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Wide Nonlinear Chirp Scaling Algorithm for Spaceborne Stripmap Range Sweep SAR Imaging Yan Wang, Member, IEEE, Jing-Wen Li, and Jian Yang, Senior Member, IEEE Abstract— The spaceborne stripmap range sweep synthetic aperture radar (SS-RSSAR) is a new concept spaceborne SAR system that images the region of interest (ROI) with ROI-orientated strips, which, unlike the traditional spaceborne SAR, are allowed to be not parallel with the satellite orbit. The SS-RSSAR imaging is a challenging problem because echoes of a wide region have strong spatial varieties, especially in high-squint geometries, and are hard to be focused by a single swath. The traditional imaging algorithms could solve this problem by costineffectively dividing an ROI into many subswaths for separate processing. In this paper, a new wide nonlinear chirp scaling (WNLCS) algorithm is proposed to efficiently image the SS-RSSAR data in a single swath. Comparing with the traditional nonlinear chirp scaling algorithm, the W-NLCS algorithm is superior in three major aspects: the nonlinear bulk range migration compensation (RMC), the interpolation-based residual RMC, and the modified azimuth frequency perturbation. Specifically, the interpolation for the residual RMC, the most significant step in achieving the wide-swath imaging performance, is made innovatively in the time domain. The derivation of the W-NLCS algorithm, as well as the performance analyses of the W-NLCS algorithm in aspects of the azimuth resolution, accuracy, and complexity, are all provided. The presented approach is evaluated by the point target simulations. Index Terms— Interpolation-based residual range migration compensation (RMC), modified azimuth frequency perturbation, nonlinear bulk RMC, spaceborne stripmap range sweep synthetic aperture radar (SS-RSSAR), wide nonlinear chirp scaling (W-NLCS).
I. I NTRODUCTION
T
HE spaceborne synthetic aperture radar (SAR) is an effective microwave imaging tool to map and monitor geologic hazards in complex weather conditions [1]–[4]. The traditional spaceborne SAR antenna illuminates regions of interest (ROI) with a fixed or step-adjusted beam in elevation and hence generates parallel-to-track beam coverage strips (BCSs) [5]–[8]. However, for some types of geologic
Manuscript received November 11, 2016; revised May 27, 2017; accepted August 1, 2017. This work was supported in part by the National Natural Science Foundation of China under Grant 61601257, Grant 61371133, and Grant 61490693, in part by the China Postdoctoral Science Foundation under Grant 2016M591179, and in part by a special project under Grant 30402010203. (Corresponding author: Jian Yang.) Y. Wang and J. Yang are with the Department of Electronic Engineering, Tsinghua University, Beijing 100084, China (e-mail:
[email protected];
[email protected]). J.-W. Li is with the School of Electronic and Information Engineering, Beihang University, Beijing 100191, China (e-mail:
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TGRS.2017.2737031
Fig. 1. Geometries of (a) conventional spaceborne stripmap SAR and (b) SS-RSSAR.
hazards, such as earthquakes, their orientations are commonly not parallel with the satellite orbit [9], [10]. For this matter, the spaceborne SAR has to image a much wider swath than the ROI, as shown in Fig. 1(a), and hence undesirably receives echoes from disinterested regions, like region E. Such a wide-swath coverage strategy will not only cause increment of the data amount and computational burden, but also will lead to either increment of the data collection period (for multiple orbits of observation) or azimuth resolution degradation (for the time-divided observation modes like the terrain observation with progressive scans SAR and ScanSAR [11], [12]). In a previous paper, Wang et al. [13] introduced an ROI-orientated spaceborne SAR mode, the spaceborne stripmap range sweep SAR (SS-RSSAR), to solve the mismatching problem between the ROI orientation and the satellite track. By continuously adjusting the beam in elevation, the SS-RSSAR can generate an ROI-matched BCS as shown in Fig. 1(b). This special data collection strategy contributes to at least three major benefits: less total data amount, shorter data collection period, and higher azimuth resolution. However, the unique data collection strategy leads to two major challenges for imaging. First, due to its perpendicularto-strip beam pointing, the SS-RSSAR would operate with a high-squint angle if the ROI was highly angled with the satellite orbit, and cause serious signal coupling. Second and most importantly, such a high-squinted geometry would lead to serious Doppler centroid variation, which is easy to shift the Doppler spectrum out of the support domain restricted by the pulse repetition frequency (PRF), and cause aliasing [14]. A direct attempt in avoiding the Doppler aliasing is to increase the PRF so that it can be high enough to include inside all the Doppler spectrum. However, this is impossible because the
