structures for discrete-time systems 主講人:虞台文. content introduction block diagram...
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Structures for Discrete-Time Systems IntroductionTRANSCRIPT
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Structures forDiscrete-Time Systems
主講人:虞台文
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Content Introduction Block Diagram Representation Signal Flow Graph Basic Structure for IIR Systems Transposed Forms Basic Structure for FIR Systems Lattice Structures
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Structures for Discrete-Time Systems
Introduction
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Characterize an LTI System
Impulse Responsez-TransformDifference Equation
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Example|||| ,
1)( 1
110 az
azzbbzH
)1()()( 10 nuabnuabnh nn
)1()()1()( 10 nxbnxbnayny
)1()()1()( 10 nxbnxbnayny Computable
Noncomputable
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Basic Operations
)1()()1()( 10 nxbnxbnayny Computable
AdditionMultiplicationDelay
In fact, there are unlimited variety of computational structures.
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Why Implement Using Different Structures?
Finite-precision number representation of a digital computer.
Truncation or rounding error.
Modeling methods:– Block Diagram– Signal Flow Graph
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Block Diagram Representation
+x1(n)
x2(n)
x1(n) + x2(n)Adder
x(n)a ax(n)Multiplier
x(n) x(n1)z1Unit Delay
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Example)()2()1()( 21 nbxnyanyany
x(n) +
+
b
a1
z1
z1
a2
y(n)
y(n1)
y(n2)
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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
)()()(
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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)
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Block Diagram Representation(Direct Form I)
+
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
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Block Diagram Representation(Direct Form I)
+
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)(
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Block Diagram Representation(Direct Form I)
+
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)()()(
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Block Diagram Representation(Direct Form I)
+
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?
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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
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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
)()()(1
nxknwanwN
kk
)()()(0
nwknxbnyM
kk
M
k
kk zbzH
01 )(
N
k
kk za
zH
1
2
1
1)(
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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
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
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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
How many Adders?How many multipliers?How many delays?
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Block Diagram Representation (Canonic Direct Form)
+
+
+
b0
b1
bN1
bN
x(n)+
z1
z1
+
z1
+
a1
aN1
aN
y(n)
AssumeM = N
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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)
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Structures for Discrete-Time Systems
Signal Flow Graph
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Nodes And Branches
wj(n)wk(n)
Associated with each node is a variable or node value.
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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.
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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.
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Source NodesNodes without entering branches
xj(n) wk(n)
Source node j
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Sink Nodes Nodes that have only entering branches
yk(n)wj(n)
Sink node k
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Example
x(n) y(n)w1(n)w2(n)
a
b
c
d
e
SourceNode
SinkNode
)()()()( 221 nbwnawnxnw
)()( 12 ncwnw
)()()( 2 newndxny
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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)
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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)
)()()( 41 nawnxnw )()( 12 nwnw
)()()( 41203 nwbnwbnw )1()( 24 nwnw
)()( 3 nwny
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Block Diagram vs. Signal Flow Graph
)()()( 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
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Structures for Discrete-Time Systems
Basic Structure for IIR Systems
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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.
