Omega-K Algorithm for Spaceborne Spotlight SAR Imaging
IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 9, NO. 3, MAY 2012
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Omega-K Algorithm for Spaceborne Spotlight SAR Imaging Hee-Sub Shin and Jong-Tae Lim
Abstract—Since the spaceborne spotlight synthetic aperture radar (SAR) has the long slant range and the high altitude, it generates the curved flight path due to the orbit curvature effect and the time delay due to the long round-trip time of the chirp pulse. Thus, to obtain the SAR image with high quality, we extend the omega-K algorithm. First, we convert the curved path into the straight-line path based on the resampling and the rotation of the transmitted-pulse path and the received pulse path. Then, using the modified matched filter and the modified Stolt interpolation based on the principle of the stationary phase, we compensate the phase error induced by the start–stop assumption of the omega-K algorithm. Index Terms—Omega-K algorithm, spaceborne spotlight synthetic aperture radar (SAR).
I. I NTRODUCTION
C
URRENT technologies in the spaceborne spotlight synthetic aperture radar (SAR) demand the accurate compensation of the phase errors to obtain the SAR images with high resolution [1]. In classical SAR imaging, most of the papers use the geometry model based on the straight-line flight path and the start–stop assumption [2], [3]. That is, it is assumed that the aircraft travels along a straight line with a constant velocity, then the aircraft stops to transmit and receive, and the aircraft moves to its next position. In particular, the omega-K algorithm based on the Fourier transform analysis and the principle of the stationary phase (PSP) needs the straight-line flight path and the uniform sampling interval along the flight path [2]. However, in practice, as the platform altitude increases in the SAR system, it generates the curvature effect due to the curved orbit and breaks the start–stop assumption due to the time delay between the transmitted pulse and the received pulse. Moreover, the curvature effect generates the position variation of the platform, and the break of the start–stop assumption produces the velocity variation in the azimuthal direction. In particular, the yaw angle shifting in the position variation of the platform is very sensitive to the phase error, and the velocity variation in the azimuthal direction generates the Doppler shift. Thus, since it is difficult to obtain the hyperbolic range history and the stationary point in the PSP, it increases the phase errors. Therefore, some critical error factors such as the curved orbit and the delay between the
Manuscript received February 20, 2010; revised March 4, 2011 and August 1, 2011; accepted August 31, 2011. Date of publication October 21, 2011; date of current version March 7, 2012. This research was supported by Agency Defense Development through Radiowave Detection Research Center at Korea Advanced Institute of Science and Technology. H.-S. Shin is with Intelligence, Surveillance, and Reconnaissance Research Center, LIG Nex1, Yongin 449-910, Korea. J.-T. Lim is with the Department of Electrical Engineering and Computer Science, Korea Advanced Institute of Science and Technology, Daejeon 305 701, Korea (e-mail: [email protected]). Digital Object Identifier 10.1109/LGRS.2011.2168380
Fig. 1. Spaceborne spotlight SAR model.
transmitted pulse and the received pulse should be compensated carefully and correctly to obtain the high-precision spaceborne SAR image [1], [4]–[6]. To compensate the curved path and the phase error, several methods have been developed [4]–[7]. The method in [6] based on the straight-orbit approximation has been developed, and the compensation techniques such as the phase gradient autofocus and the range alignment have been developed to compensate the phase error induced by the start–stop assumption [5], [7]. However, they compensate the small phase error. Thus, extending the omega-K algorithm, we derive the solution for the spaceborne spotlight SAR imaging. After we convert the curved flight path into the straight-line flight path based on the resampling and rotation of the transmitted and received pulse paths, we compensate the error induced by the start–stop assumption in the frequency domain. II. S PACEBORNE S POTLIGHT SAR M ODEL As shown in Fig. 1, we consider a spaceborne spotlight SAR model. The SAR sensor travels a curved flight path during a synthetic aperture length L and moves at the constant velocity vh . To maintain uniform spatial sampling, the radar uses a fixed interpulse sample spacing Δxr = vh /PRF where PRF is the pulse repetition frequency. After the transformation between the ellipsoid coordinate and the Cartesian coordinate, let the positions of the transmitted pulse and the received pulse be (xte , yte , zte ) and (xre , yre , zre ), respectively. Then, from the transformation for the target coordinate, we obtain (xtp , ytp , ztp ) and (xrp , yrp , zrp ), respectively. Moreover, for the target coordinate, let a target position in the 2-D plane surface be (xt , yt ). In addition, we obtain the error induced by the start–stop assumption asΔxrp = vh · 2Rslant /c with the light speed c and Rslant = ys2 + zs2 . The errors induced by the curved orbit are (Δytp = ytp − ys , Δztp = ztp − zs ) and (Δyrp = yrp − ys , Δzrp = zrp − zs ), where the perigee is the center position of the synthetic aperture (0, ys , zs ) for the target coordinate. Let Rtp =
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2 (xtp − xt )2 + (ytp − yt )2 + ztp
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IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 9, NO. 3, MAY 2012
be the distance between the transmitter and the target when the pulse is transmitted at (xtp , ytp , ztp ), and let
2 be the distance beRrp = (xrp − xt )2 + (yrp − yt )2 + zrp tween the receiver and the target when the pulse is received at (xrp , yrp , zrp ). If we use the chirp pulse st (τ ) = 2 ej(2πfc τ +πγτ ) with the duration time Tp , the time variable τ , the center transmit frequency fc , and the chirp rate γ, we obtain the received signal with the reflectivity function of the target ρ(xt , yt ) as follows: τ − Rd /c j2πfc τ − Rcd jπγ τ − Rcd 2 e e sr (τ ) = ρ(·)rect Tp (1) where Rd = Rtp + Rrp [3].
