Transfer Function Preserving Transformations on Equal-Ripple RC Polyphase Filters for Reducing Design Efforts

Transcription

1 IEICE TRANS. FUNDAMENTALS, VOL.E90 A, NO. FEBRUARY PAPER Special Section on Analog Cicuit Techniques and Related Topics Tansfe Function Peseving Tansfomations on Equal-Ripple RC Polyphase Filtes fo Reducing Design Effots Hioaki TANABE and Hioshi TANIMOTO a, Membes SUMMARY Element value spead of an equal-ipple RC polyphase filte depends heavily on the ode of eo assignment. To find the optimum design, we must conduct exhaustive design fo all the possible eo assignments. This pape descibes two cicuit tansfomations on equal-ipple RC polyphase filtes, which peseve thei tansfe functions, fo educing cicuit design effots. Poposed Method I exchanges (R, C valuesto (/C, /R fo each stage. This gives a new cicuit with diffeent eo assignment fo each stage of its oiginal cicuit. Method II flips ove the oiginal cicuit and exchanges the esulting (R i, C i values fo (C n i+, R n i+ fo each i-th stage. Those cicuit tansfomations can educe a numbe of cicuit designs down to /4 of the staight-fowad method. This consideably educes a buden fo exhaustive design fo seaching the minimum element value spead condition. Some design examples ae given to illustate the poposed methods. key wods: RC polyphase filte, equal ipple, tansfe function peseving tansfomation, element value spead. Intoduction The RC polyphase filte is a kind of analog complex coefficient filtes, and is becoming vey popula fo its capability of image ejection in wieless communication systems []. The RC polyphase filtes ae often used in cascaded sections fo obtaining wide stopband and/o passband widths. In such cascaded RC polyphase filtes, to have simultaneously equal ipple in both stopband and passband may be a majo concen. Such equal ipple RC polyphase filte has been intoduced and analyed by Gingell in tansfe function level [], and its element value design methods have been epoted ecently [3], [4]. In a multi-stage RC polyphase filte, each stage has its own tansmission eo, and we can feely assign tansmission eos among the stages in any sequence. Taditionally, the eo sequence has been designed in such a way that time constants of the eos ae placed in descending o ascending ode []. Howeve, the authos have shown that despite changes in sequence of the tansmission eo assignment, those cicuits may have exactly the same tansfe function [4]. The authos also found that ode of cascading sections in an equal-ipple multi-stage RC polyphase filte may cause dastic changes in element value spead [4]. Manuscipt eceived July 3, 006. Manuscipt evised August 5, 006. Final manuscipt eceived Octobe 0, 006. The authos ae with the Dept. of Electical and Electonic Eng., Kitami Institute of Technology, Kitami-shi, Japan. Pesently, with the Toppan Technical Design Cente Co., LTD. a tanimoto@elec.kitami-it.ac.jp DOI: 0.093/ietfec/e90 a..333 Thus, it is highly desiable to check all the cascading odes of the RC polyphase filte and find out the optimum cicuit design with minimal element-value spead. Because an n-stage RC polyphase filte has n!diffeent cascading odes, howeve, it is vey difficult and cumbesome to design and check all the diffeent cicuits fo lage n. Also, the design technique intoduced in [4] involves a numeical method, which will not convege in some cases. It may be expected that less numbe of designs cause fewe nonconvegence issues to be encounteed. This pape pesents two cicuit tansfomations which peseve a tansfe function of an equal-ipple RC polyphase filte while keeping its element value spead unchanged. Those tansfomations allow systematic ways to educe the numbe of diffeent cicuits to be designed to about /4, by using symmety natues of equal ipple RC polyphase filte cicuits. Some design examples and consideations ae pesented fo cascading ode of each RC polyphase filte section.. Design Pocedue of Equal-Ripple RC Polyphase Filte This section gives a bief summay of a design pocedue poposed in efeence [4], fo late use in the pesent pape.. Tansfe Function and Time Constants fo Equal