5th Europan Signal Procing Confrnc (EUSIPCO 7), Poznan, Pol, Sptmbr 3-7, 7, copyright by EURASIP NEW INSIGHS IN ADAPIVE CASCADED FIR SRUCURE: APPLICAION O FULLY ADAPIVE INERPOLAED FIR SRUCURES Eduardo L. O. Batita, Orlo J. obia, Rui Sara LINSE Circuit Signal Procing Laboratory Dpartmnt of Elctrical Enginring Fdral Univrity of Santa Catarina 884-9 Florianópoli SC Brazil E-mail: {dudu, orlo, ara}@lin.ufc.br ABSRAC hi papr prnt th dvlopmnt of th lat-man-quar (LMS) normalizd LMS (NLMS) algorithm for adapting cacadd FIR filtr th application of uch algorithm to th whol adaptation of intrpolatd FIR filtr. h obtaind xprion ar gnral can b xtndd to any tructur compod of th cacad of two FIR filtr. h conidrd approach allow vrifying th main charactritic of th adaptiv proc a wll a th limitation of th xiting adaptiv intrpolatd FIR tructur uing adaptiv intrpolator. Numrical imulation rult ar prntd aiming to confirm th ffctivn of th obtaind advanc.. INRODUCION Intrpolatd finit impul rpon (IFIR) filtr ar computationally fficint tructur, bing an intrting altrnativ to implmnt FIR filtr. h firt work on thi ubjct i du to Nuvo t al. []. Sinc thn, much rarch ffort in IFIR filtr ha bn carrid out, aiming to u thm in a larg numbr of cofficint-dming application, uch a lin cho cancling []-[4], activ control [5], audio procing on a digital haring aid [6]. h ida bhind IFIR filtr i to u a par FIR filtr (with a rducd numbr of cofficint) cacadd with an intrpolator filtr that rcrat th rmovd cofficint in an approximat form. h adaptiv vrion of an IFIR () filtr alo rprnt an intrting altrnativ to implmnting cofficint-dming adaptiv FIR () filtr []. An filtr i carrid out by jut adapting th par filtr cofficint whra th intrpolator i maintaind fixd [7]. h poition of th filtr in th cacad can b xchangd, placing th intrpolator at th input [8] or at th output [7] of th tructur. h intrpolator poition impact on diffrnt updat proc bcau of th tim-varying natur of th tructur [7], [8]. Dpit that, th am tady-tat prformanc i obtaind for both ralization. h computational complxity rduction in an tructur i obtaind at th xpn of a highr tady-tat man-quar rror () valu a compard with that of th tard filtr. Such dgradation i du to th u of a fixd intrpolator filtr, which gnrally lad to an inadquat rcration of cofficint. A poibl olution to thi problm i to conidr a fully adaptd IFIR (F) tructur, in which both th par intrpolator filtr Orlo J. obia i alo with th Elctrical Enginring lcom. Dpt., Rgional Univrity of Blumnau (FURB), Blumnau, SC, Brazil. hi work wa upportd in part by th Brazilian National Rarch Council for Scintific chnological Dvlopmnt (CNPq). ar adaptd. hi cla of tructur wa originally introducd in [9] applid to an adaptiv lin nhancr. In [] [], a F tructur ha bn introducd dicud in an ad hoc mannr. Rgarding th mathmatical tratmnt, thi papr prnt th drivation of th lat-man-quar (LMS) normalizd LMS (NLMS) algorithm for adapting FIR filtr in cacad a wll a it application to F filtr. h prntd framwork i gnral, prmitting to b applid in any tructur formd by a cacad of two adaptiv FIR filtr. In particular, th providd xprion ar applicabl to F tructur rgardl of th intrpolator par filtr poition. Morovr, through th propod approach mor inight on th adaptation proc of uch filtr i gaind, prmitting to a th approximation ud in [9]-[]. hi papr i organizd a follow. In Sction, th gnral mathmatical dcription of th cacadd FIR filtr i introducd. Sction 3 prnt th dvlopmnt of th LMS algorithm for uch a tructur. Sction 4 conidr th ca of th NLMS algorithm. h dvlopd thory for th F ca i dicud in Sction 5. Sction 6 how om imulation rult aiming to vrify th prformanc of a F tructur. Finally, rmark concluion of thi work ar prntd in Sction 7.. MAHEMAICAL REAMEN OF CASCADED FIR FILERS Fig. illutrat th block diagram of a cacad of two FIR filtr, whr g [ g g L gm ] h [ h h L hn ] ar th input output FIR filtr, rpctivly. Variabl x( n ) dnot th input ignal, yn ˆ( ) rprnt an intrmdiary ignal, i th output ignal of th cacad, alo obtaind by combining th cacad of filtr into an quivalnt filtr, rulting in x g w g h. () q yn ˆ( ) xn ( ) w q h Figur Block diagram of a cacadd FIR tructur it quivalnt filtr. Not from () that th quivalnt tructur i obtaind by a convolution opration. o facilitat th mathmatical tratmnt, uch an opration i xprd a a product of matric [9]. For uch, lt u dfin th following matric: 7 EURASIP 37
