Digital Signal Processing Reference
In-Depth Information
All terms on the right side of this equation must be included, because combining
terms of the right side yields the left side. The coefficient for the simple-root term is
calculated from (F.6), and the coefficients of the repeated-root terms are calculated
from the equation
(r - j)!
d
r - j
1
ds
r - j
[(s - p
2
)
r
F(s)]
`
k
2j
=
(F.8)
s =p
2
G(s), d
0
G(s)/ds
0
with
0! = 1,
and for any function
= G(s).
This equation is given
without proof [1].
The preceding developments apply to complex poles as well as real poles. Sup-
pose that
F(s)
has a single pair of complex poles at
s = a ; jb.
If we let
p
1
= a - jb
and
p
2
= a + jb,
then, with the numerator order of
F(s)
less than that of the denomi-
nator, (F.4) can be written as
k
1
s - a + jb
+
k
2
s - a - jb
+
k
3
s - p
3
k
n
s - p
n
.
Á
F(s) =
+
+
(F.9)
The coefficients
k
1
and
k
2
can be evaluated by (F.6), as before. It is seen, however,
that
k
1
and
k
2
are complex valued and that
k
2
is the conjugate of
k
1
.
All of the
remaining coefficients are real.
