + (x − xT − vtaz ) + 2
(1)
R2
2 2 2 + (x − x R ) + ( y (R1 , h ) − y R ) + (h − hR )
y (R1 , h ) = yT + R12 − (h − hT )
2
I.
INTRODUCTION TO BISTATIC FIXED RECEIVER SAR
This paper continues the research started in [1] and [2] aimed at the development of a focusing algorithm for the bistatic SAR. In this case, the transmitter follows a rectilinear trajectory, while the receiver remains in a fixed position at the top of a high tower or mountain looking down to the illuminated scene. This configuration offers some challenging possibilities as current existing SAR satellites might be used as non-cooperative transmitters, safely locating the expensive hardware of the receiver at the ground. In the same way, no downlink capability is needed and more signal to noise ratio is available due to the fact that the emitted echo must not come back to the transmitter. Such a system could perform interferometric applications by placing several receiving subsystems at slightly different positions allowing for instance urban monitoring for subsidence detection and prediction.
0-7803-9050-4/05/$20.00 ©2005 IEEE.
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It is very important to notice that the range history of a point target R(taz) does not depend any more only on the zeroDoppler distance R1 and the relative distance from the target to the transmitter (x-xT), but also on the absolute distance to the receiver. Therefore, as can be observed in Fig. 1b, there is a range ambiguity that does not exist in the Monostatic case, which is crucial for the understanding of this paper: that is, two or more targets located at different positions can have the same range delay at zero-Doppler but will have different range histories (curvature). In the following sections we will present the modifications that have been added to the Monostatic backpropagation algorithm in order to precisely focus bistatic images for two situations: flat-earth and DEM topography.
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II.
BACK-PROJECTION ALGORITHM
An efficient algorithm [3] based on non-linear chirp scaling (NLCS) already exists for the flat-earth bistatic fixed receiver case. Even though it has been extended in [4] to compensate the Quadratic Range-Cell Migration, it has not deeply studied yet. The effect of the approximations used and the limitations in the geometry and instrument parameters has not been shown. In the same way it does not consider in any way topography issues which clearly limit it usefulness. However, the approach taken by the NLCS Algorithm is very interesting in terms of a trade-off between efficiency and precision and is being studied and analyzed.
It can also be noticed in Eq (2) that for each point of the output image it is required to know the values R1 and R2. Depending on the scenario, different solutions may arise. This is considered in the following two sections. Fig. 4 shows the diagram of the proposed algorithm. First of all it is necessary to calculate R1 and R2 values which are related to the topography of the illuminated scene, and then range-compression is performed. In order to reduce the number of interpolations, the algorithm is re-arranged to do all the interpolations of the same line at the same time.
Back-propagation algorithms are the most precise but imply large computational burden. The back-projection algorithm for the bistatic case [5] relies on performing, for each point of the final image, a 1D-convolution of the interpolated rangecompressed data. The computational requirements have been reduced in [6]. As it is shown in the below Figure, for each target of the output image, we must follow the range-curvature that would suffer that target and interpolate the input raw image to obtain a one-dimensional image (sequence) of the propagated range-compressed pulse.
Figure 4. Bistatic back-propagation algorithm diagram
III. Figure 2. Interpolation of range-compressed raw data
The interpolated sequence is then convoluted with the expected value (2) to compress and focus in the cross-range (azimuth) direction: 2π 2 (2) g I (t az ) = exp − j R12 + (x − xT − vt az ) + R2 λ Fig. 3 represents the core of the algorithm in pseudo-code. It can be noticed that the term R,(i,r)+R2(i,r) is used to maintain the propagation phase in order to allow interferometric processing.
target Flat Earth : (x, y, z = 0) R = R1 + R2 R2 =
(3)
(x − xR )2 + ( y − yR )2 + hR2
By expanding Eq. (1) we can relate the Y-coordinate of a target with its zero-Doppler distance to the transmitter:
For each line a… prepareInterpolationLine(a) for each sample r… for each echo n…
y = yT + R12 − hT2
2
nv + R2 (i, r ) i = a + n → R = R12 (i, r ) + prf R out (i, j ) = out (i, j ) + getInterpolatedValue t = c 2π (R − R1 (i, r ) − R2 (i, r )) ·exp j λ
(4)
This substitution leads to the following awkward equation: 2 R22 = α + (R − R2 ) − h 2 − ∆ α = (x − xR )2 + hR2
2
(5)
∆ = yT − y R , hT = h
Figure 3. Pseudo-code back-propagation core
0-7803-9050-4/05/$20.00 ©2005 IEEE.
