Generalised maximally flat approximation M.de P. Barros, MSc LF. Lind, PhD, CEng, MIEE Indexing terms: Filters and filtering, Mathematical techniques Abstract: An analytical method to generalise the classical Butterworth approximation into a maxi- mally flat approximation function possessing finite and asymmetric transmission zeros is presented. The passband of this general function is no longer restricted to the usual — 3 dB at ±1 rad/s co- ordinates, and therefore extremely asymmetric characteristics can be generated, such as those required by pseudohighpass or pseudolowpass filters. The final network realisation can be conve- niently absorbed into narrowband crystal filter design techniques. Two examples are given. 1 Introduction The classical Butterworth approximation is probably the most well known and simple amplitude approximation, and is often the first to be presented in text books and courses. Its simplicity relies on the fact that all its reflec- tion zeros are located at the origin, while the transmis- sion zeros are concentrated at infinity. Some methods including finite transmission zeros, always in complex- conjugate pairs on the imaginary axis, can be found in the literature [1, 2], but they preserve the real character of the related rational functions and, consequently, their symmetric amplitude characteristics. There are occasions, however, when asymmetrically disposed transmission zeros are welcome in the design [3, 4]. An immediate example is single-sideband filters. Other applications can be found also, such as those requiring 'pseudo-' lowpass and highpass characteristics. In these cases the stopband attenuation is to be main- tained at high levels over a certain frequency band, but can be allowed to decay far away from the stopband of the network. These are typical examples where the pre- sence of transmission zeros at infinity is pointless, as it is certainly more economical to place transmission zeros in the rejection band of the filter. Still other applications can require highly asymmetric amplitude characteristics having, at the same time, a certain degree of group delay equalisation over part of the passband. In this case the inclusion of the complex paraconjugate asymmetric transmission zero pair ±a 0 +7 w o is °f ten useful for the partial correction or predistortion of the group delay. A notable example of the generalisation of a classical approximation function has been presented by Cameron [5, 6], who introduced a method permitting the inclusion Paper 5577G (E10), first received 27th May 1986 and in revised form 8th July 1987 The authors are with the Department of Electronic Systems Engineer- ing, University of Essex, Wivenhoe Park, Colchester CO4 3SQ, United Kingdom of asymmetric transmission zeros in equiripple (Chebychev) functions. In this paper it will be shown how the classical Butterworth allpole function can also be extended to include such transmission zeros. The pre- sence of asymmetric zeros makes the associated rational functions no longer real-coefficient ones (driving-point impedances degenerate from positive real functions into generally positive functions). Complex functions, however, can still be realised if frequency-invariant (FI) immittances are accepted as elementary components in the lowpass networks. Over narrow frequency ranges, FI- immittances can be conveniently approximated by lumped capacitors or inductors, a technique often used in the design of narrowband crystal filters. It will also be shown in this paper how the passband co-ordinates can be specified in advance, thus allowing a more relaxed approximation of pseudohighpass and -lowpass charac- teristics. 2 Theory The squared magnitude of the transmission coefficient of a lossless and reciprocal Nth-degree 2-port can be written as \F(jco)\ 2 = 1 K 2 (co) (1) where K(co) is the so-called characteristic or filtering func- tion, given by K(co) = H(s) (2) where P^s) and H(s) are the numerators of the reflection and transmission coefficients G(s) = P(s)/E(s) and F(s) = H(s)/E(s), respectively. For the final network to be realisable, K(co) must contain only real coefficients, and therefore P(jco) and H(jco) can be simply written as P(a>) and H(co). In the allpole maximally flat Butterworth case, K(a>) = P(co) = a> N , i.e. all reflection zeros are coincident at the origin, whereas all transmission zeros are located at infinity. Now consider the case where the transmission coefficient is required to possess finite transmission zeros. To cope with that, the polynomial H(co) is generalised into CO (3) which contains a total of Z prespecified finite transmis- sion zeros, the co,- s. If the transmission zeros are symmetrically disposed about the real axis of the s-plane, i.e. occurring in complex-conjugate pairs, the function F(s) will be formed IEE PROCEEDINGS, Vol. 134, Pt. G, No. 5, OCTOBER 1987 225
Transcript
Generalised maximally flat approximation
M.de P. Barros, MScLF. Lind, PhD, CEng, MIEE
Indexing terms: Filters and filtering, Mathematical techniques
Abstract: An analytical method to generalise theclassical Butterworth approximation into a maxi-mally flat approximation function possessing finiteand asymmetric transmission zeros is presented.The passband of this general function is no longerrestricted to the usual — 3 dB at ± 1 rad/s co-ordinates, and therefore extremely asymmetriccharacteristics can be generated, such as thoserequired by pseudohighpass or pseudolowpassfilters. The final network realisation can be conve-niently absorbed into narrowband crystal filterdesign techniques. Two examples are given.
