Sun, Y.

IEEE Transactions on Circuits & Systems II Express Briefs

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University of Hertfordshire Research Archive

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 53, NO. 9, SEPTEMBER 2006 961Synthesis of Leap-Frog Multiple-LoopFeedback OTA-C FiltersYichuang Sun, Senior Member, IEEEAbstract—Operational transconductance amplifier and ca-pacitor filters based on the leap-frog (LF) structure are studiedfrom the viewpoint of multiple-loop feedback (MLF) and coeffi-cient matching. The LF configuration is obtained from an MLFmodel, which has the minimum number of components and onlygrounded capacitors. Explicit and iterative design formulas arederived for the synthesis of arbitrary filter characteristics. All-polefilters are realized using the basic LF structure, and filters witharbitrary transmission zeros are synthesized by adding the inputdistributor or output summer.Index Terms—Active filters, analog circuits, continuous-timefilters, operational transconductance amplifier and capacitor(OTA-C) filters.I. INTRODUCTIONTHE leap-frog (LF) structure is one of the most popularchoices in active-RC filter design [1], [2] due to its lowersensitivity than the cascade method. LF operational transcon-ductance amplifier and capacitor (OTA-C) filters have been de-rived by simulating passive RLC ladder prototypes [2]–[5] andusing a systematic multiple-loop feedback (MLF) approach [2],[6], [7]. The difference between the LF OTA-C filter structuresbased on the two approaches is that circuit component values areobtained from simulated RLCs in the former and determined bycoefficient matching equations in the latter. Active filter designbased on the RLC ladder simulation approach needs knowledgeof passive RLC filters and treatment of input and output termi-nated impedances, while the MLF approach does not have theseproblems. More important is that the LF MLF approach can re-alize any transmission zeros, whilst the LF simulation methodcan only realize imaginary zeros. One of the most important ap-plications of OTA-C filters is the linear phase filtering in com-puter hard-disk-drive systems. In this application, real zeros arenormally required, which may only be realized by the MLF ap-proach when using the LF structure. In communication receiversand video systems, equalizers with certain transmission zerosare often required for stringent phase requirements, for whichthe LF simulation approach may not be suitable.Compared with other MLF configurations, such as thefollow-the-leader feedback (FLF) and inverse-follow-the-leader-feedback (IFLF) structures that have similar sensitivity,the LF MLF configuration has the minimum sensitivity [1], [2].Unlike the FLF and similar to IFLF, it does not have a multiplefeedback input summation node, which may simplify the ICimplementation in practice because the summing node usuallyManuscript received April 6, 2005; revised August 3, 2005, and November 4,2005. This paper was recommended by Associate Editor A. Korotkov.The author is with the School of Electronic, Communication and ElectricalEngineering, University of Hertfordshire, Hatfield Herts AL10 9AB, U.K.(e-mail: y.sun@herts.ac.uk).Digital Object Identifier 10.1109/TCSII.2006.879095has high-frequency parasitic effects. When the OTA parasiticsare considered, in particular, the input capacitance in the dif-ferential applications, the FLF and IFLF configurations maysuffer more serious performance degradation because of longfeedback paths from the th node to the input node. Becauseeach path is capacitive coupling, the effects on circuit poles andzeros appear at high frequencies. Obviously, the extent of thoseeffects can be considered as proportional to the ratio of parasiticcapacitance to circuit integrating capacitance. The LF structureis also advantageous over the cascade configuration [1], [2].Recent investigations have demonstrated that LF MLF OTA-Cfilters offer better passband magnitude sensitivity, maximuminput voltage, and magnitude frequency response consideringOTA nonideality effects over IFLF and cascade OTA-C filters[8]. In particular, the noise performance of the LF filters hasbeen shown to be much better than that of the IFLF filters.Furthermore, promising results from recent research of use ofthe LF MLF OTA-C filters for read channel applications showthat the LF MLF design method could be an attractive alter-native to other methods [9]. However, the synthesis of the LFMLF structure is not as simple as the FLF and IFLF structures.While simple explicit formulas exist for the FLF [10], [11] andIFLF [12], [13] configurations that can be easily used [14], aniterative process is required for LF synthesis. As well as generaliterative equations, straightforward formulas for the most oftenused orders of filters should be derived for ready use.The synthesis of LF OTA-C filters for realizations of bothtransmission poles and zeros can be conducted using the generalMLF approach [2], [6], [7]. This synthesis method, however, isbased on matrix manipulation and requires knowledge of thegeneral MLF theory. In this paper, we present an alternativeiterative approach for synthesis of LF MLF OTA-C filters basedon coefficient matching.II. ALL-POLE LF FEEDBACK OTA-C FILTERSA. Formulation of Circuit Transfer FunctionThe general all-pole LF OTA-C filter configuration [2], [6] isdepicted in Fig. 1. It has the minimum number of componentsand uses only grounded capacitors. With time constant, we can write the equations of the circuit as(1)where is the complex frequency, is the output voltage ofthe th integrator, and can be imagined as a voltage sourceinserted between the output of the th inte-grator and the noninverting terminal of the th OTA. Note thatand can be viewed1057-7130/$20.00 © 2006 IEEE962 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 53, NO. 9, SEPTEMBER 2006Fig. 1. General all-pole LF configuration.equivalently as