Dynamic RCS Statistical Characterization of Aircraft Target Using Gaussian Mixture Density Model (GMDM)
Ya-Qiang Zhuang, Chen-Xin Zhang, Xiao-Kuan Zhang
Air and Missile Defense College, Air Force Engineering University, Xi’an 710051, China, [email protected]
Abstract
The dynamic RCS of an aircraft target varies dramatically with target geometry and motion information (position and velocity), the values fluctuate and have no pattern. So the fluctuation models are often adopted to describe the statistical nature of dynamic RCS. In this paper, the dynamic RCS is obtained through simulation method firstly. Then, a brief introduction of Chi-Square distribution and Log-Normal distribution model is given, and Gaussian Mixture Density Model (GMDM) is proposed in order to describe the fluctuation more accurately. As an example, the dynamic RCS data are analyzed with Chi-Square distribution, Log-Normal distribution and GMDM. The fitting results show GMDM can provide a better approximation of the dynamic RCS than the other two models from Kolmogorov-Smirnov test of fitting goodness .
1. Introduction
RCS (Radar Cross Section) is an important parameter reflects target scattering ability for incident EM waves. It often applied in the field of radar detection and target recognition. The dynamic RCS of aircraft in the movement has practical significance which can be acquired by field measurement [1] and simulation method [2]. RCS of aircraft depends on geometry and orientation whose values are often time-varying during its movement. It has been advantageous to treat the target RCS as a statistic items for the dynamic RCS with random fluctuation. Statistical distributions are used for RCS characterization. Much work has been done over the years in statistical modeling of RCS, including conventional distribution such as Chi-Square distribution, Rice distribution, Log-Normal distribution model and new distribution such as Weibull distribution, NCGG (Non Central Gamma Gamma) distribution [3] and Two State Rayleigh-Chi distribution [4]. The EM calculation model of an aircraft is established and all-attitude static RCS database is calculated firstly in this paper. After the dynamic RCS for different flight paths were obtained, statistical characterization using three fluctuation models has been investigated. The comparison results show that GMDM can provide a better approximation of the dynamic RCS.
2. Aircraft dynamic RCS
2.1 Target Model and All-attitude Static RCS Database
In this section, the EM calculation model was constructed and the all-attitude static RCS database was calculated in FEKO as shown in Figure 1.The target coordinate center is located at the geometrical center of the target. The attitude angle was defined as shown in Figure 1. The pitch angle is from X axis round to Z axis in the XOZ plane, and the azimuth angle is from X axis round to Y axis in the XOY plane.
2.2 Dynamic RCS Series After the all-attitude static RCS database was calculated, the following steps were adopted to obtain the dynamic RCS series of the special flight path. Firstly, we predestined yaw path and level path the aircraft fly and Given fly height is 5km, and radar station at original point (0km, 0km, 0km). Both paths are starting from the point (300km, 10km, 5km). In the level track, the shortcut flight course is 10km, and end at the point (10km, 10km, 5km). The yaw path is a parabola flight path and end at the point (10km, 100km, 5km ). From the flight path, we can calculate the coordinate of radar line of sight in radar coordinate system easily. A coordinate transformation need to be done in order to get the coordinate in target coordinate system. The time-varying attitude angles of radar line of sight in target coordinate were calculated. We can obtain the dynamic RCS according to attitude angle using linear interpolation method. Plot of dynamic RCS for the aircraft in different paths are provided in Figure 2.
1 0 le v e l 5 y a w
0
-5
-1 0
-1 5 R C S / d B s m -2 0
-2 5
-3 0
-3 5 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 x / k m
Figure 1. Aircraft EM model and all-attitude static Figure 2. Dynamic RCS series in different paths
RCS database
3. Target Fluctuation Models
Among the existing RCS distribution models, Chi-Square distribution and Log-Normal distribution are adopted. The parameter analysis has been discussed detailedly in [5,6].The statistical modeling method of dynamic RCS using GMDM is proposed .
