Two-dimensional spectrum for MEO SAR processing using a modified advanced hyperbolic range equation
with ki (i ¼ 1, 2, 3, and 4) corresponding to ci (i ¼ 1, 2, 3, and 4) in [1], respectively. 2D spectrum using MAHRE: After the demodulation of an echo to the baseband, a range Fourier transform (FT) is performed. Then an azimuth FT is operated and the phase term of the transform is expressed as
M. Bao, M.D. Xing, Y. Wang and Y.C. Li To improve further the accuracy in the approximation of the range history for a synthetic aperture radar (SAR) onboard a mediumearth-orbit (MEO) satellite a fourth power term has been added to the advanced hyperbolic range equation (AHRE), and the new one is called a modified AHRE (MAHRE). Then, a two-dimensional spectrum using the MAHRE was derived, and the accuracy of the spectrum analysed. Promising results were obtained.
Introduction: The integration time of one synthetic aperture of a medium-earth-orbit (MEO) synthetic aperture radar (SAR) can be long or a fine spatial resolution can be achieved. However, the long integration time means that the approximation in range history with a straight flight path within one synthetic aperture could become invalid. Thus, the use of a typical hyperbolic range equation can be questionable. An accurate but potentially complicated equation is needed for the processing of the MEO SAR data. Eldhuset [1] approximated the relative earth and satellite motion using a fourth-order Taylor expansion of the equation in azimuth time. Although Eldhuset achieved accurate range approximation, one drawback was the complicated algebraic operations involved in the development of imaging algorithms for data processing with the approximation. Then, Huang et al. [2] studied an advanced hyperbolic range equation (AHRE) consisting of terms up to the third order of the range equation in [1] and a linear term. They simplified the range equation of [1]. Using the AHRE, Huang et al. derived the 2D spectrum and then developed an advanced nonlinear chirp scaling (ANLCS) algorithm [3]. Closely examining the AHRE, one notices that the AHRE only compensates for the range history up to the cubic term. Under some circumstances, the compensation might not be enough or the level of accuracy might be inadequate. To increase the accuracy level, we modify the AHRE with an addition of a fourthorder term or modified AHRE (MAHRE). This addition is not simply a roll-back to the original range equation in [1]. Instead, the addition is directly applied to the AHRE. Thus, the MAHRE is still concise in expression. MAHRE: As the azimuth integration time increases in one synthetic aperture, the error caused by the quartic term of the range equation becomes large. Ignoring the fourth-order term in the AHRE could be problematic. To compensate for the error, we modify the AHRE as R(tm ) = AHRE + btm4 = Vr2 tm2 + R20 − 2Vr tm R0 sin usq (1) + Dl tm + btm4 where tm is the azimuth time, R0 the range at the beam centre at tm = 0, Vr the equivalent velocity, usq the squint angle, and Dl the coefficient of the linear component. b is the new fourth-order component. Expanding (1) as a Taylor series and keeping terms up to the fourthorder, one has
4p(fc + fr ) Vr2 tm2 + R20 − 2Vr tm R0 sin usq c +Dl tm + btm4 − 2pfa tm
f( f r , t m ) = −
where fr is the range frequency, fa the azimuth frequency, fc the carrier frequency, and c the speed of light. Due to the presence of the high-order terms of tm in (4), azimuth stationary point, tm m of the MAHRE is not available. We approximate tm m with the stationary point of the AHRE, tm a [3] or
tm
b = k4 −
k3 R0 , k2 Vr4 cos2 usq (4 sin2 usq − cos2 usq ) 8R30
≃ tm
a
s(fr , fa ) ≃ ar (fr )aa (fa − fdc ) exp {jw1 ( fr , fa ) + jw2 ( fr , fa )}
(5)
(6)
where ar (fr )(aa (fa )) is the spectrum envelope in the range (azimuth) direction. fdc is the Doppler centroid frequency. The first phase term in (6) related to bt4m is
w1 ( fr , fa ) = −4p