5th Europan Signal Procing Confrnc (EUSIPCO 7), Poznan, Pol, Sptmbr 3-7, 7, copyright by EURASIP g L g g L g g g L M M M O M G gm gm gm 3 L g () gm gm L g gm L g M M M O M g L M with dimnion ( N + M ) N, h L h h L h h h L M M M O M H hn hn hn 3 L h (3) hn hn L h hn L h M M M O M h L N with dimnion ( N + M ) M. hn, () i now rwrittn a w q Gh Hg. (4) Du to th convolution opration in both () (4), th quivalnt filtr ha a mmory iz of N + M. Now, dfining a nw input vctor with th am dimnion of th quivalnt filtr (4) a [ ] x x x( n ) L x( n N M + ) (5) th input-output rlationhip of th cacadd tructur i writtn in trm of matrix-vctor product a w x h G x g H x. (6) q 3. LMS ALGORIHM IN HE CASCADED SRUCURE In thi ction, th adaptiv vrion of th cacad of two FIR filtr ( Fig. ) i drivd. For uch, w aum that th cofficint of both filtr ar adjutd according to th LMS algorithm. Fig. how th block diagram of th tructur in qution, in which dn ( ) dnot th ignal to b timatd (dird ignal) n ( ), th rror ignal givn by n ( ) dn ( ). (7) h rmaining ignal in thi figur ar th am a in Fig.. dn ( ) xn ( ) yn ˆ( ) g h Adaptiv algorithm + Σ n ( ) Figur Block diagram of th adaptiv cacadd tructur. By ubtituting (6) into (7) conidring that now th cofficint ar tim-varying for both FIR filtr, w gt n ( ) dn ( ) g H x (8) dn ( ) h G x. Now, dfining a cot function bad on th intantanou rror, on can writ ˆ J. (9) hn conidring th tup of Fig., th filtr cofficint ar updatd by uing th gradint of th cot function (9) a givn in []. For uch, th rquird xprion ar obtaind in th nxt ction. 3. Cofficint updat of th input filtr By conidring th input filtr g th updating proc i dcribd by th gradint rul. hu, g( n+ ) g μ g () with μ rprnting th tp-iz paramtr for adapting th input filtr. By applying th chain rul in (8), th gradint vctor i writtn a g. () g g hn, from (8), th right h id trm of () ar ( n) () n ( ) n ( ) H x (3) g whr H i th tim-varying vrion of (3). hrfor, th LMS-updat quation for th input filtr of th tup of Fig. i givn by g( n+ ) g + μ H x. (4) 3. Cofficint updat of th output filtr For th output filtr, th updat proc i again givn by h( n+ ) h μ h (5) whr μ i th tp-iz paramtr for adjuting th output filtr. hn from (8), now conidring vctor h(n), according to th chain rul w writ (6) h h rulting in () n ( ) G x (7) h whr G i th tim-varying vrion of (). Finally, th LMS-updat quation for th cofficint of th output filtr i givn by h( n+ ) h + μ G x. (8) 3.3 Particulariti of cacadd adaptiv filtr Som important qution ari in th updating proc of cacadd adaptiv filtr: i) Cofficint initialization. A mntiond in [9], if both cofficint vctor g h ar initializd with zro according to common practic, from (4) (8) on vrifi that uch cofficint vctor ar unchangd during th updating proc. On poibl olution i to initializ th cofficint vctor g h (or at lat on of thm) with nonzro valu. ii) Computational complxity. From (4) (8) on notic that an valuation of two matrix product H x G x i rquird at ach itration. Howvr, undr low adaptation condition conidring th particular tructur of G H matric, th matrix product can b implifid to an innr vctor product, rducing thu th computational burdn. Such procdur i dicud in th following ction. 7 EURASIP 37
5th Europan Signal Procing Confrnc (EUSIPCO 7), Poznan, Pol, Sptmbr 3-7, 7, copyright by EURASIP iii) Algorithm tability. Bcau of th imultanou adaptation of both filtr th u of two diffrnt tp-iz paramtr, to obtain analytical xprion for th tability bound i omwhat difficult. hn, an altrnativ olution i to u a conrvativ tp-iz valu for both filtr, obtaind xprimntally. With uch tratgy, good practical rult ar achivd a will b hown in Sction 6. A mathmatical analyi of th cacadd tructur i vry complx, thu th dtrmination of tability bound rmain an opn problm in th litratur of th ara. 4. CASCADED NLMS SRUCURE In thi ction, th xprion allowing to adapt th cofficint of th cacadd tructur by uing th NLMS algorithm ar drivd. Similarly a in [], th NLMS updating quation i obtaind by minimizing th Euclidan norm of δ g( n+ ) g( n+ ) g (9) ubjct to th contraint g ( n+ ) H x d. () Not that th contraint xprion () i lightly diffrnt