III. M AIN R ESULTS To obtain the spaceborne spotlight SAR image, we convert the curved path into the straight-line path using the straight-line approximation. Then, at the frequency domain, we compensate the phase error induced by the start–stop assumption of the omega-K algorithm. Let the total number of pulses be N . First, 1 2 N we define xtp ∈ {x1tp , x2tp , . . . , xN tp }, ytp ∈ {ytp , ytp , . . . , ytp }, 1 2 N i i , ztp , . . . , ztp }, where (xitp , ytp , ztp ) is the posiand ztp ∈ {ztp tion of the ith transmitted pulse for i = 1, . . . , N . Analogously, i i for the received pulse, (xirp , yrp , zrp ) denotes the position of the ith received pulse. A. Conversion of the Curved Path Into a Straight-Line Path 1) Resampling of the Transmitted-Pulse Path in the Range Direction: First, as shown in Fig. 2(a), for i = 1, . . . , N , we introduce the following linear fit: i i ytp = ay xitp + dy + ei1 = y˜tp + ei1 , xitp < 0 (2) i i = −ay xitp + dy + ei1 = y˜tp + ei1 , xitp > 0 ytp where ay and dy are the positive constants. Let 1 2 N y˜tp ∈ {˜ ytp = ay x1tp + dy , y˜tp , . . . , y˜tp = −ay xN and tp + dy } 1 2 N e1 ∈ {e1 , e1 , . . . , e1 }. Then, y˜tp is the model value estimated from the linear fit, and e1 is the error value obtained from the linear fit. Therefore, to increase the accuracy of the trajectory approximation, we compensate the error value induced by the 2 ˜ tp = (xtp − xt )2 + (˜ linear fit. Then, let R ytp − yt )2 + ztp be the distance between the positions of the transmitted pulse by the linear fit and the target. If we perform the extended ˜ tp around y˜tp = ytp , we Taylor approximation (ETA) [8] of R obtain 2 ˜ tp (xtp − xt )2 + (ytp − ((m − 1)/m) e1 )2 + ztp R −(1/m)e1 {ytp − ((m − 1)/m) e1 } + 2 x2tp + (ytp − ((m − 1)/m) e1 )2 + ztp Rtp + Rc11
(3) m
where Rc11 = (1/m) k=1 (−e1 {ytp − ((m − k/m))e1 }/ 2 2 ) and m is the tuning xtp + (ytp − ((m − k/m))e1 )2 + ztp parameter in the ETA. Moreover, due to the curved effect, the
Fig. 2. Resampling in range direction. (a) Linear fit. (b) Resampling of transmitted pulse and received pulse.
pulse interval on the curved path is not constant with respect to xtp and ytp . Then, to obtain the constant interval of the azimuthal position and to consider the broadside mode in the transmitted-pulse path, as shown in Fig. 2(b), we resample the azimuthal position as follows: x ¯itp = x ¯1tp + (i − 1)Δ¯ xtp
(4)
where x ¯1tp = −L cos θy /2 and Δ¯ xtp = L cos θy /(N − 1) with θy = tan−1 ay . Hence, in the transmitted pulse, from (4), we obtain the range position of the resampled pulse as follows: i i y¯tp = ay x ¯itp + dy = y˜tp + Δei1 , x ¯itp < 0 (5) i i = −ay x ¯itp + dy = y˜tp + Δei1 , x ¯itp > 0 y¯tp xitp − xitp ) for x ¯itp < 0 and Δei1 = where Δei1 = ay (¯ i i i −ay (¯ xtp − xtp ) for x ¯tp > 0. Let x ¯tp ∈ {¯ x1tp , x ¯2tp , . . . , x ¯N tp }, 1 2 N 1 2 y¯tp ∈ {¯ ytp , y¯tp , . . . , y¯tp }, and Δe1 ∈ {Δe1 , Δe1 , . . . , ΔeN 1 }. Note that Δe1 is the change in the beam direction induced by the resampling in the range direction. Thus, to focus the beam, we compensate the error value of the beam direction. Then, 2 be the distance ¯ let Rtp = (¯ xtp − xt )2 + (¯ ytp − yt )2 + ztp between the positions of the transmitted pulse on the resampled ¯ tp around path and the target. If we perform the ETA of R x ¯tp = xtp and y¯tp = y˜tp , we obtain ¯ tp R ˜ tp + Rc R (6) 12 where Rc12 = (1/m) m k=1 (a1 {xtp + ((m − k/m))a1 } + b ytp + ((m − k/m))b1 }/ 1 {˜
2 ) (xtp + ((m − k/m))a1 )2 + (˜ ytp + ((m − k/m))b1 )2 + ztp with a1 = x ¯tp − xtp and b1 = y¯tp − y˜tp = Δe1 . Thus, from (3) and (6), we obtain
¯ tp Rtp + Rc R 1
(7)
where Rc1 = Rc11 + Rc12 . 2) Resampling of Received Pulse Path in Range Direction: As shown in Fig. 2(b), to obtain the constant interval in the path of the received pulse, we resample the received pulse. Thus, to decrease the resampled distance in the received pulse, we define the resampled azimuthal position of the received pulse as follows: x ¯irp = x ¯1rp + (i − 1)Δ¯ xrp
(8)
where x ¯1rp = x1rp and Δ¯ xrp = α¯ xtp with the positive constant α to adjust the length of the received pulse path. Then, using
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pulse on the shifted path and the target, i.e., Rtps = 2 . (xtps − xt )2 + (¯ ytp − yt )2 + ztp Similarly, we shift the resampled path of the received pulse as follows: i xrps = x ¯irp − (L/4 − L cos θy /4), x ¯irp < 0 (12) i i ¯rp + (L/4 − L cos θy /4), x ¯irp > 0. xrps = x Let xrps ∈ {x1rps , x2rps , . . . , xN rps }. If we perform the ETA of Rtps around xtps = x ¯tp , we obtain ¯ tp + Rc Rtps R (13) 3 m = (1/m) k=1 (a3 {¯ xtp + ((m − k/m))a3 }/
Fig. 3. Resampling. (a) Overlap in resampled path. (b) Shifting of resampled path. (c) Azimuthal positions of the transmitted pulse and the received pulse after rotation of shifted path. (d) Resampling in altitude direction.
