12015-10-151zhongguo liu_biomedical engineering_shandong univ. biomedical signal processing chapter...
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![Page 1: 12015-10-151Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 6 structures for discrete- time system Zhongguo Liu](https://reader030.vdocuments.mx/reader030/viewer/2022020106/56649ec15503460f94bccbb1/html5/thumbnails/1.jpg)
123/4/20 1Zhongguo Liu_Biomedical Engineering_Shandong U
niv.
Biomedical Signal processingChapter 6 structures for
discrete-time systemZhongguo Liu
Biomedical Engineering
School of Control Science and Engineering, Shandong University
山东省精品课程山东省精品课程《《生物医学信号处理生物医学信号处理 (( 双语双语 )) 》》http://course.sdu.edu.cn/bdsp.htmlhttp://course.sdu.edu.cn/bdsp.html
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6.0 Introduction
6.1 Block Diagram Representation of Linear Constant-Coefficient Difference Equations
6.2 Signal Flow Graph Representation of Linear Constant-Coefficient Difference Equations
6.3 Basic Structures for IIR Systems
6.4 Transposed Forms
6.5 Basic Network Structures for FIR Systems
§6 structures for discrete-time system
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Structures for Discrete-Time Systems
6.0 Introduction
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Characterization of an LTI System:
Impulse Responsez-Transform: system functionDifference Equationconverted to a algorithm or structure
that can be realized in the desired technology, when implemented with hardware. Structure consists of an interconnection of basic operations of addition, multiplication by a constant and delay
6.0 Introduction
→Frequency response
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even if we only wanted to compute the output over a finite interval, it would not be efficient to do so by discrete convolution since the amount of computation required to compute y[n] would grow with n .
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Example: find the output of the system
10 1
1( ) , | | | |,
1
b b zH z z a
az
10 1 1 n nh n b a u n b a u n
0
n
k k
y n x n h n x k h n k h k x n k
Illustration for the IIR case by convolution
IIR Impulse Response
with input x[n].Solution1:
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Example: find the output of the system
10 1
1( ) , | | | |,
1
( )
( )
b b zH z z a
az
Y z
X z
0 11 1y n ay n b x n b x n
0 11 1y n ay n b x n b x n 0 11 1y n ay n b x n b x n
computable recursivelyThe algorithm suggested by the equation is not the only computational algorithm, there are unlimited variety of computational structures (shown later).
with input x[n].
Solution2:
![Page 7: 12015-10-151Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 6 structures for discrete- time system Zhongguo Liu](https://reader030.vdocuments.mx/reader030/viewer/2022020106/56649ec15503460f94bccbb1/html5/thumbnails/7.jpg)
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Why Implement system Using Different Structures?
Equivalent structures with regard to their input-output characteristics for infinite-precision representation, may have vastly different behavior when numerical precision is limited.
Effects of finite-precision of coefficients and truncation or rounding of intermediate computations are considered in latter sections.
Computational structures(Modeling methods):Block DiagramSignal Flow Graph
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Structures for Discrete-Time Systems
6.1 Block Diagram Representation of Linear
Constant-Coefficient Difference Equations
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6.1 Block Diagram Representation of Linear Constant-Coefficient Difference
Equations
+x1[n]
x2[n]
x1[n] + x2[n]Adder
x[n]
aax[n]Multiplier
x[n]x[n-1]z1Unit Delay
(Memory, storage)
0 11 1 y n ay n b x n b x n
Three basic elements:
M sample Delay z-Mx[n-M]
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Ex. 6.1 draw Block Diagram Representation of a Second-order Difference Equation
01 2[ ] [ 1] [ 2] [ ] a ay n y n y n b x n01 2[ ] [ 1] [ 2] [ ] a ay n y n y n b x n
x[n] +
+
b0
a1
z1
z1
a2
y[n]
y[n-1]
y[n-2]
02
1 21
( )
( ) 1
bY z
X z z a za 0
21 2
1
( )
( ) 1
bY z
X z z a za
Solution:
( )H z ( )H z
01 2[ ] [ 1] [ 2] [ ]a ay n y n y n b x n 01 2[ ] [ 1] [ 2] [ ]a ay n y n y n b x n