FLAT-EARTH TOPOGRAPHY SOLUTION
A SAR instrument is a device that measures the delay of a transmitted signal. For this bistatic configuration, the delay is caused by the path propagation (transmitter-target and targetreceiver). For flat-earth topography, we consider all targets to have Z-coordinate equal to zero. Therefore, we have the set of equations:
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The simplified solution for the flat-earth topography model (solution has been simplified using Eq. (5) substitutions): R2 =
(
(
1 αR - R∆2 + R 3 - Rh 2 + 2(R 2 - ∆2 )
∆ - 2αR 2 - 2R 2h 2 - 2αh 2 + ∆4 - 2R 2 ∆2 +
(6)
The general solution must be locally evaluated for each point of the output image, since the range-delay varies in each sample and the distance from the target to the receiver varies in each line: + R 4 + α 2 + 2h 2∆2 + 2α∆2 + h
1 4 2
)
R = ctrg = R1 + R2 (x )
(7)
To test both the focusing algorithm and the solution for flattopography a bistatic scene has been simulated using the parameters shown in Table 1. TABLE I.
SIMULATION PARAMETERS
Instrument F0 5.3 GHz PRF 1600 Hz fs 18.9 MHz v 7.2 km/s L (antenna az. length) 10m τ (pulse duration) 37µs Chirp rate -4.18·1011 Hz/s 2048 Lines 1500 Samples
Geometry TX distance 840 km TX look angle 22 º RX distance 7000 m RX look angle 84 º xR, yR, yT, h 6222 m Targets 5 1.6 km ∆ azimuth 1.5 km ∆ range
DIGITAL ELEVATION MODEL SOLUTION
In the Monostatic case, the scene topography is not considered in the focusing algorithm as the measured range delay is related with the double of the target distance and with the observed range curvature. Topography is only used to project the compressed image, which is in slant–range, to the ground-range. As it has been shown in the previous sections, it is mandatory to know both R1 and R2 values to properly focus a bistatic image, and they clearly depend on the target height. This section is dedicated to the case where there is information of the elevation of the illuminated scene, i.e. for every ground point (x,y) we are able to obtain its height – Eq. (8). Normally we start from a facet mesh that is interpolated to enhance resolution when needed h = f DEM ( x, y )
(8)
The set of equations considering that targets do not lie in the ground is more complex and in order to provide the backprojection algorithm R1 and R2 values we must solve the following equation which does not have, obviously, an analytical solution: trg c = R1 + R2 = f ( y, x = constant ) =
(x − xT )2 + ( y − yT )2 + ( f DEM (x, y ) − hT )2 +
(9)
(x − xR ) + ( y − yR ) + ( f DEM (x, y ) − hR ) 2
Fig. 5 shows R1 and R2 values for a 2048 lines x 1500 samples scene in the orbital case (rectilinear trajectory approximation used) projected at the bistatic slant-range plane. We can notice that there are some regions where there is no solution for the equation system described in Eq. (3) what means that there is no geometrical position for that bistatic distance. These complex shapes represent a great challenge towards the development of precise and efficient focusing algorithms. The situation will get more complicated with real DEM topography.
2
2
The procedure followed to solve this equation is based on simple ray-tracing techniques as shown in the below figure. For each cross-range position of the transmitter, we get an interpolated section of the Digital Elevation Model forming a plane with the transmitting antenna. Then we sweep all the points in the DEM sections in order to find those points that coincide with slant-range (R1 + R2) samples. To reduce the computational requirements an approximate solution and boundaries is initially selected.
Figure 6. Digital Elevation Model section (top), ground range Ray tracing technique (middle) and output solution example (bottom).
