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Slant-range velocity estimation based on Small-FM-Rate Chirp
ARTICLE IN PRESS Signal Processing 88 (2008) 2472– 2482
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Slant-range velocity estimation based on Small-FM-Rate Chirp Baochang Liu , Tong Wang, Zheng Bao The National Laboratory of Radar Signal Processing, Xidian University, Xi’an, Shanxi 710071, China
a r t i c l e i n f o
abstract
Article history: Received 12 December 2007 Received in revised form 18 March 2008 Accepted 21 April 2008 Available online 6 May 2008
This paper deals with the estimation of the slant-range velocity of a moving target with airborne SAR data. For the signature of a moving target, performing the Beat Frequency (BF) algorithm in the slow time and range frequency domain can generate a Small-FMRate Chirp (SFRC), which has some features of a single-frequency signal. The Fourier transform of such SFRC quite approximates a sinc function, and the maximum-value point of such Fourier transform is proportional to the slant-range velocity. Based on these principles, a methodology known as Beat Frequency Coherent Accumulation (BFCA) algorithm is developed to estimate the slant-range velocity. The prominent advantage of the BFCA algorithm is that the unambiguous velocity can be very large. Besides, the proposed methodology can deal with large range cell migration (RCM) and can work with a low signal-to-noise ratio (SNR) scene, requiring a small amount of computation but yields a high accuracy especially for the fast-moving targets. The effectiveness and efficiency of the proposed methodology are validated with both simulated and real data. & 2008 Elsevier B.V. All rights reserved.
Keywords: SAR Slant-range velocity estimation Small-FM-Rate Chirp (SFRC) Beat Frequency Coherent Accumulation (BFCA) Range cell migration (RCM)
1. Introduction As is well known, the slant-range velocity induces a Doppler-shift in Doppler domain, and a shifted azimuth location in the image domain, correspondingly. Therefore, the estimation of slant-range velocity is of great importance not only for obtaining the motion parameter, but also for re-locating the moving target. In estimating the slant-range velocity with one antenna, we must take the following three facts into account. The first one is the problem of velocity ambiguity, due to the limitation of PRF. The second one is the problem of range cell migration (RCM). As we know, most of the slant-range velocity estimation algorithms performing in the range-compressed domain are affected by large RCM, because of the loss of synthetic aperture length. The third one is the effect of residual static clutter. Generally speaking, the remaining static clutter will bias the estimates of slant-range velocity. So the algorithm used
Corresponding author.
E-mail address: [email protected] (B. Liu). 0165-1684/$ - see front matter & 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2008.04.013
to estimate the slant-range velocity must have an ability to differentiate the moving target from the remaining static clutter. Several methods have been presented to estimate the slant-range velocity with a single antenna in recent literature [1–6]. They mainly exploit both the information of the antenna radiation pattern and the information of the phase of the returned echo. In [2], a methodology is proposed which utilizes the structure of the amplitude and the phase modulations of the returned echo from a moving target in the Fourier domain. In this algorithm, a generalized likelihood ratio test approach is adopted to detect moving target and estimate the motion parameters, provided that the static clutter can be modeled as a zero mean complex white Gaussian process. The main disadvantage of this method is the heavy computation burden. The method proposed in [4] uses a non-uniform PRF to solve the problem of velocity ambiguity. However, the use of a non-uniform PRF requires a non-conventional pulse scheduling, thus introducing complexity in image reconstruction algorithms [6]. The method presented in [6] works by performing an auto-correlation algorithm in the two-dimensional (2D) frequency domain.
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In [7], an algorithm known as Multiple Look Beat Frequency (MLBF) is presented to estimate the Doppler Centroid. The algorithm is based on multiplying two range looks together to generate a beat signal and using FFT to estimate the resultant beat frequency (BF). As we know, Doppler Centroid estimation and slant-range velocity estimation share some common features. However, the MLBF algorithm cannot directly be used to estimate the slant-range velocity, because the data used for estimating the Doppler Centroid comprises a large number of static scatters, whereas the slant-range velocity estimation data that includes a point-like moving target must contain as little static clutter as possible. Moreover, it is often the case that a large RCM induced by the slant-range velocity exists, which cannot be dealt with by the MLBF algorithm. Besides, the MLBF algorithm requires the range frequency spectrum be ‘‘flat’’ [8], which is not satisfied properly in actual case. Hence, to estimate the slant-range velocity, we must modify the MLBF algorithm. Based on the principle of ‘‘beat frequency’’, a methodology called Beat Frequency Coherent Accumulation (BFCA) is proposed to estimate the slant-range velocity in this paper. Performing BF algorithm in slow time and range frequency domain can yield a Small-FM-Rate Chirp (SFRC). Such SFRC and a single-frequency signal have some common attributes as well as different features. For instance, the Fourier transform of SFRC exhibits a shape quite similar to that of the sinc function. Nevertheless, the maximum-value point of the Fourier transform of SFRC is related to its envelope, whereas the Fourier transform of a single frequency is independent of its envelope. For a large enough FM rate chirp, we can compute its Fourier transform through the stationary phase method. We cannot do so for SFRC, because the FM rate of SFRC does not meet the prerequisite of the stationary phase method. However, we will show that if this small FM rate signal satisfies certain condition, we can use another method to compute its Fourier transform. The primary aim of performing the ‘‘BF’’ is to resolve the velocity ambiguity. Because the BF algorithm works in the slow time and range frequency domain rather than in the rangecompressed domain just as the MLBF does, there is no loss of synthetic aperture length, thus making the proposed BFCA methodology free of the effect of RCM. Furthermore, the BFCA methodology can restrain the effect of noise, because of implementing coherent accumulation in performing the BFCA methodology. For the effect of residual static clutter, we adopt an algorithm called digital spotlight [6,11] to suppress the static clutter. In addition, the proposed methodology has the capability of differentiating the moving target from static clutter to some extent. The paper is organized as follows. In Section 2, we elaborate on the basic principle of the proposed methodology. The proposed methodology to estimate the slantrange velocity is given in Section 3. In Section 4, we discuss the case of multiple scatters. In Section 5, we present the experimental results illustrating the effectiveness of the proposed methodology and give the facts that affect the accuracy of the estimates. Section 6 supplies concluding remarks.
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2. The basic principle of the proposed methodology A SAR scenario in the slant-range plane is illustrated in Fig. 1, where coordinate x and y denote cross-range and slant-range, respectively. A radar travels along the cross-range direction at a constant velocity v. A moving target with slant-range velocity vr is illuminated. Suppose that the radar is located at (0, 0) and the moving target at (x0, y0) when tm ¼ 0, where tm is the slow time. The distance between the target and the radar at the instant tm is rðt m Þ ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðvt m x0 Þ2 þ ðy0 þ vr t m Þ2 .
(1)
The two-way antenna radiation pattern can be written as (2)
g½sinyðt m Þ
where tm is the slow time, y(tm) is the aspect angle, and g(x) is symmetric about x ¼ 0. In the radar synthetic aperture time, noting that y0cvrtm, we have sinyðt m Þ yðt m Þ
vtm x0 . y0
(3)
Then the received signal in range frequency and slow time domain can be written as Sðf ; t m Þ ¼ g½sinyðt m ÞPðf Þ ejð4pðf þf 0 Þ=cÞrðtm Þ
(4)
where f0 is the carrier frequency, f is range frequency corresponding to fast time t, c is the speed of light, P(f) is 2 the Fourier transform of transmitted pulse rðtÞejpgt , g and r(t) are the FM rate and envelope of the transmitted pulse, respectively. Then the range-compressed signal in terms of fast time and slow time can be expressed as 2rðt m Þ g½sinyðt m Þ ejð4pf 0 =cÞrðtm Þ Src ðt; t m Þ ¼ rrc t (5) c where rrc( ) is range envelope (after range compression), t and tm are fast time and slow time, respectively. Assuming that ð2rðt m Þ=cÞ ð2y0 =cÞ(y0 refers to the slant-range coordinate of the moving target), we can get the range-compressed Doppler spectrum of the moving
y
vr target(x0,y0)
(0,0) Fig. 1. SAR scenario.
x
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target using the stationary phase method [9] as follows c 2f f d þ 0 vr ejcðf d Þ (6) Src ðf d Þ ¼ Mg 2vf 0 c
be expanded in Taylor series about tm ¼ 0. Neglecting the small terms, we obtain jbf ðt m Þ ¼
where cðf d Þ ¼
(10)
v y pcy0 2 x0 2pf 0 y0 2 r 0 f þ vr f d þ 2p 2 2 v d 2f 0 v v v2 c þ
2pf 0 x20 4pf 0 x0 vr 4pf 0 y0 þ j0 cy0 vc c
(7)
and M and j0 are constants. From (7), it is found that the quadratic phase component of c(fd) is independent of the slant-range velocity and the information of the slantrange velocity is transformed into the linear and constant phase components. However, we cannot extract the information of the slant-range velocity from the linear phase component, since the azimuth coordinate of the moving target, x0, is unknown. Obviously, the range-compressed Doppler spectrum |Src(fd)| is a replica of the two-way antenna radiation pattern with a shifting factor of (2f0/c)vr. Therefore, we, spontaneously, will think of estimating the slant-range velocity by determining the maximum-value point of |Src(fd)| or by performing the energy balancing (DE) algorithm [10]. However, two problems are present in such algorithms: (i) The problem of velocity ambiguity: because the Doppler spectrum is subject to the limitation of PRF, the problem of velocity ambiguity will arise. (ii) The problem of RCM: in deducing (6), we neglect the effect of RCM. But in realistic case especially the case when estimating the slant-range velocity of a fastmoving target, the range-compressed signal will span over several range cells, so when we select one range cell (5) to get the Doppler spectrum, the obtained Doppler spectrum is usually distorted due to the loss of synthetic aperture length. Hence, the accuracy of estimate cannot be ensured. In order to solve above-mentioned problems, we must resort to other algorithm. Alternatively, let us beat S(f, tm) and S(f+Df, tm) ((4)), namely, multiply S(f, tm) by S*(f+Df, tm) and we get the following beat signal Sbf ðf ; t m Þ ¼ Sðf þ Df ; t m ÞS ðf ; t m Þ ¼ Pðf þ Df ÞP ðf Þg 2 ½sinyðt m Þ ejjbf ðtm Þ
(8)
where Df is a fixed range frequency interval, Df5f0, and 4pDf rðt m Þ. jbf ðt m Þ ¼ c
4pDf 4pDf 4pDf x0 2pDf v2 2 y0 vr t m þ vt m t . c c c y0 c y0 m
(9)
According to (9), it is found that jbf ðt m Þin (9) has the same structure as ð4pðf þ f 0 Þ=cÞrðt m Þin (4). The difference between jbf ðt m Þ and ð4pðf þ f 0 Þ=cÞrðt m Þ lies in that the range frequency in jbf ðt m Þ is Df, whereas the range frequency in ð4pðf þ f 0 Þ=cÞrðt m Þ is f+f0. Hence, the information of the slant-range velocity is not lost after implementing the BF algorithm. Now, let us transform Sbf(f, tm) into the Doppler domain. Assuming that x05y0, vrtm5y0, jbf(tm) (9) can
Noting that the FM rate of Sbf(f, tm), ð2Df =cÞðv2 =y0 Þ, is very small, we cannot get its Fourier transform using the stationary phase method. Alternatively, we can use the following theorem to get the Fourier transform of Sbf(f, tm). Theorem. of Small-FM-Rate Chirp (SFRC) Consider a Small-FM-Rate chirp (SFRC) sðtÞ ¼ wðt t c Þ ejðpgt
2
þ2pbtÞ
(11)
where w(ttc) is the envelope of s(t), its effective length is Dt, and w(ttc) is symmetric about tc; b is a constant frequency; g is the FM rate of s(t), and it is so small that the following expression holds1 q¼
jgjDt 2 oqT ¼ 1:181. 4
(12)
Then, the chirp s(t) can be equivalent to a singlefrequency signal as follows, from the point of view that the Fourier transforms of the equivalent signal and the original signal have the identical maximum-value point, the identical constant phase computed at the maximumvalue point and the approximately same bandwidth se ðtÞ ¼ ejj wðtÞej2pat
(13)
where a ¼ gt c þ b; j ¼
pgt 2c
2 pffiffiffiffiffiffi3 Fs 2q gDt 2 þ arctan4 pffiffiffiffiffiffi5; q ¼ 4 2q Fc
(14) Rx Rx F s ðxÞ ¼ 0 sinðpt 2 =2Þdt; F c ðxÞ ¼ 0 cosðpt 2 =2Þdt are the Fresnel functions, and e is a real constant. And the Fourier transform of s(t) reveals a shape similar to that of sin c function and has a maximum-value point at a ¼ gt c þ b.
