Coordinated Science Laboratory was formerly known as Control Systems Laborator

Kishi, G.

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Transient Responses of Butterworth and Chebyshev Filter Networks

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UNIVERSITY OF ILLINOIS -  URBANA, ILLINOISTRANSIENT RESPONSES OF BUTTERWORTH AND CHEBYSHEV FILTER NETWORKSG. KishiREPORT R -194 FEBRUARY, 1965TRANSIENT RESPONSES OF BUTTERWORTH AND CHEBYSHEV FILTER NETWORKSG. KishiIntroductionAlthough the general method for obtaining the transient response of any filter network has been established by the use of transformation calculus, a few response curves of commonly used filter networks are available in existing literature. In this report1, transient responses of the Butterworth and the Chebyshev filter networks are calculated for both suddenly impressed and suddenly removed sinusoidal input signals with angular frequency uo^ . The transient responses of these networks are treated in terms of the in-phase and the quadrature components, and these functions together with envelope functions are presented under the assumption of narrow bandwidth for the following cases:1. For the n-th degree Butterworth filter networks with mid-bandangular frequency uu and half-power (or 3db) transmission band (JU_ + U), ,u U — bthe cases of n = 3 to 7, u) = and the cases of n = 3 to 5,c 0 *(0)^ - ou0)/u)b = 0.2, 0.4, 0.6, 0.8, 1.0, and half-gain points.2. For the n-th degree Chebyshev filter networks with mid-bandangular frequency U)n and A db transmission band cjo_ + oo, , the cases ofU p  0 — bn = 3 to 7, ^  and A^ = 0.01, 0,03, 0.1, 0.3, ldb, and the cases ofn = 3 to 5, Ap = 0.03db and (u^ - u)^ /u>b = 0.2, 0.4, 0.6, 0.8, 1.0.Many words could be devoted to discussion of these response curves, but only a few general considerations will be described in this paper in order to suggest ways in which the curves might aid the designer.2PreliminariesBefore getting into details, we being with a summary of assumptions, definitions and formulas which will be used in this paper.Transfer function: Let T(s) be a transfer function whose transientresponses are to be considered. For the Butterworth and Chebyshev bandpass filter networks of the n-th degree, we can write T(s) in the following form:T(s) =H2n(s> ( 1)where H (s) is a Hurwitz polynomial of degree 2n. H (s) can be zn 2n2nexpressed in terms of its simple roots t s ] as follows-k k=l2nH (s) = K - n (s-s ) k=l k ( 2 )where s, satisfies kRe [s. ] < 0, k = 1,2,3.... 2n. (3)Transient response: Sudden impression of a sinusoidal signal canbe expressed asfx(t) =and g-^ (t) is given bygx(t) =| 0, t < 0jU> tt > 0(4)output signal g^(t) of the network defined byTo, t < 0ejv> (5)I 10““' 0  ” H2n(^ c), t > 03where2n x j(sk-ju;c)E(u> ,t) = 1 + H» (jou ) 2 ----- —  • ----------C  2 n  C  1 1 S 1 ' „ •  , Vk=l k C H_ (s )( 6 )2n k'is the response function normalized with its final value. The function E(ttc,t) is complex valued and its real and imaginary parts are calledin-phase and quadrature components, respectively.If an input signal f^Ct) expresses sudden removal of a sinusiodalsignal, it can be written init✓"N4-JCM<4-1C jo) t J ce • t < 0 (7)1 °  . t > 0The signal is also written injU)f2(t) = e C ‘ fx(t) (8)Thus the output signal g^(t) caused by f^Ct) is given byC jw tJ c e , t < 0H2n<J”c>ii4-1v—'CM60 1U) t J c(9), [1 ■ ■ t s v, t > 0It is observed that two envelope functions corresponding to turning-on and turning-off transients are generally different because of existence of the quadrature component, that is1 - |E(o> ,t)| ^ |l - E(u) ,t)| . (10)4Narrow bandwidth approximation: If the bandwidth + oo^  of the bandpassfilter network is small enough compared with its mid-band angularr , 2n nfrequency