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Nth-Order Difference Equations
1 0
[ ] [ ] [ ]N M
k kk k
y n a y n k b x n k
1 0
[ ] [ ] [ ]N M
k kk k
y n a y n k b x n k
N
k
kk
M
k
kk
za
zbzH
1
1
1)(
N
k
kk
M
k
kk
za
zbzH
1
1
1)(
0 0
[ ] [ ]
N M
k kk k
a y n k b x n k0 0
[ ] [ ]
N M
k kk k
a y n k b x n k
1 0
[ ] [ ] [ ]N M
k kk k
y n a y n k b x n k
1 0
[ ] [ ] [ ]N M
k kk k
y n a y n k b x n k
Form changed
to
a[0] normalized
to unity
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Block Diagram Representation (Direct Form I)
v[n]+
z1
z1
+
z1
+
b0
b1
bM1
bM
x[n]
x[n-1]
x[n-2]
x[n-M]
+
z1
z1
+
z1
+
a1
aN1
aN
y[n]
y[n-1]
y[n-2]
y[n-N]
1
[ ] [ ] [ ]N
kk
y n a y n k v n
1 0
[ ] [ ] [ ]
N M
k kk k
y n a y n k b x n k
0
[ ] [ ]M
kk
v n b x n k
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+
z1
z1
+
z1
+
b0
b1
bM1
bM
x[n]
x[n-1]
x[n-2]
x[n-M]
+
z1
z1
+
z1
+
a1
aN1
aN
y[n]
y[n-1]
y[n-2]
y[n-N]
Block Diagram Representation(Direct Form I)
v[n]0
[ ] [ ]M
kk
v n b x n k
0
[ ] [ ]M
kk
v n b x n k
1
[ ] [ ] [ ]N
kk
y n a y n k v n
1
[ ] [ ] [ ]N
kk
y n a y n k v n
10
( )( )
( )
Mk
kk
V zH z b z
X z
10
( )( )
( )
Mk
kk
V zH z b z
X z
2
1
( ) 1( )
( ) 1N
kk
k
Y zH z
V z a z
2
1
( ) 1( )
( ) 1N
kk
k
Y zH z
V z a z
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1
[ ] [ ] [ ]N
kk
y n a y n k v n
1
[ ] [ ] [ ]N
kk
y n a y n k v n
0
[ ] [ ]M
kk
v n b x n k
0
[ ] [ ]M
kk
v n b x n k
14
+
z1
z1
+
z1
+
b0
b1
bM1
bM
x[n]
x[n-1]
x[n-2]
x[n-M]
+
z1
z1
+
z1
+
a1
aN1
aN
y[n]
y[n-1]
y[n-2]
y[n-N]
Block Diagram Representation(Direct Form I)
v[n]
M
k
kk zbzH
01 )(
M
k
kk zbzH
01 )(
N
k
kk za
zH
1
2
1
1)(
N
k
kk za
zH
1
2
1
1)(
Implementing zeros
Implementing poles
01 2
0
1 1
1( ) ( ) ( )
1 1
Mk
kMk k
k N Nk kk
k kk k
b zH z H z H z b z
a z a z
0
1 20
1 1
1( ) ( ) ( )
1 1
Mk
kMk k
k N Nk kk
k kk k
b zH z H z H z b z
a z a z
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Block Diagram Representation (Direct Form I)
v[n]
How many Adders?How many multipliers?How many delays?
How many Adders?How many multipliers?How many delays?
+
z1
z1
+
z1
+
b0
b1
bM1
bM
x[n]
x[n-1]
x[n-2]
x[n-M]
+
z1
z1
+
z1
+
a1
aN1
aN
y[n]
y[n-1]
y[n-2]
y[n-N]
NNN+M
+M+M+1
![Page 16: 12015-10-151Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 6 structures for discrete- time system Zhongguo Liu](https://reader030.vdocuments.mx/reader030/viewer/2022020106/56649ec15503460f94bccbb1/html5/thumbnails/16.jpg)
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+
z1
z1
+
z1
+
b0
b1
bM1
bM
x[n]
x[n-1]
x[n-2]
x[n-M]
+
z1
z1
+
z1
+
a1
aN1
aN
y[n]
y[n-1]
y[n-2]
y[n-N]
Block Diagram Representation(Direct Form I)
v[n]
M
k
kk zbzH
01 )(
M
k
kk zbzH
01 )(
N
k
kk za
zH
1
2
1
1)(
N
k
kk za
zH
1
2
1
1)(
0
[ ] [ ]M
kk
v n b x n k
0
[ ] [ ]M
kk
v n b x n k
1
[ ] [ ] [ ]N
kk
y n a y n k v n
1
[ ] [ ] [ ]N
kk
y n a y n k v n
(
(
)
)
X
V
z
z
(
(
)
)
X
V
z
z
( )
( )
Y z
V z
( )
( )
Y z
V z
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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[n-1]
w[n-2]
w[n-N]
w[n]
AssumeM = N
M
k
kk zbzH
01 )(
M
k
kk zbzH
01 )(
N
k
kk za
zH
1
2
1
1)(
N
k
kk za
zH
1
2
1
1)(
(or called Canonic direct Form)
( )
( )
Y z
W z
( )
( )
Y z
W z
(
(
)
)
X
W
z
z
(
(
)
)
X
W
z
z
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+
z1
z1
+
z1
+
b0
b1
bN1
bN
x[n]+
z1
z1
+
z1
+
a1
aN1
aN
y[n]
w[n-1]
w[n-2]
w[n-N]
w[n]
AssumeM = N
Block Diagram Representation (Direct Form II)
10
( )( )
( )
Mk
kk W z
Y zH z b z
10
( )( )
( )
Mk
kk W z
Y zH z b z
2
1
1 ( )( )
( )1N
kk
k
H zX za
W z
z
2
1
1 ( )( )
( )1N
kk
k
H zX za
W z
z
0
[ ] [ ]
M
kk
y n b w n k0
[ ] [ ]
M
kk
y n b w n k1
[ ] [ ] [ ]N
kk
w n a w n k x n
1
[ ] [ ] [ ]N
kk
w n a w n k x 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[n-1]
w[n-2]
w[n-N]
w[n]
AssumeM = N
M
k
kk zbzH
01 )(
M
k
kk zbzH
01 )(
N
k
kk za
zH
1
2
1
1)(
N
k
kk za
zH
1
2
1
1)( 0
2 10
1 1
1( ) ( ) ( )
1 1
Mk
kMk k
kN Nk kk
k kk k
b zH z H z H z b z
a z a z
0
2 10
1 1
1( ) ( ) ( )
1 1
Mk
kMk k
kN Nk kk
k kk k
b zH z H z H z b z
a z a z
Implementing zeros
Implementing poles
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Block Diagram Representation (Direct Form II)
How many Adders?How many multipliers?How many delays?
How many Adders?How many multipliers?How many delays?
