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Frequency Transformations

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

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.

Figure 1: The Basic Four Ideal Frequency Responses
Figure one contains four cartesian graphs. Each graph's horizontal axis ranges in value from -5 to 5 in increments of 5, and their vertical axes are labeled Ideal response, ranging in value from -1 to 1.5 in increments of 0.5. The first graph, labeled Ideal lowpass, contains a horizontal line from (-5, 0) to (5, 0) and a vertical line from (0, 0) to (0, 1.5). The graph also contains line segments from approximately (-1, 0) to (-1, 1), (1, 0) to (1, 1) and (-1, 1) to (1, 1). The second graph, labeled Ideal highpass, contains a horizontal line from (-5, 0) to (5, 0) and a vertical line from (0, 0) to (0, 1.5). The graph also contains line segments from approximately (-5, 1) to (-1, 1), (-1, 1) to (-1, 0) (1, 0) to (1, 1), and (1, 1) to (5, 1). The third graph, labeled ideal bandpass, contains a horizontal line from (-5, 0) to (5, 0) and a vertical line from (0, 0) to (0, 1.5). The graph also contains line segments from approximately (-2, 0) to (-2, 1), (-2, 1) to (-1, 1), (-1, 1) to (-1, 0), (1, 0) to (1, 1), (1, 1) to (2, 1) and (2, 1) to (2, 0). The fourth graph, labeled ideal bandreject, contains a horizontal line from (-5, 0) to (5, 0) and a vertical line from (0, 0) to (0, 1.5). The graph also contains line segments from approximately (-5, 1) to (-2, 1), (-2, 1) to (-2, 0), (-1, 0) to (-1, 1), (1, 0) to (1, 1), (-1, 1) to (1, 1), (2, 0) to (2, 1), and (2, 1) to (5, 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.

Change the Bandedge

The classical filters have all been developed for a bandedge of ω0=1ω0=1. That is where the Butterworth filter has a magnitude squared of one half: |F|=0.5|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 ss by KsKs or: sKssKs where KK is reciprocal of the new desired bandedge. What happened to the prototype filter at ω=1ω=1 will now happen at ω=1/Kω=1/K. It is simply a linear scaling of the ωω axis. This change can be done before the conversions below or after.

The Highpass Filter

The frequency response illustrated in Figure 1b can be obtained from that in Figure 1a by replacing the complex frequency variable ss in the transfer function by 1/s1/s. This change of variable maps zero frequency to infinity, maps unity into unity, and maps infinity to zero. It turns the complex ss plane inside out and leaves the unit circle alone.

In the design procedure, the desired bandedge ω0ω0 for the highpass filter is mapped by 1/ω01/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 ss by 1/s1/s. If an elliptic-function filter approximation is used, both the passband edge ωpωp and the stopbandedge ωsωs 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 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 pp and that for the transformed analog filter by ss. The transformation is given by

p = s 2 + ω 0 2 s p = s 2 + ω 0 2 s
(1)

This change of variables doubles the order of the filter, maps the origin of the ss-plane to both plus and minus jω0jω0, and maps minus and plus infinity to zero and infinity. The entire ωω 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 Figure 2.

Figure 2: Lowpass to Bandpass Transformation
Figure two is a cartesian graph with an unlabeled vertical axis and a horizontal axis labeled Frequency, ω. The horizontal axis ranges in value from -5 to 5 in increments of 5, and the vertical axis ranges in value from 0 to 5 in increments of 1. There are two blue horizontal lines in this graph, one from (-5, 3) to (5, 3) and the other from (-5, 0) to (5, 0). Above the horizontal lines is a box containing the phrase, lowpass prototype. In between the lines is a box containing the phrase, bandpass filter. There is an arrow drawn from the lowpass prototype box to the bandpass filter box. There are three trapezoids in this graph, each with their base on one of the horizontal lines. The largest trapezoid is centered with its base from approximately (-2, 3) to (2, 3). It has a height of one. The horizontal values of the vertices of the trapezoid read from left to right, -ω_s, -ω_p, ω_p, and ω_s. There are two identical trapezoids in the lower half of the graph, with their bases from approximately (-3.5, 0) to (-1.5, 0) and (1.5, 0) to (3.5, 0) respectively. These trapezoids also have a height of one. On the base of the trapezoid to the right are labels corresponding to the horizontal values of its vertices (and including the midpoint of its base), labeled from left to right ω_1, ω_2, ω_0, ω_3, ω_4. There are arrows drawn from the  labeled horizontal values of the larger trapezoid above to these labeled points on the smaller trapezoid below and to the right.