1 Introduction
The classical Butterworth approximation is probably themost well known and simple amplitude approximation,and is often the first to be presented in text books andcourses. Its simplicity relies on the fact that all its reflec-tion zeros are located at the origin, while the transmis-sion zeros are concentrated at infinity. Some methodsincluding finite transmission zeros, always in complex-conjugate pairs on the imaginary axis, can be found inthe literature [1, 2], but they preserve the real characterof the related rational functions and, consequently, theirsymmetric amplitude characteristics.
There are occasions, however, when asymmetricallydisposed transmission zeros are welcome in the design[3, 4]. An immediate example is single-sideband filters.Other applications can be found also, such as thoserequiring 'pseudo-' lowpass and highpass characteristics.In these cases the stopband attenuation is to be main-tained at high levels over a certain frequency band, butcan be allowed to decay far away from the stopband ofthe network. These are typical examples where the pre-sence of transmission zeros at infinity is pointless, as it iscertainly more economical to place transmission zeros inthe rejection band of the filter.
Still other applications can require highly asymmetricamplitude characteristics having, at the same time, acertain degree of group delay equalisation over part ofthe passband. In this case the inclusion of the complexparaconjugate asymmetric transmission zero pair± a 0 +7wo is °ften useful for the partial correction orpredistortion of the group delay.
A notable example of the generalisation of a classicalapproximation function has been presented by Cameron[5, 6], who introduced a method permitting the inclusion
Paper 5577G (E10), first received 27th May 1986 and in revised form8th July 1987The authors are with the Department of Electronic Systems Engineer-ing, University of Essex, Wivenhoe Park, Colchester CO4 3SQ, UnitedKingdom
of asymmetric transmission zeros in equiripple(Chebychev) functions. In this paper it will be shown howthe classical Butterworth allpole function can also beextended to include such transmission zeros. The pre-sence of asymmetric zeros makes the associated rationalfunctions no longer real-coefficient ones (driving-pointimpedances degenerate from positive real functions intogenerally positive functions). Complex functions,however, can still be realised if frequency-invariant (FI)immittances are accepted as elementary components inthe lowpass networks. Over narrow frequency ranges, FI-immittances can be conveniently approximated bylumped capacitors or inductors, a technique often used inthe design of narrowband crystal filters. It will also beshown in this paper how the passband co-ordinates canbe specified in advance, thus allowing a more relaxedapproximation of pseudohighpass and -lowpass charac-teristics.
2 Theory
The squared magnitude of the transmission coefficient ofa lossless and reciprocal Nth-degree 2-port can be writtenas
\F(jco)\2 =1
K2(co) (1)
where K(co) is the so-called characteristic or filtering func-tion, given by
K(co) =H(s)
(2)
where P^s) and H(s) are the numerators of the reflectionand transmission coefficients G(s) = P(s)/E(s) andF(s) = H(s)/E(s), respectively. For the final network to berealisable, K(co) must contain only real coefficients, andtherefore P(jco) and H(jco) can be simply written as P(a>)and H(co).
In the allpole maximally flat Butterworth case,K(a>) = P(co) = a>N, i.e. all reflection zeros are coincidentat the origin, whereas all transmission zeros are locatedat infinity. Now consider the case where the transmissioncoefficient is required to possess finite transmission zeros.To cope with that, the polynomial H(co) is generalisedinto
CO(3)
which contains a total of Z prespecified finite transmis-sion zeros, the co,- s.
If the transmission zeros are symmetrically disposedabout the real axis of the s-plane, i.e. occurring incomplex-conjugate pairs, the function F(s) will be formed
IEE PROCEEDINGS, Vol. 134, Pt. G, No. 5, OCTOBER 1987 225
by real-coefficient polynomials. Therefore the gener-alisation process could start by allowing the polynomialP(co) to be an even or an odd polynomial, having all its N(reflection) zeros coincident at the origin, i.e. P(OJ) =k0coN.
But asymmetrically disposed transmission zeros are tobe included as well. They will be allowed to occur on theimaginary axis (not necessarily in complex-conjugatepairs [7]), at infinity, and also as paraconjugate complexpairs ± a, + jcOi. These complex pairs would still keepH(co) in eqn. 3 a real-coefficient polynomial, because eachparaconjugate pair s,- = ±x( +jyt would be related to acomplex-conjugate pair co, = —jsi = yi±jxi, in the co-plane.
