Chapter 7 Structures for LTI Systems Student (1)

8/13/2019 Chapter 7 Structures for LTI Systems Student (1)

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ELEC 215: Tim Woo Spring 2009/10 Chapter 7 - 1

Chapter 7: Structures for LTI systems

Spring 2009/010

Lecture: Tim Woo


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LaplaceTransform

Where we are

Differentialequations

State-space

model

CTFT

Hardware

Implementation

System

Characteristics

System

Responses

Closed-loop

Systems

Continuous-time

z-Transform

Difference

equations

State-space

model

DTFT

Hardware

Implementation

System

Characteristics

System

Responses

Closed-loop

Systems

Discrete-time

Mapping

Done in 211 To be covered In progress DoneWill be covered if available

Open-loop

Systems

Open-loop

Systems


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Expected Outcome In this chapter, you will be able to

Implement a system function with linear constant-coefficient

differential or difference equation by structures with basicelements

Adders

Multipliers

Integrators / Delays Construct the same system function with different structures for

realization of causal continuous-time (or discrete-time) LTI

system.

Compare the computational complexity of different structures Compare the robustness design in different structures


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Outline Textbook

Section 9.8.2 Block Diagram Representations for Causal LTI

systems Described by Differential Equations and RationalSystem Functions

Section 10.8.2 Block Diagram Representations for Causal LTIsystems Described by Difference Equations and RationalSystem Functions

Reference book A. V. Oppenheim, et. al., Discrete-time Signal Processing, 2nd

edition, Prentice-Hall, 1999

Section 6.1 Block Diagram representation of linear constant-

coefficient difference equations Section 6.3 Basic Structures for IIR systems

Section 6.7 The effects of coefficient quantization


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Introduction As discussed in Chapter 4, a LTI system with a rational system

function has the property that the input and output sequence satisfy

a linear constant-coefficient differential (or difference) equation.

When such systems are implemented with analog (or digital)

hardware, the differential (or difference) equation must be converted

to an algorithm or structure that can be realized in the desired

technology.

In this chapter, we will construct the system function by structures

consisting of an interconnection of the basic operations of addition,

multiplication by a scalar, and integrator (or delay).


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Consider the difference equation,

the output signal y[n] at time instant

n requires y[n-1], ., y[n - N] and

x[n], x[n-1], ., x[n - M].

Introduction Consider the differential equation,

the output signal y(t) at time instant

trequires dy(t)/dt, ., dNy(t)/dtN and

x(t), dx(t)/dt, ., dMy(t)/dtM.

][][][01

knxbknyanyM

k

k

N

k

k += ==

==

+=M

kk

k

kk

kN

k

kdt

txdb

dt

tydaty

01

)()()(

That is, we need

Multipliers for scaling

It usually has the high computation cost.

Adders for summation

In general, an adder can have any number of inputs. However, in most practical

implementations, adders have only two inputs.

Delay elements for storage

It can implemented by providing a storage register for each unit delay. Delays of M

samples can be implemented with a system with M consecutive storage registers.

Integrator

In general, integers are commonly used instead of differentiators.


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Introduction The realization of these components can be indicated either by

Block diagram representations

Signal flow graph representations

Block diagram

][2 nx

][1nx ][][ 21 nxnx +

][nx ][nax

a

][nx ]1[ nx

1z

Signal flow graph

)(2 tx

)(1 tx )()( 21 txtx +

)(tx )(tax

a

)(tx t

dx )(

s1

)(2 tx

)(1 tx )()( 21 txtx +

)(tx )(tax

)(tx

t

dx )(

s1


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Example 9.28: Consider

From the previous section, we know this

system can also be described by difference

equation:

Using 1/s to represent integrator or

, we have

Introduction

3

1)(

+=

ssH

[ ] dxyty

txty

dt

tdy

t

+=

=+

)()(3)(

)()(3)(

s

s

ssH

/31

/1

3

1)(

+=

+=


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Introduction Draw a block diagram and signal flow graph

representations of an LTI system whose difference

equation is:][]2[]1[][ 021 nxbnyanyany ++=

0b

][nx

]2[ ny

1a

1z ][ny

1z

2a

]1[ ny

Block diagram Signal flow graph


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Introduction Computations can be arranged in different ways to give

the same differential (or difference) equation, which

leads to different structures for realization of discrete-time causal LTI system.

