Research Article  Open Access
Property Analysis of MIMOBased MissileBorne ForwardLooking SAR
Abstract
As a special multipleinput multipleoutput (MIMO) radar networking mode, missileborne forwardlooking synthetic aperture radar (MFLSAR) has many potential applications. This paper describes and analyzes properties of this new configuration. Range history and Doppler history are analyzed and derived using the designed geometric configuration. Then the expressions of range and Doppler resolution are determined based on the validity of twodimensional (2D) resolution imaging capability. To help to design the proper system and motion parameters of this configuration, key parameters affecting the imaging ability are found out. Due to high velocities and accelerations of both transmitter and receiver, highorder terms in the slant range equation should be kept to reduce the approximation error. The range resolution and Doppler resolution of MFLSAR are both spacevariant and timevariant owing to the complexity of this configuration. The tiny changes of 2D resolution during the synthetic aperture time should be considered when designing the imaging algorithm of MFLSAR.
1. Introduction
Multipleinput multipleoutput (MIMO) [1–4] radar networking system is a new radar mode in which many transmitters synchronize to transmit signals and many other receivers get target echo signals simultaneously. Proposed at the beginning of 21th century, it has been attracting much attention because of its good performance in antistealth, low probability intercept and antijamming capability. Besides, application of MIMO radar networking to radar imaging [5] could improve radar imaging resolution and thus can be further applied to enhance properties of targets detection, recognition, and tracking [6, 7]. As one special MIMO radar networking mode, missileborne multistatic synthetic aperture radar (SAR) has potential applications on ground detection, battlefield reconnaissance, active homing guidance, and so forth. Electromagnetic wave transmitted by many transmitters impinges receivers after reflecting from target area and gets centralized processing [8, 9]. Such technology can achieve highresolution imaging of high speed platform for those targets in the straightahead position and realize tracking and attacking with high precision, so it can be used in terminal guidance of missile [10, 11] especially.
In order to confirm the validity of application of such MIMO radar networking system to missileborne platform, the mode of “singletransmitting and singlereceiving” is assumed, that is, missileborne bistatic forwardlooking SAR (MBFLSAR). MBFLSAR is a new bistatic SAR imaging mode [12], which has advantage of 2D imaging ability in the forwardlooking mode over monostatic SAR [13]. Owing to the “fartransmitting and nearreceiving” mode and forwardlooking mode of such configuration, transmitter can be placed in safe area far away from target area, while attack missile receives target echo signals in electromagnet silence state, acting as receiver. MBFLSAR can work to image and seek targets actively during the whole terminal attacking stage because of its low probability intercept and good stealth performance. As one mode of bistatic SARs, MBFLSAR is also subject to the problems and special requirements of synchronization of the transmitter and receiver. Some investigations were conducted and appealing approaches were suggested on synchronization of bistatic SAR [14–16]. This paper concentrates on description and analysis of the properties of this new configuration and ignores the synchronization problem.
Configuration and resolution analysis are the basis of MBFLSAR imaging study, and some researches have been published. Three different bistatic forwardlooking SAR configurations have been analyzed, and the optimal geometry configurations have been proposed in [17]. Qiu and Hu designed bistatic forwardlooking geometry with stationary transmitter and forwardlooking receiver and analyzed its 2D resolution in [18]. Experiments have already been carried out to test the bistatic forwardlooking imaging ability [19, 20]. Some imaging algorithms of bistatic forwardlooking configuration have been suggested [21–23]. But all investigations above are focused on airborne or spaceborne bistatic forwardlooking configurations; a few studies about missileborne platform have been achieved.
In this paper, we describe and analyze this special configuration. Geometric configuration and signal model of MBFLSAR are introduced in Section 2. Highorder terms appearing in the range equation cannot be ignored, because of the fact that both transmitter and receiver have high velocities and accelerations. Doppler parameters are found out and analyzed based on the Taylor series expansion of the range equation. In Section 3, we give the derivation of 2D resolution of MBFLSAR and find out key parameters affecting the imaging performance. Several examples are provided to illustrate the resolution ability in the MBFLSAR in Section 4. Finally, conclusions are drawn in Section 5.
2. Geometric Configuration and Signal Model
2.1. Geometric Configuration
MBFLSAR configuration is shown in Figure 1, in which both transmitter and receiver travel in curvilinear descending motion. (here, the height of the scene is assumed to be zero) is a target point at any position of the dashed area, that is, the imaging area. Transmitter is descending along the curve in the plane which forms an included angle with plane , in which receiver is travelling along . is the origin of coordinates, and is the projection of transmitter when the slow time . For convenience of description, all motion parameters of transmitter are described in coordinates , and the ones of receiver are described in coordinates . Assume that, at , transmitter and receiver are at and , respectively. Velocity vectors of transmitter and receiver at that moment are and , respectively; acceleration vectors are and , respectively. The components in directions and are and , respectively; the components in directions and are and , respectively.
