A Velocity Estimation Algorithm of Moving Targets ...

I. INTRODUCTION

A Velocity Estimation Algorithm of Moving Targets using Single Antenna SAR

GANG LI, Member, IEEE Tsinghua University XIANG-GEN XIA, Fellow, IEEE University of Delaware JIA XU, Member, IEEE Radar Academy of Air Force China YING-NING PENG, Senior Member, IEEE Tsinghua University

A new algorithm is proposed for velocity estimation of moving targets in single antenna synthetic aperture radar (SAR). Based on the fact that different velocity vectors cause different geometrical figures of the two-dimensional (2-D) signature in the range-Doppler (RD) domain, this algorithm estimates the azimuth and range velocities by a 2-D search such that the range cell migration correction (RCMC) and the second range compression (SRC) are correctly performed. It is shown that, using the proposed algorithm, the Doppler ambiguity problem can be avoided and satisfactory accurate velocity estimation can be obtained in high signal-to-clutter ratio (SCR) scenarios.

Manuscript received February 9, 2007; revised December 13, 2007; released for publication May 13, 2008. IEEE Log No. T-AES/45/3/933957. Refereeing of this contribution was handled by V. Chen. This work was supported by the Chuanxin Foundation from the Department of Electronic Engineering, Tsinghua University. S-G. Xia’s work was partially supported by the Air Force Office of Scientific Research (AFOSR) under Grant FA 9550-05-1-0161 and a DESPCoR Grant W911NF-07-1-0422 through ARO. Authors’ addresses: G. Li and Y-N. Peng, Dept. of Electronic Engineering, Tsinghua University, Beijing 100084, China, E-mail: ([email protected]); X-G. Xia, Dept. of Electrical and Computer Engineering, University of Delaware, Newark, DE 19715; J. Xu, Radar Academy of Air Force, Wuhan 430019, China.

c 2009 IEEE 0018-9251/09/$26.00 ° 1052

Velocity estimation of moving targets is an important issue in many synthetic aperture radar (SAR) applications. Classical algorithms of velocity estimation are mostly based on azimuth (cross-range) phase analysis [1—4], i.e., the range velocity and the azimuth velocity of a moving target are retrieved from the Doppler shift and the Doppler rate, respectively. Because the azimuth signal is sampled by the pulse repetition frequency (PRF), the estimations of the Doppler shift are contained in the interval [¡PRF=2, PRF=2). Therefore, when the range velocity of the target is fast such that the absolute value of the Doppler shift is above PRF/2, the algorithms based on azimuth phase analysis may suffer from the so-called Doppler ambiguity problem. To resolve the Doppler ambiguity, many algorithms have been proposed. An intuitive idea is to increase the PRF [2], which shortens the unambiguous range swath and meanwhile increases the memory requirements. In [5], a nonuniform PRF is suggested for the same purpose, which requires an unconventional pulse scheduling and induces a higher complexity of processing algorithm. In the dual-speed SAR [6], the platform flies with two speeds in different observation durations, and the ambiguity is solved by analyzing the two phase histories, which requires higher maneuverability of the platform. In [7], the ambiguity is avoided by tracking a target position in a sequence of multilook SAR images; this method relies on accurate measurements of the target position and amplitude. Marques and Dias have proposed several interesting algorithms to accurately estimate the velocity of a fast-moving target with single antenna SAR in [8]—[11]. In [8] and [11, ch. 5], the linear relationship between the Doppler shift and the range frequency is employed, and the velocity of the target is unambiguously estimated by computing the skew of the two-dimensional spectrum. This method requires that the background clutter has small correlation in the frequency domain. In [9] and [11, ch. 3], the structure of the amplitude and phase modulations of the target echo is exploited and the velocity of the target is unambiguously estimated by matched filtering operation. This method requires that the parameter vector candidate that is used to design the matched filter must be close enough to the true parameter vector such that the results of matched filtering have small errors. To do so, the pre-estimation of the parameter vector with moderate accuracy is required to find the eligible parameter vector candidate. In [10] and [11, ch. 4], the data along the signature curve of the moving target are sampled in the spatial domain, and the velocity is unambiguously estimated by the maximum likelihood estimation. However, the range compression errors caused by the target motion are

