New Second-Order Optimized Filter Design

30 J. POSPÍŠIL, Z. KOLKA, S. HANUS, T. DOSTÁL, J. BRZOBOHATÝ, NEW SECOND-ORDER OPTIMIZED FILTER DESIGN New Second-Order Optimized Filter Design Jiří POSPÍŠIL1, Zdeněk KOLKA1, Stanislav HANUS1, Tomáš DOSTÁL1, Jaromír BRZOBOHATÝ2 1 Dept. of Radio Electronics, Brno University of Technology, Purkyňova 118, 612 00 Brno, Czech Republic 2 Dept. of Microelectronics, Brno University of Technology, Údolní 53, 602 00 Brno, Czech Republic [email protected] Abstract. Starting from the piecewise-linear (PWL) autonomous dynamical system optimized from the eigenvalue sensitivities viewpoint the corresponding optimized nonautonomous linear (single-input single-output) system is derived. Such a design procedure gives the possibility to obtain minimum eigenvalue sensitivities with respect to the change of the individual model parameters also for nonautonomous linear systems. Two examples of the system having the complex conjugate poles and zeros, i.e. the optimized second-order band-reject and all-pass filter design, are shown. )(h TxwbxAx += & (1) where the elementary PWL feedback function (Fig. 1)      −−+= 1 1 )(h TT T 2 1xwxwxw (2) contains the regions D0 and D+1 (D-1). General block diagram corresponding to basic eqn. (1) is shown in Fig. 2. D-1 D0 D +1 1 1 h(wTx) wTx Keywords Dynamical systems, second-order systems, state models, sensitivity properties, optimized design. Fig. 1. Simple memoryless PWL feedback function. 1. Introduction The dynamical behavior of this system is determined by two characteristic polynomials associated to the individual regions, i.e. In the recent papers [1], [2], some new results in the field of linear and piecewise-linear (PWL) dynamical systems have been published. It is especially: ( ) ( ) ( )( ) ( ) = − =− − − = A1sssssPD ndet...: 210 µ µ µ (i) Generalized mutual relation between non-autonomous linear and autonomous PWL systems. ( )( n n n n nnn pspspsps 11... 1 1 2 2 1 1−+−+−+−= − + −− ) , (3) ( ) ( ) ( )( ) ( ) = − =− − − = −+ A1sssssQDD ndet...:, 2111 υ υ υ (ii) State model of autonomous PWL systems with optimized eigenvalue sensitivities. ( )( n n n n nnn qsqsqsqs 11... 1 1 2 2 1 1−+−+−+−= − + −− ) (4) These two results give a natural possibility to convey optimized properties of the autonomous PWL system to transfer function of the linear non-autonomous system, i.e. to obtain the new optimized design procedure for linear systems. As an examples two typical systems with complex conjugate poles and zeros of the corresponding transfer function, i.e. the second-order band-reject and all-pass filter, are introduced. where 1 is the unity matrix. Their roots represent the eigenvalues of the corresponding state matrices and their coefficients are the so-called equivalent eigenvalue parameters [4], [5]. & x A ∫ h(...) h( ) T wx wx T Ax w T bbwx h( ) T 2. Relation between PWL Autonomous and Linear Non-Autonomous Dynamical Systems Autonomous PWL systems of Class C [3], [4] can be described by the general state matrix form Fig. 2. General block diagram of an autonomous PWL dynamical system described by eqn. (1). RADIOENGINEERING, VOL. 12, NO 1, APRIL 2003 31 Any non-autonomous linear system with single input (v variable) and single output (y variable) can be described by the general state matrix equations 0 ""2 )""-()'-'( M 22 > + = νµ νµνµ , ( 0)"," ≠ ν µ . Choosing w1 = 1, the other parameters are obtained as vBxAx += & (5) ''b νµ −= 1, K"" )''( 2 2 µν νµ − − = b, '' K"" 2 νµ µ ν − − = w (11a,b,c) vy D+= xC . (6) The corresponding general block diagram is shown in Fig. 3. Using the Laplace transform the transfer function of the non-autonomous linear system is generally given [2] as