Digital Signal Processing Reference
In-Depth Information
2.3.2 Chebyshev Order
The order of the Chebyshev filter will be dependent on the specifications provided
by the user. The general form of the calculation for the order is the same as for the
Butterworth, except that the inverse hyperbolic cosine function is used in place of
the common logarithm function. As in the Butterworth case, the value of
n
actually calculated must be rounded to the next highest integer in order to
guarantee that the specifications will be met.
⎡
⎤
−
0
1
⋅
a
−
0
.
⋅
a
−
1
cosh
(
10
−
1
/(
10
−
1
stop
pass
⎣
⎦
n
=
(2.22)
C
−
1
cosh
(
ω
/
ω
)
stop
pass
2.3.3 Chebyshev Pole Locations
The poles for a Chebyshev approximation function are located on an ellipse
instead of a circle as in the Butterworth case. The ellipse is centered at the origin
of the
s-
plane with its major axis along the jω axis with intercepts of ± cosh(
D
),
while the minor axis is along the real axis with intercepts of ± sinh(
D
). The
variable
D
is defined as
−
1
−
1
sinh
(
ε
)
D
=
(2.23)
n
The pole locations can be defined in terms of
D
and an angle φ as shown in
(2.24). The angles determined locate the poles of the transfer function in the first
quadrant. However, we can use them to find the poles in the second quadrant by
simply changing the sign of the real part of each complex pole. The real and
imaginary components of the pole locations can now be defined as shown in
(2.25) and (2.26):
π
⋅
(
2
⋅
m
+
1
φ
=
,
m
=
0
1
…
,
(
n
/
2
−
1
(
n
even)
(2.24a)
m
2
⋅
n
π
⋅
(
2
⋅
m
+
1
φ
=
,
m
=
0
1
…
,
[(
n
−
1
)/
2
−
1
(
n
odd)
(2.24b)
m
2
⋅
n
σ
=
−
sinh(
D
)
⋅
sin(
φ
)
(2.25)
m
m
