3.5 Optimality of the four classical filter designs

It is important in designing filters to choose the particular type that is appropriate. Since in all cases, thefilters 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 ideallowpass filter magnitude at both $\omega =0$ and $\omega =\infty$ .

The Chebyshev filter gives the smallest maximum magnitude error over the entire passband of any filter that is also a Taylor'sseries approximation at $\omega =\infty$ to the ideal magnitude characteristic.

The Inverse-Chebyshev filter is a Taylor's series approximation to the ideal magnitude response at $\omega =0$ and minimizes the maximum error in the approximation to zero over thestopband. 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, thetransition-band width, the stopband ripple, and the order of the filter. For given values of any three of the four, the fourth isminimized.

It should be remembered that all four of these filter designs are magnitude approximations and do not address the phasefrequency 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 smoothphase response.

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

These basic normalized lowpass filters can have the passband edge moved from unity to any desired value by a simple change of frequencyvariable, $\omega$ replaced with $k\omega$ . They can be converted to highpass filters or bandpass or band reject filters by various changessuch as $\omega$ with $k/\omega$ or $\omega$ with $a\omega +b/\omega$ . In all of these cases the optimality is maintained, because the basic lowpassapproximation is to a piecewise constant ideal. An approximation to a nonpiecewise constant ideal, such as a differentiator, may not be optimalafter a frequency change of variables .

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