# IIR Direct method with MIP Introduction.

Cuthbert Nyack

Continuous Butterworth filters can be designed by the
modulus squared transfer function:-

## |G(s)|^{2} = 1/(1 + -1^{N}s^{2N})

The poles resulting from this expression are shown below for different N. Poles occur in mirrored pairs ie for every stable pole on the LHS of the imaginary s axis, there is a corresponding
unstable pole to the right of the imaginary s axis. The
polynomial in the denominator of the above expression can be
referred to as a mirror image polynomial.

Mirror image polynomials for the discrete case means polynomials which
have stable poles at p inside the unit circle and corresponding
unstable poles at 1/p outside the unit circle. These provide a direct method for generating poles of a discrete filter without having
to calculate the continuous poles.

The transfer function can be written as:-

## |G(z)|^{2} = 1/(1 + [f(z)]^{2N})

Different methods can be used to find f(z)^{2}. The
approach used here can be found in:- Introduction to Digital
Filters by Trevor J. Terrell and published by MacMillan.
f(z)^{2} is derived from trigonometric functions, sin, cos, etc
can be used with different ones giving low pass or high pass characteristics.

For sin one writes sin^{2}(wT/2) =
½(1 - cos(wT)) =
½(1 - ½(z + z^{-1})),
(z = e^{jwT}).

This can be expressed as f(z)^{2} = (z - 1)^{2}/-4z.
If this is substituted into the above equation for N = 1, the transfer function has 2 poles at 3 - 2Ö2 and 3 + 2Ö2. The product of these roots is 9 - 8 = 1.

The poles for different values of N and the corresponding frequency response is shown below. The poles are derived from the sin which produce low pass characteristics.

When enabled, the following gif file shows how the applet should
appear.
Because of the Butterworth frequency response, this approach is
sometimes
described as obtaining the discrete equivalent of Butterworth Filters.

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© 2007 Cuthbert A. Nyack.