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not taken into account and, therefore, the estimation of the azimuth velocity may have larger errors. Besides the above algorithms used in single antenna SAR systems, some algorithms using multiple antennas have been also proposed, e.g., multifrequency antenna array SAR [12], nonuniform linear antenna array SAR [13], dual-speed linear antenna array SAR [14], and bistatic linear antenna array SAR [15], etc., which solve the Doppler ambiguity by combining the phase analysis among multiple complex SAR images with the Chinese Remainder Theorem. In the multiantenna algorithms, higher computational complexity and hardware cost are needed. This paper focuses on the velocity estimation of moving targets using single antenna SAR. Different from existing methods, this paper implements the velocity estimation based on the geometric characteristic of the 2-D signature of a moving target in the range-Doppler (RD) domain. As shown later, different velocity vectors cause different geometrical figures of the 2-D signatures in the RD domain, which, in the SAR terminology, corresponds to different range migration tracks and remaining errors of the range compression. From this observation, we estimate the azimuth and range velocities by a 2-D search such that the range cell migration correction (RCMC) and the second range compression (SRC) are correctly done, i.e., we seek for a velocity vector such that RCMC and SRC transform the geometrical figure of a 2-D signature into a straight (or as concentrated as possible) line parallel to the Doppler axis in the RD domain. This process is not subject to the PRF limitation, and it does not rely on the clutter statistics. Numerical examples show that the proposed algorithm can estimate the velocities with an accuracy better than 2 percent when the signal-to-clutter ratio (SCR) is higher than 11 dB, which is an improvement that from the existing methods. The performance improvement comes at the cost of higher complexity from the computational point of view compared with Marques and Dias’ works [8—11]. This paper is organized as follows. In Section II, the signature of the moving target is formulated, and our algorithm is proposed. In Section III, some additional discussions are given. In Section IV, the performance of the proposed algorithm is shown using some numerical examples.

Fig. 1. Geometry of side-looking SAR.

center is passing by the target, the target is located at (0, r0 ), and this moment is defined as azimuth time (or slow-time) 0. The instantaneous distance from the radar to the target is q R(t) = (vt ¡ vx t)2 + (r0 ¡ vy t)2 (1) and the baseband echo from the target can be represented as " μ · ¶ # ¸ 2R(t) 2 4¼ s(t, ¿ ) = A ¢ exp j¼° ¿ ¡ ¢ exp ¡j R(t) c ¸ (2) where A is the amplitude, ° is the chirp rate of the chirp signal transmitted by the radar, ¿ is the range time (or fast-time), t is the azimuth time, c is the speed of light, and ¸ is the radar wavelength. Strictly speaking, the amplitude A is a slowly varying function of t and ¿ , but here we consider it as a constant for simplicity because it does not affect the phase term, which is the phase term is the most interesting aspect of the calculations. According to the stationary phase principle [17], the 2-D Fourier transform on (2) in terms of t and ¿ yields · ¸ ½ ¾ 2 S(ft , f¿ ) = A0 ¢ exp ¡j¼ ¢ exp

f¿ °

¢ exp ¡j2¼ft

(

r (v ¡ vx ) ¡j4¼ p 0 vy2 + (v ¡ vx )2

sμ

¢

f¿ 1 + c ¸

¶2

¡

ft2

9 =

4[vy2 + (v ¡ vx )2 ] ;

II. PROPOSED ALGORITHM Consider the side-looking SAR case as illustrated in Fig. 1. X and Y axes are the azimuth and range (or slant-range) directions, respectively, the radar moves with constant speed v along the azimuth direction, and a target P moves with constant azimuth speed vx and range speed ¡vy . Variables vx and vy are the parameters to be estimated. When the radar beam

r0 vy

vy2 + (v ¡ vx )2

(3)

where A0 is the amplitude, f¿ is the range frequency and f¿ 2 [¡B=2, B=2), B is the radar bandwidth, ft is the azimuth frequency and ft 2 [fd ¡ Bt =2, fd + Bt =2), fd is the Doppler centroid, Bt is the azimuth bandwidth, and they can be represented as fd = 2vy =¸ Bt = 2(v ¡ vx )=D

LI ET AL.: A VELOCITY ESTIMATION ALGORITHM OF MOVING TARGETS USING SINGLE ANTENNA SAR

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where D is the aperture of the real antenna. After the regular range compression, i.e., multiplying exp[j¼f¿2 =°] to (3), (3) becomes ½ ¾ S(ft , f¿ ) = A0 ¢ exp ¡j2¼ft ¢ exp