Figure 5. Bistatic flat-earth model topography evaluation. Transmitter to target distance (left), Receiver to target distance (right)
0-7803-9050-4/05/$20.00 ©2005 IEEE.
IV.
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Fig. 7 shows the solutions found for the Digital Elevation Model example of Fig.6. A simulation similar to the one described in Table I has been performed by locating the same number of targets at the same (X,Y)-coordinate positions but using the current DEM to set the Z-coordinate. We observe that non-linearity’s arise in the processing algorithm, making much more difficult the precise focusing of such bistatic images by means of available efficient processing algorithms.
the fact that contrary to the Monostatic case, the topography of the illuminated scene must be known a priori and therefore algorithms that lie on flat-earth assumptions may found not useful when working with real images. Future works will be centered on the characterization of the algorithm as a function of some input parameters, such as the Digital Elevation Model accuracy and resolution, the precise knowledge of the transmitter trajectory. By placing several receivers forming a baseline we can obtain single-pass high coherent inteferometric bistatic images. This possibility and its usefulness are going to be deeply analyzed.
Figure 7. Bistatic Digital Elevation Model topography evaluation. Transmitter to target distance (left), Receiver to target distance (right)
(c)
(b)
(d)
Figure 8. Interpolated IRF for all five targets of simulation: (a) range cut (b) azimuth cut (c) phase (d) two-dimensional view
V. RESULTS The presented back-projection algorithm produces very accurate bistatic focused images both in phase and in amplitude, at the expense of a large computational burden. The algorithm has been validated using simulated scenes. For instance, Fig. 8 shows the focused interpolated impulse response of the five simulated corner reflectors of the DEM topography solution example. A hardware prototype is being currently built to acquire bistatic images working at C-Band in order to validate the algorithm with real data.
ACKNOWLEDGMENT This work has been financed by the Spanish MCYT and EU FEDER funds under project TIC2002-04451-C02-01, and by the Catalan Commission for Research (CIRIT-AGAUR). REFERENCES [1]
VI. CONCLUSIONS AND FUTURE WORK For a bistatic interferometry system that uses a satellite, such as Envisat or another non-cooperative transmitter, we have a margin of 35 days to wait for the next image and this algorithm can provide very accurate results for subsidence detections. For other systems such as unmanned flight vehicles in hostile environments using the cooperative transmitted signal emitted from a transmitter in a sanctuary, the reduced computational costs of much less precise but efficient algorithms will be of course determinant.
[2]
The authors have shown that fixed-receiver bistatic systems have some limitations in terms of efficient processing due to
[6]
0-7803-9050-4/05/$20.00 ©2005 IEEE.
(a)
1059
[3]
[4]
[5]
J.Sanz-Marcos,, J.J. Mallorqui: “A Bistatic Simulator and Processor”;Proceedings of EUSAR, 2004. J. Sanz-Marcos, J,J. Mallorquí, A. Broquetas: “Bistatic parasitic SAR processor evaluation”, Proceeding of IGARSS04, 2004. F. H. Wong, and T. S. Yeo, “New Applications of Nonlinear Chirp Scaling in SAR Data Processing," in IEEE Trans. Geosci. Remote Sensing, vol. 39, pp. 946-953, May 2001. Y.L. Neo, F. H. Wong, I. Cumming: “Focusing Bistatic SAR Images using Non-Linear Chirp Scaling”, Proceedings of RADAR 2004 International Conference on Radar Systems, 2004. M. Soumekh, “Bistatic synthetic aperture radar inversion with application in dynamic object imaging,” in IEEE Trans. on Signal Processing, vol. 39, pp.2044-2055, Sep 1991. Y. Ding, D. C. Munson Jr., “Fast Back Projection Algorithm for Bistatic SAR Imaging," IEEE ICIP, vol II, pp 449-452. Sep 2002.
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processor;
SAR;
back-
There are some limitations of course, for example the reduced size of the image, synchronization issues, etc.
