structures for discrete-time systems 主講人：虞台文. content introduction block diagram...

Structures forDiscrete-Time Systems

主講人：虞台文

Content Introduction Block Diagram Representation Signal Flow Graph Basic Structure for IIR Systems Transposed Forms Basic Structure for FIR Systems Lattice Structures

Structures for Discrete-Time Systems

Introduction

Characterize an LTI System

Impulse Responsez-TransformDifference Equation

Example|||| ,

1)( 1

110 az

azzbbzH

)1()()( 10 nuabnuabnh nn

)1()()1()( 10 nxbnxbnayny

)1()()1()( 10 nxbnxbnayny Computable

Noncomputable

Basic Operations

)1()()1()( 10 nxbnxbnayny Computable

AdditionMultiplicationDelay

In fact, there are unlimited variety of computational structures.

Why Implement Using Different Structures?

Finite-precision number representation of a digital computer.

Truncation or rounding error.

Modeling methods:– Block Diagram– Signal Flow Graph

Block Diagram Representation

+x1(n)

x2(n)

x1(n) + x2(n)Adder

x(n)a ax(n)Multiplier

x(n) x(n1)z1Unit Delay

Example)()2()1()( 21 nbxnyanyany

x(n) +

+

b

a1

z1

z1

a2

y(n)

y(n1)

y(n2)

Higher-Order Difference Equations

M

kk

N

kk knxbknyany

01

)()()(

N

k

kk

M

k

kk

za

zbzH

1

1

1)(

M

kk

N

kk knxbknyany

01

)()()(

Block Diagram Representation(Direct Form I)

M

kk

N

kk knxbknyany

01

)()()(

+

z1

z1

+

z1

+

b0

b1

bM1

bM

x(n)

x(n1)

x(n2)

x(nM)

+

z1

z1

+

z1

+

a1

aN1

aN

y(n)

y(n1)

y(n2)

y(nM)

v(n)


+

z1

z1

+

z1

+

b0

b1

bM1

bM

x(n)

x(n1)

x(n2)

x(nM)

+

z1

z1

+

z1

+

a1

aN1

aN

y(n)

y(n1)

y(n2)

y(nM)

v(n)

M

kk knxbnv

0

)()(

)()()(1

nvknyanyN

kk


+

z1

z1

+

z1

+

b0

b1

bM1

bM

x(n)

x(n1)

x(n2)

x(nM)

+

z1

z1

+

z1

+

a1

aN1

aN

y(n)

y(n1)

y(n2)

y(nM)

v(n)

M

kk knxbnv

0

)()(

)()()(1

nvknyanyN

kk

M

k

kk zbzH

01 )(

N

k

kk za

zH

1

2

1

1)(


+

z1

z1

+

z1

+

b0

b1

bM1

bM

x(n)

x(n1)

x(n2)

x(nM)

+

z1

z1

+

z1

+

a1

aN1

aN

y(n)

y(n1)

y(n2)

y(nM)

v(n)

M

k

kk zbzH

01 )(

N

k

kk za

zH

1

2

1

1)(

Implementing zeros

Implementing poles

N

k

kk

M

k

kk

zazbzHzHzH

1

021

1

1)()()(


+

z1

z1

+

z1

+

b0

b1

bM1

bM

x(n)

x(n1)

x(n2)

x(nM)

+

z1

z1

+

z1

+

a1

aN1

aN

y(n)

y(n1)

y(n2)

y(nM)

v(n)

How many Adders?How many multipliers?How many delays?

Block Diagram Representation (Direct Form II)

+

z1

z1

+

z1

+

b0

b1

bN1

bN

x(n)+

z1

z1

+

z1

+

a1

aN1

aN

y(n)

w(n1)

w(n2)

w(nN)

w(n)

AssumeM = N


+

z1

z1

+

z1

+

b0

b1

bN1

bN

x(n)+

z1

z1

+

z1

+

a1

aN1

aN

y(n)

w(n1)

w(n2)

w(nN)

w(n)

AssumeM = N

)()()(1

nxknwanwN

kk

)()()(0

nwknxbnyM

kk

M

k

kk zbzH

01 )(

N

k

kk za

zH

1

2

1

1)(


+

z1

z1

+

z1

+

b0

b1

bN1

bN

x(n)+

z1

z1

+

z1

+

a1

aN1

aN

y(n)

w(n1)

w(n2)

w(nN)

w(n)

AssumeM = N

M

k

kk zbzH

01 )(

N

k

kk za

zH

1

2

1

1)(

M

k

kkN

k

kk

zbza

zHzHzH0

1

12

1

1)()()(

Implementing zeros

Implementing poles


+

z1

z1

+

z1

+

b0

b1

bN1

bN

x(n)+

z1

z1

+

z1

+

a1

aN1

aN

y(n)

w(n1)

w(n2)

w(nN)

w(n)

AssumeM = N

How many Adders?How many multipliers?How many delays?

