Fig.

B.3 gives the

signal flow graph for the general one-pole
filter. The road to the

frequency response goes as follows:

The one-pole filter has a

transfer function (hence frequency response)
which is the reciprocal of that of a one-zero. The analysis is thus quite
analogous. The frequency response in polar form is given by

A plot of the frequency response in polar form for

and
various values of

is given in Fig.

B.4.
The filter has a pole at

, in the

plane (and a zero at

= 0). Notice that the one-pole exhibits
either a lowpass or a highpass frequency response, like the
one-zero. The lowpass character occurs when the pole is near the point

(

dc), which happens when

approaches

. Conversely,
the highpass nature occurs when

is positive.
The one-pole filter section can achieve much more drastic differences
between the gain at high frequencies and the gain at low frequencies
than can the one-zero filter. This difference is achieved in the
one-pole by gain

*boost* in the

passband rather than

*attenuation* in the stopband; thus it is usually desirable when
using a one-pole filter to set

to a small value, such as

, so that the peak gain is 1 or so. When the peak gain is 1,
the filter is unlikely to overflow.

^{B.1}
Finally, note that the one-pole filter is stable if and only if

.

**Next Section:** Two-Pole**Previous Section:** One-Zero