A Fourth-Order Imaging Algorithm for Spaceborne ... - Semantic Scholar
A Fourth-Order Imaging Algorithm for Spaceborne Bistatic SAR Hua Zhong and Xingzhao Liu Department of Electronic Engineering Shanghai Jiaotong University Shanghai, China [email protected] [email protected] Abstract—Eldhuset’s research, the fourth-order Extended Exact Transfer Function (EETF4), indicates that the range history of a high spatial resolution spaceborne SAR can be modeled by a fourth-order Taylor expansion in azimuth time. In this paper, this point of view is introduced in bistatic SAR, and the transfer functions in the 2-D frequency and the range-Doppler domains are derived. Using this method, an efficient bistatic focusing solution based on squinted mode chirp scaling (CS) algorithm which accommodates both tandem and TI case is developed. Quantitative analysis and simulations are also provided.
In next section, the geometry of bistatic SAR and the parameters of “bistatic grade” are described. In section III, a fourth-order expansion model of bistatic SAR and transformation method are described in detail. In section IV, a modified bistatic CS algorithm with range-walk removal [7] is proposed as the application of the new method. The limitation of this approach is summarized in section V. Some simulation results are presented in Section VI and conclusion is in the section VII. II.
Keywords-EETF4; bistatic SAR; transfer function; focusing
I. INTRODUCTION Bistatic SAR uses separated transmitter and receiver flying on different platforms. There are several specific advantages using bistatic SAR, like flexibility, reduced vulnerability for military applications, forward looking SAR imaging, etc. However, the bistatic configuration increased processing complexity as well. Reference [1] presents a method of bistatic focusing for the tandem configuration that transmitter and receiver are following each other on the same track with equal velocities. Similar method is described in [2], [3] for Translationally Invariant (TI) and even General Case (GC). The raw data of bistatic SAR are looked like as two components, bistatic deformation and quasi-monstatic terms. Correspondingly, the processing of bistatic data is divided into two individual steps, the compensation of bistatic deformation and focusing quasimonostatic data by any traditional processor. Another approach for solving the bistatic problem are offered in [4], [5]. Modified monostatic processors like: Backprojection, Omega-k processors have been employed for bistatic data. A common disadvantage in the approaches mentioned above is lack of computational efficiency. We try to develop efficient algorithms for bistatic SAR based on Standard Doppler focusing methods, like: range-Doppler, chirp scaling (CS) algorithms. Standard Doppler focusing methods use analytically derived transfer functions in the 2-D frequency and range-Doppler domains. However, the transfer functions cannot be easily computed for bistatic case and no stationary point can be obtained analytically. In this paper, a fourth-order expansion model [6] of the bistatic phase history is introduced and the corresponding approximate transformation method is presented as well.
MODELING THE BISTATIC PROBLEM
The discussed bistatic configurations in this paper are tandem and TI. Because of tandem is a specific case of TI, we concentrate the study on TI case. The geometry of TI case is shown in Fig. 1. The transmitter and receiver are moving in parallel tracks with different velocities. t denotes azimuth time, t0T and t0R are the azimuth time instants when transmitter and receiver at closest distance to the point target, respectively. The indexes “T” and “R” denoted transmitter and receiver values. A more detailed description is presented in [2]. Z
vR
( RR , t)
(R0R , t0R )
( R0T , t0T )
vT
( RT , t ) X Y Point Target Figure 1. Bistatic geometry
The “bistatic grade” can be determined by a0 = t0T − t0 R , a1 = vT / vR ,
(1)
a2 = R0T / R0 R .
In (1), a0 expresses the azimuth time difference between the azimuth time of the closest distance of transmitter and receiver.
0-7803-9510-7/06/$20.00 (c) 2006 IEEE
a1 is
the velocity ratio of transmitter and receiver, which is first presented to describe the case of different velocities. a2 gives the ratio of slant ranges at the closest distance. III.
