Non-recirculating filter, second form

Occasionally we will use a variant of the filter above, as 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 $\overline{A}$. (If

$$A = a+bi = r{e^{i\alpha}}$$

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

$$\overline{A} = a-bi = r{e^{-i\alpha}}$$

Graphically this flips the entire complex plane across the real axis.) The transfer function of the new filter is

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

This gives rise to the same frequency response as the first form since

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

Here we use the fact that $\overline{Q} = {Z^{-1}}$, for all unit complex numbers $Q = {e^{i\omega}}$, since

$$\overline{Q} = {e^{-i\omega}} = {Q^{-1}}$$

Although the two forms of the elementary non-recirculating filter have the same frequency response, their phase responses are different, and we will occasionally use the second form for its phase response.

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