Block Diagram Representation (Canonic Direct Form)

+

+

+

b0

b1

bN1

bN

x(n)+

z1

z1

+

z1

+

a1

aN1

aN

y(n)

AssumeM = N

Block Diagram Representation (Canonic Direct Form)

+

+

+

b0

b1

bN1

bN

x(n)+

z1

z1

+

z1

+

a1

aN1

aN

y(n)

AssumeM = N

How many Adders?How many multipliers?How many delays? max(M, N)


Signal Flow Graph

Nodes And Branches

wj(n)wk(n)

Associated with each node is a variable or node value.

Nodes And Branches

wj(n)wk(n)Brach (j, k)

Each branch has an input signal and an output signal.

Input wj(n) Output: A linear transformation of input, such as constant gain and unit delay.

More on Nodes

wj(n)wk(n)

An internal node serves as a summer, i.e., its value is the sum of outputs of all branches entering the node.

Source NodesNodes without entering branches

xj(n) wk(n)

Source node j

Sink Nodes Nodes that have only entering branches

yk(n)wj(n)

Sink node k

Example

x(n) y(n)w1(n)w2(n)

a

b

c

d

e

SourceNode

SinkNode

)()()()( 221 nbwnawnxnw

)()( 12 ncwnw

)()()( 2 newndxny

Block Diagram vs. Signal Flow Graph

x(n)+

az1

+

b1

b0w(n) y(n)

x(n) w1(n)w2(n) w3(n)

a b1

b0

z1

1 2 3

4 w4(n)

y(n)


x(n)+

az1

+

b1

b0w(n) y(n)

x(n) w1(n)w2(n) w3(n)

a b1

b0

z1

1 2 3

4 w4(n)

y(n)

)()()( 41 nawnxnw )()( 12 nwnw

)()()( 41203 nwbnwbnw )1()( 24 nwnw

)()( 3 nwny


)()()( 41 nawnxnw )()( 12 nwnw

)()()( 41203 nwbnwbnw )1()( 24 nwnw

)()( 3 nwny

)()( 3 nwny )1()( 2120 nwbnwb)()( 12 nwnw )1()( 2 nawnx

)()()( 21

10 zWzbbzY

)()()( 21

2 zWazzXzW

12 1)()(

azzXzW

)(1

)()( 1

110 zX

azzbbzY

)1()()1()( 10 nxbnxbnayny


Basic Structure for IIR Systems

Criteria Reduce the number of constant multipliers

– Increase speed Reduce the number of delays

– Reduce the memory requirement Modularity: VLSI design The effects of finite register length and finite-

precision arithmetic.

Basic StructuresDirect FormsCascade FormParallel Form

Direct Forms

M

kk

N

kk knxbknyany

01

)()()(

N

k

kk

M

k

kk

za

zbzH

1

1

1)(

Direct Form I

N

kk

N

kk knxbknyany

01

)()()(

b0

b1

x(n)

x(n1)

x(n2)

x(nN)

y(n)

b2

bN-1

bN

x(nN+1)

a1

a2

aN-1

aN

y(n1)

y(n2)

y(nN)

y(nN+1)

z1

z1

z1

z1

z1

z1

v(n)

Direct Form I

M

kk

N

kk knxbknyany

01

)()()(

N

k

kk

M

k

kk

za

zbzH

1

1

1)(

b0

b1

x(n)

x(n1)

x(n2)

x(nN)

y(n)

b2

bN-1

bN

x(nN+1)

a1

a2

aN-1

aN

y(n1)

y(n2)

y(nN)

y(nN+1)

z1

z1

z1

z1

z1

z1

v(n)