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required PRF is much higher than its upper limit, which is designed to fully record echoes of a wide swath [1], [14]. If the traditional Doppler-domain-based imaging algorithms, such as the range-Doppler algorithm (RDA), the frequency scaling algorithm, and the range migration algorithm (RMA), are used for the SS-RSSAR imaging, a whole ROI should be cost-ineffectively divided into many subswaths for separate processing [2], [15]–[17]. Though the time-domain back projection algorithm can theoretically focus the SS-RSSAR echo in a single swath, it is not preferred due to its enormous O(N 3 ) computation load [18]. In a word, no existing algorithm works well for the SS-RSSAR with both high efficiency and accuracy. In this paper, an innovative wide nonlinear chirp scaling (W-NLCS) algorithm is proposed for efficient SS-RSSAR imaging. Similar to the traditional NLCS algorithm [19], [20], the W-NLCS algorithm first removes the range migration, then unifies the Doppler rate by an azimuth frequency perturbation, and lastly implements azimuth data compression. However, different from the traditional NLCS algorithm, the W-NLCS algorithm induces three major modifications to overcome the problem of the Doppler aliasing and therefore, leads to a much wider swath. These modifications are as follows. 1) Employing a nonlinear bulk range migration compensation (RMC) filter to simultaneously remove the linear range migration for all targets along the ground beam track. Comparatively, the traditional NLCS algorithm just removes the linear range migration of a single central target. 2) Employing an additional time-domain interpolation to globally remove the residual linear range migration for all targets away from the ground beam track. This is the key step to realize a wide imaging swath. 3) Employing a modified azimuth frequency perturbation filter to simultaneously unify the azimuth-dependent Doppler rate and remove the azimuth-independent bulk Doppler centroid. Comparatively, the traditional NLCS algorithm can only realize the former function. We try to present the derivation for the W-NLCS algorithm and its performance analyses, including the azimuth resolution, accuracy, and complexity, in detail in this paper. The point target simulations are provided to verify the presented approach. The paper is organized as follows. Section II gives a brief introduction to the SS-RSSAR, including its signal properties and inherent PRF-based scene swath limit (SSL). Section III gives a detailed derivation for the W-NLCS algorithm. Section IV discusses the performances of the W-NLCS algorithm. Section V validates the presented approaches by the point target simulations. The research is summarized in Section VI. II. S PACEBORNE S TRIPMAP R ANGE S WEEP SAR
Fig. 2.
Geometry of the SS-RSSAR.
the satellite velocity. H denotes the satellite height. θ denotes the tilt angle formed by the BCS and the satellite orbit. Rc and Rgref denote the central slant range and its ground projection at t = 0. S denotes an arbitrary target with the coordinate of (x s , ys , 0) within the BCS. ts denotes the time when the beam center passes target S. Rs denotes the instant slant range. βs and ϕs denote the central look angle and the central squint angle for target S. As shown in Fig. 2, a rectilinear geometry, instead of a real curved one, is used to simplify the subsequent analyses and derivations [21]. Such a simplification will not affect the analyses, methods and conclusion of this paper, but require some geometry-based adjustments to the filters of the proposed W-NLCS algorithm for practical applications. During the data collection period, the beam pointing is vertical to the BCS [13]. The continuous ROI-oriented beam tracking is implemented by continuously adjusting the beam in elevation, resulting in a continuous variation of the look angle. The instant slant range Rs for target S can be explicitly given as Rs (t; S) =
(Rg (S) − V sin θ t)2 + V 2 cos2 θ (t − ts )2 + H 2 (1)
where Rg and ts can be uniquely mapped to the target coordinates as
Rg (S) = Rgref + x s ys . ts = V cos θ
(2)
Note that targets with the same X-coordinate share the same Rg . By referring to the derivation in the Appendix, an elegant relationship among the tilt angle θ , look angle βs , and squint angle φs yields sin φs (S) = sin θ sin βs (S).
(3)
A. Signal Properties The geometry of the SS-RSSAR is shown in Fig. 2, where a Cartesian coordinate is built for the convenience of description. The y-axis direction is defined as the BCS direction. V denotes
By assuming that a traditional linear frequencymodulated (LFM) signal is adopted for transmission, the baseband backscattered signal of target S, denoted as E s ,
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Fig. 3.
yields
3
Echo folding realized by the multi-interval sampling technique.
2Rs (t; S) 2 E s (t, τ ; S) = exp j πγ τ − c 4π f c Rs (t; S) × exp − j (4) c where c is the speed of light, t and τ denote the slow and fast time, respectively, f c denotes the carrier frequency, and γ denotes the chirp rate of the LFM signal. The signal amplitude is neglected here and in the subsequent derivations as it contributes little to focusing. The range migration is presented in the first phase term. As pointed out in [13], unlike the traditional spaceborne SAR, the variation of the slant range of the SS-RSSAR can easily reach several dozens of kilometers, corresponding to several dozens of thousands of range bins, and will inevitably cause transmitting blockage [22]. To solve this problem, a multi-interval sampling technique is proposed to mitigate the global range migration in [13]. The function of the multi-interval sampling is equivalent to fold the echo in range as shown in Fig. 3. The resulted signal is 2 2Rs (t; S) − τdelay (t) E s (t, τ ; S) = exp j πγ τ − c 4π f c Rs (t; S) · exp − j (5) c where τdelay denotes the azimuth-dependent fast time delay. Please note that though the range migration has been mitigated, the Doppler history, described by the second phase term in (5), remains unchanged. Based on (3) and (5), the Doppler centroid for target S, denoted as f d , yields 2 d Rs (t; S) 2 = − V sin θ sin βs . (6) f d (S) = − λ dt λ t =ts The tilt angle θ is fixed for a given geometry. Thus, the Doppler centroid is merely determined by the look angle β, which based on the continuous range beam steering of the SS-RSSAR, varies monotonically during the data collection period. In other words, the ROI orientation along the X-dimension determines the range of the Doppler centroid variation. Unlike the traditional spaceborne stripmap SAR, the SS-RSSAR beam has an additional X-dimension velocity, resulting in a more violent Doppler centroid variation and a narrow SSL as will be shown in the following.