Ripple RC Polyphase Filte Gingell deived an analytic expession fo the voltage tansfe function of equal-ipple RC polyphase filtes and poved that its poles and eos must lay on the negative eal axis and imaginay axis, espectively, so that they can be expessed by time constants []. An equal-ipple RC polyphase filte can be chaacteied by thee paametes n, x, andε. Hee, n is a numbe of RC polyphase filte sections, x is a squae oot of a ecipocal bandwidth defined by x ω L /ω H (<, whee ω L and ω H ae angula fequencies of lowe edge and highe edge of passband and/o stopband, espectively. ε is a ipple paamete. The voltage tansfe function of the RC polyphase filte with simultaneously equal ipple in both stopband and passband is given by []: Copyight c 007 The Institute of Electonics, Infomation and Communication Enginees

2 334 H(s n s + j [( x dn K ( x n 4, ] x 4 s + [( x cs K ( x n 4, ], x 4 ( whee, K( is the complete elliptic integal of the fist kind, and dn( andcs( ae Jacobiban elliptic functions [5]. Equation ( can be factoied in the following fom: H(s ( jsτ ( jsτ ( jsτ n ( + sτ p ( + sτp ( + sτp n ( jsτ ( jsτ ( jsτ n, ( b n s n + b n s n + + b s + b 0 whee, time constants of eos and poles ae given by { [( τ x dn n { [( τ p x cs n. Cicuit Analysis by Using F Matix K ( x 4, x 4 ]}, (3 K ( x 4, x 4 ]}. (4 Cicuit topology of a single stage RC polyphase filte is shown in Fig.. The single section RC polyphase filte is descibed by [], [3] F jscr ( + scr R sc + scr. (5 If n sections ae connected in cascade, the total F matix is given by ( ( Vin Vout F I F F n (6 in I out ( A B n C D ( jsc i R i i ( Vout I out. (7 Whee, F i is an F matix of the i-th section, and R i, C i ae esisto and capacito values fo the section. Thus, a voltage tansfe function of the RC polyphase filte, H(s, is given IEICE TRANS. FUNDAMENTALS, VOL.E90 A, NO. FEBRUARY 007 by the following equation: H(s ( jsc R ( jsc R ( jsc n R n, (8 a n s n + a n s n + + a s + a 0 because, the matix enty A is an n-th ode polynomial of s, we can wite it down with eal coefficients a i (i,,...,n..3 Element Value Design by Coefficient Matching Method We can detemine the element values R i and C i, by equating Eq. ( and Eq. (8. The coefficient matching method [6] solves the poblem by equating the same ode coefficients of denominato polynomials and numeato polynomials, espectively, fom Eqs. ( and (8, i.e., solving the nonlinea simultaneous equations fo i,,...,n: { ai (R, C,...,R n, C n b i (τ p,τp,...,τp n R i C i τ. (9 Hee, the index (,,...,n inτ may not be equal to the index i in the second equation above. It is impotant to note that the ode of sections, the index i, is not necessaily aanged in ascending o descending ode of eos, the index, athe we may feely assign a paticula eo to any section. This leads to n! diffeent assignments at most; i.e., we have n! diffeent element value sets which ealie the same tansfe function. As mentioned ealie, we can take advantage of selecting a set of element values with the minimum element value spead among them. We abitaily set nominal impedance of R Ω to obtain nomalied solution, because always the elation a 0 b 0 a n b n holds. Othe impedance solutions may simply be obtained though impedance scaling as usual. 3. Symmetic Zeo-Sequences Due to the peiodic natue of Jacobian elliptic functions, the time constants given by Eqs. (3, (4 fo eos and poles exhibit symmety popety in thei values. If we assign the subscipt fo τ and τ p in descending ode of thei magnitude, we can show the following popeties among eos and poles, hee supe scipt stands fo eos and p stands fo poles: Fig. Basic single stage RC polyphase filte cicuit. τ τ n +, (0 τ p τ p n +, ( whee,,...,n. Fo thei poof, see efeence [3] fo Eq. (0, and efeence [7] fo Eq. (. Next, we define a symmetic eo-sequence. We assume that indices of eo,, ae aanged in descending ode. We say a eo sequence is symmetic if the sequence is aanged in such a way that τ is placed in k-th section fom left end of the RC polyphase filte and its ecipocal /τ τ n + is placed in k-th section fom ight end, i.e., (n k + -th section fom the left end.

