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242 Chapter 6 IMPLEMENTATION OF DISCRETE-TIME FILTERS
The equations (6.20) can also be used to determine the direct, form
coefficients {b m } from the lattice filter coefficients {K m } using a recursive
procedure [22]:
A 0 (z) =J 0 (z) =1
A m (z) =A m−1 (z)+K m z −1 J m−1 (z) , m =1, 2,...,M − 1
J m (z) =z −m A m
(
z
−1 ) , m =1, 2,...,M − 1
(6.21)
b m = K 0 α M−1 (m), m =0, 1,...,M − 1
The following MATLAB function latc2dir implements (6.21). Note
that the product K m z −1 J m−1 (z) isobtained by convolving the 2 corresponding
arrays, whereas the polynomial J m (z) isobtained by using a
fliplr operation on the A m (z) polynomial.
function [b] = latc2dir(K)
% All-Zero Lattice form to FIR Direct form Conversion
% ---------------------------------------------------
% [b] = latc2dir(K)
% b = FIR direct form coefficients (impulse response)
% K = Lattice filter coefficients (reflection coefficients)
%
M = length(K); J = 1; A = 1;
for m=2:1:M
A = [A,0]+conv([0,K(m)],J); J = fliplr(A);
end
b=A*K(1);
□ EXAMPLE 6.8 An FIR filter is given by the difference equation
y(n) =2x(n)+ 13
12 x(n − 1) + 5 4 x(n − 2) + 2 x(n − 3)
3
Determine its lattice form.
Solution
MATLAB script:
>> b=[2, 13/12, 5/4, 2/3]; K=dir2latc(b)
K =
2.0000 0.2500 0.5000 0.3333
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