2. Department of Electrical and Computer Engineering, Duke University Durham NC 27708, USA
2. 杜克大学电气与计算工程系 美国北卡罗拉州 达拉姆 27708
2. Department of Electrical and Computer Engineering, Duke University Durham NC 27708, USA
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^{[13]}. 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 ^{[46]}.
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 twodimensional polarization space, is the most common used one^{[78]}. This kind of method is time consuming; especially when fast realtime 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.[910] 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.[1112] 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 ModelIn 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) 
$P={{{(kr)}^{2}}}/{2}\;F\mathit{\boldsymbol{J}}_{r}^{\rm{T}}\mathit{\boldsymbol{KJ}}_{t}^{\rm{T}}$  (3) 
where
$F(\lambda ,\theta ,\varphi )={\lambda }/{(8\mathsf{ π} \eta )}\;{G(\theta ,\varphi )}/{{{\mathit{\boldsymbol{E}}}^{r}}{{}^{2}}}\;$  (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
${\rm{SCNR}} = \frac{{r_C^2{F_C} + r_N^2{F_N}}}{{r_T^2{F_T}}} \cdot \frac{{\mathit{\boldsymbol{J}}_{rT}^{\rm{T}}{\mathit{\boldsymbol{K}}_T}{\mathit{\boldsymbol{J}}_{tT}}}}{{\mathit{\boldsymbol{J}}_{rC}^{\rm{T}}{\mathit{\boldsymbol{K}}_C}{\mathit{\boldsymbol{J}}_{tC}} + \mathit{\boldsymbol{J}}_{rN}^{\rm{T}}{\mathit{\boldsymbol{K}}_N}{\mathit{\boldsymbol{J}}_{tN}}}}$  (5) 
where
$\mathit{\boldsymbol{J}} = \underbrace {\left[ {\begin{array}{*{20}{c}} {\sqrt {g_1^2 + g_2^2 + g_3^2} } \\ {{g_1}} \\ {{g_2}} \\ {{g_3}} \end{array}} \right]}_{{\rm{completely \ polarzed \ component}}} + \underbrace {\left[ {\begin{array}{*{20}{c}} {{g_0}  \sqrt {g_1^2 + g_2^2 + g_3^2} } \\ 0 \\ 0 \\ 0 \end{array}} \right]}_{{\rm{unpolarized \ component}}}$  (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
$ \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
Let
${\mathit{\boldsymbol{J}}_D} = \left[ {\begin{array}{*{20}{c}} 1 \\ {\vec g} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 1 \\ {{d_1}} \\ {{d_2}} \\ {{d_3}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 1 \\ {\cos (2\varepsilon )\cos (2\tau )} \\ {\cos (2\varepsilon )\sin (2\tau )} \\ {\sin (2\varepsilon )} \end{array}} \right]$  (9) 
hence,
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.
In this section, the OPCE problem based on the SCNR is created. Then a polarization states optimization method is proposed.
2.1 Problem FormulationConsidering 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 copolar 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) 
In this subsection, we recast (13) in a linear function with two variables which are polarization state
$f({\mathit{\boldsymbol{J}}_D},\lambda ) = {\rm{SCR}} \cdot \mathit{\boldsymbol{J}}_D^{\rm{T}} \cdot {\mathit{\boldsymbol{K}}_{Tp}} \cdot {\mathit{\boldsymbol{J}}_D}  \lambda \cdot \mathit{\boldsymbol{J}}_D^{\rm{T}} \cdot {\mathit{\boldsymbol{K}}_{CNp}} \cdot {\mathit{\boldsymbol{J}}_D}$  (14) 
Define the maximum value of function
${f_{\max }}(\lambda ) = \mathop {\max }\limits_{{\mathit{\boldsymbol{\vec J}}_D}} f({\mathit{\boldsymbol{J}}_D},\lambda )$  (15) 
Lemma 1
Proof Given two variables
$ \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,
$\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
Assuming the polarization state corresponding to
$\max {\rm{SCNR}} = {\rm{SCR}} \cdot \frac{{\mathit{\boldsymbol{J}}_{D\_opt}^{\rm{T}} \cdot {\mathit{\boldsymbol{K}}_{Tp}} \cdot {\mathit{\boldsymbol{J}}_{D\_opt}}}}{{\mathit{\boldsymbol{J}}_{D\_opt}^{\rm{T}} \cdot {\mathit{\boldsymbol{K}}_{CNp}} \cdot {\mathit{\boldsymbol{J}}_{D\_opt}}}}$  (18) 
According to the forward analyses, the following Theorem 1 can be summarized.
Theorem 1 The maximum SCNR equals to the
When
$f({\mathit{\boldsymbol{J}}_D}) = \mathit{\boldsymbol{J}}_D^{\rm{T}} \cdot \mathit{\boldsymbol{M}} \cdot {\mathit{\boldsymbol{J}}_D} = {\left[ {\begin{array}{*{20}{c}} 1 \\ \mathit{\boldsymbol{g}} \end{array}} \right]^{\rm{T}}} \cdot \left[ {\begin{array}{*{20}{c}} {{k_{0M}}}&{\mathit{\boldsymbol{a}}_M^{\rm{T}}} \\ {{\mathit{\boldsymbol{b}}_M}}&{{\mathit{\boldsymbol{Q}}_M}} \end{array}} \right] \cdot \left[ {\begin{array}{*{20}{c}} 1 \\ \mathit{\boldsymbol{g}} \end{array}} \right]$  (19) 
where
$L = f({\mathit{\boldsymbol{J}}_D})  v({\mathit{\boldsymbol{g}}^{\rm{T}}}\mathit{\boldsymbol{g}}  1)$  (20) 
Let the derivation of the cost function with respect to
$\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
${[{\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
$\sum\limits_{i = 1}^3 {{{[{\beta _i}/({\lambda _i}  v)]}^2}} = 1$  (23) 
The domain of Lagrange multiplier
${v_L} \leqslant {v_{\max }} \leqslant {v_U}$  (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
$ \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
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
Since the two variables in
Given the maximum value of the numerator and the minimum value of the denominator of (10) are
${\lambda _U} \le {P_{T\max }}/{P_{CN\min }}$  (29) 
Given the minimum value of the numerator and the maximum value of the denominator of (10) are
${\lambda _L} \geqslant {P_{T\min }}{\rm{ /}}{P_{CN\max }}$  (30) 
Hence the search intervals of
${\lambda _L} \leqslant \lambda \leqslant {\lambda _U}$  (31) 
The procedures for the proposed method are:
1) set
2) solve
3) if
else if
else if
4) else go to 2).
3 Numerical ExperimentsExperiments are accomplished by Matlab 2010 code running on a 32bit 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
${\mathit{\boldsymbol{K}}_T} = \left[ {\begin{array}{*{20}{c}} {13.062{\rm{ }}5}&{  5.812{\rm{ }}5}&{3\sqrt 2 }&{\sqrt 2 } \\ {  5.812{\rm{ }}5}&{9.312{\rm{ }}5}&{  \sqrt 2 }&{  3\sqrt 2 } \\ {3\sqrt 2 }&{  \sqrt 2 }&{10.25}&0 \\ {\sqrt 2 }&{  3\sqrt 2 }&0&{  6.25} \end{array}} \right]$  (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.,
Considering the target is completely polarized, i.e.,
Experiment Ⅰ: Low Polarization Ratio
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
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
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 timeconsuming and more accurately.
4 ConclusionsIn 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.
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