Digital Signal Processing Reference
In-Depth Information
Further, by carrying out the multiplication of the two sums on the lhs of (4.86)
and comparing the resulting coe
cients of the like powers of u
−1
with their
counterparts from the rhs of (4.86), we obtain
k
a
(s)−
k
b
(s)−
r
=
c
k−r+s
(0≤k≤K)
(4.87)
r=0
c
k
=−
K
b
(s)−
r
c
k−r
.
(4.88)
r=1
Introducing the integer M via
M = N−1−K−s
(4.89)
we rewrite (4.88) in the form
K
b
(s)−
r
c
K+s+m
+
c
K+s+m−r
= 0
0≤m≤M.
(4.90)
r=1
This represents an implicit system of linear equations for the unknown coe
cients b
(s)−
r
. Moreover, the same system (4.90) can be made explicit by letting
the integer m vary from 1 to M, so that
9
=
;
.
c
K+s+1
b
(s)−
+ c
K+s
b
(s)−
++ c
s+1
b
(s)−
K
= 0
0
1
c
K+s+2
b
(s)−
+ c
K+s+1
b
(s)−
++ c
s+2
b
(s)−
K
= 0
0
1
(4.91)
.
c
M +K+s
b
(s)−
+ c
M +K+s−1
b
(s)−
++ c
M +s
b
(s)−
= 0
0
1
K
We see that (4.87) will be a system of linear equations if the su
x k is varied
from 0 to K and, therefore
9
=
;
.
a
(s)−
0
= c
s
b
(s)−
0
a
(s)−
1
= c
s+1
b
(s)−
0
+ c
s
b
(s)−
1
(4.92)
.
a
(s)−
K
= c
K+s
b
(s)−
+ c
K+s−1
b
(s)−
++ c
s
b
(s)−
0
1
K
The equivalent matrix forms of the systems (4.91) and (4.92) are
0
@
b
(s)−
1
A
0
@
c
K+s
1
A
0
@
c
K+s+1
1
A
c
K−1+s
c
s+1
1
b
(s)−
2
c
K+s+1
c
K+s
c
s+2
c
K+s+2
.
c
K+s+M
=−b
(s)−
0
(4.93)
.
.
.
.
.
c
K+s+M−1
c
K+s+M−2
c
s+M
.
.
b
(s)−
K
Search WWH ::
Custom Search