ZHENG Shuyu，ZHANG Xiaokuan，ZHAO Weichen，ZHOU Jianxiong，ZONG Binfeng，and XU Jiahua

1. Air and Missile Defense College，Air Force Engineering University，Xi’an 710051，China;2. College of Economics，Jilin University of Finance and Economics，Changchun 130022，China;3. School of Electronics Science and Engineering，National University of Defense Technology，Changsha 410073，China

Abstract: The noise robustness and parameter estimation performance of the classical three-dimensional estimating signal parameter via rotational invariance techniques (3D-ESPRIT) algorithm are poor when the parameters of the geometric theory of the diffraction (GTD) model are estimated at low signal-to-noise ratio (SNR). To solve this problem，a modified 3D-ESPRIT algorithm is proposed. The modified algorithm improves the parameter estimation accuracy by proposing a novel spatial smoothing technique. Firstly，we make cross-correlation of the auto-correlation matrices; then by averaging the cross-correlation matrices of the forward and backward spatial smoothing，we can obtain a novel equivalent spatial smoothing matrix. The formula of the modified algorithm is derived and the performance of this improved method is also analyzed. Then we compare root-meansquare-errors (RMSEs) of different parameters and the locating accuracy obtained by different algorithms. Furthermore，radar cross section (RCS) of radar targets is extrapolated. Simulation results verify the effectiveness and superiority of the modified 3DESPRIT algorithm.

Keywords: parameter estimation，novel spatial smoothing，scattering center，geometric theory of diffraction (GTD) model，radar cross section (RCS) extrapolation.

1. Introduction

The geometrical theory of diffraction (GTD) model [1]is a classical scattering center model to describe the electromagnetic characteristics of radar targets at high frequencies. In the past decades，the GTD model has wide applications in many military fields，such as targets recognition[2-7]，radar cross section (RCS) extrapolation [8-10]，and three-dimensional (3D) reconstruction [11-14]. Hence，building a high-precise GTD model is vitally important for radar targets electromagnetic characteristics analysis.Obviously，accurate estimated parameters have become the key to constructing a high-precision GTD model.Therefore，researchers have applied many algorithms to extract the GTD model parameters from back-scattering data of radar targets，such as the estimating signal parameter via rotational invariance techniques (ESPRIT) algorithm [15-18]，the multiple signal classification(MUSIC) algorithm [19-22]，the matrix enhancement and matrix pencil (MEMP) algorithm [23-25]and so on.While the parameter estimation performance of the ESPRIT algorithm is poor at low signal-to-noise ratio (SNR).Though the MUSIC algorithm performs better than the ESPRIT algorithm at low SNR conditions，it needs spectral peak searching，which burdens a heavy computation.The MEMP algorithm mismatches the closed parameters and hence it needs a parameter matching process.

In this paper，to estimate the GTD model parameters in a higher accuracy at low SNR as well as to avoid mismatched parameters，a modified 3D-ESPRIT algorithm is presented. Firstly，we add a novel correlation matrix into the traditional forward-backward spatial smoothing and obtain a new total covaraince matrix. Then by squaring the total covariance matrix，we can get the final covariance matrix. Finally，based on the final covariance and spatial spectrum estimation algorithm，we can extract the GTD model parameters. The proposed algorithm fully uses the back-scattering data of radar targets as well as broadens the differences between eigenvalues of signals and eigenvalues of noises simultaneously. Simulation results verify the effectiveness and superiority of our proposed algorithm.

This paper is organized as follows: Section 2 introduces the 3D-GTD model. Section 3 presents the pro-posed modified 3D-ESPRIT algorithm to estimate the GTD model parameters. Section 4 provides the simulations and the computational analysis. Finally，Section 5 concludes this paper.

2. The 3D-GTD scattering center model

As a classical scattering center model，the GTD model can describe the electromagnetic characteristics of radar targets. At high frequencies，the GTD model of targets [12]can be expressed as

whereE(fm，θn，φk) denotes the back-scattering data of radar targets，Idenotes the total scattering centers，Idenotes the total scattering centers，，xi，yi，zidenote the scattering intensity，scattering type，transversal position parameter，longitudinal position parameter，and vertical position parameter of theith scattering center respectively. c =3×108m/s represents the propagation speed of electromagnetic waves and ω (fm，θn，φk) denotes the Gaussian white noise.

wheref0，Δf，mrepresent the initial frequency，the frequency step and the frequency index respectively. Similarly，

where θ0and φ0are the initial azimuth angle and the initial pitching angle respectively，nΔθ andkΔφ are the relative small radar rotation angles. The scattering type parameters αiof typical scattering structures are shown in Table 1 [12].

Table 1 αi values of typical scattering structures

As the operating frequency of selected radar satisfies Δf/f0?1，we can take the following approximation:

Substituting (2) into (1) and transforming the obtained formula to Cartesian coordinates，then using the resampling technique we can get the electromagnetic data of radar targets:

Fig.1 represents the 3D frequency domain data range.The cube in Fig. 1 contains the interpolated data points at equal intervals.fcrepresents the intermediate frequence betweenf0tofm.

Fig. 1 3D frequency domain data range

3. The proposed algorithm

4. Cramer-Rao bound (CRB) analysis

where σ2represents the variance of the white Gaussian noise.

