COMPLEMENTARY IIR FILTER PAIRS WITH AN ADJUSTABLE CROSSOVER FREQUENCY

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1 COPLEENTARY IIR ILTER PAIRS WITH AN ADJUSTABLE CROSSOVER REQUENCY Ljljana lć Tao Saramäk Tamere Internatonal Center for Sgnal Proceng (TICSP) Inttute of Sgnal Proceng, Tamere Unverty of Technology P. O. Box 553, IN-33 Tamere, nl, E-mal: Author ermanent afflaton Unverty of Belgrade, hajlo Pun Inttute, P.O. Box 5, Belgrade, Yugolava, Emal: ABSTRACT Th aer ntroduce three clae of comlementary recurve low-a/hgh-a flter ar wth an adjutable croover frequency n uch a way that the tob attenuaton of both flter reman the ame. or each cla, the flter ar contructed ung two alla ub-flter a buldng block. Baed on the roerte elltc mnmal Q-factor tranfer functon, mle exreon are derved for evaluatng the coeffcent n all-a ecton n order to acheve the dered croover frequency. The degn rocedure are develoed for yntheng comlementary flter ar mlemented a a arallel connecton of two all-a ub-flter for two clae of flter ar contructed a taed cacaded nterconnecton of two dentcal all-a ubflter. The drect arallel connecton ha both the ower-comlementary all-a comlementary roerty. The frt (econd) cla of flter contructed ung everal dentcal coe of the two all-a flter oee the ower-comlementary roerty (both the magntude-comlementary all-a comlementary roerte).. INTRODUCTION Comlementary nfnte-mule reone (IIR) flter ar can be yntheed to generate ower-comlementary, all-a comlementary, or magntudecomlementary flter ar (ee, e.g., [). A very attractve alternatve to generate a ower-comlementary lowa/hgh-a IIR flter ar to ue lattce wave dgtal (LWD) flter (arallel connecton of two all-a flter) [ [4. Uually, the nterecton for a lowa/hgh-a flter ar occur n the mddle of the bae-b, that, the croover frequency located exactly at /4 n term of the normaled frequency. If the tob attenuaton of both flter the ame, then the low-a hgh-a flter are both half-b IIR flter (ee, e.g., [4, [5). Th aer ntroduce a technque to change, by mean of mle formulae, the locaton of the croover frequency of the half-b flter ar to an arbtrary locaton whle tll retanng the attenuaton roerte of the ntal half-b flter. or th uroe, elltc mnmal Q-factor (EQ) tranfer functon ntroduced n [6 [8 rovde drectly the dered oluton. Th due to the fact they are ecal low-a tranfer functon that can be regarded a tranfer functon beng obtanable from half-b IIR flter ung the well known low-a to low-a tranformaton. The attractve roerty of the EQ flter that the actual ynthe can be erformed ung mle equaton, thereby enablng one to develo very mle exlct formulae for generatng a flter ar wth the gven croover frequency. In addton to the drect arallel connecton of two all-a flter, two tructure contructed a taed cacaded nterconnecton of two dentcal all-a flter are condered baed on the ue of the ynthe cheme decrbed n [9 [. The frt (econd) tructure allow one to generate ower-comlementary (magntudecomlementary) flter ar wth an adjutable croover frequency. Comared to the drect mlementaton, the advantage of the frt tructure that electve flter ar can be contructed wth the ad of very low-order all-a flter at the exene of ung addtonal coeffcent for nterconnectng everal dentcal coe of the two all-a ecton. The econd tructure allow one to generate by ung very low-order all-a ecton magntude-comlementary flter ar wth an adjutable croover frequency.. USE O EQ ILTERS OR GENERATING POWER-COPLEENTARY ILTER PAIRS WITH AN ADJUSTABLE CROSSOVER REQUENCY Th ecton how how the roerte of EQ flter [6 [8 can be exloted n a very traghtforward manner for generatng ower-comlementary IIR flter ar wth an adjutable croover frequency uch that the tob attenuaton reman the ame. Due to the fact that EQ flter comre a half-b flter a a ecal cae, we tart wth a half-b flter ar to generate