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Basic StructuresDirect FormsCascade FormParallel Form
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Direct Forms
M
kk
N
kk knxbknyany
01
)()()(
N
k
kk
M
k
kk
za
zbzH
1
1
1)(
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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)
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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)
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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
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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
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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
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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
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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)(
![Page 43: Structures for Discrete-Time Systems 主講人:虞台文. Content Introduction Block Diagram Representation Signal Flow Graph Basic Structure for IIR Systems Transposed](https://reader035.vdocuments.site/reader035/viewer/2022062219/5a4d1b687f8b9ab0599b1868/html5/thumbnails/43.jpg)
Cascade Form
2nd OrderSystem
2nd OrderSystem
2nd OrderSystem
sN
k kk
kkk
zazazbzbbzH
12
21
1
22
110
1)(
![Page 44: Structures for Discrete-Time Systems 主講人:虞台文. Content Introduction Block Diagram Representation Signal Flow Graph Basic Structure for IIR Systems Transposed](https://reader035.vdocuments.site/reader035/viewer/2022062219/5a4d1b687f8b9ab0599b1868/html5/thumbnails/44.jpg)
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
![Page 45: Structures for Discrete-Time Systems 主講人:虞台文. Content Introduction Block Diagram Representation Signal Flow Graph Basic Structure for IIR Systems Transposed](https://reader035.vdocuments.site/reader035/viewer/2022062219/5a4d1b687f8b9ab0599b1868/html5/thumbnails/45.jpg)
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)(
![Page 46: Structures for Discrete-Time Systems 主講人:虞台文. Content Introduction Block Diagram Representation Signal Flow Graph Basic Structure for IIR Systems Transposed](https://reader035.vdocuments.site/reader035/viewer/2022062219/5a4d1b687f8b9ab0599b1868/html5/thumbnails/46.jpg)
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
![Page 47: Structures for Discrete-Time Systems 主講人:虞台文. Content Introduction Block Diagram Representation Signal Flow Graph Basic Structure for IIR Systems Transposed](https://reader035.vdocuments.site/reader035/viewer/2022062219/5a4d1b687f8b9ab0599b1868/html5/thumbnails/47.jpg)
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
![Page 48: Structures for Discrete-Time Systems 主講人:虞台文. Content Introduction Block Diagram Representation Signal Flow Graph Basic Structure for IIR Systems Transposed](https://reader035.vdocuments.site/reader035/viewer/2022062219/5a4d1b687f8b9ab0599b1868/html5/thumbnails/48.jpg)
Parallel Form
sP N
k kk
kkN
k
kk zaza
zeezCzH1
22
11
110
0 1)(
z1
z1
a1k
a2k
e0k
e1k
![Page 49: Structures for Discrete-Time Systems 主講人:虞台文. Content Introduction Block Diagram Representation Signal Flow Graph Basic Structure for IIR Systems Transposed](https://reader035.vdocuments.site/reader035/viewer/2022062219/5a4d1b687f8b9ab0599b1868/html5/thumbnails/49.jpg)
Parallel Form
x(n) y(n)
sP N
k kk
kkN
k
kk zaza
zeezCzH1
22
11
110
0 1)(
![Page 50: Structures for Discrete-Time Systems 主講人:虞台文. Content Introduction Block Diagram Representation Signal Flow Graph Basic Structure for IIR Systems Transposed](https://reader035.vdocuments.site/reader035/viewer/2022062219/5a4d1b687f8b9ab0599b1868/html5/thumbnails/50.jpg)
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
![Page 51: Structures for Discrete-Time Systems 主講人:虞台文. Content Introduction Block Diagram Representation Signal Flow Graph Basic Structure for IIR Systems Transposed](https://reader035.vdocuments.site/reader035/viewer/2022062219/5a4d1b687f8b9ab0599b1868/html5/thumbnails/51.jpg)
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
![Page 52: Structures for Discrete-Time Systems 主講人:虞台文. Content Introduction Block Diagram Representation Signal Flow Graph Basic Structure for IIR Systems Transposed](https://reader035.vdocuments.site/reader035/viewer/2022062219/5a4d1b687f8b9ab0599b1868/html5/thumbnails/52.jpg)
Structures for Discrete-Time Systems
Transposed Forms
![Page 53: Structures for Discrete-Time Systems 主講人:虞台文. Content Introduction Block Diagram Representation Signal Flow Graph Basic Structure for IIR Systems Transposed](https://reader035.vdocuments.site/reader035/viewer/2022062219/5a4d1b687f8b9ab0599b1868/html5/thumbnails/53.jpg)
Signal Flow Graph Transformation
To transform signal graphs into different forms while leaving the overall system function between input and output unchanged.