Direct Form II

N

kk

N

kk knxbknyany

01

)()()(

x(n) y(n)w(n) b0

b1

b2

bN-1

bN

a1

a2

aN-1

aN

z1

z1

z1

Direct Form II

N

kk

N

kk knxbknyany

01

)()()(

N

k

kk

M

k

kk

za

zbzH

1

1

1)(

x(n) y(n)w(n) b0

b1

b2

bN-1

bN

a1

a2

aN-1

aN

z1

z1

z1

Example21

21

125.075.0121)(

zz

zzzH

x(n) y(n)z1

z1

z1

z1

0.75

0.125

2

x(n) y(n)z1

z1

0.75

0.125

2

Direct Form I

Direct Form II

Cascade Form

M

kk

N

kk knxbknyany

01

)()()(

N

k

kk

M

k

kk

za

zbzH

1

1

1)(

21

21

1

1*11

1

1

1*11

1

)1)(1()1(

)1)(1()1()( N

kkk

N

kk

M

kkk

M

kk

zdzdzc

zhzhzgzH

Cascade Form

21

21

1

1*11

1

1

1*11

1

)1)(1()1(

)1)(1()1()( N

kkk

N

kk

M

kkk

M

kk

zdzdzc

zhzhzgzH

sN

k kk

kkk

zazazbzbbzH

12

21

1

22

110

1)(

Cascade Form

2nd OrderSystem

2nd OrderSystem

2nd OrderSystem

sN

k kk

kkk

zazazbzbbzH

12

21

1

22

110

1)(

Cascade Form

sN

k kk

kkk

zazazbzbbzH

12

21

1

22

110

1)(

x(n) y(n)z1

z1

a11

a21

b11

b21

b01z1

z1

a12

a22

b12

b22

b01z1

z1

a13

a23

b13

b23

b03

Another Cascade Form

sN

k kk

kkk

zazazbzbbzH

12

21

1

22

110

1)(

sN

k kk

kk

zazazbzbbzH

12

21

1

22

11

0 1

~~1)(

Parallel Form

M

kk

N

kk knxbknyany

01

)()()(

N

k

kk

M

k

kk

za

zbzH

1

1

1)(

11

11*1

1

11

0 )1)(1()1(

1)(

N

k kk

kkN

k k

kN

k

kk zdzd

zeBzc

AzCzHP

Parallel Form

Real Poles Complex PolesPoles at zero

sP N

k kk

kkN

k

kk zaza

zeezCzH1

22

11

110

0 1)(

11

11*1

1

11

0 )1)(1()1(

1)(

N

k kk

kkN

k k

kN

k

kk zdzd

zeBzc

AzCzHP

GroupReal Poles

Parallel Form

sP N

k kk

kkN

k

kk zaza

zeezCzH1

22

11

110

0 1)(

z1

z1

a1k

a2k

e0k

e1k

Parallel Form

x(n) y(n)

sP N

k kk

kkN

k

kk zaza

zeezCzH1

22

11

110

0 1)(

Example21

21

125.075.0121)(

zz

zzzH

21

1

25.175.01878)(

zz

zzH

8

x(n) y(n)z1

z1

0.75

0.125

8

7

Example21

21

125.075.0121)(

zz

zzzH

11 25.0125

5.01188)(

zz

zH

z1

0.5

18

8

x(n) y(n)

z1

0.25

25


Transposed Forms

Signal Flow Graph Transformation

To transform signal graphs into different forms while leaving the overall system function between input and output unchanged.

Transposition of Signal Flow Graph

Reverse the directions of all arrows. Changes the roles of input and output.

x(n) y(n)z1

a

x(n)y(n) z1

a


x(n) y(n)z1

a

x(n)y(n) z1

a

Are there any relations between the two systems?

Example:

111)(

az

zHz1

a

x(n) y(n)

z1

a

x(n)y(n)

z1

a

x(n) y(n)


Reverse the directions of all arrows. Changes the roles of input and output.

x(n) y(n)z1

a

x(n)y(n) z1

a

Detail proof see reference


Basic Structure for FIR Systems

FIRFor causal FIR systems, the system

function has only zeros.