Figure 1. Bistatic system geometry with static reveicer (left) and bistatic ambiguity for two targets at the same bistatic distance (right)
Consider the geometry depicted in Fig. 1a, where instead of the usual cylindrical coordinate system, a Cartesian coordinate system has been used to define transmitter, receiver and target positions. The starting expression for the impulse response of a given target as a function of its geometry is: target : (R1 , x, h ) → g (t az , t rg ) = 1 2π = p t rg − R (t az ) exp − j R (t az ) wT (x − vt az , r1 )wR c λ R (t az ) = R12 + (x − xT − vtaz ) + 2
(1)
R2
2 2 2 + (x − x R ) + ( y (R1 , h ) − y R ) + (h − hR )
y (R1 , h ) = yT + R12 − (h − hT )
2
I.
INTRODUCTION TO BISTATIC FIXED RECEIVER SAR
This paper continues the research started in [1] and [2] aimed at the development of a focusing algorithm for the bistatic SAR. In this case, the transmitter follows a rectilinear trajectory, while the receiver remains in a fixed position at the top of a high tower or mountain looking down to the illuminated scene. This configuration offers some challenging possibilities as current existing SAR satellites might be used as non-cooperative transmitters, safely locating the expensive hardware of the receiver at the ground. In the same way, no downlink capability is needed and more signal to noise ratio is available due to the fact that the emitted echo must not come back to the transmitter. Such a system could perform interferometric applications by placing several receiving subsystems at slightly different positions allowing for instance urban monitoring for subsidence detection and prediction.
0-7803-9050-4/05/$20.00 ©2005 IEEE.
1056
It is very important to notice that the range history of a point target R(taz) does not depend any more only on the zeroDoppler distance R1 and the relative distance from the target to the transmitter (x-xT), but also on the absolute distance to the receiver. Therefore, as can be observed in Fig. 1b, there is a range ambiguity that does not exist in the Monostatic case, which is crucial for the understanding of this paper: that is, two or more targets located at different positions can have the same range delay at zero-Doppler but will have different range histories (curvature). In the following sections we will present the modifications that have been added to the Monostatic backpropagation algorithm in order to precisely focus bistatic images for two situations: flat-earth and DEM topography.
1056
II.
BACK-PROJECTION ALGORITHM
An efficient algorithm [3] based on non-linear chirp scaling (NLCS) already exists for the flat-earth bistatic fixed receiver case. Even though it has been extended in [4] to compensate the Quadratic Range-Cell Migration, it has not deeply studied yet. The effect of the approximations used and the limitations in the geometry and instrument parameters has not been shown. In the same way it does not consider in any way topography issues which clearly limit it usefulness. However, the approach taken by the NLCS Algorithm is very interesting in terms of a trade-off between efficiency and precision and is being studied and analyzed.
It can also be noticed in Eq (2) that for each point of the output image it is required to know the values R1 and R2. Depending on the scenario, different solutions may arise. This is considered in the following two sections. Fig. 4 shows the diagram of the proposed algorithm. First of all it is necessary to calculate R1 and R2 values which are related to the topography of the illuminated scene, and then range-compression is performed. In order to reduce the number of interpolations, the algorithm is re-arranged to do all the interpolations of the same line at the same time.
Back-propagation algorithms are the most precise but imply large computational burden. The back-projection algorithm for the bistatic case [5] relies on performing, for each point of the final image, a 1D-convolution of the interpolated rangecompressed data. The computational requirements have been reduced in [6]. As it is shown in the below Figure, for each target of the output image, we must follow the range-curvature that would suffer that target and interpolate the input raw image to obtain a one-dimensional image (sequence) of the propagated range-compressed pulse.
Figure 4. Bistatic back-propagation algorithm diagram
III. Figure 2. Interpolation of range-compressed raw data
The interpolated sequence is then convoluted with the expected value (2) to compress and focus in the cross-range (azimuth) direction: 2π 2 (2) g I (t az ) = exp − j R12 + (x − xT − vt az ) + R2 λ Fig. 3 represents the core of the algorithm in pseudo-code. It can be noticed that the term R,(i,r)+R2(i,r) is used to maintain the propagation phase in order to allow interferometric processing.
