For the butterworth filter, in Matlab command:
[B,A]=butter(2,0.5,'low')
I find this result:
B = [0.29289 0.58579 0.29289]
A = [1 -3.33067e-16 0.17157]
From here I cannot reach the Normalized Butterworth polynomial:
G = [1 1.4142 1]
Nor the poles:
P = [-0.7071+i0.7071 -0.7071-i0.7071]
Which is the relation or the proper parameter i am missing, if any?
The denominator of the transfer function of an analog Butterworth filter is a (non-normalized) Butterworth polynomial. Also the numerator is just a constant (i.e. the coefficients of higher order are 0).
You are designing a discrete filter here, so this does no longer apply. Your numerator has three non-zero coefficients, and the denominator is not a Butterworth polynomial.
You can check that if you design an analog Butterworth adding the parameter 's' when calling the function, you will indeed get a one non-zero element array B and the coefficients of a non-normalized Butterworth polynomial in A.