M

kk knxbny

0

)()(

M

k

knxkhny0

)()()(

othrewise

Mnbnh n

0,,1,0

)(

Direct Form

othrewise

Mnbnh n

0,,1,0

)(

x(n)

y(n)

z1 z1 z1

h(0) h(1) h(2) h(M1) h(M)

x(n)

y(n)

z1 z1 z1

h(0) h(1) h(2) h(M1) h(M)

Direct Form

x(n)

y(n) z1 z1 z1

h(0) h(1) h(2) h(M1) h(M)

x(n)

y(n)

z1 z1 z1

h(0) h(1) h(2) h(M1) h(M)

Direct Form

x(n)

y(n)z1z1z1

h(0)h(1)h(2)h(M1)h(M)

Cascade Form

M

kk knxbny

0

)()(

M

n

nznhzH0

)()(

sM

kkkk zbzbbzH

1

22

110 )()(

Cascade Form

sM

kkkk zbzbbzH

1

22

110 )()(

x(n) y(n)

z1

z1

b01

b11

b21

z1

z1

b02

b12

b22

z1

z1

b1Ms

b2Ms

b0Ms

M is even M is oddh(Mn) = h(n)

h(Mn) = h(n)

Structures for Linear Phase Systems

A generalized linear phase system satisfies:

h(Mn) = h(n) for n = 0,1,…,Mh(Mn) = h(n) for n = 0,1,…,Mor

Type I

Type III

Type IIType VI

Type I

M

n

nznhzH0

)()(

M

Mn

nMM

n

n znhzMhznh12/

2/12/

0

)()2/()(

M

Mn

nMM

n

n znMhzMhznh12/

2/12/

0

)()2/()(

12/

0

2/12/

0

)()2/()(M

n

MnMM

n

n znhzMhznh

2/12/

0

)2/())(( MM

n

Mnn zMhzznh

Type I2/

12/

0

)2/())(()( MM

n

Mnn zMhzznhzH

x(n)

y(n)

z1 z1 z1

z1 z1 z1

h(M/2)h(M/21)h(0) h(1) h(2)

Type II, III and VI

Construct them in a

similar manner by

yourselves.


