Lecture 15

Structures for Discrete-Time Systems

Quote of the DayToday's scientists have substituted mathematics for experiments, and they wander off through equation

after equation, and eventually build a structure which has no relation to reality.

Nikola Tesla

Content and Figures are from Discrete-Time Signal Processing, 2e by Oppenheim, Shafer, and Buck, ©1999-2000 Prentice Hall Inc.

Copyright (C) 2005 Güner Arslan

351M Digital Signal Processing 2

Example• Block diagram representation of

nxb2nya1nyany 021



Block Diagram Representation• LTI systems with rational

system function can be represented as constant-coefficient difference equation

• The implementation of difference equations requires delayed values of the– input– output– intermediate results

• The requirement of delayed elements implies need for storage

• We also need means of – addition– multiplication



Direct Form I• General form of difference equation

• Alternative equivalent form

M

0kk

N

0kk knxbknya

M

0kk

N

1kk knxbknyany



Direct Form I• Transfer function can be written as

• Direct Form I Represents

N

1k

kk

M

0k

kk

za1

zbzH

zVza1

1zVzHzY

zXzbzXzHzV

zbza1

1zHzHzH

N

1k

kk

2

M

0k

kk1

M

0k

kkN

1k

kk

12

nvknyany

knxbnv

N

1kk

M

0kk



Alternative Representation• Replace order of cascade LTI systems

zWzbzWzHzY

zXza1

1zXzHzW

za1

1zbzHzHzH

M

0k

kk1

N

1k

kk

2

N

1k

kk

M

0k

kk21

M

0kk

N

1kk

knwbny

nxknwanw



Alternative Block Diagram• We can change the order of the cascade systems

M

0kk

N

1kk

knwbny

nxknwanw



Direct Form II• No need to store the same

data twice in previous system

• So we can collapse the delay elements into one chain

• This is called Direct Form II or the Canonical Form

• Theoretically no difference between Direct Form I and II

• Implementation wise – Less memory in Direct II– Difference when using

finite-precision arithmetic



Signal Flow Graph Representation• Similar to block diagram representation

– Notational differences

• A network of directed branches connected at nodes

• Example representation of a difference equation



Example• Representation of Direct Form II with signal flow graphs

nwny

1nwnw

nwbnwbnw

nwnw

nxnawnw

3

24

41203

12

41

1nwbnwbny

nx1nawnw

1110

11



Determination of System Function from Flow Graph

nwnwny

1nwnw

nxnwnw

nwnw

nxnwnw

42

34

23

12

41

zWzWzY

zzWzW

zXzWzW

zWzW

zXzWzW

42

134

23

12

41

zWzWzY z11zzX

zW

z11zzX

zW

42

1

1

4

1

1

2

nu1nunh

z1z

zXzY

zH

1n1n

1

1



Basic Structures for IIR Systems: Direct Form I



Basic Structures for IIR Systems: Direct Form II



Basic Structures for IIR Systems: Cascade Form• General form for cascade implementation

• More practical form in 2nd order systems

21

21

N

1k

1k

1k

N

1k

1k

M

1k

1k

1k

M

1k

1k

zd1zd1zc1

zg1zg1zf1AzH

1M

1k2

k21

k1

2k2

1k1k0

zaza1zbzbb

zH



Example

• Cascade of Direct Form I subsections

• Cascade of Direct Form II subsections

1

1

1

1

11

11

21

21

z25.01z1

z5.01z1

z25.01z5.01z1z1

z125.0z75.01zz21

zH



Basic Structures for IIR Systems: Parallel Form• Represent system function using partial fraction expansion

• Or by pairing the real poles

P PP N

1k

N

1k1

k1

k

1kk

1k

kN

0k

kk zd1zd1

ze1Bzc1

AzCzH

SP N

1k2

k21

k1

1k1k0

N

0k

kk zaza1

zeezCzH



Example• Partial Fraction Expansion

• Combine poles to get

1121

21

z25.0125

z5.0118

8z125.0z75.01

zz21zH

21

1

z125.0z75.01z87

8zH



Transposed Forms• Linear signal flow graph property:

– Transposing doesn’t change the input-output relation

• Transposing:– Reverse directions of all branches– Interchange input and output nodes

• Example:

– Reverse directions of branches and interchange input and output

1az1

1zH



Example

Transpose

• Both have the same system function or difference equation

2nxb1nxbnxb2nya1nyany 21021



Basic Structures for FIR Systems: Direct Form• Special cases of IIR direct form structures

• Transpose of direct form I gives direct form II • Both forms are equal for FIR systems

• Tapped delay line



Basic Structures for FIR Systems: Cascade Form• Obtained by factoring the polynomial system function

M

0n

M

1k

2k2

1k1k0

nS

zbzbbznhzH



Structures for Linear-Phase FIR Systems• Causal FIR system with generalized linear phase are

symmetric:

• Symmetry means we can half the number of multiplications• Example: For even M and type I or type III systems:

IV)or II (type M0,1,...,n nhnMh

III)or I (type M0,1,...,n nhnMh

2/Mnx2/MhkMnxknxkh

kMnxkMh2/Mnx2/Mhknxkh

knxkh2/Mnx2/Mhknxkhknxkhny

12/M

0k

12/M

0k

12/M

0k

M

12/Mk

12/M

0k

M

0k



Structures for Linear-Phase FIR Systems• Structure for even M

• Structure for odd M

Lecture 15

Documents

Transcript of Lecture 15