Digital Signal Processing Reference
In-Depth Information
Employing (4.89), we can express (4.104) in the explicit form as
9
=
;
.
c
s
b
(s)+
+ c
s+1
b
(s)+
+ c
s+2
b
(s)+
+ + c
s+K
b
(s)+
K
= 0
0
1
2
c
s+1
b
(s)+
+ c
s+2
b
(s)+
+ c
s+3
b
(s)+
+ + c
s+K+1
b
(s)+
K
= 0
0
1
2
.
c
M +s
b
(s)+
+ c
M +s+1
b
(s)+
+ c
M +s+2
b
(s)+
++ c
M +s+K
b
(s)+
= 0
0
1
2
K
(4.105)
These equations have their matrix representations given by
0
@
1
A
0
@
1
A
0
@
1
A
b
(s)+
1
b
(s)+
2
b
(s)+
3
c
s
b
(s)+
c
s+1
c
s+2
c
s+3
c
s+K
0
c
s+1
b
(s)+
c
s+2
c
s+3
c
s+4
c
s+K+1
0
c
s+2
b
(s)+
c
s+3
c
s+4
c
s+5
c
s+K+2
=−
.
0
.
.
.
.
.
.
.
.
b
(s)+
K
.
c
s+M
b
(s)+
c
s+M +1
c
s+M +2
c
s+M +3
c
s+K+M
0
(4.106)
The expansion coe
cients{a
(s)+
}of the numerator polynomial A
(s)
K
(u) can
be obtained from the inhomogeneous part of the positive powers of the ex
pansion variable from (4.103)
r
9
=
;
(4.107)
c
s
b
(s)+
1
+ c
s+1
b
(s)+
2
+ c
s+3
b
(s)+
3
+ + c
s+K−1
b
(s)+
K
= a
(s)+
1
c
s
b
(s)+
2
+ c
s+1
b
(s)+
3
+ + c
s+K−2
b
(s)+
K
= a
(s)+
2
.
.
.
.
.
c
s
b
(s)+
= a
(s)+
K
K
or through the equivalent matrix form
0
@
1
A
0
@
1
A
0
@
1
A
a
(s)+
1
a
(s)+
2
a
(s)+
3
b
(s)+
1
b
(s)+
2
b
(s)+
3
c
s
c
s+1
c
s+2
c
s+K−1
0 c
s
c
s+1
c
s+K−2
0
0 c
s
c
s+K−3
=
.
(4.108)
.
.
.
.
.
.
.
.
a
(s)+
K
.
b
(s)+
K
0
0
0 c
s
A detailed derivation from Ref. [5, 17] gives the following general relation
G
FPT(s)+
n
(u) =G
LCF(s)
e,n
(u)
(u) =G
CF(s)
2n
=G
CCF(s)
e,n
(u)
(n = 1, 2, 3 ...)
(4.109)
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