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Relation of p and $\rho $.

The signal covariances W1, W2, and W represent the powers of the radar signal in orthogonal polarizations and the magnitude and phase of the cross-covariance or correlated power of the two polarizations. They can be combined into one entity called the coherency matrix J,

 \begin{displaymath}J = \left[ \begin{array}{cc} W_1 & W \\ W^\ast & W_2 \end{array} \right]
\end{displaymath} (20)

The coherency matrix can be decomposed into its polarized and unpolarized components, as

 \begin{displaymath}J =
\left[ \begin{array}{cc} W_1 & W \\ W^\ast & W_2 \end{arr...
... \begin{array}{cc} B & D \\ D^\ast & C \end{array} \right] \ .
\end{displaymath} (21)

Here, A is the unpolarized power of the signal, which is equal for the two polarizations, B and C are the polarized powers of the orthogonal polarization components, and D is cross-power of the polarized component. Because the components of the unpolarized part of the signal are uncorrelated with each other (and with the polarized components), D = W. The polarized matrix has the property that its determinant is zero, namely that BC = |D|2.

The covariance measurements determine four quantities: W1, W2, and |W| and $\angle W = \phi$ (or, equivalently, $\rm Re[W]$ and $\rm Im[W]$). The decomposed polarization matrices (20) are described by five quantities (A, B, C, |D|, and $\angle D$]), whose values can be obtained from the covariance measurements and from the polarization constraint BC = |D|2. This gives

2A = $\displaystyle (W_1+W_2)-\sqrt{(W_1-W_2)^2+4\vert W\vert^2)}$  
2B = $\displaystyle (W_1-W_2)+\sqrt{(W_1-W_2)^2+4\vert W\vert^2)}$  
2C = $\displaystyle (W_2-W_1)+\sqrt{(W_1-W_2)^2+4\vert W\vert^2)}$  
D = $\displaystyle W \ .$ (22)

The decomposed values depend only on the sum and difference of W1 and W2 and not on W1 or W2 individually.

The above quantities have the interpretation that (W1 + W2) is the total signal power, (B + C) is the total polarized power, and 2Ais the total unpolarized power. The degree of polarization p is defined as the ratio of the polarized power to the total power, namely

$\displaystyle p = \frac{B+C}{W_1+W_2} = \frac{B+C}{B+C+2A} \; .$     (23)

From (22), the total polarized power is given by

 \begin{displaymath}B+C = \sqrt{(W_1-W_2)^2 + 4 \vert W\vert^2} \ .
\end{displaymath} (24)

From this one obtains that
p = $\displaystyle \sqrt{1-\frac{4(W_1 W_2 -\vert W\vert^2)}{(W_1+W_2)^2}}$ (25)
  = $\displaystyle \sqrt{1-\frac{4\,{\rm Det(J)}}{({\rm Tr(J)})^2}}$  

The above formulations apply to any pair of orthogonal polarizations, or polarization basis. Different polarization bases are related by unitary transformations; the determinant and trace of J are invariant to such transformations so that the degree of polarization is independent of the basis used for the formulations. Similarly, the total polarized and unpolarized powers are independent of the basis.

Expression (25) can be rewritten in the form

p = $\displaystyle \sqrt{1-\frac{W_1 W_2}{(\frac{W_1+W_2}{2})^2}\left ( 1-\frac{\vert W\vert^2}{W_1W_2}\right )}$  
  = $\displaystyle \sqrt{1-\left ( \frac{\overline W_{\rm geom}}{\overline W_{\rm arith}}\right )^2 ( 1-\rho^{\,2})}\;\; ,$ (26)

where $\overline W_{\rm geom} = \sqrt{W_1 W_2}$ is the geometric mean of the orthogonal powers and $\overline W_{\rm arith} = (W_1+W_2)/2$is their arithmetic mean. Rearranging terms gives
$\displaystyle (1 - p^2) = \left ( \frac{\overline W_{\rm geom}}{\overline W_{\rm arith}}\right )^2 ( 1-\rho^{\,2})\; .$     (27)

The above is a fundamental result that is found in general treatments of polarization (e.g., Born and Wolf, 1975; Mott, 1986). It relates the degree of polarization p to the correlation coefficient $\rho $ and holds in any polarization basis.

From the fact that $\overline W_{\rm geom}/\overline W_{\rm arith}
\le 1$, it can be shown that

$\displaystyle 0 \le \vert\rho \vert \le p \le 1 \; .$     (28)

Expressed in terms of the polarization ratio W1/W2,
$\displaystyle \left ( \frac{\overline W_{\rm geom}}{\overline W_{\rm arith}}\ri...{4}{\left ( \sqrt{\frac{W_1}{W_2}} + \sqrt{\frac{W_2}{W_1}} \right ) ^2} \; ,$     (29)

When the polarization ratio W2/W1 is unity, the means ratio reaches its maximum value, also unity. In this case,
p = $\displaystyle \rho \, .$ (30)

For non-unity polarization ratios, the means ratio varies as a $\sin^2( )$function, going to zero when all the power is contained in W1 or W2. The degree of polarization is by definition then unity. At intermediate polarization ratios,
$\displaystyle p = \sqrt{1-\frac{4 ( 1 - \rho^{\,2}) }
{\left ( \frac{W_1}{W_2}+ 2 + \frac{W_2}{W_1}\right ) }\; } \;\;\;.$     (31)

In the above expressions, $\rho $ and the means ratio are functions of the particular basis in which the polarization measurements are made, but p is independent of basis. In an H-V basis, $\rho = \rho_{HV}$, and

$\displaystyle p = \sqrt{1-\frac{4 ( 1 - {\rho_{HV}}^{\;2}) }
{\left ( \frac{W_H}{W_V}+ 2 + \frac{W_V}{W_H}\right ) }\; } \;\;\;.$     (32)

A similar relation would hold in a circular polarization basis. Equating the two would enable the correlation coefficients in the two bases to be related.

next up previous
Next: Relation of g and Up: Appendix Previous: Appendix
Bill Rison