Separation Matrix

The matrix M is such that for acquisitions $D_1(\tilde{k})$, $D_2(\tilde{k})$ and $D_3(\tilde{k})$ the components are separated as

\[ \begin{pmatrix} C_{0}(\tilde{k}) \\ C_{-1}(\tilde{k}) \\ C_{+1}(\tilde{k}) \end{pmatrix} = \begin{pmatrix} H(\tilde{k})S(\tilde{k}) \\ H(\tilde{k})S(\tilde{k} - \tilde{\nu}) \\ H(\tilde{k})S(\tilde{k} + \tilde{\nu}) \end{pmatrix} = \begin{pmatrix} 1 & \tfrac{\mu_1}{2} e^{-i\phi_1} & \tfrac{\mu_1}{2} e^{i \phi_1} \\ 1 & \tfrac{\mu_2}{2} e^{-i\phi_2} & \tfrac{\mu_2}{2} e^{i \phi_2} \\ 1 & \tfrac{\mu_3}{2} e^{-i\phi_3} & \tfrac{\mu_3}{2} e^{i \phi_3} \end{pmatrix}^{-1} \begin{pmatrix} D_{1}(\tilde{k}) \\ D_{2}(\tilde{k}) \\ D_{3}(\tilde{k}) \end{pmatrix}\]