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.
this expression can be written as follows
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. 
, where 
stands for the odd number, roots 
of (3.3) can be obtained as solutions to the equation
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.
.
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 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
and 
must be obtained from system (3.9). The order of the filter 
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 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.
, and it doesn't depend on the order of the filter.
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
is determined from (3.12). If these inequalities are satisfied, then Butterworth filter meets or exceeds the specification requirements.
and 
, it is easy to see from (3.17) that the second inequality in (3.14) is satisfied.
is expressed by (3.18).
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.
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
and scaling frequency 
is as follows
in these expressions are determined by (3.11).
.
.
and 
. In both cases one of the edge requirements is not met.
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.
are poles of transfer function.
occur in complex conjugate pairs. The product 
of every pair of complex conjugate poles 
can be expressed using formulas (3.5) for Butterworth poles
can be expressed as follows
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
in (3.25) and (3.26).
can be presented in the form of serially linked blocks of the first and second orders:
of the filter is even, the transfer function of the Butterworth low-pass filter can be submitted in the form
of the filter is odd, the transfer function of the Butterworth low-pass filter can be submitted in the form
