CHAPTER 6 structures for discretetime system
6.1 signal flow graph representation of linear constantcoefficient difference equations
6.2 basic structures for IIR system
6.2.1 direct forms
6.2.2 cascade forms
6.2.3 parallel forms
6.2.4.lattice structure
6.3 basic structures for FIR system
6.3.1 direct forms
6.3.2 cascade forms
6.3.3 structures for linear-phase FIR system
6.2.4.lattice structure
6.1 signal flow graph representation of linear
constant-coefficient difference equations
Figure 6.8
Figure 6.9
node：source、sink、network
branch：constant、z-1、1、-1
The value at each node in a graph is the sum of the outputs of all the branches
entering the node.
EXAMPLE
z-1
w3[n]
w1[n]
2
y[n]
x[n]
z-1
w2[n]
-1
w4[n]
6
W1 ( z )  X ( z )  W2 ( z )
W2 ( z )  6 z 1W5 ( z )  W4 ( z )
W3 ( z )  W1 ( z )
W4 ( z )  z 1W3 ( z )
W5 ( z )  3W4 ( z )
Y ( z )  z 1W1 ( z )  2W3 ( z )  W5 ( z )
3
w5[n]
z-1
(2  4 z 1 )
Y ( z)  X ( z)
1  z 1  18 z  2
Y ( z)
(2  4 z 1 )
H ( z) 

X ( z ) 1  z 1  18 z 2
y[n]  y[n  1]  18 y[n  2]  2 x[n]  4 x[n  1]
transpose：
reverse the directions of all branches
reverse the roles of the input and output
The relationship between the input and output does not change.
transposed
form
z-1
w3[n]
w1[n]
2
x[n]
y[n]
-1
w2[n]
z-1
w4[n]
6
3
w5[n]
z-1
6.2 basic structures for IIR system
6.2.1 direct forms
1.direct I
M

H z  
bk z  k
k 0
N
1

M

ak z  k

k 0
k 1
1
bk z  k
N
1

 H1 ( z ) H 2 ( z )
ak z  k
k 1
W ( z )  X ( z ) H1 ( z ), Y ( z )  W ( z ) H 2 ( z ),
N
M
k 1
k 0
y[n]   ak y[n  k ]   bk x[n  k ]
M
N
k 0
k 1
w[n]   bk x[n  k ], y[n]  w[n]   ak y[n  k ]
M
w[n] 
N
b x[n  k ], y[n]  w[n]   a y[n  k ]
k
k
k 0
k 1
Figure 6.14
strongpoint：simple；
shortcoming：more delay；be sensitive to word length ；be inconvenient to
adjust zeros and poles
y[n] 
2
3
k 1
k 0
 a[k ]y[n  k ]   b[k ]x[n  k ]
double x[3],y[2];
while(!eof(in_file))
{
for(k=3;k>0;k++)
//M=3
x[k]=x[k-1];
x[0]=getc(in_file)-128;
for(k=2;k>0;k++)
/N=2
y[k]=y[k-1];
for(k=0,y[0]=0;k<=3;k++)
y[0]+=b[k]*x[k];
for(k=1;k<=2;k++)
y[0]+=a[k]*y[k]
putc(out_file,y[0]+128);
}
realization of direct
form in c program
2.direct II（canonic direct form）
M
b z
k
H z  
k 0
N
1

k
 H1 ( z ) H 2 ( z )  H 2 ( z ) H1 ( z )
ak z  k
k 1
where : H 2 z  
, H1 z  
1
N
1

ak z  k
k 1
W ( z)  X ( z)H 2 ( z)
Y ( z )  W ( z ) H1 ( z )
N
 w[ n]  x[ n] 
a
k w[ n
k 1
M
y[ n] 
 b w[n  k ]
k
k 0
 k]
M

k 0
bk z  k
N
w[ n]  x[ n] 
M
 a w[n  k ]，y[n]   b w[n  k ]
k
k
k 1
k 0
b0
w[n]
x[n]
y[n]
a1
a2
z-1
z-1
z-1
b1
z-1
b2
aN-1
aN
bM-1
z-1
z-1
bM
Figure 6.15
strongpoint：delay is reduced half
6.2.2 cascade forms
M
H z  

bk z  k
k 0
N
1


ak z  k
k 1
[ MAX ( M , N ) 1] / 2


k 1
[ MAX ( M , N ) 1] / 2
 H z   b
k
k 1
0
[ MAX ( M , N ) 1] / 2

k 1
1  1k z 1   2 k z  2
1  1k z 1   2 k z  2
b0 k  b1k z 1  b2 k z  2
1  1k z 1   2k z  2
reason of being nonuniform：pairing manner of real zeros and poles；
order of cascade connection;
pairing manner of zeros and poles。
Figure 6.18
strongpoint：lower sensitivity to coefficient quantization than that of direct form;
search the least-error ones because of the effects of limited word length;
pairing manner of real zeros and poles；
order of cascade connection;
be convenient to adjust zeros and poles;
time division multiplexing using a second order loop.
shortcoming：not as fast as parallel form
6.2.3 parallel forms
M
H z  