Lattice Structures

FIR LatticeConsider x(n)=(n), one will see

N

m

mm zazHzA

1

1)()(

)(nx )(0 ne )(1 ne )(2 ne )(neN

)(~0 ne )(~

1 ne )(~2 ne )(~ neN

1k1k

2k2k

NkNk

)(ny

1z 1z 1z

FIR Lattice

)()(~)( 00 nxnene

)1(~)()( 11 neknene iiii

)1(~)()(~11 nenekne iiii

)()( neny N

)(nei

)(~ nei

ikik

1z

)(1 nei

)(~1 nei

)(nx )(0 ne )(1 ne )(2 ne )(neN

)(~0 ne )(~

1 ne )(~2 ne )(~ neN

1k1k

2k2k

NkNk

)(ny

1z 1z 1z

Consider x(n)=(n), one will see

N

m

mm zazHzA

1

1)()(

FIR Lattice

)()(~)( 00 zXzEzE

)(~)()( 11

1 zEzkzEzE iiii

)(~)()(~1

11 zEzzEkzE iiii

)()( zEzY N


)()(~)( 00 nxnene



)()( neny N

)(nx )(0 ne )(1 ne )(2 ne )(neN

)(~0 ne )(~

1 ne )(~2 ne )(~ neN

1k1k

2k2k

NkNk

)(ny

1z 1z 1z

N

m

mm zazHzA

1

1)()(

FIR LatticeDefine

)()()(

0 zEzEzA i

i

)(~)(~

)(~

0 zEzEzA i

i

1)(~)( 00 zAzA

i

m

mimi zazA

1

)(1)(


)(nx )(0 ne )(1 ne )(2 ne )(neN

)(~0 ne )(~

1 ne )(~2 ne )(~ neN

1k1k

2k2k

NkNk

)(ny

1z 1z 1z

)()(~)( 00 zXzEzE

)(~)()( 11

1 zEzkzEzE iiii

)(~)()(~1

11 zEzzEkzE iiii

)()( zEzY N

N

m

mm zazHzA

1

1)()(

FIR Lattice)(~)()( 1

11 zAzkzAzA iiii

)(~)()(~1

11 zAzzAkzA iiii

Show that

)()(~ 1 zAzzA ii

i

)()()( 111

zAzkzAzA i

iiii

Define)()()(

0 zEzEzA i

i

)(~)(~

)(~

0 zEzEzA i

i

1)(~)( 00 zAzA

i

m

mimi zazA

1

)(1)(

)(nx )(0 ne )(1 ne )(2 ne )(neN

)(~0 ne )(~

1 ne )(~2 ne )(~ neN

1k1k

2k2k

NkNk

)(ny

1z 1z 1z

FIR LatticeFIR Lattice)(~)()( 1

11 zAzkzAzA iiii

)(~)()(~1

11 zAzzAkzA iiii

1)(~)( 00 zAzA

i

m

mimi zazA

1

)(1)(

i=1:1

101

101 1)(~)()( zkzAzkzAzA

110

1011 )(~)()(~ zkzAzzAkzA

zkzA 11

1 1)(

)( 11

1 zAz

)()( 10

110

zAzkzA

Show that

)()(~ 1 zAzzA ii

i

)()()( 111

zAzkzAzA i

iiii

FIR LatticeFIR Lattice)(~)()( 1

11 zAzkzAzA iiii

)(~)()(~1

11 zAzzAkzA iiii

1)(~)( 00 zAzA

i

m

mimi zazA

1

)(1)(

i = n: Assumed true)(~)()( 1

11 zAzkzAzA nnnn

)(~)()(~ 111 zAzzAkzA nnnn

)()( 1)1(

1

zAzzAk nn

nn

)()( 1)1(1

zAzkzA n

nnn

Show that

)()(~ 1 zAzzA ii

i

)()()( 111

zAzkzAzA i

iiii

)()()( 11

111 zAzkzAzA n

nnnn

)( 11

)1(

zAz nn

i = n+1 also true.Prove

FIR LatticeFIR Lattice

1)(~)( 00 zAzA

i

m

mimi zazA

1

)(1)(

)()(~ 1 zAzzA ii

i

)()()( 111

zAzkzAzA i

iiii

1

1

)1(1 1)(

i

m

mimi zazA

1

1

)1(

1

1

)1(11

)(

i

m

mimii

ii

i

m

miimi

iii

ii

zakzk

zakzkzAzk=

ii

i ka )(

)1()1()(

imii

im

im akaa im


N

m

mm zazA

1

1)(

1)(~)( 00 zAzA

i

m

mimi zazA

1

)(1)(

)()( zAzA N

)( Nmm aa N

NNN kaa )(

ii

i ka )(

)1()1()(

imii

im

im akaa im

)(nx )(0 ne )(1 ne )(2 ne )(neN

)(~0 ne )(~

1 ne )(~2 ne )(~ neN

1k1k

2k2k

NkNk

)(ny

1z 1z 1z


N

m

mm zazA

1

1)()()( zAzA N

)( Nmm aa N

NNN kaa )(

m=0 1 1 1 1 1 1 1

k1

k2

k3

k4

k5

k6

)0(ma )1(

ma )2(ma )3(

ma )4(ma )5(

ma )6(ma

m=1m=2

m=3m=4

m=5m=6

)2(1a )3(

1a)3(

2a

)4(1a

)4(2a

)4(3a

)5(1a

)5(2a

)5(3a

)5(4a

)6(1a

)6(2a

)6(3a

)6(4a

)6(5a

Given the lattice, to find A(z).

ii

i ka )(

)1()1()(

imii

im

im akaa im


N

m

mm zazA

1

1)()()( zAzA N

)( Nmm aa N

NNN kaa )(

Given A(z), to find the lattice.

)(iii ak

)1()()1(

imii

im

im akaa

ii

i ka )(

)1()1()(

imii

im

im akaa im

m=0 1 1 1 1 1 1 1

)0(ma )1(

ma )2(ma )3(

ma )4(ma )5(

ma )6(ma

m=1m=2

m=3m=4

m=5m=6

)1()()1(

i

miimi

imi akaa

)( )1()()()1(

imi

imii

im

im akakaa

2

)()()1(

1 i

imii

imi

m kakaa

)6(1a

)6(2a

)6(3a

)6(4a

)6(5a

)6(6a

m=0 1 1 1 1 1 1 1

)0(ma )1(

ma )2(ma )3(

ma )4(ma )5(

ma )6(ma

m=1m=2

m=3m=4

m=5m=6

)6(1a

)6(2a

)6(3a

)6(4a

)6(5a

)6(6a


N

m

mm zazA

1

1)()()( zAzA N

)( Nmm aa N

NNN kaa )(

Given A(z), to find the lattice.

