Due to a wideband pulse incident covering the fundamental resonance region of the radar target under examination, the SEM theory approximates the transient oscillatory response of the original signal, which begins after some late-time onset, T_L, as a sum of exponentially decaying sinusoids, expressed as y(t):

$y\left(t\right)=\underset{m=1}{\overset{M}{{\displaystyle \sum }}}\left|c_m\right|{\text{e}}^{-{\delta }_{m}t}\cdot \sin \left({\omega }_{m}t+\arg \left(c_m\right)\right)$ (1)

The parameter M defines the modal order of the signature (by default, set it equal to the number of cardinal target substructures); c_m defines the complex residues; δ_m defines the damping factor; ω_m defines the angular resonant frequencies. Notably, the residue and damping factors are essential parameters used to estimate the energy of the resonance. As a rule of thumb, the late-time onset commences after the incident wave transits the farthest tip of the target along the incident plane. The farthest tip appears as the farthest pulse peak in the return response and before the commencement of the oscillatory return (see Equation (2) in [12] ). Another way is to take the reciprocal of the first fundamental frequency (if well excited) and add the pulse width duration.

The matrix pencil of function (MPOF) algorithm may be used to extract and estimate the three resonance parameters in two steps: first, the solution of an ill-conditioned matrix of y(t) formed by delays of y(t) and then find the roots of a polynomial by the singular value decomposition (SVD) [21] .

The variance amplitude figure (VAF) metric quantifies the degree of uncertainty in the estimated resonance set by measuring the similarity of the approximated signal, y(t), to the original signal, x (t > T_L), after the selected onset, T_L [22] .

$\text{VAF}=100\ast \left(1-\frac{\mathrm{var}\left(x\right)-\mathrm{var}\left(y\right)}{\mathrm{var}\left(x\right)}\right)$ (2)

The “var” defines the variance.

Henceforth, the signature signal y(t) must fulfill two constraints to derive a reliable CPS set: have consistent M and sufficient VAF merit in all polarization channels.

For quad-polarization directions, the backscattered responses in orthogonal linear basis (h, v) of transmission and reception directions form a real-time dependent matrix S^{2 × 2} as follows:

$S=\left[\begin{array}{cc}{y}_{hh}\left(t\right)& y_{hv}\left(t\right)\\ y_{vh}\left(t\right)& y_{vv}\left(t\right)\end{array}\right]$ (3)

The subscripts hh and vv denote the co-polarized scattering directions or channels, while hv and vh denote the cross-polarized scattering channels (reciprocal for monostatic case). Since the resonance residue is the only polarization-dependent term, a reside polarization scattering matrix (RPSM), namely C^{2 × 2}, can thus describe the target polarization scattering at a single resonance by dropping the time dependence term, i.e. ${\text{e}}^{-\left({\delta }_m+j{\omega }_m\right)t}$, as follows:

$C=\left[\begin{array}{cc}{c}_{xx}& c_{yx}\\ c_{xy}& c_{yy}\end{array}\right]$ (4)

Therefore, constructing the RPSM requires that the associated residue of this resonance be extractable from the temporal response by the MPOF in all polarization channels.

On reception, the signal is split into co- and cross-polarized channels, with the power at the antenna terminals established separately for each channel. The complex residues, c_hh, and c_vv, denote the co-polarized scattering coefficients, with c_yx and c_xy denoting the cross-polarized scattering coefficients, and are equal for the reciprocal (monostatic) case. The constraint of having consistent modal order in the matrix means that the co- and cross-polarized terms must have the same modal order, i.e. $M_{hh}=M_{yx}=M_{xy}=M_{vv}$ for every extraction trial.

Next, the related power in the co-polarized channels is a function of the antenna polarization state, represented by Stokes vector g, as follows:

$P\left(g_1,g_2,g_3\right)=\frac{1}{2}g\cdot \text{diag}\left(1,1,1,-1\right)\cdot \left(R\cdot \left(\left[C\right]\text{U}{\left[C\right]}^{*}\right)\cdot R^{-1}\right)\cdot g^{\text{T}}$ (5)

where $R=\left(\begin{array}{cccc}1& 0& 0& 1\\ 1& 0& 0& -1\\ 0& 1& 1& 0\\ 0& j& -j& 0\end{array}\right)$, the Stokes vector $g=\left[g_0,g_1,g_2,g_3\right]$ defines the antenna’s polarization state in terms of g₀ as total, g₁ as linear h-v, g₂ as linear at ±45˚, and g₃ as circular power portions, respectively. The product ([C] $\otimes$ [C]^*) defines the Kronecker product of matrix C and its conjugate. Applying the Lagrangian multiplier method to the power function in (5) subject to the constraint function ( $\Phi \left(g_1,g_2,g_3\right)=\sqrt{g_1^2+g_2^2+g_3^2}-g_0$) to optimize the received power yield the following three simultaneous equations:

$\frac{\partial P}{\partial g_n}-\mu \frac{\partial \Phi }{\partial g_n}=0,n=1,2,3.$ (6)

where, $\mu$ is the Lagrangian multiplier, which optimizes the power quantity rate of change as a function of the constraint variables. Solving (6) produces two pairs of co-polarized CPS, the orthogonal co-pol max pair (cm_{1, 2})and the co-pol null pair (cn_{1, 2}), which engulf the copol max of minor power, forming a polarization fork around a great circle, as depicted in Figure 1 . The variableg₃ of the major co-pol max reflects the symmetry (0 for the symmetrical target, ±1 for the nonsymmetrical target). The co-pol null pair set is not orthogonal but defines a characteristic anglerelated to the depolarization degree, γ, as follows:

$\gamma =\frac{\text{dot}\left(g_{\text{cn}1},g_{\text{cn}2}\right)}{\Vert g_{\text{cn1}}\Vert \Vert g_{\text{cn2}}\Vert }$ (7)

Assuming that the targets have identical substructures and dimensions, as in the case of same-class aircraft targets, the problem is to discriminate or identify these same-class targets by their dihedral shape feature per resonance mode. Therefore, it is desirable to quantify the shape variant degree ( $\Delta \lambda$) of a substructure (X), e.g. wings, for i'th target, say q_i, compared to a reference target, q_ref, as follows:

$\Delta {\lambda }_X\left(q_i\right)=\frac{{\lambda }_X}{{\Omega }_{\lambda }}\frac{\left|{\theta }_i-{\theta }_{ref}\right|}{{\Omega }_{\theta }}$ (8)

where the angle $\theta$ denotes the substructure swept angle about the main body direction ranges from parallel, that is, zero, to perpendicular, i.e. max degree ${\Omega }_{\theta }$ (=90˚); the term ${\lambda }_X$ denotes the dimension of the substructure of interest; the term ${\Omega }_{\lambda }$ denotes the total sum of substructure dimensions of interest.

Then, for the number of training targets (Q) with prior probabilities of P(q_i), $i=\text{1},\cdots ,Q$, express the average shape variant ( $\Delta {\lambda }_X$) of a substructure X as follows:

$\Delta {\lambda }_X=\frac{1}{Q-1}{\displaystyle \underset{n=1}{\overset{Q}{\sum }}P\left(q_n\right)\cdot \Delta {\lambda }_X\left(q_n\right)\Delta }$ (9)

For multiple substructures $X_{\text{1}},\cdots ,X_{M^{\prime }}$ that are excited by M resonances, then the weighting of the depolarization degree, γ, per mth resonance by the averages ( $\Delta {\lambda }_{X1,\cdots ,M^{\prime }}$) forms the proposed feature set F in M-dimensional space as follows:

$F=\left[\Delta {\lambda }_{X_1}\cdot {\gamma }_1,\cdots ,\Delta {\lambda }_{X_M}\cdot {\gamma }_M\right]$ (10)

To make the feature set as compact as possible, and hence, fewer distance computations and more separable feature patterns of different targets, select the 1,…., M* resonances with ( $\Delta {\lambda }_X$) above a certain upper threshold or discard the ones below a certain lower threshold.

In the training stage, the algorithm begins by estimating the expected (mean) feature set, namely the training prototype, by generating, per resonance, a sample distribution of the feature set with noise perturbation per target. Then, in the testing stage, the algorithm assigns the test sample corrupted by noise from a target under test to the stored target database with the minimum distance aggregated across the M' resonances of interest. The following steps summarize the identification approach for trained targets (Q_tr) and test targets (Q_ts) as follows:

1) Training stage: For N realizations or trials, generate the noisy sample distribution of the set as $F_i=F_{0,i};\cdots ;F_{N-1,i}$, then the training prototype ( $\tilde{F}$) is the expected value, i.e. the first moment, expressed as follows:

$\tilde{F}=\frac{1}{N-1}{\displaystyle \underset{n=0}{\overset{N-1}{\sum }}{F}_n}$ (11)

2) Test stage: Taking one of the stored targets as the test target q_j, where its feature set samples are considered the test sample set (F_j), then across M' modes of interest, measure the aggregate distance (D) between the nth test sample (F_{n, j}) from target q_j and the training prototype ${\tilde{F}}_i$ of the i'th target q_i by a distance metric ||.|| as follows:

$D_{n,M^{\prime }}\left(i,j\right)=\frac{1}{M^{\prime }-1}\underset{m=1}{\overset{M^{\prime }}{{\displaystyle \sum }}}\Vert {\tilde{F}}_i\left(m\right)-F_{n,j}\left(m\right)\Vert$ (12)

3) Repeating the previous step for $i=\text{1},\cdots ,Q_{tr}$ cataloged targets will generate (N.Q_tr) distances, i.e. $D_{n,M^{\prime }}\left(i,j\right)$, $n=0,\cdots ,N-\text{1}$ and $i=\text{1},\cdots ,Q_{tr}$, for a single test target q_j.

4) Assign sample F_{n, j} to i'th target q_i if the minimum distance D_{n, M'}(i, j)_min > D_{n, M'}(≠i, j)_min + distance threshold (h), or else declare unable to assign and discard this result if D_{n, M'}(i, j)_min ≤ D_{n, M'}(≠i, j)_min + h. The user selects the threshold, h, as a fraction of the maximum distance $D_{\max }=\sqrt{r_1+\cdots +r_{M^{\prime }}}$ where r = max(γ), i.e. r = 1.

5) If the j'th and i'th targets of the minimum distance are the same, i.e. $D_{n,M^{\prime }}\left(i,j=i\right)=D_{\min }$ then declares able to assign to the correct target (correct decision), incrementing the number of correct target identification (A_j) for the j'th target, else declare unable to assign to the correct target (incorrect decision).

6) Repeat the previous step for each test target; then, the ROC is the ratio of the total number of correct target identifications (A) to the product (Q_ts. Q_tr. N), expressed as a percentage.

$\text{ROC}=\frac{A_1+\cdots +A_{Q_{tr}}}{Q_{ts}\times Q_{tr}\times N}\times 100$ (13)

7) Repeat previous steps with different SNRs to establish the curve of the ROC vs. SNR levels. For Q_ts targets with the same prior probability, the lower bound of the ROC curve, at least theoretically, will set approximately around 100/Q_ts, e.g. 50% for Q = 2, %33 for Q = 3.

Henceforth, the results will show how well the identification algorithm performs under modal order and VAF constraints, with shape variant weighting the feature set per resonance mode.