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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