+
z1
z1
+
z1
+
b0
b1
bN1
bN
x[n]+
z1
z1
+
z1
+
a1
aN1
aN
y[n]
w[n-1]
w[n-2]
w[n-N]
w[n]
AssumeM = N
NNN+M
+M+M+1
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Block Diagram Representation (Canonic Direct Form or direct
Form II)
How many Adders?How many multipliers?How many delays? max(M, N)
How many Adders?How many multipliers?How many delays? max(M, N)
+
+
+
b0
b1
bN1
bN
x[n] +
z1
z1
+
z1
+
a1
aN1
aN
y[n]
AssumeM = N
NN
+M+M+1
N
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Ex. 6.2 draw Direct Form I and Direct Form II implementation of an LTI
system
1
1 21.5
1 2( )
1 0.9
zH z
z z
1
1 21.5
1 2( )
1 0.9
zH z
z z
x[n]+
z1
z1
+1.5
0.9
y[n]
w[n-1]
w[n-2]
+1
2
w[n]
+
z1
1
2
x[n]
x[n-1]
+
z1
z1
+1.5
0.9
y[n]
y[n-1]
y[n-2]
v[n]Solution:
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Structures for Discrete-Time Systems
6.2 Signal Flow Graph( 信号流图 ) Representation of Linear Constant-Coefficient Difference Equations
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6.2 Signal Flow Graph Representation of Linear Constant-Coefficient Difference Equations
Associated with each node is a variable or node value, being denoted wj[n].
A Signal Flow Graph is a network of directed branches (有向支路 )that connect at nodes(节点 ).
wj[n] wk[n]
Node j Node k
梅森(Mason) 信号流图
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Nodes And Branches
wj[n] wk[n]
a or z-1
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.
We will only consider linear Signal Flow Graph
An internal node serves as a summer, i.e., its value is the sum of outputs of all branches entering the node.
An internal node serves as a summer, i.e., its value is the sum of outputs of all branches entering the node.
Node j Node k
if omitted, it indicates unity
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Source Nodes ( 源点 )
Nodes that have no entering branches
xj[n] wk[n]
Source node j
Sink Nodes ( 汇点 )Nodes that have only entering
branches
yk[n]wj[n]Sink node k
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Example : determine Linear Constant-Coefficient Difference Equations of SFG
x[n] y[n]w1[n]
w2[n]
a
b
c
d
e
SourceNode
SinkNode
1 2 2[ ] [ ] [ ] [ ]w n x n aw n bw n
2 1[ ] [ ]w n cw n
2[ ] [ ] [ ]y n dx n ew n
[ ] [ ]1
ce
y n d x nac bc
2 2 2[ ] [ ] [ ] [ ] w n cx n acw n bcw n
Solution:
2
[ ][ ]
1
cex nw n
ac bc
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Block Diagram vs. Signal Flow Graphx[n]
+
az1
+
b1
b0w[n] y[n]
x[n] w1[n]w2[n] w3[n]
ab1
b0
1
2 3
4 w4[n]
y[n]
10 1
1( ) , | | | |
1
b b zH z z a
az
Delay branch cannot be represented in time domain by a branch gain
z1
Delay branch
Canonic direct Form
Source Node Sink Node
=w2[n-1]
2 ( )W z
14 2( ) ( )W z z W z
by z-transform, a unit delay branch has a gain of z-l.
By convention, variables is represented as sequences rather than as z-transforms
branching point
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Determine the difference equation (System Function) from the Flow Graph.
30
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]
1 4[ ] [ ] [ ]w n x n aw n
2 1[ ] [ ]w n w n
3 0 2 1 4[ ] [ ] [ ]w n b w n b w n 4 2[ ] [ 1]w n w n
3[ ] [ ]y n w nSolution:
![Page 30: 12015-10-151Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 6 structures for discrete- time system Zhongguo Liu](https://reader030.vdocuments.mx/reader030/viewer/2022020106/56649ec15503460f94bccbb1/html5/thumbnails/30.jpg)
Block Diagram vs. Signal Flow Graph
31
3[ ] [ ]y n w n 0 2 1 2[ ] [ 1]b w n b w n
2 1[ ] [ ]w n w n 2[ ] [ 1]x n aw n
)()()( 21
10 zWzbbzY )()()( 21
10 zWzbbzY
)()()( 21
2 zWazzXzW )()()( 21
2 zWazzXzW
12 1
)()(
az
zXzW 12 1
)()(
az
zXzW
)(1
)()(
1
110 zX
az
zbbzY
)(1
)()(
1
110 zX
az
zbbzY
0 1[ ] [ 1] [ ] [ 1]y n ay n b x n b x n 0 1[ ] [ 1] [ ] [ 1]y n ay n b x n b x n
1 4[ ] [ ] [ ]w n x n aw n
2 1[ ] [ ]w n w n
3 0 2 1 4[ ] [ ] [ ]w n b w n b w n 4 2[ ] [ 1]w n w n
3[ ] [ ]y n w n
Determine difference equation
difficult in time-domain
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32
1 4 -w n w n x n 1 4 -W z W z X z
Ex. 6.3 Determine the System Function from Flow Graph
4 3 -1w n w n
3 2w n w n x n
2 1w n w n
2 4 y n w n w n
3 2W z W z X z
2 4Y z W z W z
2 1W z W z
-14 3W z W z z
Solution:
![Page 32: 12015-10-151Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 6 structures for discrete- time system Zhongguo Liu](https://reader030.vdocuments.mx/reader030/viewer/2022020106/56649ec15503460f94bccbb1/html5/thumbnails/32.jpg)
33
Ex. 6.3 Determine the System Function from Flow Graph
1 4
2 1
3 2
-14 3
2 4
-W z W z X z
W z W z
W z W z X z
W z W z z
Y z W z W z
-1
2 -1
-
1-
z X z X zW z
z
-1
-1
-
1-
Y z zH z
X z z
2 4 -W z W z X z
-1 -14 2W z z W z z X z
-1 -12 2W z z W z z X z X z
-1 -1 -14 4 -W z z W z z X z z X z
-1 -1
4 -1
-
1-
z X z z X zW z
z
-1
-1
-
1-
X z zY z
z
-1 1-1 -n nh n u n u n for causal system :
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34
Ex. 6.3 compare two implementation
-
x[n]
z-1 z-1
y[n]
-1
-1
-
1-
Y z zH z
X z z
-1 1-1 -n nh n u n u n
direct form I implementation
requires only one multiplication and one delay (memory) element
two multiplication and two delay
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35
Structures for Discrete-Time Systems
6.3 Basic Structure for IIR Systems
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36
