RCS Diagnostic Imaging Using Parameter Extraction Technique of State Space Method

The signature extraction and processing are the key aspects of enhancing recognition ability for radar target characteristics. In this paper, to reveal the intrinsic property of the target and improve the accuracy of radar cross section (RCS), we proposed RCS diagnostic imaging technique, extracting the parameters of scattering center model by state space method (SSM), to analyze the signature of GTD‐based returns. To demonstrate the effectiveness of SSM, the comparison with estimating signal parameter via rotational invariance techniques method by the analysis of goodness‐of‐fit and root mean square error, reconstructing the RCS profile through original signal model, and employing SSM to extract the signature of specular and creeping wave from the PEC sphere. Furthermore, the RCS diagnostic imaging by SSM extracting position and amplitude is discussed, and range‐isolated pole technique specific to the creeping wave extraction is addressed the advantage of SSM.

2 of 13 classification and recognition, but this method was mainly focus on the image segmentation rather than RCS analysis. In Ding et al. (2017), the ASC model derived from the term (jk) α of GTD and the sinc function of physical optics (PO) was contained both surface and edge contributions, but the author didn't demonstrate the RCS value that ASC model extracted was lower accuracy than the proposed method based on PO and PTD.
To address this problem, several researchers have extracted EM parameters from scattering filed models (Naishadham & Piou, 2008), such as geometrical theory of diffraction (GTD) model (Cui et al., 1997;Potter et al., 1995) and attributed scattering center (ASC) model (Ding et al., 2017;Potter & Moses, 1997) as well as inclusion of the intrinsic properties of scatters. Moreover, to improve the RCS accuracy effectively, consider the estimation of a robust, coherent-processing to the frequency domain data, we studied the RCS imaging based on the parameter extraction of GTD model and proposed state space method (SSM) method to estimate the position parameters and strength.
In this paper, Section 2 briefly presents the GTD scattering center model. Section 3 presents the SSM method to estimate the model parameters. Section 4 is the simulation of parameter extraction and the RCS comparison, whose recovered by estimating signal parameter via rotational invariance techniques (ESPRIT) and SSM. Finally, we concluded in Section 5.

Scattering Center Model
Scattering center model is commonly used in the area of automatic target recognition (ATR), radar image interpretation, RCS extrapolation, and geometry reconstruction (Zheng et al., 2020). GTD-based model (Potter & Moses, 1997) is more closely to the EM scattering than CE model. It can be seen as the coherent superposition  3 of 13 reconstruction of total EM scattering field. The incident field of plane wave propagating at location can be described by where is the propagating direction. According to the EM diffraction theory, the backscattered filed of a set discrete scattering center under the condition, the wavelength of incident excitation is smaller than object extent, can be approximated as where = 2 is the wavenumber, is the location of ith scattering center. ϕ i,k is a frequency dependent factor. Here, we assume far-field backscatter and scattering center is at the origin, to obtain the normalized field (Potter & Moses, 1997) by approximating | | −̂| | ≈ | | ,the polarization is normalized as The GTD-based model predicts that EM scattering follows a ( ) frequency dependence, where is one type of scattering center. Therefore, ϕ i,k can be represented as ( ) to the amplitude of GTD, and let =̂⋅ ⃗ denotes the zero-phase reference with respect to the range of scattering center. k c is the reference wavenumber of normalization. Hence, we can parametrize the backscattering model as follow To fully represent the return signal, the additive complex white Gaussian noise is indispensable, then where ( ) represents the scattering echo of target. I represents the number of scattering centers. { , } denotes scattering intensity, scattering type factor determined by scattering mechanism, respectively.
( ) denotes the radar frequency, (0) denotes the initial frequency, m is the number of sampling frequency, and Δf is the frequency interval. c denotes the wave velocity.
( ) denotes the complex Gaussian white noise.

