Graphical Computation of

Amplitude Response from

Poles and Zeros

Now consider what happens when we take the factored form of the general transfer function, Eq. (8.2), and set to to get the frequency response in factored form:

In the complex plane, the number
is plotted at the
coordinates [83]. The difference of two vectors
and
is
, as shown in Fig.8.1. Translating the origin of the
vector to the tip of shows that is an arrow drawn
from the tip of to the tip of . The length of a vector is
unaffected by translation away from the origin. However, the angle of
a translated vector must be measured relative to a translated copy of
the real axis. Thus the term
may be drawn as an
arrow from the th zero to the point
on the unit
circle, and
is an arrow from the th
pole. Therefore, *each term in Eq. (8.3) is the length
of a vector drawn from a pole or zero to a single point on the unit
circle*, as shown in Fig.8.2 for two poles and two zeros.

For example, the dc gain is obtained by multiplying the lengths of the lines drawn from all poles and zeros to the point . The filter gain at half the sampling rate is the product of the lengths of these lines when drawn to the point . For an arbitrary frequency Hz, we draw arrows from the poles and zeros to the point . Thus, at the frequency where the arrows in Fig.8.2 join, (which is slightly less than one-eighth the sampling rate) the gain of this two-pole two-zero filter is . Figure 8.3 gives the complete amplitude response for the poles and zeros shown in Fig.8.2. Before looking at that, it is a good exercise to try sketching it by inspection of the pole-zero diagram. It is usually easy to sketch a qualitatively accurate amplitude-response directly from the poles and zeros (to within a scale factor).

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