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Optimality of the Four Classical Filter Designs

Module by: C. Sidney Burrus. E-mail the author

It is important in designing filters to choose the particular type that is appropriate. Since in all cases, the filters are optimal, it is necessary to understand in what sense they are optimal.

The classical Butterworth filter is optimal in the sense that it is the best Taylor's series approximation to the ideal lowpass filter magnitude at both ω=0ω=0 and ω=ω=.

The Chebyshev filter gives the smallest maximum magnitude error over the entire passband of any filter that is also a Taylor's series approximation at ω=ω= to the ideal magnitude characteristic.

The Inverse-Chebyshev filter is a Taylor's series approximation to the ideal magnitude response at ω=0ω=0 and minimizes the maximum error in the approximation to zero over the stopband. This can also be stated as maximizing the minimum rejection of the filter over the stopband.

The elliptic-function filter (Cauer filter) considers the four parameters of the filter: the passband ripple, the transition-band width, the stopband ripple, and the order of the filter. For given values of any three of the four, the fourth is minimized.

It should be remembered that all four of these filter designs are magnitude approximations and do not address the phase frequency response or the time-domain characteristics. For most designs, the Butterworth filter has the smoothest phase curve, followed by the inverse-Chebyshev, then the Chebyshev, and finally the elliptic-function filter having the least smooth phase response.

Recall that in addition to the four filters described in this section, the more general Taylor's series method allows an arbitrary zero locations to be specified but retains the optimal character at ω=0ω=0. A design similar to this can be obtained by replacing ωω by 1/ω1/ω, which allows setting |F(w)|2|F(w)|2 equal unity at arbitrary frequencies in the passband and having a Taylor's series approximation to zero at ω=ω=[1].

These basic normalized lowpass filters can have the passband edge moved from unity to any desired value by a simple change of frequency variable, ωω replaced with kωkω. They can be converted to highpass filters or bandpass or band reject filters by various changes such as ωω with k/ωk/ω or ωω with aω+b/ωaω+b/ω. In all of these cases the optimality is maintained, because the basic lowpass approximation is to a piecewise constant ideal. An approximation to a nonpiecewise constant ideal, such as a differentiator, may not be optimal after a frequency change of variables .

In some cases, especially where time-domain characteristics are important, ripples in the frequency response cause irregularities, such as echoes in the time response. For that reason, the Butterworth and Chebyshev II filters are more desirable than their frequency response alone might indicate. A fifth approximation has been developed [1] that is similar to the Butterworth. It does not require a Taylor's series approximation at ω=0ω=0, but only requires that the response monotonically decrease in the passband, thus giving a narrower transition region than the Butterworth, but without the ripples of the Cheybshev.

References

  1. Parks, T. W. and Burrus, C. S. (1987). Digital Filter Design. New York: John Wiley & Sons.

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