www.ietdl.org Published in IET Radar, Sonar and Navigation Received on 28th June 2007 Revised on 12th February 2008 doi: 10.1049/iet-rsn:20070101
ISSN 1751-8784
Inverse synthetic aperture radar imaging of ship target with complex motion Y. Li1 R. Wu2 M. Xing1 Z. Bao1 1
State Key Lab for Radar Signal Processing, Xidian University, Shaanxi 710071, People’s Republic of China Tianjin Key Laboratory for Advanced Signal Processing, Civil Aviation University of China, Tianjin 300300, People’s Republic of China E-mail:
[email protected] 2
Abstract: High-resolution inverse synthetic aperture radar (ISAR) imaging and recognition of ship target is very important for many applications. Although the principle of ISAR imaging of ship target on the sea is the same as that of flying target in the sky, the former usually has more complex motion (fluctuation with the oceanic waves) than the latter, which makes the motion compensation very difficult. However, the change in phase chirp rate caused by the complex motion of ships will deteriorate the azimuth focusing quality. In this paper, we first model the complex motion of ship target with cubic phase terms (parameterised on chirp rate and its change rate), then a new ISAR imaging method, referred to as TC-DechirpClean, is proposed, which estimates the chirp rate and the change rate of chirp rate of all scatters in the time – chirp distribution plane. Both numerical and experimental results are provided to demonstrate the performance of the proposed method.
1
Introduction
High-resolution inverse synthetic aperture radar (ISAR) imaging and recognition of ship target on the sea has attracted the attention of many radar researchers [1 – 3] in the past decade. The principle of high-resolution ISAR imaging of ship target is the same as that of spatial target [4 – 6]. However, ship target usually fluctuates with the oceanic wave and its pose has three-dimensional motion (yaw, pitch and roll) [7], which causes high-order azimuth phase terms. The azimuth echo signal has curve distribution in the time – frequency plane, which will degrade the azimuth focusing performance. Conventional range-Doppler (RD) method [8] will be invalid due to the azimuth high-order phase term. Time – frequency analysis method based on the Wigner – Ville distribution is proposed in [9] for instantaneous radar imaging, which suffers from the well-known cross-term interference. Second-order phase term is considered in [10 – 12]. A Clean-based parameter estimation method (referred to as DechirpClean) is presented in [10] and Chirplet decomposition based method is discussed in IET Radar Sonar Navig., 2008, Vol. 2, No. 6, pp. 395 – 403 doi: 10.1049/iet-rsn:20070101
[11, 12]. The methods presented in [10 – 12] first obtain the estimates of the amplitude, centre frequency and chirp rate of each scatterer on the target according to the maximum peak value of the compressed signal acquired by applying the fast Fourier transform (FFT) after compensating the optimum matching value of the chirp rate, then reconstruct the echo signal and process the reconstructed signal to obtain the ISAR image of ship target. All of them can avoid the cross-term interference prolem. However, when high-order phase term (cubic and above) exists, the methods presented in [10 –12] will suffer performance degradation. In this paper, we first model the complex motion of ship target with cubic phase terms (parameterised on chirp rate and its change rate), then a new ISAR imaging method, referred to as TC-DechirpClean, is proposed. It is an extension of our previously proposed DechirpClean method [10] from quadratic to cubic phase term. TCDechirpClean first estimates the chirp rate and the change rate of chirp rate of all scatters in the time –chirp distribution plane, then applies the FFT to estimate the centre frequency and magnitude of each scatterer signal 395
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www.ietdl.org according to the maximum magnitude criteria after compensating for higher-order phase term, and finally ISAR image of ship target is obtained by applying the FFT to the signal constructed with the estimates of the amplitude and centre frequency of all scatterers. Like the DechirpClean method [10], the proposed method does not suffer from the cross-term interference. The remainder of this paper is organised as follows. Cubic phase term data model of ship target with complex motion is presented in Section 2. The feature of time – chirp rate distribution of signals with cubic phase term is discussed in Section 3. The proposed ISAR imaging method (TCDechirpClean) is presented in detail in Section 4. Both numerical and experimental results are given in Section 5. Finally, Section 6 concludes the whole paper.
