The problem of optimally selecting polarization states of the transmitted waveform is extensively studied as it can enhance the performance in target detection, tracking and identification^[1-3]. Scattering properties of targets and clutter are polarization- sensitive; hence, the benign applications of polarization can enhance the polarimetric power contrast. They are known as the optimization polarimetric contrast enhancement (OPCE) problem ^[4-6].

In partially polarized condition, Sinclair matrix cannot provide the whole polarization information. To cope with the OPCE problem in this condition, Kennaugh matrix which provides the whole polarization information is applied. Usually, there are not analytic solutions to the OPCE problem, and numerical methods are applied. Among those methods, the global search method (GSM), which searches the overall two-dimensional polarization space, is the most common used one^[7-8]. This kind of method is time- consuming; especially when fast real-time signal processing is required such as in an accurate tracking of high maneuvering target scenario. Aiming to expedite the process of enhancing the desired targets versus the clutter and noise, various fast methods are proposed. Ref.[9-10] proposed iterative numerical methods for the completely polarized condition 0 and the partially polarized condition 0. Those methods are faster than the GSM. However, the method in 0 can only be used in the condition in which the relationship between the transmitted and received polarization states is not constrained. Ref.[11-12] proposed methods based on polarization ellipse parameters. Those methods require for the information of the angle consisting of the target and the clutter on the Poincare sphere frame as well as the sphere center.

In the paper, assuming the Kennaugh matrices is measured, we emphasize on how to fast solve the OPCE problem in partially polarized condition with constrained transmitted and received polarization. We first introduce a general signal to clutter plus noise ratio (SCNR) model for the partially polarized condition. Then an OPCE problem is deduced and a fast method for the OPCE problem is proposed. The method always converges to the optimal result. It expedites the computation of the OPCE problem by converting the problem into an equivalent maximization linear function.

1 Signal to Clutter Plus Noise Ratio Model

In this section, a model of signal to clutter plus noise ratio (SCNR) defined by the Stokes vector and Kennaugh matrix is created. Stokes vector is defined as

$\mathit{\boldsymbol{J}} = {\left[ {\begin{array}{*{20}{c}} {{g_0}}&{{g_1}}&{{g_2}}&{{g_3}} \end{array}} \right]^{\rm{T}}}$

(1)

where,

$ \left\{ \begin{gathered} {g_0} = \left\langle {{{\left| {{E_H}} \right|}^2}} \right\rangle + \left\langle {{{\left| {{E_V}} \right|}^2}} \right\rangle ,{g_1} = \left\langle {{{\left| {{E_H}} \right|}^2}} \right\rangle - \left\langle {{{\left| {{E_V}} \right|}^2}} \right\rangle \hfill \\ {g_2} = 2\left\langle {\left| {{E_H}} \right| \cdot \left| {E_V^*} \right|} \right\rangle \cos (\phi ),{g_3} = 2\left\langle {\left| {{E_H}} \right| \cdot \left| {E_V^*} \right|} \right\rangle \sin (\phi ) \hfill \\ \end{gathered} \right. $

(2)

$\left| {{E_i}} \right|$($i = H,V$) denotes the amplitude of electric filed in horizontal and vertical directions. $\left\langle \cdot \right\rangle $ denotes the ensemble mean. $\phi $ is the phase difference. In the backward scattering alignment (BSA) convention, polarization received power is defined as 0,

$P={{{(kr)}^{-2}}}/{2}\;F\mathit{\boldsymbol{J}}_{r}^{\rm{T}}\mathit{\boldsymbol{KJ}}_{t}^{\rm{T}}$

(3)

where $k$ is the wave number, $r$ is the distance between the target and the transceiver, $\mathit{\boldsymbol{K}}$ is the Kennaugh matrix, and ${\mathit{\boldsymbol{J}}_r}$ and ${\mathit{\boldsymbol{J}}_t}$ are the received and transmitted Stokes vectors, respectively. The superscript "T" denotes the transpose matrix. The factor $F$ is expressed as

$F(\lambda ,\theta ,\varphi )={\lambda }/{(8\mathsf{ π} \eta )}\;{G(\theta ,\varphi )}/{|{{\mathit{\boldsymbol{E}}}^{r}}{{|}^{2}}}\;$

(4)

where λ is the received wavelength, θ and φ are the spherical coordinates of the antenna pointing direction, η is the free space impedance, G and E are the antenna gain and received electric field strength, respectively.