and the complete state equations of the optimized secondorder PWL autonomous system can be written in the form [ ] )()( 221222111 h'"h' xwxxxwxxx ++ + − − = µ ν ν &, (12) () )( )( - ... )(D )( )( K1sQ sP s sV sY s==−+== BA1C (7) )( 2212212 h'" xwxbxxx + + + = ν ν & (13) if the following conditions are valid where the parameters b2 and w2 are given by the formulas (11b,c). The corresponding integrator-based circuit block diagram, suitable as the prototype for the practical realization, is shown in Fig. 4. bB = , and D . (8) T wC=1−= & x A ∫ C D Cx Ax Dv BB v ∫ ∫ PWL h(...) (x +w x ) 122 +1 b 2 x 2 x 1 -1 +1 ν ’ ν ’ ν ’’ -’’ ν w 2 µ ’ Fig. 3. General block diagram of a non-autonomous linear dynamical system described by eqns. (5) and (6). 3. Optimized State Model of the 2nd-Order Dynamical Systems 3.1 PWL Autonomous System Considering the complex conjugate eigenvalues in the outer regions D+1, D-1 (ν1,2=ν’±jν’’) as well as in the inner region D0 (µ1,2 = µ’±jµ’’), the optimized state matrices corresponding to the outer and inner regions can be chosen in simplified and decomposed complex form [3], [7], i.e. Fig. 4. Integrator-based circuit structure of the 2nd-order PWL state model with minimized sensitivities. 3.2 Linear Non-Autonomous System Starting from optimized PWL autonomous system with complex conjugate eigenvalues, utilizing the conditions (8), the complete optimized form of the state equations (5) and (6) for the corresponding linear system is obtained, i.e.       ='" "-' νν νν A and , (9a,b)       ='K" K"-' 0 µµ µµ 1A respectively. These state matrices can mutually be expressed by the relation [4] ( ) vxvxx '"' 211 µ ν ν + − − = & , (15) T 0wbAA += , (10) vbxxx 2212 '" + + = ν ν & , (16) where and w .       = 2 1 b b b      = 2 1 w w v-xwxy 221 + = . (17) The corresponding transfer function (7) is (18a) The optimizing coefficient K in eqn. (9b) is expressed as the real root of the quadratic equation () 2121 2 2121 2 )( )( )(Q )(P )(V )(Y K νννν µµµµ ++− ++− −=−== ss ss s s s s s 01)1(MK2K2=++− , i.e. 2)(MMM1K +±+= that corresponds to general second-order form and the auxiliary parameter M is given in the form 32 J. POSPÍŠIL, Z. KOLKA, S. HANUS, T. DOSTÁL, J. BRZOBOHATÝ, NEW SECOND-ORDER OPTIMIZED FILTER DESIGN () [] [] 2 000 2 22 ωω ωω ++ ++ =∞sQs sQs KsK zzz (18b) where the parameters b2 and w2 can be expressed as 2 2 ' w b ν − =, ν ν µ ′ ′′ − ′ ′ =K w2 . (25) where the individual parameters are "' '2 1 , "' 22 0 22 0 νν ν ννω + − =+= Q, (19a,b) ν ’ ν ’’ −ν ’’ x x o o Re Im 0 −ω 0 ω 0 µ ’’= ω 0 ω z > ω z "' '2 1 , "' 2222 z µµ µ µµω + − =+= z Q. (20a,b) The corresponding integrator-based circuit block diagram, suitable as the prototype for the practical realization, is shown in Fig. 5. ∫ ∫ +1 b 2 y vx 2 x 1 -1 -1 +1 ν ’ ν ’ ν ’’ -’’ ν w 2 µ ’ ν ’ ν ’’ −ν ’’ x x Re Im 0 o o µ ’’= ω 0 ω z = −ω 0 ω z = ν ’ ν ’’ −ν ’’ x x Re Im 0 o o −ω 0 ω 0 ω z µ ’’= ω 0 ω z < c) Fig. 5. Integrator-based circuit structure of the 2nd-order linear state model with minimized sensitivities. 4. Applications to 2nd-Order Filters with Optimized Sensitivities 4.1 Band-Reject Filters As the first example the second-order band-reject filter with complex conjugate poles and imaginary conjugate zeros of the transfer function (18a) is introduced. Here the parameter Qz → ∞, i.e. Fig. 6. Zeros and poles of the 2nd-order band-reject filter. a) ω z > ω 0 , b) ω z = ω 0 , c) ω z < ω 0. The corresponding integrator-based circuit block diagram, suitable also as the prototype for the practical band-reject filter realization, is shown in Fig. 7. Its transfer function (18) has the following special form 0'= µ , and z, jj ω µ µ ± = ±= " 21 (21) as follows from eqns. (20a,b). It