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Therefore we can expand out the phase evolution to be  \begin{equation}\label{eq:phaseevo}  \phi(t) \approx 2\pi \sum_{l=0} \frac{f^{(l)}}{(l+1)!} \left(t\left(1+\dot{\delta}_k + \frac{\ddot{\delta}_k}{2}t\right)-t_0+\delta_k+t_k\right)^{(l+1)}. \frac{\ddot{\delta}_k}{2}t\right)-t_0+\delta_k+t_k\right)^{l+1}.  \end{equation}  We want to have equation~\ref{eq:phaseevo} in the following form  \begin{equation}  \phi(t) = \phi_k + 2\pi\left(f_kt + \frac{\dot{f}_k}{2}t^2 + \frac{\ddot{f}_k}{6}t^3\right),  \end{equation}  so to find $\phi_k$, $f_k$, $\dot{f}_k$ and $\ddot{f}_k$ we need to Taylor expand equation~\ref{eq:phaseevo} about $t=0$. This gives the following values \begin{eqnarray}  \phi_k &= 2\pi \sum_{l=0} \frac{f^{(l)}}{(l+1)!}\left(t_k - t_0 +\delta_k \right)^{l+1} \\  \end{eqnarray}