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 .

The Basic Four Ideal Frequency Responses 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.

Change the Bandedge The classical filters have all been developed for a bandedge of ω0=1. That is where the Butterworth filter has a magnitude squared of one half: |F|=0.5 or the Chebyshev filter has its passband edge or the Inverse Chebyshev has its stopband edge or the Elliptic filter has its passband edge. To scale the bandedge, simply replace s by Ks or: s→Ks where K is reciprocal of the new desired bandedge. What happened to the prototype filter at ω=1 will now happen at ω=1/K. It is simply a linear scaling of the ω axis. This change can be done before the conversions below or after.

The Highpass Filter The frequency response illustrated in b can be obtained from that in a by replacing the complex frequency variable s in the transfer function by 1/s. This change of variable maps zero frequency to infinity, maps unity into unity, and maps infinity to zero. It turns the complex s plane inside out and leaves the unit circle alone. In the design procedure, the desired bandedge ω0 for the highpass filter is mapped by 1/ω0 to give the bandedge for the prototype lowpass filter. This lowpass filter is next designed by one of the optimal procedures already covered and then converted to a highpass transfer function by replacing s by 1/s. If an elliptic-function filter approximation is used, both the passband edge ωp and the stopbandedge ωs are transformed. Because most optimal lowpass design procedures give the designed transfer function in factored form from explicit formulas for the poles and zeros, the transformation can be performed on each pole and zero to give the highpass transfer function in factored form.

The Bandpass Filter In order to convert the lowpass filter of a into that of c, 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 p and that for the transformed analog filter by s. The transformation is given by p = s 2 + ω 0 2 s This change of variables doubles the order of the filter, maps the origin of the s-plane to both plus and minus jω0, and maps minus and plus infinity to zero and infinity. The entire ω axis of the prototype response is mapped between zero and plus infinity on the transformed responses. It is also mapped onto the left-half axis between minus infinity and zero. This is illustrated in .

Lowpass to Bandpass Transformation Figure 7-22. Lowpass to Bandpass Frequency Transformation In order that the transformation give -ωp=(ω22-ω02)/ω2 and ωp=(ω32-ω02)/ω3, the “center" frequency ω0 must be ω 0 = ω 2 ω 3 However, because -ωs=(ω12-ω02)/ω1 and ωs=(ω42-ω02)/ω4, the center frequency must also be ω 0 = ω 1 ω 4 This means that only three of the four bandedge frequencies ω1, ω2, ω3, and ω4 can be independently specified. Normally, ω0 is determined by ω2 and ω3 which then specifies the prototype passband edge by ω p = ω 3 2 - ω 0 2 ω 3 and, using the same ω0, the stopband edge is set by either ω1 or ω4, whichever gives the smaller ωs. ω s = ω 4 2 - ω 0 2 ω 4 or ω 0 2 - ω 1 2 ω 1 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 ωp and/or ωs are calculated, the filter is designed by one of the optimal approximation methods discussed in this section or any other means. Because most of these methods give the pole and zero locations directly, they can be individually transformed to give the bandpass filter transfer function in factored form. This is accomplished by solving s2-ps+ω02 from the original transformation to give for the root locations s = p ± p 2 - 4 ω 0 2 2 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.

The Band-Reject Filter To design a filter that will reject a band of frequencies, a frequency transformation of the form p = s s 2 + ω 0 2 is used on the prototype lowpass filter. This transforms the origin of the p-plane into both the origin and infinity of the s-plane. It maps infinity in the p-plane into jω0 in the s-plane. Similar to the bandpass case, the transformation must give -ωp=ω4/(ω02-ω42) and ωp=ω1/(ω02-ω12). A similar relation of ωs to ω2 and ω3 requires that the center frequency ω0 must be ω 0 = ω 1 ω 4 = ω 2 ω 3 As before, only three of the four bandedge frequencies can be independently specified. Normally, ω0 is determined by ω1 and ω4 which then specifies the prototype passband edge by ω p = ω 1 ω 0 2 - ω 1 2 and, using the same ω0, the stopband edge is set by either ω2 or ω3, whichever gives the smaller ωs. ω s = ω 2 ω 0 2 - ω 2 2 or ω 3 ω 4 2 - ω 0 2 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 ωp and/or ωs are calculated, the filter is designed. The poles and zeros of this filter are individually transformed to give the bandreject filter transfer function in factored form. This is carried out by solving s2-(1/p)s+ω02 to give for the root locations s = 1 / p ± ( 1 / p ) 2 - 4 ω 0 2 2 A more complicated set of transformations could be developed by using a general map of s=f(s) with a higher order. Several pass or stopbands could be specified, but the calculations become fairly complicated. 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.