Numerical Computation of Simulated Polarimetric Radar Imagery for Electrically-large Aircrafts

Xia Wu 1，2 1 School of Electronics and Information Engineering, Tongji University

201804, Shanghai, P. R. China 2State Key Laboratory of Millimeter Waves,

210096, Nanjing, P. R .China wuxiaeecs00@gmail.com

Abstract—In this paper, we generate numerical results of simulated scattering-imaging-reconstruction for large complex aircrafts. Physical Optics (PO) method is employed for high-frequency electromagnetic scattering calculations of complex-shaped electrically large bodies. Polarimetric radar images in real-time are formed with possible polarizations (hh, hv, vh and vv) to allow a more detailed examination of the scattering mechanisms. Keywords—Physical Optics method, Polarimetric radar, electrically large target, electromagnetic scattering

I. INTRODUCTION The development of fast geometry- electromagnetic

modelling has been of great interest in both civilian and military applications. A variety of computational electromagnetics algorithms have been proposed in the past few decades [1, 2]. Multilevel Fast Multipole Algorithm (MLFMA) was implemented on basis of the architecture of a parallel platform and properties of extremely large targets [3]. However, the personal computer is still the workhorse computing device in homes and offices, and the most exciting innovations were centered on large-scale electromagnetics computing. Physical Optics (PO) method is a current-based high frequency asymptotic method intended for modelling of electrically very large structures [4].

There are difficulties in achieving real-time simulations for forming bistatic image. To address this problem, we have introduced a fast back projection (FBP) algorithm for radar images generation from the scattering field results [5]. The effect of CAD model complexity is considered as well. It is naturally the most visible that this study provides a highly efficient and effective theoretical and mathematic model for the development of remote sensing technology.

II. PRINCIPLE OF NUMERICAL SCATTERING MECHANISMS AND IMAGING ALGORITHM

A. Physical Optics Model PO method can be used for scattering analysis. Hereby, we

introduce Fresnel reflection coefficient Rh and Rv, in the case

the polarizations are labeled h for horizontal and v for vertical. Assume incident angles are θi and φi, scattering angles are θs and φs, hence the arbitrary polarization of electric field can be expressed as the sum of incident field and scattering field as following,

vEs r( ) = 1

4π rjke jkr E0 I - ksks( )∫ ⋅e jk ( ki−ks )⋅ ′r

⋅[O1 +O2 +O3 +O4]dr (1)

where,

O1 = (ei ⋅ qi )(ks × (n ⋅ qi ))(1+ Rh ) (2a)

O2 = (ei ⋅ pi )(n ⋅ qi )(1+ Rv ) (2b)

O3 = −(ei ⋅ qi )(n ⋅ ki )(1− Rh ) (2c)

O4 = (ei ⋅ pi )(n ⋅ ki )(ks × qi )(1− Rv ) (2d)

in which, n is element normal vector， ˆ ip and ˆiq set up a

local incident coordinate system. ie represents polarized

vector of incident field, which ihe and ive are polarizations of horizontal and vertical incidence respectively. Additionally

( ihe , ive , ik ) define an orthogonal coordinate system.

B. Fast Back Projection Imaging Algorithm The essence of imaging is recovery of function from the

echo signal by using signal processing. We analyzed radar images generating from the scattering field results.

f (xm , yn ) = sM tmn(u),u⎡⎣ ⎤⎦dut∫ (3)

where (xm , yn ) is a pixel, tmn(u) is the time delay between each position with

( )2 2( ) 2 ( ) /mn m nt u x y u c= + − (4)

sM (t,u) = s(t,u)⊗ p∗(−t) (5)

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where, p(t) is the transmit signal, ⊗ denotes convolution and the star denotes complex conjugate.

From Eq. (4), each pixel (xm , yn ) in the imaging region is calculated with respect to a synthetic aperture time and the time delay tmn(u) . The corresponding cumulative curve can be found in the data matrix after range compression, and the image of each pixel, that is the total energy of each pixel, can be obtained from the coherent superpositon of all the signals along the curve.

A fast back projection (FBP) algorithm accelerates the image reconstruction, making the numerical complication of the formulation mathematically equivalent to O(N2logN) [5].

III. GEOMETRIC MODELLING AND NUMERICAL RESULTS The first step is the preprocessing by GMSH—a three-

dimensional modelling tool [6]. Polarimetric radars can transmit and receive two orthogonal

polarizations, and four transmit-receive combinations are possible (hh, hv, vh and vv).

Fig. 1 F-16 3D lighting effects model

Figure 1 shows the 3D lighting effects of geometric

modelling of F-16 fighting falcon, with 16.1 m length, 5.5 m height, and a wingspan of 8.7 m. The incident field propagates along the direction with a 0º or 60º angle towards positive y direction. With receiver positions locate in the semicircle of the negative y-axis (radius R = 100 m).

The co-polarization and cross-polarization radar images were illustrated in Figure 2 and Figure 3 for given bandwidth (BW) 1 GHz. The run time of the procedures is 1415.42 second. Figure 4 and Figure 5 show the results from incident field propagates along the direction with a 60º angle towards positive y direction.

Fig. 2 F-16 co-polarization radar image BW=1GHz

Fig. 3 F-16 cross-polarization radar image BW=1GHz

Fig. 4 F-16 co-polarization radar image BW=1GHz (with a 60º angle towards

positive y direction)

Fig. 5 F-16 cross-polarization radar image BW=1GHz (with a 60º angle

towards positive y direction)

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IV. CONCLUSIONS An efficient imaging reconstruction method is discussed on

calculation of polarized scattering fields of complex target using PO method. For a given transmitter position, radar frequency and polarization of transmission, scattering results of canonical military case F-16 Fighting Falcon was calculated, and the co-polarization and cross-polarization radar images were generated as well. For future work, Graphics Process Unit (GPU) can accelerate the approach in hardware system.

ACKNOWLEDGMENT This work was supported by the Open Research Fund of

Key Laboratory of Digital Earth Science, Center for Earth Observation and Digital Earth, Chinese Academy of Sciences (Grant No. 2011LDE013); the National Natural Science Foundation of China (Grant No. 41001194); and the Fundamental Research Funds for the Central Universities (Grant No. 2011KJ049).

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“A hybrid time-domain technique that combines the finite element, finite difference and method of moment techniques to solve complex electromagnetic problems,” IEEE Trans. Antennas Propag., Vol.52, pp. 2666-2673, 2004.

[3] W. C. Chew, J. M. Jin, E. Michielssen, and J. Song, Fast and Efficient Algorithms in Computational Electromagnetics, Artech House, 2001.

[4] R. S. Longhurst, Geometrical and Physical Optics , 3rd ed., London: Longman, 1973.

[5] S. Xiao, D. C. Munson, S. Basu, and Y. Bresler, “An N2logN back-projection algorithm for SAR image formation,” the Thirty-Fourth Asilomar Conference on Signals, Systems and Computers, Pacific Grove , CA, USA, vol. 1, pp. 3-7, Oct. 2000.

[6] C. Geuzaine, J. F. Remacle, “Gmsh: A 3-D finite element mesh generator with built-in pre- and postprocessing facilities”, Int. J. Numer. Meth. Engng ., vol. 79, pp. 1309-1331, 2009.C. Geuzaine pp. 1309-1331, 2009.

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