Let co_ and co+ be the upper and lower passbandedges, respectively. There are two conditions to be met:K(co.) = K. = C1 and K(co+) = K+=C2. K(co) thenneeds two degrees of freedom, which can be providedwith the maximally flat numerator
P(co) = k0(co - coF)N (4)
where coF is the new maximally flat frequency in the pass-band, also the location of all reflection zeros.
It can be demonstrated that the approximation rep-resented by eqn. 1 with H(co) and P{co) given by the poly-nomials in eqns. 3 and 4 is maximally flat at co = coF. Forallpole functions, and for those possessing finite transmis-sion zeros occurring in conjugate pairs on the ̂ 'co-axis, coF
is trivially null. But when asymmetry is introduced in thecharacteristic, the maximally flat frequency locates itselfconveniently within the passband. This effect is illustratedin Fig. 1.
-1.25 -0.75 -0.25 0.25
frequency, rad/s
0.75 1.25
Fig. 1 'Slipping' of the maximally flat frequency due to asymmetricallylocated transmission zeros
(i) transmission zeros at —jl.50 and oo, flat at —0.38 rad/s(ii) two transmission zeros at oo, flat at 0 rad/s
(iii) transmission zeros at j 1.05 and oo, flat at 0.73 rad/s
The procedure then starts by defining the passbandco-ordinates, and it will be assumed hereafter that|F()co_)| and \F(jco+)\ are given in dB. Their relatedvalues /C_ = K(co.) and K+ = K(a>+) can be retrievedfrom eqn. 1 and, again noting that \F(jco)\ is in fact afunction of K2(co), X_ and K+ can be multiplied by ± 1without affecting the resulting \F(jco)\. The expressionsfor k0 and coF can be found by recalling eqns. 2, 3 and 4and substituting the known parameters co_, K_, co+,
and K + , leading to the following two equations:
k0{co+ - coF)N k0(co+ - coF)N
K(co) K
cot
(5)
*(,M-¥m/~m<-*";,: f^-c-ir*-,= 1
H(co_)
(6)
where H(coJ) and H(co+) can be calculated from eqn. 3,and the reason for the factor (— 1)N in eqn. 6 will beexplained soon.
Dividing eqns. 5 by eqn. 6 gives
(co+-coF)NH(co.) _ K +{ ' K(co. -coF)NH(co+)
and, by defining a parameter R as
(7)
R = ^CO 4. — CO, l/JV
CO. — COF
the maximally flat frequency coF can be calculated from
(8)
co+ —COp =
1 -R(9)
The reason for the factor ( — 1)N in eqn. 6 is now appar-ent : the value of R as defined in eqn. 8 must be negative ifthe maximally flat frequency coF is to be located withinthe passband, i.e. co_ < coF < co+ .
Finally, k0 can be calculated from eqn. 5, once wf isknown:
K+H(co+)
(co+ - coF)N (10)
When H(co) and P(co) are determined, H{s) and P{s) areformed by analytic continuation (co = —js) and theircommon denominator E(s), containing the poles of thefunction, can be obtained from the unitary energy condi-tion for lossless (and complex) networks:
F(s)FJs) + G(s)Gm(s) = (IDwhere F(s) = H(s)/E(s) and G(s) = P(s)/E(s), and the sub-script asterisk denotes the paraconjugate polynomial.
Of course both the transmission and the reflectioncoefficients F(s) and G(s) are no longer real functions [7],and their network realisation will be possible only byaccepting frequency-invariant (FI) immittances jXs aselementary network components, alongside Rs, Ls andCs. Those Fl-immittances can be conveniently approx-imated, over a narrow frequency band, by lumped induc-tors or capacitors, and that is indeed a usual procedureas far as the design of narrowband crystal filters isconcerned [8].
The generalised asymmetrical approximation degener-ates into the classical Butterworth approximation bydefining the passband co-ordinates as (— 1 rad/s, —3 dB),(+ 1 rad/s, — 3 dB), and by shifting all transmission zerosto infinity. This will make R = — 1, cof = 0, and k0 = 1.