the effective rotating vector. The line velocity of scatterer P is ve rP, and the corresponding radial component is (ve rP)R(ve rP) † R. By taking the translational component vr into account, we can obtain the Doppler frequency of the scatterer P, which has the form 2 fd ¼ (vr þ (ve r P ) t R) l
where ‘’ and ‘†’ represent the outer and inner product, respectively, and l denotes the wavelength. Since the ISAR imaging time is usually very short, we can assume that the projection plane remains unchanged within the imaging period, then the Doppler frequency can be further written as 2 fd ¼ [vr þ vex (rPy Rz rPz Ry ) l þ vey (rPz Rx rPx Rz ) þ vez (rPx Ry rPy Rx )]
2 Data model and problem formulation The ISAR imaging geometry of ship target with complex motion is shown in Fig. 1. In Fig. 1, point O is the origin point representing the rotating centre of the target, point P is a random scatterer on the target whose coordinate is denoted as xp , yp , zp , rP is a vector from the origin to the position vector point P and v is a vector representing the three-dimensional angular speed of the rotating target. Vector v can be decomposed into two components vR and ve in the plane determined by v and radar line of sight (whose unit vector is R), where vR is the component parallel to R and ve is the component perpendicular to R. vR does not cause radial motion and thus has no effect on the phase of the echo signal, whereas the radial motion caused by ve does affect the phase and cause the change of the Doppler frequency, which can be utilised to realise high-resolution azimuth imaging. Therefore ve is called
(1)
(2)
where vex , vey and vez denote the components of ve along each of the three axes, respectively, rPx , rPy and rPz are the components of rP along each of the three axes, respectively, Rx , Ry and Rz represent the components of R along each of the three axes, respectively. Since the ship target is in relatively complex motion, vex , vey and vez are usually time-variant, which can be approximated as 1 vex ’ vx þ ax tm þ gx tm2 2
(3)
1 vey ’ vy þ ay tm þ gy tm2 2
(4)
1 vez ’ vz þ az tm þ gz tm2 2
(5)
and
where tm represents the azimuth time (or slow time), vx , vx , vz , ax , ay , az and gx , gy , gz , are the constant term, firstorder term and quadratic term coefficients of the angular speeds decomposing on the three axes, respectively. By substituting (3) –(5) into (2), we have 2 1 fd ¼ (vr þ v1 t r þ (a1 t r)tm þ (g1 t r)tm2 ) l 2
(6)
v1 ¼ [vx , vy , vz ]
(7)
a1 ¼ [ax , ay , az ]
(8)
g1 ¼ [gx , gy , gz ]
(9)
where
and r ¼ [(rPy Rz rPz Ry ), (rPz Rx rPx Rz ), (rPx Ry Figure 1 Three-dimensional ISAR imaging geometry 396 & The Institution of Engineering and Technology 2008
rPy Rx )]
(10)
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www.ietdl.org From (6), it can be noted that the distance from scatterer P to radar can be expressed as ð tm l fd dtm R(tm ) ¼ 2 t0 ð tm 1 ¼ (vr þ v1 t r þ (a1 t r)tm þ (g1 t r)tm2 )dtm 2 t0
phase makes the above-mentioned ISAR imaging methods [8– 12] encounter many new technical problems, such as the severe cross-term interference and inaccurate parameter estimation, which will heavily deteriorate the imaging quality. Moreover, the complex motion of target will also render the migration through resolution cells (MTRCs).