the same linear fit in (2) and (5), we obtain i i y¯rp = ay x ¯irp + dy = y˜rp + Δei2 , i i i ¯rp + dy = y˜rp + Δei2 , y¯rp = −ay x
x ¯irp < 0 x ¯irp > 0
(9)
xirp − xirp ) for x ¯irp < 0 and Δei2 = where Δei2 = ay (¯ i i i 1 2 N −ay (¯ xrp − xrp ) for x ¯rp > 0. Let y˜rp ∈ {˜ yrp , y˜rp , . . . , y˜rp } 1 2 N and e2 ∈ {e2 , e2 , . . . , e2 }. Then, y˜rp is the model value estimated from the linear fit. Thus, we obtain e2 = yrp − y˜rp , i.e., the error value obtained from the linear fit in the received pulses. Let x ¯rp ∈ 1 2 N {¯ x1rp , x ¯2rp , . . . , x ¯N } and y ¯ ∈ {¯ y , y ¯ , . . . , y ¯ }. Then, rp rp rp rp rp 2 2 2 ¯ rp = (¯ xrp − xt ) + (¯ yrp − yt ) + zrp be the distance let R between the positions of the received pulse on the resampled path and the target. Thus, using the same method in the transmitted pulse, we obtain ¯ rp Rrp + Rc R 2
(10)
1 2 N where Rc2 = Rc21 + Rc22 . Let Δe2 ∈ 2 , Δe2 , . . . , Δe2 }. {Δe m Then, we show that Rc21 = (1/m) k=1 (−e2 {yrp − ((m −
2 ) and R k/m))e2 }/ x2rp +(yrp −((m − k/m))e2 )2 +zrp c22 = m (1/m) k=1 (a2 {xrp +((m−k/m))a2 }+b2 {yrp +((m−k/m)) 2 ) b2}/ (xrp +((m−k/m))a2 )2 +(yrp +((m−k/m))b2 )2 +zrp ¯rp − xrp and b2 = y¯rp − y˜rp = Δe2 . with a2 = x 3) Shifting of Resampled Path: To convert the curved path into the straight-line path in the range direction, we need to rotate the resampled path. However, as shown in Fig. 3(a), since the rotation of the linear path generates the overlap in the flight path, we need to shift the resampled path. As shown in Fig. 3(b), if we shift the resampled path of the transmitted pulse, we obtain the azimuthal position as follows: i xtps = x ¯itp − (L/4 − L cos θy /4), x ¯itp < 0 (11) i i ¯tp + (L/4 − L cos θy /4), x ¯itp > 0. xtps = x
Let xtps ∈ {x1tps , x2tps , . . . , xN tps }. Then, let Rtps be the distance between the positions of the transmitted
where Rc3 2 + z2 ) (¯ xtp + ((m − k/m))a3 )2 + y¯tp with a3 = tp xtps − x ¯tp . Analogously, let Rrps be the distance between the positions of the received pulse on the shifted path and the 2 . If we yrp − yt )2 + zrp target, i.e., Rrps = (xrps − xt )2 + (¯ perform the ETA of Rrps around xrps = x ¯rp , we obtain ¯ rp + Rc Rrps R (14) 4 = (1/m) m xrp + ((m − k/m))a4 }/ k=1 (a4 {¯
where Rc4 2 + z2 ) (¯ xrp + ((m − k/m))a4 )2 + y¯rp with a4 = rp ¯rp . xrps − x 4) Rotation of Resampled Path: As shown in Fig. 3(b), to obtain the straight-line path in the range direction, we rotate the shifted path with respect to the constant center position yac = (dy + (dy − L sin θy /2))/2 based on the maximum value and the minimum value in the linear fit. Then, as shown in Fig. 3(c), rotating the shifted path of the transmitted pulse, we define the azimuthal position of the transmitted pulse on the rotated path as follows: xia = −L/2 + (i − 1)Δxr .
(15)
In addition, rotating the shifted path of the received pulse, we define the azimuthal position of the received pulse on the rotated path as follows: xir = −Ld /2 + (i − 1)Δ¯ xr where Ld = −2(x1rp − (L/4 − L cos θy /4)/ cos θy ) Δ¯ xr = αΔxr . Thus, from (15) and (16), we obtain xir = αxia + β
(16) and (17)
where β = −Ld /2 + αL/2. Let xa ∈ {x1a , x2a , . . . , xN a } and xr ∈ {x1r , x2r , . . . , xN }. Then, let R be the distance between tpr r the positions of the transmitted pulse on the rotated path and
2 . If the target, i.e., Rtpr = (xa − xt )2 + (yac − yt )2 + ztp we perform the ETA of Rtpr around xa = xtps and yac = y¯tp , we obtain
Rtpr Rtps + Rc5 (18) where Rc5 = (1/m) m k=1 (a5 {xtps + ((m − k/m))a5 } + b ytp + ((m − k/m))b5 }/ 5 {¯
2 ) (xtps +((m−k/m))a5 )2 +(¯ ytp +((m − k/m))b5 )2 + ztp with a5 = xa − xtps and b5 = yac − y¯tp . Analogously, let
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2 Rrpr = (xr − xt )2 + (yac − yt )2 + zrp be the distance between the positions of the received pulse on the rotated path and the target. If we perform the ETA of Rrpr around xr = xrps and yac = y¯rp , we obtain Rrpr Rrps + Rc6 (19) m where Rc6 = (1/m) k=1 (a6 {xrps + ((m − k/m))a6 } + b yrp + ((m − k/m))b6 }/ 6 {¯
2 ) (xrps +((m−k/m))a6 )2 + (¯ yrp + ((m − k/m))b6 )2 + zrp with a6 = xr − xrps and b6 = yac − y¯rp . 5) Resampling in Altitude Direction: To convert the curved path into a straight-line path in the altitude direction, we introduce the following linear fit for the altitude position of the resampled flight path: i ztp az xia + dz , xia < 0 (20) i i ztp −az xa + dz , xia > 0
where az and dz are the positive constants. Then, as shown in Fig. 3(d), we move the altitude position of the resampled flight path to the constant center position zac = −1 (d z + (dz − L tan θz /2))/2 with θz = tan az . Let Ra = 2 2 2 (xa − xt ) + (yac − yt ) + zac be the distance between the positions of the transmitted pulse on the constant altitude path and the target. If we perform the ETA of Ra around zac = ztp , we obtain Ra Rtpr + Rc7 (21) where Rc7 = (1/m) m k=1 (a7 {ztp + ((m − 2 2 with k/m))a7 }/ xa + yac + (ztp + ((m − k/m))a7 )2 ) 2 a7 = zac − ztp . Let Rr = (xr − xt )2 + (yac − yt )2 + zac be the distance between the positions of the received pulse on the constant altitude path and the target. Analogously, if we perform the ETA of Rr around zac = zrp , we obtain (22) Rr Rrpr + Rc8 where Rc8 = (1/m) m k=1 (a8 {zrp + ((m − k/m))a8 }/ 2 + (z 2 x2r + yac rp + ((m − k/m))a8 ) ) with a8 = zac − zrp . Thus, from (21) and (22), we obtain Ra Rtp + Rc1 + Rc3 + Rc5 + Rc7 (23) Rr Rrp + Rc2 + Rc4 + Rc6 + Rc8 .
