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Appendix A: Catalog of Matrices

Here I repeat Noordam's description of the various matrices:

• G_i represents complex gain (amplitude, phase) per IF channel. It includes the effects of all the electronics after the feed (amplifiers, mixers, LO, cables etc), excluding the correlator of course.

  $$\left(\vphantom{\begin{array}{cc}g_{i\rm p} & 0\\ 0 & g_{i\rm q} \end{array}}\right. \begin{array}{cc}g_{i\rm p} & 0\\ 0 & g_{i\rm q} \end{array} \left.\vphantom{\begin{array}{cc}g_{i\rm p} & 0\\ 0 & g_{i\rm q} \end{array}}\right)$$ (20)

• D_i represents the signal `leakage' between the two receptors in a feed, due to deviations from nominal. The real part corresponds roughly to a dipole alignment error, and the imaginary part to ellipticity.

  $$\left(\vphantom{\begin{array}{cc}1 & d_{i\rm p}\\ -d_{i\rm q} & 1 \end{array}}\right. \begin{array}{cc}1 & d_{i\rm p}\\ -d_{i\rm q} & 1 \end{array} \left.\vphantom{\begin{array}{cc}1 & d_{i\rm p}\\ -d_{i\rm q} & 1 \end{array}}\right)$$ (21)

  NB: Other processes may also contribute to leakage. For example, cross-talk between the two IF-channels can be described by adding a non-zero mutual coupling factor c_i to the off-diagonal terms.

• C_i represents the nominal feed configuration, which includes rotation of the dipoles w.r.t. the antenna (e.g. WSRT) , and conversion from linear to circular polarisation (e.g. VLA).
• E_i represents the feed beam response. For the important case of a dominating compact source in the centre of the field (and an axially symmetric antenna-feed combination), it is the unit matrix.
• P_i represents parallactic angle $\phi$, i.e. rotation of the antenna w.r.t. the sky. For an equatorial antenna like used in the WSRT, it is a unit matrix. For an alt-az antenna, $\phi$ varies smoothly with Hour Angle (HA), as a function of LATitude and DEClination:

  $$\left(\vphantom{\begin{array}{cc}\cos\phi(t) & \sin\phi(t)\\ -\sin\phi(t) & \cos\phi(t) \end{array}}\right. \begin{array}{cc}\cos\phi(t) & \sin\phi(t)\\ -\sin\phi(t) & \cos\phi(t) \end{array} \left.\vphantom{\begin{array}{cc}\cos\phi(t) & \sin\phi(t)\\ -\sin\phi(t) & \cos\phi(t) \end{array}}\right)$$ (22)

• F_i represents (ionospheric) Faraday rotation.

  $$\left(\vphantom{\begin{array}{cc}\cos\psi & \sin\psi\\ -\sin\psi & \cos\psi \end{array}}\right. \begin{array}{cc}\cos\psi & \sin\psi\\ -\sin\psi & \cos\psi \end{array} \left.\vphantom{\begin{array}{cc}\cos\psi & \sin\psi\\ -\sin\psi & \cos\psi \end{array}}\right)$$ (23)