fc + fr 4 btm c
m
(7)
and is weakly dependent on range. It can be generally compensated using the phase value at the centre of an image swath. If the swath is very wide such as under the MEO case, the swath can be subdivided into contiguous segments along the range direction such that the variation of w1 ( fr , fa ) crossing each segment in the range is ≤0.25p, a typical threshold value. Then, w1 ( fr , fa ) can be corrected using the phase value at the centre in each segment. w2 ( fr , fa ) is the same phase term derived from the AHRE [2]. Thus, after the compensation of w1 ( fr , fa ) the MAHRE is readily implemented into the ANLCS algorithm [3] that is used for the MAHRE. Accuracy analysis of 2D spectrum: To analyse the accuracy of (6), one needs to use the analytical solution that is unfortunately not available. Alternatively, we use the 2D spectrum derived from the method of series reversion (MSR) [4] to assess the accuracy. Expanding terms within (. . .) of (4) and keeping the expansion up to the fourth-order terms, one has
f( fr , tm ) ≃ − (2)
4p(fc + fr ) (R0 + k1 tm + k2 tm2 + k3 tm3 + k4 tm4 ) c
(8)
− 2pfa tm Thus, with the MSR one represents tm
MSR
as the power series,
1 cfa − k1 − tm MSR ( fa ) = 2k2 2(fc + fr ) 2 3k3 cfa − k1 − 3 − 2(fc + fr ) 8k2 3 2 9k − 4k2 k4 cfa − k1 + 3 − 5 2(fc + fr ) 16k2
Letting the coefficients of the first, second, third, and fourth terms of (2) equal those in the expansion of the range history up to the fourth-order terms of [1], respectively, one obtains R0 k3 2 R0 k3 +2R0 k2 , usq = asin , Vr = k2 Vr k2 Dl = k 1 +
m
cfa Dl + R0 cos usq R0 sin usq 2Vr (fc + fr ) Vr = − 2 + Vr cfa Dl + Vr 1 − 2Vr (fc + fr ) Vr
Substituting (5) into (4), one approximates the 2D spectrum as
R(tm ) ≃ R0 + (Dl − Vr sin usq )tm V 2 cos2 usq 2 Vr3 cos2 usq sin usq 3 + r tm + tm 2R0 2R20 4 V cos2 usq (4 sin2 usq − cos2 usq ) + r + b tm4 8R30
(4)
(3)
(9)
Also, we have used azimuth stationary point tm a to approximate tm m in (5). The only difference between the MAHRE and AHRE is btm4 as shown in (1). Thus, replacing k4 in (9) with (k4 2 b), one can expand
ELECTRONICS LETTERS 1st September 2011 Vol. 47
No. 18
as 1 cfa − k1 − tm m ( fa ) ≃ 2(fc + fr ) 2k2 2 3k3 cfa − k1 − 3 − 2(fc + fr ) 8k2 3 2 9k − 4k2 (k4 − b) cfa − k + 3 − 1 2(fc + fr ) 16k25 m
(10)
peak sidelobe ratio (PSLR) and integrated sidelobe ratio (ISLR) in Table 2 clearly indicates the deterioration in performance (broadening of the resolution, and departing from an ideal PSLR or LSLR value). Thus, the 2D spectrum was better after the inclusion of bt4m. Finally, to ensure that the 2D spectrum was acceptable for a given segment width in range, we limited the width so that the variation of w1 ( fr , fa ) is between +0.25p. With the simulated parameters, the width was 75 km.
f2 (fc + fr ) R0 w(fr , fa ) = −p r − 4p g c 2 c 2(fc + fr ) fa + k1 +p 4k2 (fc + fr ) c 3 2 k3 c 2(fc + fr ) k1 +p f + 2 a 3 c 16k2 (fc + fr ) 4 (9k 2 − 4k2 k4 )c3 2(fc + fr ) k +p 3 5 f + a 1 c 256k2 (fc + fr )3
range direction
Thus, substituting (10) into (8), we get the expanded phase term in (6) as
950
950
900
900
range direction
and approximate tm
850 800 750
Similarly, substituting (9) into (8), one can obtain another phase term that is identical to the MSR-derived phase in [4]. Therefore, the 2D spectrum of (6) with the phase term of (11) should analytically have the same level of accuracy as the MSR-derived spectrum [4]. In both cases, the accuracy is up to the fourth-order of series expansion. Simulated results: To evaluate the MAHRE and possible improvement in accuracy over the AHRE, we simulated a MEO SAR at three altitudes (Table 1). The satellite orbit was circular with an inclination angle at 908. The SAR was an L-band with a look angle of 15.958. The range and azimuth resolutions were both 2 m. The integration time and Doppler bandwidth were also tabulated (Table 1). There was a target located on the equator of the earth.