B. Mixing Stage To reconstruct the SAR image for the spaceborne spotlight SAR, we use the reference signal with the compensation term ΔR as follows: 2(Rs +ΔR) 2(Rs +ΔR) 2 j2πfc τ − jπγ τ − c c sref (n, τ ) = e e (24) 8 2 + z 2 and ΔR = − where Rs = yac ac k=1 Rck /2. Then, if we mix the received signal (1) with the complex conjugated signal of (24) and the range deskew scom (fτ ) = 2 e−j(πfτ /γ−4πΔRfτ /c) where fτ is the frequency variable,
whereas τ is the time variable [2], from (23), we obtain the resulting signal s(n, kr ; xa , yac , zac ) as follows: s(·) ρ(·)rect(·)e−jkr (Ra +Rr 2−Rs )
(25)
where kr = (4πγ/c)(fc /γ + τ − 2Rs /c). C. Fourier Transform Analysis If we perform the Fourier transform of s(·) with respect to xa , we obtain the phase function Φ(n, kx , kr ) as follows: 1 Φ(·) = − kr (xa − xt )2 + Rb2 + kr Rs 2 1 − kr (αxa + β − xt )2 + Rb2 − kx xa (26) 2 2 , and k is the azimuthal spawhere Rb = (yac − yt )2 + zac x tial frequency. Now, we derive the stationary point of its phase using the PSP. Thus, equating the first derivative of (26) with respect to xa to zero, i.e., ∂Φ(·)/∂xa = 0 and using some algebraic manipulations, we obtain 1 2 2 2 ky2 + 14 α2 kr2 G2 H 2 kx2 Rb2 2 4 kr α − kx = + G + H2 1 2 1 2 1 2 2 2 2 Rb2 4 kr − kx 4 kr − kx 4 kr − kx 1 2 G2 + Rb2 H 2 + Rb2 2 kr GαH + 1 2 (27) 2 Rb2 4 kr − kx
where ky = kr2 /4 − kx2 , G = xa − xt , and H = αxa + β − xt . Moreover, in the spaceborne SAR, we obtain Rb max{α, β, xt , yt , xa } by the high altitude and the long slant distance, and kr kx by the definitions of kx and kr . Thus, we obtain √ 2 2√ 2 2 ⎧ 2 G +Rb H +Rb ⎨ G2 H 0, 1 Rb2 Rb2 (28) ⎩ 14 kr2 α2 −kx2 α2 , 12 kr2 GαH 2GαH. 1 2 2 4 kr −kx
1 2 2 4 kr −kx
Thus, from (27) and (28), we obtain (G + αH)2
kx2 R2 . 1 2 2 b 4 kr − kx
(29)
Therefore, solving (29) for xa , from G and H, we obtain 1 αβ 1+α 1+α2 kx Rb x∗a − − + xt 2 1+α 1 + α2 1 2 2 4 kr − kx
α ¯ k¯x Rb αβ − − +α ¯x ¯t 2 2 ¯ ¯ 1 + α2 kr − kx
(30)
where k¯x = αk ¯ x , k¯r = (1 + α)kr /2, and x ¯t = α ¯ xt with α ¯= (1 + α/1 + α2 ) as the approximated stationary point. Moreover, from (26), we obtain 1 2 2 Φ(·) = − kr (xa − xt ) + Rb − Rs − kx xa 2
xt 2 Rb2 1 β − kr α (xa + − ) + 2 − Rs . (31) 2 α α α
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TABLE I S IMULATION PARAMETERS
Then, substituting the stationary point x∗a for xa in (31), we approximate the phase term as follows: k¯r Rb 1 1 k¯r Rb Φ(·) − kr − kr α 2 2 k¯r2 − k¯x2 k¯r2 − k¯x2 + k¯r Rs + = k¯r Rs −
Fig. 4. Case 1 (argument of perigee : ω = 85o ): Point target responses. (a) PFA with 20 × 20 subpatches in [3]. (b) Method with 20 × 20 subpatches using [5] and [6]. (c) Proposed method with 10 × 10 subpatches.
k¯x2 Rb αβ + kx − k¯x x ¯t 2 2 ¯ ¯ 1 + α2 kr − kx
αβkx − k¯x x ¯t . k¯r2 − k¯x2 Rb + 1 + α2
(32)
D. Stolt Interpolation Stage: Matched Filtering and Stolt Interpolation Analysis The modified matched filter Φmf (n, k¯x , k¯r ) is αβ k + (33) k¯r2 − k¯x2 Rs . Φmf (·) = −k¯r Rs − x 1 + α2
Fig. 5. Case 2 (argument of perigee: ω = 90o ): Point target responses. (a) PFA with 20 × 20 subpatches in [3]. (b) Method with 20 × 20 subpatches using [5] and [6]. (c) Proposed method with 10 × 10 subpatches. TABLE II R ESOLUTION AND PSLR
Moreover, from the definition of the Stolt interpolation [2], we obtain the modified Stolt interpolation as k¯y = k¯r2 − k¯x2 . Then, using the modified matched filter and the modified Stolt interpolation, we obtain Φrma (n, k¯x , k¯y ) −k¯y (Rb − Rs ) − k¯x x ¯t .
(34)
From (34), if we take the 2-D inverse Fourier transform for the distributed targets, we reconstruct the SAR images [2]. IV. S IMULATION R ESULTS We simulate five point targets with the squint angle 10◦ in a 2-D plane with 10 km x 5 km and use the parameters shown in Table I. To obtain the curved path, we use the orbit curvature with maximum 2 m in the range direction and maximum 2 m in the altitude direction. First, to remove the aliasing, we perform the aliasing removal and the subpatch technique [2], [3]. Next, we perform the proposed method for each subpatch and add independently the subpatch together to form an image of the full scene. As shown in Figs. 4 and 5 and Table II, the polar format algorithm (PFA) in [2] and the method using [5] and [6] produce the shifted point targets and the ghost points, and degrade the quality of the reconstructed point targets such as a 3-dB width resolution and a peak-to-sidelobe ratio (PSLR). However, the proposed method generates the desired results for two latitudes. V. C ONCLUSION For the spaceborne spotlight SAR imaging, we have modified the omega-K algorithm. To compensate the curved path induced
by the large slant range and the high altitude, we have proposed the straight-line approximation based on the resampling and the rotation of the transmitted-pulse path and the received pulse path. Then, using the PSP, we have analyzed and compensated the phase error induced by the start–stop assumption of the omega-K algorithm. R EFERENCES [1] K. Eldhuset, “Ultra high resolution spaceborne SAR processing,” IEEE Trans. Aerosp. Electron. Syst., vol. 40, no. 1, pp. 370–378, Jan. 2004. [2] W. G. Carrara, R. S. Goodman, and R. M. Majewski, Spotlight Synthetic Aperture Radar Signal Processing Algorithms. Norwood, MA: Artech House, 1995. [3] M. Soumekh, Synthetic Aperture Radar Signal Processing with MATLAB Algorithms. Hoboken, NJ: Wiley, 1999. [4] L. B. Neronskiy, S. G. Likhansky, I. V. Elizavetin, and D. V. Sysenko, “Phase and altitude histories model adapted to the spaceborne SAR survey,” Proc. Inst. Elect. Eng.—Radar Sonar Navig., vol. 150, no. 3, pp. 184–192, Jun. 2003. [5] W. L. Van Rossum, M. P. G. Otten, and R. J. P. Van Bree, “Extended PGA for range migration algorithms,” IEEE Trans. Aerosp. Electron. Syst., vol. 42, no. 2, pp. 478–488, Apr. 2006. [6] D. D’Aria and A. Monti Guarnieri, “High-resolution spaceborne SAR focusing by SVD-Stolt,” IEEE Geosci. Remote Sens. Lett., vol. 4, no. 4, pp. 639–643, Oct. 2007. [7] W. Q. Wang, C. B. Ding, and X. D. Liang, “Time and phase synchronisation via direct-path signal for bistatic synthetic aperture radar systems,” IET Radar Sonar Navig., vol. 2, no. 1, pp. 1–11, Feb. 2008. [8] H. S. Shin and J. T. Lim, “Omega-k algorithm for airborne spatial invariant bistatic spotlight SAR imaging,” IEEE Trans. Geosci. Remote Sens., vol. 47, no. 1, pp. 238–250, Jan. 2009.
