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240 Chapter 6 IMPLEMENTATION OF DISCRETE-TIME FILTERS
FIGURE 6.18
All-zero lattice filter
x(n), then the output of the (M − 1) stage lattice filter corresponds to
the output of an (M − 1) order FIR filter; that is,
f 0 (n) =g 0 (n) =K 0 x(n)
y(n) =f M−1 (n)
If the FIR filter is given by the direct form
H(z) =
M−1
∑
m=0
M−1
∑
b m z −m = b 0
(1+
m=1
)
b m
z −m
b 0
(6.17)
(6.18)
and if we denote the polynomial A M−1 (z) by
A M−1 (z) =
(
1+
M−1
∑
m=1
α M−1 (m)z −m )
; (6.19)
α M−1 (m) = b m
b 0
,m=1,...,M − 1
then the lattice filter coefficients {K m } can be obtained by the following
recursive algorithm [23]:
K 0 = b 0
K M−1 = α M−1 (M − 1)
J m (z) =z −m A m
(
z
−1 ) ; m = M − 1,...,1
A m−1 (z) = A m(z) − K m J m (z)
1 − Km
2 , m = M − 1,...,1
K m = α m (m), m = M − 2,...,1
(6.20)
Note that this algorithm will fail if |K m | =1for any m =1,...,M − 1.
Clearly, this condition is satisfied by linear-phase FIR filters since
∣ b 0 = |b M−1 |⇒|K M−1 | = |α M−1 (M − 1)| =
b M−1 ∣∣∣
∣ =1
b 0
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