Explicit Lagrange Coefficient Formulas

Given a desired fractional delay of $\Delta$ samples, the Lagrange fraction-delay impulse response can be written in closed form as

$$h_\Delta(n) = \prod_{\stackrel{k=0}{k\neq n}}^N \frac{\Delta-k}{n-k}, \quad n=0,1,2,\ldots,N. \protect$$ (5.7)

The following table gives specific examples for orders 1, 2, and 3:

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

Note that, for $N=1$ , Lagrange interpolation reduces to linear interpolation, i.e., the interpolator impulse response is $h = [1-\Delta,\Delta]$ . Also, remember that, for order $N$ , the desired delay should be in a one-sample range centered about $\Delta=N/2$ .