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Omega-K Algorithm for Spaceborne Spotlight SAR Imaging Hee-Sub Shin and Jong-Tae Lim
Abstract—Since the spaceborne spotlight synthetic aperture radar (SAR) has the long slant range and the high altitude, it generates the curved flight path due to the orbit curvature effect and the time delay due to the long round-trip time of the chirp pulse. Thus, to obtain the SAR image with high quality, we extend the omega-K algorithm. First, we convert the curved path into the straight-line path based on the resampling and the rotation of the transmitted-pulse path and the received pulse path. Then, using the modified matched filter and the modified Stolt interpolation based on the principle of the stationary phase, we compensate the phase error induced by the start–stop assumption of the omega-K algorithm. Index Terms—Omega-K algorithm, spaceborne spotlight synthetic aperture radar (SAR).
I. I NTRODUCTION
C
URRENT technologies in the spaceborne spotlight synthetic aperture radar (SAR) demand the accurate compensation of the phase errors to obtain the SAR images with high resolution [1]. In classical SAR imaging, most of the papers use the geometry model based on the straight-line flight path and the start–stop assumption [2], [3]. That is, it is assumed that the aircraft travels along a straight line with a constant velocity, then the aircraft stops to transmit and receive, and the aircraft moves to its next position. In particular, the omega-K algorithm based on the Fourier transform analysis and the principle of the stationary phase (PSP) needs the straight-line flight path and the uniform sampling interval along the flight path [2]. However, in practice, as the platform altitude increases in the SAR system, it generates the curvature effect due to the curved orbit and breaks the start–stop assumption due to the time delay between the transmitted pulse and the received pulse. Moreover, the curvature effect generates the position variation of the platform, and the break of the start–stop assumption produces the velocity variation in the azimuthal direction. In particular, the yaw angle shifting in the position variation of the platform is very sensitive to the phase error, and the velocity variation in the azimuthal direction generates the Doppler shift. Thus, since it is difficult to obtain the hyperbolic range history and the stationary point in the PSP, it increases the phase errors. Therefore, some critical error factors such as the curved orbit and the delay between the
Manuscript received February 20, 2010; revised March 4, 2011 and August 1, 2011; accepted August 31, 2011. Date of publication October 21, 2011; date of current version March 7, 2012. This research was supported by Agency Defense Development through Radiowave Detection Research Center at Korea Advanced Institute of Science and Technology. H.-S. Shin is with Intelligence, Surveillance, and Reconnaissance Research Center, LIG Nex1, Yongin 449-910, Korea. J.-T. Lim is with the Department of Electrical Engineering and Computer Science, Korea Advanced Institute of Science and Technology, Daejeon 305 701, Korea (e-mail: [email protected]). Digital Object Identifier 10.1109/LGRS.2011.2168380
Fig. 1. Spaceborne spotlight SAR model.
transmitted pulse and the received pulse should be compensated carefully and correctly to obtain the high-precision spaceborne SAR image [1], [4]–[6]. To compensate the curved path and the phase error, several methods have been developed [4]–[7]. The method in [6] based on the straight-orbit approximation has been developed, and the compensation techniques such as the phase gradient autofocus and the range alignment have been developed to compensate the phase error induced by the start–stop assumption [5], [7]. However, they compensate the small phase error. Thus, extending the omega-K algorithm, we derive the solution for the spaceborne spotlight SAR imaging. After we convert the curved flight path into the straight-line flight path based on the resampling and rotation of the transmitted and received pulse paths, we compensate the error induced by the start–stop assumption in the frequency domain. II. S PACEBORNE S POTLIGHT SAR M ODEL As shown in Fig. 1, we consider a spaceborne spotlight SAR model. The SAR sensor travels a curved flight path during a synthetic aperture length L and moves at the constant velocity vh . To maintain uniform spatial sampling, the radar uses a fixed interpulse sample spacing Δxr = vh /PRF where PRF is the pulse repetition frequency. After the transformation between the ellipsoid coordinate and the Cartesian coordinate, let the positions of the transmitted pulse and the received pulse be (xte , yte , zte ) and (xre , yre , zre ), respectively. Then, from the transformation for the target coordinate, we obtain (xtp , ytp , ztp ) and (xrp , yrp , zrp ), respectively. Moreover, for the target coordinate, let a target position in the 2-D plane surface be (xt , yt ). In addition, we obtain the error induced by the start–stop assumption asΔxrp = vh · 2Rslant /c with the light speed c and Rslant = ys2 + zs2 . The errors induced by the curved orbit are (Δytp = ytp − ys , Δztp = ztp − zs ) and (Δyrp = yrp − ys , Δzrp = zrp − zs ), where the perigee is the center position of the synthetic aperture (0, ys , zs ) for the target coordinate. Let Rtp =
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2 (xtp − xt )2 + (ytp − yt )2 + ztp
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be the distance between the transmitter and the target when the pulse is transmitted at (xtp , ytp , ztp ), and let
2 be the distance beRrp = (xrp − xt )2 + (yrp − yt )2 + zrp tween the receiver and the target when the pulse is received at (xrp , yrp , zrp ). If we use the chirp pulse st (τ ) = 2 ej(2πfc τ +πγτ ) with the duration time Tp , the time variable τ , the center transmit frequency fc , and the chirp rate γ, we obtain the received signal with the reflectivity function of the target ρ(xt , yt ) as follows: τ − Rd /c j2πfc τ − Rcd jπγ τ − Rcd 2 e e sr (τ ) = ρ(·)rect Tp (1) where Rd = Rtp + Rrp [3].