is well known that three different cases can exist as shown in Fig. 6, i.e. a) 0 ω ω > z - (Fig. 6a), () )"'('2 " )(Q )(P )(V )(Y K222 22 ννν µ ++− + −=−== ss s s s s s s b) 0 ω ω = z - (Fig. 6b), c) 0 ω ω < z - (Fig. 6c) . which corresponds to general form Then the complete optimized form of the state equations is () [] 2 000 2 22 ωω ω ++ + =∞sQs s KsK z (26b) () 211 "' xvxx ν ν −−= & , (22) where the individual parameters are: vbxxx 2212 '" ++= ν ν & , (23) 1 − = ∞ K, " µ ω = z, 22 0 22 0"' '2 1 ,"' νν ν ννω + − =+= Q v-xwxy 221 += (24) RADIOENGINEERING, VOL. 12, NO 1, APRIL 2003 33 vxxx '2"' 211 µ µ µ + − − = &, (28) ∫ ∫ +1 b 2 y vx 2 x 1 -1 -1 +1 ν ’ ν ’ ν ’’ -’’ ν w 2 vbxxx 2212 '" + − = µ µ & , (29) v-xwxy 221 + = (30) where the parameters b2 and w2 can be expressed as 22 2wb µ ′ = , ( ) 2 21mm +±−=w, µ µ ′ ′ ′ = m . −µ ’’ µ ’’ −µ ’ µ ’ x x o o Re Im 0 Fig. 7. Integrator-based circuit structure of the 2nd-order bandreject filter with minimized sensitivities. Typical magnitude characteristics in frequency domain for all three relations between ω 0 and ω z (Fig 6a, b, c) are introduced in Fig. 8. |K| |K| |K| ω ω ω ω z ω0=ωz ωz ω0 ω0 a) b) c) Fig. 9. Zeros and poles of the 2nd-order all-pass filter. The corresponding integrator-based circuit block diagram, suitable also as the prototype for the practical allpass filter realization, is shown in Fig. 10. Its transfer function (18) ∫ ∫ +1 b 2 y vx 2 x 1 -1 −µ ’’ µ ’’ w 2 2µ ’ −µ ’ −µ ’ Fig. 10. Integrator-based circuit structure of the 2nd-order all-pass filter with minimized sensitivities. has the following special form Fig. 8. Computer simulated magnitude characteristics of the optimized 2nd-order band-reject filter. a) ω z > ω 0 , b) ω z = ω 0 , c) ω z < ω 0. () )"'('2 )"'('2 )(Q )(P )(V )(Y K222 222 µµµ µµµ +++ ++− −=−== ss ss s s s s s 4.2 All-Pass Filters which corresponds to general form As another example the second-order all-pass filter with symmetric complex conjugate poles and zeros of the transfer function (18) is introduced. Here, in accordance with Fig. 9, the complete state model can be described by using of the real and imaginary parts of zeros only (µ’>0, µ’’>0). Considering the root symmetry () [ ] [] 2 00 2 2 00 2 ωω ωω ++ +− =∞sQs sQs KsK (32b) where the individual parameters are . "' '2 1 , "' ,1 2222 0 µµ µ µµω +=+=−= ∞QK (33) "",' ' µνµν =−= , (27) The phase characteristic in frequency domain is introduced in Fig. 11. the complete optimized form of the state equations is 34 J. POSPÍŠIL, Z. KOLKA, S. HANUS, T. DOSTÁL, J. BRZOBOHATÝ, NEW SECOND-ORDER OPTIMIZED FILTER DESIGN [6] KOLKA, Z. Using similarity transformation for nonlinear system synthesis. In Proc. Rádioelektronika’ 2001, Brno, 2001, pp. 5-7. 0.1 1 10 -150 -100 -50 0 50 100 150 ϕ ω / ω 0 Q=2 1 0.5 [7] POSPISIL, J., BRZOBOHATY, J., KOLKA ,Z.,. HORSKA, J., DOSTAL, T. Dynamical systems with low eigenvalue sensitivities. In Proc. MIC’2001, Innsbruck, 2001, pp. 217-219. [8] M. S. SCHAUMAN, M. S. et al. Design of Analog Filters. Passive, Active RC, and Switched Capacitor. Engelwood Cliffs, NJ: PrenticeHall, 1990. [9] HANUS, S. Realization of third-order chaotic systems using their elementary canonical state models. In Proc. Rádioelektronika’97, Bratislava, 1997, pp. 44-45. [10] POSPISIL, J., BRZOBOHATY, J., KOLKA, Z., HANUS, S., MICHALEK, V. Optimized state model of piecewise-linear dynamical systems. Radioengineering, 2003, vol. 12, no.1, pp. 27-29. Fig. 11. Computer simulated phase characteristics of the 2nd-order all-pass filter with optimized sensitivities. About Authors... 5. Conclusion Jiří POSPÍŠIL was born in Brno, Czechoslovakia, in 1939. M.Sc. and Ph.D. (equiv. degrees): 1963 and 1973, respectively; DSc. (equiv. degree.): 1988, all in el. engg, TU Brno, Czechoslovakia. 