1 1 ¼ R0 (vr þ v1 t r)tm (a1 t r)tm2 (g1 t r)tm3 2 6
3 Time–chirp rate distribution of signals with cubic phase term
(11) where t0 denotes the initial time of ISAR imaging, R0 denotes the corresponding initial distance from radar to the target centre and the scope of tm is [2T/2,T/2], where T denotes the azimuth imaging time. The quadratic and cubic term in range induced by the complex motion of target introduces the nonlinear-modulated phase term, which have severe effect on azimuth Doppler imaging. Assume that the transmitted signal is 1 2 ~ ~ ~ ~ (12) s(t ) ¼ ar (t ) exp j2p fc t þ gt 2
Denote the cubic polynomial phase of the azimuth signal given in (14) as 1 1 w(t) ¼ a þ ft þ gt 2 þ kt 3 2 6
(15)
where a is a constant, f the centre frequency, g the chirp rate and k the change rate of the chirp rate. It can be noted from [13] that the shift of signal along the time axis and chirp rate axis can be written as Tt0 (x(t)) ¼ x(t t0 )
(16)
and where t~ denotes the fast time, fc denotes the carrier frequency and g denotes the chirp rate. Then the baseband echo signal of scatterer P becomes ! 2R(tm ) 2R(tm ) 2 aa (tm ) exp j pg t~ s(t~, tm ) ¼ ar t~ c c 4p R(tm ) (13) exp j l After range-matched filtering, (13) becomes 2R(tm ) s(t~, tm ) ¼ sm sinc Dfr t~ c 4p R(tm ) exp j l
(14)
From (11) and (14), it can be seen that the azimuth data after envelope alignment has a time – frequency (t 2 f ) distribution as shown in Fig. 2 because of the complex motion of the target. The cubic term of the azimuth
gg0 (x(t)) ¼ x(t) exp
g t2 j 0 2
! (17)
where x(t) ¼ exp( jw(t)), Tt0 is the operation of shift along the time axis and gg0 denotes the operation of shift along the chirp rate axis. If the phase of x(t) is a quadratic polynomial, it can be seen from [14] that t t ¼ tw0 (t) w tþ w t (18) 2 2 where t covers the whole time interval as t, w0 (t) denotes the first-order differential of w(t). Therefore the Wigner time – frequency distribution of x(t) is ð t t Wx (t, v) ¼ x t þ x t exp ( j vt)dt 2 2 ð ¼ exp ( j t(v w0 (t)))dt ¼ 2pd(v w0 (t)) (19) where d(.) denotes the infinite impulse response. Because w0 (t) is a first-order polynomial function of t, the signal has linear distribution in the time – frequency plane. For the signal with cubic polynomial phase term, we have
w(t þ t) 2w(t) þ w(t t) ¼ t2 w00 (t)
(20)
where w00 (t) is a quadratic differential of w(t).
Figure 2 t– f distribution for signal with cubic phase term IET Radar Sonar Navig., 2008, Vol. 2, No. 6, pp. 395 – 403 doi: 10.1049/iet-rsn:20070101
From the analysis of the above equations, we can obtain the distribution feature of signal with respect to time and 397
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www.ietdl.org as shown in (23) " max jsi (ti )j ti
M X
T
# jsm (tm )j
i ¼ 1, 2, . . . , M
(23)
m¼1,m=i
where M denotes the number of coherent integration points and s is a vector which represents the range data of every azimuth time. The complex motion of target will result in the MTRCs of each scatterer. We can use the keystoneformatting method [17] to perform the envelope correction on the received data, as shown in (24) Figure 3 t– c distribution for signal with cubic phase term
ð Wx (t, g) ¼ Zx (t, t) exp ( j gt)dt ð
¼ exp ( j t(g w00 (t)))dt ¼ 2pd(g w00 (t))
(21)
where
pffiffiffi pffiffiffi x(t þpffiffiffiffiffiffi t )x (t)x (t)x(t pffiffiffiffiffiffi t) x (t þ t )x(t)x(t)x (t t )
m ¼ 1, 2, . . . , M
(24)
where fa denotes the azimuth frequency, tm denotes the azimuth time and tm denotes the new time axis after interpolation.