TRANSGER FUNCTION IN THE 2-D FREQUENCY DOMAIN
The bistatic phase history is the sum of the individual phase histories of transmitter and receiver and the point response in time domain can be expressed as follows: 2 R ( t ) 2π pp (τ , t ; r ) = A (τ , t ) ⋅ exp − jKπ τ − bi ⋅ exp − j Rbi ( t ) .(2) λ c
In (2), τ and t are range time and azimuth time respectively , K is the chirp rate of the transmitted signal, c is the light speed and λ is the wave length. A (τ , t ) is a function responsible for all the variation of amplitude. The function of Rbi ( t ) is the bistatic range history, expressed as: Rbi ( t ) = R02T + vT2 ⋅ (t + t0T )2 + R02R + vR2 ⋅ (t + t0 R )2
,
(3) t0T = R0T ⋅ tan (θT ) / vT , t0 R = R0R ⋅ tan (θ R ) / v R . (4) Where R0T and R0R are the closest distance for transmitter and receiver, respectively, vT and vR are the speed of platforms, θT and θ R denote the squint angles. The transfer function of (2) in the 2-D frequency domain can’t be easily calculated and no stationary point can be obtained analytically. To get the bistatic focussing solutions, several approximations must be used. The first one is that the range history is approximated by a fourth-order Taylor expansion in azimuth time t , Rbi ( t ) ≈ Rbi 4 ( t ) = c0 R0T + c1vT t + c2
vT2 t 2 v3 t 3 v4 t 4 + c3 T2 + c4 T3 R0T R0T R0T
. (5)
Where the c-coefficients are given by c0 = c1 = c2 =
sec (θ R ) a2
c3 = −
+
cos3 (θT ) , 2
a22 cos 4 (θ R ) sin (θ R ) 2a13
cos 4 (θT ) sin (θT ) − , 2
(
(6)
)
3 7 2 1 a2 cos (θ R ) −1 + 4 tan (θ R ) + c4 = 4 8 a1
(
Substitute the cubic stationary point (5) to calculate the transfer function in the 2-D frequency domain. The resultant form of the expansion of PP (⋅) can be easily expressed as a series in range frequency, and the transfer function in rangeDoppler domain can be computed. In the next section, a modified CS algorithm for bistatic SAR is described using the transformation method mentioned above. IV. BI-STATIC CHIRP SCALING ALGORITHM Here, we select a CS algorithm with the range-walk removal [7] as the application to show how a monostatic processor is modified to accommodate the bistatic SAR data. In the viewpoint of range-walk removal, the range migration can be divided into two parts, the linear part and the non-linear part. As the range-walk is independent of range and uniform to the whole scene, it can be easily removed by a shear operation in the time domain. With the range-walk removed, the residual range migration is much small, the coupling between range and azimuth and difference of azimuth signal at various range cells are also reduced. All of those make the squinted SAR data easy to be processed. A more detailed description is presented in [7]. The range-walk can be removed by shearing the image with the slope of −c1 ⋅ vT . With the range-walk removed and amplitude variation function omitted, (2) can be rewritten as: 2 R ( t ) − c1 ⋅ vT ⋅ t ppwr (τ , t; r ) = exp − jKπ τ − bi 4 c . 2π R ( t ) ⋅ exp − j λ bi 4
(8)
Using principle of stationary point, the range Fourier transform of (8) can then be written
To obtain transfer function in the 2-D frequency domain, we apply the method mentioned in the section III. And then, the transfer function can be extended up to the second order term of the range frequency and expressed as:
sin (θ R ) + sin (θT ) , a1 2a12
(7) * tbi 3 in
f2 c ⋅ v ⋅ t f 1 Pp ( f , t; r ) = exp jπ − j 2π Rbi 4 ( t ) + + j 2π 1 T f .(9) K c c λ
+ sec (θT ) ,
a2 cos3 (θ R )
* * tbi 3 ≈ tbi 4 .
f2 PP ( f , f a ; R0T ) = exp jπ − j 2πτ d f + j 2π R0T Y , Km
where f and
f a is the azimuth
(10)
and range frequency and
1 1 = + B ⋅ R0T , Km K B = B1 + B2 ⋅ c4 ,
)
1 cos 7 (θT ) −1 + 4 tan 2 (θT ) . 8
The transfer function in the 2-D frequency domain PP (⋅) can be calculated by using the approach described in [6]. However, the expansion of PP (⋅) in range frequency is too complicated because of the quartic time term in (5). To simplify the further derivation, the second approximation is applied to the stationary point, considering that the quartic time term is small enough that the stationary point submits to the following approximate expression,
c0 ⋅ R0T X, c X = X1 + X 2 ⋅ c4 ,
τd =
(11)
Y = Y1 + Y2 ⋅ c4 .
