J. Cent. South Univ. (2012) 19: 1548-1557

DOI: 10.1007/s11771-012-1175-2

Micro-motion effect in inverse synthetic aperture radar imaging of ballistic mid-course targets

ZOU Fei(邹飞), FU Yao-wen(付耀文), JIANG Wei-dong(姜卫东)

Institute of Space Electronic Technology, National University of Defense Technology, Changsha 410073, China

? Central South University Press and Springer-Verlag Berlin Heidelberg 2012

Abstract: For ballistic mid-course targets, in addition to constant orbital motion, the target or any structure on the target undergoes micro-motion dynamics, such as spin, precession and tumbling. The micro-motion characteristics of the ballistic mid-course targets were discussed. The target motion model and inverse synthetic aperture radar (ISAR) imaging model for this kind of targets were  built. Then, the influence of micro-motion on ISAR imaging based on the established imaging model was presented. The computer simulation to get mid-course target echoes from static darkroom electromagnetic scattering data based on the established target motion model was realized. The imaging results of computer simulation show the validity of ISAR imaging analysis for micro-motion targets.

Key words: micro-motion, ballistic mid-course targets, inverse synthetic aperture radar imaging (ISAR)

1 Introduction

The inverse synthetic aperture radar (ISAR) is a technique for forming high resolution images by utilizing the information inherent in the differential target dopplers. It has drawn wide attention during the past decades [1-3], due to its excellent capability of producing high-resolution target images over long ranges under all weather conditions. The resolution in range is obtained by transmitting large bandwidth signal and the resolution in cross-range is achieved by utilizing the relative motion between target and radar.

For ballistic mid-course targets, in addition to constant orbital motion, the target or any structure on the target undergoes micro-motion dynamics, such as spin, precession and tumbling. The backscatter of target with micro-motions induces doppler modulations on the returned signal, which affects the inverse synthetic aperture radar (ISAR) imaging. For a target that has only bulk translation with a constant velocity, the doppler frequency shift of echoes is a time-invariant function. If the target also undergoes micro-motion, then the doppler frequency shift is a time-varying frequency function and imposes a periodic time-varying modulation onto the carrier frequency [1]. Spin, precession and tumbling make the ballistic mid-course targets have complex motion and imaging characteristics, which are discussed in this work.

The contributions of this work are as follows. Firstly, a model of the micro-motion effect for mid-course targets is developed; Secondly, mathematical formulas of ISAR echoes with micro-motion modulations are derived and verified by simulation studies; Thirdly, computer simulation is realized to get dynamic mid-course target echoes from static darkroom electromagnetic scattering data based on the established target motion model; Finally, micro-motion effect on ISAR imaging is demonstrated by using real darkroom electromagnetic scattering data.

2 Data model

2.1 Target motion model

2.1.1 Orbital motion

The mid-course target center rotates around the earth core with elliptical orbit by the effect of earth gravity, when the rarefied atmosphere resistance and  star gravity are ignored, as shown in Fig. 1. So, the acceleration of mid-course target can be expressed as  g=-9.8 m/s². Let the initial target center location and velocity be P_o and V_o, respectively, then the motion of target center can be expressed as

                           (1)

2.1.2 Micro-motion

The ballistic mid-course targets undergo micro- motion dynamics, such as spin, precession and tumbling, in addition to bulk orbital motion. Ballistic targets are under the effect of aerodynamic moment when flying inside aerosphere. To avoid warhead turnover by the aerodynamic moment, the missile keeps spin around its central axis with high velocity and precession around the target center forward direction, which makes its warhead uniformly go forward and keep fly direction. When targets fly in free section (the middle part of ballistic trajectory), it will keep the spin and precession characteristics, though the target is not under the effect of aerodynamic moment and control moment. The spin and precession characteristics are inherent attribute of warhead and could be important characteristic quantity for target recognition, just as the orbit and electromagnetic scattering characteristics [4-6]. As shown in Fig. 2, target spin and precession would make the relative azimuth angle and elevation angle between target and radar change with time.

Fig. 1 Orbital motion of mid-course target

Fig. 2 Micro-motion of mid-course target

2.1.3 Relative motion of radar to target

In order to describe the relative motion of radar to target, the radar coordinate system Q-xyz, target coordinate system O-xyz and reference coordinate system O-x′y′z′ are introduced. Radar is located in the origin of radar coordinate. Target coordinate has the same translation and rotation to the target. With the origin lying at the origin of coordinate, x axis always points the target motion direction and z axis is perpendicular to trajectory plane. The relationship of the three coordinate systems is shown in Fig. 3.

