One-Pole

Fig.10.4 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.10.5.

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.^{11.1}

Finally, note that the one-pole filter is stable if and only if .

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