6.3 Basic Structure for IIR Systems
Reduce the number of constant multipliersIncrease speed
Reduce the number of delaysReduce the memory requirement
others: VLSI design;Modularity; multiprocessor implementations; effects of a finite register length and finite-precision arithmetic
for a rational system function, many equivalent difference equations or network structures exists. one criteria in the choice among these different structures is computational complexity:
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37
Basic Structures for IIR Systems
Direct FormsCascade FormParallel Form
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38
6.3.1 Direct Forms
1 0
[ ] [ ] [ ]N M
k kk k
y n a y n k b x n k
1 0
[ ] [ ] [ ]N M
k kk k
y n a y n k b x n k
N
k
kk
M
k
kk
za
zbzH
1
1
1)(
N
k
kk
M
k
kk
za
zbzH
1
1
1)(
v[n]+
z1
z1
+
z1
+
b0
b1
bM1
bM
x[n]
x[n-1]
x[n-2]
x[n-M]
+
z1
z1
+
z1
+
a1
aN1
aN
y[n]
y[n-1]
y[n-2]
y[n-N]
![Page 38: 12015-10-151Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 6 structures for discrete- time system Zhongguo Liu](https://reader030.vdocuments.mx/reader030/viewer/2022020106/56649ec15503460f94bccbb1/html5/thumbnails/38.jpg)
39
Direct Form
I
v[n]+
z1
z1
+
z1
+
b0
b1
bM1
bM
x[n]
x[n-1]
x[n-2]
x[n-M]
+
z1
z1
+
z1
+
a1
aN1
aN
y[n]
y[n-1]
y[n-2]
y[n-N]
b0
b1
x[n]x[n-1]
x[n-2]
x[n-M]
y[n]
b2
bN-1
bN
x[nM+1]
a1
a2
aN-1
aN
y[n-1]
y[n-2]
y[n-N]
y[nN+1]
z1
z1
z1
z1
z1
z1
v[n]
Block Diagram
Signal Flow
Graph
![Page 39: 12015-10-151Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 6 structures for discrete- time system Zhongguo Liu](https://reader030.vdocuments.mx/reader030/viewer/2022020106/56649ec15503460f94bccbb1/html5/thumbnails/39.jpg)
40
Direct Form I Signal Flow Graph
1 0
[ ] [ ] [ ]N M
k kk k
y n a y n k b x n k
1 0
[ ] [ ] [ ]N M
k kk k
y n a y n k b x n k
N
k
kk
M
k
kk
za
zbzH
1
1
1)(
N
k
kk
M
k
kk
za
zbzH
1
1
1)(
b0
b1
x[n]x[n-1]
x[n-2]
x[n-M]
y[n]
b2
bN-1
bN
x[nM+1]
a1
a2
aN-1
aN
y[n-1]
y[n-2]
y[n-N]
y[nN+1]
z1
z1
z1
z1
z1
z1
v[n]
![Page 40: 12015-10-151Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 6 structures for discrete- time system Zhongguo Liu](https://reader030.vdocuments.mx/reader030/viewer/2022020106/56649ec15503460f94bccbb1/html5/thumbnails/40.jpg)
41
Direct Form II
1 0
[ ] [ ] [ ]N N
k kk k
y n a y n k b x n k
1 0
[ ] [ ] [ ]N N
k kk k
y n a y n k b x n k
1
[ ] [ ] [ ]N
kk
w n a w n k x n
1
[ ] [ ] [ ]N
kk
w n a w n k x n
0
[ ] [ ]
M
kk
y n b w n k0
[ ] [ ]
M
kk
y n b w n k
x[n] y[n]w[n] b0
b1
b2
bN-1
bN
a1
a2
aN-1
aN
z1
z1
z1
2
1
( )( )
( )
1
1N
kk
k
W zH z
X z
a z
2
1
( )( )
( )
1
1N
kk
k
W zH z
X z
a z
1
0
( )( )
( )M
kk
k
Y zH z
W z
b z
1
0
( )( )
( )M
kk
k
Y zH z
W z
b z
N
k
kk
M
k
kk
za
zbzH
1
1
1)(
N
k
kk
M
k
kk
za
zbzH
1
1
1)(
![Page 41: 12015-10-151Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 6 structures for discrete- time system Zhongguo Liu](https://reader030.vdocuments.mx/reader030/viewer/2022020106/56649ec15503460f94bccbb1/html5/thumbnails/41.jpg)
42
Direct Form II
N
k
kk
M
k
kk
za
zbzH
1
1
1)(
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
1 0
[ ] [ ] [ ]N M
k kk k
y n a y n k b x n k
1 0
[ ] [ ] [ ]N M
k kk k
y n a y n k b x n k
![Page 42: 12015-10-151Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 6 structures for discrete- time system Zhongguo Liu](https://reader030.vdocuments.mx/reader030/viewer/2022020106/56649ec15503460f94bccbb1/html5/thumbnails/42.jpg)
43
Ex. 6.4 draw Direct Form I and Direct Form II structures of
system
21
21
125.075.01
21)(
zz
zzzH 21
21
125.075.01
21)(
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
Solution:
![Page 43: 12015-10-151Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 6 structures for discrete- time system Zhongguo Liu](https://reader030.vdocuments.mx/reader030/viewer/2022020106/56649ec15503460f94bccbb1/html5/thumbnails/43.jpg)
44
6.3.2 Cascade Form( 串联形式 )
N
k
kk
M
k
kk
za
zbzH
1
1
1)(
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
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
1 0
[ ] [ ] [ ]N M
k kk k
y n a y n k b x n k
1 0
[ ] [ ] [ ]N M
k kk k
y n a y n k b x n k
when all the coefficients are real
1st-order factors represent real zeros at gk and real poles at ck , and the 2nd-order factors represent complex conjugate pairs of zeros at hk and h*
k and poles at dk ,d*
k
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45
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
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
zaza
zbzbbzH
12
21
1
22
110
1)(
sN
k kk
kkk
zaza
zbzbbzH
12
21
1
22
110
1)(
2nd OrderSystem
2nd OrderSystem
2nd OrderSystem
2nd OrderSystem
2nd OrderSystem
2nd OrderSystem
A modular structure
![Page 45: 12015-10-151Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 6 structures for discrete- time system Zhongguo Liu](https://reader030.vdocuments.mx/reader030/viewer/2022020106/56649ec15503460f94bccbb1/html5/thumbnails/45.jpg)
46
Cascade Form
sN
k kk
kkk
zaza
zbzbbzH
12
21
1
22
110
1)(
sN
k kk
kkk
zaza
zbzbbzH
12
21
1
22
110
1)(
x[n] y[n]z1
z1
a11
a21
b11
b21
b01
z1
z1
a12
a22
b12
b22
b02
z1
z1
a13
a23
b13
b23
b03
1 2 3For example, assume Ns=3
It is used when implemented with fixed-point arithmetic, the structure can control the size of signals at various critical points because they make it possible to distribute the overall gain of the system.
