3 Butterworth approximation
The classical method of analog filters design is Butterworth approximation. The Butterworth filters are also known as maximally flat filters. Squared magnitude response of a Butterworth low-pass filter is defined as follows
where 
- radian frequency, 
- constant scaling frequency, 
- order of the filter.Some properties of the Butterworth filters are:
gain at DC is equal to 1;
has a maximum at 
- The first
derivatives of (3.1) are equal to zero at 
. This is why Butterworth filters are known as maximally flat filters.
Poles locations
Using property (1.26) expression (3.1) can be rearranged to the form
Given 
this expression can be written as follows
Function 
has 
poles and doesn't have any finite zeros. It is easy to see that if 
is a pole of (3.2), then 
is also a pole of (3.2). In order to find the poles of transfer function 
that satisfy (3.2), we have to select one pole from each pair 
of the poles of expression (3.2). As it was mentioned before, the poles of a valid filter have to have negative real parts. The poles of (3.2) can be found as roots of equation
Observing that 
, where 
stands for the odd number, roots 
of (3.3) can be obtained as solutions to the equation
The solution of (3.4) can be presented in the form
So, roots (3.5) are the poles of .All
poles lie on a circle of radius 
in the complex s-plane. Since the difference 
has the same value for all roots, it can be concluded that the poles are equally spaced on the circumference.
Fig 3.1 Pole locations of the squared magnitude response for the
Butterworth low-pass filters with orders N=6 and N=5. Scaling frequency
.It is easy to see that
for the 
. Therefore, N roots of (3.5) have the negative real parts; these are the poles of 
. The remaining roots are the poles of 
.
Real parts in (3.5) are also never zero, so poles never fall to the
imaginary axis. For the Butterworth low-pass filters with odd orders,
two of the poles have zero imaginary parts, so they fall on the real
axis in the s-plane. For the filters with even orders, all imaginary and
real parts are nonzero.The poles of Butterworth filter lie on left half of the s-plane, and they can be given as follows
Minimum order determination
The practical low-pass filter specification is determined by four parameters:
.
The first step to design a filter with these parameters is to determine
the minimum order of the filter that meets this specification. The
signal attenuation for the Butterworth approximation can be expressed as
follows
Applying (3.7) to the pass-band and stop-band edges results in the following system of two equations
These equations can be rearranged as 
Variables 
and 
must be obtained from system (3.9). The order of the filter is an integer variable, and scaling frequency is the real variable. Parameters 
are real and they are set in the specification (parameters 
were replaced with 
).
In general, system of equations (3.9) doesn't have a precise solution
for those kinds of variables. But it can be solved if integer variable 
is replaced with the real variable 
.
The precise solution of system (3.10) can be easily found as
(3.11) is a precise solution to the equations (3.10). It can be
used to determine the approach to the solution of the system (3.9).
Minimum order of the Butterworth filter that meets specification can be
found as
where brackets [] stand for the nearest integer exceeding 
.Since
,
the solution of the equations (3.9) will be exceeding the specification
requirements. In order to complete the Butterworth filter design, the
scaling frequency 
must be determined.Cutoff frequency determination
Cutoff frequency for the ideal low-pass filters is determined as a frequency up to which the signals pass well and the signals beyond it are rejected. Due to the transition band in the practical filters, the definition of cutoff frequency is unclear. It is a common practice to characterize practical filters with a natural cutoff frequency, which is determined as a frequency at which the power gain is 1/2 of the power gain at the passband frequencies. Since the power of the signal is proportional to the amplitude, the magnitude response at the natural cutoff frequency is as follows
Converting gain to decibels results in
Therefore, the natural cutoff frequency can be determined as a
frequency where the signal loss through the filter is approximately 3
dB.It is easy to see from (3.1), that for Butterworth filters, the natural cutoff frequency
, and it doesn't depend on the order of the filter.The minimum required order for the Butterworth filter could be computed using (3.12). The scaling frequency
cannot be determined to precisely satisfy both of the edge conditions
(3.9). The scaling frequency can be determined in such a way as to
satisfy the system of inequalities
where the order of the filter 
is determined from (3.12). If these inequalities are satisfied, then
Butterworth filter meets or exceeds the specification requirements.System of inequalities (3.14) has multiple solutions. Let us consider some practical solutions.
-
Case 1. Specification requirements at the pass-band edge are met precisely.
In this case, the first inequality in (3.14) should be replaced with equality, and scale frequency can be found as follows
Inserting scale frequency (3.15) to (3.7), the attenuation at the stop-band edge can be computed
This attenuation must satisfy the second inequality in (3.14).