In order that the transformation give -ωp=(ω22-ω02)/ω2-ωp=(ω22-ω02)/ω2 and ωp=(ω32-ω02)/ω3ωp=(ω32-ω02)/ω3, the “center" frequency ω0ω0 must be

ω 0 = ω 2 ω 3 ω 0 = ω 2 ω 3
(2)

However, because -ωs=(ω12-ω02)/ω1-ωs=(ω12-ω02)/ω1 and ωs=(ω42-ω02)/ω4ωs=(ω42-ω02)/ω4, the center frequency must also be

ω 0 = ω 1 ω 4 ω 0 = ω 1 ω 4
(3)

This means that only three of the four bandedge frequencies ω1ω1, ω2ω2, ω3ω3, and ω4ω4 can be independently specified. Normally, ω0ω0 is determined by ω2ω2 and ω3ω3 which then specifies the prototype passband edge by

ω p = ω 3 2 - ω 0 2 ω 3 ω p = ω 3 2 - ω 0 2 ω 3
(4)

and, using the same ω0ω0, the stopband edge is set by either ω1ω1 or ω4ω4, whichever gives the smaller ωsωs.

ω s = ω 4 2 - ω 0 2 ω 4 or ω 0 2 - ω 1 2 ω 1 ω s = ω 4 2 - ω 0 2 ω 4 or ω 0 2 - ω 1 2 ω 1
(5)

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 ωpωp and/or ωsωs 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+ω02s2-ps+ω02 from the original transformation to give for the root locations

s = p ± p 2 - 4 ω 0 2 2 s = p ± p 2 - 4 ω 0 2 2
(6)

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 p = s s 2 + ω 0 2
(7)

is used on the prototype lowpass filter. This transforms the origin of the pp-plane into both the origin and infinity of the ss-plane. It maps infinity in the pp-plane into jω0ω0 in the ss-plane.

Similar to the bandpass case, the transformation must give -ωp=ω4/(ω02-ω42)-ωp=ω4/(ω02-ω42) and ωp=ω1/(ω02-ω12)ωp=ω1/(ω02-ω12). A similar relation of ωsωs to ω2ω2 and ω3ω3 requires that the center frequency ω0ω0 must be

ω 0 = ω 1 ω 4 = ω 2 ω 3 ω 0 = ω 1 ω 4 = ω 2 ω 3
(8)

As before, only three of the four bandedge frequencies can be independently specified. Normally, ω0ω0 is determined by ω1ω1 and ω4ω4 which then specifies the prototype passband edge by

ω p = ω 1 ω 0 2 - ω 1 2 ω p = ω 1 ω 0 2 - ω 1 2
(9)

and, using the same ω0ω0, the stopband edge is set by either ω2ω2 or ω3ω3, whichever gives the smaller ωsωs.

ω s = ω 2 ω 0 2 - ω 2 2 or ω 3 ω 4 2 - ω 0 2 ω s = ω 2 ω 0 2 - ω 2 2 or ω 3 ω 4 2 - ω 0 2
(10)

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 ωpωp and/or ωsωs 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+ω02s2-(1/p)s+ω02 to give for the root locations

s = 1 / p ± ( 1 / p ) 2 - 4 ω 0 2 2 s = 1 / p ± ( 1 / p ) 2 - 4 ω 0 2 2
(11)

A more complicated set of transformations could be developed by using a general map of s=f(s)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.

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