5 Inverse Chebyshev approximation

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 stop-band. The pass-band 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 stop-band 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 low-pass filters can be determined as follows

At the stop-band 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 pass-band 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 pass-band, 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 stop-band, 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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Pole and zero locations

The zeros of the inverse Chebyshev filter may be determined from the equation

A Chebyshev polynomial of degree has simple roots

Therefore all roots of (5.14) can be presented as

When is an even number, zeros may be written in the form:

When is an odd number, one of the roots (5.15) is zero( when ). Therefore one of the zeros (5.16) is an infinite number. This simply means that the frequency response is zero when the frequency is infinity. Other zeros of the inverse Chebyshev filter are finite numbers. They may be given in the form

Consequently, we may conclude that all finite zeros of the inverse Chebyshev filter are purely imaginary complex conjugate pairs. The number of complex conjugate zeros is equal to for the odd order of the filter and is equal to for the even order.

The poles of the inverse Chebyshev filters are determined by the equation

Compairing equations (5.19) and (4.16), we can conclude that the poles of the inverse Chebyshev filter are inverse of the poles of the Chebyshev filter. By analogy with (4.23), the poles of the inverse Chebyshev filter may be expressed as follows:

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How to obtain the transfer function of Inverse Chebyshev filters

To obtain the transfer function of the low-pass inverse Chebyshev filter that meets the specification, the following procedure can be used:

  • Determine ripple factor

  • Determine minimum filter order that meets the specification.

  • Compute filter poles

  • Compute zeros using one of the following expressions

  • Obtain transfer function .

    When the filter order is an even number, the transfer function of the inverse Chebyshev filter has finite purely complex conjugate pairs of zeros and complex conjugate pairs of poles. In this case, the transfer function can be represented as follows

    When the filter order is an odd number, the inverse Chebyshev filter has finite zeros and finite poles. The zeros are finite purely complex conjugate pairs and the poles are complex conjugate pairs. In this case, the transfer function can be represented as

    Consequently, we can see that the transfer function of the low-pass inverse Chebyshev filter consists from the blocks .

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