chirp rate
Zx (t, t) ¼
fc tm ¼ ( fc þ fa )tm
t0 t0 (22)
From (21), it can be noted that the time – chirp (t– c) distribution of signal with cubic phase term is a linear function with respect to time, as shown in Fig. 3. Like the quadratic curve function in the t – f distribution plane, this feature is really helpful for the estimation of instantaneous chirp rate and its change rate and hence for the ISAR imaging of complex moving target. Of course, the effect of the cross-terms also exists in the t– c distribution plane when multiple signals exist at the same time. Therefore the energy of signal should be integrated in the same way as used in the Radon– Wigner transform [15] to eliminate the effect of cross-terms in practice, which will be discussed later on.
Step 2. Initial phase error correction. The initial phase error can be corrected by the multiple well-isolated scatterer synthesis method [18]. However, this becomes very difficult due to the complex motion of the target. The least-square fitting method presented in [19] can be used for initial phase error correction. Because instantaneous imaging is of our interest, the linear and quadratic phase errors obtained through the least-square fitting will not have any influence upon the quality of imagery other than a position shift of the image. Its details will not be discussed here and the readers can refer to [19]. Step 3. Parameter estimation via TC-DechirpClean. Now we proceed the azimuth imaging process after completing Steps 1 and 2. Assume that the s(t~, tm ) in (14) turns into s0 (t~, tm ) after Steps 1 and 2. Assume that there are P scatterers in some range cell, then the echo signal corresponding to this range cell can be written as s(m) ¼
Step 1. Envelope alignment and MTRCs mitigation.The translational component of each scatterer should be compensated after the signal being range-compressed. In consideration of the jump error and drift error existing in some conventional method of range alignment, we use the global optimisation method in [16] to align the envelope, 398 & The Institution of Engineering and Technology 2008
si exp ( j(ai þ fi mTa
i¼1
1 1 þ gi (mTa )2 þ ki (mTa )3 )) þ ei (mTa ) 2 6
(25)
where
4 TC-DechirpClean ISAR imaging algorithm The TC-DechirpClean algorithm is an extension of our previously proposed DechirpClean method [10] from quadratic to cubic phase term, which consists of the following steps.
P X
fi ¼
4p (v1i r i ) l
(26)
gi ¼
4p (a r ) l 1i i
(27)
ki ¼
4p (g r ) l 1i i
(28)
and
where ai denotes the constant phase term of the ith scatterer, fi , gi and ki denote the centre frequency, chirp rate and change rate of the chirp rate of the echo signal of the ith scatterer, respectively, Ta denotes the pulse repetition time, IET Radar Sonar Navig., 2008, Vol. 2, No. 6, pp. 395– 403 doi: 10.1049/iet-rsn:20070101
www.ietdl.org si denotes the amplitude of the ith scatterer and ei(mTa) denotes the noise. The parameter estimation process can be summarised as follows.
