Karakci et al. (2013)
For given input CMB power spectra, simulate the Stokes visibilities.
- <math>V_{jk} = J_k \left[ \int A_k(r) S A_j^H(r) e^{-2\pi i u_{jk}.r} d^2r \right] J_j^H</math>
- <math>S = \begin{pmatrix}
I+Q & U+iV \\ U-iV & I-Q \end{pmatrix}</math>
J is DIE (instrumental) and A id DDE (beam).
In ideal case: <math>J=I</math> (identity matrix) and <math>A_k^s(r) = A(r) \times I</math>, where <math>A(r)</math> is the actual primary beam.
But errors are introduced as matrices:
- <math>J_k = \begin{pmatrix}
1+g_k^1 & \epsilon_k^1 \\ \epsilon_k^2 & 1+g_k^2 \end{pmatrix}</math>
- <math>A_k^s = A_k^0 (\rho,\phi) \begin{pmatrix}
1+\frac{1}{2}\mu_k\frac{\rho^2}{\sigma^2}\cos{2\phi} & \frac{1}{2}\mu_k\frac{\rho^2}{\sigma^2}\sin{2\phi} \\ \frac{1}{2}\mu_k\frac{\rho^2}{\sigma^2}\sin{2\phi} & 1-\frac{1}{2}\mu_k\frac{\rho^2}{\sigma^2}\cos{2\phi} \\ \end{pmatrix}</math>
- <math>g_k</math> - antenna gain
- <math>\epsilon_k</math> - coupling error
- <math>\rho, \phi, \sigma</math> fixes size and shape of the beam.
- <math>\mu_k</math> - cross polarization parameter
Effect of errors are calculated as:
<math>\alpha = \frac{\langle (C_l^{real}-C_l^{ideal})^2 \rangle^{1/2}}{\sigma_{error}^{ideal}}</math>