Fig. 3 Relative motion of ballistic target to radar

Let the range between target scatter point i to radar be  the range between target center to radar be  radar line of sight (LOS) in reference coordinate be  and the position of scatter point i in reference coordinate be  Then,  can be approximately expressed as [7]

          (2)

To analyze the motion of ballistic target center, let be the position of target center in radar coordinate system at the moment of t, then  can be expressed as

        (3)

Ballistic targets have various forms of micro-motion dynamics, such as spin, precession and tumbling. The rotation angular velocity produced by the micro-motion is expressed as

             (4)

Let r_i(t_m) be the position of scatter point i in reference coordinate system at the moment of t_m. According to the geometry transformation theory [7],

                       (5)

Let  be the rotation matrix in target coordinate system from moment t_m to moment t. As the ballistic target is in the same moving plane, the target coordinate system only rotates by z axis. Let the change of target velocity direction angle be  so  can be written as [8]

               (6)

According to Rodrigues function [7],

                      (7)

where

                     (8)

Substituting Eqs. (6) and (7) into Eq. (5), then  can be rewritten as

                     (9)

Then,

           (10)

Substituting Eqs. (3) and (10) into Eq. (2), then

            (11)

 in Eq. (11) is approximated by two order Taylor expansion at t=t_m [9]:

     (12)

where

                (13)

              (14)

Let  and   be the position and velocity of the target center in radar coordinate system at moment t respectively:

       (15)

As the target in the middle part of the ballistic trajectory is mainly with the characteristic of turbulence, the influence of earth rotation and earth oblateness can be ignored. The target acceleration in radar coordinate system can be approximately expressed as a_o=  (g=-9.8 m/s²) and

     (16)

where

(17)

Substituting Eqs. (17) and (16) into Eqs. (13) and (14),

                     (18)

                       (19)

Then  can be rewritten as

(20)

Asthe Eq. (20) can be simplified as

    (21)

For  in Eq. (11), as  , then

                      (22)

As   there is

                  (23)

By doing Taylor expansion for Eq. (23), there

                        (24)

Substituting Eq. (21) and (24) into Eq. (11), then the second order approximation of  can be expressed as

                   (25)

Therefore,

        (26)

where

2.2 ISAR imaging model of ballistic mid-course targets

In practice, in order to increase transmit power and decrease the receiver bandwidth, the radar usually transmits LFM signal, and then dechirps the received signal [10]. Suppose the transmitting signal as

     (27)

where γ=B/T_p, is chirp rate; T_p is the time width of transmit signal; B is the bandwidth of transmit signal; R_rect(·) is the rectangular enclosure of transmit signal; f_c is carrier frequency;  is the range fast time; t_m=mT, is the azimuth slow time, and m is the  sequence number of the transmit signal. In high frequency region, radar target can be approximately expressed as point scatter model. Suppose the target contains Q scatter points, δ_i is scattering intensity of scatter point i, and the range of scatter point i to radar is  then radar echo signal can be expressed as

                                     (28)

where w(t) is white Gaussian noise, with zero mean value and a variance of σ².

Let R_ref(t_m) be the reference range, which describes the range from target center to radar. As shown in Fig. 3, R_ref(t_m) can be expressed as

                 (29)

Then, the reference signal can be written as

            (30)

After dechirping [10],

(31)

where

Substituting Eq. (26) into Eq. (32), yields

           (33)

where

3 Analysis of micro-motion effect on ISAR imaging

The rotation frequency of mid-course ballistic targets is generally less than 10 Hz (with the typical value of 2 Hz) [11]. For intercontinental ballistic missile, the warhead and missile body are separated after boost phase, so mid-course targets are usually with small size. For example, the bottom radius and warhead height of American minuteman missile are about 0.3 and 2 m, respectively. From the target group allocation of the eighth American integrated flight test (IFT-8) [12], it can be seen that baits are usually with small size, too. The diameter of large balloons is about 2.2 m in IFT-8. Assuming the rotation angular velocity Ω=4π rad/s, target size ||r_max||<10 m, then  As   and  Eq. (35) can be approximated as

           (34)

where

According to Eq. (33), it can be found that

So, function (34) can be further simplified as

(35)

where

By the above analysis, for typical mid-course targets (with a size less than 10 m and an effective rotation angular velocity less than 10 Hz) [11-12], the radar echo signal can be approximated to multi- component LFM signal with same chirp rate, where the chirp rate depends on the effective rotation angular velocity of target center [13].

To anayze the influence by micro-motion, the middle course radar target echo can be rewritten as

      (36)

where

After making Fourier transformation for Eq. (36) in time  there is

    (37)

The frequency spectrum of LFM signal can be approximated as

        (38)

where  

The second term in Eq. (37) can be approximated as the multiple points frequency signal with the frequency  and then the convolution in Eq. (37) can be approximate as

(39)

Getting the amplitude spectrum of Eq. (38), we obtain the 1-D high-resolution range (HHR) profile as

     (40)

When the radial velocity v_l(t_m)=0, there is

       (41)

Doing a comparison between Eqs. (40) and (41), we can find that when v_l(t_m)≠0, the HRR changes from point frequency f=-(2γ/c)?r_i(t_m) to rectangular with a center frequency of

 

and bandwidth ?B=[4v_l(t_m)·B]/c. The HRR spectral resolution by FFT ?f is 1/T_p, so the HRR center frequency translation Δ_m and spectrum broadening Δ_b induced by radial velocity are

     (42)

(43)

Equation (43) can be expressed as

,

where the denominator denotes the HRR resolution, and the molecular denotes the distance change in the fast time. So, the spectrum broadening represents the range resolution cells undergoing in fast time, and expresses the effect induced by micro-motion, which results in the decrease of imaging quality.