Step 3.1. Set the initial value of the change rate of the chirp rate to be k0 according to (29), where g0 and r0 denote the parameters of the whole target, which can be obtained via tracking the target with radar. Moreover, the initial value k0 can be chosen to be the same for all scatterers, which has the form
k0 ¼
4p (g r ) l 0 0
(29)
Step 3.2. Set a step size, denoted as step, to search around k0 in the upper and lower regions according to (30), where k0 (n) is the searching range of the change rate of the chirp rate and 2N is the number of searchs needed
k0 (n) ¼ k0 þ n step
Step 3.7. When the optimum kiopt of the ith scatterer is obtained, the t– c distribution of the echo signal of this scatterer will be a line parallel to the time axis at the same time. The corresponding value of this line at the chirp-rate axis is the optimum chirp rate giopt of this scatterer. Then, the azimuth quadratic and cubic phase compensation function can be constructed according to kiopt and giopt , which has the form 1 1 sref iopt (m) ¼ exp j giopt (mTa )2 þ kiopt (mTa )3 2 6 (34) Step 3.8. By multiplying the signal of this range cell by (34) and applying the FFT, we can obtain the optimum peak value of this scatterer in the frequency domain, from whose position we can obtain the centre frequency of the signal. Then shift the signal to zero frequency and extract the amplitude of the peak value via adding narrow window, that is
(30)
n ¼ N =2, . . . , 0, . . . , N =2 1
Step 3.3. Construct the corresponding phase compensation function for searching according to k0 (n), which has the form 1 w(m, n) ¼ k0 (n)(mTa )3 6 n ¼ N =2, . . . , 0, . . . , N =2 1
fiopt ¼ arg max {jFFT{s(m) sref iopt (m)}j}
(35)
and
siopt ¼ Wini ( fa ) ðjFFT{s(m) 1 exp ( j( fiopt mTa þ giopt (mTa )2 2 1 þ kiopt (mTa )3 ))} 6
(31)
(36)
where Step 3.4. Multiply (25) with (31), we can have
Wini ( fa ) ¼
s0 (m, n) ¼ s(m) exp ( j w(m, n))
1 fLi , fa , fRi 0 other
(32)
n ¼ N =2, . . . , 0, . . . , N =2 1
where s 0 (m, n) represents the data of this range cell obtained via the nth search. Step 3.5. Calculate Ws(m, g; k) from s 0 (m, n) in (32) via the time – chirp rate transform as defined in (21). Step 3.6. If the estimated k0 is equal to the true change rate of the chirp rate (i.e. kiopt), then the t – c distribution of the signal s 0 (m, n) will be a line parallel to the time axis. Hence, the maximum signal energy can be obtained via integration along this line. On the basis of this feature, kiopt can be determined as follows (
kiopt ¼ arg max arg max kn
gi
( M X
)) jWs (m, gi ; kn )j
(33)
m¼1
We can choose a smaller value for the step size to make it more accurate (say 0.01). IET Radar Sonar Navig., 2008, Vol. 2, No. 6, pp. 395 – 403 doi: 10.1049/iet-rsn:20070101
Figure 4 Flow chart of the proposed TC-DechirpClean mehod 399
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www.ietdl.org the initial energy as
is the window function of the ith component in the frequency domain, fRi . 0, fLi , 0, whose values are determined according to the spectrum width around zero frequency. Step 3.9. The echo signal of the ith scatterer now can be reconstructed by the estimated parameters. Then subtract this signal from the original signal, as defined in (37), and we can obtain the new signal si21(m) for the next search si1 (m) ¼ si (m) siopt exp ( j( fiopt mTa 1 1 þ giopt (mTa ) 2 þ kiopt (mTa )3 )) 2 6
j¼
PM 2 ksi1 (m)k m¼1 jsi1 (m)j ¼ P M 2 ks(m)k m¼1 js(m)j
(38)
Repeat Steps 3.1– 3.9 until j is smaller than a preset threshold (say 0.01). We can reconstruct the echo signal of the complex moving target and finally obtain its instantaneous image through the above three steps. The flow chart of the proposed TCDechirpClean ISAR imaging is as shown in Fig. 4.
