4 Chebyshev approximation

Chebyshev polynomials of the first kind are polynomials least deviating from zero. Chebyshev polynomials of the first kind can be determined recursively:

Also, Chebyshev polynomials of the first kind can be expressed via trigonometric functions

The squared magnitude response of a Chebyshev low-pass filter is defined as follows

where

  • - Angular frequency

  • - Constant scaling frequency

  • - Order of the filter

  • - Parameter that characterizes ripple

  • - Chebyshev function of degree

The scaling frequency for Chebyshev filters is defined as equal to the pass-band edge frequency; the squared magnitude response can be represented as

The main feature of the Chebyshev filter is a ripple in the pass-band. The magnitude response in the pass-band, where , varies from 1 to , following cosine function. When , the Chebyshev function increases rapidly with no ripples, and monotonically decreases and tends to zero.

Fig 4.1 Magnitude response of a typical Chebyshev filter.

Signal attenuations of the low-pass Chebyshev filters can be expressed as

Pass-band attenuation determines the maximum allowable attenuation in the pass-band. The magnitude response varies in the pass-band in the range from to 1. Observing that at the pass-band edge , and , the connection between parameters and can be determined as

Chebyshev filters are more efficient than Butterworth filters of the same order. A disadvantage of Chebyshev filters is that they have a ripple in the pass-band.


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Minimum order determination

The attenuation at the stop-band edge of the Chebyshev filter can be expressed as

The order of the filter, that meets precisely the specification requirements at the stop-band, 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 .


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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.


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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 s-plane. The major axis of the ellipse are and .


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How to obtain transfer function of the low-pass Chebyshev filters

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

  • Determine ripple factor

  • Determine minimum filter order , that meets specification

  • Compute filter poles

  • Obtain transfer function .

    When the filter order is an odd number, the transfer function is expressed as

    When the filter order is an even number, the Chebyshev function and the gain .

    Therefore, in order to achieve zero minimum pass-band loss at DC, the transfer function of the Chebyshev filter of any even order has to have the factor . For this case, the transfer function can be obtained as follows

So, we can see that the transfer function of the low-pass Chebyshev filter consists of the same the blocks as Butterworth filters, plus block , which represents a constant.


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