Fig. 4. PRF-based SSL of the SS-RSSAR if processed by the traditional Doppler-domain-based algorithms. (a) SSL. (b) Explains the reason for the SSL using the spectrums of the nearest, central, and farthest targets.
B. Scene Swath Limit If focused by the Doppler-domain-based algorithms, the SS-RSSAR has a PRF-based inherent SSL: the PRF should be both higher than the Doppler bandwidth to avoid aliasing and be lower than a certain threshold for full echo recording. The problem is that for a common high-squint case of the SS-RSSAR; the upper bound of the PRF is far less enough to counteract the Doppler centroid variation. This problem is explained in Fig. 4, where three targets, Pn , Pc , and P f , are used to represent the nearest, central, and farthest targets within the swath. Fig. 4(a) and (b) shows their distributions and range-azimuth spectrums, respectively. ft and f τ denote the Doppler and the range frequencies, respectively. Ba and Br denote the Doppler bandwidth and the bandwidth of the LFM signal, respectively. f t denotes the maximum allowable range for the Doppler centroid variation. By assuming that β f and βn denote the look angles of target P f and Pn , respectively, the following relationship based on (6) can be
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TABLE I S IMULATION PARAMETERS FOR SSL A NALYSIS
Fig. 6. RMC of the W-NLCS algorithm. (a) Target distribution. (b) Original range migrations. (c) Range migrations after the nonlinear bulk RMC. (d) Range migrations after the residual RMC. Fig. 5.
IFP with track-beam coordinate and BCS-beam coordinate.
achieved as
⎧ 2
ft ⎪ ⎨ V sin θ sin β f − f d = λ 2 (7) 2
ft ⎪ ⎩ f d − V sin θ sin βn = . λ 2 By assuming that the slant range for target Pc is Rc , the maximum ground swath Wr can be estimated by
λ f t Rc . (8) 2V sin θ Note that Wr denotes the distance between two parallelto-track iso-Doppler lines marked in red in Fig. 4(a). Due to the tilt angle θ between the iso-Doppler lines and the BCS direction, the effective imaging swath is restricted within the green diamond region, which is the very PRF-based SSL for the SS-RSSAR. Wa denotes the SSL along the BCS direction as Wr =
found to be inspiring: first, it can deal with echoes of a highsquint geometry; second and most importantly, it converts the image formation plane (IFP) from the traditional track-beam coordinate to the BCS-beam coordinate as shown in Fig. 5. Thus, in generating a final focused image no zero padding is required to fulfill the blank regions, marked by data E in gray. Thus, the BCS-beam coordinates lead to less data amount and hence contribute to a higher computational efficiency. However, the accuracy of the traditional NLCS algorithm is limited by its coarse assumption that all targets within an ROI share the same range migration. In this paper, a novel W-NLCS algorithm is proposed to effectively expand the SSL of the traditional NLCS algorithm by accurately removing the spatially variant range migrations for all targets within the ROI. A. Nonlinear Bulk RMC and Range Compression
III. W IDE N ONLINEAR C HIRP S CALING A LGORITHM
The first step of the W-NLCS algorithm is to compress echoes in range and simultaneously compensate the bulk range migration. Nine targets with different locations, marked from A to I as shown in Fig. 6(a), are used to give an intuitive explanation for the RMC of the W-NLCS algorithm. Both these two manipulations are conducted in the range frequency domain. By making a range Fourier transform to the signal in (5), we have 4π f c Rs (t; S) E s (t, f τ ) = exp − j c 4π f τ c · exp − j Rs (t; S) + τdelay (t) c 2 π f τ2 . (10) × exp − j γ
Due to the special data collection method of the SS-RSSAR, the processing strategy of the NLCS algorithm is
The first range-independent term denotes the Doppler phase and the second term denotes the range migration. The third
Wa = Wr cot θ.
(9)
In practical applications, it is found that both Wr and Wa are far less than enough to satisfy the wide-swath geometrical hazard monitoring requirement. A simple simulation can demonstrate this: by assuming a C-band SS-RSSAR with the parameters in Table I; Wr and Wa are calculated to be both 7.8 km. Comparatively, the typical range swath of a C-band stripmap spaceborne SAR is around 50 km, which is much wider than the 7.8-km limit. Thus, an effective SS-RSSAR imaging algorithm should be able to expand the PRF-based narrow SSL.
