This section develops the properties of the Butterworth
filter which has as its basic concept a Taylor's series
approximation to the desired frequency response. The measure of
the approximation is the number of terms in the Taylor's series
expansion of the actual frequency response that can be made equal
to those of the desired frequency response. The optimal or best
solution will have the maximum number of terms equal. The
Taylor's series is a power series expansion of a function in the
form of
F
(
ω
)
=
K
0
+
K
1
ω
+
K
2
ω
2
+
K
3
ω
3
+
⋯
F
(
ω
)
=
K
0
+
K
1
ω
+
K
2
ω
2
+
K
3
ω
3
+
⋯
(1)where
K
0
=
F
(
0
)
,
K
1
=
d
F
(
ω
)
d
ω
|
ω
=
0
,
K
2
=
(
1
/
2
)
d
2
F
(
ω
)
d
ω
2
|
ω
=
0
,
e
t
c
.
,
K
0
=
F
(
0
)
,
K
1
=
d
F
(
ω
)
d
ω
|
ω
=
0
,
K
2
=
(
1
/
2
)
d
2
F
(
ω
)
d
ω
2
|
ω
=
0
,
e
t
c
.
,
(2)with the coefficients of the Taylor's
series being proportional to the various order derivatives of
F(ω)F(ω) evaluated at ω=0ω=0. A basic characteristic of
this approach is that the approximation is all performed at one
point, i.e., at one frequency. The ability of this approach to give
good results over a range of frequencies depends on the analytic
properties of the response.
The general form for the squared-magnitude response is an even
function of ωω and, therefore, is a function of ω2ω2
expressed as
F
F
(
j
ω
)
=
d
0
+
d
2
ω
2
+
d
4
ω
4
+
.
.
.
+
d
2
M
ω
2
M
c
0
+
c
2
ω
2
+
c
4
ω
4
+
.
.
.
c
2
N
ω
2
N
F
F
(
j
ω
)
=
d
0
+
d
2
ω
2
+
d
4
ω
4
+
.
.
.
+
d
2
M
ω
2
M
c
0
+
c
2
ω
2
+
c
4
ω
4
+
.
.
.
c
2
N
ω
2
N
(3)In order to obtain a solution that is a lowpass filter, the Taylor's
series expansion is performed around ω=0ω=0, requiring that
FF(0)=1FF(0)=1 and that FF(j∞)=0FF(j∞)=0, (i.e., d0=c0d0=c0, N>MN>M,
and c2N≠0c2N≠0). This is written as
F
F
(
j
ω
)
=
1
+
E
(
ω
)
F
F
(
j
ω
)
=
1
+
E
(
ω
)
(4)Combining Equation 3 and Equation 4
gives
d
0
+
d
2
ω
2
+
⋯
+
d
2
M
w
=
c
0
+
c
2
w
+
⋯
+
c
2
N
ω
2
N
+
E
(
ω
)
[
c
0
+
c
2
ω
+
⋯
]
d
0
+
d
2
ω
2
+
⋯
+
d
2
M
w
=
c
0
+
c
2
w
+
⋯
+
c
2
N
ω
2
N
+
E
(
ω
)
[
c
0
+
c
2
ω
+
⋯
]
(5)The best Taylor's approximation requires that FF(jω)FF(jω) and the
desired ideal response have as many terms as possible equal in their
Taylor's series expansion at a given frequency. For a lowpass
filter, the expansion is around ω=0ω=0, and this requires
E(ω)E(ω) have as few low-order ωω terms as possible. This
is achieved by setting
c
0
=
d
0
,
c
2
=
d
2
,
⋯
c
2
M
=
d
2
M
,
⋯
c
2
M
+
2
=
0
,
c
2
N
-
2
=
0
,
c
2
N
≠
0
c
0
=
d
0
,
c
2
=
d
2
,
⋯
c
2
M
=
d
2
M
,
⋯
c
2
M
+
2
=
0
,
c
2
N
-
2
=
0
,
c
2
N
≠
0
(6)Because the ideal response in the passband is a constant, the
Taylor's series approximation is often called “maximally flat".