(15)
The effective width of the Fourier transform of s(t) approximately equals B
1 . Dt
(16)
(The proof of this theorem lies in Appendix A) Let sin y(tm) ¼ 0 (3) and we have tc ¼
x0 . v
(17)
Note that the FM rate of Sbf(f, tm) (8) is gbf ¼ ð2Df =cÞðv2 =y0 Þ (see (10)) and the frequency of the linear phase component of Sbf(f, tm) is bbf ¼ ð2Df =cÞvr þ ð2Df =cÞðx0 =y0 Þv (see (10)). If the selected Df satisfies jgbf j ðL=vÞ2 =4o1:181 (L is the effective synthetic aperture 1
This threshold of q, 1.181, comes from Appendix A.
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Multiply S(f, tm) ((4)) by P*(f) and we get
length), according to the theorem of SFRC, substitution of tc ¼ (x0/v) into gbftc+bbf leads to an equivalent single frequency a¼
2Df vr . c
Sp ðf ; t m Þ ¼ g½sinyðt m ÞjPðf Þj2 ejð4pðf þf 0 Þ=cÞrðtm Þ . Thereby, (19) can be rewritten as vtm jF j2patm . e e Spbf ðf ; t m Þ ¼ CjPðf þ Df Þj2 jPðf Þj2 g 2 y0
(18)
Then Sbf(f, tm) (expression (8)) can be written as vtm jF j2patm Sbf ðf ; t m Þ ¼ CPðf þ Df ÞP ðf Þg 2 e e y0
After coherent accumulation, we obtain Z f2 vt m jF j2patm Spbf ðt m Þ ¼ Spbf ðf ; t m Þ df ¼ Q 0 g 2 e e y0 f1
(19)
where C is a constant, and F¼
2pDfx20 4pDf y0 þ þ j0 c cy0
where Z f2 Q¼ C 1 jPðf þ Df Þj2 jPðf Þj2 df .
(22)
C1 is a constant. From (21), we can see that the slant-range velocity is proportional to the maximum-value point of |Sbf(f, fd)|. This fact is the basic principle of our methodology.
(28)
(a) From (27), it is very clear that Spbf(fd) exhibits a much sharper shape than Src(fd) (6). Fig. 2 shows the comparison of Src(fd) and Spbf(fd). (b) From the derivation of above expressions, it is found that the information of the azimuth coordinate of the moving target is removed automatically, during the
14
1200
12
1000
10
800
8
x 107
power
power
1400
600
6
400
4
200
2
0 fd[Hz]
100
(27)
Now, let us make some explanations for the BF algorithm as follows.
It is often the case that an amount of noise exists in the SAR data. In order to estimate the slant-range velocity accurately, we must restrain the effect of the noise. We can do so by accumulating Sbf(f, fd) with respect to f.
-100
(25)
f1
3. The proposed methodology
0 -200
(24)
f1
Then (21) can be rewritten as pL 2Df fd þ vr Spbf ðf d Þ ¼ Q ejF sin c v c
where L is the effective synthetic aperture length, and sinðxÞ x
(23)
where f1 and f2 are the beginning and ending range frequency of accumulation, respectively, and Z f2 Q0 ¼ CjPðf þ Df Þj2 jPðf Þj2 df . (26)
(20)
j0 being a constant phase. The Fourier transform of Sbf(f, tm) can be expressed as pL 2Df fd þ vr (21) Sbf ðf ; f d Þ Kðf ÞejF sin c v c
Kðf Þ ¼ C 1 Pðf þ Df ÞP ðf Þ; sin cðxÞ ¼
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200
0 -200
-100
0 fd[Hz]
Fig. 2. Comparison of Src(fd) and Spbf(fd): (a) Src(fd) and (b) Spbf(fd).
100
200
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process of the equivalence of Sbf(f, tm) (expression (8)) to a single-frequency signal (expression (19)), leaving the information of the slant-range velocity only. (c) As a matter of fact, from the noise suppression point of view, the effect of coherent accumulation is equivalent to that of range compression. However, the RCM exerts no effect on our algorithm, because there is no loss of synthetic aperture length during transforming Spbf(tm) (expression (25)) into Doppler domain, since Spbf(tm) is a one-dimensional function about tm and independent of fast time. (d) It should be noted that Spbf(fd) (27) is not a replica of Src(fd) (6), with a shape compression ratio f0/Df, because the Fourier transformation of Src(t, tm) is a ‘‘local’’ transformation, or in other words, Src(fd) is an approximate replica of Src(t, tm) (5) with a certain transformation scale, whereas the Fourier transformation of Spbf(tm) is a ‘‘global’’ transformation, having in mind that a SFRC can be approximately equivalent to a single-frequency signal. We can also comprehend this fact from the following example. As we know, the shape of the Fourier transform of a chirp whose FM rate is quite large with a rectangle envelope is also an approximate rectangle, but the shape of the Fourier transform of a SFRC with a rectangle envelope is quite similar to that of a sin c function. In fact, the maximum-value point of Spbf(fd) contains the information of the whole Spbf(tm) and we can consider the maximumvalue point of Spbf(fd) as the ‘‘gravity center’’ of Src(fd) scaled by a ratio Df/f0. This fact is helpful for the improvement of the estimation accuracy of our algorithm, having in mind that the noise, multiplicative speckle and static clutter exist in the Doppler spectrum. (e) In addition, from expression (24), we can find that the phase of Spbf(f, tm) is independent of range frequency f, thus avoiding the dependency of the shape of range frequency spectrum |P(f)|, noting that such dependency is present in MLBF algorithm [7,8].
unamb unamb It is clear that the ratio between v^ r_bf and v^ r_rc is
Z¼
f0 . Df
(32)
unamb Because Df5f0, Zc1 which means that v^ r_bf is much unamb larger than v^ r_rc . Normally, the static clutter can bias the estimate of slantrange velocity, so we must suppress the static clutter. The digital spotlight algorithm is a well method for suppressing static clutter and noise. This method works by separating the moving target in the image domain using a window and then re-constructing its signature to the required domain. This window must entirely cover the moving target; otherwise the re-constructed signature will be distorted. To suppress static clutter, we perform the digital spotlight algorithm in the image domain via cropping up the moving target in the image using a window [6,11]. In summary, we propose the BFCA methodology to estimate the slant-range velocity of a moving target. As follows are the estimation procedures:
(i) Use a high-pass filter in the 2D frequency domain with stop-band adjusted to filter out static targets, i.e., use the external-band-filtering algorithm, and then implement imaging using static parameters and the moving targets are detected [6]. (ii) For each detected moving target, digitally spotlight the moving target image in image domain using a window and re-synthesize its signature back to range frequency and slow time domain and we get Sp(f, tm)2. (iii) Perform the BF algorithm on Sp(f, tm) and we get Spbf (f, tm). (iv) Accumulate Spbf(f, tm) with f and we get Spbf(tm). (v) Transform Spbf(tm) into Doppler domain and we get Spbf(fd). (vi) Find the maximum-value point fbf of |Spbf(fd)|. Through expression (29), we acquire the estimate of the slant-range velocity finally.
4. The case of multiple scatters Let fbf denote the maximum-value point of |Spbf(fd)|. Then, we can get the estimate of the slant-range velocity through the expression v^ r ¼
c f bf . 2 Df
(29)
Hence the maximum unambiguous velocity can be found, by letting fbf ¼ PRF/2, as unamb v^ r_bf ¼
c PRF. 4Df
(30)
unamb Because the coefficient c/(4Df) is large, v^ r_bf can be very large for a given PRF. Obviously, the maximum unambiguous velocity corresponding to the range-compressed Doppler spectrum Src(fd) (6) is unamb v^ r_rc ¼
cPRF . 4f 0
(31)
In above sections, we only consider the case of one moving target. However, as discussed in Section 3, in estimating the slant-range velocity, we must suppress the static clutter. Even if after performing the digital spotlight algorithm, there is a little amount of static clutter. Hence, it is necessary to analyze the effect of static clutter on the estimation. We first consider the case of two moving targets for sake of simplicity and then generalize to the case when multiple static scatters and a moving target exist together. Consider two moving targets whose slant-range velocities are vr1 and vr2. Their respective coordinates are (x1, y0) (x2, y0) (Because x5y, we assume the two targets have the same slant-range coordinate.), and their respective back reflectivity coefficients are A1 and A2. 2 Here we need not performing the matching algorithm with P*(f), because the matching has been performed during the imaging.
