Digital Signal Processing Reference
In-Depth Information
−
1
−
1
CEI
(
rt
)
⋅
sc
(
ε
,
kn
)
v
o
=
(2.66)
n
⋅
CEI
(
kn
)
Next, the pole's real and imaginary components are determined as
⎛
⎞
⎛
⎞
[
] [
]
2
2
cn
f
(
m
),
rt
⋅
dn
f
(
m
),
rt
⋅
sn
v
,
1
−
rt
⋅
cn
v
,
1
−
rt
⎝
⎠
⎝
⎠
o
o
(2.67)
σ
=
−
[
]
⎛
⎞
2
2
2
1
−
dn
f
(
m
),
rt
⋅
sn
v
,
1
−
rt
⎝
⎠
o
[
]
⎛
⎞
2
sn
f
(
m
),
rt
⋅
dn
v
,
1
−
rt
⎝
⎠
o
ω
=
(2.68)
[
]
⎛
⎞
2
2
2
1
−
dn
f
(
m
),
rt
⋅
sn
v
,
1
−
rt
⎝
⎠
o
where
CEI
(
rt
)
⋅
(
2
⋅
m
+
1
f
(
m
)
=
,
m
=
0
1
…
,
(
n
/
2
−
1
(
n
even)
(2.69a)
n
CEI
(
rt
)
⋅
(
2
⋅
m
+
2
f
(
m
)
=
,
m
=
0
1
…
,
[(
n
−
1
)/
2
−
1
(
n
odd)
(2.69b)
n
Note the negative sign for σ
m
, which effectively moves the pole location from the
first quadrant to the second quadrant.
In the case of odd-order approximations, the first-order denominator pole will
be located on the negative real axis at
⎛
⎞
⎛
⎞
2
2
sn
v
,
1
−
rt
⋅
cn
v
,
1
−
rt
⎝
⎠
⎝
⎠
o
o
σ
=
−
(2.70)
⎛
⎞
2
2
1
−
sn
v
,
1
−
rt
⎝
⎠
o
And finally, the location of the zeros that will be purely imaginary on the
j
ω
axis are given by
σ
=
0
(2.71)
zm
