Minimum order determination
The attenuation at the stopband edge of the Chebyshev filter can be expressed as
The order of the filter, that meets precisely the specification requirements at the stopband, must satisfy equation
In general, this equation cannot be satisfied because is the integer number. Therefore, this equation can be replaced with inequality
Since , Chebyshev function can be expressed by the hyperbolic cosine, and inequality (4.7) can be written in trigonometric form
Using expression (4.6) for ripple factor , the solution of this inequality can be found
So, the minimum filter order, which meets specification, can be found as
where brackets [] stand for the nearest integer exceeding .
Matheonics Technology Inc, 2009
Natural cutoff frequency determination
The natural cutoff frequency for Chebyshev filters can be determined from the equation
To analyze this equation, two cases can be considered

Case 1.
In this case natural cutoff frequency .
Outside of the passband, the Chebyshev function increases rapidly with no ripples, so equation (4.11) has a single solution. To find that solution, the Chebyshev function can be expressed in terms of hyperbolic functions, and equation
must be solved. Replacing ripple parameter with expression (4.6) and solving this equation the natural cutoff frequency can be determined as

Case 2.
In this case, the cutoff frequency varies in the range . Due to the ripples in the pass band, the gain of the Chebyshev filter varies from a minimum equal to , to the maximum of 1( in linear scale ). Therefore, the signal attenuation equal to must occur at multiple frequencies( see plot ).
The number of frequencies at which the signal attenuation is equal to , depends on the filter order.
For analytic representation of the natural cutoff frequency, the trigonometric form of Chebyshev polynomials can be used. The natural cutoff frequency can be expressed as
Since , all solutions of (4.13) are real, and they can be presented in the form
If , then (4.14) has different solutions for . Half of them are negative and the other half is positive. By observing that only positive frequencies are of interest, we can conclude that in the considering case the number of natural cutoff frequencies are equal to the order of the filter.
Matheonics Technology Inc, 2009
Pole locations
The squared magnitude response of the Chebyshev filter (4.4) can be rearranged into the following form
To obtain (4.15) from (4.4), frequency must be replaced with , and must be replaced with according to (1.26).
The poles of (4.15) are the roots of the equation
By using the trigonometric form of the Chebyshev function, this equation can be rearranged into the form
To find the solution of (4.17), a new complex variable is introduced as follows
Inserting (4.18) into (4.17) gives
Equating real parts of this equation results in
Since , the solutions of (4.20) are
Equating imaginary parts of equation (4.19) and using (4.21) results in
So, real variables are expressed via parameters .
Now, the poles of the squared magnitude response may be found from (4.18)
The poles of the Chebyshev filter are amongst . Noting that for the stable filter the real parts of the poles must be negative and avoiding redundancy, the poles of the filter can be presented as follows:
Real and imaginary parts of the poles may be expressed as
Since , real and imaginary parts of the poles are connected with the equation
This equation demonstrates that all poles of the Chebyshev filter are arranged on the ellipse centered at s = 0 on the complex splane. The major axis of the ellipse are and .
Matheonics Technology Inc, 2009
How to obtain transfer function of the lowpass Chebyshev filters
To obtain the transfer function of the lowpass Chebyshev filter, that meets the specification, the following procedure can be used:
So, we can see that the transfer function of the lowpass Chebyshev filter consists of the same the blocks as Butterworth filters, plus block , which represents a constant.
Matheonics Technology Inc, 2009