4 Simulation of micro-motion dynamics effected on ISAR imaging

4.1 Echo generation

By comprehensive consideration of the target orbital motion and around center rotation motion, the middle course target ISAR echo can be produced as follows:

1) Setting the initial moving state (V_k, R_k, θ_k) in free phase trajectory of ballistic missile center, and the around target center rotation angular velocity (θ, Ω, ω);

2) Getting the target center location R_m(t) from (V_k, R_k, θ_k) and the radar target moving model;

3) Calculating the distance ||R_0m(t)|| and angular (α(t), β(t)) of radar to target in the moment t according to R_m(t) and the radar location R₀;

4) Calculating the radar line of sight (γ(t), f(t)) in the target coordinate system according to (θ, Ω, ω) and (α(t), β(t));

5) Getting the echo E_s(t) under (γ(t), f(t)) using the static darkroom radar echoes of all attitude angles;

6) Obtaining the echo data of the middle course ballistic target E(t)=E_s(t)exp(-jπ4f||R_0m(t)||/c).

4.2 Numerical examples

1) Simulation of orbital motion

The simulation radar parameters are: radar position (-74.1° 40.75°), carrier frequency f_c=10 GHz, pulse width T=100 μs, transmit signal bandwidth B=1 GHz, rang resolution 0.15 m, chirp rate K=1×10¹³, data sampling rate f_s=10 MHz, and pulse repetition frequency F_PRF=100 Hz. Supposing the launching point of longitude latitude is (2.43°  48.88°) and the impact point of longitude latitude is (-74.1°  40.75°), the position and velocity of ballistic target in earth centered inertial (ECI) are shown in Fig. 4.

2) Simulation of micro-motion dynamics in ISAR imaging.

As the azimuth angle changes between 0°-48° from shutdown point to impact point [6], we select 0°, 20° and 40° azimuth angles to obtain imaging results. Using static darkroom electromagnetic scattering data to simulate real mid-course target echo, the ISAR  imaging results are shown in Figs. 5-6. Figures 5(a₁-a₃) show the ISAR imaging results of ballistic target with orbital motion at different azimuth angles. In       Figs. 5(b₁-b₃), the imaging results with both orbital motion and spin are shown. Because the darkroom electromagnetic scattering model is rotational symmetry metal cone, the imaging results with both orbital motion and spin are basically the same as the imaging results with only orbital motion. Figures 5(c₁-c₃) and (d₁-d₃), and Fig. 6 show the ISAR imaging results with both orbital motion and precession of various frequency. It can be noted that the imaging quality decreases with the increase of precession frequency, when the precession frequency exceed 1 Hz, the images are defocusing seriously.

Fig. 4 Position and velocity of ballistic target in earth centered inertial (ECI): (a) Position; (b) Velocity

Fig. 5 ISAR imaging results on azimuth angles of 0° (a₁-d₁), 20° (a₂-d₂) and 40° (a₃-d₃): (a) Orbital motion; (b) Orbital motion with 0.8 Hz spin; (c) Orbital motion with 0.2 Hz precession; (d) Orbital motion with 0.5 Hz precession

Fig. 6 ISAR imaging results of orbital motion with 1 Hz and 2 Hz precessions: (a) and (a′) Azimuth angle of 0°; (b) and (b′) Azimuth angle of 20°; (c) and (c′) Azimuth angle of 40°

5 Conclusions

1) The micro-motion dynamics in ballistic mid-course targets is discussed. The micro-motion model and ISAR echo model for mid-course targets are presented. Mathematical formulas for micro-motion modulation effect of radar echo is derived. By simulation dynamic mid-course target echoes from static darkroom electromagnetic scattering data with ISAR imaging results, the effectiveness of this theoretical analysis is confirmed.

2) The usage of micro-motion model for improving ISAR imaging quality, and applications of micro-motion characteristics in space target recognition should be studied in future work.

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(Edited by DENG Lü-xiang)

Foundation item: Project(61360020102) supported by the National Basic Research Development Program of China

Received date: 2011-07-01; Accepted date: 2011-11-12

Corresponding author: FU Yao-wen, Associate Professor, PhD; Tel: +86-731-4575716; E-mail: fuyaowen@sina.com