The received power includes the power of the targets, the clutter and the noise, expressed separately as ${P_T}$, ${P_C}$ and ${P_N}$. SCNR can be defined as

${\rm{SCNR}} = \frac{{r_C^2{F_C} + r_N^2{F_N}}}{{r_T^2{F_T}}} \cdot \frac{{\mathit{\boldsymbol{J}}_{rT}^{\rm{T}}{\mathit{\boldsymbol{K}}_T}{\mathit{\boldsymbol{J}}_{tT}}}}{{\mathit{\boldsymbol{J}}_{rC}^{\rm{T}}{\mathit{\boldsymbol{K}}_C}{\mathit{\boldsymbol{J}}_{tC}} + \mathit{\boldsymbol{J}}_{rN}^{\rm{T}}{\mathit{\boldsymbol{K}}_N}{\mathit{\boldsymbol{J}}_{tN}}}}$

(5)

where ${\mathit{\boldsymbol{J}}_{ri}}$($i = T,C,N$) and ${\mathit{\boldsymbol{J}}_{ti}}$($i = T,C,N$) denote the received and transmitted Stokes vectors of the targets, clutter, and noise, respectively. A general scattering polarization echo may include the completely polarized component and the unpolarized component; the Stokes vector in (1) can be rewritten as

$\mathit{\boldsymbol{J}} = \underbrace {\left[ {\begin{array}{*{20}{c}} {\sqrt {g_1^2 + g_2^2 + g_3^2} } \\ {{g_1}} \\ {{g_2}} \\ {{g_3}} \end{array}} \right]}_{{\rm{completely \ polarzed \ component}}} + \underbrace {\left[ {\begin{array}{*{20}{c}} {{g_0} - \sqrt {g_1^2 + g_2^2 + g_3^2} } \\ 0 \\ 0 \\ 0 \end{array}} \right]}_{{\rm{unpolarized \ component}}}$

(6)

The ratio of the completely polarized power to the total power is defined as the polarization ratio

$p={\sqrt{g_{1}^{2}+g_{2}^{2}+g_{3}^{2}}}/{{{g}_{0}}}\;$

(7)

where $0 \leqslant p \leqslant 1$. Hence, (6) can be normalized as

$ \mathit{\boldsymbol{J}} = {g_0}\left\{ {p\underbrace {\left[ {\begin{array}{*{20}{c}} 1 \\ {\cos (2\varepsilon )\cos (2\tau )} \\ {\cos (2\varepsilon )\sin (2\tau )} \\ {\sin (2\varepsilon )} \end{array}} \right]}_{{\rm{completely \ polarzed \ component}}} + \underbrace {\left[ {\begin{array}{*{20}{c}} {1 - p} \\ 0 \\ 0 \\ 0 \end{array}} \right]}_{{\rm{unpolarized \ component}}}} \right. $

(8)

where $\varepsilon $ and $\tau $ are the polarization ellipse angle and the polarization inclination angle, respectively.

Let ${\mathit{\boldsymbol{J}}_D}$ represent a unit polarization state,

${\mathit{\boldsymbol{J}}_D} = \left[ {\begin{array}{*{20}{c}} 1 \\ {\vec g} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 1 \\ {{d_1}} \\ {{d_2}} \\ {{d_3}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 1 \\ {\cos (2\varepsilon )\cos (2\tau )} \\ {\cos (2\varepsilon )\sin (2\tau )} \\ {\sin (2\varepsilon )} \end{array}} \right]$

(9)

hence, $d_1^2 + d_2^2 + d_3^2 = 1$.

In the detection period, the polarization direction of the antenna to the target, clutter, and noise are the same. Hence, SCNR defined in (5) can be rewritten as

$ {\rm{SCNR}}={\rm{SCR}}\cdot {(\mathit{\boldsymbol{J}}_{rD}^{\rm{T}}\cdot {{\mathit{\boldsymbol{K}}}_{Tp}}\cdot {{\mathit{\boldsymbol{J}}}_{tD}})}/{(\mathit{\boldsymbol{J}}_{rD}^{\rm{T}}\cdot {{\mathit{\boldsymbol{K}}}_{CNp}}\cdot {{\mathit{\boldsymbol{J}}}_{tD}})}\; $

(10)

where SCR denotes signal to clutter ratio, K_{CNP} is the sum of the clutter Kennaugh matrix and the noise Kennaugh matrix. ${\mathit{\boldsymbol{J}}_{rD}}$ and ${\mathit{\boldsymbol{J}}_{tD}}$ denote the received and the transmitted polarization states, respectively. The OPCE problem is to select the optimal polarization states to maximize the SCNR in (10).

