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246 Chapter 6 IMPLEMENTATION OF DISCRETE-TIME FILTERS
FIGURE 6.22
Lattice-ladder structure for realizing a pole-zero IIR filter
where {C m } are called the ladder coefficients that determine the zeros of
the system function H(z). It can be shown [23] that {C m } are given by
M∑
B M (z) = C m J m (z) (6.26)
m=0
where J m (z) isthe polynomial in (6.20). From (6.26) one can obtain a
recursive relation
B m (z) =B m−1 (z)+C m J m (z);
m =1, 2,...,M
or equivalently,
M∑
C m = b m + C i α i (i − m); m = M, M − 1,...,0 (6.27)
i=m+1
from the definitions of B m (z) and A m (z).
6.4.6 MATLAB IMPLEMENTATION
To obtain a lattice-ladder structure for a general rational IIR filter, we
can first obtain the lattice coefficients {K m } from A N (z) using the recursion
(6.20). Then we can solve (6.27) recursively for the ladder coefficients
{C m } to realize the numerator B M (z). This is done in the following
MATLAB function dir2ladr.Itcan also be used to determine the all-pole
lattice parameters when the array b is set to b=[1].
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