)(iii ak

)1()()1(

imii

im

im akaa

)5(1a

)5(2a

)5(3a

)5(4a

)5(5a

)1()()1(

i

miimi

imi akaa

)( )1()()()1(

imi

imii

im

im akakaa

2

)()()1(

1 i

imii

imi

m kakaa

)4(1a

)4(2a

)4(3a

)4(4a

)3(1a

)3(2a

)3(3a

)2(1a

)2(2a

)1(1a

Example)9.01)(8.01)(8.01()( 111 zjzjzzA

321 576.064.09.01 zzz

)(iii ak

2

)()()1(

1 i

imii

imi

m kakaa

1 1 1 1m=0m=1

m=2m=3

)0(ma )1(

ma )2(ma )3(

ma

0.6728 0.79520.1820

0.90.640.576

Example)(nx )(ny

1z 1z 1z

0.576

0.5760.1820

0.1820 0.6728

0.6728

1 1 1 1m=0m=1

m=2m=3

)0(ma )1(

ma )2(ma )3(

ma

0.6728 0.79520.1820

0.90.640.576

Inverse Filter

)()(~)( 00 nxnene



)()( neny N

)(1 nei )(nei

)(~1 nei )(~

2 neik

ik

1z

)(nx )(0 ne )(1 ne )(2 ne )(neN

)(~0 ne )(~

1 ne )(~2 ne )(~ neN

1k1k

2k2k

NkNk

)(ny

1z 1z 1z

)(ny )(0 ne )(1 ne )(2 ne )(neN

)(~0 ne )(~

1 ne )(~2 ne )(~ neN

1k1k

2k2k

NkNk

)(nx

1z 1z 1z

)(nx )(0 ne )(1 ne )(2 ne )(neN

)(~0 ne )(~

1 ne )(~2 ne )(~ neN

1k1k

2k2k

NkNk

)(ny

1z 1z 1z

All-Pole Filter

N

m

mm zazA

zXzY

1

1)()()(

)()(~)( 00 nxnene



)()( neny N

)(1 nei )(nei

)(~1 nei )(~

2 neik

ik

1z

)()( nxneN



)()(~)( 00 nynene

N

m

mm zazA

zYzX

1

1)()()(

N

m

mm za

zAzXzY

1

1

1

1)()()(

All-Pole Filter

)(1 nei )(nei

)(~1 nei )(~

2 neik

ik

1z

)(1 nei )(nei

)(~1 nei )(~ nei

ik

1zik

)(nei )(1 nei

)(~ nei )(~1 nei

ik

1zik

)(ny )(0 ne )(1 ne )(2 ne )(neN

)(~0 ne )(~

1 ne )(~2 ne )(~ neN

1k1k

2k2k

NkNk

)(nx

1z 1z 1z)()( nxneN



)()(~)( 00 nynene

N

m

mm zazA

zYzX

1

1)()()(

N

m

mm za

zAzXzY

1

1

1

1)()()(

All-Pole Filter

)(nei )(1 nei

)(~ nei )(~1 nei

ik

1zik

)(ny )(0 ne )(1 ne )(2 ne )(neN

)(~0 ne )(~

1 ne )(~2 ne )(~ neN

1k1k

2k2k

NkNk

)(nx

1z 1z 1z)()( nxneN



)()(~)( 00 nynene

N

m

mm zazA

zYzX

1

1)()()(

N

m

mm za

zAzXzY

1

1

1

1)()()(

)()( nenx N )(1 neN

)(~ neN )(~1 neN )(~

0 ne

NkNk

1 Nk1Nk

1k1k

)(ny

1z 1z

)(0 ne)(2 neN

)(~1 ne

1z

All-Pole Filter

)(nei )(1 nei

)(~ nei )(~1 nei

ik

1zik

)(1 zA

)(nx )(0 ne )(1 ne )(2 ne )(neN

)(~0 ne )(~

1 ne )(~2 ne )(~ neN

1k1k

2k2k

NkNk

)(ny

1z 1z 1z)(zA

)()( nenx N )(1 neN

)(~ neN )(~1 neN )(~

0 ne

NkNk

1 Nk1Nk

1k1k

)(ny

1z 1z

)(0 ne)(2 neN

)(~1 ne

1z

Example)9.01)(8.01)(8.01()( 111 zjzjzzA

321 576.064.09.01 zzz

321 576.064.09.011)(

zzz

zA

)(nx )(ny

1z 1z1z

0.6728

0.6728

0.1820

0.1820

0.576

0.576

)(nx )(ny

1z 1z 1z

0.576

0.5760.1820

0.1820 0.6728

0.6728

Example

321 576.064.09.011)(

zzz

zA

)(nx )(ny

1z 1z1z

0.6728

0.6728

0.1820

0.1820

0.576

0.576

)(nx

0.91z

1z

1z

)(ny

0.64

0.576

)9.01)(8.01)(8.01()( 111 zjzjzzA321 576.064.09.01 zzz

Stability of All-Pole Filter

N

m

mm zazA

zH

1

1

1)(

1)( All zeros of A(z) have to lie within the unit circle.

Necessary and sufficient conditions:

All of k-parameters ki’s satisfy |ki| < 1.

