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404 Chapter 8 IIR FILTER DESIGN
which returns zeros in z array, poles in p array, and the gain value k.
However, we need an unnormalized Butterworth filter with arbitrary Ω c .
From Example 8.1 we observe that there are no zeros and that the poles
of the unnormalized filter are on a circle with radius Ω c instead of on a
unit circle. This means that we have to scale the array p of the normalized
filter by Ω c and the gain k by Ω N c .Inthe following function, called
U buttap(N,Omegac), wedesign the unnormalized Butterworth analog
prototype filter.
function [b,a] = u_buttap(N,Omegac);
% Unnormalized Butterworth Analog Lowpass Filter Prototype
% --------------------------------------------------------
% [b,a] = u_buttap(N,Omegac);
% b = numerator polynomial coefficients of Ha(s)
% a = denominator polynomial coefficients of Ha(s)
% N = Order of the Butterworth Filter
% Omegac = Cutoff frequency in radians/sec
%
[z,p,k] = buttap(N);
p = p*Omegac;
k = k*Omegac^N;
B = real(poly(z));
b0 = k; b = k*B; a = real(poly(p));
This function provides a direct form (or numerator-denominator) structure.
Often we also need a cascade form structure. In Chapter 6 we have
already studied how to convert a direct form into a cascade form. The
following sdir2cas function describes the procedure that is suitable for
analog filters.
function [C,B,A] = sdir2cas(b,a);
% DIRECT-form to CASCADE-form conversion in s-plane
% -------------------------------------------------
% [C,B,A] = sdir2cas(b,a)
% C = gain coefficient
% B = K by 3 matrix of real coefficients containing bk’s
% A = K by 3 matrix of real coefficients containing ak’s
% b = numerator polynomial coefficients of DIRECT form
% a = denominator polynomial coefficients of DIRECT form
%
Na = length(a)-1; Nb = length(b)-1;
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