(

r0 vy

vy2

+ (v ¡ vx )2

r (v ¡ vx ) ¡j4¼ p 0 vy2 + (v ¡ vx )2

sμ

¢

f¿ 1 + c ¸

¶2

¢

9 =

ft2 ¡ 4[vy2 + (v ¡ vx )2 ] ;

= A0 ¢ expfjÁ(ft , f¿ )g:

(5)

Taking the Taylor series expansion of the phase term in terms of f¿ , we have Á(ft , f¿ ) = Á0 (ft ) + Á1 (ft ) ¢ f¿ + Á2 (ft ) ¢ f¿2 + ¢ ¢ ¢ where 4¼r (v ¡ vx ) Á0 (ft ) = ¡ q 0 vy2 + (v ¡ vx )2

s

(6)

1 ft2 ¡ 2 2 ¸ 4[vy + (v ¡ vx )2 ]

r0 vy ¡ j2¼ft 2 vy + (v ¡ vx )2 4¼r0 (v ¡ vx )

(7)

Á1 (ft ) = ¡ q c vy2 + (v ¡ vx )2 ¡ ft2 ¸2 =4 Á2 (ft ) =

¼r0 (v ¡ vx )ft2 ¸3 : 2c2 [vy2 + (v ¡ vx )2 ¡ ft2 ¸2 =4]3=2

The term Á0 (ft ) is independent of f¿ and it is responsible for azimuth focusing. The term Á1 (ft ) implies that the range migration follows r0 (v ¡ vx ) R(ft ) = q : vy2 + (v ¡ vx )2 ¡ ft2 ¸2 =4

(8)

From (8) one can see that different velocity vectors (vx , vy ) cause different range migration tracks, though they cause the same baseband Doppler shift if mod[2(vy,1 ¡ vy,2 )=¸, PRF] = 0, where vy,1 and vy,2 belong to two possible targets, respectively. The goal of RCMC is to compensate Á1 (ft ) for ft 2 [fd ¡ Bt =2, fd + Bt =2) such that the hyperbola in (8) is adjusted to a straight line at the range cell r0 . The term Á2 (ft ) indicates that the range signal is still a chirp signal with the chirp rate °m = ¡2c2 [vy2 + (v ¡ vx )2 ¡ ft2 ¸2 =4]3=2 =r0 (v ¡ vx )ft2 ¸3 , though the regular range compression is already done. This indicates that the signature exhibits a spread in the range direction, and the spread width is ¡ (ft ) =

c B Br0 (v ¡ vx )ft2 ¸3 : ¢ = 2 2 j°m j 4c[vy + (v ¡ vx )2 ¡ ft2 ¸2 =4]3=2 (9)

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Fig. 2. Signal of moving target in RD domain. (a) After regular range compression. (b) After RCMC and SRC with correct (vx , vy ). (c) After RCMC and SRC with wrong (v¯ x , v¯ y ).

The goal of SRC is to compensate Á2 (ft ) for ft 2 [fd ¡ Bt =2, fd + Bt =2) such that the range spread is compressed to a line, such that ¡ (ft ) in (9) is decreased as much as possible. Now we consider the scenario in which vy is large enough that jfd j > PRF=2. We then explain the geometrical change of the signature in the RD domain using several illustrations. Before RCMC and SRC, the signature of the target in the RD domain is shown in Fig. 2(a), the range migration track and the range spread follow (8) and (9), respectively. If we perform RCMC and SRC with correct (vx , vy ), both Á1 (ft ) and Á2 (ft ) will be completely compensated, so the signature of the target will be concentrated into a horizontal line, as shown in Fig. 2(b). Contrarily, if we perform RCMC and SRC with incorrect (v¯ x , v¯ y ), the signature of the target will still be spread, not concentrated into a line, as shown in Fig. 2(c). The reason for this is that, on the one hand, wrong v¯ y and v¯ x cause the wrong f¯d and B¯ t , respectively, i.e., the range of ft is mismatched; and on the other hand, wrong (v¯ x , v¯ y ) induces the remaining phase error in the process of compensating Á1 (ft ) and Á2 (ft ). This gives us the idea to estimate (vx , vy ) by a 2-D search such that the RCMC and SRC are correctly done. Next we prove that there is only one solution of (vx , vy ) such that the compensation of Á1 (ft ) and Á2 (ft ) concentrates the signature of one target into a straight line parallel to the Doppler axis. Firstly, the sign of vy can be determined by observing the target signature in the RD domain before RCMC and SRC. For example, in Fig. 2(a) the derivative of the range migration is minus in the RD domain, which implies

