Digital Signal Processing Reference
In-Depth Information
ω
=
cosh(
D
)
⋅
cos(
φ
)
(2.26)
m
m
If the function has an odd-order, there will be a real pole located in the LHP
as indicted by (2.27):
(2.27)
σ
=
−
sinh(
D
)
R
2.3.4 Chebyshev Transfer Functions
Using the results of (2.27), we know that an odd-order Chebyshev transfer
function will have a factor of the form illustrated in (2.28):
sinh(
D
)
H
o
(
S
)
=
(2.28)
S
+
sinh(
D
)
The quadratic factors for the Chebyshev transfer function will take on exactly
the same form as the Butterworth case, as shown below:
B
2
m
H
(
S
)
=
(2.29)
m
2
S
+
B
⋅
S
+
B
1
m
2
m
B
= 2
⋅
σ
(2.30)
1
m
m
2
2
B
=
σ +
ω
(2.31)
2
m
m
m
We are now just about ready to define the general form of the Chebyshev
transfer function. However, one small detail still must be considered. Because
there is ripple in the passband, Chebyshev even and odd-order approximations do
not
have the same gain at ω = 0. As seen in Figure 2.13 (a result of a future
example), each approximation has a number of half-cycles of ripple in the
passband equal to the order of the filter. This forces even-order filters to have a
gain of
a
pass
at ω = 0. However, the first-order and quadratic factors we have
defined are all set to give 0 dB gain at ω = 0. Therefore, if no adjustment of gain is
made to even-order Chebyshev approximations, they would have a gain of 0 dB at
ω = 0 and a gain of −
a
pass
(that is, a gain greater than 1.0) at certain other
frequencies where the ripple peaks. A gain constant must therefore be included for
even-order transfer functions with the value of