2 Polarization Optimization Method

In this section, the OPCE problem based on the SCNR is created. Then a polarization states optimization method is proposed.

2.1 Problem Formulation

Considering the radar system receives the echoes of the targets embedded in clutter and noise background, the maximization of the SCNR is choosing the optimization criterion to design antenna. For simplicity, we assume a co-polar condition, i.e., the transmitted Stokes vector is the same to the received Stokes vector (other transmitted and received polarization relationship can be realized by matrix rotation). Hence,

${\mathit{\boldsymbol{J}}_{rD}} = {\mathit{\boldsymbol{J}}_{tD}} = {\mathit{\boldsymbol{J}}_D}$

(11)

The fundamental principle of optimal reception is to adaptively adjust the polarization states to maximize the SCNR. The OPCE problem is then converted to be the optimization problem,

$\begin{array}{l} {\rm{max \ \ \ \ SCNR}}({\mathit{\boldsymbol{J}}_\mathit{D}})\\ {\rm{s}}{\rm{.t}}{\rm{. }}\ \ \ \ \left\| {d_1^2 + d_2^2 + d_3^2} \right\| = 1 \end{array}$

(12)

Constituting (9) and (10) into (12), the optimization problem is transformed to be,

$ \begin{align} & {\rm{max}}\ \ \ \ \ \ {\rm{SCR}}\cdot {(\mathit{\boldsymbol{J}}_{D}^{\rm{T}}\cdot {{\mathit{\boldsymbol{K}}}_{Tp}}\cdot {{\mathit{\boldsymbol{J}}}_{D}})}/{(\mathit{\boldsymbol{J}}_{D}^{\rm{T}}\cdot {{\mathit{\boldsymbol{K}}}_{CNp}}\cdot {{\mathit{\boldsymbol{J}}}_{D}})}\; \\ & {\rm{s}}{\rm{.t}\rm{.}}\ \ \ \ \ \ \ \ \ d_{1}^{2}+d_{2}^{2}+d_{3}^{2}=1 \\ \end{align} $

(13)

2.2 A Fast Polarization Optimization Method

In this subsection, we recast (13) in a linear function with two variables which are polarization state ${\mathit{\boldsymbol{J}}_D}$ and supplementary parameter $\lambda $. Then we solve the problem numerically. Let us first define the following function,

$f({\mathit{\boldsymbol{J}}_D},\lambda ) = {\rm{SCR}} \cdot \mathit{\boldsymbol{J}}_D^{\rm{T}} \cdot {\mathit{\boldsymbol{K}}_{Tp}} \cdot {\mathit{\boldsymbol{J}}_D} - \lambda \cdot \mathit{\boldsymbol{J}}_D^{\rm{T}} \cdot {\mathit{\boldsymbol{K}}_{CNp}} \cdot {\mathit{\boldsymbol{J}}_D}$

(14)

Define the maximum value of function $f({\mathit{\boldsymbol{J}}_D},\lambda )$ with respect to ${\mathit{\boldsymbol{J}}_D}$ as

${f_{\max }}(\lambda ) = \mathop {\max }\limits_{{\mathit{\boldsymbol{\vec J}}_D}} f({\mathit{\boldsymbol{J}}_D},\lambda )$

(15)

Lemma 1 ${f_{\max }}(\lambda )$ in (15) is a monotone decreasing function with respect to $\lambda $.

Proof Given two variables ${\lambda _1}$, ${\lambda _2}$ and ${\lambda _1} > {\lambda _2}$, their corresponding maximum values are assumed to be ${f_{\max 1}}$ and ${f_{\max 2}}$. Their corresponding polarization states are ${\mathit{\boldsymbol{J}}_{D1}}$ and ${\mathit{\boldsymbol{J}}_{D2}}$, respectively. We obtain

