Lagrange Interpolation Coefficients
Orders 1, 2, and 3

$$\begin{array}{\vert\vert r\vert\vert c\vert c\vert c\vert c\vert} \hline h_\Delta{Order} & h_\Delta(0) & h_\Delta(1) & h_\Delta(2) & h_\Delta(3) \\ \hline \hline N=1 & 1-\Delta & \Delta & & \\ \hline N=2 & \frac{(\Delta-1)(\Delta-2)}{2} & -\Delta(\Delta-2) & \frac{\Delta(\Delta-1)}{2} & \\ \hline N=3 & -\frac{(\Delta-1)(\Delta-2)(\Delta-3)}{6} & \frac{\Delta(\Delta-2)(\Delta-3)}{2} & -\frac{\Delta(\Delta-1)(\Delta-3)}{2} & \frac{\Delta(\Delta-1)(\Delta-2)}{6} \\ \hline \end{array}$$

• For $N=1$ , Lagrange interpolation reduces to linear interpolation $h = [1-\Delta,\Delta]$ , as before

• For order $N$ , desired delay should be in a one-sample range centered about $\Delta=N/2$