Where B , X and Y , all of them can be seen as the sum of two components. B1 , X1 and Y1 respond to range component of c0 , c1 , c2 , c3 , and B2 , X 2 and Y2 respond to that of c4 . B1 and B2 can be calculated as follows:
0-7803-9510-7/06/$20.00 (c) 2006 IEEE
B1 =
B2 = −
(
X2 =
(
9 A3 / 2c 2c32vT2 X
Y2 =
(12)
)) λ ,
(
)
9c32 f f2 + 3c22c3 ⋅ f f vT + A 2c23 + 27c0c32 vT3 / 2 − 2c24vT2 27c0 Ac32vT3 / 2
(
)
Azimuth FFT
,
(
2c2 vT 9c32 f f2 + 6c22c3 ⋅ f f vT − 4c24 vT2 + A −9c32 f f2 + 2c24vT2 81c0 Y
Ac34vT2
where A and
ff
(
),
(13) chirp scaling
is calculated as:
(
9c2c3 ⋅ f f vT + 2 Ac22vT3 / 2 − 2c23vT2 − 3c3 9c0c3vT2 + 2 f f −9c32 f f2 + 8c23
27c32vT2 λ
)
AvT
(
81c34vT2 λ
Range FFT
),
A − c2 vT vT3 / 2 − 12c2c3 ⋅ f f −2c2vT + AvT
),
Range IFFT
A = −3c3 f f + c22vT , f f = c1 ⋅ vT + f a ⋅ λ.
Note that an important modification to the monastatic CS algorithm is the correspondence relationship of bistatic delay range to slant range of image. In the bistatic case, R0T R0 R + , cos (θT ) cos (θ R )
R0 R , u0 can
blx u0 = a2 1 + R02Tref − HT2
R0R can
be
(17)
R0 R R0 Rref + u0 ⋅ ∆R,
is the reference range of
be calculated
,
(18)
where bl x is the across-track baseline between transmitter and receiver, R0Tref is the reference range of R0T , HT is the height of transmitter and R0T = R0Tref + ∆R, R0T
can be calculated by
R0T = R0Tref + ∆R = R0Tref +
Image Shearing
Final Image Figure 2. The flow diagram of the bistatic CS algorithm
V. LIMITATIONS The proposed method of computing transfer function based on four-order Taylor expansion of range history and corresponding standard Doppler imaging algorithms for bistatic SAR data is effective and computational efficient. The limitation is that this approach is lack of flexibility. First, because of the bistatic range history of GC is hardly expressed by (5), there are difficulties to apply this approach to processing bistatic data of GC. Secondly, although we use several methods to simplify the procedure of derivation, the algorithm is still complicated. Overcoming this limitation is the emphasis of our research in the future work.
(19)
R0 Rref = R0Tref / a2 .
Therefore,
Azimuth IFFT
(16)
we select R0T as the slant range of bistatic image. So, calculated as follows: R0 Rref
azimuth focus and phase correction
(15)
The following procedure is the same as that of classic CS algorithm. A shear operation may be employed to recover the geometry relationship of the image.
where as:
bulk RCMC and range focus
(14)
can be expressed as follows:
RT + RR =
Bistatic SAR data in time domain Range-Walk Removal
can be calculated by:
and then the components of Y1 =
,
2 A ⋅ c 2vT3 / 2
f f2 2 A3 / 2 + c2 vT 9c3 ⋅ f f − 2c22vT
and the components of X1 =
ff 2λ
RT + RR − R0Tref ⋅ c0 u0 1 cos (θ ) + cos (θ ) T R
(20)
The flow chat of the bistatic CS algorithm with range-walk removal is illustrated by Fig. 2.
VI.
SIMULATION
Some simulation results are presented in this section to show the performance of the bistatic CS algorithm with rangeremoval in processing bistatic SAR data. Three point targets are placed in the scene at the same azimuth position, and the distance between the center point and the two at each side in slant range direction is 20 km.
0-7803-9510-7/06/$20.00 (c) 2006 IEEE
A. Tanden configuration In the first simulation, the bistatic configuration is tandem case. The parameters used are listed in Table I.
