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

for which

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

to

and from

to

.

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 *plots show

and

, respectively, plotted with

so that the frequency

ranges from

up to

, which are the minimum and maximum frequencies representable in a digital system.

[1] R. Allred,

*Digital Filters for Everyone: Third Edition*, Creative Arts & Sciences House, 2015.