![Page 54: Structures for Discrete-Time Systems 主講人:虞台文. Content Introduction Block Diagram Representation Signal Flow Graph Basic Structure for IIR Systems Transposed](https://reader035.vdocuments.site/reader035/viewer/2022062219/5a4d1b687f8b9ab0599b1868/html5/thumbnails/54.jpg)
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
![Page 55: Structures for Discrete-Time Systems 主講人:虞台文. Content Introduction Block Diagram Representation Signal Flow Graph Basic Structure for IIR Systems Transposed](https://reader035.vdocuments.site/reader035/viewer/2022062219/5a4d1b687f8b9ab0599b1868/html5/thumbnails/55.jpg)
Transposition of Signal Flow Graph
x(n) y(n)z1
a
x(n)y(n) z1
a
Are there any relations between the two systems?
![Page 56: Structures for Discrete-Time Systems 主講人:虞台文. Content Introduction Block Diagram Representation Signal Flow Graph Basic Structure for IIR Systems Transposed](https://reader035.vdocuments.site/reader035/viewer/2022062219/5a4d1b687f8b9ab0599b1868/html5/thumbnails/56.jpg)
Example:
111)(
az
zHz1
a
x(n) y(n)
z1
a
x(n)y(n)
z1
a
x(n) y(n)
![Page 57: Structures for Discrete-Time Systems 主講人:虞台文. Content Introduction Block Diagram Representation Signal Flow Graph Basic Structure for IIR Systems Transposed](https://reader035.vdocuments.site/reader035/viewer/2022062219/5a4d1b687f8b9ab0599b1868/html5/thumbnails/57.jpg)
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
Detail proof see reference
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Structures for Discrete-Time Systems
Basic Structure for FIR Systems
![Page 59: Structures for Discrete-Time Systems 主講人:虞台文. Content Introduction Block Diagram Representation Signal Flow Graph Basic Structure for IIR Systems Transposed](https://reader035.vdocuments.site/reader035/viewer/2022062219/5a4d1b687f8b9ab0599b1868/html5/thumbnails/59.jpg)
FIRFor causal FIR systems, the system
function has only zeros.
M
kk knxbny
0
)()(
M
k
knxkhny0
)()()(
othrewise
Mnbnh n
0,,1,0
)(
![Page 60: Structures for Discrete-Time Systems 主講人:虞台文. Content Introduction Block Diagram Representation Signal Flow Graph Basic Structure for IIR Systems Transposed](https://reader035.vdocuments.site/reader035/viewer/2022062219/5a4d1b687f8b9ab0599b1868/html5/thumbnails/60.jpg)
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)
![Page 61: Structures for Discrete-Time Systems 主講人:虞台文. Content Introduction Block Diagram Representation Signal Flow Graph Basic Structure for IIR Systems Transposed](https://reader035.vdocuments.site/reader035/viewer/2022062219/5a4d1b687f8b9ab0599b1868/html5/thumbnails/61.jpg)
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)
![Page 62: Structures for Discrete-Time Systems 主講人:虞台文. Content Introduction Block Diagram Representation Signal Flow Graph Basic Structure for IIR Systems Transposed](https://reader035.vdocuments.site/reader035/viewer/2022062219/5a4d1b687f8b9ab0599b1868/html5/thumbnails/62.jpg)
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)
![Page 63: Structures for Discrete-Time Systems 主講人:虞台文. Content Introduction Block Diagram Representation Signal Flow Graph Basic Structure for IIR Systems Transposed](https://reader035.vdocuments.site/reader035/viewer/2022062219/5a4d1b687f8b9ab0599b1868/html5/thumbnails/63.jpg)
Cascade Form
M
kk knxbny
0
)()(
M
n
nznhzH0
)()(
sM
kkkk zbzbbzH
1
22
110 )()(
![Page 64: Structures for Discrete-Time Systems 主講人:虞台文. Content Introduction Block Diagram Representation Signal Flow Graph Basic Structure for IIR Systems Transposed](https://reader035.vdocuments.site/reader035/viewer/2022062219/5a4d1b687f8b9ab0599b1868/html5/thumbnails/64.jpg)
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
![Page 65: Structures for Discrete-Time Systems 主講人:虞台文. Content Introduction Block Diagram Representation Signal Flow Graph Basic Structure for IIR Systems Transposed](https://reader035.vdocuments.site/reader035/viewer/2022062219/5a4d1b687f8b9ab0599b1868/html5/thumbnails/65.jpg)
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
![Page 66: Structures for Discrete-Time Systems 主講人:虞台文. Content Introduction Block Diagram Representation Signal Flow Graph Basic Structure for IIR Systems Transposed](https://reader035.vdocuments.site/reader035/viewer/2022062219/5a4d1b687f8b9ab0599b1868/html5/thumbnails/66.jpg)
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
![Page 67: Structures for Discrete-Time Systems 主講人:虞台文. Content Introduction Block Diagram Representation Signal Flow Graph Basic Structure for IIR Systems Transposed](https://reader035.vdocuments.site/reader035/viewer/2022062219/5a4d1b687f8b9ab0599b1868/html5/thumbnails/67.jpg)
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)
![Page 68: Structures for Discrete-Time Systems 主講人:虞台文. Content Introduction Block Diagram Representation Signal Flow Graph Basic Structure for IIR Systems Transposed](https://reader035.vdocuments.site/reader035/viewer/2022062219/5a4d1b687f8b9ab0599b1868/html5/thumbnails/68.jpg)
Type II, III and VI
Construct them in a
similar manner by
yourselves.
