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248 Chapter 6 IMPLEMENTATION OF DISCRETE-TIME FILTERS
J = 1; a = 1; A = zeros(N,N);
for m=1:1:N
a = [a,0]+conv([0,K(m)],J);
A(m,1:m) = -a(2:m+1); J = fliplr(a);
end
b(N+1) = C(N+1);
for m = N:-1:1
A(m,1:m) = A(m,1:m)*C(m+1);
b(m) = C(m) - sum(diag(A(m:N,1:N-m+1)));
end
The lattice-ladder filter is implemented using (6.23) and (6.25).
This is done in the following MATLAB function ladrfilt. Itshould
be noted that, due to the recursive nature of this implementation along
with the feedback loops, this MATLAB function is neither an elegant
nor an efficient method of implementation. It is not possible to exploit
MATLAB’s inherent parallel processing capabilities in implementing this
lattice-ladder structure.
function [y] = ladrfilt(K,C,x)
% LATTICE/LADDER form realization of IIR filters
% ----------------------------------------------
% [y] = ladrfilt(K,C,x)
% y = output sequence
% K = LATTICE (reflection) coefficient array
% C = LADDER coefficient array
% x = input sequence
%
Nx = length(x); y = zeros(1,Nx);
N = length(C); f = zeros(N,Nx); g = zeros(N,Nx+1);
f(N,:) = x;
for n = 2:1:Nx+1
for m = N:-1:2
f(m-1,n-1) = f(m,n-1) - K(m-1)*g(m-1,n-1);
g(m,n) = K(m-1)*f(m-1,n-1) + g(m-1,n-1);
end
g(1,n) = f(1,n-1);
end
y = C*g(:,2:Nx+1);
□ EXAMPLE 6.10 Convert the following pole-zero IIR filter into a lattice-ladder structure.
H(z) =
1+2z−1 +2z −2 + z −3
1+ 13
24 z−1 + 5 8 z−2 + 1 3 z−3
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