5. Simulation experiments

To assess the parameter estimation performance of the modified 3D-ESPRIT algorithm，we first compare the root-mean-square-error (RMSE) among the classical 3DESPRIT algorithm，the algorithm in [15]and the modified algorithm proposed in this paper. Afterwards，we compare the positioning accuracy of the three algorithms.Finally，we apply the parameters estimated by the three algorithms to the 3D-GTD model，which can extrapolate the RCS in the frequency domain versus SNR. All simulations are performed by Matlab 2017A.

Suppose the radar target consists of four scattering centers and the values of parameters are shown in Table 2.We set the initial frequencyf0as 10 GHz. The frequency step Δfis 16 MHz，the frequency indexMis 11，the initial azimuth angle θ0is 90°，the angular step Δ θ is 0.01°，the frequency indexNis 11，the initial pitching angleφ0is 90°，the angular step Δ φ is 0.01°，the frequency indexKis 11，and the paring parameter β is 0.5.

Table 2 Parameters of the four scattering centers

We add two dimensional Gaussian white noise to the back-scattering data. The SNR is defined as follows:

Define the RMSE of the GTD model parameter estima-tion fromDMonte Carlo trials as

where ?i，? andDrepresent the estimated parameters of theith run of the simulation，the true value，and the total trials of Monte Carlo at each SNR respectively.

5.1 Simulation settings

Example 1RMSE versus SNR

In the first simulation，we investigate the performance of the modified algorithm with respect to the SNR. Based on the four scattering centers shown in Table 2，the matrix beam parametersP，Q，andLare set as 6，SNR varies from 0 dB to 30 dB with interval 2 dB，and 200 Monte Carlo trials are performed at every fixed SNR.Here we only compare the mean RMSE of the four scattering center parameters. The results are shown in Fig. 2-Fig. 6.

Fig. 2 Mean RMSE of x versus SNR

Fig. 3 Mean RMSE of y versus SNR

Fig. 4 Mean RMSE of z versus SNR

Fig. 5 Mean RMSE of α versus SNR

Fig. 6 Mean RMSE of A versus SNR

Example 2Positioning accuracy analysis

In this example，we verify the positioning accuracy of the proposed algorithm. The simulation conditions are the same as that of Example 1 except that we set SNR as 0 dB and 10 dB. The positioning accuracy of different algorithms at two different values of SNR is shown in Fig.7 and Fig.8.

Fig. 7 Positioning accuracy between different algorithms at SNR=0 dB

Fig. 8 Positioning accuracy between different algorithms at SNR=10 dB

Example 3RCS extrapolation accuracy analysis

In this example，we verify the performance of the proposed algorithm by the comparisons of the RCS extrapolation accuracy. The simulation conditions are the same with those of Example 2. Based on the estimated parameters and the relations between electric field and RCS at far fields shown in (62)，the RCS of radar targets can be extrapolated in the angular domain. Here we use the backscattering data of 10 GHz to 10.16 GHz at θ =φ=90°to extrapolate the back-scattering data of 10 GHz to 11.6 GHz at θ =φ=90°. The RCS extrapolation accuracy of different algorithms at two different values of SNR is shown in Fig.9 and Fig.10.

whereEsandEirepresent the scattering electric field and the incident electric field respectively，Rrepresents the far field distance.

Fig. 9 Comparison between RCS frequency fitting and extrapolation at 90° azimuth and pitching angle (SNR=0 dB)

Fig. 10 Comparisons between RCS frequency fitting and extrapolation at 90° azimuth and pitching angle (SNR=10 dB)

Example 4Computational complexity analysis

The main increase of computational complexity is constructing the novel covariance matrix. Hence we merely compare the computational burden of constructing the covariance matrix，which is shown in Table 3.

Table 3 Comparison of computational burden among different algorithms

5.2 Discussion and analysis

As shown in Fig.2-Fig.6，the RMSE of the GTD model parameters decreases as SNR increases，which verifies the validity of the proposed algorithm. Furthermore，the mean RMSE curve of the proposed algorithm is lower than that of the classical 3D-ESPRIT algorithm，the classical 3D-MUSIC algorithm and the MEMP algorithm，which indicates the proposed algorithm has a better parameter estimation performance and noise robustness than the other three algorithms for different SNRs.

Additionally，F(xiàn)ig.7 and Fig.8 show that the positioning accuracy of the proposed algorithm is higher than that of the other two algorithms. Similarly，we can observe from Fig.9 and Fig.10 that the reconstructed RCS of the proposed method has a better fitting degree with the theoretical RCS than that of the other two methods. The two simulation results verify the superiority and effectiveness of the proposed algorithm from two different aspects.From Table 3 it is noticed that the proposed algorithm burdens a heavier computational complexity than the other two algorithms.

6. Conclusions

A modified 3D-ESPRIT algorithm with a better parameter estimation performance and more stable noise robustness ability is developed in this paper. The modified algorithm proposes a novel spatial smoothing method and squares the total covariance matrix，which can broaden the differences between eigenvalues of signals and eigenvalues of noises. Simulation results verify the superiority and effectiveness of the proposed algorithm. Furthermore，simulation results indicate that the parameter estimation performance and noise robustness of the modified algorithm are better than that of the classical 3D-ESPRIT algorithm，the classical 3D-MUSIC algorithm and the MEMP algorithm.