III. M AIN R ESULTS To obtain the spaceborne spotlight SAR image, we convert the curved path into the straight-line path using the straight-line approximation. Then, at the frequency domain, we compensate the phase error induced by the start–stop assumption of the omega-K algorithm. Let the total number of pulses be N . First, 1 2 N we define xtp ∈ {x1tp , x2tp , . . . , xN tp }, ytp ∈ {ytp , ytp , . . . , ytp }, 1 2 N i i , ztp , . . . , ztp }, where (xitp , ytp , ztp ) is the posiand ztp ∈ {ztp tion of the ith transmitted pulse for i = 1, . . . , N . Analogously, i i for the received pulse, (xirp , yrp , zrp ) denotes the position of the ith received pulse. A. Conversion of the Curved Path Into a Straight-Line Path 1) Resampling of the Transmitted-Pulse Path in the Range Direction: First, as shown in Fig. 2(a), for i = 1, . . . , N , we introduce the following linear fit: i i ytp = ay xitp + dy + ei1 = y˜tp + ei1 , xitp < 0 (2) i i = −ay xitp + dy + ei1 = y˜tp + ei1 , xitp > 0 ytp where ay and dy are the positive constants. Let 1 2 N y˜tp ∈ {˜ ytp = ay x1tp + dy , y˜tp , . . . , y˜tp = −ay xN and tp + dy } 1 2 N e1 ∈ {e1 , e1 , . . . , e1 }. Then, y˜tp is the model value estimated from the linear fit, and e1 is the error value obtained from the linear fit. Therefore, to increase the accuracy of the trajectory approximation, we compensate the error value induced by the 2 ˜ tp = (xtp − xt )2 + (˜ linear fit. Then, let R ytp − yt )2 + ztp be the distance between the positions of the transmitted pulse by the linear fit and the target. If we perform the extended ˜ tp around y˜tp = ytp , we Taylor approximation (ETA) [8] of R obtain 2 ˜ tp (xtp − xt )2 + (ytp − ((m − 1)/m) e1 )2 + ztp R −(1/m)e1 {ytp − ((m − 1)/m) e1 } + 2 x2tp + (ytp − ((m − 1)/m) e1 )2 + ztp Rtp + Rc11
(3) m
where Rc11 = (1/m) k=1 (−e1 {ytp − ((m − k/m))e1 }/ 2 2 ) and m is the tuning xtp + (ytp − ((m − k/m))e1 )2 + ztp parameter in the ETA. Moreover, due to the curved effect, the
Fig. 2. Resampling in range direction. (a) Linear fit. (b) Resampling of transmitted pulse and received pulse.
pulse interval on the curved path is not constant with respect to xtp and ytp . Then, to obtain the constant interval of the azimuthal position and to consider the broadside mode in the transmitted-pulse path, as shown in Fig. 2(b), we resample the azimuthal position as follows: x ¯itp = x ¯1tp + (i − 1)Δ¯ xtp
(4)
where x ¯1tp = −L cos θy /2 and Δ¯ xtp = L cos θy /(N − 1) with θy = tan−1 ay . Hence, in the transmitted pulse, from (4), we obtain the range position of the resampled pulse as follows: i i y¯tp = ay x ¯itp + dy = y˜tp + Δei1 , x ¯itp < 0 (5) i i = −ay x ¯itp + dy = y˜tp + Δei1 , x ¯itp > 0 y¯tp xitp − xitp ) for x ¯itp < 0 and Δei1 = where Δei1 = ay (¯ i i i −ay (¯ xtp − xtp ) for x ¯tp > 0. Let x ¯tp ∈ {¯ x1tp , x ¯2tp , . . . , x ¯N tp }, 1 2 N 1 2 y¯tp ∈ {¯ ytp , y¯tp , . . . , y¯tp }, and Δe1 ∈ {Δe1 , Δe1 , . . . , ΔeN 1 }. Note that Δe1 is the change in the beam direction induced by the resampling in the range direction. Thus, to focus the beam, we compensate the error value of the beam direction. Then, 2 be the distance ¯ let Rtp = (¯ xtp − xt )2 + (¯ ytp − yt )2 + ztp between the positions of the transmitted pulse on the resampled ¯ tp around path and the target. If we perform the ETA of R x ¯tp = xtp and y¯tp = y˜tp , we obtain ¯ tp R ˜ tp + Rc R (6) 12 where Rc12 = (1/m) m k=1 (a1 {xtp + ((m − k/m))a1 } + b ytp + ((m − k/m))b1 }/ 1 {˜
2 ) (xtp + ((m − k/m))a1 )2 + (˜ ytp + ((m − k/m))b1 )2 + ztp with a1 = x ¯tp − xtp and b1 = y¯tp − y˜tp = Δe1 . Thus, from (3) and (6), we obtain
¯ tp Rtp + Rc R 1
(7)
where Rc1 = Rc11 + Rc12 . 2) Resampling of Received Pulse Path in Range Direction: As shown in Fig. 2(b), to obtain the constant interval in the path of the received pulse, we resample the received pulse. Thus, to decrease the resampled distance in the received pulse, we define the resampled azimuthal position of the received pulse as follows: x ¯irp = x ¯1rp + (i − 1)Δ¯ xrp
(8)
where x ¯1rp = x1rp and Δ¯ xrp = α¯ xtp with the positive constant α to adjust the length of the received pulse path. Then, using
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pulse on the shifted path and the target, i.e., Rtps = 2 . (xtps − xt )2 + (¯ ytp − yt )2 + ztp Similarly, we shift the resampled path of the received pulse as follows: i xrps = x ¯irp − (L/4 − L cos θy /4), x ¯irp < 0 (12) i i ¯rp + (L/4 − L cos θy /4), x ¯irp > 0. xrps = x Let xrps ∈ {x1rps , x2rps , . . . , xN rps }. If we perform the ETA of Rtps around xtps = x ¯tp , we obtain ¯ tp + Rc Rtps R (13) 3 m = (1/m) k=1 (a3 {¯ xtp + ((m − k/m))a3 }/
Fig. 3. Resampling. (a) Overlap in resampled path. (b) Shifting of resampled path. (c) Azimuthal positions of the transmitted pulse and the received pulse after rotation of shifted path. (d) Resampling in altitude direction.
