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 + -1Ns2N)

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 sin2(wT/2) = ½(1 - cos(wT)) = ½(1 - ½(z + z-1)), (z = ejwT).
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.