The classical filters have all been developed for a bandedge of

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

In addition to the lowpass frequency response, other basic ideal responses are often needed in practice. The ideal highpass filter rejects signals with frequencies below a certain value and passes those with frequencies above that value. The ideal bandpass filter passes only a band of frequencies, and the ideal band reject filter completely rejects a band of frequencies. These ideal frequency responses are illustrated in Figure 1.

This section presents a method for designing the three new filters by using a frequency transformation on the basic lowpass design. When used on the four classical IIR approximations (e.g. Butterworth, Chebyshev, inverse-Chebyshev, and Elliptic Function), the optimality is preserved. This procedure is used in the FREQXFM() subroutine of Program 8 in the appendix.

The classical filters have all been developed for a bandedge of

The frequency response illustrated in Figure 1b can be
obtained from that in Figure 1a by replacing the complex
frequency variable

In the design procedure, the desired bandedge

In order to convert the lowpass filter of Figure 1a into
that of Figure 1c, a more complicated frequency
transformation is required. In order to reduce confusion, the complex
frequency variable for the prototype analog filter transfer function will
be denoted by

This change of variables doubles the order of the filter, maps the origin
of the

In order that the transformation give

However, because

This means that only three of the four bandedge frequencies

and, using the same

The finally designed bandpass filter will meet both passband edges and one transition band width, but the other will be narrower than originally specified. This is not a problem with the Butterworth or either of the Chebyshev approximation because they only have passband edges or stopband edges, but not both. The elliptic-function has both.

After the bandedges for the prototype lowpass filter

This gives two transformed roots for each prototype root which doubles the order as expected.

The roots that result from transforming the real pole of an odd- order prototype cause some complication in programming this procedure. Program 8 should be studied to understand how this is carried out.

To design a filter that will reject a band of frequencies, a frequency transformation of the form

is used on the prototype lowpass filter. This transforms the origin of
the

Similar to the bandpass case, the transformation must give

As before, only three of the four bandedge frequencies can be
independently specified. Normally,

and, using the same

The finally designed bandpass filter will meet both passband edges and one transition-band width, but the other will be narrower than originally specified. This does not occur with the Butterworth or either Chebyshev approximation, only with the elliptic-function.

After the bandedges for the prototype lowpass filter

A more complicated set of transformations could be developed
by using a general map of

Although this method of transformation is a powerful and simple way for designing bandpass and bandreject filters, it does impose certain restrictions. A Chebyshev bandpass filter will be equal-ripple in the passband and maximally flat at both zero and infinity, but the transformation forces the degree of flatness at zero and infinity to be equal. The elliptic-function bandpass filter will bave the same number of ripples in both stopbands even if they are of very different widths. These restrictions are usually considered mild when compared with the complexity of alternative design methods.

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