The squared magnitude response of the inverse Chebyshev filters is defined as
The inverse Chebyshev approximation method is closely related to the Chebyshev approximation.The main feature of the inverse Chebyshev approximation method is a ripple in the stopband. The passband loss in this type of approximation is a monotonic function of frequency, as in the Butterworth type of filters.
Fig. 5.1 Magnitude response of a typical inverse Chebyshev filter.
The scaling frequency for inverse Chebyshev filters is defined equal to the stopband edge frequency. The squared magnitude response can be presented in the form
Note that in (5.2) and (5.1), the Chebyshev function is introduced as a function of inverse frequency.
Signal attenuations for the inverse Chebyshev lowpass filters can be determined as follows
At the stopband edge when , the Chebyshev function for any integer . Observing that at the stop edge the attenuation (in order to meet specification requirment), the connection between parameters and are as follows
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Minimum order determination
The attenuation at the passband edge of the inverse Chebyshev filter can be expressed in the form
The order of the filter that meets precisely the specification requirements at the passband, must satisfy equation
In general, this equation cannot be satisfied because variable is an integer number. This equation must therefore be replaced with inequality
The solution to this inequality is as follows:
Since , Chebyshev function can be expressed by the hyperbolic cosine, and the inequality (5.6) can be written as
Inequality (5.7) can be solved
The minimum filter order which meets the specification can be found as
Brackets [] in (4.10) stand for the nearest integer exceeding .
Note, that expression (5.8) is exactly identical to (4.9). So Chebyshev filters and inverse Chebyshev filters have to have the same minimum order to satisfy the specification requirements.
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Natural cutoff frequency determination
The natural cutoff frequency for inverse Chebyshev filters can be determined from the equation
To analyze equation (5.10), two cases can be considered

Case 1.
This is a practical case. In this case, natural cutoff frequency . Beyond the stopband, the Chebyshev function increases rapidly with no ripples, so equation (5.10) has a single solution. To find that solution, the Chebyshev function can be expressed in terms of hyperbolic functions, and equation
must be solved. To find the solution of this equation, the ripple parameter must be replaced with expression (5.4). The solution is

Case 2.
In this case, the natural cutoff frequency varies in the range . Due to the ripples in the stop band, the gain of the inverse Chebyshev filter varies from minimum to infinity, and the signal attenuation equal to must occur at multiple frequencies. The number of frequencies at which the signal attenuation is equal to depends on the filter order.
For an analytic representation of the natural cutoff frequency, the trigonometric form of Chebyshev polynomial can be used. The natural cutoff frequency can be found as
Since , all solutions of (4.13) are real and can be presented in the form
If , (5.13) has different solutions for . Half of them are negative and the other half are positive. Since only positive solutions are of interest, we can conclude that the Chebyshev filter of order has different frequencies at which the attenuation of the filter has the value of .
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How to obtain the transfer function of Inverse Chebyshev filters
To obtain the transfer function of the lowpass inverse Chebyshev filter that meets the specification, the following procedure can be used:
Matheonics Technology Inc, 2009