Normalized Lattice

)(nx )(0 ne )(1 ne )(2 ne )(neN

)(~0 ne )(~

1 ne )(~2 ne )(~ neN

1k1k

2k2k

NkNk

)(ny

1z 1z 1z

)(nx )(0 ne )(1 ne )(2 ne )(neN

)(~0 ne )(~

1 ne )(~2 ne )(~ neN

1k1k

2k2k

NkNk

)(ny

1z 1z 1z

N

m

mm zazA

zXzY

1

1)()()(

)()(~)( 00 nxnene



)()( neny N

)()( nxneN



)()(~)( 00 nynene

N

m

mm zazA

zYzX

1

1)()()(

N

m

mm za

zAzXzY

1

1

1

1)()()(

)(nx )(0 ne )(1 ne )(2 ne )(neN

)(~0 ne )(~

1 ne )(~2 ne )(~ neN

1k1k

2k2k

NkNk

)(ny

1z 1z 1z

)(nx )(0 ne )(1 ne )(2 ne )(neN

)(~0 ne )(~

1 ne )(~2 ne )(~ neN

1k1k

2k2k

NkNk

)(ny

1z 1z 1z

N

m

mm zazA

zXzY

1

1)()()(

)()(~)( 00 nxnene



)()( neny N

)()( nxneN



)()(~)( 00 nynene

N

m

mm zazA

zYzX

1

1)()()(

N

m

mm za

zAzXzY

1

1

1

1)()()(

Normalized Lattice)(1 nei )(nei

)(~1 nei )(~

2 neik

ik

1z

)(nei )(1 nei

)(~ nei )(~1 nei

ik

1zik

Normalized Lattice

)(nx )(0 ne )(1 ne )(2 ne )(neN

)(~0 ne )(~

1 ne )(~2 ne )(~ neN

1k1k

2k2k

NkNk

)(ny

1z 1z 1z)()( nxneN



)()(~)( 00 nynene

N

m

mm zazA

zYzX

1

1)()()(

N

m

mm za

zAzXzY

1

1

1

1)()()(

)1(~)1()()(~1

2 neknekne iiiii


)(~)()( 11

1 zEzkzEzE iiii

)(~)1()()(~1

12 zEzknEkzE iiiii Section

i

)(1 nei)(nei

)1(~1 nei)(~ nei

Normalized Lattice

SectionN

)()( nenx N

)(~ neN

SectionN1

Section1

)(0 ne

)(~0 ne

)(ny

Sectioni

)(1 nei)(nei

)1(~1 nei)(~ nei

)(~)()( 11

1 zEzkzEzE iiii

)(~)1()()(~1

12 zEzknEkzE iiiii

)1(~)1()()(~1

2 neknekne iiiii


Normalized Lattice

)(1 nei)(nei

)(~1 nei)(~ nei 1z

ikik

21 ik

Sectioni

)(1 nei)(nei

)1(~1 nei)(~ nei

)(~)()( 11

1 zEzkzEzE iiii

)(~)1()()(~1

12 zEzknEkzE iiiii

)1(~)1()()(~1

2 neknekne iiiii


Three-Multiplier Form

Normalized Lattice

Four-Multiplier, Normalized Form

)(1 nei)(nei

)(~1 nei)(~ nei 1z

ikik

ik1

ik1

Four-Multiplier, Kelly-Lochbaum Form

)(1 nei)(nei

)(~1 nei)(~ nei 1z

isini sin

icos

icos

1|| ik ii ksin21cos ii k

)(1 nei)(nei

)(~1 nei)(~ nei 1z

ikik

21 ik


Normalized Lattice

SectionN

)()( nenx N

)(~ neN

SectionN1

Section1

)(0 ne

)(~0 ne

)(ny

N

m

mm zazA

zH

1

1

1)(

1)()(1 nei)(nei

)(~1 nei)(~ nei 1z

ikik

21 ik


Normalized Lattice

