Digital Signal Processing Reference
In-Depth Information
The radius of the circle for our normalized case is a function of the passband
gain and is given in (2.9):
−
= ε
/
n
R
(2.9)
Once the radius of the circle is known, the pole positions are determined by
calculating the necessary angles. Equation 2.10 can be used to determine the
angles for those complex poles in the second quadrant:
π
⋅
(
2
⋅
m
+
n
+
1
…
(2.10a)
θ
=
,
m
=
0,
1,
,
(
n
/2)
−
1
(
n
even)
m
2
⋅
n
π
⋅
(
2
⋅
m
+
n
+
1
θ
=
,
m
=
0
1
…
,
[(
n
−
1
)/
2
]
−
1
(
n
odd)
(2.10b)
m
2
⋅
n
It is important to remember that in this equation θ
m
represents only the angles
in the second quadrant
that have complex conjugates in the third quadrant
. In
other words, θ
m
does
not
include the pole on the real axis for odd-order functions.
For this reason, (2.10b) is valid only for odd-order filters where
n
≥ 3 since a first-
order filter would have no complex conjugate poles. (We'll see that this definition
allows a cleaner algorithm for the C code that is discussed in Appendix D.) The
precise pole locations can then be determined from (2.11) and (2.12):
σ
=
R
⋅
cos(
m
θ
)
(2.11)
m
ω
=
R
⋅
sin(
m
θ
)
(2.12)
m
In the case of odd-order transfer functions, the first-order pole will be located
at a position σ
R
equal to the radius of the circle as indicated in (2.13):
σ
=
−
R
(2.13)
R
2.2.4 Butterworth Transfer Functions
The Butterworth transfer function can be determined from the pole locations in the
LHP as we saw in the first section of this chapter. Since most of these poles are
complex conjugate pairs (except for the possible pole on the real axis for odd
orders), we can get all of the information we need from the poles in the second
quadrant. The complete approximation transfer function can be determined from a
combination of a first-order factor (for odd orders) and quadratic factors. Each of