uon, the roots is, j, . can be classified into two groups 1 s, j, „ u k k=l k k=lnand is , , satisfyingn+k k=l J °sk - jlV  <<: I \  + ^ o 1 (ii)andSn+k ' j“V  >=> I Sn+k + ^ O 1 ( 12)Thus we obtain1 1 »  1 1Sk • s , - jU)n+k c(13)unless large amount of detuning appears. Under this assumption, we can neglect one half of terms appearing in the summation in (6).Normalization: For convenience, we adopt the following normalizationformulas:s - ju)z = 0(JO  -  (JUzk ■(JO,Sk - Jmo(JO,X = 0(JO,(JO - (JO c 0 x = -------C (JO,H0 (s) = D (z) , 2n • n v ' ’HI (s) = —  D' (z) .2nv • (JO, nv '  b(14)JThus (6) can be written in5E(cjuc> t) = 1(15)under the assumption of narrow bandwidth.Transient Responses of Butterworth FiltersFor the Butterworth filter network of n-th degree, Dn(z) is given bynDn(z) = n (z - zk)(16)z . 2k-1 _ . 2k-1, = - sin — —  tt + j cos — —  k 2n J 2n Jwhere the normalization factor aob gives the half-power (or 3db) transmission band. Thus the normalized transient response is easily obtained by the use of Eqs. (15) and (16). For the case of no detuning, that is xc = 0, transient responses of networks of n = 1 to 7 are shown in Fig. 1. This set of curves is also available, for example, in [2] but slightly different in the normalization of the time axis. The curves show that the transient response of the Butterworth filter networks has con­siderable amount of overshoot and ringing even for the case of no detuning.For the detuned case, in-phase and quadrature components for networks of degree n = 3 to 5 are shown in Fig. 2 to 7, each of which is corresponding to the detuning factor xc = 0.2, 0.4, 0.6, 0.8, 1, and half-gain points, respectively. It can be observed that as the detuning factor x^ is increased from 0.2 to 0.8, positive smear in in-phase components becomes greater. This6is closely correlated to the group delay characteristic of networks.It has been shown in [3] that(1) the group delay characteristic t (x ) of the n-th degreeBnButterworth function has two maximum peaks at + x and one— nlocal minimum at x = 0 wherever n > 2.(2) x satisfies 0 < x <1.n n(3) x satisfies x > 0.8 whenever n > 3.n n —Thus the increase in smear of in-phase components can be interpreted asthe result of the fact that the group delay at the impressed frequency xis a monotone increasing function of x whenever 0 < x <0.8.c cThe last set of curves may be useful in the application of I.F. amplifier design for the television receiver. Approximate values of half­gain points for n = 3, 4, and 5 are x = 1.201, 1.147 and 1.116,crespectively.Turning-on and turning-off envelope functions, | E (twc , t) | and | 1 - E(0)c, t) | , are shown in Fig. 8 to 13. The cross points of turning-on and turning-off envelope functions of the same network have the amplitude coordinate greater than 0.5 because of asymmetry caused by the existence of quadrature components. Considerable length of tails of turning-off envelope functions is an important feature of the set of curves. This is sometimes an essential defect in the practical application of the Butterworth filter networks. The synchronous detection can be used to improve the transient response, but it is not so powerful because of poor response in in-phase components.7Transient Responses of Chebyshev FilterFor the Chebyshev filter network of the n-th degree, Dn(z) is givenbyD (z) = ^  a - 1 • 2zk = j cos (Ck + jT|)n-1 nn (z-zk) k=l2k-1 „ -n 1 , -1 a +1£k = - —  n, H = —  cosh2na Z-1 7P ^(17)where the normalization factor U), gives A = 20 lognrta db transmissionb p 10 pband. Substituting (17) into (15), we can obtain in-phase and quadrature components.For the case of no detuning, xc = 0, transient responses of networksof n = 3 to 7 with = 0.01, 0.03, 0.1, 0.3 and ldb are shown in Fig. 14to 18. Part