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the plus sign for vy , i.e., the target is moving toward the radar. Otherwise, the plus derivative of the range migration in the RD domain adds the minus sign to vy . Then we assume that there are two solutions (vx1 , vy1 ) and (vx2 , vy2 ). This implies that (vx1 , vy1 ) and (vx2 , vy2 ) generate the same Á1 (ft ) and Á2 (ft ) in (7), i.e., 4¼r0 (v ¡ vx1 ) q 2 c vy1 + (v ¡ vx1 )2 ¡ ft2 ¸2 =4 4¼r0 (v ¡ vx2 ) = q 2 + (v ¡ v )2 ¡ f 2 ¸2 =4 c vy2 t x2 ¼r0 (v ¡ vx1 )ft2 ¸3 2 + (v ¡ v )2 ¡ f 2 ¸2 =4]3=2 2c2 [vy1 t x1 =

2 2c2 [vy2

¼r0 (v ¡ vx2 )ft2 ¸3 : + (v ¡ vx2 )2 ¡ ft2 ¸2 =4]3=2

Then we have vx1 = vx2 and jvy1 j = jvy2 j. Since the sign of vy has been determined, we have (vx1 , vy1 ) = (vx2 , vy2 ), i.e., there is only one solution of (vx , vy ) such that the compensation of Á1 (ft ) and Á2 (ft ) concentrates the signature of one target into a straight line parallel to the Doppler axis. Herein the problem of interest is to seek for a parameter/quantity to measure whether the RCMC and SRC are correctly done. After the RCMC and SRC operations according to a candidate (v¯ x , v¯ y ), let us integrate the signal power along the Doppler axis at every range cell n and generate a sequence z(n); in other words, the nth element of sequence z is the power integration result at the nth range cell in the RD domain, for n = 1, 2, : : : , N, where N is the range cell number of the extracted image region. The contrast of z is defined as [20] hz 2 i (10) Cz (v¯ x , v¯ y ) = hzi2 where h¢i denotes the averaging operator. If the candidate (v¯ x , v¯ y ) is equal to the correct (vx , vy ), the RCMC and SRC concentrate the signature in a horizontal line, so a dominant element appears in z and a larger Cz is obtained, as shown in Fig. 2(b). Contrarily, if the candidate (v¯ x , v¯ y ) is apart from the correct (vx , vy ), the RCMC and SRC still yield a 2-D spread signature, so z has multiple elements with similar values and the Cz becomes smaller, as shown in Fig. 2(c). This implies that Cz obtained by a correct (vx , vy ) is larger than the one obtained by a wrong (v¯ x , v¯ y ). Moreover, the uniqueness of the solution (vx , vy ) has been proved in the previous paragraph. Therefore, the correct (vx , vy ) can be uniquely obtained by a 2-D search on (v¯ x , v¯ y ) such that Cz (v¯ x , v¯ y ) reaches the maximum. In summary, the proposed algorithm can be carried out by the following steps.

Step 1 Extract the image region containing the moving target of interest from the scene image using the spotlighting technique [16]. (Note: the regular range compression is already operated when the background scene is imaged, and the range compress errors may exist for the moving target.) Step 2 Transform the extracted data into RD domain by azimuth Fourier transform and determine the sign of vy , then transform the data into 2-D frequency domain by range Fourier transform. Step 3 For a candidate (v¯ x , v¯ y ), implement RCMC and SRC by sequentially multiplying H1 (ft , f¿ ) and H2 (ft , f¿ ), where H1 (ft , f¿ ) = expf¡jÁ1 (ft )f¿ g H2 (ft , f¿ ) = expf¡jÁ2 (ft )f¿2 g

(11)