$ \left\{ \begin{gathered} {f_{\max 1}} = {\rm{SCR}} \cdot {\mathit{\boldsymbol{J}}_{D1}}^{\rm{T}} \cdot {\mathit{\boldsymbol{K}}_{Tp}} \cdot {\mathit{\boldsymbol{J}}_{D1}} - {\lambda _1} \cdot {\mathit{\boldsymbol{J}}_{D1}}^{\rm{T}} \cdot {\mathit{\boldsymbol{K}}_{CNp}} \cdot {\mathit{\boldsymbol{J}}_{D1}} \hfill \\ {f_{\max 2}} = {\rm{SCR}} \cdot {\mathit{\boldsymbol{J}}_{D2}}^{\rm{T}} \cdot {\mathit{\boldsymbol{K}}_{Tp}} \cdot {\mathit{\boldsymbol{J}}_{D2}} - {\lambda _2} \cdot {\mathit{\boldsymbol{J}}_{D2}}^{\rm{T}} \cdot {\mathit{\boldsymbol{K}}_{CNp}} \cdot {\mathit{\boldsymbol{J}}_{D2}} \hfill \\ \end{gathered} \right. $

(16)

Since, ${P_C}$ and ${P_N}$ are the received power of the clutter and the noise, there are ${P_C} \geqslant 0$ and ${P_N} > 0$, which makes sure ${\mathit{\boldsymbol{J}}_D}^{\rm{T}} \cdot {\mathit{\boldsymbol{K}}_{CNp}} \cdot {\mathit{\boldsymbol{J}}_D} > 0$. Hence,

$\begin{gathered} {f_{\max 1}} - {f_{\max 2}} = f({\mathit{\boldsymbol{J}}_{D1}},{\lambda _1}) - f({\mathit{\boldsymbol{J}}_{D2}},{\lambda _2}) \leqslant \hfill \\ {\rm{ }}f({\mathit{\boldsymbol{J}}_{D1}},{\lambda _1}) - f({\mathit{\boldsymbol{J}}_{D1}},{\lambda _2}) = \hfill \\ {\rm{ }}({\lambda _2} - {\lambda _1}) \cdot {\mathit{\boldsymbol{J}}_{D1}}^{\rm{T}} \cdot {\mathit{\boldsymbol{K}}_{CNp}} \cdot {\mathit{\boldsymbol{J}}_{D1}} < 0 \hfill \\ \end{gathered} $

(17)

That is ${f_{\max 1}} < {f_{\max 2}}$, hence, the Lemma 1. Q.E.D

Assuming the polarization state corresponding to ${f_{\max }} = 0$ is ${\mathit{\boldsymbol{J}}_{D\_opt}}$; then, the maximum SCNR is

$\max {\rm{SCNR}} = {\rm{SCR}} \cdot \frac{{\mathit{\boldsymbol{J}}_{D\_opt}^{\rm{T}} \cdot {\mathit{\boldsymbol{K}}_{Tp}} \cdot {\mathit{\boldsymbol{J}}_{D\_opt}}}}{{\mathit{\boldsymbol{J}}_{D\_opt}^{\rm{T}} \cdot {\mathit{\boldsymbol{K}}_{CNp}} \cdot {\mathit{\boldsymbol{J}}_{D\_opt}}}}$

(18)

According to the forward analyses, the following Theorem 1 can be summarized.

Theorem 1 The maximum SCNR equals to the $\lambda $ when ${f_{\max }} = 0$ and the corresponding polarization state is ${\mathit{\boldsymbol{J}}_{D\_{\rm{opt}}}}$.

When $\lambda $ is ascertained, the problem of (14) is transformed to be the one in 0. The problem has the form,

$f({\mathit{\boldsymbol{J}}_D}) = \mathit{\boldsymbol{J}}_D^{\rm{T}} \cdot \mathit{\boldsymbol{M}} \cdot {\mathit{\boldsymbol{J}}_D} = {\left[ {\begin{array}{*{20}{c}} 1 \\ \mathit{\boldsymbol{g}} \end{array}} \right]^{\rm{T}}} \cdot \left[ {\begin{array}{*{20}{c}} {{k_{0M}}}&{\mathit{\boldsymbol{a}}_M^{\rm{T}}} \\ {{\mathit{\boldsymbol{b}}_M}}&{{\mathit{\boldsymbol{Q}}_M}} \end{array}} \right] \cdot \left[ {\begin{array}{*{20}{c}} 1 \\ \mathit{\boldsymbol{g}} \end{array}} \right]$