![Page 69: Structures for Discrete-Time Systems 主講人:虞台文. Content Introduction Block Diagram Representation Signal Flow Graph Basic Structure for IIR Systems Transposed](https://reader035.vdocuments.site/reader035/viewer/2022062219/5a4d1b687f8b9ab0599b1868/html5/thumbnails/69.jpg)
Structures for Discrete-Time Systems
Lattice Structures
![Page 70: Structures for Discrete-Time Systems 主講人:虞台文. Content Introduction Block Diagram Representation Signal Flow Graph Basic Structure for IIR Systems Transposed](https://reader035.vdocuments.site/reader035/viewer/2022062219/5a4d1b687f8b9ab0599b1868/html5/thumbnails/70.jpg)
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
![Page 71: Structures for Discrete-Time Systems 主講人:虞台文. Content Introduction Block Diagram Representation Signal Flow Graph Basic Structure for IIR Systems Transposed](https://reader035.vdocuments.site/reader035/viewer/2022062219/5a4d1b687f8b9ab0599b1868/html5/thumbnails/71.jpg)
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)()(
![Page 72: Structures for Discrete-Time Systems 主講人:虞台文. Content Introduction Block Diagram Representation Signal Flow Graph Basic Structure for IIR Systems Transposed](https://reader035.vdocuments.site/reader035/viewer/2022062219/5a4d1b687f8b9ab0599b1868/html5/thumbnails/72.jpg)
FIR Lattice
)()(~)( 00 zXzEzE
)(~)()( 11
1 zEzkzEzE iiii
)(~)()(~1
11 zEzzEkzE iiii
)()( zEzY N
Consider x(n)=(n), one will see
)()(~)( 00 nxnene
)1(~)()( 11 neknene iiii
)1(~)()(~11 nenekne iiii
)()( 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)()(
![Page 73: Structures for Discrete-Time Systems 主講人:虞台文. Content Introduction Block Diagram Representation Signal Flow Graph Basic Structure for IIR Systems Transposed](https://reader035.vdocuments.site/reader035/viewer/2022062219/5a4d1b687f8b9ab0599b1868/html5/thumbnails/73.jpg)
FIR LatticeDefine
)()()(
0 zEzEzA i
i
)(~)(~
)(~
0 zEzEzA i
i
1)(~)( 00 zAzA
i
m
mimi zazA
1
)(1)(
Consider x(n)=(n), one will see
)(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)()(
![Page 74: Structures for Discrete-Time Systems 主講人:虞台文. Content Introduction Block Diagram Representation Signal Flow Graph Basic Structure for IIR Systems Transposed](https://reader035.vdocuments.site/reader035/viewer/2022062219/5a4d1b687f8b9ab0599b1868/html5/thumbnails/74.jpg)
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
![Page 75: Structures for Discrete-Time Systems 主講人:虞台文. Content Introduction Block Diagram Representation Signal Flow Graph Basic Structure for IIR Systems Transposed](https://reader035.vdocuments.site/reader035/viewer/2022062219/5a4d1b687f8b9ab0599b1868/html5/thumbnails/75.jpg)
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
![Page 76: Structures for Discrete-Time Systems 主講人:虞台文. Content Introduction Block Diagram Representation Signal Flow Graph Basic Structure for IIR Systems Transposed](https://reader035.vdocuments.site/reader035/viewer/2022062219/5a4d1b687f8b9ab0599b1868/html5/thumbnails/76.jpg)
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
![Page 77: Structures for Discrete-Time Systems 主講人:虞台文. Content Introduction Block Diagram Representation Signal Flow Graph Basic Structure for IIR Systems Transposed](https://reader035.vdocuments.site/reader035/viewer/2022062219/5a4d1b687f8b9ab0599b1868/html5/thumbnails/77.jpg)
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
