Elementary non-recirculating filter

We generalize the non-recirculating comb filter to the design shown in figure 8.7, called the non-recirculating elementary filter, of the first form.

Figure 8.7: A delay network with a single-sample delay and a complex gain $Q$. This is the non-recirculating elementary filter, first form. Compare the simpler non-recirculating comb filter shown in Figure 7.3, which corresponds to choosing $Q=-1$ here.

To find the frequency response, as in Chapter 7 we feed the delay network a complex sinusoid $1, Z, {Z^2}, \ldots$ whose frequency is $\omega=\arg(Z)$, so that as before, $Z={e^{i\omega}}$. The $n$th sample of the input is $Z^n$ and that of the output is

$$(1 - Q{Z^{-1}}){Z^n}$$

so the transfer function is

$$H(Z) = 1 - Q{Z^{-1}} = 1 - Q{e^{i\omega}}.$$

This can be represented graphically as shown in Figure 8.8. Suppose we write the coefficient $Q$ in polar form:

$$Q = q {e^{i\alpha}}$$

Then the gain of the filter is the distance from the point $Q$ to the point $Z$ in the complex plane. Analytically we can see this because

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

Graphically, the number $Q{Z^{-1}}$ is just the number $Q$ rotated backwards (clockwise) by the angular frequency $\omega$ of the incoming sinusoid. The value $\vert 1 - Q{Z^{-1}}\vert$ is the distance from $Q{Z^{-1}}$ to $1$ in the complex plane, which is equal to the distance from $Q$ to $Z$.

Figure 8.8: Diagram for calculating the frequency response of the non-recirculating elementary filter (Figure 8.7). The frequency response is given by the length of the segment connecting $Z$ to $A$ in the complex plane.

As the frequency of the input sweeps from 0 to $2\pi$, the point $Z$ travels couterclockwise around the unit circle. At the point where $\omega = \alpha$, the distance is at a minimum, equal to $q-1$. The maximum occurs which $Z$ is at the opposite point of the circle. Figure 8.9 shows the transfer function for three different values of $q$.

Figure 8.9: Frequency response of the elementary non-recirculating filter Figure 8.7. Three values of $Q$ are used, all with the same argument (-2 radians), but with varying absolute value.