(19)

where $\mathit{\boldsymbol{M}} = {\rm{SCR}} \cdot {\mathit{\boldsymbol{K}}_{Tp}} - \lambda \cdot \mathit{\boldsymbol{J}}{\mathit{\boldsymbol{K}}_{CNp}}$. A faster numerical solver will be proposed for maximizing (19). Assuming a Lagrange multiplier is $v$, the cost function is,

$L = f({\mathit{\boldsymbol{J}}_D}) - v({\mathit{\boldsymbol{g}}^{\rm{T}}}\mathit{\boldsymbol{g}} - 1)$

(20)

Let the derivation of the cost function with respect to $\mathit{\boldsymbol{g}}$ equal to zero, we obtain

$\begin{gathered} \mathit{\boldsymbol{g}} = - {({\mathit{\boldsymbol{Q}}_M} + \mathit{\boldsymbol{Q}}_M^{\rm{T}} - 2{\mathit{\boldsymbol{I}}_3} \cdot v)^{ - 1}}({\mathit{\boldsymbol{a}}_M} + {\mathit{\boldsymbol{b}}_M}) \triangleq \\ - {(\mathit{\boldsymbol{M}} - {\mathit{\boldsymbol{I}}_{3}} \cdot v)^{ - 1}}\mathit{\boldsymbol{V}} \\ \end{gathered} $

(21)

According to matrix decomposition theory, there is a unitary matrix $\mathit{\boldsymbol{U}}$ and a diagonal matrix $\mathit{\boldsymbol{ \boldsymbol{\varLambda} }}$ such that $\mathit{\boldsymbol{M}} = {\mathit{\boldsymbol{U\varLambda U}}^{\rm{T}}}$. Let

${[{\beta _1},{\beta _2},{\beta _3}]^{\rm{T}}} = \mathit{\boldsymbol{UV}},\ \ \mathit{\boldsymbol{ \boldsymbol{\varLambda} }} = {\rm{diag}}\{ {\lambda _1},{\lambda _2},{\lambda _3}\} $

(22)

According to $\left\| \mathit{\boldsymbol{g}} \right\| = 1$, we obtain

$\sum\limits_{i = 1}^3 {{{[{\beta _i}/({\lambda _i} - v)]}^2}} = 1$

(23)

The domain of Lagrange multiplier ${v_{\max }}$ that maximizes (19) is 0,

${v_L} \leqslant {v_{\max }} \leqslant {v_U}$

(24)

where,

$\left\{ \begin{array}{l} {v_L} = \max \left\{ {\left\{ {\min \left\{ {{\lambda _i} + \sqrt 3 \left| {{\beta _i}} \right|} \right\},\max \left\{ {{\lambda _i} + \left| {{\beta _i}} \right|} \right\}} \right\}} \right\}\\ {v_U} = \max \left\{ {{\lambda _i} + \sqrt 3 \left| {{\beta _i}} \right|} \right\},\ \ \ {\rm{ }}i = 1,2,3 \end{array} \right.$

(25)

and,

$\left\{ \begin{gathered} {[\begin{array}{*{20}{c}} {{\beta _1}}&{{\beta _2}}&{{\beta _3}} \end{array}]^{\rm{T}}} = \frac{1}{4}({\mathit{\boldsymbol{Q}}_M} + \mathit{\boldsymbol{Q}}_M^{\rm{T}}) \cdot ({\mathit{\boldsymbol{a}}_M} + {\mathit{\boldsymbol{b}}_M}) \hfill \\ {\rm{diag}}\{ \begin{array}{*{20}{c}} {{\lambda _1}}&{{\lambda _2}}&{{\lambda _3}} \end{array}\} = {\rm{eigs}}\left( {\frac{1}{2}({\mathit{\boldsymbol{Q}}_M} + \mathit{\boldsymbol{Q}}_M^{\rm{T}})} \right) \hfill \\ \end{gathered} \right.$

(26)

"eigs" denotes the eigenvalues of matrix.