3 TANABE and TANIMOTO: TRANSFER FUNCTION PRESERVING CIRCUIT TRANSFORMATIONS 335 Table Examples of symmetic eo-sequences. All symmetic sequences fo n All symmetic sequences fo n Fo example, eo sequences --3-4, -3--4, and ae symmetic sequences, while --4-3, -3-4-, and ae not. If the numbe of total sections, n, isan odd numbe, the cente section must always be assigned to τ (n+/. Table shows all the possible symmetic eo sequences fo n 4andn 5 cases. 4. Reduction in Numbe of Zeo Sequences We intoduce two cicuit tansfomations on the equal-ipple RC polyphase filte, which peseves its tansfe function. By using those tansfomations, a numbe of diffeent cicuits to be designed and checked educes to about /4. In this section, the index i is exclusively used fo denote i-th section fom the input side of an RC polyphase filte; i.e., C i and R i belong to the i-th section. On the othe hand, the index is used fo indexing the eo time constants, τ, which ae aanged in descending ode of thei values. This implies a one-to-one mapping fom i to can be defined, depending on an actual assignment of eo time constants to a paticula ode of sections. 4. Method I (Element Value Exchange This method exchanges esisto values fo capacito values of an equal-ipple RC polyphase filte. Tansfe function of an n-section RC polyphase filte is given by the following expession: n ( H(s + sci R i R i jsc i i R i sc i + sc i R i ( whee, [X] denotes the (, element of the matix X. Eliminating C i by using the elation τ R i C i and extacting the common facto, the time constant, fom the matix, we have n ( H(s + sτ R i jsτ i sτ /R i + sτ (3 n τ ( /τ + s R i /τ jsτ i s/r i /τ + s (4 n ( /τ + s /C i jsτ i s/r i /τ + s. (5 In the above modification fom Eq. (4 to (5, the identity fo equal-ipple design, n i τ i [3], has been used. Thus, the following exchanges in element values Table Fig. Illustation of Method I. Time constants of eos fo n 5 ω H /ω L 00 design. τ τ τ 3 τ 4 τ R i C i, C i R i (6 on Eq. ( peseve the time constant τ and do not change the tansfe function. Figue illustates those exchanges. We have assumed that τ (,,...,n ae aanged in descending ode in the above calculation; howeve, this is not indeed a estiction, because we assumed only the symmety elation of Eq. (0. Hence, the index of τ may un though fom to n in any ode. It should be noted that the esulting element-value set will coespond to a new eo sequence of which eos ae eplaced by thei ecipocals, so that we need to design only a halfof n!diffeent cicuits and the est can be obtained by using the Method I. Also, the element value spead is peseved in the pocess of Method I, because an element value spead is equal to an element value spead fo its ecipocals. Fo illustative example, we designed two elementvalue sets fo eo sequences and by the coefficient matching method. Table shows the time constants of eos used in this example. The esult is shown in Table 3. We can obtain an element value set fo , fom that of the sequence by the poposed Method I. Table 4 shows how the Method I can be applied to an actual case. The second column of Table 4, which is labeled Value, is an exact copy of the element-value set fo the eo sequence given in the Table 3. The thid column is a list of inveted and swapped element values fo the second column, and the last column is an impedance scaled vesion of the thid column. The impedance scaling is done in such a way that R value becomes Ω as in the oiginal design. We see that the esulting element-value set exactly matches with the diectly designed element-value set fo the