target Flat Earth : (x, y, z = 0) R = R1 + R2 R2 =
(3)
(x − xR )2 + ( y − yR )2 + hR2
By expanding Eq. (1) we can relate the Y-coordinate of a target with its zero-Doppler distance to the transmitter:
For each line a… prepareInterpolationLine(a) for each sample r… for each echo n…
y = yT + R12 − hT2
2
nv + R2 (i, r ) i = a + n → R = R12 (i, r ) + prf R out (i, j ) = out (i, j ) + getInterpolatedValue t = c 2π (R − R1 (i, r ) − R2 (i, r )) ·exp j λ
(4)
This substitution leads to the following awkward equation: 2 R22 = α + (R − R2 ) − h 2 − ∆ α = (x − xR )2 + hR2
2
(5)
∆ = yT − y R , hT = h
Figure 3. Pseudo-code back-propagation core
0-7803-9050-4/05/$20.00 ©2005 IEEE.
FLAT-EARTH TOPOGRAPHY SOLUTION
A SAR instrument is a device that measures the delay of a transmitted signal. For this bistatic configuration, the delay is caused by the path propagation (transmitter-target and targetreceiver). For flat-earth topography, we consider all targets to have Z-coordinate equal to zero. Therefore, we have the set of equations:
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1057
The simplified solution for the flat-earth topography model (solution has been simplified using Eq. (5) substitutions): R2 =
(
(
1 αR - R∆2 + R 3 - Rh 2 + 2(R 2 - ∆2 )
∆ - 2αR 2 - 2R 2h 2 - 2αh 2 + ∆4 - 2R 2 ∆2 +
(6)
The general solution must be locally evaluated for each point of the output image, since the range-delay varies in each sample and the distance from the target to the receiver varies in each line: + R 4 + α 2 + 2h 2∆2 + 2α∆2 + h
1 4 2
)
R = ctrg = R1 + R2 (x )
(7)
To test both the focusing algorithm and the solution for flattopography a bistatic scene has been simulated using the parameters shown in Table 1. TABLE I.
SIMULATION PARAMETERS
Instrument F0 5.3 GHz PRF 1600 Hz fs 18.9 MHz v 7.2 km/s L (antenna az. length) 10m τ (pulse duration) 37µs Chirp rate -4.18·1011 Hz/s 2048 Lines 1500 Samples
Geometry TX distance 840 km TX look angle 22 º RX distance 7000 m RX look angle 84 º xR, yR, yT, h 6222 m Targets 5 1.6 km ∆ azimuth 1.5 km ∆ range
DIGITAL ELEVATION MODEL SOLUTION
In the Monostatic case, the scene topography is not considered in the focusing algorithm as the measured range delay is related with the double of the target distance and with the observed range curvature. Topography is only used to project the compressed image, which is in slant–range, to the ground-range. As it has been shown in the previous sections, it is mandatory to know both R1 and R2 values to properly focus a bistatic image, and they clearly depend on the target height. This section is dedicated to the case where there is information of the elevation of the illuminated scene, i.e. for every ground point (x,y) we are able to obtain its height – Eq. (8). Normally we start from a facet mesh that is interpolated to enhance resolution when needed h = f DEM ( x, y )
(8)
The set of equations considering that targets do not lie in the ground is more complex and in order to provide the backprojection algorithm R1 and R2 values we must solve the following equation which does not have, obviously, an analytical solution: trg c = R1 + R2 = f ( y, x = constant ) =
(x − xT )2 + ( y − yT )2 + ( f DEM (x, y ) − hT )2 +
(9)
(x − xR ) + ( y − yR ) + ( f DEM (x, y ) − hR ) 2
Fig. 5 shows R1 and R2 values for a 2048 lines x 1500 samples scene in the orbital case (rectilinear trajectory approximation used) projected at the bistatic slant-range plane. We can notice that there are some regions where there is no solution for the equation system described in Eq. (3) what means that there is no geometrical position for that bistatic distance. These complex shapes represent a great challenge towards the development of precise and efficient focusing algorithms. The situation will get more complicated with real DEM topography.
2
2
The procedure followed to solve this equation is based on simple ray-tracing techniques as shown in the below figure. For each cross-range position of the transmitter, we get an interpolated section of the Digital Elevation Model forming a plane with the transmitting antenna. Then we sweep all the points in the DEM sections in order to find those points that coincide with slant-range (R1 + R2) samples. To reduce the computational requirements an approximate solution and boundaries is initially selected.
