Complex Example

To illustrate an example involving complex poles, consider the filter

$$H(z) \eqsp \frac{g}{1+z^{-2}},$$

where $g$ can be any real or complex value. (When $g$ is real, the filter as a whole is real also.) The poles are then $p_1=j$ and $p_2=-j$ (or vice versa), and the factored form can be written as

$$H(z) \eqsp \frac{g}{(1-jz^{-1})(1+jz^{-1})}.$$

Using Eq.(6.8), the residues are found to be

$$\begin{eqnarray*} r_1 &=& \left.(1-jz^{-1})H(z)\right\vert _{z=j} \eqsp \left.\frac{g}{1+jz^{-1}}\right\vert _{z=j} \eqsp \frac{g}{2}\,,\mbox{ and}\\ r_2 &=& \left.(1+jz^{-1})H(z)\right\vert _{z=-j} \eqsp \left.\frac{g}{1-jz^{-1}}\right\vert _{z=-j} \eqsp \frac{g}{2}\,. \end{eqnarray*}$$

Thus,

$$H(z) \eqsp \frac{g/2}{1-jz^{-1}} + \frac{g/2}{1+jz^{-1}}.$$

More examples are given in §3.12.