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The range frequency matched signature of the two targets in (f, tm) can be expressed as Sp ðf ; t m Þ ¼ Sp1 ðf ; t m Þ þ Sp2 ðf ; t m Þ vtm x1 jj1 ðf ;tm Þ ¼ A1 jPðf Þj2 g e y 0 vtm x2 jj2 ðf ;tm Þ þ A2 jPðf Þj2 g e y0
Q 11 ¼ jAj21 C 11 F11 ¼
(33)
where 2pðf þ f 0 Þ 2pðf þ f 0 Þ y0 2 vr1 t m c c 2pðf þ f 0 Þ x1 2pðf þ f 0 Þ v2 2 þ2 vtm t c c y0 y0 m
and 2pðf þ f 0 Þ 2pðf þ f 0 Þ y0 2 vr2 t m j2 ðf ; t m Þ ¼ 2 c c 2pðf þ f 0 Þ x2 2pðf þ f 0 Þ v2 2 þ2 vtm t . c c y0 y0 m
Z
Q 12 ¼ A1 A2 C 12
Dvr ¼ v1 v2 ;
Performing the BF algorithm, we obtain Spbf ðf d Þ ¼ Spbf 11 ðf d Þ þ Spbf 22 ðf d Þ þ Spbf 12 ðf d Þ þ Spbf 21 ðf d Þ pL 2Df fd þ vr1 ¼ Q 11 ejF11 sin c v c pL 2Df fd þ vr2 þ Q 22 ejF22 sin c v c
pL 2Df 2 2 Dx fd þ vr1 þ f 0 Dvr f 0 v v c c c y0 pL 2Df 2 2 Dx jF21 fd þ vr2 f 0 Dvr þ f 0 v sin c þ Q 21 e v c c c y0
þ Q 12 ejF12 sin c
(36) where Table 1 Simulation parameters Parameter
Value
Carrier frequency Chirp bandwidth Swath central slant-range Platform velocity Antenna radiation pattern PRF Df Effective synthetic aperture length
9 GHz 130 MHz 10 km 120 m/s Gaussian function 300 Hz 37.5 MHz 300 m
f2
f1
(37)
Pðf þ Df Þ 2 Pðf Þ 2 df ,
2pDfx22 4pDf y0 þ ¼ þ j0 c cy0
Q 21 ¼ A1 A2 C 21
(35)
jPðf þ Df Þj2 jPðf Þj2 df ,
f1
j1 ðf ; t m Þ ¼ 2
(34)
f2
2pDfx21 4pDf y0 þ þ j0 c cy0
Q 22 ¼ jAj22 C 22 F22
Z
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Z
f2
(38)
jPðf þ Df Þj2 jPðf Þj2 df
(39)
jPðf þ Df Þj2 jPðf Þj2 df
(40)
f1
Z
f2 f1
Dx ¼ x1 x2
(41)
C11, C22, C12, C21, and j0 are constants, and F12 and F21 are the cross-beat spectra phase of Spbf12(fd) and Spbf21(fd), respectively. There are four terms in (36). The first two terms are the auto-beat spectra generated by the two targets of themselves and the last two terms are cross-beat spectra generated by the cross beating between the two targets. From (36), it is found that for moving targets with different slant-range velocities, their auto-beat spectra appear at different positions in the fd axis, so the proposed method can differentiate multiple moving targets, as long as their slant-range velocities are different. Suppose we use a rectangle window of area Dx Dy centered at (x0, y0) to digitally spotlight the moving target. Suppose that the residual static scatters contained in the window are uniformly distributed and none of which is predominant. Then the signal after performing the spotlight algorithm can be expressed as Spbf ðf d Þ ¼ Spbf_vr ðf d Þ þ Spbf_c1 ðf d Þ þ Spbf_c2 ðf d Þ þ Spbf_c_vr ðf d Þ (42) where Spbf_vr ðf d Þ is the auto-beat spectrum about the moving target, Spbf_c1 ðf d Þ denotes the sum of the auto-beat spectra of the static scatters, Spbf_c2 ðf d Þ denotes the sum of the cross-beat spectra among the static scatters, and Spbf_c_vr ðf d Þ refers to the sum of the cross-beat spectra between the moving target and the static scatters.
Table 2 Estimation results of Case 1 (range frequency spectrum is flat) Algorithm
Real velocity (m/s) 0.26
1.93
15.24
26.65
43.76
MLBF Estimate (m/s) Absolute error (m/s) Relative error (%)
0.2681 0.0081 3.1242
1.9553 0.0253 1.3127
15.0868 0.1532 1.0052
26.4526 0.1974 0.7407
43.5340 0.2260 0.5165
BFCA Estimate (m/s) Absolute error (m/s) Relative error (%)
0.2655 0.0055 2.1181
1.9318 0.0018 0.0929
15.2438 0.0038 0.0247
26.6514 0.0014 0.0053
43.7537 0.0063 0.0144
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For Spbf_c_vr ðf d Þ, according to (36), Spbf_c_vr ðf d Þ is quite far away from Spbf_vr ðf d Þ along the fd axis. So, we can remove Spbf_c_vr ðf d Þ easily. But Spbf_c1 ðf d Þ and Spbf_c2 ðf d Þ are quite close with Spbf_vr ðf d Þ, we cannot remove them conveniently. jSpbf_c1 ðf d Þj computed at fd ¼ 0 can be expressed as 2pDfx0 2pDf Dy jSpbf_c1 ðf d Þjf d ¼0 ¼ DxDyjQ j sin c Dx sin c c cy0 (43) where 2
Q ¼ s C1
Z
f2
2
jPðf þ Df Þj jPðf Þj df , sinðxÞ . x
(44)
(The derivation of (43) lies in Appendix B.) From (43), we can see that the magnitude of jSpbf_c1 ðf d Þj at zero Doppler is smaller than the sum of the magnitude of the auto-beat spectra of the static scatters, DxDyjQ j, because of the interference among the auto-beat spectra. In fact, jSpbf_c1 ðf d Þj at zero Doppler is modulated by j sin cðð2pDfx0 =cy0 ÞDxÞ sin cðð2pDf =cÞDyÞj which is smaller than unity. Noting that the static-scatter-echo reflectivity coefficients An ; n ¼ 1; . . . ; N are mutually independent and have phase uniformly distributed in a 2p interval Table 3 Estimation results of Case 2 (range frequency spectrum is not flat, vr ¼ 0.26 m/s) MLBF
Estimate (m/s) Absolute error (m/s) Relative error (%)
0.2158 0.0442 16.9976
BFCA
Estimate (m/s) Absolute error (m/s) Relative error (%)
0.2655 0.0055 2.1181
5. Experimental results 5.1. Simulated data In this subsection, first, we compare the proposed BFCA algorithm with the MLBF algorithm, and then analyze the performance of the proposed algorithm when noise exists. In order to compare the MLBF algorithm and the proposed algorithm, we consider two cases. In Case 1, the range frequency spectrum is flat and in Case 2, the range frequency spectrum is not flat. For the sake of simplicity, we assume there is only one moving target and no noise and static clutter exists in both cases. The simulation parameters are listed in Table 1. The estimation results of Cases 1 and 2 are shown in Tables 2 and 3, respectively. From Table 2, we can see that when the slant-range velocity is small, (i.e. the RCM is small), the estimates are slightly biased in both algorithms, but when the slantrange velocity (or the RCM) become larger, the results given by the MLBF algorithm are biased significantly, whereas the results given by the proposed algorithm are still very accurate. We can also see this fact in Fig. 3. The beat spectra of vr ¼ 15.24 m/s given by the MLBF algorithm and the proposed BFCA algorithm are shown in Fig. 3, where (a) and (b) show the beat spectra given by the MLBF algorithm and by the BFCA algorithm, respectively. It is clear that the beat spectrum given by the MLBF is wider than that given by the proposed BFCA algorithm, due to the loss of synthetic aperture length, which tends to bias the estimate of slant-range velocity. Note that we probably cannot use the proposed method to solve the
2
f1
s2 ¼ E½jAn j2 ; sin cðxÞ ¼
independent of its magnitude, interference also exists in the components of Spbf_c2 ðf d Þ. In fact, the interference is advantageous for the slant-range velocity estimation. But, of course, in order to raise the estimation accuracy, we still need suppress the static clutter.
14
12000
x 107
12
10000
10 power
power
8000 6000 4000
6 4
2000 0 -150 -100
8
2
-50
0 fd[Hz]
50
100
150
0 -150 -100
-50
0 fd[Hz]
50
100
150
Fig. 3. Beat spectra of vr ¼ 15.24 m/s given by the MLBF algorithm and the proposed BFCA algorithm, (a) and (b) show the beat spectra given by the MLBF algorithm and by the BFCA algorithm, respectively.
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problem of RCM in estimating the Doppler Centroid, because we cannot apply the external-band filtering algorithm and spotlight algorithm to the static clutter, since all the scatters are static. Cumming and Li [8] give some approaches to solve the problem of RCM in estimating the Doppler Centroid. From Table 2, we can also see that the proposed algorithm can, indeed, estimate a relatively larger slantrange velocity unambiguously. In fact, the maximum unambiguous velocity given by (30) under the parameters listed in Table 1 is 600 m/s. In order to accentuate the effect of the shape of range frequency spectrum on the two algorithms, we estimate a relatively small slant-range velocity, vr ¼ 0.26 m/s. Table 3 gives the estimation results of Case 2 when the range frequency spectrum is not flat. From Table 3, it is found that when the range frequency spectrum is not flat, the result given by the MLBF is quite biased, whereas that given by the proposed BFCA remains the same as the case when the range frequency spectrum is flat.
standard deviation of estimates[m/s]
0.07
Table 4 Mission parameters Parameter
Value
Carrier frequency Chirp bandwidth Swath central slant-range Platform velocity PRF Df Effective synthetic aperture length
8.85 GHz 40 MHz 9090 m 115 m/s 1000 Hz 3.52 MHz 485 m
Now let us analyze the performance of proposed algorithm when noise exists. The variation of standard deviation of estimate of the slant-range velocity versus signal-to-noise ratio (SNR)3 is given in Fig. 4. Fig. 5 shows the variation of standard deviation of estimate of the slant-range velocity versus the number of accumulation. From Fig. 4, it is found that the proposed algorithm can work with a low SNR scene. Moreover, from Fig. 5, it is seen that the more number of accumulation, the more accurate the estimate is.
0.06
5.2. Real data
0.05
In this subsection, experiments are performed with real SAR data to test the performance of the proposed methodology when static clutters together with noise exist. The clutter and noise background is real data, and four simulated moving targets with different velocities are inserted in the raw SAR data. In re-synthesizing the signatures of the moving targets from image domain to (f, tm) domain, we take steps as follows: (1) cross-range direction FFT, (2) multiply the signal with the cross-range compress function to restore the cross-range phase of the moving target, (3) cross-range IFFT, and (4) range FFT. The signal-to-static clutter plus noise ratio (SCNR) of the experiments is 14 dB. The Mission Parameters and estimates of the slantrange velocity of four moving targets are shown in Tables 4 and 5, respectively. The image after clutter suppression is shown in Fig. 6 and the four moving targets are labeled in the image. The re-synthesized (after performing spotlight algorithm) signature of the third moving target in slow time and range frequency domain is presented in Fig. 7. The beat spectrum of the third moving target is illustrated in Fig. 8. From Table 5, we see that the proposed method works well with real static clutter background. Now let us analyze the facts affecting the accuracy of the proposed methodology. Here, we discuss two major factors as follows
0.04 0.03 0.02 0.01 0
0
5
10 15 SNR[dB]
20
25
Fig. 4. Variation of standard deviation of estimate versus SNR.
0.07 standard deviation of estimates[m/s]
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SNR=10dB
0.06 0.05 0.04 0.03 0.02 0.01
0
100
200 300 400 number of accumulation
500
600
Fig. 5. Variation of standard deviation of estimate versus the number of accumulation.
