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:
As usual for the frequency response, we prefer the polar form for this
expression. Consider first the amplitude response
.
In the
complex plane, the number
is plotted at the
coordinates
[
84]. 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.
In summary:
Figure 8.2:
Measurement of amplitude response from a
polezero diagram. A pole is represented in the complex plane by `X';
a zero, by `O'.

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 oneeighth the
sampling rate) the gain of this
twopole twozero 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 polezero diagram. It is usually easy to sketch a
qualitatively accurate amplituderesponse directly from the poles and
zeros (to within a scale factor).
Figure:
Amplitude response obtained by traversing the
entire upper semicircle in Fig.8.2 with the point
.
The point of the amplitude obtained in that figure is marked by a heavy dot.
For real filters, precisely the same curve is obtained if the
lower half of the unit circle is traversed, since
. Thus, plotting the response over positive frequencies only is
sufficient for real filters.

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