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quadratic term denotes the phase that should be removed for range compression. Thus, the range compression filter FRC can be generated as πf2 FRC ( f τ ) = exp j τ . (11) γ Different from the traditional linear bulk RMC that considers merely the central target, the W-NLCS algorithm employs a nonlinear bulk RMC filter, denoted as FBRMC , to simultaneously remove the linear range migration for all targets along the y-axis as FBRMC (t, τ) f 2 2 4π f τ R − V sin θ t + H gref = exp j 2 + H2 + cτ c − Rgref 2 delay (t) 4π f c ( (Rgref − V sin θ t)2 + H 2 · exp j c 2 2 − Rgref + H ) . (12)
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NLCS algorithm overlooks the residual linear range migration for small patches, it cannot be neglected for the wide-swath SS-RSSAR imaging. Based on the definition in (14), Rs1 is found to be spatially variant and hence can be corrected by a time-domain interpolation. The core strategy of the interpolation is to relocate all targets with the same X-coordinate in the same range bin. The range bin array at t = 0 is used as the referential interpolation array. By assuming that Sc1 and Sc2 denote the central slant ranges of target S before and after the interpolation and that x 1 and x 2 denote the X-coordinates of target S before and after the interpolation, we have ⎧ ⎨x 1 (S) = R 2 (S) − H 2 + V sin θ ts − Rgref c1 (17) ⎩x (S) = R 2 (S) − H 2 − R . 2
c2
gref
The function of the interpolation is to realize the follow mapping: x 1 (S) → x 2 (S).
(18)
The second term is to compensate the Doppler phase. By multiplying FRC , FBRMC , and E s , the resulting signal E s1 yields 4π f c Rs1 (t; S) E s1 (t, f τ ) = exp − j c 4π f τ × exp − j Rs1 (t; S) (13) c
Additionally, it is known that for the data before and after the interpolation, the target central slant ranges Rc1 and Rc2 yield ⎧ c 2 + H2 ⎪ ⎨ Rc1 (S) = τs + (Rgref −V sin θ ts )2 + H 2 − Rgref 2 c ⎪ ⎩ Rc1 (S) = τs 2 (19)
where
where τs and τs denote the fast time corresponding to the central slant range of target S before and after the interpolation as ⎧ ⎪ 2 (Rg (S) − V sin θ ts )2 + H 2 ⎪ ⎪ ⎪ τs = ⎪ c ⎪ ⎨ 2 + H2 − (Rgref − V sin θ ts )2 + H 2 + Rgref (20) ⎪ ⎪ ⎪ 2 2 2 Rg (S) + H ⎪ ⎪ ⎪ ⎩τs = . c As the interpolation is not designed merely for the specific target S, ts , τs , and τs should be replaced by t, τ , and τ , respectively. By substituting (17) and (19) into (18), the time-domain mapping can be realized by the following interpolation: ⎞ ⎛ 2 2 c ⎟ − H 2 − V sin θ t + H2 τ 2⎜ ⎟ ⎜ 2 τ→ ⎜ ⎟ ⎠ c⎝ 2 2 2 2 − (Rgref − V sin θ t) + H + Rgref + H
Rs1 (t; S) = Rs (t; S) − (Rgref − V sin θ t)2 + H 2 2 + H 2 . (14) + Rgref
It is easy to verify that for any target locating at the y-axis with Rg = Rgref (x s = 0), the linear component of Rs1 always equals to zero as d Rs1 (t; S) = 0. (15) dt t =ts Thus, for the targets along the y-axis, the linear range migrations have been completely removed. By converting E s1 back to the range time domain, the signal yields 2 E s1 (t, τ ) = sinc Br τ − Rs1 (t; S) c 4π f c Rs1 (t; S) × exp − j (16) c where Br denotes the signal bandwidth. Fig. 6(b) and (c) illustrates the target range migrations before and after the nonlinear bulk RMC in exaggeration. But note that Fig. 6(b) does not represent any practical intermediate output of the W-NLCS algorithm. B. Interpolation-Based Residual RMC For a more general case with Rg = Rgref , (15) is no longer valid. Thus, for the targets away from the y-axis there will exist residual linear range migration. Though the traditional
(21) where τ and τ denote the original and the new range time. By employing the mapping in (21) to (16), the interpolated signal, denoted as E s2 , yields 4π f c Rs1 (t; S) . E s2 (t, τ ) = sinc Br δ(τ , t; S) exp − j c (22) The δ function in the first term of (22) yields δ(τ , t; S)
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⎛ ⎞ 2 c 2 2⎜ ⎟ τ = ⎝ − H 2 − V sin θ t + H 2 − Rs (t; S)⎠. c 2 (23) The corrected range migration for target S after the interpolation should satisfy δ(τ , t; S) = 0.