4 336 IEICE TRANS. FUNDAMENTALS, VOL.E90 A, NO. FEBRUARY 007 Table 3 Nomalied element values designed by coefficient matching method (n 5 ω H /ω L 00 case. Zeo sequence Element R.0000E E E E+00 C E E E 0.849E 0 R E 0.649E E E+00 C.6760E E E E 0 R E E E E+0 C E E E E 0 R 4.064E E E E+0 C 4.34E 0.837E E 03.34E 0 R E+0.768E E+0.768E+0 C E E E E 03 Table 4 Element values obtained by Method I (inveted and exchanged. Element Value [Ω],[F] Inveted and exchanged Scaled τ R.0000E E E+00 C E E E 0 τ R E E E+00 3 C.6760E E E 0 τ R E 0.56E E+0 4 C E 0.579E E 0 τ R 4.064E E E+0 5 C 4.34E E 0.34E 0 τ R E+0.034E+0.768E+0 C E 0.536E E 03 eo sequence , which is given in Table Method II (Cicuit Flipping Method II consists of two stages: Tansposition of the oiginal cicuit followed by exchanges of esisto values fo capacito values. As (F F n T F T n F T holds fo ectangula matices F i (i,,...,n, cascaded matices of F i satisfy a elation fo its (, element: [F F F n ] [F T n F T n FT ] (7 Hee, supescipt T denotes a matix tansposition. This suggests how to deive a flipped ove RC polyphase filte which has the same voltage tansfe function of an oiginal RC polyphase filte with a given eo-sequence. Now, we stat fom Eq. ( with its matix tansposed and use the elation Eq. (7. The total voltage tansfe function H(s can be descibed by the following expessions. n ( H(s + sτ T R i jsτ i sc i + sτ (8 ( + sτ sc i jsτ in R i + sτ (9 Note that the index i uns invesely, fom n to, in Eq. (9. Because the diagonal enties of the i-th section have a time constant τ,theoff-diagonal enties in each section of Eqs. (8 and (9 must satisfy a elation C i R i τ. Next, we apply an impedance scaling on Eq. (9 by a Fig. 3 Illustative explanation of Method II. scalingfacto of /(s. It should be noted that while diagonal elements of a chain matix have no physical dimensions, off-diagonal elements have physical dimensions; i.e., (, element has an impedance dimension and (, element has an admittance dimension. Then, only the off-diagonal elements ae affected by the impedance scaling. We have an impedance scaled vesion of Eq. (9: ( H(s + sτ C i jsτ sr i + sτ in. (0 Equation (0 and Eq. ( have vey simila expessions, and have exactly the same tansfe function. Compaing those expessions, we obseve the following facts:. The index i of poducts invesely uns in Eq. (0. This means that a cascading ode of sections in Eq. (0 is a evesed ode that of the oiginal Eq. ( case; that is, Eq. ( epesents the cascading sections in an ascending ode of i, o, in a coesponding ode of, while Eq. (0 epesents the cascading sections in a descending ode of i, o, in a coesponding ode of (n +.. In Eq. (0, an element value exchange R i C i takes place fo each section; howeve, the eo location of each section emains unchanged because the symmety popety R i C i τ must hold in an equal-ipple RC polyphase filte. 3. In the tansfomation pocess of the Method II, element value spead emains unchanged because R i and C i values do not change. In conclusion, a tansfomed RC polyphase filte by the Method II has exactly the same tansfe function as its oiginal RC polyphase filte. Figue 3 shows an illustation of the pocedue fo the Method II. Table 5 shows an example of the Method II. In this example, the oiginal eo sequence is and the esulting eo sequence is The esulting element