Model Transformation
Before estimating the parameters of the position and amplitude, scattering center model need to be transformed. Then, the model of Equation 5 can be simplified as where B i and P i corresponds to the following equations The derived equation P i contains the down range and aspect dimension information between radar and target, the estimated parameter can be obtained as

State Space Approach
According to the theory of radar target characteristic, we suppose the attitude angle of target is stable and the target can be regarded as the shock response frequency of a discrete linear time-invariant (LTI) system, and incident wave can be seen as rational approximation of LTI system which was derived by auto regressive and moving average (ARMA) model (Naishadham & Piou, 2005). Hence, the SSM equations can be characterized as where ( ) ∈ ×1 is the state variable; A ∈ C M×M is the open-loop matrix, B ∈ C M×1 and C ∈ C 1×M are constant matrices (Potter & Moses, 1997). We assume that the initial condition (0) = 0 , the output unit impulse response equation can be obtained as where the eigenvalue of A represents the pole of ARMA model. According to Equation 6, we assume ( ) = ( ) , Hankel matrix can be constructed as Here, H matrix is composed of signal and noise subspace, the singular value decomposition (SVD) of H can be expressed as where subscript sn, n denotes signal and noise component, superscript H represents conjugate transpose. U sn , U n represents left unitary matrix, V sn , V n represents right unitary matrix. sn ,  represents diagonal matrix. Thus, H matrix truncated by noise and its eigenvector can be expressed as where Ω = sn 1∕2 sn , state matrix A can be expressed as where Ω − 1 , Ω − 1 denote the matrix Ω removed the first row and last row respectively. Matrix C can be expressed as =Ω(1, ∶) , and matrix B can be derived from least square method by equation Ω = , Therefore, matrix In the processing we employed SSM, the matrices B and C are not the key information since the poles of ARMA transfer function H(z) (Naishadham & Piou, 2005;Piou, 2005) obtained by taking z-transform of Equation 11 are determined from matrices A. For brevity, we deduce H(z) by the ratio of output and input from Equation 11 ( ) = ( − ) −1 + 1 Formula 18 shows the zero-pole of H(z) is refer to the open-loop matrix A, and the modal parameter A can be computed using eigenvalue decomposition method, one has Next, the phase θ i and position r i of the GTD-based poles can be deduced by where ϕ i is the phase of the eigenvalue λ i , and Equation 9 can be written into Equation 21. Then we conclude that the parameter P i in Equation 9 can be obtained by the eigenvalue from eigenvalue decomposition of matrix A, the position parameter r i and scattering type factor α i are both solved by substituting P i into Equations 9 and 10. Equations 20 and 21 denote that range-isolated pole technique (RIPT) can be used to solve parameter estimation, which is a significant advantage of SSM.

Simulation and Discussion
In this section, we discuss RCS diagnostic imaging relative to the GTD-based parameter extraction as well as the analysis of PEC sphere scattering mechanism. First, we consider the parameter estimation of GTD-based return from target using SSM in the absence of noise. Extract the information of range location and amplitude, and analyze its accuracy of the estimates by comparison with ESPRIT algorithm. Next, we consider the Mie series example pertinent to peak-value estimation of specular and creeping wave from PEC target using the SSM. According to the interpretation of radar image (Skinner et al., 1998), to study how RCS image of PEC sphere is formed and good approximation of elementary target by isotropic and nondispersive, we obtain scattering brightness form by one-dimensional downrange image since its RCS is approximately nondispersive.

Parameter Extraction for RMSE Analysis
We suppose the initial frequency f 0 = 10 GHz, the initial bandwidth B = 1.6 GHz, the number of sampling N = 512, the interval frequency radar inter-pulse is Δf = B/N, respectively. The radial range resolution unit of returns = 2 , define the signal-noise ratio (SNR) 10-30 dB, denote c for the wave velocity. To validated the extraction ability of SSM, the initial random test data of location ( ) and strength A i listed in Table 1 need to be calculated under the higher radial range resolution unit, that is to change the bandwidth from 1.6 to 2 GHz.
The simulation of root mean square error (RMSE) results, which is used to compare with the parameters under different frequency, performed by 100 times of Monte Carlo test can be defined as where z i , z, K represent the estimated parameter of extraction scattering center, the true value, the number times of Monte Carlo test, respectively. The results are showed in Figures 1 and 2. By means of the data profiles, we can summarize as 1. Noise has a great influence on the extraction of scattering position parameter, and little influence on intensity parameter. In Figure 1, the greater of SNR, the higher accuracy of extracted position r i is. On the other hand, the strength A i showed in Figure 2 is different with the position.