(24)
By solving (24), we have τ −
2 Rnew (t; S) = 0 c
(25)
where Rnew denotes the corrected range migration for target S as (26), as shown at the bottom of the page. Now, it is easy to verify that for a target with any available Rg , the linear range migration has been completely removed as
Doppler rate fr can be calculated as ⎞ ⎧ ⎛ −V sin θ Rg (S) − V sin θ ts ⎪ ⎪ ⎪ ⎟ ⎜ ⎪ 2 ⎪ ⎜ ⎪ ⎪ Rg (S) − V sin θ ts + H 2 ⎟ ⎪ ⎟ ⎜ ⎪ 2 fc ⎜ ⎪ ⎪ ⎟ f d (S) = − ⎟ ⎜ ⎪ ⎪ V sin θ Rgref − V sin θ ts ⎟ ⎪ c ⎜ ⎪ ⎟ ⎜ + ⎪ ⎪ 2 ⎪ ⎠ ⎝ ⎪ 2 ⎪ R − V sin θ t + H gref s ⎪ ⎨ ⎞ ⎛ 2 ⎪ V 2 cos2 θ Rg (S) − V sin θ ts + V 2 H 2 ⎪ ⎪ ⎪ 3/2 ⎟ ⎜ ⎪ 2 ⎪ ⎟ ⎜ ⎪ 2 ⎪ R (S) − V sin θ t + H ⎟ ⎜ g s ⎪ ⎪ 2 fc ⎜ ⎟ ⎪ ⎪ f (S) = ⎟. ⎜ ⎪ r 2 sin2 θ H 2 ⎪ ⎟ ⎜ c ⎪ V ⎪ ⎟ ⎜ ⎪ − 3/2 ⎪ ⎠ ⎝ ⎪ 2 ⎪ 2 ⎪ Rgref − V sin θ ts + H ⎩ (30)
C. Modified Azimuth Frequency Perturbation
The fact that fr depends on target locations (Rg and ts ) causes a depth of focusing problem, which means that targets of a given range bin suffer from spatially variant Doppler rates and hence cannot be compressed coherently. Furthermore, the seriously varied Doppler centroid can easily shift the Doppler spectrum out of the PRF-determined data support domain and causes azimuth aliasing. Thus, both the spatial variations of the Doppler rate and the Doppler centroid should be effectively suppressed. Fig. 8(a) illustrates the azimuth time frequency-domain (TFD) signals after the residual RMC interpolation. The Doppler centroid and the Doppler rate can be analyzed further by being expanded with respect to the target azimuth position ts . A second-order approximation to the Doppler centroid fd and a first-order approximation to the Doppler rate fr are assumed to have enough precision. Thus, (30) can be approximately rewritten as ⎧ 2f ⎪ ⎨ f d (S) ≈ c l0 (S) + l1 (S)ts + l2 (S)ts2 c (31) 2f ⎪ ⎩ fr (S) ≈ c (q0 (S) + q1 (S)ts ) c where ⎧ V sin θ Rgref V sin θ Rg (S) ⎪ ⎪ l0 (S) = − ⎪ ⎪ 2 + H2 ⎪ Rg2 (S) + H 2 Rgref ⎪ ⎪ ⎪ ⎪ 2 2 2 ⎪ V sin θ H V 2 sin2 θ H 2 ⎪ ⎪ ⎪ l (S) = − + 1 ⎪ ⎪ 2 + H 2 3/2 2 (S) + H 2 3/2 ⎪ R R ⎪ g gref ⎪ ⎪ 3 3 3 2 3 2 ⎪ ⎪ ⎪l (S) = − 3V sin θ H Rg (S) + 3V sin θ H Rgref ⎪ 2 5/2 ⎪ 5/2 ⎨ 2 2 H 2 + Rg2 (S) 2 H 2 + Rgref V 2 cos2 θ Rg2 (S) + V 2 H 2 ⎪ V 2 sin 2 θ H 2 ⎪ ⎪ q (S) = − ⎪ 0 2 + H 2 )3/2 ⎪ 2 3/2 ⎪ (Rgref Rg2 (S) + H ⎪ ⎪ ⎪ ⎪ ⎪ 2 sin2 θ H 2 + cos2 θ Rg2 (S) + H 2 ⎪ 3 ⎪ q1 (S) = V sin θ Rg (S) ⎪ 5/2 ⎪ ⎪ Rg2 (S) + H 2 ⎪ ⎪ ⎪ ⎪ ⎪ 3V 3 sin3 θ H 2 Rgref ⎪ ⎪ − ⎪ 2 5/2 . ⎩ Rgref + H 2 (32)
The second term in (29) denotes the Doppler phase. Based on the definition of Rs1 in (14), the Doppler centroid f d and
Note that as all the parameters l0 , l1 , l2 , q0 , and q1 depend on the X-coordinates of targets, rather than the
d Rnew (t; S) = 0. dt t =ts
(27)
Thus, the interpolated signal in (22) can be rewritten in the most familiar form as 2 E s2 (t, τ ) = sinc Br τ − Rnew (t; S) c 4π f c Rs1 (t; S) . (28) × exp − j c Fig. 6(d) shows the target range migrations after the time-domain residual RMC interpolation. By neglecting the second- and higher order terms of t in Rnew for an observation with a moderate resolution level, the signal E s2 can be further simplified as 2 2 Rg (S) + H 2 E s2 (t, τ ) ≈ sinc Br τ − c 4π f c Rs1 (t; S) . (29) × exp − j c Comment 1: One should note that based on (21), the original time τ and the new range time τ do not have the same support domain as shown in Fig. 7, where Fig. 7(a) and (b) denotes the minimum and the maximum range time of the data with respect to transmission. Both the original range time τ and the new range time τ before and after the interpolation are presented for comparison. It can be seen that both the maximum and minimum values of τ exceed the upper and lower bounds of τ when t is larger than 0. Thus, at the marginal regions with t > 0 the interpolation-based residual RMC will inevitably lead to fake data, which should be discarded before moving into the next processing step.
