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The Butterworth filter does not give a sufficiently good
approximation across the complete passband in many cases. The
Taylor's series approximation is often not suited to the way
specifications are given for filters. An alternate error measure is
the maximum of the absolute value of the difference between the
actual filter response and the ideal. This is considered over the
total passband. This is the Chebyshev error measure and was defined
and applied to the FIR filter design problem. For the IIR filter,
the Chebyshev error is minimized over the passband and a Taylor's
series approximation at
The design of Chebyshev filters is particularly interesting, because the results of a very elegant theory insure that constructing a frequency-response function with the proper form of equal ripple in the error will result in a minimum Chebyshev error without explicitly minimizing anything. This allows a straightforward set of design formulas to be derived which can be viewed as a generalization of the Butterworth formulas [4], [7].
The form for the magnitude squared of the frequency-response function for the Chebyshev filter is
where
The Chebyshev polynomial is a powerful function in approximation theory. Although the function is a polynomial, it is best defined and developed in terms of trigonometric functions by[4], [5], [1], [7].
where
where
Although this definition of
From (Equation 2), it is clear that
etc.
Other relations useful for developing these polynomials are
where M and N are coprime.
These are remarkable functions [7]. They oscillate
between +1 and -1 for
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The filter frequency-response function for
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The approximation parameters must be clearly understood. The
passband ripple is defined to be the difference between the maximum
and the minumum of
The Chebyshev theory states that the maximum error over that
band is minimum and that this optimal approximation function has
equal ripple over the pass band. It is easy to see that e in
(Equation 1) determines the ripple in the passband and the order
Pole Locations
A method for finding the pole locations for the Chebyshev filter transfer function is next developed. The details of this section can be skipped and the results in (Equation 22,Equation 24) used if desired.
From (Equation 1), it is seen that the poles of
or
From (Equation 4), define
This gives,
which implies the real part of
which implies
which in turn implies that
For these values of
which requires
Using
which gives the location of the
where
for
A partially factored form for F(s) can be derived using the same approach as for the Butterworth filter. For N even, the form is
for
for
A single formula for both even and odd N is
for
Note the similarity to the pole locations for the Butterworth filter. Cross multiplying, squaring, and adding the terms in (Equation 28,Equation 29) gives
This is the equation for an ellipse and shows that the poles of a Chebyshev filter lie on an ellipse similar to the way the poles of a Butterworth filter lie on a circle[4], [5], [3], [6], [1], [7].
Summary
This section has developed the classical Chebyshev filter approximation which minimizes the maximum error over the passband and uses a Taylor's series approximation at infinity. This results in the error being equal ripple in the passband. The transfer function was developed in terms of the Chebyshev polynomial and explicit formulas were derived for the location of the transfer function poles. These can be expressed as a modification of the pole locations for the Butterworth filter and are implemented in the appendix.
It is possible to develop a theory for Chebyshev passband approximation and arbitrary zero location similar to the Taylor's series result in ((Reference)).
The Chebyshev filter has a passband optimized to minimize
the maximum error over the complete passband frequency range, and a
stopband controlled by the frequency response being maximally flat
at
The form for the specifications that is most consistent with
the Chebyshev filter formulation is a maximum allowed error in the
passband and a desired degree of “flatness" at
As stated earlier, the design parameters must be clearly
understood to obtain a desired result. The passband ripple is defined to
be the difference between the maximum and the minimum of
In some cases, stopband performance is not given in terms of
degree of flatness at
The design of a Chebyshev filter involves the following steps:
This process is easily programmed for computer aided design as illustrated in Program 8 in the appendix.
If the design procedure uses (Equation 34) to determine the order
and the right-hand side of the equation is not exactly an integer,
it is possible to improve on the specifications. Direct use of the
order with
Example 7-2. The Design of a Chebyshev Lowpass Filter.
The design specifications require a maximum passband ripple
of
Given
Given
These multipliers are used to scale the root locations of the example third-order Butterworth filter to give
The frequency response is shown in Figure 3
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A second form of the mixture of a Chebyshev approximation and a Taylor's series approximation is called the Inverse Chebyshev filter or the Chebyshev II filter. This error measure uses a Taylor's series for the passband just as for the Butterworth filter and minimizes the maximum error over the total stopband. It reverses the types of approximation used in the preceding section. A fifth-order example is illustrated in (Reference)c and Figure 4c.
Rather than developing the approximation directly, it is
easier to modify the results from the regular Chebyshev filter. First, the
frequency variable
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This highpass characteristic is subtracted from unity to give the desired lowpass inverse-Chebyshev frequency response illustrated in Figure 4c. The resulting magnitude-squared frequency- response function is given by
Zero Locations
The zeros of the Chebyshev polynomial
which requires
for
for
The zeros of the inverse-Chebyshev filter transfer function are derived from (Equation 41) and (Equation 43) to give
The zero locations are not a function of
Pole Locations
The pole locations are the reciprocal of those for the regular Chebyshev filter. If the poles for the inverse filter are denoted by
the locations are
Although this gives a straightforward formula for calculating the location of the poles and zeros of the inverse- Chebyshev filter, they do not lie on a simple geometric curve as did those for the Butterworth and Chebyshev filters. Note that the conditions for a Taylor's series approximation with preset zero locations are satisfied.
A partially factored form for the Butterworth filter and for the Chebyshev filter can be written for the inverse-Chebyshev filter using the zero locations from (Equation 45) and the pole locations from the regular Chebyshev filter. For N even, this becomes
for
for
Because of the relationships between the locations of the poles of the Butterworth, Chebyshev, and inverse-Chebyshev filters, it is easy to write a design program with many common calculations. That is illustrated in the program in the appendix.
The natural form for the specifications of an inverse-Chebyshev
filter is in terms of the flatness of the response at
The stopband ripple d is simply defined as the maximum value
that
In some cases passband performance is not given in terms of
degree of flatness at
The design of an inverse-Chebyshev filter is summarized in the following steps:
Example Design of an Inverse-Chebyshev Filter
A third-order inverse-Chebyshev lowpass filter is desired
with a maximum-allowed stopband ripple of
The zeros are calculated from (Equation 45), and the poles of the prototype are inverted to give, from (Equation 50), the desired inverse- Chebyshev filter transfer function of