the input voltage (to the noninverting terminal)of the th OTA. Introducing is mainly for the convenienceof defining functions below and facilitating derivations inSection III. For noise analysis, are input-referral noisesources and functions are the intermediate functions [2], [8].Defining the transfer function from the filter input to theoutput of integrator as(2)it can be shown from (1) that(3)(4)Note that (3) and (4) show that and can beobtained in an iterative way.Defining the transfer function from the input of integratorto the filter output as(5)from (1) we can obtain(6)(7)Again, note that and in (6) and (7) can be deter-mined iteratively.It can be shown that the and functions have the followingrelation:(8)For any order, using the iterative formulas in (3), we can ob-tain all associated numerator polynomials of the functions.For order and , it can be shown that(9)(10)(11)(12)Also, using the iterative formulas in (6), we can obtain corre-sponding for order and , given as(13)(14)(15)n(16)Note that the formulated numerator polynomials of the re-spective and functions are useful for filter sensitivity, max-imum input voltage, and noise analysis [8].SUN: SYNTHESIS OF LEAP-FROG MULTIPLE-LOOP FEEDBACK OTA-C FILTERS 963If is taken as the overall input and as the overalloutput, the overall transfer function can be obtained usingeither the or functions, given by(17)Using (4) and (7) and corresponding and functions, one canobtain the denominator polynomial of respective overalltransfer functions as(18)(19)(20)(21)B. Formulas for Determination of Parameter ValuesTo realize the general unity gain all-pole transfer function(22)the explicit expressions for determining parameter values can beobtained from the coefficient matching equations based on thecomparison between the coefficients of and those in (22).The following are the derived design formulas:(23)(24)(25)Fig. 2. Universal LF structure with input distribution OTA network.(26)III. LF MLF OTA-C FILTERS WITH TRANSMISSION ZEROSA. Functions and Formulas for Input Distributor TypeBy adding the input distribution OTA network to the all-poleLF structure in Fig. 1 to produce zeros, the desired generaltransfer function in (27) can be realized:(27)The resultant general structure is shown in Fig. 2.Denoting and ,the circuit equations can be written from Fig. 2 as(28)where, equivalent to (1), we have .The set of equations in (28) can be rearranged withand as(29)From the last equation in (29), we can solve for toobtain(30)964 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 53, NO. 9, SEPTEMBER 2006The expression of the overall transfer function is obtainedfrom the last relation in (28) as(31)By substituting (30) into (31), we obtain the transfer function as(32)where can be obtained from (6) iteratively and those for– have been given in Section II-A.Using (6) and (32), we can derive the numerators ofthe transfer functions and the corresponding design formulasto determine the distribution parameters when realizing thegeneral function in (27) for – as(33)(34)(35)(36)(37)(38)Fig. 3. Universal LF structure with output summation OTA network.(39)(40)Note that the denominators of the corresponding transferfunctions have already been derived in Section II-A and thedesign formulas for the pole parameters of the respectiveorders have also been given in (23)–(26).B. Functions and Formulas for Output Summer TypeSimilarly, by adding the output summation OTA network tothe all-pole LF structure in Fig. 1 the general transfer function in(27) can be realized and the resulting circuit structure is shownin Fig. 3.Defining , the circuit equations for theintegrator OTAs can be written from Fig. 3 as(41)which corresponds to (1) with .Note that (41) shows that, by a complete feedforward sub-stitution, we can obtain . Then, beginning from ,we can compute consec-utively.Finally, by weighted summation and denoting ,we can attain the overall transfer function of the circuit as(42)Note that can be obtained iteratively as shown in (3)and given in Section II-A for – .Using (3) and (42), we can formulate the numerators ofthe transfer functions and the corresponding design formulas ofSUN: SYNTHESIS OF LEAP-FROG MULTIPLE-LOOP FEEDBACK OTA-C FILTERS 965summation parameters when realizing the general functionin (27) for – , given as(43)(44)(45)(46)(47)(48)(49)(50)IV. CONCLUSIONThe synthesis of low-sensitivity LF MLF OTA-C filters basedon coefficient matching has been presented. The filter structuresare canonical, and every node has a grounded capacitor. All-polefilters are realized using the basic LF structure, and filters witharbitrary transmission zeros are synthesized using either inputdistribution or output summation OTA networks. The explicitand iterative design formulas derived (up to ) are simpleand useful for quick design of LF-based all-pole and arbitrarytransmission zero filters without the need for reformulation.REFERENCES[1] R. Schaumann, M. S. Ghausi, and K. R. Laker, Design of Analog Fil-ters: Passive, Active-RC, and Switched Capacitor. Englewood Cliffs,NJ: Prentice-Hall, 1990.[2] T. Deliyannis, Y. Sun, and J. K. Fidler, Continuous-Time Active FilterDesign. Boca Raton, FL: CRC, 1999.[3] R. Nawrocki, “Electronically tunable all-pole low-pass leapfrog ladderfilter with operational transconductance amplifier,” Int. J. Electron.,vol. 62, pp. 667–672, 1987.[4] A. C. M. de Queiroz, L. P. Caloba, and E. Sanchez-Sinencio, “Signalflow graph OTA-C integrated filters,” in Proc. IEEE ISCAS, 1988, pp.2165–68.[5] R. Schaumann, “Simulating lossless ladders with transconductance-Ccircuits,” IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process.,vol. 45, no. 3, pp. 407–410, Mar. 1998.[6] Y. Sun and J. K. 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Synthesis of leapfrog multiple loop feedback OTA-C filters

Synthesis of leapfrog multiple loop feedback OTA-C filters

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