the same linear fit in (2) and (5), we obtain i i y¯rp = ay x ¯irp + dy = y˜rp + Δei2 , i i i ¯rp + dy = y˜rp + Δei2 , y¯rp = −ay x
x ¯irp < 0 x ¯irp > 0
(9)
xirp − xirp ) for x ¯irp < 0 and Δei2 = where Δei2 = ay (¯ i i i 1 2 N −ay (¯ xrp − xrp ) for x ¯rp > 0. Let y˜rp ∈ {˜ yrp , y˜rp , . . . , y˜rp } 1 2 N and e2 ∈ {e2 , e2 , . . . , e2 }. Then, y˜rp is the model value estimated from the linear fit. Thus, we obtain e2 = yrp − y˜rp , i.e., the error value obtained from the linear fit in the received pulses. Let x ¯rp ∈ 1 2 N {¯ x1rp , x ¯2rp , . . . , x ¯N } and y ¯ ∈ {¯ y , y ¯ , . . . , y ¯ }. Then, rp rp rp rp rp 2 2 2 ¯ rp = (¯ xrp − xt ) + (¯ yrp − yt ) + zrp be the distance let R between the positions of the received pulse on the resampled path and the target. Thus, using the same method in the transmitted pulse, we obtain ¯ rp Rrp + Rc R 2
(10)
1 2 N where Rc2 = Rc21 + Rc22 . Let Δe2 ∈ 2 , Δe2 , . . . , Δe2 }. {Δe m Then, we show that Rc21 = (1/m) k=1 (−e2 {yrp − ((m −
2 ) and R k/m))e2 }/ x2rp +(yrp −((m − k/m))e2 )2 +zrp c22 = m (1/m) k=1 (a2 {xrp +((m−k/m))a2 }+b2 {yrp +((m−k/m)) 2 ) b2}/ (xrp +((m−k/m))a2 )2 +(yrp +((m−k/m))b2 )2 +zrp ¯rp − xrp and b2 = y¯rp − y˜rp = Δe2 . with a2 = x 3) Shifting of Resampled Path: To convert the curved path into the straight-line path in the range direction, we need to rotate the resampled path. However, as shown in Fig. 3(a), since the rotation of the linear path generates the overlap in the flight path, we need to shift the resampled path. As shown in Fig. 3(b), if we shift the resampled path of the transmitted pulse, we obtain the azimuthal position as follows: i xtps = x ¯itp − (L/4 − L cos θy /4), x ¯itp < 0 (11) i i ¯tp + (L/4 − L cos θy /4), x ¯itp > 0. xtps = x
Let xtps ∈ {x1tps , x2tps , . . . , xN tps }. Then, let Rtps be the distance between the positions of the transmitted
where Rc3 2 + z2 ) (¯ xtp + ((m − k/m))a3 )2 + y¯tp with a3 = tp xtps − x ¯tp . Analogously, let Rrps be the distance between the positions of the received pulse on the shifted path and the 2 . If we yrp − yt )2 + zrp target, i.e., Rrps = (xrps − xt )2 + (¯ perform the ETA of Rrps around xrps = x ¯rp , we obtain ¯ rp + Rc Rrps R (14) 4 = (1/m) m xrp + ((m − k/m))a4 }/ k=1 (a4 {¯
where Rc4 2 + z2 ) (¯ xrp + ((m − k/m))a4 )2 + y¯rp with a4 = rp ¯rp . xrps − x 4) Rotation of Resampled Path: As shown in Fig. 3(b), to obtain the straight-line path in the range direction, we rotate the shifted path with respect to the constant center position yac = (dy + (dy − L sin θy /2))/2 based on the maximum value and the minimum value in the linear fit. Then, as shown in Fig. 3(c), rotating the shifted path of the transmitted pulse, we define the azimuthal position of the transmitted pulse on the rotated path as follows: xia = −L/2 + (i − 1)Δxr .
(15)
In addition, rotating the shifted path of the received pulse, we define the azimuthal position of the received pulse on the rotated path as follows: xir = −Ld /2 + (i − 1)Δ¯ xr where Ld = −2(x1rp − (L/4 − L cos θy /4)/ cos θy ) Δ¯ xr = αΔxr . Thus, from (15) and (16), we obtain xir = αxia + β
(16) and (17)
where β = −Ld /2 + αL/2. Let xa ∈ {x1a , x2a , . . . , xN a } and xr ∈ {x1r , x2r , . . . , xN }. Then, let R be the distance between tpr r the positions of the transmitted pulse on the rotated path and
2 . If the target, i.e., Rtpr = (xa − xt )2 + (yac − yt )2 + ztp we perform the ETA of Rtpr around xa = xtps and yac = y¯tp , we obtain
Rtpr Rtps + Rc5 (18) where Rc5 = (1/m) m k=1 (a5 {xtps + ((m − k/m))a5 } + b ytp + ((m − k/m))b5 }/ 5 {¯
2 ) (xtps +((m−k/m))a5 )2 +(¯ ytp +((m − k/m))b5 )2 + ztp with a5 = xa − xtps and b5 = yac − y¯tp . Analogously, let
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2 Rrpr = (xr − xt )2 + (yac − yt )2 + zrp be the distance between the positions of the received pulse on the rotated path and the target. If we perform the ETA of Rrpr around xr = xrps and yac = y¯rp , we obtain Rrpr Rrps + Rc6 (19) m where Rc6 = (1/m) k=1 (a6 {xrps + ((m − k/m))a6 } + b yrp + ((m − k/m))b6 }/ 6 {¯
2 ) (xrps +((m−k/m))a6 )2 + (¯ yrp + ((m − k/m))b6 )2 + zrp with a6 = xr − xrps and b6 = yac − y¯rp . 5) Resampling in Altitude Direction: To convert the curved path into a straight-line path in the altitude direction, we introduce the following linear fit for the altitude position of the resampled flight path: i ztp az xia + dz , xia < 0 (20) i i ztp −az xa + dz , xia > 0
where az and dz are the positive constants. Then, as shown in Fig. 3(d), we move the altitude position of the resampled flight path to the constant center position zac = −1 (d z + (dz − L tan θz /2))/2 with θz = tan az . Let Ra = 2 2 2 (xa − xt ) + (yac − yt ) + zac be the distance between the positions of the transmitted pulse on the constant altitude path and the target. If we perform the ETA of Ra around zac = ztp , we obtain Ra Rtpr + Rc7 (21) where Rc7 = (1/m) m k=1 (a7 {ztp + ((m − 2 2 with k/m))a7 }/ xa + yac + (ztp + ((m − k/m))a7 )2 ) 2 a7 = zac − ztp . Let Rr = (xr − xt )2 + (yac − yt )2 + zac be the distance between the positions of the received pulse on the constant altitude path and the target. Analogously, if we perform the ETA of Rr around zac = zrp , we obtain (22) Rr Rrpr + Rc8 where Rc8 = (1/m) m k=1 (a8 {zrp + ((m − k/m))a8 }/ 2 + (z 2 x2r + yac rp + ((m − k/m))a8 ) ) with a8 = zac − zrp . Thus, from (21) and (22), we obtain Ra Rtp + Rc1 + Rc3 + Rc5 + Rc7 (23) Rr Rrp + Rc2 + Rc4 + Rc6 + Rc8 .
