To implement a causal IIR filter implemented in the Direct Form II structure, the function

direct2 given below can be employed.

function [y,sf] = direct2(p,d,x,si);

% Y = DIRECT2(P,D,X) filters input data vector X with

% the filter described by vectors P and D to create the

% filtered data Y. The filter is a "Direct Form II"

% implementation of the difference equation:

% y(n) = p(1)*x(n) + p(2)*x(n-1) + ... + p(np+1)*x(n-np)

% - d(2)*y(n-1) - ... - d(nd+1)*y(n-nd)

% [Y,SF] = DIRECT2(P,D,X,SI) gives access to initial and

% final conditions, SI and SF, of the delays.

dlen = length(d); plen = length(p);

N = max(dlen,plen); M = length(x);

sf = zeros(1,N-1); y = zeros(1,M);

if nargin ~= 3,

sf = si;

end

if dlen <>

d = [d zeros(1,plen - dlen)];

else

p = [p zeros(1, dlen - plen)];

end

p = p/d(1); d = d/d(1);

for n = 1:M;

wnew = [1 -d(2:N)]*[x(n) sf]’;

K = [wnew sf];

y(n) = K*p’;

sf = [wnew sf(1:N-2)];

end

The structure being simulated can be verified in MATLAB by computing its transfer function with the aid of the function strucver given below.

function [p,d] = strucver(ir,N);

H = zeros(2*N+1,N+1);

H(:,1) = ir’;

for n = 2:N+1;

H(:,n) = [zeros(1,n-1) ir(1:2*(N+1)-n)]’;

end

H1 = zeros(N+1,N+1);

for k = 1:N+1;

H1(k,:) = H(k,:);

end

H3 = zeros(N,N+1);

for k = 1:N;

H3(k,:) = H(k+N+1,:);

end

H2 = H3(:,2:N+1);

hf = H3(:,1);

% Compute the denominator coefficients

d = -(inv(H2))*hf;

% Compute the numerator coefficients

p = H1*[1;d];

d = [1; d];

Program to illustrate the design of a causal IIR filter and its simulation in Direct Form

II. It employs the functions strucver and direct2 described earlier.

% Program

Wp = [0.4 0.5]; Ws = [0.1 0.8]; Rp = 1; Rs = 30;

[N1, Wn1] = buttord(Wp, Ws, Rp, Rs);

[num,den] = butter(N1,Wn1);

disp(’Numerator coefficients are ’);disp(num);

disp(’Denominator coefficients are ’);disp(den);

impres = direct2(num,den,[1 zeros(1,4*N1)]);

[p,d] = strucver(impres,2*N1);

disp(’Actual numerator coeffs are ’); disp(p’);

disp(’Actual denominator coeffs are ’); disp(d’);