Typically, there are many basic forms of realization, but

we focus on four of them.

Direct form I

Canonic Direct form (or Direct form II)

Cascade form

Parallel form


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9.8.2 Block Diagram Representation Consider a linear constant coefficient differential equation of an LTI

system

For simplicity, we assume

N= M. If N M, some of the coefficients will be zero.

All the coefficients are normalized such that

By taking the Laplace transform on both sides, we have

=

=

=

=

=

=

=

=

==N

k

k

k

N

k

k

k

N

k

k

k

N

k

k

k

N

k

kN

k

N

N

k

kN

k

sa

sb

sa

sb

sas

sb

sXsYsH

1

0

1

0

1

0

11

1

1)()()(

kN

kNN

kk

M

kkM

kM

kN

N

dt

tyda

dt

txdb

dt

tyda

==

+=

)(~)(~)(~

100

000

00 ~

~

,~

~

,1~

~

a

bba

aaa

aa kkkk ====


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9.8.2 Block Diagram Representation Direct form I

Decompose the system function such that H(s) = H1(s) H2(s)

=

=

=

=

==

==

==N

k

k

k

N

k

k

kN

k

k

k

N

k

k

k

sa

sV

sYsHand

sb

sX

sVsH

sa

sb

sX

sYsH

1

2

0

1

1

0

11

1

)(

)()(

1

)(

)()(

11

1

)(

)()(

=

+=N

kkk

s

sYasVsY

1

)()()(

=

=N

kkk

s

sXbsV

0

)()(

)(2 sH)(1 sH

)(tv

s1

)(tx

s1

)(tys1

s1

s1

s1

t

dx )(

t

dy )(

t

dx )(

t

dx )(

t

dy )(

t

dy )(1Nb

Nb


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9.8.2 Block Diagram Representation Canonic form (Direct form II)

Decompose the system function such that H(s) = H1(s) H2(s)

=

==

=

==

==

==N

k

k

kN

k

k

k

N

k

k

k

N

k

k

k

sb

sW

sYsHand

sa

sX

sWsH

sa

sb

sX

sYsH0

2

1

1

1

0 1

)(

)()(1

1

1

)(

)()(1

1

1

)(

)()(

=

=N

kkk

s

sXbsV

0

)()(

)(2 sH

)(1 sH

)(tw

s1

)(tx

s1

)(ty

s1

s1

s1

s1

t

dw )(

t

dw )(

t

dw )(

)(tw

)(tx )(ty

s1

s1

s1

=

+=N

kkk

s

sWasXsW

1

)()()(


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9.8.2 Block Diagram Representation Example 9.31 Use block diagram

to draw the direct form I and direct

form II for an LTI system with

system function

Rewrite system function as

This gives

23

642)(

2

2

++

+=

ss

sssH

2

2

231

642

)(

ss

sssH

++

+=

Direct form I

Direct form II2,3,6,4,2 21210 ===== aabbb


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9.8.2 Block Diagram Representation Cascade form

Without loss of generality, the system function be factorized into a

cascade of Ns second order sub-systems as follows,

Parallel form

Without loss of generality, the system function can be expressed as a

partial fraction expansion in the form,

Each sub-system in cascade and parallel forms can be realized in

either direct form I and the direct II.

=

++=

sN

k kk

kk

sasa

sbsbGsH

12

2

1

1

2

2

1

1

1

1)(

=

=

++=

sp N

k kk

kk

N

k

k

ksasa

seesBsH

12

2

1

1

1

10

0 1)(


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Example 9.28 : A system function

9.8.2 Block Diagram Representation

1

1

1

1

1

1

1

1

21

2

2

211211231

2

1

1

1

2

1

1

1

23

1)(

+

+=

+

+=

++=

+

+=

+

+=

++=

s

s

s

s

s

s

s

s

ss

s

sssssssH

Direct form I

Cascade form

Parallel formDirect form II


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Reference book: 6.1 Block Diagram Representation

Consider a linear constant coefficient difference equation of an LTI

system

For simplicity, we assume

N= M. If N M, some of the coefficients will be zero.

