Snapshot 1: first-order all-pass filter
Snapshot 2: all-pole filter reducing amplitude of high frequencies
Snapshot 3: bandpass filter
The transfer function of the general fifth-order digital IIR filter as shown in this Demonstration is represented by
Given an input
, the output of the filter,
, is determined by
. Thus, the transfer function for the filter specifies the change in amplitude and phase on an input with frequency distribution defined by the unitary complex number
. For example, if the magnitude of the transfer function at frequency
is 0.5, then the filter output amplitude at that frequency will be half the amplitude of the input.
In filter design, it is helpful to know which complex valued inputs to the transfer function produce zero output amplitude and which inputs produce infinite output amplitude. This information is readily available by examining the numerator and denominator of the transfer function. Specifically, if for a given input value
the numerator is zero, then the entire transfer function evaluates to zero. Likewise, if for
the denominator is zero, then the value of the transfer function is infinite. Complex values of
are called zeros
of the transfer function, while values for which
is infinite are called poles
When we plot the transfer function of a digital filter, we let
and generate the plot by varying
in the range
. In this way, we cover the entire range of frequencies from 0 up to the Nyquist frequency, which is half the sampling rate. Hence the value
indicates the maximum frequency at which the filter operates without aliasing.
In this Demonstration, the poles and zeros plot shows the location of the poles and zeros in the complex plane. For a benchmark, the unit circle is also shown with a dashed line. The solid line shows the absolute value of the transfer function plotted in polar coordinates as the angle varies from
The impulse response plot shows the individual sample values that would result from applying the digital filter to an input containing a single one followed by zeros:
. This is useful for illustrating how the filter causes impulsive inputs to spread out in the time domain.
The magnitude and phase
, respectively, plotted with
so that the frequency
, which are the minimum and maximum frequencies representable in a digital system.
 R. Allred, Digital Filters for Everyone: Third Edition
, Creative Arts & Sciences House, 2015.