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Lattice Filter Structures 239
1.0000 1.8478 1.0000
1.0000 1.9616 1.0000
1.0000 -1.0000 0
1.0000 1.0000 0
Note that only 4 gain coefficients are nonzero. Hence the frequency sampling
form is
⎡
−0.9952 + 0.9952z−1
H (z) = 1 − z−32
32
⎢
⎣
⎤
0.9808 − 0.9808z−1
2 +2
1 − 1.9616z −1 + z−2 1 − 1.8478z −1 + z + −2 ⎥
⎦
−0.9569 + 0.9569z −1
1 − 1.6629z −1 + z + 1
−2 1 − z −1
To determine the computational complexity, note that since H (0) = 1, the 1storder
section requires no multiplication, whereas the three 2nd-order sections
require 3 multiplications each for a total of 9 multiplications per output sample.
The total number of additions is 13. To implement the linear-phase structure
would require 16 multiplications and 31 additions per output sample. Therefore
the frequency sampling structure of this FIR filter is more efficient than the
linear-phase structure.
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6.4 LATTICE FILTER STRUCTURES
The lattice filter is extensively used in digital speech processing and in
the implementation of adaptive filters. It is a preferred form of realization
over other FIR or IIR filter structures because in speech analysis and in
speech synthesis the small number of coefficients allows a large number of
formants to be modeled in real time. The all-zero lattice is the FIR filter
representation of the lattice filter, while the lattice ladder is the IIR filter
representation.
6.4.1 ALL-ZERO LATTICE FILTERS
An FIR filter of length M (or order M − 1) has a lattice structure with
M −1 stages as shown in Figure 6.18. Each stage of the filter has an input
and output that are related by the order-recursive equations [23]:
f m (n) =f m−1 (n)+K m g m−1 (n − 1), m =1, 2,...,M − 1
g m (n) =K m f m−1 (n)+g m−1 (n − 1), m =1, 2,...,M − 1
(6.16)
where the parameters K m , m = 1, 2,...,M − 1, called the reflection
coefficients, are the lattice filter coefficients. If the initial values of f m (n)
and g m (m) are both the scaled value (scaled by K 0 )ofthe filter input
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