2 comlementary flter ar whoe croover frequency can be arbtrarly choen. A G.. Proerte of EQ flter for Generatng Power-Comlementary lter Par Comlementary IIR flter ar contructed a a arallel connecton of two all-a ub flter hown n gure. or half-b IIR flter, the comlementary flter ar gven by (ee, e.g., [4, [5 ) G = A ± A ( N ) = 3,5, , ( N ) + = =,4,... + < +. ± () Here, N, the flter order, retrcted to be odd T G ( ) = G G a vector denotng the low-a (wth the lu gn) hgh-a (wth the mnu gn) half-b flter tranfer functon G () G (). The uercrt ued to emhae that the flter under conderaton are halfb flter. One of ole of the above tranfer functon located at the orgn the remanng ole are comlex-conjugate ar beng located on the magnary ax at = jr l for l =,, L, ( N ) /, gvng = ( r ). In Eq. (), the notaton < + ndcate how the ole are hared between the two all-a ecton. Th flter ar when generated by mean of an oddorder elltc flter charactered by the followng attractve roerte. rt, the um of the quaredmagntude reone of the two flter dentcally equal to unty. Hence, the flter ar ha the owercomlementary roerty. Second, the 3-dB croover frequency (the frequency where the quared-magntude reone of both G () G () acheve the value of /) located at f = / 4 n term of the normaled frequency. Thrd, the ab tob edge, denoted by f f n term of the normaled frequency, atfy = / f, that, the dtance of f the edge from the center of the baeb are the ame. ourth, the attenuaton of the low-a hgh-a half-b flter the ame. Snce the um of the tranfer functon G ) () A ( G, the reultng flter ar alo an all-a comlementary flter ar, thereby called a a double comlementary flter ar [. The low-a/hgh-a EQ flter ar generated accordng to the half-b flter ar, a gven by Eq (), a follow: l A G gure. Double-comlementary IIR flter ar contructed a a arallel connecton of two all-a flter. G = [ A ± A = ( N ) + α( + ) + ± =,4,... + α( + ) + ( N ) α + + α( + ) + + α + α + + = 3,5,... ( ) T, < +. where G( ) = G G agan a vector denotng the reultng low-a hgh-a flter tranfer functon G () G () that can be mlemented a hown n gure. The mle formulae for convertng the ure delay the econd-order all-a ecton n Eq. () to the frt-order the econd-order ecton n Eq. () wll be gven later on n th ecton. It hould be onted out that when ung LWD flter for the mlementaton, the adator coeffcent γ = α for the frt-order all-a ecton. or the econd-order all-a ecton, the correondng adator coeffcent are gven by γ = γ = α (ee, e.g., [4). There ext the followng three very attractve roerte of EQ flter guaranteeng that for the reultng EQ flter ar, a hown by Eq (), the croover frequency can be changed whle keeng the attenuaton of the low-a hgh-a flter the ame a for the tart-u half-b flter ar. urthermore, the double comlementary roerty reerved. rt, thee flter are generated ung half-b IIR flter a tart-u oluton [6 [8. Second, ntead of drectly alyng the well known low-a to low-a tranformaton or low-a to hgh-a tranformaton, the flter wth the gven a-b to-b edge are generated ung mle formulae. Thrd, f a low-a or a hgha flter generated frt, then t guaranteed that the flter n the reultng low-a/hgh-a flter ar have exactly the ame attenuaton n the to-b regon. Th due to the fact the tart-u flter an IIR halfb-flter... Degn Procedure () Baed on the roerte of EQ flter, a owercomlementary flter ar wth the gven odd order N, 3- db croover frequency, mnmum to-b A n decbel can be carred out n the fol- attenuaton lowng te: f