By taking the z-transform on both sides, we have

=

=

==N

k

k

k

M

k

k

k

za

zb

zX

zYzH

1

0

1)(

)()(

][][][01

knxbknyanyM

k k

N

k k

+=

==


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Direct form I

Decompose the system function such that H(z) = H1(z) H2(z)

Difference equation

System function

][][][

][][

1

0

nvknyany

knxbnv

N

k

k

M

k

k

+=

=

=

=

)()()(

1

1)(

)()()()(

2

1

10

zVzHzV

za

zY

zXzHzXzbzV

N

k

k

k

M

k

k

k

=

=

=

=

=

=

z-transform

z-transform

=

=

==N

k

k

k

M

k

k

k

za

zb

zX

zYzH

1

0

1)(

)()(

][][][01

knxbknyanyM

k

k

N

k

k += ==


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Direct form II

=

=

==N

k

k

k

M

k

k

k

za

zb

zX

zYzH

1

0

1)(

)()(

)()()()(

)()()(

1

1)(

1

0

2

1

zWzHzWzbzY

zXzHzX

za

zW

M

k

k

k

N

k

k

k

=

=

=

=

=

=

z-transform

z-transform

][][][01

knxbknyanyM

k

k

N

k

k += ==

][][

][][][

0

1

knwbny

nxknwanw

M

k

k

N

k

k

=

+=

=

=


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Example: Use block diagram to draw the direct form I and direct form II for

an LTI system with system function

21

1

9.05.11

21)(

+

+= zz

zzH

Direct form I Direct form II


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Reference book: 6.3 Basic Structures for Infinite Impulse

Response (IIR) systems

Cascade form

Without loss of generality, the system function be factorized into a

cascade of Ns second order sub-systems as follows,

Parallel form

Without loss of generality, the system function can be expressed as a

partial fraction expansion in the form,

Each sub-system in cascade and parallel forms can be realized in

either direct form I and the direct II.

=

++=

sN

k kk

kk

zaza

zbzbGzH

12

2

1

1

2

2

1

1

1

1)(

=

=

++=

sp N

k kk

kk

N

k

k

kzaza

zeezCzH

12

2

1

1

1

10

0 1)(


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Reference book: 6.3 Basic Structures for IIR systems

Example: A system function

1

41

31

1

21

32

1

411

212

811

41 111

1

1

1

1

1)(

+

+=

+=

+=

zzzzzzsH

Direct form I and II

Cascade form

Parallel form


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Criteria of the choice of realization The criteria of our choice of a specific realization are

Computational complexity : Number of multipliers and adders

Memory requirements : Number of delay (storage) unit Finite-word-length effects : Zero-pole diagram in signal quantization.

Another advantage of the cascade and parallel realizations in system

function (IIR filters) is that the system stability can be easily monitoredby investigating the pole locations in each second order subsystem. If

at least one of the poles have magnitudes larger than one, the system

will become unstable.

In all real applications, the system should be causal.


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Criteria of the choice of realization Computational complexity of IIR system function realizations:

Parallel form

Cascade form

Canonic form

Direct form INumber of 2-input addersNumber of multipliersStructure (

N=M

)

( )

12/14 ++N

( ) 12/14 ++N

12 +N

( ) 2/14 +N( ) 12/13 ++N

12 +N

N2

N2


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Reference book: 6.7 The effects of Quantization effect

Under filter coefficient quantization, the cascade or parallel realizations are

more robust than the direct forms, i.e.

Their frequency responses are more closer to the desired responses.

