Having established the convolution theorem is very
important since digital filters can be described in terms of
convolutions. The Z transform version allows us to work with these
filters by merely manipulating polynomials. In order to see that
this is so, I have to reveal one neat fact about the Z transform.
The Z transform that we have describe above can be written as,

(9) |

If this is evaluated on the unit circle in the complex plane, that
is we let
, then we get

(10) |

In general we can write a digital filter in the form,

(11) |

or written out in the time domain,

(12) |

If represents the measurement data and and are the (somehow) known filter coefficients then the above equation can be reorganized in a way that is more obviously useful,

(13) |

This equation is a general formulation of a digital filter. If the coefficients beyond are nonzero, it is called an Infinite Impluse Response (IIR) filter because an input series consisting of all zeros except at one point will result in an output series that will always contain some amount of the original input (for real computers with finite

For the special
case where the coefficients beyond are all zero, the
filter is called a Finite Impluse Response (FIR) filter because an
impulsive input will cause an output that will eventually settle down
to a steady state. A simple example is the *running average*,

Everything that one can know about a digital filter is contained in
the ratio of elements the of and ,

(14) |

(15) |

Recall that in an earlier column, I pointed out the the power transfer
function is only part of the story. One should also consider what
the filter does to the phase of the input. In general a filters
effect on the phase is frequency dependent and so there is a
*phase transfer function*,

(16) |

The filter can also have an effect on *packets* of waves. These
packets are what you get, for example, when you modulate a one
frequency by another. The frequency of the modulation envelope
itself becomes a filterable component with its own transfer
function, the *group transfer function*,

(17) |

The characteristic function for the leaky integrator is,

Converting this to the time domain gives the impulse response of the filter,

For the running average filter, the function is,

and its impulse response is

Certain forms for the polynomials and have proven to be
useful and are widely used. For example the bandpass Butterworth
filter has the coefficients in the form,

(18) | |||

The Butterworth filter chooses to optimize the flatness of the bandpass region of and in doing so gives up simple forms for and .

Chebyshev filters are designed to give the sharpest possible
transition between the bandpass and bandstop regions. The filter
gets its name because the filter characteristic contains a special
polynomial known as a Chebyshev polynomial, .

(19) |

Elliptical or Cauer filters get even sharper transitions than
Chebyshev filters, but achieve this at the expense of ripples in
*both* the pass and stop bands. The filter characteristic
looks like the Chebyshev filter, except the Chebyshev polynomial
gets replaced by a Chebyshev rational function.

Bessel filters are designed with the goal of achieving the
flattest possible group delay. This results in a filter
has no ringing when it receives a step or impulse input.
The characteristic function for the th order Bessel filter is
given by,

(20) |

where,

(21) | |||