Á1 (ft ) and Á2 (ft ) are represented in (7), transform the data into RD domain by range inverse Fourier transform, integrate the data power along the Doppler axis for each range cell n and generate the sequence z(n), then compute Cz (v¯ x , v¯ y ) according to (10). Step 4 Solve for the correct (vx , vy ) by searching all the possible (v¯ x , v¯ y ) such that the maximum of Cz (v¯ x , v¯ y ) is reached, i.e., (vx , vy ) = arg max(v¯x ,v¯y ) [Cz (v¯ x , v¯ y )]. III. SOME DISCUSSIONS About the proposed algorithm, we have the following discussions. A. Efficient Implementation To reduce the complexity of the 2-D search, in this subsection we introduce an efficient implementation approach by combining the phase analysis methods with the proposed algorithm. It is known that the fd can be written as Doppler centroid fd = fdb + m¢ PRF = 2vy =¸ and the Doppler rate fr = ¡2(v ¡ vx )2 = ¸r0 , where fdb is the baseband Doppler centroid and m is called the Doppler ambiguity number. If the SCR is higher, fdb and fr can be estimated by the existing phase analysis methods, such as the correlation Doppler estimator (CDE) [18] and phase difference (PD) algorithm [19], respectively. Using the estimates of fdb and fr , one can determine the correct (m, vxp ) by 2-D search on ¯ v¯ x ) with the initial v¯ x = v ¡ ¡fr ¸r0 =2 such (m, that Cz reaches the maximum, then compute the correct vy by vy = (fdb + m ¢ PRF)¸=2. The reduction of the computational complexity relies on the following facts: 1) the search range in terms of vy is converted from the real interval to the integer set; 2) the initial v¯ x is helpful for finding the maximum of Cz quickly. There are two issues with the efficient implementation approach. 1) In the phase analysis

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methods, the range migration needs to be neglected to guarantee the integrality of the Doppler phase history. However, the range migration is indispensable for the proposed algorithm since the RCMC is employed. To solve this problem, we derive two channels from the range compressed data: the high range resolution channel and the low range resolution channel, where the former is actually the range compressed data and the latter is obtained by smoothing the former until the range migration can be neglected. Then we estimate fdb and fr at the low range resolution channel, and match them to the 2-D search at high range resolution channel. 2) If the SCR is not high enough, the estimation results of fdb and fr will be degraded due to the interference from the strong clutter, and accordingly, the efficiency of the presented approach will decrease. B. Maximal and Minimal Determinable Velocities In this subsection we discuss the maximal and minimal determinable velocities of the proposed algorithm. With the consideration of the presence of stationary clutter and the practical range-directional sizes of moving targets, some necessary conditions are needed in order to distinguish moving targets from stationary targets, and these necessary conditions will be reflected from the moving target velocities, i.e., the maximal and the minimal determinable velocities. To do so, let us first calculate the range migration amounts of the stationary scatterers and the moving target, respectively. For the stationary targets located at range cell r0 , from (8) we have their common range migration amount −s = [R(fd + Bt =2) ¡ R(fd )]jvx =vy =0 ! Ã r ¸2 = r0 1= 1 ¡ ¡1 : 4D 2

(12)

For the moving target, without loss of generality, we assume vy > 0 and accordingly fd > 0 in (4). There are two cases regarding to the relationship between fd and Bt . 1) If fd · Bt =2, i.e., the target is moving so slowly that vy · ¸(v ¡ vx )=2D ¼ ¸v=2D, we have its range migration amount from (8) −1 = R(fd + Bt =2) ¡ R(fd ) r0 ¡ r0 =s ¸vy ¸2 1¡ ¡ 4D2 D(v ¡ vx ) ¼

¶¡3=2 μ r0 ¸vy ¸2 ¢ 1¡ 2Dv 4D 2

(13)

where the approximation holds because vx ¿ v. 2) If fd > Bt =2, i.e., vy > ¸(v ¡ vx )=2D ¼ ¸v=2D, we have its 1056

range migration amount also from (8) −2 = R(fd + Bt =2) ¡ R(fd ¡ Bt =2) r0 r0 ¡s =s ¸vy ¸vy ¸2 ¸2 1¡ ¡ 1 ¡ + 4D 2 D(v ¡ vx ) 4D 2 D(v ¡ vx ) r0 ¸vy ¼ Dv

¶¡3=2 μ ¸2 1¡ 4D 2

(14)

where vx ¿ v is also used for the approximation. These two cases correspond to the slow and fast moving targets, respectively. The first necessary condition is that all the scatterers of range-spread moving targets should have the same range migration. From (13) and (14)one can see that −1 and −2 vary linearly with r0 . Thus, the first necessary condition implies that the changes of −1 and −2 versus range is smaller than the range resolution ½r . Assume that the range-directional size of the moving target is ¢r, by substituting it in (14) and letting ¢−2 · ½r since −2 > −1 , then the maximal determinable velocity can be represented as ¶3=2 μ ½r Dv ¸2 jvy,max j = : (15) 1¡ ¸¢r 4D 2 The second necessary condition is that the range migration of the moving target differs from one of the stationary scatterers. This condition implies that the range migration amount of the moving target is larger than −s in (12). Since −2 > −1 (see (13) and (14)), by letting −1 ¸ −s + ½r , we have the minimal determinable velocity from (12) and (14) μ ¶3=2 2½r Dv ¸2 jvy,min j = : (16) 1¡ ¸r0 4D 2 Now let us look at an example. Assume D = 0:49 m, v = 100 m/s, r0 = 10 km, ¢r = 10 m, Fig. 3 plots the maximal and the minimal determinable velocities versus various range resolutions under three specified wavelengths. This indicates that the proposed algorithm may cope with most of the familiar ground/ocean moving targets, if the wavelength is not too long and the range resolution not too high. C.

