Digital Signal Processing Reference
In-Depth Information
fraction are present only in (2.135) for determination of the c
n
's from c
0
to
c
K−1
. Starting from c
K
, any coe
cient c
m
is always expressed by means
of the K preceding coe
cients c
m−1
,c
m−2
,c
m−3
,...,c
m−K
, each of which is
individually multiplied by the respective coe
cients b
K−1
,b
K−2
,b
K−3
,...,b
0
of the denominator polynomial of the generating fraction. Finally, the sum
of the latter products is divided by b
K
. In particular, we shall single out the
last equation from the string (2.136) as
b
0
c
n
+ b
1
c
n+1
+ b
2
c
n+2
++ b
K
c
n+K
= 0.
(2.137)
Since here the constant coe
cients{b
r
}(0≤r≤K) are known, we can rec
ognize (2.137) as a difference equation whose solution is given by the complete
integral as the geometric sequence
K
d
k
u
k
.
c
n
=
(2.138)
k=1
The u
k
's are the roots of the characteristic polynomial L
K
(u) due to (2.138)
L
K
(u) = b
0
+ b
1
u + b
2
u
2
++ b
K
u
K
.
(2.139)
Here, the d
k
's are some arbitrary integration constants which can be deter
mined by imposing the K boundary conditions to (2.137). These K initial
conditions can always be reduced to the request of passing the continuous
curve c(t) through a fixed number K of given points t
n
= nt≡nτ. In this
way, the values of t
0
,t
1
,t
2
,...,t
K
will be associated with c
0
,c
1
,c
2
,...,c
K
,
respectively. This procedure gives K linear equations
c
0
= d
1
+ d
2
+ d
3
++ d
K
c
1
= d
1
u
1
+ d
2
u
2
+ d
3
u
3
++ d
K
u
K
c
3
= d
1
u
1
+ d
2
u
2
+ d
3
u
3
++ d
K
u
2
K
.
c
K
= d
1
u
1
+ d
2
u
2
+ d
3
u
3
++ d
K
u
K
=
;
.
(2.140)
From here, we can extract the values of the d
k
's that will acquire their forms
d
1
= c
0
(2.141)
9
=
−u
2
c
0
+ c
1
u
1
d
1
=
−u
2
K = 2
(2.142)
−u
2
c
0
+ c
1
u
2
;
d
2
=
−u
1
9
=
d
1
=
u
2
u
3
c
0
−(u
2
+ u
3
)c
1
+ c
2
(u
1
−u
2
)(u
1
−u
3
)
d
2
=
u
1
u
3
c
0
−(u
1
+ u
3
)c
1
+ c
2
(u
2
−u
1
)(u
2
−u
3
)
d
3
=
u
1
u
2
c
0
−(u
1
+ u
2
)c
1
+ c
2
(u
3
K = 3
(2.143)
;
−u
1
)(u
3
−u
2
)
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