3.1 Chi-Square Distribution Model
This model was introduced by Swerling and assumes that the PDF (Probability Density Function) of the target RCS is a generalized Chi-Square distribution is as Eq.(1) k 1 k k k p exp , 0 (1) k
Where is RCS variable, is the mean of , k is the gamma function and parameter k (called
2 double-degrees of freedom) is equal to /variance of .
3.2 Log-Normal Distribution Model
The fluctuation RCS of targets such as ships and missiles is often well modeled as a log-normal random variable. The PDF of Log-Normal distribution can be written as Eq.(2) 1 (In )2 p exp , 0 (2) 2 s 2 2s The normal and Log-Normal distributions are closely related. If the RCS value is distributed log normally with parameters and s , then In is distributed normally with mean and standard deviation s .
3.3 GMDM
GMDM is a semi parametric density estimation method, and it combines the advantages of parameter estimation method and non-parametric estimation method [7]. GMDM has been used widely in the field of voice recognition, image processing, micro-array gene expression data and so on. The PDF of GMDM is as Eq.(3)
2 M a x f x, i exp i i1 2s 2s i i (3)
Where (a1,a2 ,...aM ;b1,b2 ,...bM ;s1, s2 ,...sM ) , ai is the weight of number i component, and it
M i meets ai 1. The variable i and si are the mean and variance of number component respectively. If the i1
component number M is enough, the GMDM will approximates any continuous distribution with any precision.
4 Results
4.1 Fitting Results
The statistical modeling of dynamic RCS distribution and fitting method has been discussed for many years . We choose the Chi-Square distribution, Log-Normal distribution and two-order GMDM to fit the distribution of the dynamic RCS. The fitting parameters of Chi-Square distribution and Log-Normal distribution can be estimated by non-linear least squares method [8]. The distribution parameters of GMDM are estimated by EM method. This plot of fitting results as figure 3 which provide the fitting results of PDF.
1 .3 0 .1 0 Statistical 1 .2 Statistical C h i-2 0 .0 9 C h i-2 1 .1 L o g -n o rm a l L o g -n o rm a l 1 .0 GMDM 0 .0 8 GMDM 0 .9 0 .0 7 0 .8 0 .0 6 0 .7 0 .0 5 PDF
0 .6 PDF 0 .5 0 .0 4
0 .4 0 .0 3 0 .3 0 .0 2 0 .2 0 .1 0 .0 1
0 .0 0 .0 0 -3 2 -2 9 -2 6 -2 3 -2 0 -1 7 -1 4 -1 1 -8 -3 0 -2 5 -2 0 -1 5 -1 0 -5 0 5 1 0 1 5 R C S / d B s m R C S / d B s m (a) (b) Figure 3. PDF fitting results
4.2 Result Analysis
The non-parametric goodness of fit test procedure is investigated to evaluate the probability model of the RCS sample data for three models applied above. And K-S test[8] is used to test the fitting goodness. The test function is shown in Eq. (5), in which F(x) is the statistical CDF of dynamic RCS data and F'(x) is CDF of the fitting results. The larger of test value indicates the worse applicable of distribution model. The results of K-S test are shown in Table 1. And critical value of K-S test is 0.13403 for the sample number larger than 100 at the significance of 0.05.
D max Fx F'x (6)
Table 1. K-S test result in level track and maneuver track Chi-Square Log-Normal GMDM Level 0.3807 0.3595 0.2963 Yaw 0.1434 0.0805 0.0664 From the fitting curves and K-S test results, we can conclude that GMDM and Log-Normal distribution are better than Chi-Square distribution for dynamic RCS of this target in both flight paths. The fitting result of GMDM is a little better than Log-Normal distribution from errors-of-fit and K-S test results.
5. Conclusion
A statistical modeling method of dynamic RCS based on GMDM is investigated. The comparison on statistical characterization of dynamic RCS under typical paths using Chi-Square distribution, Log-Normal distribution and GMDM is analyzed. The fitting results show that GMDM is more applicable than Chi-Square distribution and Log-Normal distribution to describe the dynamic RCS, especially under yaw path. The result of this paper describes exactly the fluctuation characteristic of target dynamic RCS, and can provide a reliable support to accurate simulation and examining of maneuver target's echo.
6. References
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