Substituting (21) into (19), there is:

$\max f(v) = \frac{1}{2}\left[ {{k_{0M}} - \sum\limits_{i = 1}^3 {\frac{{{\beta _i}^2}}{{{\lambda _i} - v}}} + \sum\limits_{i = 1}^3 {\frac{{{\beta _i}^2v}}{{{{({\lambda _i} - v)}^2}}}} } \right]$

(27)

Given two variables ${v_1}$ and ${v_2}$, ${v_1} > {v_2}$, according to Cauchy-Schwartz inequality theory, there is:

$ \begin{array}{l} \max f({v_1}) - \max f({v_2}) = \\ {\rm{ }}\frac{1}{2}\left\{ {({v_1} - {v_2}) + \sum\limits_{i = 1}^3 {\left[ {\frac{{{\beta _i}^2({v_1} - {v_2})}}{{({\lambda _i} - {v_2})({\lambda _i} - {v_1})}}} \right]} } \right\} > \\ {\rm{ }}\frac{1}{2}({v_1} - {v_2}) \cdot \left\{ {1 - \sum\limits_{i = 1}^3 {\left[ {\frac{{{\beta _i}^2}}{{2{{({\lambda _i} - {v_2})}^2}}} + \frac{{{\beta _i}^2}}{{2{{({\lambda _i} - {v_1})}^2}}}} \right]} } \right\} = 0 \end{array} $

(28)

Hence, the function (19) is monotone increasing with respect to the variable $v$. The conclusion can be established by the following theorem 2.

Theorem 2 The maximization of (19) can be calculated by the Lagrange method in (20). The function is a monotone increasing one with respect to the Lagrange multiplier $v$ in (20).

Since the two variables in $f({\mathit{\boldsymbol{J}}_D},\lambda )$, i.e., ${\mathit{\boldsymbol{J}}_D}$ and $\lambda $, are mutual independent, the problem of searching the optimal polarization state that maximizing the SCNR can be divided into the following two steps: 1) selecting $\lambda $ to construct the function (14); 2) selecting ${\mathit{\boldsymbol{J}}_D}$to obtain ${f_{\max }}$ of (15). According to Lemma 1, (15) is monotone decreasing with respect to $\lambda $. According to theorem 2, the (19) is monotone increasing with respect to Lagrange multiplier $v$. Hence, bisection method can be applied to both the two steps.

Given the maximum value of the numerator and the minimum value of the denominator of (10) are ${P_{T\max }}$ and ${P_{CN\min }}$, respectively, the upper bound of $\lambda $ is:

${\lambda _U} \le {P_{T\max }}/{P_{CN\min }}$

(29)

Given the minimum value of the numerator and the maximum value of the denominator of (10) are ${P_{T\min }}$ and ${P_{CN\max }}$, respectively, the low bound of $\lambda $ is:

${\lambda _L} \geqslant {P_{T\min }}{\rm{ /}}{P_{CN\max }}$

(30)

Hence the search intervals of $\lambda $ is:

${\lambda _L} \leqslant \lambda \leqslant {\lambda _U}$

(31)

The procedures for the proposed method are:

1) set $\lambda = ({\lambda _L} + {\lambda _U})/2$ (see (29) and (30));

2) solve ${f_{\max }}$ in (15) by bisection search in $[{v_L},{v_U}]$ (cf. (25)).

3) if ${f_{\max }} > 0$, $\lambda = ({\lambda _U} + \lambda )/2$;

else if ${f_{\max }} < 0$, $\lambda = ({\lambda _L} + \lambda )/2$;

else if ${f_{\max }} = 0$, the maximum SCNR is$\lambda $, end.

4) else go to 2).

3 Numerical Experiments

Experiments are accomplished by Matlab 2010 code running on a 32-bit computer with CPU AMD Athlon 3.0GHz, RAM 4G. Monte Carlo simulation time is 100. Let us consider the following Kennaugh matrix: the target Kennaugh matrix ${\mathit{\boldsymbol{K}}_T}$ in 0 and the clutter Kennaugh matrix ${\mathit{\boldsymbol{K}}_C}$ in 0.

${\mathit{\boldsymbol{K}}_T} = \left[ {\begin{array}{*{20}{c}} {13.062{\rm{ }}5}&{ - 5.812{\rm{ }}5}&{3\sqrt 2 }&{\sqrt 2 } \\ { - 5.812{\rm{ }}5}&{9.312{\rm{ }}5}&{ - \sqrt 2 }&{ - 3\sqrt 2 } \\ {3\sqrt 2 }&{ - \sqrt 2 }&{10.25}&0 \\ {\sqrt 2 }&{ - 3\sqrt 2 }&0&{ - 6.25} \end{array}} \right]$

(32)

