Digital Signal Processing Reference
In-Depth Information
then the reciprocal of
p
gives the inverse Chebyshev pole position as
σ
−
j
ω
σ
ω
−
1
p
=
=
−
j
(2.38)
2
2
2
2
(
σ
+
j
ω
)
⋅
(
σ
−
j
ω
)
σ
+
ω
σ
+
ω
Notice that if a pole's distance from the origin is greater than one, the
reciprocal's distance will be less than one, and vice versa. In addition, the position
of the pole is reflected across the real axis, so although the original pole position
may be in the second quadrant, the reciprocal is located in the third quadrant.
Consequently, if we are able to determine pole positions for the standard
Chebyshev approximation function as discussed in the previous section, we should
have little problem finding the inverse Chebyshev pole locations.
Let's derive the mathematical equations necessary to determine the pole
locations for the inverse Chebyshev approximation function along the same lines
as we did for the standard Chebyshev case. First,
D
i
will be defined in terms of ε
i
in (2.39).
−
1
−
1
sinh
(
ε
)
i
D
=
(2.39)
i
n
Next, we can define the pole locations in the second quadrant in the manner
of the previous section as shown in (2.40)-(2.42), remembering that these primed
values must still be inverted.
′
σ
=
−
sinh(
D
)
⋅
sin(
φ
)
(2.40)
m
i
m
′
ω
=
cosh(
D
)
⋅
cos(
φ
)
(2.41)
m
i
m
π
⋅
(
2
⋅
m
+
1
φ
=
,
m
=
0
1
…
,
(
n
/
2
−
1
(
n
even)
(2.42a)
m
2
⋅
n
π
⋅
(
2
⋅
m
+
1
φ
=
,
m
=
0
1
…
,
[(
n
−
1
)/
2
−
1
(
n
odd)
(2.42b)
m
2
⋅
n
We can determine the final pole locations by inverting these poles as
indicated in (2.43) and (2.44):
′
σ
m
(2.43)
σ
=
m
2
2
σ
′
+
ω
′
m
m