The locations of transmitter and receiver at are and ; then we have
The position of transmitter in coordinates , according to the rotation relationship between coordinates and , can be expressed as
Then we can obtain the expression of instantaneous bistatic slant range as where is the position of in coordinates ; it can be obtained using the method similar to (2) and with
2.2. Signal Model
Suppose that the transmitted waveform is the linear frequency modulation (LFM), and scattering from can be written as where is the range time, is the wavelength, is the speed of light, and is the chirp rate. and are the range and azimuth envelopes, respectively.
To design imaging algorithms for MBFLSAR more conveniently, it is important to take efficient approximation of ; some simulations are given using the parameters listed in Table 1 and the results are shown in Figure 2.

(a) Up to quadratic term
(b) Up to the third term
It can be seen that the maximum approximation error when keeping the terms up to the quadratic term is about 0.02 m, nearly the value of wavelength, so it is necessary to keep the higher term. When keeping the terms up to the third term, the maximum approximation error is only 2.44 × 10^{−6} m, which is much less than the value of wavelength. Therefore, expanding (3) as a Taylor series of needs to keep the terms up to the thirdorder with
2.2.1. Doppler Frequency and Doppler Centroid
Doppler frequency can be obtained through derivation of slant range and is expressed as
Doppler centroid represents echo Doppler frequency at the moment beam center cross target, given by
Doppler centroid represents correlativity only to motion parameters and irrelevance to location of targets in the traditional case where monostatic SAR moves with invariant velocity along a straight line or the case where transmitter and receiver move with equal velocities along parallel trajectories in traditional airborne bistatic SAR system. However, it can be seen from (9) that Doppler centroid in MBFLSAR conducts different performance that varies with such parameters correlative to location of targets as , , , and , so is spacevariant in MBFLSAR configuration.
2.2.2. Doppler Frequency Rate
Doppler frequency rate is the change rate of Doppler frequency and can be expressed as
We can find from (2.1) and (10) that Doppler frequency rate not only depends on the coordinates of targets but also relates strongly to original velocities and accelerations of platforms in this special configuration. Given such consumption that transmitter and receiver have equivalent velocities and invariant moving heights and accelerations are ignored and that the included angle is consumed as 0°, we can get the traditional bistatic parallel and velocityequivalent mode. Therefore, the traditional bistatic parallel and velocityequivalent mode is the special case of MBFLSAR to some extent.
2.2.3. Range Walk Ratio (RWR)
Range walk ratio represents the range walk increment of echo signal in per time unit; it can be obtained as
Similar to Doppler centroid, range walk ratio in MBFLSAR configuration is also spacevariant. Such spacevariance leads to the result that MBFLSAR does not have azimuthinvariant property, which is important to conventional algorithms in monostatic SAR and airborne bistatic SAR in parallel and velocityequivalent mode. So spacevariance must be considered when the range cell migration correction is achieved.
3. Analysis of 2D Resolution
In this section, 2D imagingresolution of MBFLSAR is analyzed. The expressions of range resolution and Doppler resolution are determined using gradient method [24]. Then key parameters affecting the 2D imaging ability are found out, which is helpful to design system parameters and motion parameters.
3.1. Range Resolution
Assume that and represent slant range vectors from transmitter and receiver to target at , respectively, and can be written as where , , and are unit vectors in the directions axis, axis, and axis.
According to the geometrical relationship in Figure 1, we can determine the coefficients in (12) such that where and are squint angle and look angle of transmitter, respectively, and are angle deviating from the beam center and look angle of receiver, respectively, and the angles are as follows:
Substituting (13) into (12) and rearranging the result, we can obtain the unit vectors and from transmitter and receiver to , respectively, expressed as
According to gradient method in [24], we can get the range resolution of MBFLSAR: where is bandwidth of transmitted signal and is the range resolution which reflects the capability of distinguishing two targets in horizontal plane of imaging area. Given a certain bandwidth , is related to such angles between radars and target as , , , and and included angle (collectively called “space angles”). Although heights of transmitter and receiver do not appear in (16) explicitly, they can also affect through such “space angles.” In addition, the values of are different because of different space angles caused by different coordinates of targets and different slow time . Thus, is not only spacevariant but also timevariant.
3.2. Doppler Resolution
As shown in the coordinate system in Figure 1, velocity vectors of transmitter and receiver platforms are expressed as where and represent speed components of transmitter and receiver in horizontal plane, respectively, and and denote speed components of transmitter and receiver in the vertical plane. Because the directions of motion of transmitter and receiver in the vertical plane are downward, speed components in vertical direction are negative.
Again, using the gradient method, the Doppler resolution can be obtained as with
Similarly, is also dependent on “space angles.” Besides, synthetic aperture time and signal wavelength can also have influence on . Similar to , is spacevariant and timevariant too.
4. Numerical Results
In this section, several examples are provided to illustrate the resolution ability in the MBFLSAR. Below are some simulation results of the resolution capability and 2D resolution. Parameters used in the simulations are listed in Table 1. Due to the features of spacevariance and timevariance, we give the results at a certain time (here, ). Figure 3 depicts MBFLSAR range contours and Doppler frequency contours, and MBFLSAR range resolution and Doppler resolution are given in Figure 4. To test the timevariant property, some simulations of the 2D resolution of the point target located at (0, 4500, 0) at different slow time are also implemented, and the results are shown in Figure 5.