${\mathit{\boldsymbol{K}}_C} = \left[ {\begin{array}{*{20}{c}} {2.100{\rm{ }}0}&{0.254{\rm{ }}5}&{0.379{\rm{ }}8}&{0.152{\rm{ }}8} \\ {0.052{\rm{ }}4}&{1.436{\rm{ }}4}&{0.866{\rm{ }}4}&{ - 0.023} \\ {0.179{\rm{ }}8}&{0.666{\rm{ }}4}&{ - 0.560{\rm{ }}4}&{0.219{\rm{ }}2} \\ { - 0.047{\rm{ }}2}&{ - 0.223}&{0.019{\rm{ }}2}&{0.823{\rm{ }}8} \end{array}} \right]$

(33)

Some assumptions are: 1) the distances of the target and the clutter to the antenna are the same, i.e., ${r_T} = {r_C}$. 2) the antenna gains are the same, i.e., ${G_T} = {G_C}$. Hence, ${\rm{SCR}} = {{{{\left| {{E_T}} \right|}^2}}/ {{{\left| {{E_C}} \right|}^2}}}$ and ${\rm{CNR}} = $ ${{{{\left| {{E_C}} \right|}^2}}/{{{\left| {{E_N}} \right|}^2}}}$. Equation (10) shows the SCR does not affect the selection of the optimal polarization state, we choose SCR = 10 dB. Three different clutter to noise ratios(CNRs), i.e., CNR = 10 dB, 0 dB and −10 dB are tested to validate the proposed method.

Considering the target is completely polarized, i.e., ${p_T} = 1$. To test the performance of proposed method in the partially polarized condition, three conditions are operated: 1) Low polarization ratio ${p_C} = 0.01$; 2) Middle polarization ratio ${p_C} = 0.5$; 3) High polarization ratio ${p_C} = 0.99$. The results obtained by the GSM in a small search-step, i.e., ${\rm{ \mathsf{ π} }}/720$, is considered to be the real ones.

Experiment Ⅰ: Low Polarization Ratio ${p_C} = 0.01$

The maximum SCNRs corresponding to CNR=10 dB, 0 dB and −10 dB are 22.333 9 dB, 19.824 3 dB and 12.509 3 respectively. Their corresponding polarization states are (−0.754 4, 0.633 0, 0.173 6), (−0.771 5, 0.608 2, 0.186 5) and (−0.771 5, 0.608 2, 0.186 5), respectively.

The average time consumed by the proposed method is about 5% of that consumed by the GSM with the similar calculation accuracy. The optimal polarization sates to different CNRs are similar to each other.

Experiment Ⅱ: Middle Polarization Ratio ${p_C} = 0.5$

The maximum SCNRs corresponding to CNR = 10 dB, 0 dB and -10 dB are 22.2646 dB, 19.7747 dB and 12.497 3, respectively. Their corresponding polarization states are (-0.674 8, 0.730 0, 0.108 9), (-0.709 3, 0.691 0, 0.139 2), and (-0.756 6, 0.629 2, 0.177 9) respectively.

The average time consumed by the proposed method is about 5% of that consumed by the GSM with the similar calculation accuracy. The optimal polarization states to different CNRs are different.

Experiment Ⅲ: High Polarization Ratio ${p_C} = 0.99$

The maximum SCNRs corresponding to CNR = 10 dB, 0 dB and −10 dB are 22.910 8 dB, 20.045 6 dB, and 12.504 3, respectively. Their corresponding polarization states are (−0.488 6, 0.872 4, −0.013 1), (−0.536 9, 0.842 7, 0.039 3), and (−0.693 2, 0.705 4, 0.147 8) respectively.

The average time consumed by the proposed method is about 3% of that consumed by the GSM with the same calculation accuracy. The optimal polarization states to different CNRs are greatly different.

The proposed method has been proved to be able to obtain the optimal polarization states for all the partially polarized conditions. Compared with the GSM, the proposed method is less time-consuming and more accurately.

4 Conclusions

In the paper, the OPCE problem with constrained transmitted and received polarization state relationship in partially polarized condition is discussed. A general SCNR model is first created to contain the partially polarized condition. A fast method for the OPCE problem is proposed based on the SCNR model. The method has converted the OPCE problem into the maximization problem of a linear function. Hence, the computational burden is greatly reduced. The numerical experiments have demonstrated the proposed method is better and has higher efficiency than the GSM. This method is easily extended to other polarization states conditions, such as the cross- polarize condition, by matrix rotation.

In the following work, we will research on the fast polarization optimization methods for the OPCE problem with unconstrained relationship between the transmitted polarization state and the received polarization state.