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 .