Equation 6 states that the numerator of the transfer
function may be chosen arbitrarily. Then by setting the denominator
coefficients of FF(s) equal to the numerator coefficients plus one
higher-order term, an optimal Taylor's series approximation is
achieved [2].
Since the numerator is arbitrary, its coefficients can be
chosen for a Taylor's approximation to zero at ω=∞ω=∞.
This is accomplished by setting d0=1d0=1 and all other d's equal
zero. The resulting magnitude-squared function is[2]
F
F
(
j
ω
)
=
1
1
+
c
2
N
ω
2
N
F
F
(
j
ω
)
=
1
1
+
c
2
N
ω
2
N
(7)The value of the constant c2Nc2N determines at which value of
ωω the transition of passband to stopband occurs. For this
development, it is normalized to c2N=1c2N=1, which causes the
transition to occur at ω=1ω=1. This gives the simple form for
what is called the Butterworth filter
F
F
(
j
ω
)
=
1
1
+
ω
2
N
F
F
(
j
ω
)
=
1
1
+
ω
2
N
(8)This approximation is sometimes called “maximally flat" at both ω=0ω=0 and ω=∞ω=∞, since it is simultaneously a Taylor's series
approximation to unity at ω=0ω=0 and to zero at ω=∞ω=∞. A
graph of the resulting frequency response function is shown in
Figure 1 for several NN.
The characteristics of the normalized Butterworth filter
frequency response are:
- Very close to the ideal near ω=0ω=0 and
ω=∞ω=∞,
- Very smooth at all frequencies with a
monotonic decrease from ω=0ω=0 to ∞∞, and
- Largest
difference between the ideal and actual responses near the
transition at ω=1ω=1 where |F(j1)|2=1/2|F(j1)|2=1/2.
Although not part of the approximation addressed, the phase curve
is also very smooth.
An important feature of the Butterworth filter is the closed-
form formula for the solution, F(s)F(s). The
expression for FF(s)FF(s) may be determined as
F
(
s
)
F
(
-
s
)
=
1
1
+
(
-
s
2
)
N
F
(
s
)
F
(
-
s
)
=
1
1
+
(
-
s
2
)
N
(9)This function has 2N2N poles evenly spaced around a unit radius circle and
2N2N zeros at infinity. The determination of F(s)F(s) is very simple. In
order to have a stable filter, F(s)F(s) is selected to have the NN
left-hand plane poles and NN zeros at infinity; F(-s)F(-s) will necessarily
have the right-hand plane poles and the other N zeros at infinity. The
location of these poles on the complex ss plane for N=1,2,3N=1,2,3, and
4 is shown in Figure 2.
Because of the geometry of the pole positions, simple
formulas are easy to derive for the pole locations. If the real
and imaginary parts of the pole location are denoted as
s
=
u
+
j
w
s
=
u
+
j
w
(10)the locations of the NN poles are given by
u
k
=
-
cos
(
k
π
/
2
N
)
u
k
=
-
cos
(
k
π
/
2
N
)
(11)
ω
k
=
sin
(
k
π
/
2
N
)
ω
k
=
sin
(
k
π
/
2
N
)
(12)for NN values of kk where
k
=
±
1
,
±
3
,
±
5
,
.
.
.
,
±
(
N
-
1
)
for
N
even
k
=
±
1
,
±
3
,
±
5
,
.
.
.
,
±
(
N
-
1
)
for
N
even
(13)
k
=
0
,
±
2
,
±
4
,
.
.
.
,
±
(
N
-
1
)
for
N
odd
k
=
0
,
±
2
,
±
4
,
.
.
.