To demonstrate it, the following equations for the precise solution and
for the solution of the system that precisely meets the specification
requirements at the pass-band edge can be considered.
Since 
and 
, it is easy to see from (3.17) that the second inequality in (3.14) is satisfied.
For the system that precisely meets the specification requirements at the pass-band edge, the scale frequency can be computed using (3.15), and the attenuation at the stop-band will be exceeding the specification requirements.
-
Case2. Specification requirements at the stop-band edge are met precisely.
In this case, the second inequality in (3.14) should be replaced with equality. The scale frequency can be expressed as follows
Attenuation at the pass-band edge
where scale frequency 
is expressed by (3.18).
Equations for the precise solution and for the solution of the system that precisely meets the specification requirements at the stop-band edge are
Since 
and 
, it is easy to see from (3.20) that 
,
and the first inequality in (3.14) is satisfied. For the system that
precisely meets the specification requirements at the stop-band edge,
the scale frequency can be computed using (3.18), and attenuation at the
pass-band will be exceeding the specification requirements.
-
Case3. Specification requirements on both edges are exceeded.
At some conditions, the scale frequency
which was determined for the precise solution (3.11) can be used to
satisfy the system of inequalities (3.14). Attenuation for the precise
solution of the system is given by
Attenuation of the Butterworth filter with order 
and scaling frequency 
is as follows
Parameters 
in these expressions are determined by (3.11).
Fig 3.2 displays the precise solution (3.21) and solution (3.22) for the case when
.
Fig. 3.2 Case 
.
It is clear that in this case the specification requirements are exceeded on both pass-band and stop-band edges.
Fig 3.3 and Fig 3.4 display solutions (3.21) and (3.22) for the cases when
and 
. In both cases one of the edge requirements is not met.
Fig 3.3 
Requirements at the pass-band are exceeded but requirements at the stop-band are not met.
Fig 3.4 
Requirements at the stop-band are exceeded but requirements at the pass-band are not met.
Consequently, the scaling frequency
can be used to meet or exceed specification requirements on both edges only in the case when 
.
For most practical filters, these conditions are met. If there is the
need to design a filter with different conditions, it is recommended to
use expressions (3.15) or (3.18) to compute scaling frequency for the
Butterworth filters.
How to obtain transfer function of the low-pass Butterworth filters
The transfer function of the low-pass Butterworth filter can be expressed as follows
where 
are poles of transfer function.As it was mentioned before, all poles of the filter with even order
occur in complex conjugate pairs. The product 
of every pair of complex conjugate poles 
can be expressed using formulas (3.5) for Butterworth poles
Therefore, the transfer function of the low-pass Butterworth filter with even order 
can be expressed as follows
In the case when the order 
is an odd number, the Butterworth filters have a single real pole equal to 
and 
complex conjugate pairs. Therefore, the transfer function of the
Butterworth low-pass filter with an odd order can be expressed as
follows
Note that the normalized form of transfer functions can be obtained by setting 
in (3.25) and (3.26).As it was shown earlier in (1.16), the transfer function of a serial system is determined as a product of transfer functions of the linked blocks. Therefore, Butterworth filters of any order
can be presented in the form of serially linked blocks of the first and second orders:
When the order 
of the filter is even, the transfer function of the Butterworth low-pass filter can be submitted in the form
When the order 
of the filter is odd, the transfer function of the Butterworth low-pass filter can be submitted in the form
Design samples using FAZA
-
Sample 1:
Apass = 0.5 db
Astop = 3.0 db
Fpass = 0.5 rad/sec
Fstop = 1.5 rad/sec
Design results: min order = 1; Transfer function:
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Sample 2:
Apass = 0.5 db
Astop = 10.0 db
Fpass = 0.5 rad/sec
Fstop = 1.5 rad/sec
Design results: min order = 2; Transfer function:
-
Sample 3:
Apass = 0.5 db
Astop = 15.0 db
Fpass = 0.5 rad/sec
Fstop = 1.5 rad/sec
Design results: min order = 3; Transfer function:
-
Sample 4:
Apass = 0.5 db
Astop = 20.0 db
Fpass = 0.5 rad/sec
Fstop = 1.5 rad/sec
Design results: min order = 4; Transfer function:
-
Sample 5:
Apass = 0.5 db
Astop = 30.0 db
Fpass = 0.5 rad/sec
Fstop = 1.5 rad/sec
Design results: min order = 5; Transfer function:
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