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6.1 signal flow graph representation of linear constant- coefficient difference equations6.2 basic structures for IIR system
6.2.1 direct forms6.2.2 cascade forms6.2.3 parallel forms6.2.4.lattice structure
6.3 basic structures for FIR system 6.3.1 direct forms6.3.2 cascade forms6.3.3 structures for linear-phase FIR system6.2.4.lattice structure
CHAPTER 6 structures for discrete-time system
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.
w2[n]
w1[n] w3[n]
- 1 w4[n]
6
w5[n] 3
2
z-1
y[n] x[n]
z-1
z-1
EXAMPLE
)()(2)()(
)(3)(
)()(
)()(
)()(6)(
)()()(
5311
45
31
4
13
451
2
21
zWzWzWzzY
zWzW
zWzzW
zWzW
zWzWzzW
zWzXzW
21
1
181
)42()()(
zz
zzXzY
21
1
181
)42(
)(
)()(
zz
z
zX
zYzH
]1[4][2]2[18]1[][ nxnxnynyny
y[n]
w2[n]
w1[n] w3[n]
- 1 z-1
w4[n]
6 z-1
w5[n] 3
2
z-1
x[n]
transposed form
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.
6.2 basic structures for IIR system 6.2.1 direct forms
][][][],[][
][][][
10
01
knyanwnyknxbnw
knxbknyany
N
kk
M
kk
M
kk
N
kk
1.direct I
),()()(),()()(
)()(
1
1
1
21
21
1
0
1
0
zHzWzYzHzXzW
zHzH
za
zb
za
zb
zHN
k
kk
kM
k
kN
k
kk
kM
k
k
Figure 6.14
strongpoint ： simple ；shortcoming ： more delay ； be sensitive to word length ； be inconvenient to adjust zeros and poles
][][][],[][10
knyanwnyknxbnwN
k
k
M
k
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
][][][][][3
0
2
1
knxkbknykanykk
M
k
k
N
k
k
M
k
kkN
k
kk
N
k
kk
kM
k
k
knwbny
knwanxnw
zHzWzY
zHzXzW
zbzH
za
zHwhere
zHzHzHzH
za
zb
zH
0
1
1
2
0
1
1
2
1221
1
0
][][
][][][
)()()(
)()()(
,
1
1:
)()()()(
1
2.direct II （ canonic direct form ）
b0 w[n]
z-1
z-1
z-1
y[n]
a1
a2
aN
z-1
z-1
z-1
aN-1
bM
bM-1
b2
b1
x[n]
M
k
k
N
k
k knwbnyknwanxnw01
][][][][][ ，
Figure 6.15
strongpoint ： delay is reduced half
2/]1),([
12
21
1
22
110
2/]1),([
12
21
1
22
11
0
2/]1),([
1
1
0
1
1
1
1
NMMAX
k kk
kkk
NMMAX
k kk
kkNMMAX
k
kN
k
kk
kM
k
k
zz
zbzbb
zz
zzbzH
za
zb
zH
6.2.2 cascade forms
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
NM
k
N
k kk
kokkkN
k
kk
kM
kk
zz
zzC
za
zb
zH0
]2
1[
12
21
1
11
1
0
11
6.2.3 parallel forms
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.
α 2[(N+1)/2]
z-1
α 1[(N+1)/2] γ 1[(N+1)/2]
z-1
γ 0[(N+1)/2]
α 21
z-1
γ 11 α 11
z-1
γ 01
C0
y[n] x[n]
Figure 6.20
6.2.4. supplement: lattice structure
321
3
1
8
5
24
131
1)(
zzzzH
y[n]
x[n]
-z-1 -z-1 -z-1
-1/4
1/4
-1/2
1/2
-1/3
1/3
EXAMPLE
strongpoint ： be not sensitive to limited word length
knxkhnyzkhzHM
k
kM
k
00
)(,
6.3 basic structures for FIR system
6.3.1 direct forms
Figure 6.31
transversal filter structure
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);
}
realization of direct form in c program][][][
3
0
knxkhnyk
]2
1[
1
22
110
]2
1[
1
22
11
0
'''1]0[
M
kkkk
M
kkk
kM
k
zbzbbzbzbhzkhzH
Figure 6.33
6.3.2 cascade forms
strongpoint ： be convenient to adjust zeros; time division multiplexing using a second order loop.
12/
0
12/
0
12/
0
12/
0 12/
0
]2/[]2/[])[][]([
][][]2/[]2/[][][
][][]2/[]2/[][][
][][][
.1
M
k
M
k
M
k
M
k
M
Mk
M
k
MnxMhkMnxknxkh
kMnxkMhMnxMhknxkh
knxkhMnxMhknxkh
knxkhny
evenisM
6.3.3 structures for linear-phase FIR system
Figure 6.34
2/)1(
0
2/)1(
0
2/)1(
0
2/)1(
0 2/)1(
0
])[][]([
][][][][
][][][][
][][][
:.2
M
k
M
k
M
k
M
k
M
Mk
M
k
kMnxknxkh
kMnxkMhknxkh
knxkhknxkh
knxkhny
oddisM
Figure 6.35
strongpoint ： multiplication operation is reduced half
6.3.4 supplement: lattice structure
321
3
1
8
5
24
131)( zzzzH
1/4
1/4
y[n] x[n]
-z-1 -z-1 -z-1
1/3
1/3
1/2
1/2
EXAMPLE
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 forms6.2.2 cascade forms6.2.3 parallel forms
6.3 basic structures for FIR system 6.3.1 direct forms6.3.2 cascade forms6.3.3 structures for linear-phase FIR syste
m
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
6.226.256.29