bk z  k
k 0
N
1

M N

ak z  k
C z
k
k 0
[
k

N 1
]
2

k 1
 ok   1k z 1
1  1k z 1   2 k z  2
k 1
strongpoint：lower sensitivity to coefficient quantization than that of direct form;
less error because of the effects of limited word length;
be convenient to adjust poles;
fast hardware realization.
shortcoming：can not adjust zeros; can not be used in the filters with high precision
requirement of zero location, such as notch filter and narrowband
bandstop filter.
C0
γ01
x[n]
y[n]
z-1
γ11
α11
z-1
α21
γ0[(N+1)/2]
z-1
α1[(N+1)/2]
γ1[(N+1)/2]
z-1
α2[(N+1)/2]
Figure 6.20
6.2.4. supplement: lattice structure
EXAMPLE
x[n]
-1/3
-1/2
-1/4
1/3
1/2
1/4
-1
-z
-1
-z
-1
-z
1
H ( z) 
13 1 5  2 1 3
1 z  z  z
24
8
3
strongpoint ：be not sensitive to limited word length
y[n]
6.3 basic structures for FIR system
6.3.1 direct forms
H z  
M

hk z  k , y (n) 
k 0
M
 hk xn  k 
k 0
Figure 6.31
transversal filter structure
3
y[n] 
 h[k ]x[n  k ]
realization of direct
form in c program
k 0
while(!eof(in_file))
{
for(k=3;k>0;k++)
//M=3
x[k]=x[k-1];
x[0]=getc(in_file)-128;
for(k=0,y[0]=0;k<=3;k++)
y[0]+=h[k]*x[k];
putc(out_file,y[0]+128);
}
6.3.2 cascade forms
H z  
M

k 0
[
hk z  k  h[0]
M 1
]
2
 1  b
M 1
]
2
  b
1
2
z

b
z

1k
2k
k 1
[
1
2
'

b
'
z

b
'
z
0k
1k
2k
k 1
Figure 6.33
strongpoint：be convenient to adjust zeros;
time division multiplexing using a second order loop.

6.3.3 structures for linear-phase FIR system
1.M is even
M
y[n] 
 h[k ]x[n  k ]
k 0
M / 2 1

M

h[k ]x[n  k ]  h[ M / 2]x[n  M / 2] 

h[k ]x[n  k ]  h[ M / 2]x[n  M / 2] 
k 0
M / 2 1

k 0
M / 2 1

 h[k ]x[n  k ]
k  M / 2 1
M / 2 1
 h[M  k ]x[n  M  k ]
k 0
 h[k ]( x[n  k ]  x[n  M  k ])  h[M / 2]x[n  M / 2]
k 0
Figure 6.34
2.M is odd :
M
y[n] 
 h[k ]x[n  k ]
k 0
( M 1) / 2


M
h[k ]x[n  k ] 
k 0
k  ( M 1) / 2
( M 1) / 2


k 0
( M 1) / 2

 h[k ]x[n  k ]
( M 1) / 2
h[k ]x[n  k ] 
 h[M  k ]x[n  M  k ]
k 0
 h[k ]( x[n  k ]  x[n  M  k ])
k 0
Figure 6.35
strongpoint：multiplication operation is reduced half
6.3.4 supplement: lattice structure
EXAMPLE
x[n]
1/4
1/2
1/3
1/4
1/2
1/3
-z-1
H ( z)  1 
-z-1
13 1 5  2 1 3
z  z  z
24
8
3
-z-1
y[n]
6.4 effects of limited word length
different infinite-precision realization structures：the same result,
different operation quantity, speed, storage space;
different finite-precision realization structures：different results,
different error of frequency response, different difficulties to adjust
frequency response。
Reasons of error：
1. coefficient quantity of filter’s ：frequency response alters，even is
instable; the more dense zeros and poles are, the more sensitive to
effects of limited word length
2. round in operations
High-order IIR should try to avoid using direct form;
FIR（generally, zeros distribute uniformly）use linear-phase direct
form widely.
summary
6.1 signal flow graph representation
6.2 basic structures for IIR system
6.2.1 direct forms
6.2.2 cascade forms
6.2.3 parallel forms
6.3 basic structures for FIR system
6.3.1 direct forms
6.3.2 cascade forms
6.3.3 structures for linear-phase FIR system
consideration：
complexity of realization;
operation quantity and storage quantity ;
sensitivity to effects of limited word length.
requirement：
transformation between system function and flow graph representation;
strongpoints and shortcomings of different structures.
exercises
6.22
6.25
6.29