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Y-coordinates, they can be conveniently updated in range. The W-NLCS algorithm employs a modified azimuth frequency perturbation to simultaneously unify the Doppler rate and remove the azimuth-independent bulk Doppler centroid. First, the modified azimuth frequency perturbation filter FMAFP is assumed to have the following form: FMAFP (t, τ ) = exp{ j π(μ(τ )t + α(τ )t 3 )}.
(33)
By multiplying FMAFP and E s2 and neglecting the constant terms and the terms that have little contribution to focusing, the filtered signal, denoted as E s3 , yields E s3 (t, τ ; S)
2 2 2 Rg (S) + H = sin c Br τ − c 2π f c 2 · exp − j q0 (S)(t − ts ) c 4 fc · exp j π μ(τ ) + l0 (S) (t − ts ) c 2 fc 2 · exp j π 3α(τ ) − q1 (S) ts (t − ts ) c 4 fc l1 (S)ts + 3α(τ )ts2 (t − ts ) · exp j π c × exp{ j πμ(τ )ts }. (34)
The second and the third phase terms denote the azimuthindependent bulk Doppler centroid and the spatial variant component of the Doppler rate that should be eliminated, respectively. Thus, the coefficients of both the second and the third terms should always be zeroes at the range time τs as ⎧ 4f ⎪ ⎨μ(τs ) = − c l0 (S) c (35) ⎪α(τ ) = 2 fc q (S). ⎩ 1 s 3c By considering the arbitrariness of target S, τs should be replaced by τ , resulting in the coefficients of the FMAFP as ⎧ 4f ⎪ ⎨μ(τ ) = − c l0 (τ ) c (36) ⎪α(τ ) = 2 f c q (τ ) ⎩ 1 3c where l0 and q1 in (32) should be updated as ⎧ √ ⎪ V sin θ c2 τ 2 − 4H 2 V sin θ Rgref ⎪ ⎪l0 (τ ) = − ⎪ ⎪ ⎪ cτ 2 + H2 ⎪ Rgref ⎪ ⎪ √ ⎪ ⎪ ⎨ q1 (τ ) = 4V 3 sin θ c2 τ 2 − 4H 2 (12 sin2 θ H 2 + c2 cos2 θ τ 2 ) ⎪ ⎪ × ⎪ ⎪ c5 τ 5 ⎪ ⎪ 3 sin3 θ H 2 R ⎪ 3V ⎪ gref ⎪ ⎪ − 5/2 . ⎪ ⎩ 2 2 Rgref + H
By substituting (36) into (33), the modified azimuth frequency perturbation filter FMAFP yields 4 fc 2 fc l0 (τ )t + q1 (τ )t 3 FMAFP (t, τ ) = exp j π − . c 3c (38) By substituting (36) into (34), E s3 can be further simplified as E s3 (t, τ ; S) 2 2 2 = sin c Br τ − Rg (S) + H c · exp j πμ(τ )ts 2π f c · exp − j q0 (S)(t − ts )2 c 2π f c 2 2 · exp j 2l1 (S)ts + 2l2 (S)ts + q1 (τ )ts (t − ts ) . c (39) Note that in (39) though the Doppler rate has been unified, the spatial variation of the Doppler centroid still exists in the last phase term with the influences shown in Fig. 8(b): at the x-axis where ts = 0, the Doppler centroids of the targets are zeroes. At other positions away from the x-axis, the targets turn to have nonzero position-determined residual Doppler centroids. However, unlike the azimuth TFD in Fig. 8(a), the variation of such residual Doppler centroid has been effectively suppressed in Fig. 8(b) and the corresponding Doppler spectrum shift will no longer cause azimuth aliasing. However, the residual spatially variant Doppler centroid will induce an additional azimuth geometrical distortion, the correction of which is to be discussed in the following. Comment 2: The above discussion neglects a phase term that is the last term of (34) and the second term of (39). This range-dependent term will induce an additional range frequency modulation. By expanding l0 in (37) to a refer to the first order, the phase can be ential range time τref approximated as πμ(τ )ts ≈ 2π f (ts ) τ − τref (40) where f denotes an additional range frequency centroid induced by the modified azimuth frequency perturbation as f (ts ) = −
(37)
8 f c H 2 V sin θ ts . 2 2 − 4H 2 c2 τref c2 τref
(41)
As the additional azimuth-dependent range frequency centroid is only involved by the linear range phase term as shown in (40), it will not lead to image quality degradation, but will cause additional deformations to the impulse response function (IRF) of focused targets.
! 2 Rnew (t; S) = (Rg (S) − V sin θ t)2 + V 2 cos2 θ (t − ts )2 + V sin θ t + H 2
(26)
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Fig. 7.
Illustration of (a) minimum and (b) maximum range time before and after the residual RMC interpolation.
Fig. 8.
Target Doppler features in the TFD (a) before and (b) after the modified azimuth frequency perturbation.