1964: Assist. Prof., Military Acad. of Brno, Dept of El. Engg; 1970-1972: Visit. Prof., Military Tech. College, Cairo, Egypt; since 1974: TU Brno, Dept of Radioelectronics; 1980: Assoc. Prof.; 1989: Prof.; Research and pedag. interest: Circuits and Systems Theory, PWL Dynam. Networks, Dynam. Systems Modelling. IEEE: M.-1992, S.M.- 1995. The general optimization condition for the second-order autonomous PWL dynamical system is utilized for the design of their state models with low eigenvalue sensitivities [7]. Then the state model of the corresponding secondorder non-autonomous linear dynamical system is derived. The results achieved are applied to optimized band-reject and all-pass filter design that can be realized in the form of simple electronic circuits having separately adjustable parameters. It has been also proved numerically by simulation and also by the starting laboratory experiments. Zdeněk KOLKA was born in Brno, Czechoslovakia, in 1969. He received the M.S. (92) and Ph.D. (97) degrees in electrical engineering, both from the Faculty of Electrical Engineering and Computer Science, Brno University of Technology. At present he is an Assistant Professor at the Institute of Radio Electronics. He is interested in PWL modeling, circuit simulation, and nonlin. dynam. systems. Acknowledgement This research is partially supported by the Grant Agency of the Czech Republic, Grant projects No. 102/02/1312/A and No. 102/01/0229. It represents the part of the Research Program of the Czech Ministry of Education – CEZ: J22/98: 262200011. Jaromír BRZOBOHATÝ was born in Brno, in 1935. M.Sc. and Ph.D. (equiv. degrees): 1960 and 1980, respectively, both in el. engg, TU Brno, Czechoslovakia. 1960-1963: researcher in Metra Blansko; 1963: Assoc. Prof.; 1987: Prof.; 1985-1989: Dean of Faculty of El. Engg; 1980-1993: Head of Dept of Microelectronics. Research interests: Circuits Theory, Microelectronics, PWL Dynam. Netw. IEEE: M.-1988, S.M.-1990. References [1] POSPISIL, J., BRZOBOHATY, J., HORSKA, J.Mutual relation between multiple-input linear and multiple-feedback piecewise-linear dynamical systems. Radioengineering, 2000, vol. 9, no.4, pp. 28-32. Stanislav HANUS was born in Brno, Czechoslovakia, in 1950. He received the Ing. (M.Sc.) and CSc. (Ph.D.) degrees from the Brno University of Technology. He is Associated Professor at the Institute of Radio Electronics, FEEC, BUT Brno. His research is concentrated on Circuit theory and Wireless and Mobile Communications. [2] POSPISIL, J., KOLKA, Z., HORSKA, J. Synthesis of optimized piecewise-linear system using similarity transformation – part II: second-order systems. Radioengineering, 2001, vol. 10, no.3, pp. 810. [3] POSPISIL, J., BRZOBOHATY, J. Elementary canonical state models of Chua’s circuit family. IEEE Trans. Circ. Syst.-I: Fundamentals .. , 1996, 43(8), pp. 702-705. Tomáš DOSTÁL was born in Brno in 1943. CSc. and DrSc. (equiv. degrees): 1976 and 1989, respectively, both in el. engg, TU Brno, Czechoslovakia. 1973-1978 and 1980-84 with Military Academy Brno, 1978-80 with Military Technical College Baghdad. Since 1984 with the Brno University of Technology as Professor of Radio Electronics. Research interests: circuit theory, filters, switched capacitor networks, circuits in current mode. [4] POSPISIL, J., BRZOBOHATY, J., KOLKA. Z.,. HORSKA, J. Simplest ODE equivalents of Chua’s equations. Intern. Journ. of Bifurcation & Chaos, 2000, 10(1), pp. 1-23 (Tutorial & Review paper). [5] WU, C. W., CHUA, L. O. On linear topological conjugacy of Lur'e systems. IEEE Trans. Circ. Syst. - I: Fundamentals..., 1996, 43(2), pp. 158-161.