,
±
(
N
-
1
)
for
N
odd
(14)Because the coefficients of the numerator and denominator
polynomials of F(s)F(s) are real, the roots occur in complex
conjugate pairs. The conjugate pairs in Equation 11,Equation 12 can be
combined to be the roots of second-order polynomials so that for NN
even, F(s)F(s) has the partially factored form of
F
(
s
)
=
∏
k
1
s
2
+
2
cos
(
k
π
/
2
N
)
s
+
1
F
(
s
)
=
∏
k
1
s
2
+
2
cos
(
k
π
/
2
N
)
s
+
1
(15)for k=1,3,5,...,N-1k=1,3,5,...,N-1. For NN odd,
F(s)F(s) has a single real pole and, therefore, the form
F
(
s
)
=
1
s
+
1
∏
k
1
s
2
+
2
cos
(
k
π
/
2
N
)
s
+
1
F
(
s
)
=
1
s
+
1
∏
k
1
s
2
+
2
cos
(
k
π
/
2
N
)
s
+
1
(16)for k=2,4,6,⋯,N-1k=2,4,6,⋯,N-1
This is a convenient form for the cascade and parallel realizations
discussed in elsewhere.
A single formula for the pole locations for both even and odd
NN is
u
k
=
-
sin
(
(
2
k
+
1
)
π
/
2
N
)
u
k
=
-
sin
(
(
2
k
+
1
)
π
/
2
N
)
(17)
ω
k
=
cos
(
(
2
k
+
1
)
π
/
2
N
)
ω
k
=
cos
(
(
2
k
+
1
)
π
/
2
N
)
(18)for NN values of kk where k=0,1,2,...,N-1k=0,1,2,...,N-1
One of the important features of the Butterworth filter design
formulas is that the pole locations are found by independent
calculations which do not depend on each other or on factoring a
polynomial. A FORTRAN program which calculates these values is
given in the appendix as Program 8. Mathworks has a powerful command
for designing analog and digital Butterworth filters.
The classical form of the Butterworth filter given in Equation 8
is discussed in many books [3], [1], [4], [5], [2]. The less
well-known form given in Equation 6 also has many useful
applications [2]. If the frequency location of unwanted
signals is known, the zeros of the transfer function given by the
numerator can be set to best reject them. It is then possible to
choose the pole locations so as to have a passband as flat as the
classical Butterworth filter by using Equation 6. Unfortunately,
there are no formulas for the pole locations; therefore, the
denominator polynomial must be factored.
This section has derived design procedures and formulas for
a class of filter transfer functions that approximate the ideal
desired frequency response by a Taylor's series. If the
approximation is made at ω=0ω=0 and ω=∞ω=∞, the
resulting filter is called a Butterworth filter and the response is
called maximally-flat at zero and infinity. This filter has a very
smooth frequency response and, although not explicitly designed for,
has a smooth phase response. Simple formulas for the pole locations
were derived and are implemented in the design program in the
appendix of this book.
This section considers the process of going from given
specifications to use of the approximation results derived in the
previous section. The Butterworth filter is the simplest of the four
classical filters in that all the approximation effort is placed at
two frequencies: ω=0ω=0 and ω=∞ω=∞. The transition
from passband to stopband occurs at a normalized frequency of
ω=1ω=1. Assuming that this transition frequency or bandedge
can later be scaled to any desired frequency, the only parameter to
be chosen in the design process is the order NN.
The filter specifications that are consistent with what is
optimized in the Butterworth filter are the degree of “flatness" at
ω=0ω=0 (DC) and at ω=∞ω=∞. The higher the order,
the flatter the frequency response at these two points. Because of
the analytic nature of rational functions, the flatter the response
is at ω=0ω=0 and ω=∞ω=∞, the closer it stays to the
desired response throughout the whole passband and stopband. An
indirect consequence of the filter order is the slope of the
response at the transition between pass and stopband. The slope of
the squared-magnitude frequency response at ω=1ω=1 is
s
=
F
F
'
(
j
1
)
=
-
N
/
2
s
=
F
F
'
(
j
1
)
=
-
N
/
2
(19)The effects of the increased flatness and increased transition
slope of the frequency response as N increases are illustrated in
Figure 1 from Design of Infinite Impulse Response (IIR) Filters by Frequency Transformations.