(1) The signal-to-clutter-noise ratio (SCNR) SCNR affects the accuracy of estimates most. As far as our methodology is concerned, the method performs well as long as the SCNR is more than 11 dB (after the 3 We define SNR as the ratio between the squared magnitude of the correctly focused moving target and the variance of the noise.
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Table 5 Estimation results, SCNR ¼ 14 dB Moving targets
Real velocity (m/s)
Estimates (m/s)
Absolute error (m/s)
Relative error (%)
1 2 3 4
6.3 11 15.7 28.88
6.1745 11.0309 15.6098 28.9301
0.1255 0.0309 0.0902 0.0501
1.9914 0.2811 0.5746 0.1736
x 1013 9 8 7
power
6 5 4 3 2 1 Fig. 6. Image after clutter suppression. The four moving targets are labeled in the image.
0 -10
-8
-6
-4
-2
0 2 fd[Hz]
4
6
8
10
Fig. 8. Beat spectrum of the third moving target.
-1.5 -1
tm[s]
-0.5 0 0.5 1 1.5 -3
-2
-1
0 f[Hz]
1
2
3 x 107
Fig. 7. Signature of the third moving target in range frequency and slow time domain.
elimination of the clutter and the implementation of the digital spotlight algorithm). (2) The number of accumulation In fact, the more number of independent samples used to accumulate Spbf(f, tm), the more accurate the estimate is.
The aim of doing so is to resolve the slant-range velocity ambiguity. The ‘‘global’’ Fourier transformation of such SFRC can guarantee the estimation accuracy. The proposed BFCA algorithm is not susceptive to the noise because we apply coherent accumulation in implementing the algorithm. Moreover, the proposed methodology can deal with large RCM since there is no loss of synthetic aperture length when we get the beat spectrum. The algorithm can gain a high accuracy especially for the fast-moving targets. The methodology proposed in this paper uses the external-band filtering algorithm in 2D frequency domain to detect the moving target. One of the disadvantages of such detection algorithm is that when the signatures of the moving target and static clutters overlap completely in 2D frequency domain or when the SCNR of the scene is low, it is difficult to detect the moving target. Probably we can adopt more sophisticated detection algorithm, such as the detection algorithm proposed in [2], to detect the moving target, at the expense of higher computational complexity.
Acknowledgement 6. Conclusion A SFRC can be produced by performing the BF algorithm in the slow time and range frequency domain.
The authors are grateful to the National Laboratory of Radar Signal Processing for providing the real SAR data. This work was supported in part by the National Nature Science Foundation of China under Grant 60402039.
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Appendix A. The proof of the theorem of SFRC
we can have
In this section, we will prove the theorem of SFRC. Proof: For simplicity, w(ttc) can be equivalent to a rectangle with width of Dt, magnitude of unity and tc as the center. Then, the Fourier transform of s(t) (11) can be written as
Z pffiffiffiffi 2 2q 1 2 pffiffiffiffi ejpx =2 dx , 2g 2q Z pffiffiffiffiffiffiffiffi pffiffiffiffi 2 1=2qþ 2q 1 2 jpx2 =2 dx , jSðf 1 Þj ¼ pffiffiffiffiffiffiffiffi pffiffiffiffi e 2g 1=2q 2q Z pffiffiffiffiffiffiffiffi pffiffiffiffi 2 1 1=2qþ 2q 2 jSðf 3 Þj2 ¼ pffiffiffiffiffiffiffiffi pffiffiffiffi ejpx =2 dx . 2g 1=2q 2q
Sðf Þ ¼
Z
þ1
sðtÞ ej2pft dt ¼
Z
t c þDt=2
ej½pgt
2
þ2pðbf Þt
dt.
(A.1)
tc Dt=2
1
jSðf 2 Þj2 ¼
(A.5)
Let u ¼ t þ ðb f Þ=g and we obtain
jSðf Þj2 ¼
"Z
t c þDt=2þðbf Þ=g
#2 cosðpgu2 Þ du
t c Dt=2þðbf Þ=g
"Z
t c þDt=2þðbf Þ=g
þ
#2 sinðpgu2 Þ du
.
(A.2)
From digital calculation by computer, it is found that, when q changes from smaller values to larger values, f1 and f3 are the two extreme points to which the value of |S(f)|2 corresponding exceeds |S(f2)|2 the first time. We define F(q, r) as
t c Dt=2þðbf Þ=g
By letting d½jSðf Þj2 =df ¼ 0, we get three key extreme points among all the extremes of |S(f)|2 as follows 1 ; Dt
f 2 ¼ b þ gt c ;
f 3 ¼ b þ gt c þ
1 Dt
(A.3)
where f2 is no other than the central frequency of s(t). Now let us determine under what condition |S(f2)|2 is the maximum value of |S(f)|2. Letting gDt 2 4
(A.4)
Letting F(q, g) ¼ 0, we can acquire the threshold of q, at which |S(f1)|2 the first time exceeds |S(f2)|2 when q changes from smaller values to larger values through digital calculation, as qT ¼ 1:181.
(A.7)
0.9
0.8
0.9
0.8
0.7
0.8
magnitude of the Fourier transform of SFRC
magnitude of the Fourier transform of SFRC
q¼
(A.6)
0.7 0.6 0.5 0.4 0.3 0.2
magnitude of the Fourier transform of SFRC
f 1 ¼ b þ gt c
Fðq; gÞ ¼ jSðf 2 Þj2 jSðf 1 Þj2 2 pffiffiffiffi 2 Z pffiffiffiffiffiffiffiffi pffiffiffiffi 2 3 Z 2q 1=2qþ 2q 1 4 jpx2 =2 jpx2 =2 ¼ dx pffiffiffiffiffiffiffiffi pffiffiffiffi e dx 5. pffiffiffiffi e 2g 2q 1=2q 2q
0.6
0.5
0.4
0.3
0.2
0.1
0.1 0 -5
0 f[Hz]
5
0 -5
0.7 0.6 0.5 0.4 0.3 0.2 0.1
0 f[Hz]
5
0 -5
0 f[Hz]
Fig. 9. Variation of |S(f)|2 versus q: (a) q ¼ 0.5, (b) q ¼ qT ¼ 1.181, and (c) q ¼ 1.21.
5
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And g can have arbitrary values. Therefore, to make |S(f2)|2 be the maximum value of |S(f)|2, q must satisfy
q¼
gDt 2 oqT ¼ 1:181. 4
(A.8)
The variation of jSðf Þj2 versus q is shown in Fig. 9. If q satisfy (A.8), s(t) can be equivalent to a singlefrequency signal as follows se ðtÞ ¼ ejj wðtÞej2pat
(A.9)
Noting that the following expression generally holds under the stripe SAR data parameters 2Df Dx2 pop. cy0 4
(B.18)
According to the theorem of SFRC, the quadratic term ð2pDf =cy0 Þm2 can be omitted. Then we have Spbf_c1 ðf d Þ f
2
d ¼0
where e is a real constant, and
¼ Q ej½j0 ð4pDf =cÞy0 þð2pDfx0 =cy0 Þ Z Dx=2 ejð4pDfx0 =cy0 Þm dm Dx=2
a ¼ f 2 ¼ b þ gt c .
Z
(A.10)
Dy=2
ejð4pDf =cÞn dn
Obviously, the Fourier transform of se(t) is
Dy=2 2
Se ðf Þ ¼ Dt sinc½pDtðf aÞejj . Substitution of a into (A.1) yields R 9 2 R Dt=2 2 2 Dt=2 SðaÞ ¼ ejpgtc Dt=2 ejpgt dt ¼ Dt=2 ejpgt dt ejZ > > = pffiffiffiffi F 2q s 2 >
pffiffiffiffi > Z ¼ pgt c þ arctan ; Fc
(A.12)
Rx where F s ðxÞ ¼ 0 sinðpt 2 =2Þ dt; F c ðxÞ ¼ 0 cosðpt 2 =2Þ dt are the Fresnel functions. Comparing (A.11) and (A.12), we have 2 pffiffiffiffiffiffi3 Fs 2q 2 (A.13) j ¼ Z ¼ pgt c þ arctan4 pffiffiffiffiffiffi5. 2q Fc And, the equivalent width of S(f) approximately equals 1 . B ðf 2 f 1 Þ ¼ Dt
(A.14)
Appendix B. . Derivation of expression (43) In this section, we will derive the expression (43). According to (36), we have Z x0 þDx=2 Z y0 þDy=2 Q ejFðx;yÞ dxdy (B.15) Spbf_c1 ðf d Þ f ¼0 ¼ x0 Dx=2
y0 Dy=2
where Q ¼ s2 C 1
R f2 f1
9 jPðf þ Df Þj2 jPðf Þj2 df = 2
2pDfx þ j0 Fðx; yÞ ¼ 4pDf c y þ cy 0
;
(B.16)
C1 and j0 are constants, and s2 ¼ E½jAn j2 is the average echo magnitude of the static clutter. Let m ¼ x x0 ; n ¼ y y0 and we obtain 2 Spbf_c1 ðf d Þ f ¼0 ¼ Q ej½j0 ð4pDf =cÞy0 þð2pDfx0 =cy0 Þ d Z Dx=2 2 ej½ð2pDf =cy0 Þm þð4pDfx0 =cy0 Þm dm Dx=2
Z
Dy=2
ejð4pDf =cÞn dn.
Dy=2
where sin cðxÞ ¼ sinðxÞ=x. We finally get
2q
Rx
d
¼ DxDyQ ej½j0 ð4pDf =cÞy0 þð2pDfx0 =cy0 Þ 2pDfx0 2pDf Dy (B.19) sin c Dx sin c c cy0
(A.11)
(B.17)
jSpbf_c1 ðf d Þjf d ¼0 ¼ DxDyjQ j 2pDfx0 2pDf Dy . sin c Dx sin c c cy0 (B.20)
References [1] P. Marques, J. Dias, Optimal detection and imaging of moving objects with unknown velocity, In Proceedings of the Third European Conference on SAR(EUSAR 2000), 2000, pp. 561–564. [2] J.M.B. Dias, P.A.C. Marques, Multiple moving target detection and trajectory estimation using a single SAR sensor, IEEE Trans. Aerosp. Electron. Syst. 39 (2) (April 2003) 604–623. [3] S. Barbarossa, Detection and imaging of moving objects with synthetic aperture radar, IEE Proc. Pt. F 139 (February 1992) 79–88. [4] J. Legg, A. Bolton, D. Gray, SAR moving target detection using nonuniform PRI, In Proceedings of the First European Conference on SAR (EUSAR’96), 1996, pp. 423–442. [5] A.C.M. Paulo, M.B.D. Jose, SAR moving objects trajectory estimation: solving the blind angle ambiguity with a single sensor, 2000 IEEE. [6] A.C.M. Paulo, M.B.D. Jose, Velocity estimation of fast moving targets suing a single SAR sensor, IEEE Trans. Aerosp. Electron. Syst. 41 (1) (January 2005) 75–89. [7] F. Wong, I.G. Cumming, A combined SAR Doppler centroid estimation scheme based upon signal phase, IEEE Trans. Geosci. Remote Sensing 34 (3) (1996). [8] I.G. Cumming, S. Li, Adding sensitivity to the MLBF Doppler centroid estimator, IEEE Trans. Geosci. Remote Sensing 45 (2) (2007) 279–292. [9] M. Born, E. Wolf, Principles of Optics, sixth ed., Pergamon, Elmsford, NY, 1983. [10] S.N. Madsen, Estimating the Doppler centroid of SAR data, IEEE Trans. Aerosp. Electron. Syst. 25 (1989) 134–140. [11] M. Soumekh, Moving target detection and imaging using an X band along-track monopulse SAR, IEEE Trans. Aerosp. Electron. Syst. 38 (1) (2002).