(37)
Step 3.10. Repeat the above process until the parameter estimates of all the P scatterers are obtained. Usually, P is not known in advance and must be estimated at the same time. Define the ratio of the residual energy and
5 Numerical and experimental results First we present a simulation example. Assume that the radar system parameters are as follows: the carrier frequency f0 ¼ 5.2 GHz, the bandwidth of the transmitted signal
Figure 5 Simulated results a b c d e f
Original ship target t–f distribution of an isolated scatterer t–c distribution of an isolated scatterer t–c distribution of an isolated scatterer before cubic phase correction RD imaging result TC-DechirpClean imaging result
400 & The Institution of Engineering and Technology 2008
IET Radar Sonar Navig., 2008, Vol. 2, No. 6, pp. 395– 403 doi: 10.1049/iet-rsn:20070101
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Figure 6 Experimental results a Range– azimuth image after range comprssion b Range – azimuth image after keystone transform c Search diagram of the changing rate of the chirp rate d t– f distribution of an isolated scatterer before cubic phase correction e t– c distribution of an isolated scatterer before cubic phase correction f t– f distribution of an isolated scatterer after cubic phase correction g t–c distribution of an isolated scatterer after cubic phase correction h Conventional RD imaging result i TC-DechirpClean imaging result
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www.ietdl.org B ¼ 80 MHz, the sampling rate fs ¼ 100 MHz, the pulse repetition frequency PRF ¼ 1000 Hz, the initial distance of the target to radar is R0 ¼ 37 km. The simulated target is a ship whose speed and acceleration are vr ¼ 30 m/s and ar ¼ 5 m/s2 respectively. Fig. 5a shows the simulated ship target model. Fig. 5b shows the distribution result of an isolated scatterer in the t – f plane, from which it can be noted clearly that the third-order phase term apparently exists and the signal has quadratic curved line feature. Fig. 5c shows the distribution of this isolated scatterer in the t – c plane, from which it can be noted that it is in linear distribution as (14) shows. Fig. 5d shows the t –c distribution of this scatterer after the cubic phase is corrected. The ISAR images obtained via the conventional RD method and the proposed TC-DechirpClean method are shown in Figs. 5e and 5f, respectively, from which it can be noted that TC-DechirpClean exhibits much better performance. Now we present the experimental results to illustrate the performance of the proposed method. The stationary radar parameters used for ISAR imaging of real ship target are as follows: the carrier frequency f0 ¼ 9.25 GHz, the bandwidth of the transmitted signal B ¼ 500 MHz and the pulse repetition frequency PRF ¼ 200 Hz. Fig. 6a shows the range-compressed image in the range – azimuth plane. Fig. 6b shows the result after range walk correction and keystone transform, from which it can be noted that the translational component and MTRCs have been corrected and the range envelope of the echo signal has been well aligned. Fig. 6c shows the searching diagram for the change rate of chirp rate of an isolated prominent scatterer in a range cell, from which it can be noted that the greatest integrated energy occurs at the optimum point via the 500 searches in the t – c plane. Figs. 6d and 6e show the distribution diagram of this scatterer in the t – f and t– c plane before third-order phase correction, respectively, from which it can be noted that the data is of curve distribution in the t – f plane and of linear distribution in the t – c plane due to the existence of the cubic phase term. Figs. 6f and 6g show the distribution diagram of this prominent scatterer in the t– f plane and t –c plane after cubic phase term correction, from which it can be noted that the data now are of linear distribution in the t – f plane and distributes along a beeline parallel to the time axis in the t – c plane. The ISAR images obtained via the conventional RD method and the proposed TC-DechirpClean method are shown in Figs. 6h and 6i, respectively, from which it can be noted once again that TC-DechirpClean outperforms the conventional RD method without cubic phase correction. In Fig. 6i, the bow is on the right-hand side, the stern is on the left-hand side and the mast is in the middle.
6
Conclusions
In this paper, we first model the complex three-dimensional motions (yaw, pitch and roll) of the ship target by cubic phase term, which is valid for a short integration period. 402 & The Institution of Engineering and Technology 2008
Then a new ISAR imaging method, referred to as TCDechirpClean, is proposed, which estimates the chirp rate and the change rate of chirp rate of all scatters in the time – chirp distribution plane. The numerical and experimental results indicate that the proposed method outperforms its counterparts established for quadratic phase model.
7
Acknowledgments
This work was supported by the National Natural Science Foundation of China under grant 60325102, 60736009, and 60502044. The authors would like to thank the No. 38 Research Institute of China for providing us with the real ship data.
8
References
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