![Page 78: Structures for Discrete-Time Systems 主講人:虞台文. Content Introduction Block Diagram Representation Signal Flow Graph Basic Structure for IIR Systems Transposed](https://reader035.vdocuments.site/reader035/viewer/2022062219/5a4d1b687f8b9ab0599b1868/html5/thumbnails/78.jpg)
FIR LatticeFIR Lattice
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
![Page 79: Structures for Discrete-Time Systems 主講人:虞台文. Content Introduction Block Diagram Representation Signal Flow Graph Basic Structure for IIR Systems Transposed](https://reader035.vdocuments.site/reader035/viewer/2022062219/5a4d1b687f8b9ab0599b1868/html5/thumbnails/79.jpg)
FIR LatticeFIR Lattice
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
![Page 80: Structures for Discrete-Time Systems 主講人:虞台文. Content Introduction Block Diagram Representation Signal Flow Graph Basic Structure for IIR Systems Transposed](https://reader035.vdocuments.site/reader035/viewer/2022062219/5a4d1b687f8b9ab0599b1868/html5/thumbnails/80.jpg)
FIR LatticeFIR Lattice
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
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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
FIR LatticeFIR Lattice
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
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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
![Page 83: Structures for Discrete-Time Systems 主講人:虞台文. Content Introduction Block Diagram Representation Signal Flow Graph Basic Structure for IIR Systems Transposed](https://reader035.vdocuments.site/reader035/viewer/2022062219/5a4d1b687f8b9ab0599b1868/html5/thumbnails/83.jpg)
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
![Page 84: Structures for Discrete-Time Systems 主講人:虞台文. Content Introduction Block Diagram Representation Signal Flow Graph Basic Structure for IIR Systems Transposed](https://reader035.vdocuments.site/reader035/viewer/2022062219/5a4d1b687f8b9ab0599b1868/html5/thumbnails/84.jpg)
Inverse Filter
)()(~)( 00 nxnene
)1(~)()( 11 neknene iiii
)1(~)()(~11 nenekne iiii
)()( 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
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)(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
)1(~)()( 11 neknene iiii
)1(~)()(~11 nenekne iiii
)()( neny N
)(1 nei )(nei
)(~1 nei )(~
2 neik
ik
1z
)()( nxneN
)1(~)()( 11 neknene iiii
)1(~)()(~11 nenekne iiii
)()(~)( 00 nynene
N
m
mm zazA
zYzX
1
1)()()(
N
m
mm za
zAzXzY
1
1
1
1)()()(
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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
)1(~)()( 11 neknene iiii
)1(~)()(~11 nenekne iiii
)()(~)( 00 nynene
N
m
mm zazA
zYzX
1
1)()()(
N
m
mm za
zAzXzY
1
1
1
1)()()(
![Page 87: Structures for Discrete-Time Systems 主講人:虞台文. Content Introduction Block Diagram Representation Signal Flow Graph Basic Structure for IIR Systems Transposed](https://reader035.vdocuments.site/reader035/viewer/2022062219/5a4d1b687f8b9ab0599b1868/html5/thumbnails/87.jpg)
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
)1(~)()( 11 neknene iiii
)1(~)()(~11 nenekne iiii
)()(~)( 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
![Page 88: Structures for Discrete-Time Systems 主講人:虞台文. Content Introduction Block Diagram Representation Signal Flow Graph Basic Structure for IIR Systems Transposed](https://reader035.vdocuments.site/reader035/viewer/2022062219/5a4d1b687f8b9ab0599b1868/html5/thumbnails/88.jpg)
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
