Example

For the example of Eq.(6.12), we obtain

$$\begin{eqnarray*} r_{11} &=& \left.\left(1-\frac{1}{2}z^{-1}\right)^3H(z)\right\vert _{z=\frac{1}{2}}\\ &=& \left.7 - 5z^{-1}+ z^{-2}\right\vert _{z=\frac{1}{2}} = 7 - 5\cdot 2 + 2^2 = 1\\ [10pt] r_{12} &=& \left.-2\frac{d}{dz^{-1}} (7 - 5z^{-1}+ z^{-2})\right\vert _{z^{-1}=2}\\ &=& \left.-2(- 5 + 2z^{-1})\right\vert _{z^{-1}=2} = 2\\ [10pt] r_{13} &=& \left.\frac{1}{2\left(-\frac{1}{2}\right)^2}\frac{d}{dz^{-1}} (-5 + 2z^{-1})\right\vert _{z^{-1}=2} = 2\cdot 2 = 4. \end{eqnarray*}$$