![Page 46: 12015-10-151Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 6 structures for discrete- time system Zhongguo Liu](https://reader030.vdocuments.mx/reader030/viewer/2022020106/56649ec15503460f94bccbb1/html5/thumbnails/46.jpg)
47
Ex. 6.5 draw the Cascade structures
1 2
1 2
1 2( )
1 0.75 0.125
z zH z
z z
1 2
1 2
1 2( )
1 0.75 0.125
z zH z
z z
x[n] y[n]z1
z1
0.75
0.125
2Direct Form II
1 1
1 1
(1 ) (1 )
(1 0.5 )(1 0.25 )
z z
z z
1 1
1 1
(1 ) (1 )
(1 0.5 )(1 0.25 )
z z
z z
1st-order Direct Form II
1st-order Direct Form I
Solution:
![Page 47: 12015-10-151Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 6 structures for discrete- time system Zhongguo Liu](https://reader030.vdocuments.mx/reader030/viewer/2022020106/56649ec15503460f94bccbb1/html5/thumbnails/47.jpg)
48
Another Cascade Form
sN
k kk
kkk
zaza
zbzbbzH
12
21
1
22
110
1)(
sN
k kk
kkk
zaza
zbzbbzH
12
21
1
22
110
1)(
sN
k kk
kk
zaza
zbzbbzH
12
21
1
22
11
0 1
~~1
)(
sN
k kk
kk
zaza
zbzbbzH
12
21
1
22
11
0 1
~~1
)(
x[n]y[n]z1
z1
a11
a21
b11
b21
z1
z1
a12
a22
b12
b22
z1
z1
a13
a23
b13
b23
b0
~
~
~
~
~
~
implemented with fixed-point arithmetic
when floating-point arithmetic is used and dynamic range is not a problem.
used to decrease the amount of computation,
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49
6.3.3 Parallel Form
N
k
kk
M
k
kk
za
zbzH
1
1
1)(
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
zeB
zc
AzCzH
P
11
11*1
1
11
0 )1)(1(
)1(
1)(
N
k kk
kkN
k k
kN
k
kk zdzd
zeB
zc
AzCzH
P
1 0
[ ] [ ] [ ]N M
k kk k
y n a y n k b x n k
1 0
[ ] [ ] [ ]N M
k kk k
y n a y n k b x n k
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50
Parallel Form
Real Poles Complex PolesPoles at zero
10 1
1 20 1 1 2
( )1
sP NNk k k
kk k k k
e e zH z C z
a z a z
1
0 11 2
0 1 1 2
( )1
sP NNk k k
kk k k k
e e zH z C z
a z a z
11
11*1
1
11
0 )1)(1(
)1(
1)(
N
k kk
kkN
k k
kN
k
kk zdzd
zeB
zc
AzCzH
P
11
11*1
1
11
0 )1)(1(
)1(
1)(
N
k kk
kkN
k k
kN
k
kk zdzd
zeB
zc
AzCzH
P
GroupReal Poles in pairs
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51
Parallel Form1
0 11 2
0 1 1 2
( )1
sP NNk k k
kk k k k
e e zH z C z
a z a z
1
0 11 2
0 1 1 2
( )1
sP NNk k k
kk k k k
e e zH z C z
a z a z
z1
z1
a1k
a2k
e0k
e1k
x[n] y[n]
Ckz-k
C0
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52
Ex. 6.6 draw parallel-form structures of system
21
21
125.075.01
21)(
zz
zzzH 21
21
125.075.01
21)(
zz
zzzH
21
1
25.175.01
878)(
zz
zzH 21
1
25.175.01
878)(
zz
zzH
8
x[n] y[n]z1
z10.75
0.125
8
7
Solution 1: If we use 2nd –order sections,
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53
11 25.01
25
5.01
188)(
zz
zH 11 25.01
25
5.01
188)(
zz
zH
z1
0.5
18
8
x[n] y[n]
z1
0.25
25
Solution 2: If we use 1st –order sections,
Ex. 6.6 draw parallel-form structures of system
21
21
125.075.01
21)(
zz
zzzH 21
21
125.075.01
21)(
zz
zzzH
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54
6.3.4 feedback in the IIR systems
[ ] [ 1] [ ] y n ay n x n[ ] [ 1] [ ] y n ay n x n
1
1( )
1H z
az
z1
z1
ax[n] y[n]
-a2
2 2 1 11
1 1
1 (1 )(1 )( ) (1 )
1 1
a z az azH z az
az az
[ ] [ ] [ 1] h n n a n[ ] [ ] [ 1] h n n a n
z1ax[n] y[n]
ax[n] y[n]
[ ] [ ] [ 1] y n x n ax n[ ] [ ] [ 1] y n x n ax n
[ ] [ ] [ ] y n ay n x n[ ] [ ] [ ] y n ay n x n
( )( )
1
x ny n
a
systems with feedback may be FIR
Noncomputable network
z1 ax[n] y[n]
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55
Structures for Discrete-Time Systems
6.4 Transposed Forms
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There are many procedures for transforming signal flow graphs into different forms while leaving the overall system function between input and output unchanged.
