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Elementary nonrecirculating filter
The nonrecirculating comb
filter may be generalized to yield the design shown in Figure 8.7.
This is the
elementary nonrecirculating filter,
of the first form. Its single, complexvalued parameter controls the
complex gain of the delayed signal subtracted from the original one.
Figure 8.7:
A delay network with a singlesample delay and a complex
gain . This is the nonrecirculating elementary filter, first form. Compare
the nonrecirculating comb filter shown in Figure 7.3,
which corresponds to choosing here.

To find its frequency response, as in Chapter 7 we feed the delay network
a complex sinusoid
whose frequency is
.
The th sample of the input is and that of the output
is
so the transfer function is
This can be analyzed graphically as shown in Figure 8.8.
The real numbers
and are the magnitude and argument of the complex number :
The gain of the filter is the distance from the point to the point
in the complex plane. Analytically we can see this because
Graphically, the number is just the number rotated backwards
(clockwise) by the angular frequency of the incoming sinusoid. The
value
is the distance from to in the complex
plane, which is equal to the distance from to .
Figure 8.8:
Diagram for calculating the frequency response of the
nonrecirculating elementary filter
(Figure 8.7). The frequency response is given by the length of the
segment connecting to in the complex plane.

As the frequency of the input sweeps from 0 to , the point travels
couterclockwise around the unit circle. At the point where
,
the distance is at a minimum, equal to . The maximum occurs which is
at the opposite point of the circle. Figure 8.9 shows the transfer
function for three different values of .
Figure 8.9:
Frequency response of the elementary nonrecirculating filter
Figure 8.7. Three values of are used, all with the
same argument (2 radians), but with varying absolute value (magnitude) .

Next: Nonrecirculating filter, second form
Up: Elementary filters
Previous: Elementary filters
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Miller Puckette
20061230