Velocity Transmitter Baseline along track Baseline across track
of of of
-20
-40
-60
7,450 m/s
a0
48.04 s
325.31 km
a1
1.0
0.0 km
a2
1.0
dB
-20
dB
-20 -40
75 MHz
-40
-60
Range Direction
-60
Range Direction
Range Direction
0
0
0
-20
-20
-20
-40
-40
-60
The simulation results of the three points are shown in Fig. 3 and Table III (N-Near; C-Center; F-Far).
Range Direction 0
dB
4,100 Hz
0.24 m 33 us
Range Direction 0
dB
PRF
Wave Length Pulse Width Sampling Frequency
Range Direction 0
dB
3.1994 s 30 MHz/us
Far Azimuth Direction
Azimuth Direction
Azimuth Direction
PARAMETERS OF SPACEBORNE TANDEM CONFIGURATION
Aperture Time Chirp Rate
Center
dB
TABLE I.
Near
-40
-60
Azimuth Direction
-60
Azimuth Direction
Azimuth Direction
Figure 4. Focusing results of TI configuration
Range Direction
Far Azimuth Direction
Azimuth Direction
Center
Azimuth Direction
Near
Range Direction
0
0
0
-20
-20
dB
dB
dB
N -40
-40
-60
-40
-60
Range Direction
Range Direction
-40
-60
F
dB
0 -20
dB
0 -20
dB
0 -20 -40
C
-60
Range Direction
-60
Azimuth Direction
Azimuth Direction
Figure 3. Focusing results of tandem configuration
B. Tranlationary Invariant Configuration In the second simulation, the bistatic configuration is TI case. The along-track distance and the across-track distance between both platforms are 125.31 km and 35.29 km, respectively, which is a usual configuration in constellation SAR system such as Cartwheel. The parameters used are listed in Table II. TABLE II.
PARAMETERS OF SPACEBORNE TI CONFIGURATION
Aperture Time Chirp Rate
3.4794 s 30M Hz/us
PRF Velocity Transmitter Baseline along track Baseline across track
4,000 Hz of of of
Range Resolution (m)
Azimuth Resolution (m)
Range PSLR (dB)
Azimuth PSLR (dB)
tandem
5.0533
5.0649
-13.1455
-13.2623
TI
5.0248
5.5351
-13.3362
-13.2400
tandem
5.0488
5.0650
-13.1479
-13.2710
TI
5.0413
5.5309
-13.2585
-13.2629
tandem
4.9807
5.0641
-13.7538
-13.2503
TI
4.9833
5.5367
-13.7493
-13.2231
-40
-60
Azimuth Direction
SIMULATION RESULTS
Bistatic Case
Range Direction
-20
TABLE III.
Wave Length Pulse Width Sampling Frequency
0.24 m 33 us
VII. CONCLUSIONS From the analysis and simulation results above, the new bistatic SAR algorithm and method proposed in this paper is effective and computational efficient. Additional, this algorithm can process monostatic, tandem, and TI data without any modification. REFERENCES [1]
[2]
[3] [4]
75 MHz [5]
7,450 m/s
a0
55.48 s
357.86 km
a1
1.16
[6]
121.19 km
a2
0.91
[7]
The simulation results of the three points are shown in Fig. 4 and Table III.
[8]
D. D’Aria, A. Monti Guarnieri, F. Rocca, “Focusing bistatic synthetic aperture radar using dip move out”, IEEE Trans. Geosci. Remote Sensing, vol. 42, pp.1362-1376, July 2004 O. Loffeld, H. Nies, V. Peters, S. Knedlik, “Models and useful relations for bistatic SAR processing”, IEEE Trans. Geosci. Remote Sensing, vol. 42, pp. 2031-2038, October 2004 K. Natroshvili, O. Loffeld, H. Nies, A. Ortiz, “First steps to bistatic focusing”, Proc. IGARSS 2005, Seoul, Korea, Jun, 2005 J. H. Ender, I. Walterscheid, A. Brenner, “New aspects of bistatic SAR: processing and experiments”, Proc, IGARSS 2004, Anchorage, U.S, Sep, 2004 J. H. Ender, “Bistatic SAR Processing”, EUSAR 2004, Ulm, Germany, may, 2004 G. W. Davidson, “A chirp-scaling approach for processing squint mode SAR data”, IEEE Trans on AES, vol. 32, pp. 121-133. Jan 1996 K. Wang, X. Liu, “Squint spotlight SAR imaging by sub-band combination and range-walk removal”, Proc, IGARSS 2004, Anchorage, U.S, Sep, 2004 K. Eldhuset, “A new fourth-order processing algorithm for spaceborne SAR”, IEEE trans on AES, vol. 34, pp. 824-835, July 1998
0-7803-9510-7/06/$20.00 (c) 2006 IEEE
In next section, the geometry of bistatic SAR and the parameters of “bistatic grade” are described. In section III, a fourth-order expansion model of bistatic SAR and transformation method are described in detail. In section IV, a modified bistatic CS algorithm with range-walk removal [7] is proposed as the application of the new method. The limitation of this approach is summarized in section V. Some simulation results are presented in Section VI and conclusion is in the section VII. II.