![Page 89: Structures for Discrete-Time Systems 主講人:虞台文. Content Introduction Block Diagram Representation Signal Flow Graph Basic Structure for IIR Systems Transposed](https://reader035.vdocuments.site/reader035/viewer/2022062219/5a4d1b687f8b9ab0599b1868/html5/thumbnails/89.jpg)
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
![Page 90: Structures for Discrete-Time Systems 主講人:虞台文. Content Introduction Block Diagram Representation Signal Flow Graph Basic Structure for IIR Systems Transposed](https://reader035.vdocuments.site/reader035/viewer/2022062219/5a4d1b687f8b9ab0599b1868/html5/thumbnails/90.jpg)
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
![Page 91: Structures for Discrete-Time Systems 主講人:虞台文. Content Introduction Block Diagram Representation Signal Flow Graph Basic Structure for IIR Systems Transposed](https://reader035.vdocuments.site/reader035/viewer/2022062219/5a4d1b687f8b9ab0599b1868/html5/thumbnails/91.jpg)
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.
![Page 92: Structures for Discrete-Time Systems 主講人:虞台文. Content Introduction Block Diagram Representation Signal Flow Graph Basic Structure for IIR Systems Transposed](https://reader035.vdocuments.site/reader035/viewer/2022062219/5a4d1b687f8b9ab0599b1868/html5/thumbnails/92.jpg)
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
)1(~)()( 11 neknene iiii
)1(~)()(~11 nenekne iiii
)()( neny N
)()( nxneN
)1(~)()( 11 neknene iiii
)1(~)()(~11 nenekne iiii
)()(~)( 00 nynene
N
m
mm zazA
zYzX
1
1)()()(
N
m
mm za
zAzXzY
1
1
1
1)()()(
![Page 93: Structures for Discrete-Time Systems 主講人:虞台文. Content Introduction Block Diagram Representation Signal Flow Graph Basic Structure for IIR Systems Transposed](https://reader035.vdocuments.site/reader035/viewer/2022062219/5a4d1b687f8b9ab0599b1868/html5/thumbnails/93.jpg)
)(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
)1(~)()( 11 neknene iiii
)1(~)()(~11 nenekne iiii
)()( neny N
)()( nxneN
)1(~)()( 11 neknene iiii
)1(~)()(~11 nenekne iiii
)()(~)( 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
![Page 94: Structures for Discrete-Time Systems 主講人:虞台文. Content Introduction Block Diagram Representation Signal Flow Graph Basic Structure for IIR Systems Transposed](https://reader035.vdocuments.site/reader035/viewer/2022062219/5a4d1b687f8b9ab0599b1868/html5/thumbnails/94.jpg)
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
)1(~)()( 11 neknene iiii
)1(~)()(~11 nenekne iiii
)()(~)( 00 nynene
N
m
mm zazA
zYzX
1
1)()()(
N
m
mm za
zAzXzY
1
1
1
1)()()(
)1(~)1()()(~1
2 neknekne iiiii
)1(~)()( 11 neknene iiii
)(~)()( 11
1 zEzkzEzE iiii
)(~)1()()(~1
12 zEzknEkzE iiiii Section
i
)(1 nei)(nei
)1(~1 nei)(~ nei
![Page 95: Structures for Discrete-Time Systems 主講人:虞台文. Content Introduction Block Diagram Representation Signal Flow Graph Basic Structure for IIR Systems Transposed](https://reader035.vdocuments.site/reader035/viewer/2022062219/5a4d1b687f8b9ab0599b1868/html5/thumbnails/95.jpg)
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
)1(~)()( 11 neknene iiii
![Page 96: Structures for Discrete-Time Systems 主講人:虞台文. Content Introduction Block Diagram Representation Signal Flow Graph Basic Structure for IIR Systems Transposed](https://reader035.vdocuments.site/reader035/viewer/2022062219/5a4d1b687f8b9ab0599b1868/html5/thumbnails/96.jpg)
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
)1(~)()( 11 neknene iiii
Three-Multiplier Form