Effect of SRC

The range spread caused by the SRC term has been formulated in (9). In this subsection we quantitatively observe the effect of SRC. At ft = fd , (9) becomes Br0 ¸vy2 : (17) ¡ (fd ) = c(v ¡ vx )2 Let us look at an example. Assuming B = 200 MHz, r0 = 10 km, v = 100 m/s, vx = 9 m/s, Fig. 4 illustrates the range spreads versus various velocities under

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Fig. 4. Range spread versus vy .

Fig. 3. Maximal and minimal determinable velocities versus various ½r . (a) Maximal determinable velocity. (b) Minimal determinable velocity.

three specified wavelengths, respectively. If ½r = 1 m, one can see that the range signal is obviously spread over several range resolution cells for the velocities of most vehicles, which may decrease Cz even if the RCMC is correctly done and, accordingly, degrade the velocity estimation accuracy. Therefore, not only RCMC but also SRC must be implemented to concentrate the signature to a line as completely as possible. References [10] and [11, ch. 4] show that after the regular range compression, the data along the signature curve of the moving target are sampled in the spatial domain, and the range velocity is unambiguously estimated by the maximum likelihood estimation. However, the SRC is not taken into the account and, therefore, the range spread of the target signature may cause larger errors in the estimation of the azimuth velocity.

Fig. 5. 2-D search results when only one moving target is present.

IV. EXAMPLES In this section, we analyze the performance of the proposed algorithm by some numerical examples. The system parameters are: the radar speed v = 100 m/s, the wavelength ¸ = 0:03 m, the radar bandwidth B = 200 MHz, the pulsewidth T = 10 ¹s, the aperture of the real antenna D = 0:49 m, PRF = 1 kHz, and the range r0 = 10 km at the moment t = 0. EXAMPLE 1 The Estimation Error Versus SCR for a Single Moving Target. Assume that a target is moving with constant velocity vx = 9 m/s and vy = 12 m/s. From (4) one can know that fd = 800 Hz and Bt = 370:6 Hz. Note fd > jPRF=2j, i.e., the Doppler ambiguity occurs. In the clutter-free case, the 2-D search result is shown in Fig. 5, where one can see that the true (vx , vy ) is obtained by finding the maximum of Cz . Then, with the consideration of the presence of the clutter, we assume that the background of the extracted region

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Fig. 6. Estimation errors versus SCR. (a) Estimation error of vx . (b) Estimation error of vy .

has low contrast, and we define the SCR as the ratio between the reflectivity of the moving target and the mean reflectivity of the background. The estimation error of the proposed algorithm versus various SCR is shown in Fig. 6, where vx and vy are estimated with accuracy better than 2% when SCR ¸ 11 dB. If we add an assumption that the background clutter in the extracted region exhibits small correlation in the frequency domain, then the definition of SCR in this paper is the same as the one in [8]. Compared with the algorithm in [8] that can achieve the estimation accuracy better then 3% when SCR > 14 dB, the proposed algorithm has better accuracy, while the one in [8] is more computationally efficient since it avoids the 2-D search. In the case of SCR = 11 dB, the signature in the RD domain after the regular range compression is shown in Fig. 7(a), where it is a hyperbola with little range-direction spread. With the estimates of vx and vy obtained by the proposed algorithm, RCMC and SRC transform the 2-D signature of the target into a horizontal line, as shown in Fig. 7(b), and the sequence z, which has the maximal Cz , is plotted in Fig. 7(c). EXAMPLE 2 Multiple Moving Targets. We next explain how the proposed algorithm works in the multiple moving targets scenario by considering that two moving targets are present in the extracted image region. Assume two targets have equal SCR = 20 dB and velocities vx1 = 9 m/s, vy1 = 12 m/s, vx2 = 6 m/s, and vy2 = ¡8 m/s, respectively. The 2-D search results are shown in Fig. 8, where two peaks are clearly distinguishable, and each peak corresponds to one moving target. Therefore, the velocities of the two targets can be respectively estimated by finding the local maximums of Cz . 1058

Fig. 7. Results of proposed algorithm when SCR = 11 dB. (a) Signature in RD domain after regular range compression. (b) Signature in RD domain after RCMC and RSC with the estimation results of (vx , vy ). (c) Sequence z generated by power-integration on data in (b) along Doppler axis.

