Non-recirculating filter, second form

Sometimes we will need a variant of the filter above, shown in Figure 8.10, called the elementary non-recirculating filter, second form. Instead of multiplying the delay output by $Q$ we multiply the direct signal by its complex conjugate $\overline{Q}$. If

$$A = a+bi = r \cdot (\cos(\alpha) + i \sin(\alpha))$$

is any complex number, its complex conjugate is defined as:

$$\overline{A} = a-bi = r \cdot (\cos(\alpha) - i \sin(\alpha))$$

Graphically this reflects points of the complex plane up and down across the real axis. The transfer function of the new filter is

$$H(Z) = \overline{Q} - {Z^{-1}}$$

This gives rise to the same frequency response as before since

$$\vert\overline{Q} - {Z^{-1}}\vert = \vert Q - \overline{Z^{-1}}\vert = \vert Q- Z\vert$$

Here we use the fact that $\overline{Z} = {Z^{-1}}$, for any unit complex number $Z$, as can be verified by writing out $Z\overline{Z}$ in either polar or rectangular form.

Although the two forms of the elementary non-recirculating filter have the same frequency response, their phase responses are different; this will occasionally lead us to prefer the second form.

Figure 8.10: The elementary non-recirculating filter, second form.