![Page 97: Structures for Discrete-Time Systems 主講人:虞台文. Content Introduction Block Diagram Representation Signal Flow Graph Basic Structure for IIR Systems Transposed](https://reader035.vdocuments.site/reader035/viewer/2022062219/5a4d1b687f8b9ab0599b1868/html5/thumbnails/97.jpg)
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
Three-Multiplier Form
![Page 98: Structures for Discrete-Time Systems 主講人:虞台文. Content Introduction Block Diagram Representation Signal Flow Graph Basic Structure for IIR Systems Transposed](https://reader035.vdocuments.site/reader035/viewer/2022062219/5a4d1b687f8b9ab0599b1868/html5/thumbnails/98.jpg)
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
Three-Multiplier Form
![Page 99: Structures for Discrete-Time Systems 主講人:虞台文. Content Introduction Block Diagram Representation Signal Flow Graph Basic Structure for IIR Systems Transposed](https://reader035.vdocuments.site/reader035/viewer/2022062219/5a4d1b687f8b9ab0599b1868/html5/thumbnails/99.jpg)
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
![Page 100: Structures for Discrete-Time Systems 主講人:虞台文. Content Introduction Block Diagram Representation Signal Flow Graph Basic Structure for IIR Systems Transposed](https://reader035.vdocuments.site/reader035/viewer/2022062219/5a4d1b687f8b9ab0599b1868/html5/thumbnails/100.jpg)
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
![Page 101: Structures for Discrete-Time Systems 主講人:虞台文. Content Introduction Block Diagram Representation Signal Flow Graph Basic Structure for IIR Systems Transposed](https://reader035.vdocuments.site/reader035/viewer/2022062219/5a4d1b687f8b9ab0599b1868/html5/thumbnails/101.jpg)
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
![Page 102: Structures for Discrete-Time Systems 主講人:虞台文. Content Introduction Block Diagram Representation Signal Flow Graph Basic Structure for IIR Systems Transposed](https://reader035.vdocuments.site/reader035/viewer/2022062219/5a4d1b687f8b9ab0599b1868/html5/thumbnails/102.jpg)
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
N
i
ii
i
zAzAzczH
0
1
)()()(
![Page 103: Structures for Discrete-Time Systems 主講人:虞台文. Content Introduction Block Diagram Representation Signal Flow Graph Basic Structure for IIR Systems Transposed](https://reader035.vdocuments.site/reader035/viewer/2022062219/5a4d1b687f8b9ab0599b1868/html5/thumbnails/103.jpg)
Lattice Systems with Poles and Zeros
)()(
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
)(
![Page 104: Structures for Discrete-Time Systems 主講人:虞台文. Content Introduction Block Diagram Representation Signal Flow Graph Basic Structure for IIR Systems Transposed](https://reader035.vdocuments.site/reader035/viewer/2022062219/5a4d1b687f8b9ab0599b1868/html5/thumbnails/104.jpg)
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
![Page 105: Structures for Discrete-Time Systems 主講人:虞台文. Content Introduction Block Diagram Representation Signal Flow Graph Basic Structure for IIR Systems Transposed](https://reader035.vdocuments.site/reader035/viewer/2022062219/5a4d1b687f8b9ab0599b1868/html5/thumbnails/105.jpg)
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
![Page 106: Structures for Discrete-Time Systems 主講人:虞台文. Content Introduction Block Diagram Representation Signal Flow Graph Basic Structure for IIR Systems Transposed](https://reader035.vdocuments.site/reader035/viewer/2022062219/5a4d1b687f8b9ab0599b1868/html5/thumbnails/106.jpg)
)(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