800 750
700
700 700
(11)
850
750
800 850 900 azimuth direction
950
700
a
750
800 850 900 azimuth direction b
950
Fig. 1 Contours of impulse responses using MAHRE and AHRE as range approximations a MAHRE b AHRE
Table 2: Assessment of spectrum in azimuth direction Resolution (m) PSLR (dB) ISLR (dB) MAHRE AHRE
2.00 2.11
213.28 211.51
210.07 27.55
Conclusion: A modified AHRE with an addition of the fourth-order term or the MAHRE is proposed. With the MAHRE, one compensated the error in range history up to the quartic term. Of three MEO SAR cases simulated, the modification improved the accuracy. The impulse responses of the 2D spectrum were well focused. Also the fourthorder term was directly added to the AHRE and the stationary phase point from the AHRE was used as the point for the MAHRE. Thus, a minimal effort would be required in the implementation of the MAHRE into the ANLCS.
Table 1: Phase errors caused by range approximations Altitude Integration Doppler bandwidth Phase error Phase error of (km) time (s) (Hz) of MAHRE (rad) AHRE (rad) 5000 49.23 1483.9 0.0006318p 0.2585p 5750 59.24 1351.1 0.0012p 0.3557p 6500
70.07
1237.3
0.0023p
0.5075p
Phase errors caused by two range approximations: Compared with the true distance between the SAR and target, the use of the MAHRE caused a small difference in range or an azimuth phase error occurred (Table 1). Errors in absolute value were much smaller than the threshold value of 0.25p. Thus, the performance of the MAHRE was satisfactory. As comparison, the error using the AHRE was also provided. Errors were ≥0.25p or unsatisfactory results were obtained. Assessment of 2D spectrum: With the orbit altitude of 6500 km, integration time of 70.07 s, and Doppler bandwidth of 1237.3 Hz, we simulated the 2D spectrum from the point target. The MAHRE and AHRE were used for range approximations, respectively. Contours of the impulse responses are shown in Fig. 1. When the MAHRE was used, the responses were not only well focused but also there was clear delineation between the main-lobe and first and sub-sequential sidelobes (Fig. 1a). When the AHRE was employed, the responses in range were also well focused and the main-lobe and first and sub-sequential sidelobes were clearly separated. However, the responses in azimuth were not focused well and there was a lack of separation among the main-lobes and sidelobes (Fig. 1b). Data of the azimuth resolution,
Acknowledgment: This work was supported by the Chinese ‘973’ Program under grant 2010CB731903 to the Xidian University, China. # The Institution of Engineering and Technology 2011 4 May 2011 doi: 10.1049/el.2011.1322 One or more of the Figures in this Letter are available in colour online. M. Bao, M.D. Xing and Y.C. Li (National Lab of Radar Signal Processing, Xidian University, Xi’an, Shaanxi 710071, People’s Republic of China) E-mail:
[email protected] Y. Wang (Department of Geography, East Carolina University, Greenville, NC 27858, USA) References 1 Eldhuset, K.: ‘A new fourth-order processing algorithm for spaceborne SAR’, IEEE Trans. Aerosp. Electron. Syst., 1998, 34, (3), pp. 824–835 2 Huang, L.J., Qiu, X.L., Hu, D.H., and Ding, C.B.: ‘An advanced 2-D spectrum for high-resolution and MEO spaceborne SAR’. Proc. APSAR, Xi’an, China, October 2009, pp. 447– 450 3 Huang, L.J., Qiu, X.L., Hu, D.H., and Ding, C.B.: ‘Focusing of mediumearth-orbit SAR with advanced nonlinear chirp scaling algorithm’, IEEE Trans. Geosci. Remote Sens., 2011, 49, (1), pp. 500–508 4 Neo, Y.L., Wong, F.H., and Cumming, I.G.: ‘A two-dimentional spectrum for bistatic SAR processing using series reversion’, IEEE Geosci. Remote Sens. Lett., 2007, 4, (1), pp. 93–96
ELECTRONICS LETTERS 1st September 2011 Vol. 47 No. 18
