Structures for Discrete-Time Systems
Quote of the Day
Today'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.
Example
• Block diagram representation of
y n   a1 y n  1   a2 y n  2   b 0 x n 
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
2
Block Diagram Representation
• LTI systems with rational
system function can be
represented as constantcoefficient 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
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
3
Direct Form I
• General form of difference equation
N
M
 ˆa y n  k    bˆ x n  k 
k
k
k 0
k 0
• Alternative equivalent form
y n  
N
 a y n  k    b x n  k 
k
k 1
Copyright (C) 2005 Güner Arslan
M
k
k 0
351M Digital Signal Processing
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Direct Form I
• Transfer function can be written as
M

H z  
N
• Direct Form I Represents
N
ak z


Y z   H 2 z V z   

1 

Copyright (C) 2005 Güner Arslan
k
k 1
 M
k
V z   H 1 z X z     b k z
 k 0
1

k 1

 M
k
 
b
z
k
  k
0



 X z 

N
ak z

ak z
k
k 1
1

k
k 0
1


H z   H 2 z H 1 z   

1 

bkz
k



v n  
M
 b x n  k 
k
k 0
y n  


 V z 



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N
 a y n  k   v n 
k
k 1
5
Alternative Representation
• Replace order of cascade LTI systems
 M
k
H z   H 1 z H 2 z     b k z
 k 0


W z   H 2 z X z   

1 

1
N
a
k 1
 M
k
Y z   H 1 z W z     b k z
 k 0
Copyright (C) 2005 Güner Arslan




 1 


k
z
k
1
N

ak z
k 1


 X z 



k






w n  
N
 a w n  k   x n 
k
k 1
y n  
M
b
k
w n  k 
k 0

 W z 

351M Digital Signal Processing
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Alternative Block Diagram
• We can change the order of the cascade systems
w n  
N
 a w n  k   x n 
k
k 1
y n  
M
b
k
w n  k 
k 0
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
7
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
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
8
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
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
9
Example
• Representation of Direct Form II with signal flow graphs
w 1 n   aw 4 n   x n 
w 2 n   w 1 n 
w 3 n   b 0 w 2 n   b 1 w 4 n 
w 4 n   w 2 n  1 
y n   w 3 n 
w 1 n   aw 1 n  1   x n 
y n   b 0 w 1 n   b 1 w 1 n  1 
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
10
Determination of System Function from Flow Graph
w 1 n   w 4 n   x n 
w 2 n    w 1 n 
w 3 n   w 2 n   x n 
w 4 n   w 3 n  1 
y n   w 2 n   w 4 n 
W1 z   W 4 z   X z 
W 2 z    W1 z 
W 3 z   W 2 z   X z 
W 4 z   W 3 z z
1
Y z   W 2 z   W 4 z 
Copyright (C) 2005 Güner Arslan
W 2 z  
W 4 z  

 X z  z
1
1  z
X z z
1
1

1
1   
1  z
1
Y z   W 2 z   W 4 z 
H z  
Y z 
X z 
h n   
351M Digital Signal Processing
n 1

z
1
 
1  z
1
u n  1   
n 1
u n 
11
Basic Structures for IIR Systems: Direct Form I
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
12
Basic Structures for IIR Systems: Direct Form II
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
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Basic Structures for IIR Systems: Cascade Form
• General form for cascade implementation
M1
H z   A
z
k
 1  c
k
k
k 1
N1
k
k 1
•
M2
 1  f z  1  g
1
z
1
k 1
N2
 1  d
z
1
1  g
1
1  d

k
z

k
z
1

1

k 1
More practical form in 2nd order systems
H z  
M1

k 1
Copyright (C) 2005 Güner Arslan
b 0 k  b 1k z
1  a1 k z
1
1
 b 2k z
 a2k z
2
2
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Example
H z  
1  2z
1  0 . 75 z
1
1  z 
1

1
 z
2
1  z 1  z 
1  0 . 5 z 1  0 . 25 z 
1
 0 . 125 z
2
1  z 

1
1
1
1
1  0 . 5 z  1  0 . 25 z 
1
1
• Cascade of Direct Form I subsections
• Cascade of Direct Form II subsections
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
15
Basic Structures for IIR Systems: Parallel Form
• Represent system function using partial fraction expansion
H z  
NP
C
k
k 0
z
k
NP
NP
Ak

k 1 1  c k z
1

 1  d
k 1

Bk 1  ek z
z
k
1
1
1  d


k
z
1

• Or by pairing the real poles
H z  
NP
C
k 0
k
z
k
NS

k 1
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e 0 k  e 1k z
1  a1 k z
1
1
 a2k z
2
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Example
• Partial Fraction Expansion
H z  
1  2z
1  0 . 75 z
1
1
 z
2
 0 . 125 z
2
 8 
18

25
1  0 . 5 z  1  0 . 25 z 
1
1
• Combine poles to get
H z   8 
 7  8z
1  0 . 75 z
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1
1
 0 . 125 z
2
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17
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:
H z  
1
1  az
1
– Reverse directions of branches and interchange input and output
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
18
Example
Transpose
• Both have the same system function or difference equation
y n   a1 y n  1   a2 y n  2   b 0 x n   b 1 x n  1   b 2 x n  2 
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
19
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
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
20
Basic Structures for FIR Systems: Cascade Form
• Obtained by factoring the polynomial system function
H z  
M

n0
Copyright (C) 2005 Güner Arslan
h n z
n
MS

 b
0k
 b 1k z
1
 b 2k z
2

k 1
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Structures for Linear-Phase FIR Systems
• Causal FIR system with generalized linear phase are
symmetric:
h M  n   h n 
n  0,1,..., M
(type I or III)
h M  n    h n 
n  0,1,..., M
(type II or IV)
• Symmetry means we can half the number of multiplications
• Example: For even M and type I or type III systems:
y n  

M
M / 2 1
k 0
k 0
 h k x n  k    h k x n  k   h M / 2 x n  M / 2  
M / 2 1
M / 2 1
k 0
k 0
 h k x n  k   h M / 2 x n  M / 2    h M
M / 2 1

 h k x n  k   x n  M
M
 h k x n  k 
k M / 2 1
 k x n  M  k 
 k   h M / 2 x n  M / 2 
k 0
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
22
Structures for Linear-Phase FIR Systems
• Structure for even M
• Structure for odd M
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351M Digital Signal Processing
23