RCS Reconstruction and Comparison
The parameters of SCMs, extracted by SSM method proposed in this paper, is used to reconstruct RCS. The traditional definition of RCS under far field condition is where E s denotes scattering electric field, E i denotes incident electric field. To present the amplitude and phase of target scattering information as well as the changes property of frequency, the definition of RCS can be modified as where represents the fixed phase of local scattering structure of target, which can be interpreted as SCMs (Skinner et al., 1998). Therefore, the scattering distribution function of point target can be described as RCS with different SCMs, the electrical field signal of echo can be expressed by where E r denotes the electrical field signal of echo wave. Γ(R, f) denotes scattering distribution function of point target. In addition, the RCS value, showed in  Figure 3 has interference from unknown scatters, but with the increase of SNR, the original scatters and interference scatters are almost indistinct. In Figure 4, due to the higher bandwidth than Figure 3, RCS calculation of the five scattering points is still easy to distinguish regardless of the change of SNR, and no redundant scattering point interference appears. Therefore, high-bandwidth reconstruction is superior to low-bandwidth reconstruction. 3. To distinguish the RCS accuracy of ESPRIT and SSM in recovering original signals, it can be seen that the RMSE of SSM is lower than that of ESPRIT by computing the RMSE of different scattering types showed in Figures 6 and 7. Thus, this is the reason we choose SSM method to extract parameters in this paper. 4. The ability of ESPRIT and SSM to recover original signal RCS under different SNRs is compared in Figure 5.
It can be seen that both reconstruction methods can ensure a certain degree of accuracy, and their accuracy can be expressed by the goodness-of-fit. In Figure 8, compare with the goodness of fit of the signal with and without noise under different SNR, we know, the accuracy of fitting is lower than that at high SNR at low SNR, and with the increase of SNR, the fitting effect becomes the better. As a rule of thumb, as long as the goodness-of-fit greater than 0.5 is a good result.
As for the previous comparison of these statistical data, set the location with respect to the target center and intensity of original signal as random data showed in Table S1. By comparing and analyzing intuitively, we can conclude that SSM method performs more advantages than ESPRIT in these indexes.

Example: Specular and Creeping Wave Extracted From a PEC Sphere by SSM
In this section, we consider the case of specular and creeping wave extraction to Mie solution of PEC sphere, one in the absence of noise, another is adding Gaussian noise. Meanwhile, Monte Carlo test is used to examine robustness of the SSM algorithm (Naishadham & Piou, 2005, 2008. Here, suppose the radius of PEC sphere is a = 0.12 m, the monostatic RCS showed in Figure 9 is calculated by Mie series. To obtain range profile (see Figure 10) of back-scattering field, hamming window is used to suppress the sidelobes while conducting complex electric field gating FFT of frequency samples. In Figure 10a, the main brightness line corresponds to the first peak termed as specular reflection, its reference phase is shifted to zero range, the second peak located at 0.316 m corresponds to the darkness line is creeping wave. The two red asterisks showed in Figure 10b denote the two poles, specular reflection and creeping wave, need to be extracted by using the aforementioned SSM method. On the other hand, RIPT, the specular peak extraction located at zero range and the creeping wave peak approximately located at (1 + π/2)a ≈ 0.3085 (a is the radius)range, is described in this figure.
To effectively extract the specular reflection wave and the creeping wave of the PEC sphere, two poles in the state space equation are extracted from the MIE series with Gaussian noise. We set the 20 orders frequency bandwidth for the irradiated electromagnetic wave and take the corresponding center frequency. Meanwhile, in order to evaluate the robustness of SSM to extract feature signals, 100 Monte Carlo experiments were carried   Figure 11 shows the extraction of the position poles of the metal sphere reflection and the creeping wave. It can be observed from the figure that the position extraction with different noises is basically stable, and the position of the creeping wave is located at 1.39 m outside the radius of the metal sphere, in the low frequency range of 4 GHz. As the center frequency increases, the beginning of the specular reflection wave changes slightly (see Figure 11a), but in general does not affect the extraction of the signal. The position of the creeping wave extraction, showed in (see Figure 11b), is very large fluctuations in the high frequency band, which means inaccurate information characteristics as well as the creeping wave can not be observed in the high frequency band. Similarly, Figure 12 is the extraction of the amplitude poles of the specular reflection and creeping wave of the metal sphere. As the frequency increases, it can be seen that the specular reflection wave is continuously enhanced (see Figure 12a), but the creeping wave is continuously weakened (see Figure 12b).
Since the PEC sphere we research is the simplest model, the signature of specular scattering and creeping wave are existed in the large radar detection target, some parts of their problem can still be solved by the proposed method in practical engineering. Therefore, we need to consider the actual back-scattering model, such as point spread function (PSF) model, to extract the target signatures. As we know, the RCS value is the most important characteristic of radar target, to verify the SSM effectiveness and RCS diagnostic imaging calculation, the complex image PSF is discussed in the next section.