Keywords-EETF4; bistatic SAR; transfer function; focusing
I. INTRODUCTION Bistatic SAR uses separated transmitter and receiver flying on different platforms. There are several specific advantages using bistatic SAR, like flexibility, reduced vulnerability for military applications, forward looking SAR imaging, etc. However, the bistatic configuration increased processing complexity as well. Reference [1] presents a method of bistatic focusing for the tandem configuration that transmitter and receiver are following each other on the same track with equal velocities. Similar method is described in [2], [3] for Translationally Invariant (TI) and even General Case (GC). The raw data of bistatic SAR are looked like as two components, bistatic deformation and quasi-monstatic terms. Correspondingly, the processing of bistatic data is divided into two individual steps, the compensation of bistatic deformation and focusing quasimonostatic data by any traditional processor. Another approach for solving the bistatic problem are offered in [4], [5]. Modified monostatic processors like: Backprojection, Omega-k processors have been employed for bistatic data. A common disadvantage in the approaches mentioned above is lack of computational efficiency. We try to develop efficient algorithms for bistatic SAR based on Standard Doppler focusing methods, like: range-Doppler, chirp scaling (CS) algorithms. Standard Doppler focusing methods use analytically derived transfer functions in the 2-D frequency and range-Doppler domains. However, the transfer functions cannot be easily computed for bistatic case and no stationary point can be obtained analytically. In this paper, a fourth-order expansion model [6] of the bistatic phase history is introduced and the corresponding approximate transformation method is presented as well.
MODELING THE BISTATIC PROBLEM
The discussed bistatic configurations in this paper are tandem and TI. Because of tandem is a specific case of TI, we concentrate the study on TI case. The geometry of TI case is shown in Fig. 1. The transmitter and receiver are moving in parallel tracks with different velocities. t denotes azimuth time, t0T and t0R are the azimuth time instants when transmitter and receiver at closest distance to the point target, respectively. The indexes “T” and “R” denoted transmitter and receiver values. A more detailed description is presented in [2]. Z
vR
( RR , t)
(R0R , t0R )
( R0T , t0T )
vT
( RT , t ) X Y Point Target Figure 1. Bistatic geometry
The “bistatic grade” can be determined by a0 = t0T − t0 R , a1 = vT / vR ,
(1)
a2 = R0T / R0 R .
In (1), a0 expresses the azimuth time difference between the azimuth time of the closest distance of transmitter and receiver.
0-7803-9510-7/06/$20.00 (c) 2006 IEEE
a1 is
the velocity ratio of transmitter and receiver, which is first presented to describe the case of different velocities. a2 gives the ratio of slant ranges at the closest distance. III.