Figure 6. Digital Elevation Model section (top), ground range Ray tracing technique (middle) and output solution example (bottom).
Figure 5. Bistatic flat-earth model topography evaluation. Transmitter to target distance (left), Receiver to target distance (right)
0-7803-9050-4/05/$20.00 ©2005 IEEE.
IV.
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Fig. 7 shows the solutions found for the Digital Elevation Model example of Fig.6. A simulation similar to the one described in Table I has been performed by locating the same number of targets at the same (X,Y)-coordinate positions but using the current DEM to set the Z-coordinate. We observe that non-linearity’s arise in the processing algorithm, making much more difficult the precise focusing of such bistatic images by means of available efficient processing algorithms.
the fact that contrary to the Monostatic case, the topography of the illuminated scene must be known a priori and therefore algorithms that lie on flat-earth assumptions may found not useful when working with real images. Future works will be centered on the characterization of the algorithm as a function of some input parameters, such as the Digital Elevation Model accuracy and resolution, the precise knowledge of the transmitter trajectory. By placing several receivers forming a baseline we can obtain single-pass high coherent inteferometric bistatic images. This possibility and its usefulness are going to be deeply analyzed.
Figure 7. Bistatic Digital Elevation Model topography evaluation. Transmitter to target distance (left), Receiver to target distance (right)
(c)
(b)
(d)
Figure 8. Interpolated IRF for all five targets of simulation: (a) range cut (b) azimuth cut (c) phase (d) two-dimensional view
V. RESULTS The presented back-projection algorithm produces very accurate bistatic focused images both in phase and in amplitude, at the expense of a large computational burden. The algorithm has been validated using simulated scenes. For instance, Fig. 8 shows the focused interpolated impulse response of the five simulated corner reflectors of the DEM topography solution example. A hardware prototype is being currently built to acquire bistatic images working at C-Band in order to validate the algorithm with real data.
ACKNOWLEDGMENT This work has been financed by the Spanish MCYT and EU FEDER funds under project TIC2002-04451-C02-01, and by the Catalan Commission for Research (CIRIT-AGAUR). REFERENCES [1]
VI. CONCLUSIONS AND FUTURE WORK For a bistatic interferometry system that uses a satellite, such as Envisat or another non-cooperative transmitter, we have a margin of 35 days to wait for the next image and this algorithm can provide very accurate results for subsidence detections. For other systems such as unmanned flight vehicles in hostile environments using the cooperative transmitted signal emitted from a transmitter in a sanctuary, the reduced computational costs of much less precise but efficient algorithms will be of course determinant.
[2]
The authors have shown that fixed-receiver bistatic systems have some limitations in terms of efficient processing due to
[6]
0-7803-9050-4/05/$20.00 ©2005 IEEE.
(a)
1059
[3]
[4]
[5]
J.Sanz-Marcos,, J.J. Mallorqui: “A Bistatic Simulator and Processor”;Proceedings of EUSAR, 2004. J. Sanz-Marcos, J,J. Mallorquí, A. Broquetas: “Bistatic parasitic SAR processor evaluation”, Proceeding of IGARSS04, 2004. F. H. Wong, and T. S. Yeo, “New Applications of Nonlinear Chirp Scaling in SAR Data Processing," in IEEE Trans. Geosci. Remote Sensing, vol. 39, pp. 946-953, May 2001. Y.L. Neo, F. H. Wong, I. Cumming: “Focusing Bistatic SAR Images using Non-Linear Chirp Scaling”, Proceedings of RADAR 2004 International Conference on Radar Systems, 2004. M. Soumekh, “Bistatic synthetic aperture radar inversion with application in dynamic object imaging,” in IEEE Trans. on Signal Processing, vol. 39, pp.2044-2055, Sep 1991. Y. Ding, D. C. Munson Jr., “Fast Back Projection Algorithm for Bistatic SAR Imaging," IEEE ICIP, vol II, pp 449-452. Sep 2002.
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