of the sets of curves is also available in [4], but differentin the normalization of the amplitude scale for networks of even degree.It can be observed that as the pass-band ripple A is increased, time delayand ringing become greater. It is interesting that the responses of oddand even degree networks contain positive (or increasing) and negative(or decreasing) smear, respectively.For the detuned case, in-phase and quadrature components for networksof A = 0.03db and n = 3 to 5 are shown in Fig. 19 to 23, each of which is Pcorresponding to the detuning factor x^ = 0.2, 0.4, 0.6, 0.8 and 1,2respectively. It has been shown in [3] that the group delay characteristic TTn(x) °f fche Chebyshev function of n-th degree is a monotone increasing8function whenever 0 < x < 1 and n = 3 to 5. So the increase in smear in in-phase components can be interpreted in the same way as the Butterworth filter.The envelope function |e (u) , t)| and |l - E(u) ,t)| are shown in Fig.c c24 to 28, each of which is corresponding to = 0.2, 0.4, 0.6, 0.8 and 1, respectively. Remarkable smear in turning-on envelope functions and long masking tail in turning-off envelope functions are observed in these figures.Relative amplitudeFig. 1 Transient response of Butterworth filter, x£ = 0*Relative amplitudeRelative amplitudeO 2 3 4 5 6 7 8 9 10 11Fig. 3 In-phase and quadrature components of Butterworth filtertransient, x = 0.4. ccobt (radian)1 12Relative amplitude1.21.0-0.21--------- -^-------- 1--------- ^ ^ _____ I I I l I0 V 2 3 4 5 6 7 8 9Fig. 5 In-phase and quadrature components of Butterworth filtertransient, x = 0.8. c10 11 o j j i  (radian)-ï>Relative amplitude1.2aibt (radian)Fig. 8 Envelope function of Butterworth filter transient, = 0.2.H-»O'Relative amplitude1.20 1 2 3 4 5 6 7 8 9 10 11 1?cobt (radian)Fig. 9 Envelope function of Butterworth filter transient, x = 0.4.Relative amplitude1.2(radian)Fig. 10 Envelope function of Butterworth filter transient, x =0.6.cooVONJORelative amplitudeRelative amplitudeX =cFig. 14 Transient response of Chebyshev filter, n = 3, 0.Relative amplitudecobt (radian)Fig. 15 Transient response of Chebyshev filter, n = 4, x = c 0.LORelative amplitude-Fig. 16 Transient response of Chebyshev filter, n = 5, xc 0.N>-F-Relative amplitudecobt (radian)Fig. 17. Transient response of Chebyshev filter, n = 6, x^ = 0.N3CnRelative amplituderoONRelative amplitude1.2- 0.20 Fi|>. 19 Iti0?hase an^ quadrature compotfents of Çhebyshevgjîilter 0 transient, A = 0,03c[b, x = (f. 2. 10 11 u>bt (radian) 12roRelative amplitudeRelative amplitudei- • \(jON3cLOLOcto-P>U>LnRelative amplitudeooACKNOWLEDGMENTThe author is grateful to Professor M. E. Van Valkenburg for many valuable suggestions and criticisms.Footnotes1. Part of this report is published in Japanese. See [l].2. A set of group delay curves have been published recently. See [5 ]'.39REFERENCES' X. Kishi, G., "Transient Responses of Butterworth and Chebyshev FilterNetworks", Jour. Inst. Elec. Comm. Engr., Japan, Vol. 41, pp. 979-984, October 1958.2. Valley, G. E. and Wallman, H., Vacuum Tube Amplifiers, McGraw-Hill Book Co., Inc., New York, 1948.3. Kishi, G. , "Group Delay Characteristics of Butterworth and Chebyshev Filter Networks", Jour. Inst. Elec. Comm. Engrs., Japan, Vol. 38, pp. 691-696, September 1955.4. Henderson, K. W. and Kautz, W. H., "Transient Responses of Conventional Filters", IRE Trans. Circuit Theory. Vol. CT-5, pp. 333-347, December 1958.5. Lim, Jim-Twan and Scanlan, J. 0., "Group Delay Characteristics of Chebyshev Filters", IEEE Trans, on Circuit Theory, Vol. CT-11, pp. 427-430, September 1964.

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Transient Responses of Butterworth and Chebyshev Filter Networks

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