TRANSGER FUNCTION IN THE 2-D FREQUENCY DOMAIN
The bistatic phase history is the sum of the individual phase histories of transmitter and receiver and the point response in time domain can be expressed as follows: 2 R ( t ) 2π pp (τ , t ; r ) = A (τ , t ) ⋅ exp − jKπ τ − bi ⋅ exp − j Rbi ( t ) .(2) λ c
In (2), τ and t are range time and azimuth time respectively , K is the chirp rate of the transmitted signal, c is the light speed and λ is the wave length. A (τ , t ) is a function responsible for all the variation of amplitude. The function of Rbi ( t ) is the bistatic range history, expressed as: Rbi ( t ) = R02T + vT2 ⋅ (t + t0T )2 + R02R + vR2 ⋅ (t + t0 R )2
,
(3) t0T = R0T ⋅ tan (θT ) / vT , t0 R = R0R ⋅ tan (θ R ) / v R . (4) Where R0T and R0R are the closest distance for transmitter and receiver, respectively, vT and vR are the speed of platforms, θT and θ R denote the squint angles. The transfer function of (2) in the 2-D frequency domain can’t be easily calculated and no stationary point can be obtained analytically. To get the bistatic focussing solutions, several approximations must be used. The first one is that the range history is approximated by a fourth-order Taylor expansion in azimuth time t , Rbi ( t ) ≈ Rbi 4 ( t ) = c0 R0T + c1vT t + c2
vT2 t 2 v3 t 3 v4 t 4 + c3 T2 + c4 T3 R0T R0T R0T
. (5)
Where the c-coefficients are given by c0 = c1 = c2 =
sec (θ R ) a2
c3 = −
+
cos3 (θT ) , 2
a22 cos 4 (θ R ) sin (θ R ) 2a13
cos 4 (θT ) sin (θT ) − , 2
(
(6)
)
3 7 2 1 a2 cos (θ R ) −1 + 4 tan (θ R ) + c4 = 4 8 a1
(
Substitute the cubic stationary point (5) to calculate the transfer function in the 2-D frequency domain. The resultant form of the expansion of PP (⋅) can be easily expressed as a series in range frequency, and the transfer function in rangeDoppler domain can be computed. In the next section, a modified CS algorithm for bistatic SAR is described using the transformation method mentioned above. IV. BI-STATIC CHIRP SCALING ALGORITHM Here, we select a CS algorithm with the range-walk removal [7] as the application to show how a monostatic processor is modified to accommodate the bistatic SAR data. In the viewpoint of range-walk removal, the range migration can be divided into two parts, the linear part and the non-linear part. As the range-walk is independent of range and uniform to the whole scene, it can be easily removed by a shear operation in the time domain. With the range-walk removed, the residual range migration is much small, the coupling between range and azimuth and difference of azimuth signal at various range cells are also reduced. All of those make the squinted SAR data easy to be processed. A more detailed description is presented in [7]. The range-walk can be removed by shearing the image with the slope of −c1 ⋅ vT . With the range-walk removed and amplitude variation function omitted, (2) can be rewritten as: 2 R ( t ) − c1 ⋅ vT ⋅ t ppwr (τ , t; r ) = exp − jKπ τ − bi 4 c . 2π R ( t ) ⋅ exp − j λ bi 4
(8)
Using principle of stationary point, the range Fourier transform of (8) can then be written
To obtain transfer function in the 2-D frequency domain, we apply the method mentioned in the section III. And then, the transfer function can be extended up to the second order term of the range frequency and expressed as:
sin (θ R ) + sin (θT ) , a1 2a12
(7) * tbi 3 in
f2 c ⋅ v ⋅ t f 1 Pp ( f , t; r ) = exp jπ − j 2π Rbi 4 ( t ) + + j 2π 1 T f .(9) K c c λ
+ sec (θT ) ,
a2 cos3 (θ R )
* * tbi 3 ≈ tbi 4 .
f2 PP ( f , f a ; R0T ) = exp jπ − j 2πτ d f + j 2π R0T Y , Km
where f and
f a is the azimuth
(10)
and range frequency and
1 1 = + B ⋅ R0T , Km K B = B1 + B2 ⋅ c4 ,
)
1 cos 7 (θT ) −1 + 4 tan 2 (θT ) . 8
The transfer function in the 2-D frequency domain PP (⋅) can be calculated by using the approach described in [6]. However, the expansion of PP (⋅) in range frequency is too complicated because of the quartic time term in (5). To simplify the further derivation, the second approximation is applied to the stationary point, considering that the quartic time term is small enough that the stationary point submits to the following approximate expression,
c0 ⋅ R0T X, c X = X1 + X 2 ⋅ c4 ,
τd =
(11)
Y = Y1 + Y2 ⋅ c4 .