Contents lists available at ScienceDirect
Signal Processing journal homepage: www.elsevier.com/locate/sigpro
Slant-range velocity estimation based on Small-FM-Rate Chirp Baochang Liu , Tong Wang, Zheng Bao The National Laboratory of Radar Signal Processing, Xidian University, Xi’an, Shanxi 710071, China
a r t i c l e i n f o
abstract
Article history: Received 12 December 2007 Received in revised form 18 March 2008 Accepted 21 April 2008 Available online 6 May 2008
This paper deals with the estimation of the slant-range velocity of a moving target with airborne SAR data. For the signature of a moving target, performing the Beat Frequency (BF) algorithm in the slow time and range frequency domain can generate a Small-FMRate Chirp (SFRC), which has some features of a single-frequency signal. The Fourier transform of such SFRC quite approximates a sinc function, and the maximum-value point of such Fourier transform is proportional to the slant-range velocity. Based on these principles, a methodology known as Beat Frequency Coherent Accumulation (BFCA) algorithm is developed to estimate the slant-range velocity. The prominent advantage of the BFCA algorithm is that the unambiguous velocity can be very large. Besides, the proposed methodology can deal with large range cell migration (RCM) and can work with a low signal-to-noise ratio (SNR) scene, requiring a small amount of computation but yields a high accuracy especially for the fast-moving targets. The effectiveness and efficiency of the proposed methodology are validated with both simulated and real data. & 2008 Elsevier B.V. All rights reserved.
Keywords: SAR Slant-range velocity estimation Small-FM-Rate Chirp (SFRC) Beat Frequency Coherent Accumulation (BFCA) Range cell migration (RCM)
1. Introduction As is well known, the slant-range velocity induces a Doppler-shift in Doppler domain, and a shifted azimuth location in the image domain, correspondingly. Therefore, the estimation of slant-range velocity is of great importance not only for obtaining the motion parameter, but also for re-locating the moving target. In estimating the slant-range velocity with one antenna, we must take the following three facts into account. The first one is the problem of velocity ambiguity, due to the limitation of PRF. The second one is the problem of range cell migration (RCM). As we know, most of the slant-range velocity estimation algorithms performing in the range-compressed domain are affected by large RCM, because of the loss of synthetic aperture length. The third one is the effect of residual static clutter. Generally speaking, the remaining static clutter will bias the estimates of slant-range velocity. So the algorithm used
Corresponding author.
E-mail address: [email protected] (B. Liu). 0165-1684/$ - see front matter & 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2008.04.013
to estimate the slant-range velocity must have an ability to differentiate the moving target from the remaining static clutter. Several methods have been presented to estimate the slant-range velocity with a single antenna in recent literature [1–6]. They mainly exploit both the information of the antenna radiation pattern and the information of the phase of the returned echo. In [2], a methodology is proposed which utilizes the structure of the amplitude and the phase modulations of the returned echo from a moving target in the Fourier domain. In this algorithm, a generalized likelihood ratio test approach is adopted to detect moving target and estimate the motion parameters, provided that the static clutter can be modeled as a zero mean complex white Gaussian process. The main disadvantage of this method is the heavy computation burden. The method proposed in [4] uses a non-uniform PRF to solve the problem of velocity ambiguity. However, the use of a non-uniform PRF requires a non-conventional pulse scheduling, thus introducing complexity in image reconstruction algorithms [6]. The method presented in [6] works by performing an auto-correlation algorithm in the two-dimensional (2D) frequency domain.
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In [7], an algorithm known as Multiple Look Beat Frequency (MLBF) is presented to estimate the Doppler Centroid. The algorithm is based on multiplying two range looks together to generate a beat signal and using FFT to estimate the resultant beat frequency (BF). As we know, Doppler Centroid estimation and slant-range velocity estimation share some common features. However, the MLBF algorithm cannot directly be used to estimate the slant-range velocity, because the data used for estimating the Doppler Centroid comprises a large number of static scatters, whereas the slant-range velocity estimation data that includes a point-like moving target must contain as little static clutter as possible. Moreover, it is often the case that a large RCM induced by the slant-range velocity exists, which cannot be dealt with by the MLBF algorithm. Besides, the MLBF algorithm requires the range frequency spectrum be ‘‘flat’’ [8], which is not satisfied properly in actual case. Hence, to estimate the slant-range velocity, we must modify the MLBF algorithm. Based on the principle of ‘‘beat frequency’’, a methodology called Beat Frequency Coherent Accumulation (BFCA) is proposed to estimate the slant-range velocity in this paper. Performing BF algorithm in slow time and range frequency domain can yield a Small-FM-Rate Chirp (SFRC). Such SFRC and a single-frequency signal have some common attributes as well as different features. For instance, the Fourier transform of SFRC exhibits a shape quite similar to that of the sinc function. Nevertheless, the maximum-value point of the Fourier transform of SFRC is related to its envelope, whereas the Fourier transform of a single frequency is independent of its envelope. For a large enough FM rate chirp, we can compute its Fourier transform through the stationary phase method. We cannot do so for SFRC, because the FM rate of SFRC does not meet the prerequisite of the stationary phase method. However, we will show that if this small FM rate signal satisfies certain condition, we can use another method to compute its Fourier transform. The primary aim of performing the ‘‘BF’’ is to resolve the velocity ambiguity. Because the BF algorithm works in the slow time and range frequency domain rather than in the rangecompressed domain just as the MLBF does, there is no loss of synthetic aperture length, thus making the proposed BFCA methodology free of the effect of RCM. Furthermore, the BFCA methodology can restrain the effect of noise, because of implementing coherent accumulation in performing the BFCA methodology. For the effect of residual static clutter, we adopt an algorithm called digital spotlight [6,11] to suppress the static clutter. In addition, the proposed methodology has the capability of differentiating the moving target from static clutter to some extent. The paper is organized as follows. In Section 2, we elaborate on the basic principle of the proposed methodology. The proposed methodology to estimate the slantrange velocity is given in Section 3. In Section 4, we discuss the case of multiple scatters. In Section 5, we present the experimental results illustrating the effectiveness of the proposed methodology and give the facts that affect the accuracy of the estimates. Section 6 supplies concluding remarks.
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2. The basic principle of the proposed methodology A SAR scenario in the slant-range plane is illustrated in Fig. 1, where coordinate x and y denote cross-range and slant-range, respectively. A radar travels along the cross-range direction at a constant velocity v. A moving target with slant-range velocity vr is illuminated. Suppose that the radar is located at (0, 0) and the moving target at (x0, y0) when tm ¼ 0, where tm is the slow time. The distance between the target and the radar at the instant tm is rðt m Þ ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðvt m x0 Þ2 þ ðy0 þ vr t m Þ2 .
(1)
The two-way antenna radiation pattern can be written as (2)
g½sinyðt m Þ
where tm is the slow time, y(tm) is the aspect angle, and g(x) is symmetric about x ¼ 0. In the radar synthetic aperture time, noting that y0cvrtm, we have sinyðt m Þ yðt m Þ
vtm x0 . y0
(3)
Then the received signal in range frequency and slow time domain can be written as Sðf ; t m Þ ¼ g½sinyðt m ÞPðf Þ ejð4pðf þf 0 Þ=cÞrðtm Þ
(4)
where f0 is the carrier frequency, f is range frequency corresponding to fast time t, c is the speed of light, P(f) is 2 the Fourier transform of transmitted pulse rðtÞejpgt , g and r(t) are the FM rate and envelope of the transmitted pulse, respectively. Then the range-compressed signal in terms of fast time and slow time can be expressed as 2rðt m Þ g½sinyðt m Þ ejð4pf 0 =cÞrðtm Þ Src ðt; t m Þ ¼ rrc t (5) c where rrc( ) is range envelope (after range compression), t and tm are fast time and slow time, respectively. Assuming that ð2rðt m Þ=cÞ ð2y0 =cÞ(y0 refers to the slant-range coordinate of the moving target), we can get the range-compressed Doppler spectrum of the moving
y
vr target(x0,y0)
(0,0) Fig. 1. SAR scenario.