3 ) Synthee a ower-comlementary half-b IIR flter ar, a gven by Eq. (), uch that the mnmum attenuaton for both the low-a hgh-a flter tranfer functon G () G () exactly A. Let the correondng normaled cutoff frequence be f f = / f. ) Determne the o-called electvty factor ξ = ( πf ) tan( πf ) tan. 3) Synthee the dered ower-comlementary IIR flter ar, a gven by Eq. (), by determnng t arameter accordng to the formulae gven n Table. Table : Parameter for a ower-comlementary flter ar wth the gven cutoff frequency rt-order Secton Secondorder Secton tan α = + tan α = + f ( πf ) ( πf ) ( tan( πf )) ( tan( πf )) =.3. Proerte of the Reultng lter Par + α α + The above rocedure reult n the owercomlementary flter ar wth the followng roerte: ) The croover frequency beng related to the common contant α n Eq. () or n Table a follow: ( α ) π f co = (3) located exactly at the gven frequency ont. ) The normaled cutoff frequence are gven by f f ( = tan tan( πf ) ξ ) (4a) π ( ξ tan( πf ). = tan ) (4b) π 3) The mnmum to-b attenuaton for both G () on the nterval [ f, G () on the nterval, f exactly. 4) The arameter α the common contant for all the econd-order all-a ecton n Eq. ()..3. Numercal Examle Conder ower-comlementary flter ar of order N = 7 havng A = 6 db. The order of the all-a ecton are thu 4 3. gure (a) how a famly of the reultng low-a flter that are derved from the halfb flter (thck lne). or each new flter only 5 contant value have to be comuted ung the exreon of Table. gure (b) how the reone of the overall bank for ome value of the croover frequency. A Attenuaton Attenuaton Normaled frequency (a) Normaled frequency (b) gure. amly of eventh-order ower-comlementary flter ar for A = 6 db. (a) Low-a flter. (b) Overall flter ar. 3. GENERATING POWER-COPLEENTARY IIR ILTER PAIRS USING A TAPPED CASCADED INTERCONNECTION O TWO IDENTICAL ALL-PASS ILTERS Th ecton how how the ynthe technque decrbed n [9, [ can be ued for generatng owercomlementary IIR flter ar wth an adjutable croover frequency. In th cae, the overall tructure generated a a taed cacaded nterconnecton of two dentcal all-a flter agan the roerte of EQ flter are exloted. 3. Prooed lter Par Effcent Imlementaton In th cae, the low-a hgh-a tranfer functon are gven by H m m = a[ m [ A [ A (5a)

4 H m m m = ( ) a[ m [ A [ A, (5b) where an odd nteger A () A () are the alla flter n Eq. () or (), that, G = [ A + A a low-a flter of an odd order N. The above ower-comlementary flter ar can be mlemented ung the lattce tructure hown n g. 3. The detal on how to convert the a[ m to the k can be found, e.g., n [. 3. Degn Procedure Gven the mnmum tob attenuaton, the frt te n the overall ynthe to fnd the + coeffcent a m the Nth-order low-a half-b flter [ A A ( ) G ( ) = +, a gven by Eq. (), uch that the followng condton are met: Condton : The mnmum attenuaton for both the lowa hgh-a flter tranfer functon H () H () exactly A. Condton : The normaled cutoff frequence are related va f = / f f f maxmed. a[ m A f The dered coeffcent can be found a follow: A ) Determne =. ) Otme the a m to maxme f uch that the amltude reone of wthn the lmt m a[ m = tay n the normaled a-b regon [, f the maxmum amltude value [ f, Note that. n the normaled to-b regon the tranfer functon of a mnmumhae fnte-mule reone (IR) flter. The above rocedure can be accomlhed by lghtly modfyng the degn cheme decrbed n [9. What left to degn the ower-comlementary half-b flter ar, a gven by Eq. (), uch that the above-mentoned two condton are met, then, to tranfer the croover frequency f to the dered frequency accordng to the dcuon of Subecton.. Th can be accomlhed by ung the degn rocedure decrbed n Subecton.. The man dfference that now, due to the ue of everal coe of the two dentcal all-a ecton, the mnmum to-b attenuaton, In th cae, the all-a ecton n g. 3 are A A. A A [ A () k k3 k [ A ( ) k 3 [ A () [ A ( ) H k k H gure 3. Lattce tructure for the rooed owercomlementary flter ar. Attenuaton Attenuaton Normaled frequency (a) Normaled frequency (b) gure 4. A famly of ower-comlementary flter ar contructed ung 5 dentcal coe of econdorder frt-order all-a ecton for A = 6 db. (a) Low-a flter. (b) Overall flter ar. ntead of the orgnal detal): A A 3.3 Numercal Examle, gven a follow (ee [9 for ( ( f = log π )). (6) Conder ower-comlementary flter ar that are contructed ung = 5 dentcal coe of two all-a flter of order, that, N = 3. The dered attenuaton A = 6 db. gure 4(a) llutrate a famly of the reultng low-a flter. The thck lne ued for the ntal half-b flter. or each new flter only 3 contant value have to be comuted. The value of the ad-

5 dtonal ta coeffcent tay unchanged. The requred attenuaton of the arallel connecton of the frt-order econd-order all-a flter only 9.37 db. The reone of the overall bank for ome value of the croover frequency are dected n gure 4(b). 4. GENERATING APLITUDE-COPLE- ENTARY IIR ILTER PAIRS USING A TAPPED CASCADED INTERCONNECTION O TWO IDENTICAL ALL-PASS ILTERS Th ecton how how the ynthe technque decrbed n [9 can be ued for generatng amltudecomlementary IIR flter ar wth an adjutable croover frequency. A n the revou ecton, the overall tructure generated a a taed cacaded nterconnecton of two dentcal all-a flter the roerte of EQ flter are exloted. 4.. Prooed lter Par Effcent Imlementaton [ [ A ( ) [ A ( ) A A A A [ A ( ) a [ a[ a[ a A [ A ( ) H H gure 5. Structure for the rooed magntudecomlementary (all-a comlementary) flter ar. 8 A In th cae, the low-a hgh-a tranfer functon leadng to the effcent mlementaton are gven by Attenuaton 6 4 H H m a[ m [ A [ A = m m b[ m [ A [ A, = m (7a) (7b) Normaled frequency (a) where an nteger that two tme an odd nteger, a [ m = a[ m, b [ m = b[ m, b[ m = a[ m for m =,, L, ( ) /. urthermore, a [ = b[ = a [ m = b[ m = for the remanng odd value of m. An effcent mlementaton for the above magntude-comlementary flter ar hown n g. 5. The um of the above two tranfer functon [ A [ A o that they form alo an all-a comlementary flter ar. 4.. Degn Procedure In th cae, the dered coeffcent a follow: A ) Determne =. a[ m can be found ) Otme the a m to maxme f uch that the ero-hae frequency reone of = a [ m m tay wthn the lmt ( ) n the normaled a-b regon [, f wthn the lmt n the normaled tob regon [ f,. Attenuaton Normaled frequency (b) gure 6. A famly of magntude-comlementary (all-a comlementary) flter ar contructed ung 6 dentcal coe of econd-order frt-order all-a ecton. (a) A =6dB. (b) A =db. ( ) Note that the tranfer functon of a lnear-hae IR half-b flter. The above rocedure can be accomlhed by lghtly modfyng the degn cheme decrbed n [. Otherwe, the overall ynthe the ame a n the revou ecton.