D. Azimuth Compression and Geometric Distortion Correction The azimuth compression can be conveniently made in the range-Doppler domain. By neglecting the constant terms, the signal converted to the range-Doppler domain yields E s3 ( f t , τ ; S)
2 2 Rg (S) + H 2 c · exp{j πμ(τ )ts } c 2 f · exp j π 2 f c q0 (S) t 2l1 (S)ts + 2l2 (S)ts2 + q1 (τ )ts2 · exp − j 2π ts + ft . 2q0 (S) (42)
= sin c
Br
τ −
The second phase term denotes the quadratic phase that needs to be compensated by the azimuth compression filter FAC , which yields c 2 f FAC ( f t , τ ) = exp − j π (43) 2 f c q0 (τ ) t where q0 in (32) should be updated as q0 (τ ) =
2c2 V 2 cos2 θ τ 2 +8V 2 sin2 θ H 2 V 2 sin2 θ H 2 − . 3 2 + H 2 3/2 (cτ ) Rgref (44)
By first multiplying E s3 with FAC and then converting the signal back to the time domain, the focused amplitude signal
yields E s4 (t, τ ; S) 2 2 2 Rg (S) + H = sin c Br τ − c 2l1 (S)ts + 2l2 (S)ts2 + q1 (τ )ts2 · sin c Ba t − ts − 2q0 (S) (45) where Ba denotes the Doppler bandwidth for target S. Based on (45), target S will be rebuilt at an erroneous azimuth location away from its real azimuth position represented by ts . In other words, we have an undesired geometrical distortion. Considering that the position-sensitive feature of the geometrical distortion, the interpolation is an ideal tool for the geometrical distortion correction (GDC) as t − ts −
2l1 (τ )ts + 2l2 (S)ts2 + q1 (τ )ts2 → t − ts 2q0 (τ )
(46)
where t denotes the new azimuth time, l1 and l2 in (32) should be updated as ⎧ 2 2 2 V 2 sin 2 θ H 2 ⎪ ) = − 8V sin θ H + ⎪ l (τ ⎪ 1 ⎪ 2 + H 2 3/2 ⎪ c3 τ 3 ⎪ Rgref ⎪ ⎪ ⎪ ⎨ 3V 3 sin3 θ H 2 l2 (τ ) = (47) ⎪ ⎞ ⎛ 2√ ⎪ ⎪ ⎪ ⎪ Rgref 16 c2 τ 2 − 4H 2 ⎪ ⎪ ⎪ + × ⎝− 5/2 ⎠ . ⎪ 5 5 ⎩ c τ H 2 + R2 gref
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9
The fact that the resulted Doppler rate fr T is independent of ts means that all the targets having the same azimuth coordinates share the same Doppler rate. Comparatively, based on the geometry and the instant target slant range in (1), the original target Doppler rate, denoted as fr0 , yields 2 fc d 2 fr0 (S) = R (t; S) s c dt 2 t =ts
Fig. 9.
Flowchart of the W-NLCS.
If a slant range IFP is required, the final focused image, denoted as E s51, yields 2 2 2 E s51 (t , τ ; S) = sin c Br τ − Rg (S) + H c × sin c(Ba (t − ts )). (48) Comparatively, if a ground range IFP is expected, an additional interpolation is required to convert the focused image from the slant range IFP to the ground range IFP as 2 2 2 Rg (S) + H 2 → τ − Rg (S) (49) τ − c c where τ denotes the range time after the interpolation. In this way, the final focused image, denoted as E s52, yields 2 E s52 (t , τ ; S) = sin c Br τ − Rg (S) (50) c × sin c(Ba (t − ts )). The flowchart of the proposed W-NLCS algorithm is displayed in Fig. 9. IV. W-NLCS P ERFORMANCE A NALYSES A. Azimuth Resolution The azimuth resolution of an SAR system is determined by the sensor velocity and the Doppler bandwidth. For the SS-RSSAR, the azimuth resolution ρa for target S is defined along the y-axis as V cos θ (51) Ba (S) where the Doppler bandwidth Ba can be approximated by multiplying the central target Doppler rate fr and the beam dwell time Td as ρa (S) =
2 f c (V 2 cos2 θ (Rg (S) − V sin θ ts )2 + V 2 H 2 ) = . (54) c((Rg (S) − V sin θ ts )2 + H 2)3/2 The inconsistency of the final Doppler rate fr T and the original Doppler rate fr0 is caused by the additional quadratic Doppler phases of the nonlinear bulk RMC and the modified azimuth frequency perturbation filters. By assuming ψ as the azimuth beamwidth, the beam dwell time for target S can be approximated by (Rg (S) − V sin θ ts )2 + H 2 ψ. (55) Td (S) = V cos θ The larger the X-coordinate and the smaller the Y-coordinate of a target, the longer the beam dwell time can be achieved. By substituting (52), (53), and (55) into (51), the theoretical azimuth resolution for the SS-RSSAR yields V cos θ ⎛ ⎞. ρa (S) = V 2 cos2 θ R 2 (S)+V 2 H 2
2 fc c
g ⎜ 3.2 − V sin θ H3/2 2 +H 2 ⎜ Rg2 (S)+H 2 Rgref ⎝ √ 2 2 (Rg (S)−V sin θts ) +H ψ · V cos θ 2
2
2
⎟ ⎟ ⎠
(56) Thus, ρa is the spatially variant as it relies both on the target X- and Y-coordinates. B. Accuracy The derivation of the W-NLCS algorithm employs two major approximations, which will affect the accuracy of the algorithm when applied to different observation tasks. 1) Approximation of Range Walk Only: This approximation is used in deriving (29) from (28), in which the second- and higher-order components of Rnew , corresponding to the range curve and other residual range migrations, are neglected. Such an approximation is made based on the fact that in a highsquint geometry the linear component accounts for a major part of the total range migration. As the linear component of Rnew has been eliminated by the interpolation-based residual RMC, the quadratic component, denoted as RnewQ , accounts for the major part. Based on the definition of Rnew in (26), RnewQ can be calculated as RnewQ (t; S) =
V 2 cos2 θ Rg (S) (t − ts )2 . 2 2 2(Rg (S) − V sin θ ts ) H + Rg (S)
(52)
(57)
Based on (32) and (43), the Doppler rate used for the target rebuilding, denoted as fr T , is
Thus, the maximum value of RnewQ , denoted as RnewQ_MX , yields
Ba (S) = fr (S)Td (S).
