Polarization Synthesis

The representation of the elements of the [S]-matrix are dependent on the polarization basis. Using the results from section 4.9, we can now derive from a given [S]-matrix in { $\vec{e}_h,\vec{e}_v$} basis, the [S]-matrix for an arbitrary basis { $\vec{e}_x , \vec{e}_y$} by an unitary congruence transformation [Lüneburg96]

$$[S]_{xy}=[U]_{2 \times 2}^T [S]_{hv} [U]_{2 \times 2} \quad where... ...ho \rho^*}} \left[\begin{array}{cc} 1 & - \rho^* \rho & 1 \end{array}\right]$$ (3.8)

The scattering matrix components $S_{xy}$ in the { $\vec{e}_x , \vec{e}_y$} basis, characterised by the polarisation ratio $\rho$, are then given by

$$S_{xx} = \frac {1}{1 + \rho \rho^*}[S_{hh} + 2 \rho S_{x} +\rho^{2}S_{vv}]$$ (3.9)

$$S_{xy} = \frac {1}{1 + \rho \rho^*}[\rho S_{hh} + (1 - \rho \rho^*)S_{x} -\rho^{*} S_{vv}]$$

$$S_{yx} = \frac {1}{1 + \rho \rho^*}[\rho S_{hh} - (\rho \rho^* - 1)S_{x} -\rho^{*} S_{vv}]$$

$$S_{yy} = \frac {1}{1 + \rho \rho^*}[\rho^{2}S_{hh} + 2 \rho S_{x} + S_{vv}]$$

Even though the elements of [S] change under the transformation some properties of the [S]-matrix are invariant.

a)

The span of the matrix, defined as the quadratic sum of its elements, corresponds to the total power, and is constant.

b)

[S] is symmetric in any arbitrary polarization bases as long as the BSA convention [IEEE79] is used, hence the reciprocity theorem yields than that the cross-polar components are equal.

c)

det([S]) is invariant because $det([U]_{2 \times 2}) = 1$.

Therefore, given a [S]-matrix for an arbitrary combination of transmit and receive polarizations, it is possible to generate the [S]-matrix for any other arbitrary combination of transmit and receive polarizations [Boerner90]. One of the common applications of this relation is the transformation from the linear { $\vec{e}_h,\vec{e}_v$} basis to the circular { $\vec{e}_R,\vec{e}_L$} basis. For example, for the investigations in [Krogager95a] the cirular basis is better suited and therefore, the [S] - matrix, measured with two linear polarized antennas $[S]_{HV}$ is transformed in the cirular basis $[S]_{RL}$. The elements of the $[S]_{RL}$ basis for the backscatter case are given by

$$S_{RR} = iS_{x} + \frac {1}{2}(S_{hh} - S_{vv})$$ (3.10)

$$S_{LL} = iS_{x} - \frac {1}{2}(S_{hh} - S_{vv})$$

$$S_{RL} = \frac {i}{2}(S_{hh} + S_{vv})$$

If the polarization transformation properties of a target change moderately, one could envisage a radar which dynamically adjusts the polarization in order to receive maximum power from a given target all the times [Boerner93], [Cysz91]. Such polarization tracking radars are, for instance, applied to achieve significant signal-to-clutter enhancement and interference suppression [Poelman84].