Post on 20-Jul-2020
transcript
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Lattice-Structure forFIR filters
Spring 2009
© Ammar Abu-Hudrouss -Islamic University Gaza
Slide ٢Digital Signal Processing
Lattice Structures
m
k
kmm mzkzA
11)(1)(
Where by definition
1,......,2,1,0)()( MmzAH mm
Lattice filter is used in extensively in digital speech processing and in implementing adaptive filters
Which leads to
mnn
nnh
m ,....,2,1)(01
)(
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Slide ٣Digital Signal Processing
Lattice Structures
This can be expressed by the direct form as
m
km knxknxny
1
)()()()(
So the output can be expressed as
1 αm(1) αm(2) αm(3) αm(m -1) αm(m)
x(n)
y(n)
z -1 z -1z -1z -1
+ + + + +
Slide ٤Digital Signal Processing
Lattice Structures
The lattice filter is generally described by the following set of equations
)()()( 00 nxngnf 1,......,2,1)1()()( 11 MmngKnfnf mmmm
1,......,2,1)1()()( 11 MmngnfKng mmmm
)()( 1 nfny M
The output of the (M-1) stage filter
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Slide ٥Digital Signal Processing
Lattice Structures
fm(n)
gm(n)
Km
Km
gm-1(n)
fm-1(n)
First Stage
f0(n)
g0(n)
Second Stage
f1(n)
g1(n)
Second Stage
f2(n)
g2(n)
fM-2(n)
gM-2(n)
y(n)=fM-1(n)
gM-1(n)
z -1
+
+
Slide ٦Digital Signal Processing
Lattice Structures
Suppose we have a filter of order one (m = 1). The output of this filter can be expressed as
x(n)
K1
K1
)1()1()()( 1 nxnxny f1(n) =y(n)
g1(n)
f0(n)
g0(n)
)1()()( 11 nxKnxnf
)1()()( 11 nxnxKng
)1(11 K
z -1
+
+
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Slide ٧Digital Signal Processing
Lattice Structures
Suppose we have a filter of order one m = 2. The output of this filter can be expressed as
)2()2()1()1()()( 22 nxnxnxny
x(n)K1
K1
f2(n) =y(n)
g2(n) g0(n)
K2
K2
g1(n)
f1(n)f0(n)
z -1 z -1
+
+
+
+
Slide ٨Digital Signal Processing
Lattice Structures
The output from the lattice implementation is )1()()( 1212 ngknfnf
)2()1()1()()( 21212 nxKnxKKnxKnxnfSubstituting for g1(n - 1) and f1(n)
)2()1()()()( 21212 nxKnxKKKnxnf
)2()2()1()1()()( 22 nxnxnxny Comparing that with
)1()1( 212 KK 22 )2( K
)2(1)1(
2
21
K )2(21 K
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Slide ٩Digital Signal Processing
Lattice Structures
The output from the lattice implementation is
)1()()( 1122 ngnfKng
)2()1()1()()( 11222 nxnxKnxKKnxKngSubstituting for g1(n - 1) and f1(n)
)2()1()1()()( 2122 nxnxKKnxKng
)2()1()1()()2()( 222 nxnxnxng
m
kmm knxkng
0
)()()(
mkkmk mm ,......,1,0)()(
Slide ١٠Digital Signal Processing
Lattice Structures
)(*)()()()(0
nxnknxkng m
m
kmm
)(*)()()()(0
kxkknxknf m
m
kmm
Convert to z-transform
)()()( zXzAzG mm )()()( zXzBzF mm
Then if we convert the recursive lattice equation to z domain
)()()( 00 zXzGzF
)()()( 11
1 zGzKzFzF mmmm
)()()( 11
1 zGzzFKzG mmmm
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Slide ١١Digital Signal Processing
Lattice Structures
Divide the previous equation by X (z)
1)()( 00 zBzA
1,.....,3,2,1)()()( 11
1
MmzBzKzAzA mmmm
1,.....,3,2,1)()()( 11
1
MmzzzAKzB mmmm
Divide the previous equation by X(z)
)(
)(1
1)()(
111
zBzzA
KK
zBzA
m
m
m
m
m
m
Slide ١٢Digital Signal Processing
Lattice to Direct Form
To get the direct form coefficients from the lattice constants we have
1)()( 00 zBzA
)()()( 11
1 zBzKzAzA mmmm
m
l
mlm
m
k
kmm zlzkzB
00)()()(
Solve the previous equation recursively to get Am(z)
)()( 1 zAzzB mm
m
m
l
m
l
lm
mmlmm zlzzlzB
0 0)()()(
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Slide ١٣Digital Signal Processing
Lattice to Direct Form
Example: Given a three stage lattice filter with coefficients K1 = 0.25, K2 = 0.5 and K3 = 1/3, determine the FIR filter coefficients for the direct-form structure.
10
1101
)4/1(1
)()()(
zzBzKzAzA
11 4/1)( zzB
By reversing the order of A1(z), we get
2nd stage
211
1212
)2/1()8/3(1
)()()(
zzzBzKzAzA
212 )8/3()2/1()( zzzB
Slide ١٤Digital Signal Processing
Lattice to Direct Form
The third stage
2212
1323
)3/1(8/5)24/13(1
)()()(
zzz
zBzKzAzA
1)0(3 By performing the inverse z transform
24/13)1(3
8/5)2(3
3/1)3(3
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Slide ١٥Digital Signal Processing
To get the lattice constants from the direct form coefficients
)()()( 11
1 zBzKzAzA mmmm
)()()()( 11 zAKzBKzAzA mmmmmm
Solve the previous equation to get Am-1(z)
Direct Form to Lattice
1,....,2,11
)()()( 21
MMmK
zBKzAzAm
mmmm
This is a step-down recursion
Slide ١٦Digital Signal Processing
ExampleExample: Determine the lattice coefficients corresponding to the FIR filter
3213 )3/1()8/5()24/13(1)()( zzzzAzH
3213 )24/13()8/5()3/1()( zzzzB
The step down relationship with m = 3
Direct Form to Lattice
23
3332 1
)()()(K
zBKzAzA
212 )2/1()8/3(1)( zzzA
3/13 K
2/12 K
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Slide ١٧Digital Signal Processing
212 )8/3()2/1()( zzzB
The step down relationship with m = 2
Direct Form to Lattice
22
2221 1
)()()(K
zBKzAzA
11 )4/1(1)( zzA
4/11 K