This result may directly be generalized to the case of multiple moving targets. From the observation of this example we have the following two considerations. 1) If the two velocity vectors are too close, the two peaks in Fig. 8 will become superposed and indistinct, which may cause the degradation of the velocity estimation. Therefore, how to sharpen the peaks needs to be further investigated. 2) If the second target is stationary, i.e., the extracted image region contains a

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relies on the detectability of the moving targets. From Fig. 7(c) one can see that the moving target is clearly detectable, or one can combine the proposed algorithm with the existing detection methodologies. 2) With the increase of the velocity of the moving target, in (6) the cubic and higher terms of f¿ can not be neglected, i.e., besides the regular range compression and SRC, a more elaborated range compression function is necessary for fine concentration of the signature. 3) Reducing the computational complexity of the algorithm needs to be further investigated although we have given a way to do so in Section III. ACKNOWLEDGMENTS Fig. 8. 2-D search results when two moving targets are present.

dominant stationary scatterer, the corresponding peak will appear at the position vx2 = vy2 = 0. However, this peak is undesired here since we are interested in the moving targets. We can suppress this kind of dominant stationary scatterer according to the following strategy: if the 2-D search outputs a dominant stationary scatterer, then remove the row, corresponding to the maximal value of z, from the RCMC implemented- and SRC implemented-data block; repeat this process until all the dominant stationary scatterers are removed. By application of the additional strategy, the proposed algorithm can cope with not only the low-contrast background case (as described in Example 1) but also the high-contrast background case such as that of Example 2. V. CONCLUSION In this paper, we propose a velocity estimation algorithm for moving targets in single antenna SAR. We show that different velocity vectors induce different geometric figures of the target signature in the RD domain. Based on this fact, we estimate the azimuth and range velocities by a 2-D search such that the RCMC and the SRC are correctly done. In addition, some practical problems are discussed, including the efficient implementations of the proposed 2-D search, the maximal and minimal determinable velocities and the effect of the SRC. The advantages of the proposed algorithm are that it works well even when not subject to the PRF limitation, and it does not have any requirements on the clutter statistics. The numerical examples show that the proposed algorithm can achieve an estimation accuracy of better than 2 percent when SCR ¸ 11 dB, which comes at the cost that the proposed algorithm has higher complexity from the computational point of view compared with Marques and Dias’ works [8—11]. As final remarks, three issues should be noted. 1) Like the method in [8—11], the proposed algorithm

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Werness, S. A. S., Carrara, W. G., Joyce, L. S., and Franczak, D. B. Moving target imaging algorithm for SAR data. IEEE Transactions on Aerospace and Electronic Systems, 26, 1 (Jan. 1990), 57—67. Barbarossa, S. Detection and imaging of moving objects with synthetic aperture radar. IEE Proceedings, Pt. F., 139, 1 (Feb. 1992), 79—88. Perry, R. P., DiPietro, R. C., and Fante, R. L. SAR imaging of moving targets. IEEE Transactions on Aerospace and Electronic Systems, 35, 1 (Jan. 1999), 188—200. Freeeman, A., and Currie, A. Synthetic aperture radar (SAR) images of moving targets. GEC Journal of Research, 5, 2 (1987), 106—115. Legg, J., Bolton, A., and Gray, D. SAR moving target detection using non-uniform PRI. In Proceedings of the 1st European Conference on Synthetic Aperture Radar (EUSAR’96), 1996, 423—426. Wang, G., Xia, X-G., and Chen, V. C. Dual-speed SAR imaging of moving targets. IEEE Transactions on Aerospace and Electronic Systems, 42, 1 (Jan. 2006), 368—379. Kirscht, M. Detection and imaging of arbitrarily moving targets with single-sensor SAR. IEE Proceedings–Radar Sonar & Navigation, 150, 1 (Feb. 2003), 7—11. Marques, P., and Dias, J. Velocity estimation of fast moving targets using a single SAR sensor. IEEE Transactions on Aerospace and Electronic Systems, 41, 1 (Jan. 2005), 75—89. Dias, J., and Marques, P. Multiple moving target detection and trajectory estimation using a single SAR sensor. IEEE Transactions on Aerospace and Electronic Systems, 39, 2 (Apr. 2003), 604—624. Marques, P., and Dias, J. Moving targets processing in SAR spatial domain. IEEE Transactions on Aerospace and Electronic Systems, 43, 3 (July 2007), 864—874.