fr T (S) =
2 fc c
V 2 cos2 θ Rg2 (S)
+ 3/2
Rg2 (S) + H 2
V2H2
−
V 2 sin2 θ H 2 . 2 + H 2 3/2 Rgref (53)
RnewQ_MAX(S) =
V 2 cos2 θ Rg (S) Td2 (S). 8(Rg (S) − V sin θ ts ) H 2 + Rg2 (S) (58)
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TABLE II C OMPUTATION L OAD OF BASIC O PERATIONS
It is known that if the maximum range migration is less than half of a single range bin, the range migration can be neglected [2]. Thus, if the second- and high-order components can be safely neglected, RnewQ_MAX should be limited within a given threshold as c RnewQ_MAX(S) < (59) 4Fs where Fs is the sampling rate. By synthetizing (55), (58), and (59), it is found that the lower the sampling frequency, the narrower the beamwidth and the smaller the ROI swath, the more accurate the approximation could be. 2) Approximation of Quadratic Doppler Phase: This approximation is used in (31). During the derivation of the modified azimuth frequency perturbation filter FMAFP , the cubic and higher-order components of f d and the quadratic and high-order components of fr have been neglected. It is known that the Doppler rate error contribute to the image quality deterioration and the Doppler centroid error causes the geometric distortion. If the previously omitted cubic component of the Doppler centroid and the quadratic component of the Doppler rate were added back, (31) could be rewritten as ⎧ 2f ⎪ ⎨ fd (S) ≈ c l0 (S) + l1 (S)ts + l2 (S)ts2 + l3 (S)ts3 c (60) ⎪ f (S) ≈ 2 f c q (S) + q (S)t + q (S)t 2 ⎩ r 0 1 s 2 s c where the new components l3 and q2 yield ⎧ 2H 2 Rg2 (S)V 4 sin4 θ H 4 V 4 sin4 θ ⎪ ⎪ ⎪l3 (S) = − 7/2 ⎪ ⎪ 2(H 2 + Rg2 (S))7/2 ⎪ H 2 + Rg2 (S) ⎪ ⎪ ⎪ 2 V 4 sin4 θ ⎪ 2H 2 Rgref ⎪ H 4 V 4 sin 4 θ ⎪ ⎪ − + 7/2 7/2 ⎪ ⎪ 2 2 ⎪ H 2 + Rgref 2 H 2 + Rgref ⎪ ⎨ −2H 4 sin2 θ − H 4 + H 2 Rg2 (S) ⎪ V 4 sin2 θ ⎪ 2 R 2 (S) sin2 θ +2R 4 (S) cos2 θ ⎪ +11H ⎪ g g ⎪ ⎪ q2 (S) = 7/2 ⎪ ⎪ 2 2 ⎪ 2 H + Rg (S) ⎪ ⎪ 4 ⎪ 2 ⎪ V 4 sin4 θ 3 H − 4H 2 Rgref ⎪ ⎪ ⎪ + . ⎪ ⎩ 7/2 2 2 H 2 + Rgref (61) By keeping the modified azimuth frequency perturbation , different filter in (38) unchanged, the new filtered signal E s3 from E s3 in (39), yields 4π f c l3 (S)ts3 (t − ts ) E s3 (t, τ ; S) = E s3 (t, τ ; S) · exp j c 2π f c 2 2 × exp − j q2 (S)ts (t − ts ) . (62) c The first and the second phase terms denote the linear and the quadratic phases that are previously neglected by
the approximation, respectively. Based on [2], if the absolute value of the maximum quadratic phase error (QPE) is lower than a threshold, for example, π/2, the resulting image quality degradation can be neglected. According to the second term of (62), the maximum QPE for target S, denoted as ϕQPE , yields π fc Td2 (S)q2 (S)ts2 . (63) 2c Thus, the condition to make the first-order approximation valid for the Doppler rate is π (64) ϕQPE (S) < . 2 By assuming that the condition in (64) has been satisfied, the additional target shift υ, represented by the first term of (62), yields V cos θl3 (S) 3 ts . υ(S) = (65) q0 (S) ϕQPE (S) =
The condition to make the second-order approximation valid for the Doppler centroid is to keep the additional distortion within half of the azimuth resolution. Thus, υ should satisfy ρa (S) . (66) 2 Equations (64) and (66) give the restrictions for the W-NLCS algorithm. υ(S)