Where B , X and Y , all of them can be seen as the sum of two components. B1 , X1 and Y1 respond to range component of c0 , c1 , c2 , c3 , and B2 , X 2 and Y2 respond to that of c4 . B1 and B2 can be calculated as follows:
0-7803-9510-7/06/$20.00 (c) 2006 IEEE
B1 =
B2 = −
(
X2 =
(
9 A3 / 2c 2c32vT2 X
Y2 =
(12)
)) λ ,
(
)
9c32 f f2 + 3c22c3 ⋅ f f vT + A 2c23 + 27c0c32 vT3 / 2 − 2c24vT2 27c0 Ac32vT3 / 2
(
)
Azimuth FFT
,
(
2c2 vT 9c32 f f2 + 6c22c3 ⋅ f f vT − 4c24 vT2 + A −9c32 f f2 + 2c24vT2 81c0 Y
Ac34vT2
where A and
ff
(
),
(13) chirp scaling
is calculated as:
(
9c2c3 ⋅ f f vT + 2 Ac22vT3 / 2 − 2c23vT2 − 3c3 9c0c3vT2 + 2 f f −9c32 f f2 + 8c23
27c32vT2 λ
)
AvT
(
81c34vT2 λ
Range FFT
),
A − c2 vT vT3 / 2 − 12c2c3 ⋅ f f −2c2vT + AvT
),
Range IFFT
A = −3c3 f f + c22vT , f f = c1 ⋅ vT + f a ⋅ λ.
Note that an important modification to the monastatic CS algorithm is the correspondence relationship of bistatic delay range to slant range of image. In the bistatic case, R0T R0 R + , cos (θT ) cos (θ R )
R0 R , u0 can
blx u0 = a2 1 + R02Tref − HT2
R0R can
be
(17)
R0 R R0 Rref + u0 ⋅ ∆R,
is the reference range of
be calculated
,
(18)
where bl x is the across-track baseline between transmitter and receiver, R0Tref is the reference range of R0T , HT is the height of transmitter and R0T = R0Tref + ∆R, R0T
can be calculated by
R0T = R0Tref + ∆R = R0Tref +
Image Shearing
Final Image Figure 2. The flow diagram of the bistatic CS algorithm
V. LIMITATIONS The proposed method of computing transfer function based on four-order Taylor expansion of range history and corresponding standard Doppler imaging algorithms for bistatic SAR data is effective and computational efficient. The limitation is that this approach is lack of flexibility. First, because of the bistatic range history of GC is hardly expressed by (5), there are difficulties to apply this approach to processing bistatic data of GC. Secondly, although we use several methods to simplify the procedure of derivation, the algorithm is still complicated. Overcoming this limitation is the emphasis of our research in the future work.
(19)
R0 Rref = R0Tref / a2 .
Therefore,
Azimuth IFFT
(16)
we select R0T as the slant range of bistatic image. So, calculated as follows: R0 Rref
azimuth focus and phase correction
(15)
The following procedure is the same as that of classic CS algorithm. A shear operation may be employed to recover the geometry relationship of the image.
where as:
bulk RCMC and range focus
(14)
can be expressed as follows:
RT + RR =
Bistatic SAR data in time domain Range-Walk Removal
can be calculated by:
and then the components of Y1 =
,
2 A ⋅ c 2vT3 / 2
f f2 2 A3 / 2 + c2 vT 9c3 ⋅ f f − 2c22vT
and the components of X1 =
ff 2λ
RT + RR − R0Tref ⋅ c0 u0 1 cos (θ ) + cos (θ ) T R
(20)
The flow chat of the bistatic CS algorithm with range-walk removal is illustrated by Fig. 2.
VI.
SIMULATION
Some simulation results are presented in this section to show the performance of the bistatic CS algorithm with rangeremoval in processing bistatic SAR data. Three point targets are placed in the scene at the same azimuth position, and the distance between the center point and the two at each side in slant range direction is 20 km.
0-7803-9510-7/06/$20.00 (c) 2006 IEEE
A. Tanden configuration In the first simulation, the bistatic configuration is tandem case. The parameters used are listed in Table I.
Velocity Transmitter Baseline along track Baseline across track
of of of
-20
-40
-60
7,450 m/s
a0
48.04 s
325.31 km
a1
1.0
0.0 km
a2
1.0
dB
-20
dB
-20 -40
75 MHz
-40
-60
Range Direction
-60
Range Direction
Range Direction
0
0
0
-20
-20
-20
-40
-40
-60
The simulation results of the three points are shown in Fig. 3 and Table III (N-Near; C-Center; F-Far).