B. Mixing Stage To reconstruct the SAR image for the spaceborne spotlight SAR, we use the reference signal with the compensation term ΔR as follows: 2(Rs +ΔR) 2(Rs +ΔR) 2 j2πfc τ − jπγ τ − c c sref (n, τ ) = e e (24) 8 2 + z 2 and ΔR = − where Rs = yac ac k=1 Rck /2. Then, if we mix the received signal (1) with the complex conjugated signal of (24) and the range deskew scom (fτ ) = 2 e−j(πfτ /γ−4πΔRfτ /c) where fτ is the frequency variable,
whereas τ is the time variable [2], from (23), we obtain the resulting signal s(n, kr ; xa , yac , zac ) as follows: s(·) ρ(·)rect(·)e−jkr (Ra +Rr 2−Rs )
(25)
where kr = (4πγ/c)(fc /γ + τ − 2Rs /c). C. Fourier Transform Analysis If we perform the Fourier transform of s(·) with respect to xa , we obtain the phase function Φ(n, kx , kr ) as follows: 1 Φ(·) = − kr (xa − xt )2 + Rb2 + kr Rs 2 1 − kr (αxa + β − xt )2 + Rb2 − kx xa (26) 2 2 , and k is the azimuthal spawhere Rb = (yac − yt )2 + zac x tial frequency. Now, we derive the stationary point of its phase using the PSP. Thus, equating the first derivative of (26) with respect to xa to zero, i.e., ∂Φ(·)/∂xa = 0 and using some algebraic manipulations, we obtain 1 2 2 2 ky2 + 14 α2 kr2 G2 H 2 kx2 Rb2 2 4 kr α − kx = + G + H2 1 2 1 2 1 2 2 2 2 Rb2 4 kr − kx 4 kr − kx 4 kr − kx 1 2 G2 + Rb2 H 2 + Rb2 2 kr GαH + 1 2 (27) 2 Rb2 4 kr − kx
where ky = kr2 /4 − kx2 , G = xa − xt , and H = αxa + β − xt . Moreover, in the spaceborne SAR, we obtain Rb max{α, β, xt , yt , xa } by the high altitude and the long slant distance, and kr kx by the definitions of kx and kr . Thus, we obtain √ 2 2√ 2 2 ⎧ 2 G +Rb H +Rb ⎨ G2 H 0, 1 Rb2 Rb2 (28) ⎩ 14 kr2 α2 −kx2 α2 , 12 kr2 GαH 2GαH. 1 2 2 4 kr −kx
1 2 2 4 kr −kx
Thus, from (27) and (28), we obtain (G + αH)2
kx2 R2 . 1 2 2 b 4 kr − kx
(29)
Therefore, solving (29) for xa , from G and H, we obtain 1 αβ 1+α 1+α2 kx Rb x∗a − − + xt 2 1+α 1 + α2 1 2 2 4 kr − kx
α ¯ k¯x Rb αβ − − +α ¯x ¯t 2 2 ¯ ¯ 1 + α2 kr − kx
(30)
where k¯x = αk ¯ x , k¯r = (1 + α)kr /2, and x ¯t = α ¯ xt with α ¯= (1 + α/1 + α2 ) as the approximated stationary point. Moreover, from (26), we obtain 1 2 2 Φ(·) = − kr (xa − xt ) + Rb − Rs − kx xa 2
xt 2 Rb2 1 β − kr α (xa + − ) + 2 − Rs . (31) 2 α α α
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TABLE I S IMULATION PARAMETERS
Then, substituting the stationary point x∗a for xa in (31), we approximate the phase term as follows: k¯r Rb 1 1 k¯r Rb Φ(·) − kr − kr α 2 2 k¯r2 − k¯x2 k¯r2 − k¯x2 + k¯r Rs + = k¯r Rs −
Fig. 4. Case 1 (argument of perigee : ω = 85o ): Point target responses. (a) PFA with 20 × 20 subpatches in [3]. (b) Method with 20 × 20 subpatches using [5] and [6]. (c) Proposed method with 10 × 10 subpatches.
k¯x2 Rb αβ + kx − k¯x x ¯t 2 2 ¯ ¯ 1 + α2 kr − kx
αβkx − k¯x x ¯t . k¯r2 − k¯x2 Rb + 1 + α2
(32)
D. Stolt Interpolation Stage: Matched Filtering and Stolt Interpolation Analysis The modified matched filter Φmf (n, k¯x , k¯r ) is αβ k + (33) k¯r2 − k¯x2 Rs . Φmf (·) = −k¯r Rs − x 1 + α2
Fig. 5. Case 2 (argument of perigee: ω = 90o ): Point target responses. (a) PFA with 20 × 20 subpatches in [3]. (b) Method with 20 × 20 subpatches using [5] and [6]. (c) Proposed method with 10 × 10 subpatches. TABLE II R ESOLUTION AND PSLR
Moreover, from the definition of the Stolt interpolation [2], we obtain the modified Stolt interpolation as k¯y = k¯r2 − k¯x2 . Then, using the modified matched filter and the modified Stolt interpolation, we obtain Φrma (n, k¯x , k¯y ) −k¯y (Rb − Rs ) − k¯x x ¯t .
(34)
From (34), if we take the 2-D inverse Fourier transform for the distributed targets, we reconstruct the SAR images [2]. IV. S IMULATION R ESULTS We simulate five point targets with the squint angle 10◦ in a 2-D plane with 10 km x 5 km and use the parameters shown in Table I. To obtain the curved path, we use the orbit curvature with maximum 2 m in the range direction and maximum 2 m in the altitude direction. First, to remove the aliasing, we perform the aliasing removal and the subpatch technique [2], [3]. Next, we perform the proposed method for each subpatch and add independently the subpatch together to form an image of the full scene. As shown in Figs. 4 and 5 and Table II, the polar format algorithm (PFA) in [2] and the method using [5] and [6] produce the shifted point targets and the ghost points, and degrade the quality of the reconstructed point targets such as a 3-dB width resolution and a peak-to-sidelobe ratio (PSLR). However, the proposed method generates the desired results for two latitudes. V. C ONCLUSION For the spaceborne spotlight SAR imaging, we have modified the omega-K algorithm. To compensate the curved path induced
by the large slant range and the high altitude, we have proposed the straight-line approximation based on the resampling and the rotation of the transmitted-pulse path and the received pulse path. Then, using the PSP, we have analyzed and compensated the phase error induced by the start–stop assumption of the omega-K algorithm. R EFERENCES [1] K. Eldhuset, “Ultra high resolution spaceborne SAR processing,” IEEE Trans. Aerosp. Electron. Syst., vol. 40, no. 1, pp. 370–378, Jan. 2004. [2] W. G. Carrara, R. S. Goodman, and R. M. Majewski, Spotlight Synthetic Aperture Radar Signal Processing Algorithms. Norwood, MA: Artech House, 1995. [3] M. Soumekh, Synthetic Aperture Radar Signal Processing with MATLAB Algorithms. Hoboken, NJ: Wiley, 1999. [4] L. B. Neronskiy, S. G. Likhansky, I. V. Elizavetin, and D. V. Sysenko, “Phase and altitude histories model adapted to the spaceborne SAR survey,” Proc. Inst. Elect. Eng.—Radar Sonar Navig., vol. 150, no. 3, pp. 184–192, Jun. 2003. [5] W. L. Van Rossum, M. P. G. Otten, and R. J. P. Van Bree, “Extended PGA for range migration algorithms,” IEEE Trans. Aerosp. Electron. Syst., vol. 42, no. 2, pp. 478–488, Apr. 2006. [6] D. D’Aria and A. Monti Guarnieri, “High-resolution spaceborne SAR focusing by SVD-Stolt,” IEEE Geosci. Remote Sens. Lett., vol. 4, no. 4, pp. 639–643, Oct. 2007. [7] W. Q. Wang, C. B. Ding, and X. D. Liang, “Time and phase synchronisation via direct-path signal for bistatic synthetic aperture radar systems,” IET Radar Sonar Navig., vol. 2, no. 1, pp. 1–11, Feb. 2008. [8] H. S. Shin and J. T. Lim, “Omega-k algorithm for airborne spatial invariant bistatic spotlight SAR imaging,” IEEE Trans. Geosci. Remote Sens., vol. 47, no. 1, pp. 238–250, Jan. 2009.
