12015-10-151Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 6 structures for discrete- time system Zhongguo Liu

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

2

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

3

Structures for Discrete-Time Systems

6.0 Introduction

4

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

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 .

5

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:

6

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:

7

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

8


6.1 Block Diagram Representation of Linear

Constant-Coefficient Difference Equations

9

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]

10

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

11

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

12

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

13

+

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

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]


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

15

Block Diagram Representation (Direct Form I)

v[n]

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

16

+

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]


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

17

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

19

+

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


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

20


+

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

21




+

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

22

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

23

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:

24


6.2 Signal Flow Graph( 信号流图 ) Representation of Linear Constant-Coefficient Difference Equations

25

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

26

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

27

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

28

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

29

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

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:

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

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:

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 :

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

35


6.3 Basic Structure for IIR Systems

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:

37

Basic Structures for IIR Systems

Direct FormsCascade FormParallel Form

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]

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

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]

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)(

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

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:

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

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

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.

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:

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,

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

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

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

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,

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

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]

55



56

There are many procedures for transforming signal flow graphs into different forms while leaving the overall system function between input and output unchanged.


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

57

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:

58

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:

59

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

60

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]

61

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

62


6.5 Basic Structure for FIR Systems

63

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

64

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

65

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]

66

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 .

67

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

68

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

69

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

70

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

71

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]

72

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

73

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]

74

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

75

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.

76

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.

77

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


The fractional part of can be represented with the positional notation

78

For the case of two's-complement rounding, - Δ/2 < e <Δ/2, and for two's-complement truncation, - Δ< e <0


truncation

rounding

For B =2

79

6.6.2Quantization in Implementing Systems

Consider the following system

A more realistic model would be

1

1

bH z

az

80

6.6.2 Quantization in Implementing Systems

In order to analyze it we would prefer

1ˆ

ˆ1

bH z

az

81

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.

82

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

6.7.1 Effects of Coefficient Quantization in IIR Systems

83

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

84

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

6.7.2 Example of Coefficient Quantization in an Elliptic Filter

85

An IIR bandpass elliptic filter was designed to meet the following specifications:

6.7.2 Example of Coefficient Quantization in an bandpass Elliptic

Filter

86

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

6.7.2 Example of Coefficient Quantization in an bandpass Elliptic

Filter

87

the cascade form is much less sensitive to coefficient quantization

Magnitude in passband for 16-bit quantization of the cascade form

6.7.2 Example of Coefficient Quantization in an Elliptic Filter

88

89

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

90


3-bits

The pole locations after quantization will be on the grid point

91


7-bits

The pole locations after quantization will be on the grid point

92

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.

93

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

94

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

95


M

0n

nznhzH

zHzHznhzHM

0n

n

M

0n

nznhzH

No poles to worry about only zeros

Direct form is commonly used for FIR systems

96


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

97

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

98

6.8 EFFECTS OF ROUND-OFF NOISE IN DIGITAL FILTERS

Model with quantization effectDensity function error terms for rounding

99

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

100


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

101


n

2

ef

B22f nh

122

N1M

zA/1zHef

The variance of the output error term f[n] is

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

103

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

104

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

105

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

106

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

108

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

110

Avoiding Limit-Cycles

Trade-off between limit-cycle

avoidance and complexity

FIR systems cannot support

zero-input limit cycles

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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12015-10-151Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 6 structures for discrete- time system Zhongguo Liu