Continuous-Time (CT) Analog Filters are used for : 1. Interface Applications

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Susan Benson
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3 Continuou-Time CT Analog Filter are ued for :. Interface Application Anti-Aliaing or Preconditioning, and, Smoothing or Recontruction function. The real ignal proceing i performed in the Digital domain.. Direct Filtering At high frequencie, CT filter perform all of the ignal proceing in the analog domain to realize Magnitude and/or Phae Delay Shaping. 3

4 High-Frequency HF relate to the Technology available for implementation and Application under conideration Active and Paive ; currently HF tart from Hundred of MHz, and extend up to ten of GHz. With.3um Technology and lumped element a oppoed to ditributed, limited functionality i achievable up to ~ GHz. At thee frequencie, we are forced to appreciate the interdependence between Magnitude and Phae/Delay. Thi interaction i very important in direct HF filtering unlike in Interface filtering where, in general, only magnitude haping i of prime concern. 4

5 For a tranfer function H, at real frequencie, with =j,.. j e G j e j H j H where G and are the gain-magnitude, or imply the gain, and the phae component repectively. Phae Delay Pd i defined a, Pd Group Delay i defined a, 5

6 In general, both Pd and are function of frequency. The Phae delay Pd repreent the abolute delay, and i of little ignificance. The Group Delay i ued a the criterion to evaluate phae nonlinearity. A linear phae variation with frequency over a band of frequencie implie a contant Group Delay and no phae ditortion in that frequency band. In order to preerve the integrity of the pule through a ytem, it i mandatory that the group delay of the ytem be contant up to the maximum frequency component of the pule. 6

7 For the Firt Order Low Pa Tranfer Function LPF : H j H j G arctan 7

8 G, where, Note i the value at dc. At=, i 6 db relative to it dc value. Only for the firt order function, the group delay repone i the quare of the magnitude repone. 8

9 For the Second Order Low Pa Tranfer Function LPF : Q Q H 4 4 Q Q Q, and i inverely proportional to o and Q. 9

10 peak for Q >.577, and, 4 Q Q peak From and peak /, o and Q can be calculated in real application. i a real and even function of.

11 Q Fig. Normalized Group Delay v. radian frequency for LPF [] ec radian/ec

12 Theoretically peaking, Magnitude and Phae repone of Tranfer Function are independent. However, in practice, the Magnitude and Phae function are cloely related. For a given magnitude function, the tranfer function H can be made unique by requiring that H be a Minimum Phae Function MPF. A MPF i defined a a function for which the phae, at zero radian frequency minu the phae, at infinite radian frequency, i the minimum that can be aociated with that Magnitude function. A MPF ha no RHS pole for tability reaon and no RHS zeroe.

13 A non-mpf, H n-mpf can be realized with the ame Magnitude function a the MPF, H mpf in the following manner :. A H H mpf mpf n where, A i an All-Pa Function APF defined a : Q Q A, where Q i a Hurwitz polynomial. Example of A are : A, Q Q A, and o on. For =j, the magnitude of APF i unity. Therefore A can be ued to vary the delay of H n-mpf without affecting the magnitude a pecified by H mpf. Thi i the general principle of realizing Fixed Magnitude Variable Delay filter. Example are delay haping circuit for cable equalizer. 3

14 For a given phae function, the ytem function i definitely not unique. However, for a pecified phae function, the ytem tranfer function H can be made unique, within a contant calar frequency-independent multiplier, by requiring the ytem function be a minimum phae function H mpf for that pecified magnitude function. In order to provide flexibility in magnitude haping without affecting the phae characteritic, the H mpf function i multiplied with a trictly magnitude function to yield the required function H actual a follow H actual = H mpf.m where M = Q.Q-, and Q i a Hurwitz polynomial. 4

15 Note for =j, M i Q, which i trictly a magnitude function, and the phae/group delay of H actual i ame a that of H mpf. Example of M are M Thi ha been ued for read-channel IC filter for HDD application for almot year to provide a magnitude boot around a pecified frequency. 5

16 7 tap Analog Tranveral FIR Equalizer for Read Channel [] Objective : Perform high peed ignal equalization in Analog domain. 6

17 Feature : Six identical delay cell, where the nominal delay i one read clock period. 8-bit multiplier for each tap except for center tap which ha a fixed gain of unity Adaptation ue deciion directed ign-ign LMS algorithm Delay Cell : Each Delay element i a Fourth order tranfer function with a linear-phae, flat magnitude repone given by the equation : H 7

18 Note that thi tranfer function i an example of a Fixed Delay Variable Magnitude tranfer function, and i realized a the product of a MPF phae function modified by a magnitude function of the form M = Q.Q-..64 The reulting Group Delay i, and i et to T, the clock period correponding to the data rate. The delay i maintained contant over temperature and proce and upply variation via a tandard mater-lave clock/rc tuning circuit. In order to accommodate variable data rate, the clock period i programmed within a +/- 5% range. The delay cell i realized a a cacade of two biquad uing Gm-C integrator, hown below. 8

19 Fig. 3. Gm-C Baed Biquad Deign 9

20 Fig.4. Schematic of a Gm-C cell

21 Fig.5. Magnitude and Group-Delay Repone of Analog Equalizer

22 Advantage : Analog FIR filter achieve highet peed of operation for the given technology. Equalization i achieved without requiring a ampling clock and with reduced latency. The imple analog architecture reult in lower power conumption and maller die ize.

23 Analog equalizer are becoming increaingly popular a both the field of application and the frequency of operation increae. What i required i a baic analog delay cell which i realized uing a Gm-C tructure. To a firt approximation where the pole at the output dominate, the delay i, where f o i the BW in Hz.. f For a ps delay, the BW hould be 6 GHz Small delay imply large BW. For a Gm-C cell driving a reitive load R, i ame a RC, auming that the BW of the Gm circuit i much larger than f o! A variable delay cell i realized uing a variable Gm or a variable C varactor. A major challenge i to maintain the delay contant over proce, upply and temperature fluctuation. 3

24 . C.S.Lindquit, Active Network Deign with Signal Filtering Application, Steward & Son, 977, pp.34.. A PRML Read/Write Channel IC Uing Analog Signal Proceing for Mb/ HDD, K. Pari, et al, IEEE Journal of Solid-State Circuit, Vol.3, No., pp.87-83, Nov A 5 MHz Continuou-Time Seventh-Order.5 o Equiripple Linear-Phae Filter, N.Rao et al, ISCAS Proceeding, vol.ii, pp , June L.Weinberg, Network Analyi and Synthei, McGraw Hill, 96, pp.3. 4

25 A very general property of the Group Delay i : H Real., for = j H H' Real, for = j H H S Real, for = j where S H i the Senitivity of H with repect to. 5

Mixed Method of Model Reduction for Uncertain Systems

SERBIAN JOURNAL OF ELECTRICAL ENGINEERING Vol 4 No June Mixed Method of Model Reduction for Uncertain Sytem N Selvaganean Abtract: A mixed method for reducing a higher order uncertain ytem to a table reduced