Inside Collection (Textbook): Digital Signal Processing and Digital Filter Design (Draft)

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

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

The Inverse-Chebyshev filter is a Taylor's series
approximation to the ideal magnitude response at

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

These basic normalized lowpass filters can have the passband
edge moved from unity to any desired value by a simple change of frequency
variable,

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

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