(a) Range resolution
(b) Doppler resolution
(a) Range resolution
(b) Doppler resolution
To further analyze the signal properties of MBFLSAR, several simulations about airborne bistatic forwardlooking SAR (ABFLSAR) are also realized, where the velocities of the transmitter and receiver are 150 m/s and 100 m/s with included angle of 20 degrees, and the heights are 5 km and 2 km, respectively. Doppler frequency and RWR comparisons of fivepoint targets 150 m apart from each other in the imaging area are conducted between ABFLSAR and MBFLSAR during the same synthetic aperture time. The results are given in Figures 6 and 7, respectively. Besides, experiments on range cell curvature (RCC) of onepoint target are finished and results are in Figure 8.
(a) ABFLSAR
(b) MBFLSAR
(a) ABFLSAR
(b) MBFLSAR
(a) ABFLSAR
(b) MBFLSAR
MBFLSAR range contours and Doppler frequency contours when are shown in Figure 3, where range contours and Doppler frequency contours are intersectant but all of them are not perpendicular. It can be seen that MBFLSAR configuration has the capability of 2D imaging resolution. In addition, range contours and Doppler frequency contours are nearly perpendicular in some places of the imaging area but are nearly parallel in some other places, which describe the spacevariant feature of 2D resolution as discussed previously.
Figure 4 represents the results of MBFLSAR range resolution and Doppler resolution when . Range resolution is shown in Figure 4(a), where we can find that the values of range resolution vary with the location of the targets in the imaging area. And because of the particularity of MBFLSAR configuration, there exist isolines among different targets located in different location and location; for example, two targets located in (1350, 1510) and (2220, 3810) have equal value of 6.5759 m. Unlike range resolution, the value of Doppler resolution is very low, which can be seen from Figure 4(b). The spacevariant property of Doppler resolution can also be found from the same value of 0.9113 m of two targets located in (810, 3130) and (3640, 2720), respectively. In addition, targets located in smaller location conduct better performance than ones located in greater location in MBFLSAR configuration.
Some simulations are also achieved to analyze range resolution and Doppler resolution of MBFLSAR at different , and the results are like Figure 5. Range resolution and Doppler resolution vary with but the changes are tiny during the synthetic aperture time, 0.044 m of range resolution and 0.016 m of Doppler resolution, respectively. This characteristic can be considered when designing the imaging algorithms of this configuration, and some approximations could be taken.
Because of the presence of high velocities and accelerations in MBFLSAR configuration, Doppler frequencies of targets in the imaging area have greater changes than ABFLSAR. And range cell migrations (RCMs) are also more severe during the same synthetic aperture time; that is to say, RWRs are greater. As shown in Figure 6, Doppler frequency curves in both configurations are nearly parallel linear lines, but the maximum changes of Doppler frequency in ABFLSAR are only 131 Hz, while the one in MBFLSAR is much greater, about 9600 Hz. Results given in Figure 7 show that RWRs in both configurations are spacevariant, but the one in MBFLSAR is more grievous. The maximum RCM in ABFLSAR is about 16 m, while the one during the same synthetic aperture time in MBFLSAR is 45 m, which results from the fact that the range in MBFLSAR is much further and changes more rapidly than ABFLSAR. Figure 8 depicts RCC comparison between ABFLSAR and MBFLSAR, where we can find that RCC in MBFLSAR is more serious than ABFLSAR, about 20 m and 2 m, respectively. The greater RCC could not be neglected in MBFLSAR when designing imaging algorithms, while the one in ABFLSAR can be ignored if the influence on the resolution is small enough.
5. Conclusions
Applications of MIMO radar networking technology to radar imaging can improve the performance of radar imaging resolution, targets detection, recognition, and tracking. As a special MIMO radar networking system, MBFLSAR is a special mode of bistatic forwardlooking imaging by combining geometry configuration of bistatic motion with forwardlooking mode and has potential of being applied to missile precision terminal guidance. Properties of MBFLSAR are analyzed in this paper. The high velocities and the two squareroot terms lead to the presence of highorder terms in slant range history and Doppler history which cannot be ignored. Simulations have been done to illustrate the resolution ability of this configuration, and both range resolution and Doppler resolution are spacevariant and timevariant. The tiny changes of the values of 2D resolution in this configuration during short synthetic aperture time and comparisons with ABFLSAR can be used to design imaging algorithms in this special configuration. Research on the properties of MBFLSAR configuration validates the efficiency of the application of MIMO radar networking to radar imaging, which also provides theoretical groundwork for the whole system parameter design and imaging algorithms of MBFLSAR.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was supported by the National Natural Science Foundation of China under Grant nos. 61001211 and 61303035, the Fundamental Research Funds for the Central Universities (K5051202016), and the Science Foundation for Navigation (20110181004).
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Copyright
Copyright © 2014 Yachao Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.