x
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target using the stationary phase method [9] as follows c 2f f d þ 0 vr ejcðf d Þ (6) Src ðf d Þ ¼ Mg 2vf 0 c
be expanded in Taylor series about tm ¼ 0. Neglecting the small terms, we obtain jbf ðt m Þ ¼
where cðf d Þ ¼
(10)
v y pcy0 2 x0 2pf 0 y0 2 r 0 f þ vr f d þ 2p 2 2 v d 2f 0 v v v2 c þ
2pf 0 x20 4pf 0 x0 vr 4pf 0 y0 þ j0 cy0 vc c
(7)
and M and j0 are constants. From (7), it is found that the quadratic phase component of c(fd) is independent of the slant-range velocity and the information of the slantrange velocity is transformed into the linear and constant phase components. However, we cannot extract the information of the slant-range velocity from the linear phase component, since the azimuth coordinate of the moving target, x0, is unknown. Obviously, the range-compressed Doppler spectrum |Src(fd)| is a replica of the two-way antenna radiation pattern with a shifting factor of (2f0/c)vr. Therefore, we, spontaneously, will think of estimating the slant-range velocity by determining the maximum-value point of |Src(fd)| or by performing the energy balancing (DE) algorithm [10]. However, two problems are present in such algorithms: (i) The problem of velocity ambiguity: because the Doppler spectrum is subject to the limitation of PRF, the problem of velocity ambiguity will arise. (ii) The problem of RCM: in deducing (6), we neglect the effect of RCM. But in realistic case especially the case when estimating the slant-range velocity of a fastmoving target, the range-compressed signal will span over several range cells, so when we select one range cell (5) to get the Doppler spectrum, the obtained Doppler spectrum is usually distorted due to the loss of synthetic aperture length. Hence, the accuracy of estimate cannot be ensured. In order to solve above-mentioned problems, we must resort to other algorithm. Alternatively, let us beat S(f, tm) and S(f+Df, tm) ((4)), namely, multiply S(f, tm) by S*(f+Df, tm) and we get the following beat signal Sbf ðf ; t m Þ ¼ Sðf þ Df ; t m ÞS ðf ; t m Þ ¼ Pðf þ Df ÞP ðf Þg 2 ½sinyðt m Þ ejjbf ðtm Þ
(8)
where Df is a fixed range frequency interval, Df5f0, and 4pDf rðt m Þ. jbf ðt m Þ ¼ c
4pDf 4pDf 4pDf x0 2pDf v2 2 y0 vr t m þ vt m t . c c c y0 c y0 m
(9)
According to (9), it is found that jbf ðt m Þin (9) has the same structure as ð4pðf þ f 0 Þ=cÞrðt m Þin (4). The difference between jbf ðt m Þ and ð4pðf þ f 0 Þ=cÞrðt m Þ lies in that the range frequency in jbf ðt m Þ is Df, whereas the range frequency in ð4pðf þ f 0 Þ=cÞrðt m Þ is f+f0. Hence, the information of the slant-range velocity is not lost after implementing the BF algorithm. Now, let us transform Sbf(f, tm) into the Doppler domain. Assuming that x05y0, vrtm5y0, jbf(tm) (9) can
Noting that the FM rate of Sbf(f, tm), ð2Df =cÞðv2 =y0 Þ, is very small, we cannot get its Fourier transform using the stationary phase method. Alternatively, we can use the following theorem to get the Fourier transform of Sbf(f, tm). Theorem. of Small-FM-Rate Chirp (SFRC) Consider a Small-FM-Rate chirp (SFRC) sðtÞ ¼ wðt t c Þ ejðpgt
2
þ2pbtÞ
(11)
where w(ttc) is the envelope of s(t), its effective length is Dt, and w(ttc) is symmetric about tc; b is a constant frequency; g is the FM rate of s(t), and it is so small that the following expression holds1 q¼
jgjDt 2 oqT ¼ 1:181. 4
(12)
Then, the chirp s(t) can be equivalent to a singlefrequency signal as follows, from the point of view that the Fourier transforms of the equivalent signal and the original signal have the identical maximum-value point, the identical constant phase computed at the maximumvalue point and the approximately same bandwidth se ðtÞ ¼ ejj wðtÞej2pat
(13)
where a ¼ gt c þ b; j ¼
pgt 2c
2 pffiffiffiffiffiffi3 Fs 2q gDt 2 þ arctan4 pffiffiffiffiffiffi5; q ¼ 4 2q Fc
(14) Rx Rx F s ðxÞ ¼ 0 sinðpt 2 =2Þdt; F c ðxÞ ¼ 0 cosðpt 2 =2Þdt are the Fresnel functions, and e is a real constant. And the Fourier transform of s(t) reveals a shape similar to that of sin c function and has a maximum-value point at a ¼ gt c þ b.
(15)
The effective width of the Fourier transform of s(t) approximately equals B
1 . Dt
(16)
(The proof of this theorem lies in Appendix A) Let sin y(tm) ¼ 0 (3) and we have tc ¼
x0 . v
(17)
Note that the FM rate of Sbf(f, tm) (8) is gbf ¼ ð2Df =cÞðv2 =y0 Þ (see (10)) and the frequency of the linear phase component of Sbf(f, tm) is bbf ¼ ð2Df =cÞvr þ ð2Df =cÞðx0 =y0 Þv (see (10)). If the selected Df satisfies jgbf j ðL=vÞ2 =4o1:181 (L is the effective synthetic aperture 1
This threshold of q, 1.181, comes from Appendix A.
ARTICLE IN PRESS B. Liu et al. / Signal Processing 88 (2008) 2472–2482
Multiply S(f, tm) ((4)) by P*(f) and we get
length), according to the theorem of SFRC, substitution of tc ¼ (x0/v) into gbftc+bbf leads to an equivalent single frequency a¼
2Df vr . c
Sp ðf ; t m Þ ¼ g½sinyðt m ÞjPðf Þj2 ejð4pðf þf 0 Þ=cÞrðtm Þ . Thereby, (19) can be rewritten as vtm jF j2patm . e e Spbf ðf ; t m Þ ¼ CjPðf þ Df Þj2 jPðf Þj2 g 2 y0
(18)
Then Sbf(f, tm) (expression (8)) can be written as vtm jF j2patm Sbf ðf ; t m Þ ¼ CPðf þ Df ÞP ðf Þg 2 e e y0
After coherent accumulation, we obtain Z f2 vt m jF j2patm Spbf ðt m Þ ¼ Spbf ðf ; t m Þ df ¼ Q 0 g 2 e e y0 f1
(19)
where C is a constant, and F¼
2pDfx20 4pDf y0 þ þ j0 c cy0
where Z f2 Q¼ C 1 jPðf þ Df Þj2 jPðf Þj2 df .
(22)
C1 is a constant. From (21), we can see that the slant-range velocity is proportional to the maximum-value point of |Sbf(f, fd)|. This fact is the basic principle of our methodology.
(28)
(a) From (27), it is very clear that Spbf(fd) exhibits a much sharper shape than Src(fd) (6). Fig. 2 shows the comparison of Src(fd) and Spbf(fd). (b) From the derivation of above expressions, it is found that the information of the azimuth coordinate of the moving target is removed automatically, during the
14
1200
12
1000
10
800
8
x 107
power
power
1400
600
6
400
4
200
2
0 fd[Hz]
100
(27)
Now, let us make some explanations for the BF algorithm as follows.
It is often the case that an amount of noise exists in the SAR data. In order to estimate the slant-range velocity accurately, we must restrain the effect of the noise. We can do so by accumulating Sbf(f, fd) with respect to f.
-100
(25)
f1
3. The proposed methodology
0 -200
(24)
f1
Then (21) can be rewritten as pL 2Df fd þ vr Spbf ðf d Þ ¼ Q ejF sin c v c
where L is the effective synthetic aperture length, and sinðxÞ x
(23)
where f1 and f2 are the beginning and ending range frequency of accumulation, respectively, and Z f2 Q0 ¼ CjPðf þ Df Þj2 jPðf Þj2 df . (26)
(20)
j0 being a constant phase. The Fourier transform of Sbf(f, tm) can be expressed as pL 2Df fd þ vr (21) Sbf ðf ; f d Þ Kðf ÞejF sin c v c
Kðf Þ ¼ C 1 Pðf þ Df ÞP ðf Þ; sin cðxÞ ¼
2475
200
0 -200
-100
0 fd[Hz]
Fig. 2. Comparison of Src(fd) and Spbf(fd): (a) Src(fd) and (b) Spbf(fd).
100
200
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process of the equivalence of Sbf(f, tm) (expression (8)) to a single-frequency signal (expression (19)), leaving the information of the slant-range velocity only. (c) As a matter of fact, from the noise suppression point of view, the effect of coherent accumulation is equivalent to that of range compression. However, the RCM exerts no effect on our algorithm, because there is no loss of synthetic aperture length during transforming Spbf(tm) (expression (25)) into Doppler domain, since Spbf(tm) is a one-dimensional function about tm and independent of fast time. (d) It should be noted that Spbf(fd) (27) is not a replica of Src(fd) (6), with a shape compression ratio f0/Df, because the Fourier transformation of Src(t, tm) is a ‘‘local’’ transformation, or in other words, Src(fd) is an approximate replica of Src(t, tm) (5) with a certain transformation scale, whereas the Fourier transformation of Spbf(tm) is a ‘‘global’’ transformation, having in mind that a SFRC can be approximately equivalent to a single-frequency signal. We can also comprehend this fact from the following example. As we know, the shape of the Fourier transform of a chirp whose FM rate is quite large with a rectangle envelope is also an approximate rectangle, but the shape of the Fourier transform of a SFRC with a rectangle envelope is quite similar to that of a sin c function. In fact, the maximum-value point of Spbf(fd) contains the information of the whole Spbf(tm) and we can consider the maximumvalue point of Spbf(fd) as the ‘‘gravity center’’ of Src(fd) scaled by a ratio Df/f0. This fact is helpful for the improvement of the estimation accuracy of our algorithm, having in mind that the noise, multiplicative speckle and static clutter exist in the Doppler spectrum. (e) In addition, from expression (24), we can find that the phase of Spbf(f, tm) is independent of range frequency f, thus avoiding the dependency of the shape of range frequency spectrum |P(f)|, noting that such dependency is present in MLBF algorithm [7,8].
unamb unamb It is clear that the ratio between v^ r_bf and v^ r_rc is
Z¼
f0 . Df
(32)
unamb Because Df5f0, Zc1 which means that v^ r_bf is much unamb larger than v^ r_rc . Normally, the static clutter can bias the estimate of slantrange velocity, so we must suppress the static clutter. The digital spotlight algorithm is a well method for suppressing static clutter and noise. This method works by separating the moving target in the image domain using a window and then re-constructing its signature to the required domain. This window must entirely cover the moving target; otherwise the re-constructed signature will be distorted. To suppress static clutter, we perform the digital spotlight algorithm in the image domain via cropping up the moving target in the image using a window [6,11]. In summary, we propose the BFCA methodology to estimate the slant-range velocity of a moving target. As follows are the estimation procedures:
(i) Use a high-pass filter in the 2D frequency domain with stop-band adjusted to filter out static targets, i.e., use the external-band-filtering algorithm, and then implement imaging using static parameters and the moving targets are detected [6]. (ii) For each detected moving target, digitally spotlight the moving target image in image domain using a window and re-synthesize its signature back to range frequency and slow time domain and we get Sp(f, tm)2. (iii) Perform the BF algorithm on Sp(f, tm) and we get Spbf (f, tm). (iv) Accumulate Spbf(f, tm) with f and we get Spbf(tm). (v) Transform Spbf(tm) into Doppler domain and we get Spbf(fd). (vi) Find the maximum-value point fbf of |Spbf(fd)|. Through expression (29), we acquire the estimate of the slant-range velocity finally.
4. The case of multiple scatters Let fbf denote the maximum-value point of |Spbf(fd)|. Then, we can get the estimate of the slant-range velocity through the expression v^ r ¼
c f bf . 2 Df
(29)
Hence the maximum unambiguous velocity can be found, by letting fbf ¼ PRF/2, as unamb v^ r_bf ¼
c PRF. 4Df
(30)
unamb Because the coefficient c/(4Df) is large, v^ r_bf can be very large for a given PRF. Obviously, the maximum unambiguous velocity corresponding to the range-compressed Doppler spectrum Src(fd) (6) is unamb v^ r_rc ¼
cPRF . 4f 0
(31)
In above sections, we only consider the case of one moving target. However, as discussed in Section 3, in estimating the slant-range velocity, we must suppress the static clutter. Even if after performing the digital spotlight algorithm, there is a little amount of static clutter. Hence, it is necessary to analyze the effect of static clutter on the estimation. We first consider the case of two moving targets for sake of simplicity and then generalize to the case when multiple static scatters and a moving target exist together. Consider two moving targets whose slant-range velocities are vr1 and vr2. Their respective coordinates are (x1, y0) (x2, y0) (Because x5y, we assume the two targets have the same slant-range coordinate.), and their respective back reflectivity coefficients are A1 and A2. 2 Here we need not performing the matching algorithm with P*(f), because the matching has been performed during the imaging.