5 TANABE and TANIMOTO: TRANSFER FUNCTION PRESERVING CIRCUIT TRANSFORMATIONS 337 Table 5 Example of Method II. Element values afte tansposed and exchanged. τ Oiginal values Tansposed Exchanged Scaled τ R E E E+00 C E E E E+00 τ R E 0.064E+00.34E-0.649E+00 3 C.6760E+00.34E 0.064E E-0 τ R E E E E+00 4 C E E E E 0 τ R 4.064E E E E+0 5 C 4.34E E E 0.837E 0 τ R E E E+0 C E E E 0 values ae e-nomalied in such a way that R becomes Ω. The values of the last column exactly match with the values obtained via coefficient matching method (see Table Discussion 5. Numbe of Possible Cicuit Design Reduction We have intoduced two eduction methods in numbe of designs fo possible eo sequence combinations. As is known fom the above discussion, the numbe of eduction depends on each eo sequence s symmety. Figue 4 illustates how the Methods I and II tanslate oiginal eo sequences. If an oiginal eo sequence is symmetic, Methods I and II give the same cicuits so that the eduction ate will be /, because two consecutive tansfomations give the oiginal sequence as shown in Fig. 4(a. On the othe hand, a non-symmetic eo sequence will give diffeent cicuits fo Methods I and II, and the ode of two consecutive tansfomations esult in diffeent eo sequences. Fou non-symmetic eo-sequences can be tansfomed each othe by applying Methods I and II. Hence, the eduction ate will be /4 as shown in Fig. 4(b in this case. 5. Numbe of Stages vs. Possible Cicuit Design Reduction A numbe of symmetic eo-sequences depends on a numbe of stages in an RC polyphase filte, n, and whethe n is an even numbe o an odd numbe. If n is an even numbe, it suffices to specify fist n/ sequences due to the symmety constaint of Eq. (0. The latte half sequence can be detemined automatically by Eq. (0. Thee is two possibilities to assign a paticula eo fo the fist half o last half. Thus, all the possible combinations ae n/. In addition, the n/ fist half eos have a total pemutations of (n/!. Then, the total numbe of possible symmetic eo-sequences becomes n/ (n/! fo an even n. The numbe of the est of possible sequences, n! n/ (n/!, must be non-symmetic eo-sequences. Fo an odd n, the symmetic eo-sequence constaint foces the τ (n+/( to be fixed at the cente of the sequence. Then, the numbe of possible symmetic eosequences is given by (n / {(n /}! fo an odd n. Fig. 4 Possible numbe of eductions fo symmetic sequences (a, and non-symmetic sequences (b. Table 6 Minimum equied numbe of diffeent designs fo n 7. n Total Symmetic Non-symmetic Requied Reduction (a (b (c (d (d (a! ! ! ! ! ! Also, the est of possible sequences must be non-symmetic sequences. In summay, we have the following fomulae fo total equied numbe of eo-sequences. Fo even n: { ( n n! + { ( n! n n!, ( 4 fo odd n: { n ( n! + { n! n 4 ( n!. ( Table 6 summaies the elation of numbe of stages, n, total numbe of eo sequences (a, numbe of symmetic eo sequences (b, numbe of non-symmetic eo sequences (c, minimum numbe of equied eo sequences (d, and eduction ate fo n to 7. The minimum equied numbe of cicuits to be designed is calculated by (d (b + (c 4, and the eduction ate is given by (d (a. Fo example, the minimum numbe of eo sequences to be designed is 3 fo 5 stage case. The est of 88 cicuits can be obtained fom the 3 designs by using Methods I and II. Hence, the eduction ate is 0.7 in this case. Fo lage n, the eduction ate appoaches to / Meits of Poposed Methods As mentioned in Sect. 4, the poposed Methods I and II peseve an element value spead of its oiginal cicuit. This

6 338 IEICE TRANS. FUNDAMENTALS, VOL.E90 A, NO. FEBRUARY 007 implies that the poposed methods can be utilied fo classifying all the eo sequences which have the same element value spead befoe we go to actual design. This is a big economy and is an advantage of the method, because we only have to design a epesentative sequence fo each class, and the numbe of diffeent classes ae educed to about /4 of the total numbe of sequences. This eduction of design effot may also be beneficial fo actual element value design pocess. The element design pocess is assumed to use the element matching method which numeically solves a nonlinea simultaneous algebaic equations. Unfotunately, existing numeical methods sometimes fail to convege. The poposed methods may educe the chance of non-convegence issue by educing the numbe of to be solved poblems, and if a epesentative sequence fail to convege, we may ty othe sequences of the same class which must yield the same solution. 