Range Direction 0
dB
4,100 Hz
0.24 m 33 us
Range Direction 0
dB
PRF
Wave Length Pulse Width Sampling Frequency
Range Direction 0
dB
3.1994 s 30 MHz/us
Far Azimuth Direction
Azimuth Direction
Azimuth Direction
PARAMETERS OF SPACEBORNE TANDEM CONFIGURATION
Aperture Time Chirp Rate
Center
dB
TABLE I.
Near
-40
-60
Azimuth Direction
-60
Azimuth Direction
Azimuth Direction
Figure 4. Focusing results of TI configuration
Range Direction
Far Azimuth Direction
Azimuth Direction
Center
Azimuth Direction
Near
Range Direction
0
0
0
-20
-20
dB
dB
dB
N -40
-40
-60
-40
-60
Range Direction
Range Direction
-40
-60
F
dB
0 -20
dB
0 -20
dB
0 -20 -40
C
-60
Range Direction
-60
Azimuth Direction
Azimuth Direction
Figure 3. Focusing results of tandem configuration
B. Tranlationary Invariant Configuration In the second simulation, the bistatic configuration is TI case. The along-track distance and the across-track distance between both platforms are 125.31 km and 35.29 km, respectively, which is a usual configuration in constellation SAR system such as Cartwheel. The parameters used are listed in Table II. TABLE II.
PARAMETERS OF SPACEBORNE TI CONFIGURATION
Aperture Time Chirp Rate
3.4794 s 30M Hz/us
PRF Velocity Transmitter Baseline along track Baseline across track
4,000 Hz of of of
Range Resolution (m)
Azimuth Resolution (m)
Range PSLR (dB)
Azimuth PSLR (dB)
tandem
5.0533
5.0649
-13.1455
-13.2623
TI
5.0248
5.5351
-13.3362
-13.2400
tandem
5.0488
5.0650
-13.1479
-13.2710
TI
5.0413
5.5309
-13.2585
-13.2629
tandem
4.9807
5.0641
-13.7538
-13.2503
TI
4.9833
5.5367
-13.7493
-13.2231
-40
-60
Azimuth Direction
SIMULATION RESULTS
Bistatic Case
Range Direction
-20
TABLE III.
Wave Length Pulse Width Sampling Frequency
0.24 m 33 us
VII. CONCLUSIONS From the analysis and simulation results above, the new bistatic SAR algorithm and method proposed in this paper is effective and computational efficient. Additional, this algorithm can process monostatic, tandem, and TI data without any modification. REFERENCES [1]
[2]
[3] [4]
75 MHz [5]
7,450 m/s
a0
55.48 s
357.86 km
a1
1.16
[6]
121.19 km
a2
0.91
[7]
The simulation results of the three points are shown in Fig. 4 and Table III.
[8]
D. D’Aria, A. Monti Guarnieri, F. Rocca, “Focusing bistatic synthetic aperture radar using dip move out”, IEEE Trans. Geosci. Remote Sensing, vol. 42, pp.1362-1376, July 2004 O. Loffeld, H. Nies, V. Peters, S. Knedlik, “Models and useful relations for bistatic SAR processing”, IEEE Trans. Geosci. Remote Sensing, vol. 42, pp. 2031-2038, October 2004 K. Natroshvili, O. Loffeld, H. Nies, A. Ortiz, “First steps to bistatic focusing”, Proc. IGARSS 2005, Seoul, Korea, Jun, 2005 J. H. Ender, I. Walterscheid, A. Brenner, “New aspects of bistatic SAR: processing and experiments”, Proc, IGARSS 2004, Anchorage, U.S, Sep, 2004 J. H. Ender, “Bistatic SAR Processing”, EUSAR 2004, Ulm, Germany, may, 2004 G. W. Davidson, “A chirp-scaling approach for processing squint mode SAR data”, IEEE Trans on AES, vol. 32, pp. 121-133. Jan 1996 K. Wang, X. Liu, “Squint spotlight SAR imaging by sub-band combination and range-walk removal”, Proc, IGARSS 2004, Anchorage, U.S, Sep, 2004 K. Eldhuset, “A new fourth-order processing algorithm for spaceborne SAR”, IEEE trans on AES, vol. 34, pp. 824-835, July 1998
0-7803-9510-7/06/$20.00 (c) 2006 IEEE