Note that, besides Z ^ N, no other restrictions wereimposed on the total number Z of finite transmissionzeros. It is possible to locate all transmission zeros atfinite frequencies (i.e. Z = N), and in that case | F(jco) \will present a finite value at infinite frequencies, given by
\F(joo)\ = -10 log 1+fc8 ft"?]dB (12)
226 IEE PROCEEDINGS, Vol. 134, Pt. G, No. 5, OCTOBER 1987
Another interesting point consists in creating an equi-ripple stopband by choosing the transmission zeros suchthat the amplitude lobes between them present equalisedlevels. By investigating the behaviour of the first deriv-ative of the magnitude function | F(j(o) \ in eqn. 1, the lobefrequencies are found to be the roots of
N
a> — co
z
-I1
= 0=x co —
(13)
which is a Zth-degree equation, with the a>,s being theprescribed transmission zeros. For higher values of Z sayZ > 2, it is better to solve that equation numerically, aspart of a computer program procedure that will iter-atively shift the transmission zeros into the convenientlocations.
3 Examples
An immediate application for the approximation functionjust introduced can be better seen through numericalexamples. The first one will deal with the design of apseudohighpass crystal filter having a cutoff (— 6 dB) fre-quency fc = 21.4 MHz. Over a frequency interval cover-ing 50 kHz below fc, the attenuation level must be above20 dB. Over the passband itself, which shall span 50 kHzabove 21.4 MHz, the insertion loss must not exceed0.5 dB.
The design starts in the lowpass domain, and it is seenthat the above specifications can be met by a 3rd-degreepseudohighpass filter. First, the maximally flat passbandhas co-ordinates: — 6 dB at — 1 rad/s (the cutoff fre-quency in the lowpass domain), and —0.5 dB at +9 rad/s. Next, three finite transmission zeros are posi-tioned over the stopband. Their location is determined bycontinually solving eqn. 13 and shifting the zeros iter-atively, until a minimum of 20 dB attenuation level isobtained. Their final lowpass positions are found to be—7*1.4, —7*2.8 and —78.2 rad/s. The transmission coeffi-cient numerator H(co) is then formed, from eqn. 3, as
H(co) = 1.0000 + 1.1934 co + 0.3858 co2 + 0.0311 co3
Eqn. 1 gives K_ = 1.7266 (at to_ = — 1 rad/s) and K+ =0.3493 (at co+ = 9 rad/s). With these values and theabove H(co), the maximally flat frequency is found fromeqns. 8 and 9 as being coF = 0.8688 rad/s, which leads tok0 = 0.0427 (eqn. 10). Therefore the reflection coefficientnumerator is completely determined:
P(co) = 0.0427 (co - 0.\
= -0.0280 + 0.0966co - 0.1112co2 + 0.0427 co3
After obtaining H(s) and P(s) from the above poly-nomials, the product E(s)Eitt(s) is formed from eqn. 11 andthen split into E(s) and £+(s), the former being
E(s) = 0.0528s3 + (0.5666 + ./0.1374)s2
+ (0.9113 +yi.l571)s + (-0.2704 +;0.9631)
which possesses all roots (the function poles) on the left-hand s-plane.
The next step involves the synthesis of the lowpassnetwork that realises the just approximated character-istic. As the final pseudohighpass network is to be rea-lised as a crystal filter, a cascade synthesis method is used[10], yielding a network (seen in Fig. 2) where thelowpass prototypes of the crystal units present equalisedmotional inductances.
The narrowband lowpass-bandpass transformation isthen effected, with f0 = 21,405 kHz and band-
width = 10 kHz. These are in fact false parameters, justto relate the lowpass and bandpass cutoff frequencies, i.e.— 1 rad/s to 21.4 MHz. Finally, the bandpass network is
jB2
(-J2.8) (-J1.4) ( - J 8 . 2 )
Fig. 2R-
Xt -B, =BA
3rd-degree lowpass prototype network1 L - 13.5271 X 2 = 6.0613 *3 - 11.85752.4540 B 2 - 1.3752 B 3 - -0.79250.2145 B, - -0.0816 B 6 = 0.0269
B 7 = - 0 . 2 1 1 3 B 8 = 0.2734 B, 0.6391
impedance scaled such that the piezoelectric crystalspresent a motional inductance of 2.66 mH. The finalladder crystal filter, is shown in Fig. 3, and its frequency
crystal units: L motional = 2.66 mHseries resonance frequencies: /5 l = 21.38736 MHz, fl2 - 21.37469 MHz, ftl
21.34571 MHz
analysis gives the pseudohighpass characteristic illus-trated in Fig. 4, clearly satisfying the initial specifications.Note that, yet a narrowband design, a relative frequency
- 4 0
DOa," -30D-•
§.-20E
-10
0
\ I: /
21.364I21 391
II 21.398
'• / \ l\• / \ J \
: -6dBmaximally flat
at 21.40934
/ -0.5 dB
21.35 21.40
frequency, MHz
21.45
Fig. 4. Pseudohighpass characteristic of the crystal filter in Fig. 3
range of nearly 0.5% (100 kHz/21.4 MHz) is being con-trolled by the crystal filter.