6. Conclusion Two cicuit tansfomations have been poposed, which systematically change the ode of eo assignment fom the existing designs while peseving voltage tansfe function and element value spead fo equal-ipple RC polyphase filtes. By using the poposed tansfomations, a equied numbe fo cicuit designs can be educed to /4 of the pevious capet bombing method. Fo example, a five stage RC polyphase filte having 5! 0 diffeent eo placements can be educed to only 3 diffeent eo placements. The est of 88 cicuits can be obtained without actual design pocedue, but can be obtained fom simple calculations on 3 cicuits. These cicuit tansfomations consideably educe design effots which ae equied fo seaching the minimum element value spead design. Acknowledgment The authos ae gateful to K. Mashiko, M. Katakua, H. Sato, M. Miyamoto, and S. Nakanishi of the Semiconducto Technology Academic Reseach Cente fo thei suppot and useful discussions. They also thank D. K. Wada of Toyohashi Univ. of Technology and D. C. Muto of Kyushu Inst. of Technology, fo thei stimulating discussions. The authos would like to expess thei gatitude to unknown eviewes fo thei constuctive comments. They ae vey helpful fo impoving the oiginal manuscipt. [4] H. Tanabe and H. Tanimoto, Design consideations fo RC polyphase filtes with simultaneously equal ipple both in stopband and passband, IEICE Tans. Fundamentals, vol.e89-a, no., pp , Feb [5] M. Abamowit and I.A. Stegun, Handbook of Mathematical Functions, Dove Publications, New Yok, 97. [6] G.C. Temes and J.W. LaPata, Intoduction to cicuit synthesis and design, Chapt., McGaw-Hill, 977. [7] H. Tanabe and H. Tanimoto, Design and element value spead considetations fo RC polyphase filtes with simultaneously equal ipple both in stopband and passband, IEICE Technical Repot, CAS005-30, Sept Hioaki Tanabe eceived B.E. and M.E. degees fom Kitami Institute of Technology in 004 and 006, espectively. Duing his mastes couse, he studied equal ipple design fo RC polyphase filte. Fom 006, he has been with the Toppan Technical Design Cente Co., LTD., Sappoo, Japan, whee, he is engaged in analog cicuit design. Hioshi Tanimoto eceived the B.E., M.E., and Ph.D. degees in Electonic Eng. fom Hokkaido Univesity, in 975, 977, and 980, espectively. In 980 he joined the Reseach & Development Cente, Toshiba Cop., Kawasaki, Japan, whee he was engaged in eseach and development of telecommunication LSIs. Since 000, he has been a Pofesso in the Dept. of Electical and Electonics Eng., Kitami Institute of Technology, Kitami, Japan. His main eseach inteests include analog integated cicuit design, analog signal pocessing, and cicuit simulation algoithms. He seved as an associate edito fo IEICE Tans. Fundamentals, IEEE TCAS- II, and is seving fo IEICE Tans. Electon. He is a vice chai of IEEE Cicuits and Systems Society Japan Chapte. D. Tanimoto is a membe of IEEJ and IEEE. Refeences [] F. Behbahani, Y. Kishigami, J. Leete, and A.A. Abidi, CMOS mixes and polyphase filtes fo lage image ejection, IEEE J. Solid-State Cicuits, vol.36, no.6, pp , June 00. [] M.J. Gingell, The Synthesis and Application of Polyphase Filtes with Sequence Asymmetic Popeties, Ph.D. Thesis in the Faculty of Engineeing, Univesity of London, 975. [3] K. Wada and Y. Tadokoo, RC polyphase filte with flat gain chaacteistic, Poc. ISCAS 003, pp.i-537 I-540, May 003.