Consider a system function

In the presence of coefficient quantization, we have

=

=

=

=

=

++=

++=

=sps N

k kk

kk

N

k

k

k

N

k kk

kk

N

k

k

k

M

k

k

k

zaza

zeezC

zaza

zbzbG

za

zb

zH1

2

2

1

1

1

10

012

2

1

1

2

2

1

1

1

0

11

1

1

)(

( )

( )

( ) ( )( ) ( )

( ) ( ) ( )( ) ( )

=

=

=

=

=

++

+++++=

++++++=

+

+

=

sp

s

N

k kkkk

kkkk

N

k

k

kk

N

k kkkk

kkkk

N

k

k

kk

M

k

k

kk

zaazaa

zeeeezcC

zaazaa

zbbzbbG

zaa

zbb

zH

12

22

1

11

1

1100

0

12

22

1

11

222

111

1

0

1

1

1

1

)( Small error in any one coefficient can cause largeshifts of the poles (zeros).

Small error in one coefficient affects acomplex conjugate poles (zeros).


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=

++=

sN

k kk

kk

zaza

zbzbGzH

12

2

1

1

2

2

1

1

1

1)(

=

=

++

=

s

p

N

k kk

kk

N

k

k

k

zaza

zee

zCzH

12

2

1

1

1

10

0

1

)(

=

=

==N

k

kk

M

k

k

k

za

zb

zX

zYzH

1

0

1)(

)()(

( ) ( ) ( )( ) ( )=

++=

sN

k kk

kk

zaqzaq

zbqzbqGqzH

12

2

1

1

2

2

1

1

11)(

( )( ) ( )( ) ( )

=

=

++

=s

p

N

k kk

kk

N

k

k

k

zaqzaq

zeqeq

zCqzH

12

2

1

1

1

10

0

1

)(

( )

( )

=

=

=N

k

kk

M

k

k

k

zaq

zbq

zH

1

0

1

)(

Unquantized system function Quantized system function

( )xqxa

( )xqxa

( )xqxa


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Passband of ellipticfilter for directstructure with 16-bitcoefficients

Passband of ellipticfilter for parallelstructure with 16-bitcoefficients

Passband of ellipticfilter for cascadestructure with 16-bitcoefficients


For a 12th order elliptic bandpassfilter Rational system function

Cascade form

32-bit floating accuracy in coefficients

Bandpass filter

Passband

Bandpass filter for cascadestructure with 32-bit coefficients


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A simulation of the quantization effect is done.

0 5 10 15 20 25 30 35-200

-150

-100

-50

0

50

Number of quantization bits

Totalsquaredmagnitudeerror(dB)

Quantization effects on coefficients of different system realizati on

Direct Form

Cascade Form

Parallel Form

-1 -0.5 0 0.5 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Pole-zeo map of unqunatized system

Real Axis

Im

aginaryAxis

-1.5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5Pole-zeo map of system for direct sturcture w ith 10-bit coefficients

Real Axis

Imagin

aryAxis

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-500

0

500

1000

Normalized Frequency (rad/sample)

Phase(degrees)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-200

-100

0

100


Magn

itude(dB)

Frequency response of system for direct sturcture with 10-bit coefficients

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-500

0

500


Phase(degrees)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-150

-100

-50

0


Magnitud

e(dB)

Frequency response of the unqunatized system


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Structures for LTI systems

Readings

Section 9.8.2 Block Diagram Representations for Causal LTI systems

Described by Differential Equations and Rational System Functions

Section 10.8.2 Block Diagram Representations for Causal LTI systems

Described by Difference Equations and Rational System Functions

Readings from reference book

A. V. Oppenheim, et. al., Discrete-time Signal Processing, 2nd edition,Prentice-Hall, 1999

Section 6.0 Introduction

Section 6.1 Block Diagram representation of linear constant-coefficient

difference equations Section 6.3 Basic Structures for IIR systems

Section 6.7 The effects of coefficient quantization


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Laplace

Transform

Where we are

Differential

equations

State-space

model

CTFT

Hardware

Implementation

System

Characteristics

SystemResponses

Closed-loop

Systems

Continuous-time

z-Transform

Differenceequations

State-space

model

DTFT

Hardware

Implementation

System

Characteristics

SystemResponses

Closed-loop

Systems

Discrete-time

Mapping

Done in 211 To be covered In progress DoneWill be covered if available

Open-loop

Systems

Open-loop

Systems

Chapter 7 Structures for LTI Systems Student (1)

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Transcript of Chapter 7 Structures for LTI Systems Student (1)