RCS Diagnostic Imaging
As aforementioned, we know that both the parameter of position and magnitude can be extracted by SSM from EM scattering model, and then substitute the extracted parameters to the reconstructed model to calculate the 12 of 13 RCS. The RCS of PEC sphere is computed by the well-known Mie series, and the image is depending on a set of scattering data. In order to highlight the advantage of SSM using RIPT to extract creeping waves as well as the RCS computation, this section will focus on demonstrating the difference between RCS and scattering imaging, we suppose PSF is the backscattering model, the extracted parameter is substituted, of complex radar target, whose scatterers with specular reflection and creeping wave. For example, G r , used to present the image function, can be defined as where σ terms as Formula 23, w (k m ) is the Kaiser-Bessel window, it can be expressed as where C is a constant, I 0 is a modified Bessel function, β is set to the range 0-4, k m is the corresponds to the wavenumber. S (k m ), the calibrated phase of scattering target, is the same as the exponential term of Formulas 24 and 25. Whereas, to completely depict the characteristic of image, the scattering brightness (Skinner et al., 1998) discriminate to the RCS is more convenient to interpret the image. Thereby, the scattering brightness can be described by In terms of Formula 29, describes the scattering center component of the approximate target element, which makes the computation of the RCS diagnostic imaging of the PEC sphere fully demonstrated. The imaging is shown in Figure 13 and the scattering brightness of each center frequency is illustrated in this figure. The zero position in the distance direction corresponds to the specular reflection wave, and the 1.39 m position of RIPT corresponds to the creeping wave. As we can see, the specular reflection wave does not disappear but the creeping wave gradually disappear as the frequency band center change from low to high. Comparing with Figure 12, the amplitude of the PEC characteristic signal obtained by the exact solution of the MIE series (see Table 2) did not change significantly under different SNRs, which also verified the correctness of the SSM extraction mode. Additionally, according to the data in Table 2, it is easy to obtain the profiles showed in Figure 14, which further demonstrate the intensity of creeping wave decreases with the increase of frequency center. From the available literature (Naishadham & Piou, 2008), the author obtained the both amplitude of specular signal and creeping wave decreased with the increasing frequency by the 1,000 Monte Carlo trials. However, since we have accumulated the 20-order signal and only extracted two poles, the intensity of the specular reflection signal obtained by our simulation results through 100 Monte Carlo extractions in Figure 12 increases with the increase of the frequency center, while it is shown in Figure 13 that its intensity is basically unchanged, which is our improvement of the literature. On the other hand, the simulation results of creeping wave are consistent with the literature.

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We consider changing the window function, does not affect both the position of specular and creeping wave, only the amplitude. From Figure 13, show that the amplitude used to form image is independent of the bandwidth, but the factor in window function, such as C and β, do affect the amplitude of creeping wave. As conclusion, the RCS value of creeping wave is not possible to assign from downrange profile, but the important point is to interpret the image from RCS data, even specular to the PEC sphere.

Conclusion
In this paper, we proposed a methodology, using state-space ARMA representation technique, to extract the scattering parameter from different GTD-based scattering types. Comparison with the ESPRIT method, we analyze the RMSE of the position and amplitude parameters extracted by the two methods, and verify the ability of SSM to recover the original signal under different SNRs and different bandwidths. The RMSE of the estimated parameters of the two methods under different scattering types is also compared. In terms of the calculation results of the goodness of fit, the accuracy of parameter extracted by SSM is better than ESPRIT. In order to prove the application ability of SSM in practical environment, two poles of 20-order MIE series modal tested with 100 Monte Carlo experiments are extracted by taking PEC sphere as an example. The simulation results show that SSM effectively extracts the position and amplitude parameters of specular reflection wave and creeping wave with RIPT. Finally, according to the relevant literature, we calculate the RCS diagnostic imaging of the PEC sphere and analyze the concept of diagnostic imaging, employing Scattering brightness as RCS imaging. In conclusion, the RCS imaging can be formed by coherent scattering data, but cannot be interpret as image level.

Data Availability Statement
This paper is a theoretical simulation presentation, the data results can be accessed through the website https:// doi.org/10.5281/zenodo.7524472.