ARTICLE IN PRESS B. Liu et al. / Signal Processing 88 (2008) 2472–2482
The range frequency matched signature of the two targets in (f, tm) can be expressed as Sp ðf ; t m Þ ¼ Sp1 ðf ; t m Þ þ Sp2 ðf ; t m Þ vtm x1 jj1 ðf ;tm Þ ¼ A1 jPðf Þj2 g e y 0 vtm x2 jj2 ðf ;tm Þ þ A2 jPðf Þj2 g e y0
Q 11 ¼ jAj21 C 11 F11 ¼
(33)
where 2pðf þ f 0 Þ 2pðf þ f 0 Þ y0 2 vr1 t m c c 2pðf þ f 0 Þ x1 2pðf þ f 0 Þ v2 2 þ2 vtm t c c y0 y0 m
and 2pðf þ f 0 Þ 2pðf þ f 0 Þ y0 2 vr2 t m j2 ðf ; t m Þ ¼ 2 c c 2pðf þ f 0 Þ x2 2pðf þ f 0 Þ v2 2 þ2 vtm t . c c y0 y0 m
Z
Q 12 ¼ A1 A2 C 12
Dvr ¼ v1 v2 ;
Performing the BF algorithm, we obtain Spbf ðf d Þ ¼ Spbf 11 ðf d Þ þ Spbf 22 ðf d Þ þ Spbf 12 ðf d Þ þ Spbf 21 ðf d Þ pL 2Df fd þ vr1 ¼ Q 11 ejF11 sin c v c pL 2Df fd þ vr2 þ Q 22 ejF22 sin c v c
pL 2Df 2 2 Dx fd þ vr1 þ f 0 Dvr f 0 v v c c c y0 pL 2Df 2 2 Dx jF21 fd þ vr2 f 0 Dvr þ f 0 v sin c þ Q 21 e v c c c y0
þ Q 12 ejF12 sin c
(36) where Table 1 Simulation parameters Parameter
Value
Carrier frequency Chirp bandwidth Swath central slant-range Platform velocity Antenna radiation pattern PRF Df Effective synthetic aperture length
9 GHz 130 MHz 10 km 120 m/s Gaussian function 300 Hz 37.5 MHz 300 m
f2
f1
(37)
Pðf þ Df Þ 2 Pðf Þ 2 df ,
2pDfx22 4pDf y0 þ ¼ þ j0 c cy0
Q 21 ¼ A1 A2 C 21
(35)
jPðf þ Df Þj2 jPðf Þj2 df ,
f1
j1 ðf ; t m Þ ¼ 2
(34)
f2
2pDfx21 4pDf y0 þ þ j0 c cy0
Q 22 ¼ jAj22 C 22 F22
Z
2477
Z
f2
(38)
jPðf þ Df Þj2 jPðf Þj2 df
(39)
jPðf þ Df Þj2 jPðf Þj2 df
(40)
f1
Z
f2 f1
Dx ¼ x1 x2
(41)
C11, C22, C12, C21, and j0 are constants, and F12 and F21 are the cross-beat spectra phase of Spbf12(fd) and Spbf21(fd), respectively. There are four terms in (36). The first two terms are the auto-beat spectra generated by the two targets of themselves and the last two terms are cross-beat spectra generated by the cross beating between the two targets. From (36), it is found that for moving targets with different slant-range velocities, their auto-beat spectra appear at different positions in the fd axis, so the proposed method can differentiate multiple moving targets, as long as their slant-range velocities are different. Suppose we use a rectangle window of area Dx Dy centered at (x0, y0) to digitally spotlight the moving target. Suppose that the residual static scatters contained in the window are uniformly distributed and none of which is predominant. Then the signal after performing the spotlight algorithm can be expressed as Spbf ðf d Þ ¼ Spbf_vr ðf d Þ þ Spbf_c1 ðf d Þ þ Spbf_c2 ðf d Þ þ Spbf_c_vr ðf d Þ (42) where Spbf_vr ðf d Þ is the auto-beat spectrum about the moving target, Spbf_c1 ðf d Þ denotes the sum of the auto-beat spectra of the static scatters, Spbf_c2 ðf d Þ denotes the sum of the cross-beat spectra among the static scatters, and Spbf_c_vr ðf d Þ refers to the sum of the cross-beat spectra between the moving target and the static scatters.
Table 2 Estimation results of Case 1 (range frequency spectrum is flat) Algorithm
Real velocity (m/s) 0.26
1.93
15.24
26.65
43.76
MLBF Estimate (m/s) Absolute error (m/s) Relative error (%)
0.2681 0.0081 3.1242
1.9553 0.0253 1.3127
15.0868 0.1532 1.0052
26.4526 0.1974 0.7407
43.5340 0.2260 0.5165
BFCA Estimate (m/s) Absolute error (m/s) Relative error (%)
0.2655 0.0055 2.1181
1.9318 0.0018 0.0929
15.2438 0.0038 0.0247
26.6514 0.0014 0.0053
43.7537 0.0063 0.0144
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For Spbf_c_vr ðf d Þ, according to (36), Spbf_c_vr ðf d Þ is quite far away from Spbf_vr ðf d Þ along the fd axis. So, we can remove Spbf_c_vr ðf d Þ easily. But Spbf_c1 ðf d Þ and Spbf_c2 ðf d Þ are quite close with Spbf_vr ðf d Þ, we cannot remove them conveniently. jSpbf_c1 ðf d Þj computed at fd ¼ 0 can be expressed as 2pDfx0 2pDf Dy jSpbf_c1 ðf d Þjf d ¼0 ¼ DxDyjQ j sin c Dx sin c c cy0 (43) where 2
Q ¼ s C1
Z
f2
2
jPðf þ Df Þj jPðf Þj df , sinðxÞ . x
(44)
(The derivation of (43) lies in Appendix B.) From (43), we can see that the magnitude of jSpbf_c1 ðf d Þj at zero Doppler is smaller than the sum of the magnitude of the auto-beat spectra of the static scatters, DxDyjQ j, because of the interference among the auto-beat spectra. In fact, jSpbf_c1 ðf d Þj at zero Doppler is modulated by j sin cðð2pDfx0 =cy0 ÞDxÞ sin cðð2pDf =cÞDyÞj which is smaller than unity. Noting that the static-scatter-echo reflectivity coefficients An ; n ¼ 1; . . . ; N are mutually independent and have phase uniformly distributed in a 2p interval Table 3 Estimation results of Case 2 (range frequency spectrum is not flat, vr ¼ 0.26 m/s) MLBF
Estimate (m/s) Absolute error (m/s) Relative error (%)
0.2158 0.0442 16.9976
BFCA
Estimate (m/s) Absolute error (m/s) Relative error (%)
0.2655 0.0055 2.1181
5. Experimental results 5.1. Simulated data In this subsection, first, we compare the proposed BFCA algorithm with the MLBF algorithm, and then analyze the performance of the proposed algorithm when noise exists. In order to compare the MLBF algorithm and the proposed algorithm, we consider two cases. In Case 1, the range frequency spectrum is flat and in Case 2, the range frequency spectrum is not flat. For the sake of simplicity, we assume there is only one moving target and no noise and static clutter exists in both cases. The simulation parameters are listed in Table 1. The estimation results of Cases 1 and 2 are shown in Tables 2 and 3, respectively. From Table 2, we can see that when the slant-range velocity is small, (i.e. the RCM is small), the estimates are slightly biased in both algorithms, but when the slantrange velocity (or the RCM) become larger, the results given by the MLBF algorithm are biased significantly, whereas the results given by the proposed algorithm are still very accurate. We can also see this fact in Fig. 3. The beat spectra of vr ¼ 15.24 m/s given by the MLBF algorithm and the proposed BFCA algorithm are shown in Fig. 3, where (a) and (b) show the beat spectra given by the MLBF algorithm and by the BFCA algorithm, respectively. It is clear that the beat spectrum given by the MLBF is wider than that given by the proposed BFCA algorithm, due to the loss of synthetic aperture length, which tends to bias the estimate of slant-range velocity. Note that we probably cannot use the proposed method to solve the
2
f1
s2 ¼ E½jAn j2 ; sin cðxÞ ¼
independent of its magnitude, interference also exists in the components of Spbf_c2 ðf d Þ. In fact, the interference is advantageous for the slant-range velocity estimation. But, of course, in order to raise the estimation accuracy, we still need suppress the static clutter.
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12
10000
10 power
power
8000 6000 4000
6 4
2000 0 -150 -100
8
2
-50
0 fd[Hz]
50
100
150
0 -150 -100
-50
0 fd[Hz]
50
100
150
Fig. 3. Beat spectra of vr ¼ 15.24 m/s given by the MLBF algorithm and the proposed BFCA algorithm, (a) and (b) show the beat spectra given by the MLBF algorithm and by the BFCA algorithm, respectively.
ARTICLE IN PRESS B. Liu et al. / Signal Processing 88 (2008) 2472–2482
problem of RCM in estimating the Doppler Centroid, because we cannot apply the external-band filtering algorithm and spotlight algorithm to the static clutter, since all the scatters are static. Cumming and Li [8] give some approaches to solve the problem of RCM in estimating the Doppler Centroid. From Table 2, we can also see that the proposed algorithm can, indeed, estimate a relatively larger slantrange velocity unambiguously. In fact, the maximum unambiguous velocity given by (30) under the parameters listed in Table 1 is 600 m/s. In order to accentuate the effect of the shape of range frequency spectrum on the two algorithms, we estimate a relatively small slant-range velocity, vr ¼ 0.26 m/s. Table 3 gives the estimation results of Case 2 when the range frequency spectrum is not flat. From Table 3, it is found that when the range frequency spectrum is not flat, the result given by the MLBF is quite biased, whereas that given by the proposed BFCA remains the same as the case when the range frequency spectrum is flat.
standard deviation of estimates[m/s]
0.07
Table 4 Mission parameters Parameter
Value
Carrier frequency Chirp bandwidth Swath central slant-range Platform velocity PRF Df Effective synthetic aperture length
8.85 GHz 40 MHz 9090 m 115 m/s 1000 Hz 3.52 MHz 485 m
Now let us analyze the performance of proposed algorithm when noise exists. The variation of standard deviation of estimate of the slant-range velocity versus signal-to-noise ratio (SNR)3 is given in Fig. 4. Fig. 5 shows the variation of standard deviation of estimate of the slant-range velocity versus the number of accumulation. From Fig. 4, it is found that the proposed algorithm can work with a low SNR scene. Moreover, from Fig. 5, it is seen that the more number of accumulation, the more accurate the estimate is.