SectionN

)()( nenx N

)(~ neN

SectionN1

Section1

)(0 ne

)(~0 ne

)(ny

N

m

mm

N

ii

N

ii

za

k

zA

kzH

1

11

1

)1(

)(

)1()(

Four-Multiplier, Normalized Form

)(1 nei)(nei

)(~1 nei)(~ nei 1z

ikik

ik1

ik1

Normalized Lattice

SectionN

)()( nenx N

)(~ neN

SectionN1

Section1

)(0 ne

)(~0 ne

)(ny

N

m

mm

N

ii

N

ii

zazAzH

1

11

1

cos

)(

cos)(

Four-Multiplier, Kelly-Lochbaum Form

)(1 nei)(nei

)(~1 nei)(~ nei 1z

isini sin

icos

icos

Lattice Systems with Poles and Zeros

)()( nenx N

)(~ neN

)(0 ne

)(~0 ne

)(ny

SectionN1

Section1

SectionN

c0c1cN2cN1cN

)(~1 neN )(~

2 neN )(~1 ne

)(1 neN )(2 neN )(1 ne

N

iii zEczY

0

)(~)(

N

i

ii zX

zEczXzYzH

0 )()(~

)()()(

)()(~)(~)(~

1

0

zAzzAzEzE

ii

ii

)(~)()(~)(~0

1 zEzAzzAzE ii

ii

)(~)()( 0 zEzAzX


)()( nenx N

)(~ neN

)(0 ne

)(~0 ne

)(ny

SectionN1

Section1

SectionN

c0c1cN2cN1cN

)(~1 neN )(~

2 neN )(~1 ne

)(1 neN )(2 neN )(1 ne

N

iii zEczY

0

)(~)(

N

i

ii zX

zEczXzYzH

0 )()(~

)()()(

)()(~)(~)(~

1

0

zAzzAzEzE

ii

ii

)(~)()(~)(~0

1 zEzAzzAzE ii

ii

)(~)()( 0 zEzAzX

N

i

ii

i

zAzAzczH

0

1

)()()(


)()(

zAzB

)(

0

zA

zbN

m

mm

1

0

)(

1

)(1)(i

k

kiki

ii

k

kiik

ii

i zazzazzAz

N

i

i

k

kikii

ii

N

m

mm zaczczb

0

1

0

)(

0

N

mi

imiimm accb

1

)(

N

i

ii

i

zAzAzczH

0

1

)()()(

N

mi

imiimm acbc

1

)(

Example

321

321

576.064.09.01331)(

zzz

zzzzH

N

mi

imiimm acbc

1

)(

)(nx

)(ny

1z 1z1z

0.6728

0.6728

0.1820

0.1820

0.576

0.576

c3 c2 c1 c0

321

321

576.064.09.01331)(

zzz

zzzzH

N

mi

imiimm acbc

1

)(

)(nx

)(ny

1z 1z1z

0.6728

0.6728

0.1820

0.1820

0.576

0.576

c3 c2 c1 c0

Example1 1 1 1m=0

m=1m=2

m=3

)0(ma )1(

ma )2(ma )3(

ma

0.6728 0.79520.1820

0.90.640.576

133 bc9.3)3(

1322 acbc

4612.5)3(23

)2(1211 acacbc

5404.4)3(33

)2(22

)1(1100 acacacbc

)(nx

)(ny

1z 1z1z

0.6728

0.6728

0.1820

0.1820

0.576

0.576

1 3.9 5.4612 4.5404

Example

321

321

576.064.09.01331)(

zzz

zzzzH

133 bc9.3)3(

1322 acbc

4612.5)3(23

)2(1211 acacbc

5404.4)3(33

)2(22

)1(1100 acacacbc

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