6 4.3. Numercal Examle Conder magntude-comlementary flter ar that are contructed ung = 6 dentcal coe of two all-a flter of order, that, N = 3. gure 6(a) how two bank wth the tob attenuaton of 6 db llutratng the flter charactertc for ome value of the croover frequency. gure 6(b) dect two flter bank for the tob attenuaton of db. The thck lne are ued for the ntal half-b flter. The requred attenuaton for the arallel connecton of the frt-order econd-order all-a flter are only 4.53 db 4.39 db for the flter of gure 6(a) 6(b), reectvely. 5. CONCLUSION Three clae of comlementary IIR low-a/hgh-a flter ar wth an adjutable croover frequency have been ntroduced. or the frt cla, the tart-u comlementary flter are low-a/hgh-a half-b flter ar contructed a arallel connecton of two all-a flter ecton (lattce wave dgtal flter). It ha been hown how thee flter ar can be ealy redegned to have an arbtrary croover frequency n uch a manner that the tob attenuaton for both the low-a hgh-a flter the ame ndeendent of the croover frequency. Th goal ha been acheved by exlotng the attractve roerte of elltc mnmal Q- factor tranfer functon. The ue of the charactertc of thee tranfer functon ha leaded to mle formulae for drectly changng the coeffcent value of the two all-a ecton to acheve the dered croover frequency. Th allow one to mlement rogrammable comlementary flter bank wth an adjutable croover frequency n a very mle traghtforward manner. The flter ar belongng to the frt cla both ower comlementary all-a comlementary. The comlementary flter ar belongng to the remanng two clae have been contructed a taed cacaded nterconnecton of two dentcal all-a ubflter. The man beneft of thee flter ar le n the fact that the all-a ub-flter are of very low order, thereby makng the adjutment of the croover frequency gnfcantly eaer. The rce to be ad for th fact that there a need for addtonal contant coeffcent for nterconnectng the all-a ub-flter. lter ar belongng to the econd (thrd) cla are ower comlementary (both magntude comlementary alla comlementary). ACKNOWLEDGEENTS Th work wa uorted by the academy of nl, roject No (nnh Centre of Excellence Program ( 5). Author greatly arecate the nrng dcuon wth Prof. arkku Renfor, Tamere Unverty of Technology. REERENCES [ S. K. tra, Dgtal Sgnal Proceng: A Comuter- Baed Aroach. cgraw-hll,. [ R. Nouta, The Jaumann tructure n wave-dgtal flter, Int. J. Crcut Theory Alcat., vol., , June 974. [3 A. ettwe H. Levn A. Sedlmeyer, Wave dgtal lattce flter, Int. J. Crcut Theory Alcat., vol.,. 3, June 974. [4 L. Ga, Exlct formula for lattce wave dgtal flter, IEEE Tran. Crcut Syt., vol. CAS-3,. 3, Jan [5. Renfor T. Saramäk, Recurve Nth-b dgtal flter Part I: Degn roerte, Part II: Degn of multtage decmator nterolator," IEEE Tran. Crcut Syt., vol. CAS-34,. 4 5, Jan [6. D. Lutovac, D. V. Tošć, B. L. Evan, lter Degn for Sgnal Proceng Ung ATLAB athematca. Uerde Rver, New Jerey: Prentce Hall,. [7. D. Lutovac Lj. D. lć, "Degn of comutatonally effcent elltc IIR flter wth a reduced number of hft--add oeraton n multler," IEEE Tran. Sgnal Proceng, vol. 45, no.,. 4 43, Oct.997. [8 Lj. D. lć. D. Lutovac, "Degn of multlerle elltc IIR flter wth a mall quantaton error," IEEE Tran. Sgnal Proceng, Vol.47, No., , eb [9 T. Saramäk. Renfor, A novel aroach for the degn IIR flter a a taed cacaded nterconnecton of dentcal alla ubflter, n Proc. 987 IEEE Int. Sym. Crcut Syt. (Phladelha, Pennylvana), ay 987, vol., [ H. Johanon T. Saramäk, A cla of comlementary IIR flter, n Proc. 999 IEEE Int. Sym. Crcut Syt. (Orlo, lorda), July 999, vol. 3, [ H. Johanon L. Wanhammar, Hgh-eed recurve flter tructure comoed of dentcal all-a ubflter for nterolaton, decmaton, Q bank wth erfect magntude recontructon, IEEE Tran. Crcut Sytem-II: Analog Dgtal Sgnal Proceng, vol. 46, no.,. 6 8, January 999.