The second example (Fig. 5) shows a typical casewhere a paraconjugate pair has been included to equalisethe group delay of a higher-degree maximally flat filter.That transmission zero pair, located at 1.00 — ;'0.15, com-pensates for the presence of two zeros on the imaginaryaxis, at +./2.00 and +./3.5O. Four transmission zeros atinfinity are also present, increasing the degree of the filterto 8. The group delay variation does not exceed 5% over50% of the passband, and is kept within 10% over 65%of the passband. Note also how the return loss peak is
1EE PROCEEDINGS, Vol. 134, Pt. G, No. 5, OCTOBER 1987 227
concentrated on a single frequency—the maximally flatfrequency, in this example at 0.0865 rad/s, the locationof the eight coincident reflection zeros.
125r -125
100
75
50
25
-100COT3
J-75T3D
1-50oE
-25
-5 -3 -1 1frequency,rad/s
o
12.5r -5
10.0
5 7.5•o
ag 5.0
2.5
I"2
E-1
magnitude
group delay
-1.25 -0.75 -0.25 0.25frequency, rad/s
b
0.75 1.25
Fig. 5 Stopband and passband asymmetric characteristics of an 8th-degree maximally flat filter, with group delay equalisation
a Stopbandb Passband
4 Conclusion
A simple and efficient generalisation of the classicalallpole Butterworth approximation has been presented.Asymmetrically prescribed transmission zeros can nowbe included in the design of networks that are to possessmaximally flat passbands.
Although the method introduced is completelygeneral, it is particularly useful for the design of narrow-band crystal filters realising extremely asymmetric char-acteristics. The related equations can be easilyimplemented as computer programmed procedures andembedded into larger approximation packages that gen-erate higher-degree filtering functions.
5 Acknowledgment
M. de P. Barros would like to acknowledge the financialsupport received from the Conselho Nacional de Desen-
volvimento Cientifico e Tecnologico—CNPq, Brasilia,Brazil.
References
1 BALABANIAN, N., and BICKART, T.A.: 'Electrical network theory'(John Wiley & Sons Inc, 1969), pp. 420-421
2 TEMES, G.C.: 'Approximation theory' in TEMES, G.C., andMITRA, S.K.: 'Modern filter theory and design' (John Wiley andSons, 1973), Chap. 2
3 BAUM, R. F.: 'Design of unsymmetrical band-pass filters', IRETrans., 1957, CT-4, (2), pp. 33-40
4 DILLON, C.R., and LIND, L.F.: 'Cascade synthesis of monolithiccrystal filters possessing finite transmission zeros', Int. J. CircuitTheory & Appi, 1975, 3, pp. 101-107
5 CAMERON, R.J.: 'Fast generation of Chebyshev filter prototypeswith asymmetrically-prescribed transmission zeros', ESA J., 1982, 6,(1), pp. 83-95
8 SZENTIRMAI, G.: 'Crystal and ceramic filters' in TEMES, G.C.,and MITRA, S.K.: 'Modern filter energy and design' (John Wiley andSons, 1973), Chap. 4
9 BARROS, M. de P., and LIND, L.F.: 'On the realisation of asym-metric transmission zeros for crystal filter design', IEE Proc. G, Elec-tron. Circuits & Syst., 1985,132, (5), pp. 217-220
Marcos de Paiva Barros was born in Riode Janeiro, Brazil, in 1951. He receivedthe degree of Engenheiro Eletronico in1974 from the Universidade Catolica doRio de Janeiro. In 1983, he obtained theMSc degree, and in 1986 the PhD degreein electrical engineering, both from theUniversity of Essex, UK. Since 1974 hehas been with Radio Cristais do BrasilS.A., in Rio de Janeiro, involved in thedesign and manufacture of polylithic and
discrete crystal filters, and of oscillating crystal units. Hisresearch interests include network theory, crystal filter synthe-sis, crystal oscillators, and computer-aided circuit design.
Larry Lind was born in Minnesota, USA,in 1941. He came to the UK in 1965, andobtained a PhD degree from Leeds Uni-versity in 1968. Since then, he has been atEssex University, where he is now aReader in the Department of ElectronicSystems Engineering. He has served onand was Vice Chairman of IEE Pro-fessional Group E10 (Circuit theory anddesign). His research interests includecircuit theory, communications theory
and logic design.
228 IEE PROCEEDINGS, Vol. 134, Pt. G, No. 5, OCTOBER 1987