6.4 Transposed Forms
Flow Graph Reversal or Transposition
x[n] y[n]z1
a
x[n]y[n] z1
a
Changes the roles of input and output. Reverse the directions of all arrows.
Transposing doesn’t change the input-output relation
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Ex. 6.7 determine Transposed Forms for a
first-order system
11
1)(
azzH 11
1)(
azzH
z1
a
x[n] y[n]
z1
a
x[n]y[n]
z1
a
x[n] y[n]
Solution:
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Both have the same system function or difference equation
1 2 0 1 2-1 - 2 -1 - 2y n a y n a y n b x n b x n b x n
Ex. 6.8 draw Transposed Forms for a basic second-order section
Transpose
Solution:
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Ex. 6.8 Transposed Forms for a basic second-order section
Transpose
b0
b1x[n] y[n]
b2
a1
a2
z1
z1
v1[n]
z1
z1
b0
b1x[n] y[n]
b2
a1
a2
z1
z1
v1[n]
x[n] y[n]b0
b1
b2
z1
z1
v2[n]
a1
a2
z1
z1
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Transposed Direct Form Ib0
b1
x[n]x[n-1]
x[n-2]
x[n-M]
y[n]
b2
bN-1
bN
x[nM+1]
a1
a2
aN-1
aN
y[n-1]
y[n-2]
y[n-N]
y[nN+1]
z1
z1
z1
z1
z1
z1
v[n]
b0
b1
x[n]y[n]
b2
bN-1
bN
a1
a2
aN-1
aN
z1
z1
z1
z1
z1
z1
v'[n]
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Transposed Direct Form II
x[n] y[n]w[n] b0
b1
b2
bN-1
bN
a1
a2
aN-1
aN
z1
z1
z1
y[n] x[n]w' [n] b0
b1
b2
bN-1
bN
a1
a2
aN-1
aN
z1
z1
z1
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Structures for Discrete-Time Systems
6.5 Basic Structure for FIR Systems
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6.5 Basic Structure for FIR Systems
For causal FIR systems, the system function has only zeros(except for poles at z = 0).
0
[ ] [ ]M
kk
y n b x n k
0
[ ] [ ]M
kk
y n b x n k
0
[ ] [ ] [ ]M
k
y n h k x n k
0,1, ,[ ]
0nb n M
h nothrewise
6.5. 1 Direct Form
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Direct Form I
0,1, ,[ ]
0nb n M
h nothrewise
x[n]
y[n]
z1 z1 z1
h[0] h[1] h[2] h[M1] h[M]
b0
b1
x[n]x[n-1]
x[n-2]
x[n-M]
y[n]
b2
bM-1
bM
x[nM+1]
a1
a2
aN-1
aN
y[n-1]
y[n-2]
y[n-N]
y[nN+1]
z1
z1
z1
z1
z1
z1
v[n]
y[n]
0
[ ] [ ]M
kk
y n b x n k
0
[ ] [ ]M
kk
y n b x n k
x[n-1] x[n-2] x[nM+1]x[n-M]
0
[ ] [ ] [ ]M
k
y n h k x n k
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Direct Form II
x[n]
y[n]
z1 z1 z1
x[n] y[n]w[n] b0
b1
b2
bM-1
bM
a1
a2
aN-1
aN
z1
z1
z1
y[n]
0
[ ] [ ]M
kk
y n b x n k
0
[ ] [ ]M
kk
y n b x n k
0
[ ] [ ] [ ]M
k
y n h k x n k
0,1, ,
[ ]0
nb n Mh n
othrewise
h[0] h[1] h[2] h[M1] h[M]
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x[n]
y[n]
z1 z1 z1
h[0] h[1] h[2] h[M1] h[M]
Traspostion of Direct Form
x[n]
y[n] z1 z1 z1
h[0] h[1] h[2] h[M1] h[M]
x[n]
y[n]z1z1z1
h[0]h[1]h[2]h[M1]h[M]
tapped delay line structure or transversal filter structure.抽头延迟线结构 or 横向滤波器结构 .