In some cases specifications state the response must stay
above or below a certain value over a given frequency band. Although
this type of specification is more compatible with a Chebyshev error
optimization, it is possible to design a Butterworth filter to meet
the requirements. If the magnitude of the frequency response of the
filter over the passband of 0<ω<ωP0<ω<ωP must remain
between unity and GG, where ωp<1ωp<1 and G<1G<1, the
required order is found by the smallest integer NN satisfying
N
≥
log
(
(
1
/
G
)
2
-
1
)
1
log
(
ω
p
)
N
≥
log
(
(
1
/
G
)
2
-
1
)
1
log
(
ω
p
)
(20)This is illustrated in Figure 3 where |F||F| must remain above 0.9
for ωω up to 0.9, i.e., GG = 0.9 and ωpωp = 0.9. These
requirements require an order of at least N=7N=7.
If stopband performance is stated in the form of requiring
that the response stay below a certain value for frequency above a
certain value, i.e., |F|<G|F|<G for ω>ωsω>ωs, the order is
determined by the same formula Equation 20 with ωpωp replaced
by ωsωs.
Note |F(j1)|=1/(2)|F(j1)|=1/(2) which is called the “half power" frequency
because |F(j1)|2=1/2|F(j1)|2=1/2. This frequency is normalized to one for the
theory but can be scaled to any value for applications.
To illustrate the calculations, a lowpass Butterworth filter
is designed. It is desired that the frequency response stay above
0.8 for frequencies up to 0.9. The formula Equation 20 for
determining the order gives a value of 2.73; therefore, the order is
three. The analytic function corresponding to the squared-magnitude
frequency response in Equation 9 is
|
F
(
j
ω
)
|
2
=
1
1
+
ω
6
|
F
(
j
ω
)
|
2
=
1
1
+
ω
6
(21)The transfer function corresponding to the left-half-plane poles of
F'(s) are calculated from Equation 11 to give
F
(
s
)
=
1
(
s
+
1
)
(
s
+
0
.
5
+
j
0
.
866
)
(
s
+
0
.
5
-
j
0
.
866
)
F
(
s
)
=
1
(
s
+
1
)
(
s
+
0
.
5
+
j
0
.
866
)
(
s
+
0
.
5
-
j
0
.
866
)
(22)
F
(
s
)
=
1
(
s
+
1
)
(
s
2
+
s
+
1
)
F
(
s
)
=
1
(
s
+
1
)
(
s
2
+
s
+
1
)
(23)
F
(
s
)
=
1
s
3
+
2
s
2
+
2
s
+
1
F
(
s
)
=
1
s
3
+
2
s
2
+
2
s
+
1
(24)The frequency response is obtained by setting s=jωs=jω which
has a plot illustrated in Figure 1 for N=3N=3. The pole locations
are the same as shown in Figure 2c.
-
Oppenheim, A. V. and Schafer, R. W. (1999). Discrete-Time Signal Processing. (Second). [Earlier editions in 1975 and 1989]. Englewood Cliffs, NJ: Prentice-Hall.
-
Parks, T. W. and Burrus, C. S. (1987). Digital Filter Design. New York: John Wiley & Sons.
-
Rabiner, L. R. and Gold, B. (1975). Theory and Application of Digital Signal Processing. Englewood Cliffs, NJ: Prentice-Hall.
-
Taylor, F. J. (1983). Digital Filter Design Handbook. New York: Marcel Dekker, Inc.
-
Valkenburg, M.E. Van. (1982). Analog Filter Design. New York: Holt, Rinehart, and Winston.