0.06
5.2. Real data
0.05
In this subsection, experiments are performed with real SAR data to test the performance of the proposed methodology when static clutters together with noise exist. The clutter and noise background is real data, and four simulated moving targets with different velocities are inserted in the raw SAR data. In re-synthesizing the signatures of the moving targets from image domain to (f, tm) domain, we take steps as follows: (1) cross-range direction FFT, (2) multiply the signal with the cross-range compress function to restore the cross-range phase of the moving target, (3) cross-range IFFT, and (4) range FFT. The signal-to-static clutter plus noise ratio (SCNR) of the experiments is 14 dB. The Mission Parameters and estimates of the slantrange velocity of four moving targets are shown in Tables 4 and 5, respectively. The image after clutter suppression is shown in Fig. 6 and the four moving targets are labeled in the image. The re-synthesized (after performing spotlight algorithm) signature of the third moving target in slow time and range frequency domain is presented in Fig. 7. The beat spectrum of the third moving target is illustrated in Fig. 8. From Table 5, we see that the proposed method works well with real static clutter background. Now let us analyze the facts affecting the accuracy of the proposed methodology. Here, we discuss two major factors as follows
0.04 0.03 0.02 0.01 0
0
5
10 15 SNR[dB]
20
25
Fig. 4. Variation of standard deviation of estimate versus SNR.
0.07 standard deviation of estimates[m/s]
2479
SNR=10dB
0.06 0.05 0.04 0.03 0.02 0.01
0
100
200 300 400 number of accumulation
500
600
Fig. 5. Variation of standard deviation of estimate versus the number of accumulation.
(1) The signal-to-clutter-noise ratio (SCNR) SCNR affects the accuracy of estimates most. As far as our methodology is concerned, the method performs well as long as the SCNR is more than 11 dB (after the 3 We define SNR as the ratio between the squared magnitude of the correctly focused moving target and the variance of the noise.
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Table 5 Estimation results, SCNR ¼ 14 dB Moving targets
Real velocity (m/s)
Estimates (m/s)
Absolute error (m/s)
Relative error (%)
1 2 3 4
6.3 11 15.7 28.88
6.1745 11.0309 15.6098 28.9301
0.1255 0.0309 0.0902 0.0501
1.9914 0.2811 0.5746 0.1736
x 1013 9 8 7
power
6 5 4 3 2 1 Fig. 6. Image after clutter suppression. The four moving targets are labeled in the image.
0 -10
-8
-6
-4
-2
0 2 fd[Hz]
4
6
8
10
Fig. 8. Beat spectrum of the third moving target.
-1.5 -1
tm[s]
-0.5 0 0.5 1 1.5 -3
-2
-1
0 f[Hz]
1
2
3 x 107
Fig. 7. Signature of the third moving target in range frequency and slow time domain.
elimination of the clutter and the implementation of the digital spotlight algorithm). (2) The number of accumulation In fact, the more number of independent samples used to accumulate Spbf(f, tm), the more accurate the estimate is.
The aim of doing so is to resolve the slant-range velocity ambiguity. The ‘‘global’’ Fourier transformation of such SFRC can guarantee the estimation accuracy. The proposed BFCA algorithm is not susceptive to the noise because we apply coherent accumulation in implementing the algorithm. Moreover, the proposed methodology can deal with large RCM since there is no loss of synthetic aperture length when we get the beat spectrum. The algorithm can gain a high accuracy especially for the fast-moving targets. The methodology proposed in this paper uses the external-band filtering algorithm in 2D frequency domain to detect the moving target. One of the disadvantages of such detection algorithm is that when the signatures of the moving target and static clutters overlap completely in 2D frequency domain or when the SCNR of the scene is low, it is difficult to detect the moving target. Probably we can adopt more sophisticated detection algorithm, such as the detection algorithm proposed in [2], to detect the moving target, at the expense of higher computational complexity.
Acknowledgement 6. Conclusion A SFRC can be produced by performing the BF algorithm in the slow time and range frequency domain.
The authors are grateful to the National Laboratory of Radar Signal Processing for providing the real SAR data. This work was supported in part by the National Nature Science Foundation of China under Grant 60402039.
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Appendix A. The proof of the theorem of SFRC
we can have
In this section, we will prove the theorem of SFRC. Proof: For simplicity, w(ttc) can be equivalent to a rectangle with width of Dt, magnitude of unity and tc as the center. Then, the Fourier transform of s(t) (11) can be written as
Z pffiffiffiffi 2 2q 1 2 pffiffiffiffi ejpx =2 dx , 2g 2q Z pffiffiffiffiffiffiffiffi pffiffiffiffi 2 1=2qþ 2q 1 2 jpx2 =2 dx , jSðf 1 Þj ¼ pffiffiffiffiffiffiffiffi pffiffiffiffi e 2g 1=2q 2q Z pffiffiffiffiffiffiffiffi pffiffiffiffi 2 1 1=2qþ 2q 2 jSðf 3 Þj2 ¼ pffiffiffiffiffiffiffiffi pffiffiffiffi ejpx =2 dx . 2g 1=2q 2q
Sðf Þ ¼
Z
þ1
sðtÞ ej2pft dt ¼
Z
t c þDt=2
ej½pgt
2
þ2pðbf Þt
dt.
(A.1)
tc Dt=2
1
jSðf 2 Þj2 ¼
(A.5)
Let u ¼ t þ ðb f Þ=g and we obtain
jSðf Þj2 ¼
"Z
t c þDt=2þðbf Þ=g
#2 cosðpgu2 Þ du
t c Dt=2þðbf Þ=g
"Z
t c þDt=2þðbf Þ=g
þ
#2 sinðpgu2 Þ du
.
(A.2)
From digital calculation by computer, it is found that, when q changes from smaller values to larger values, f1 and f3 are the two extreme points to which the value of |S(f)|2 corresponding exceeds |S(f2)|2 the first time. We define F(q, r) as
t c Dt=2þðbf Þ=g
By letting d½jSðf Þj2 =df ¼ 0, we get three key extreme points among all the extremes of |S(f)|2 as follows 1 ; Dt
f 2 ¼ b þ gt c ;
f 3 ¼ b þ gt c þ
1 Dt
(A.3)
where f2 is no other than the central frequency of s(t). Now let us determine under what condition |S(f2)|2 is the maximum value of |S(f)|2. Letting gDt 2 4
(A.4)
Letting F(q, g) ¼ 0, we can acquire the threshold of q, at which |S(f1)|2 the first time exceeds |S(f2)|2 when q changes from smaller values to larger values through digital calculation, as qT ¼ 1:181.
(A.7)
0.9
0.8
0.9
0.8
0.7
0.8
magnitude of the Fourier transform of SFRC
magnitude of the Fourier transform of SFRC
q¼
(A.6)
0.7 0.6 0.5 0.4 0.3 0.2
magnitude of the Fourier transform of SFRC
f 1 ¼ b þ gt c
Fðq; gÞ ¼ jSðf 2 Þj2 jSðf 1 Þj2 2 pffiffiffiffi 2 Z pffiffiffiffiffiffiffiffi pffiffiffiffi 2 3 Z 2q 1=2qþ 2q 1 4 jpx2 =2 jpx2 =2 ¼ dx pffiffiffiffiffiffiffiffi pffiffiffiffi e dx 5. pffiffiffiffi e 2g 2q 1=2q 2q
0.6
0.5
0.4
0.3
0.2
0.1
0.1 0 -5
0 f[Hz]
5
0 -5
0.7 0.6 0.5 0.4 0.3 0.2 0.1
0 f[Hz]
5
0 -5
0 f[Hz]
Fig. 9. Variation of |S(f)|2 versus q: (a) q ¼ 0.5, (b) q ¼ qT ¼ 1.181, and (c) q ¼ 1.21.
5
ARTICLE IN PRESS 2482
B. Liu et al. / Signal Processing 88 (2008) 2472–2482
And g can have arbitrary values. Therefore, to make |S(f2)|2 be the maximum value of |S(f)|2, q must satisfy
q¼
gDt 2 oqT ¼ 1:181. 4
(A.8)
The variation of jSðf Þj2 versus q is shown in Fig. 9. If q satisfy (A.8), s(t) can be equivalent to a singlefrequency signal as follows se ðtÞ ¼ ejj wðtÞej2pat
(A.9)
Noting that the following expression generally holds under the stripe SAR data parameters 2Df Dx2 pop. cy0 4
(B.18)
According to the theorem of SFRC, the quadratic term ð2pDf =cy0 Þm2 can be omitted. Then we have Spbf_c1 ðf d Þ f
2
d ¼0
where e is a real constant, and
¼ Q ej½j0 ð4pDf =cÞy0 þð2pDfx0 =cy0 Þ Z Dx=2 ejð4pDfx0 =cy0 Þm dm Dx=2
a ¼ f 2 ¼ b þ gt c .
Z
(A.10)
Dy=2
ejð4pDf =cÞn dn
Obviously, the Fourier transform of se(t) is
Dy=2 2
Se ðf Þ ¼ Dt sinc½pDtðf aÞejj . Substitution of a into (A.1) yields R 9 2 R Dt=2 2 2 Dt=2 SðaÞ ¼ ejpgtc Dt=2 ejpgt dt ¼ Dt=2 ejpgt dt ejZ > > = pffiffiffiffi F 2q s 2 >
pffiffiffiffi > Z ¼ pgt c þ arctan ; Fc
(A.12)
Rx where F s ðxÞ ¼ 0 sinðpt 2 =2Þ dt; F c ðxÞ ¼ 0 cosðpt 2 =2Þ dt are the Fresnel functions. Comparing (A.11) and (A.12), we have 2 pffiffiffiffiffiffi3 Fs 2q 2 (A.13) j ¼ Z ¼ pgt c þ arctan4 pffiffiffiffiffiffi5. 2q Fc And, the equivalent width of S(f) approximately equals 1 . B ðf 2 f 1 Þ ¼ Dt
(A.14)
Appendix B. . Derivation of expression (43) In this section, we will derive the expression (43). According to (36), we have Z x0 þDx=2 Z y0 þDy=2 Q ejFðx;yÞ dxdy (B.15) Spbf_c1 ðf d Þ f ¼0 ¼ x0 Dx=2
y0 Dy=2
where Q ¼ s2 C 1
R f2 f1
9 jPðf þ Df Þj2 jPðf Þj2 df = 2
2pDfx þ j0 Fðx; yÞ ¼ 4pDf c y þ cy 0
;
(B.16)
C1 and j0 are constants, and s2 ¼ E½jAn j2 is the average echo magnitude of the static clutter. Let m ¼ x x0 ; n ¼ y y0 and we obtain 2 Spbf_c1 ðf d Þ f ¼0 ¼ Q ej½j0 ð4pDf =cÞy0 þð2pDfx0 =cy0 Þ d Z Dx=2 2 ej½ð2pDf =cy0 Þm þð4pDfx0 =cy0 Þm dm Dx=2
Z
Dy=2
ejð4pDf =cÞn dn.
Dy=2
where sin cðxÞ ¼ sinðxÞ=x. We finally get
2q
Rx
d
¼ DxDyQ ej½j0 ð4pDf =cÞy0 þð2pDfx0 =cy0 Þ 2pDfx0 2pDf Dy (B.19) sin c Dx sin c c cy0
(A.11)
(B.17)
jSpbf_c1 ðf d Þjf d ¼0 ¼ DxDyjQ j 2pDfx0 2pDf Dy . sin c Dx sin c c cy0 (B.20)
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