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6.5.2 Cascade Form
M
kk knxbny
0
)()(
M
kk knxbny
0
)()(
M
n
nznhzH0
)()(
M
n
nznhzH0
)()(
sM
kkkk zbzbbzH
1
22
110 )()(
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
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Cascade Form
sM
kkkk zbzbbzH
1
22
110 )()(
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
sN
k kk
kkk
zaza
zbzbbzH
12
21
1
22
110
1)(
sN
k kk
kkk
zaza
zbzbbzH
12
21
1
22
110
1)(
x[n] y[n]z1
z1
a11
a21
b11
b21
b01
z1
z1
a12
a22
b12
b22
b01
z1
z1
a13
a23
b13
b23
b03
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M is even M is oddh[M-n]= h[n]
h[M-n]= h[n]
6.5.3 Structures for Linear Phase Systems
A causal FIR system has generalized linear phase if h[n] satisfies:
h[M-n]= h[n] for n = 0,1,…,Mh[M-n]= h[n] for n = 0,1,…,Mor
Type I
Type III
Type II
Type VI
x[n]
y[n]
z1 z1 z1
h[0] h[1] h[2] h[M1] h[M]
M+1 multiplicatio
ns
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For even M and type I or type III systems:
n 0,1,...,M h M n h n
0
-M
k
y n h k x n k
/2-1 /2-1
0 0
-2 2
- - -
M M
k k
h k x n k h M k x nM
h M kM
x n
/2-1
0 /2 1
-2 2
- -M M
k k M
M Mh xh k x n nnk h k x k
/2-1
0
- - -2 2
M
k
h k x n k xM M
hn M k x n
0 M/2 M
0 M/2
M
Symmetry means we can half the number of multiplications
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Type I and III
0
-M
k
y n h k x n k
/2-1
0
- - -2 2
M
k
M Mh k x n k x n M k h x n
x[n-1] x[n-2] x[n-M/2+1]
x[n-M/2]x[n-M]
x[n]
x[n-M+1] x[n-M+2] x[n-M/2-1]
x[n]
y[n]
z1 z1 z1
z1 z1 z1
h[M/2]h[M/21]h[0] h[1] h[2]
0 M/2 M
0 M/2
M
Type III=0
- - - -
Type I
x[n]
y[n]
z1 z1 z1
h[0] h[1] h[2] h[M1] h[M]
x[n-1] x[n-2] x[n-M]x[n] x[n-M+1]
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Type II or Type IV FIR Systems
For odd M and type II or type IV systems:
n 0,1,...,M h M n h n
0
-M
k
y n h k x n k
( -1)/2 ( -1)/2
0 0
- - -M M
k k
h k x n k h M k x n M k
( -1)/2
0 ( 1)/2
- -M M
k k M
h k x n k h k x n k
( -1)/2
0
- -M
k
h k x n k x n M k
0 M/2 M
0 M/2
M
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Type II and IVStructure for odd M
0
-M
k
y n h k x n k
( -1)/2
0
- -M
k
h k x n k x n M k
x[n-(M-1)/2]
x[n-(M+1)/2]
x[n-1] x[n-2]
x[n-M] x[n-M+1]
x[n-M+2]
x[n]
0 M/2 M
0 M/2
M
- - --
Type II Type IV
x[n]
y[n]
z1 z1 z1
h[0] h[1] h[2] h[M1] h[M]
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Type I, and II
x[n-1] x[n-2] x[n-M/2+1]
x[n-M/2]x[n-M]
x[n]
x[n-M+1] x[n-M+2] x[n-M/2-1]
x[n]
y[n]
z1 z1 z1
z1 z1 z1
h[M/2]h[M/21]h[0] h[1] h[2]
x[n-(M-1)/2]
x[n-(M+1)/2]
x[n-1] x[n-2]
x[n-M] x[n-M+1]
x[n-M+2]
x[n]
Type I
Type II
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6.6 OVERVIEW OF FINITE-PRECISION NUMERICAL EFFECTS
6.6.1 Number RepresentationsA real number can be represented with infinite precision in two's-complement form as
where Xm is an arbitrary scale factor and the bis are either 0 or 1. The quantity b0 is referred to as the sign bit. If b0= 0, then 0≤ x <Xm , and if b0= 1, then -Xm≤ x <0.
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For a finite number of bits (B +1), the equation above must be modified to
so the smallest difference between numbers is
6.6.1 Number Representations
the quantized numbers are in the range :
-Xm≤ <Xm.
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quantizing a number to (B +1) bits can be implemented by rounding or by truncation, which is a nonlinear memoryless operation. define the quantization error as
6.6.1 Number Representations
The fractional part of can be represented with the positional notation
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For the case of two's-complement rounding, - Δ/2 < e <Δ/2, and for two's-complement truncation, - Δ< e <0
6.6.1 Number Representations
truncation
rounding
For B =2
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6.6.2Quantization in Implementing Systems
Consider the following system
A more realistic model would be
1
1
bH z
az
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6.6.2 Quantization in Implementing Systems
In order to analyze it we would prefer
1ˆ
ˆ1
bH z
az
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6.7.1 Effects of Coefficient Quantization in IIR Systems
When the parameters of a rational system are quantized, The poles and zeros of the system function move.
If the system structure of the system is sensitive to perturbation of coefficients,The resulting system may no longer meet the original specifications,
and may no longer be stable.
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6.7 Effects of Coefficient Quantization in IIR Systems
Detailed sensitivity analysis for general case is complicated. Using simulation tools, in specific cases,Quantize the coefficients and analyze frequency response
Compare frequency response to original response
We would like to have a general sense of the effect of quantization
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6.7.1 Effects of Coefficient Quantization in IIR Systems
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Each root is affected by quantization errors in ALL coefficient
Tightly clustered roots are significantly effectedNarrow-bandwidth lowpass or bandpass filters can be very sensitive to quantization noise
N
1k
kk
M
0k
kk
za1
zbzH
N
1k
kk
M
0k
kk
za1
zbzH
Quantization
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Effects on Roots(poles and zeros)
The larger the number of roots in a cluster the more sensitive it becomes
So second order cascade structures are less sensitive to quantization error than higher order systemEach second order system is independent from each other
N
1k
kk
M
0k
kk
za1
zbzH
N
1k
kk
M
0k
kk
za1
zbzH
Quantization
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6.7.2 Example of Coefficient Quantization in an Elliptic Filter
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An IIR bandpass elliptic filter was designed to meet the following specifications:
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6.7.2 Example of Coefficient Quantization in an bandpass Elliptic
Filter
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Poles and zeros of H(z) for unquantized Coefficients and 16-bit quantization of the direct form
unquantized16-bit quantization
the direct form system cannot be implemented with 16-bit coefficients because it would be unstable
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6.7.2 Example of Coefficient Quantization in an bandpass Elliptic
Filter
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the cascade form is much less sensitive to coefficient quantization
Magnitude in passband for 16-bit quantization of the cascade form
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6.7.2 Example of Coefficient Quantization in an Elliptic Filter
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6.7.3 Poles of Quantized 2nd-Order Sections
1 2 2 1 1
1 1( )
1 2 cos (1 )(1 ) j j
H zr z r z re z re z
Consider a 2nd order system with complex-conjugate pole pair
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6.7.3 Poles of Quantized 2nd-Order Sections
3-bits
The pole locations after quantization will be on the grid point
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6.7.3 Poles of Quantized 2nd-Order Sections
7-bits
The pole locations after quantization will be on the grid point
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Coupled-Form Implementation of Complex-Conjugate Pair
Equivalent implementation of the 2nd order system
1
1 2 2
sin( )
1 2 cos
r zH z
r z r z
Twice as many constant multipliers are required to achieve more uniformdensity.