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Gang Li (M’08) received the B.S. and Ph.D. degrees from Tsinghua University, Beijing, China, in 2002 and 2007, respectively. In July 2007, he joined the Department of Electronic Engineering, Tsinghua University, Beijing, China, where he is currently an assistant professor. His current interests include array signal processing, SAR imaging and moving target processing. Dr. Li received the Outstanding Young Investigator Award of the Department of Electronic Engineering, Tsinghua University in 2006, and the Outstanding Graduate Award of Tsinghua University in 2007. 1060

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Xiang-Gen Xia (M’97–SM’00) received his B.S. degree in mathematics from Nanjing Normal University, Nanjing, China, and his M.S. degree in mathematics from Nankai University, Tianjin, China, and his Ph.D. degree in electrical engineering from the University of Southern California, Los Angeles, in 1983, 1986, and 1992, respectively. He was a senior/research staff member at Hughes Research Laboratories, Malibu, CA, during 1995—1996. In September 1996, he joined the Department of Electrical and Computer Engineering, University of Delaware, Newark, where he is the Charles Black Evans Professor. He was a visiting professor at the Chinese University of Hong Kong during 2002—2003, where he is an adjunct professor. Before 1995, he held visiting positions in a few institutions. His current research interests include space-time coding, MIMO and OFDM systems, and SAR and ISAR imaging. Dr. Xia has had 170 refereed journal articles published and accepted, 7 U.S. patents awarded, and is the author of the book Modulated Coding for Intersymbol Interference Channels (New York, Marcel Dekker, 2000). Dr. Xia received the National Science Foundation (NSF) Faculty Early Career Development (CAREER) Program Award in 1997, the Office of Naval Research (ONR) Young Investigator Award in 1998, and the Outstanding Overseas Young Investigator Award from the National Nature Science Foundation of China in 2001. He also received the Outstanding Junior Faculty Award of the Engineering School of the University of Delaware in 2001. He is currently an associate editor of the IEEE Transactions on Wireless Communications, the IEEE Transactions on Vehicular Technology, the (EURASIP) Signal Processing, and the Journal of Communications and Networks (JCN). He was a guest editor of Space-Time Coding and Its Applications in the EURASIP Journal of Applied Signal Processing in 2002. He served as an associate editor of the IEEE Transactions on Signal Processing during 1996 to 2003, the IEEE Transactions on Mobile Computing during 2001 to 2004, and the EURASIP Journal of Applied Signal Processing during 2001 to 2004. He is also a member of the Sensor Array and Multichannel (SAM) Technical Committee in the IEEE Signal Processing Society. He was the general cochair of the 2005 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP’05) in Philadelphia.

Jia Xu (M’05) received his B.S. and M.S. degree from Radar Academy of Air Force, Wuhan, China, and his Ph.D. degree from Navy Engineering University, Wuhan, China, in 1995, 1998, and 2001, respectively. He is currently an associate professor with the Radar Academy of Air Force and a postdoctoral fellow with the Department of Electronics Engineering, Tsinghua University, Beijing, China. His current research interests include detection and estimation theory, SAR/ISAR imaging, target recognition, array signal processing, and adaptive signal processing. He has authored or coauthored more than 60 papers. Dr. Xu is a senior member of the Chinese Institute of Electronics. He received the Outstanding Postdoctoral Researcher Award from Tsinghua University in 2004. LI ET AL.: A VELOCITY ESTIMATION ALGORITHM OF MOVING TARGETS USING SINGLE ANTENNA SAR

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Ying-Ning Peng (M’93–SM’96) received the B.S. and M.S. degrees from Tsinghua University, Beijing, China, in 1962 and 1965, respectively. In 1993, he joined the Department of Electronic Engineering, Tsinghua University, where he is currently a professor. He has worked with real-time signal processing for many years, has published more than 200, papers and has received many awards for his contributions to research and education in China. Dr. Peng is a fellow of the Chinese Institute of Electronics. He has received many awards for his contributions to research and education in China. 1062

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