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Coupled-Form Implementation of Complex-Conjugate Pair
3-bits 7-bits
Twice as many constant multipliers are required to achieve this more uniform density of quantization grid
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6.7.4 Effects of Coefficient Quantization in FIR Systems
M
0n
nznhzH
zHzHznhzHM
0n
n
M
0n
nznhzH
No poles to worry about only zeros
Direct form is commonly used for FIR systems
Suppose the coefficients are quantized
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95
6.7.4 Effects of Coefficient Quantization in FIR Systems
M
0n
nznhzH
zHzHznhzHM
0n
n
M
0n
nznhzH
No poles to worry about only zeros
Direct form is commonly used for FIR systems
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6.7.4 Effects of Coefficient Quantization in FIR Systems
Quantized system is linearly related to the quantization error
Again quantization noise is higher for clustered zeros
However, most FIR filters have spread zeros
zHzHznhzHM
0n
n
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6.8 EFFECTS OF ROUND-OFF NOISE IN DIGITAL FILTERS
Difference equations implemented with finite-precision arithmetic are non-linear systems.
Second order direct form I system
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6.8 EFFECTS OF ROUND-OFF NOISE IN DIGITAL FILTERS
Model with quantization effectDensity function error terms for rounding
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6.8.1 Analysis of the Direct Form IIR Structures
0 1 2 3 4e n e n e n e n e n e n
Combine all error terms to single location to get
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100
6.8.1 Analysis of the Direct Form IIR Structures
122
N1MB2
2e
N
1kk neknfanf
The variance of e[n] in the general case is
The contribution of e[n] to the output is
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101
6.8.1 Analysis of the Direct Form IIR Structures
n
2
ef
B22f nh
122
N1M
zA/1zHef
The variance of the output error term f[n] is
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102
Example 6.9 Round-Off Noise in a First-Order System
Suppose we want to implement the following stable system
1 ,
1
a 1
bH z
az
2
B2
0n
n2B2
n
2
ef
B22f
a1
1122
2a122
2nh122
N1M
The quantization error noise variance is
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Example 6.9 Round-Off Noise in a First-Order System
Noise variance increases as |a| gets closer to the unit circle
As |a| gets closer to 1 we have to use more bits to compensate for the increasing error
1 ,
1
a 1
bH z
az
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6.9 Zero-Input Limit Cycles in Fixed-Point Realization of IIR
Filters
For stable IIR systems the output will decay to zero when the input becomes zero
A finite-precision implementation, however, may continue to oscillate indefinitely
Nonlinear behaviour is very difficult to analyze, so we will study by example
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6.9 Zero-Input Limit Cycles in Fixed-Point Realization of IIR
FiltersExample: Limite Cycle Behavior in First-
Order Systems 1 , a 1y n ay n x n
Assume x[n] and y[n-1] are implemented by 4 bit registers
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Example Cont’d
Assume that a=1/2=0.100b and the input is
nb111.0n87
nx
n y[n] Q(y[n])0 7/8=0.111b 7/8=0.111b1 7/16=0.011100b 1/2=0.100b2 1/4=0.010000b 1/4=0.010b3 1/8=0.001000b 1/8=0.001b4 1/16=0.00010b 1/8=0.001b
1a nx1nayny
If we calculate the output for values of n
A finite input caused an oscillation with period 1
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Example: Limite Cycles due to Overflow
Consider a second-order system realized by
2nyaQ1nyaQnxny 21
b010.14/32y and b110.04/31y
Where Q() represents two’s complement rounding
Word length is chosen to be 4 bitsAssume a1=3/4=0.110b and a2=-3/4=1.010b
Also assume
ˆ 0 0.110 0.110b 1.010 1.010by b b
The output at sample n=0 is
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Example: Limite Cycles due to Overflow
-3/41.010b0.101b 0.101b0y
4/3110.01.010b 1.010b0y
Binary carry overflows into the sign bit changing the sign
When repeated for n=1
0.100100b 0.100100b
1.010b b010.1 0.110b b110.00y
The output at sample n=0 is
After rounding up we get
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Avoiding Limit-Cycles
Desirable to get zero output for
zero input: Avoid limit-cycles
Generally adding more bits would
avoid overflow
Using double-length accumulators
at addition points would decrease
likelihood of limit cycles
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Avoiding Limit-Cycles
Trade-off between limit-cycle
avoidance and complexity
FIR systems cannot support
zero-input limit cycles
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111 23/4/20111Zhongguo Liu_Biomedical Engineering_Shandong U
niv.
Chapter 6 HW
6.5, 6.6, 6.196.1, 6.3, 6.20
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