from that prntd in [] du to th particular charactritic of th cacadd tructur. By applying th mthod of Lagrang multiplir [] to (9) (), th cot function i now givn by Jg δ g ( n+ ) +λ d g ( n+ ) H x () whr λ i a ral-valud Lagrang multiplir. By diffrntiating () with rpct to g ( n + ), w gt Jg [ g( n+ ) g ] λh x. () g( n + ) By tting () qual to zro, w obtain g( n+ ) g + λh x. (3) By ubtituting (3) into () rult in dn ( ) g + λ H x H x g H x + λ H x. Now, olving (4) for λ w gt λ dn ( ) g H x ( n) H x H x (4). (5) Finally, ubtituting (5) into (3), adding a poitiv contant α for controlling th adaptation proc a wll a a mall poitiv contant ψ, prvnting a diviion by zro [], th cofficint updat quation of th input filtr conidring an NLMS-lik adaptiv algorithm i givn by α g( n+ ) g + H x. (6) H x +ψ In a imilar way a (9) (), for th output filtr w hav to minimiz δ h( n+ ) h( n+ ) h (7) ubjct to th contraint h ( n+ ) G x d. (8) Again, by applying th mthod of Lagrang multiplir [], on obtain J h δ h ( n+ ) +λ d h ( n+ ) G x. (9) Not that (9) ha th am form a (); thu, by achiving a imilar dvlopmnt a bfor, w gt th following updat quation: α h( n+ ) h + G x (3) G x +ψ whr α ψ ar mall poitiv contant aiming to control th adaptation proc to prvnt a diviion by zro, rpctivly. Rgarding practical qution, am conidration a tho givn in Sction 3-3 ar alo applicabl hr. Howvr, it i intrting to highlight that, if th implifid form i ud, it lad to mallr valu for α α than whn uing th tard NLMS algorithm to obtain a atifactory prformanc. 5. FULLY ADAPIVE IFIR FILERS In thi ction, by following th procdur prviouly dcribd, th cofficint updat quation i drivd for a F tructur. Fig. 3 how th block diagram of an IFIR tructur, whr w charactriz a par FIR filtr i [ ii L im ], th intrpolator filtr. h input ignal x( n ) it intrpolatd vrion ar rlatd by M x i i x( n j) (3) j j th par filtr output i givn by yn ˆ( ) xn %( ) w (3) whr * dnot th convolution oprator. An IFIR filtr may alo b implmntd by xchanging th ordr of th block. h rult prntd hr ar alo valid for uch a ca, providd corrponding modification ar mad. x i w Figur 3 Block diagram of an IFIR filtr. h factor dtrmining th parity in w i trmd intrpolation factor, which i dnotd by L [7]. h par filtr i obtaind by tting to zro ( L ) ampl from ach L concutiv on from th N-dimnional modl filtr w [ w w L wn ]. hu, th corrponding N -dimnional par vctor i w [ w L wl L w L L w( N ) L L ] (33) with an input vctor givn by [ ( n ) ( n ) L ( n N + )]. (34) In (33), N dnot th numbr of nonzro cofficint, givn by N N + L. (35) with rprnting th truncation opration. Not that an IFIR filtr i a particular ca of th tructur prntd in Sction. hu, th quivalnt cofficint vctor for th tructur of Fig. 3 i givn by wi Iw Wi (36) whr th matric I W ar dfind in a imilar way a in () (3), rpctivly. h fully adaptiv updat quation of th IFIR filtr ar obtaind by uing th xprion from Sction 3 4. hu, by conidring th LMS algorithm, w gt i( n+ ) i + μ W x (37) 7 EURASIP 37
5th Europan Signal Procing Confrnc (EUSIPCO 7), Poznan, Pol, Sptmbr 3-7, 7, copyright by EURASIP w ( n+ ) w + μ I x. (38) Exprion (37) (38) ar valid for an IFIR tructur conidring th intrpolator in any poition, ithr at th front or nd. h implmntation of th rcurion (37) (38) rquir a conidrabl computational burdn bcau of th matrix product. Howvr, by conidring a low adaptation condition, uch filtr can b implmntd with l complxity by admitting om implification. hi procdur i carrid out taking into account that th product W x, givn by w L w x x W x [ w x w x ( n ) ( n M + )] can b rplacd by it approximatd vrion x w [ w x w ( n ) x( n ) L w ( n M + ) x( n M + )] whr x in (39) (4) i [ ] (39) (4) x x x( n ) L x( n N + ). (4) o obtain th lmnt from (39), th valuation of all M innr vctor product ar rquird at ach itration n. W can rduc th computational complxity of (39) by uing th approximation (4), rquiring now th computation of only th firt lmnt w x at ach itration. With uch a rduction th applicability of th cacadd tructur i ignificantly nhancd. h am conidration i alo ud for computing th product I x, rulting in vctor which i alrady availabl from th filtring opration. h implifid updat xprion of th F-LMS filtr ar i( n+ ) i + μ x w (4) w( n+ ) w + μ. (43) A bfor, th am car mut b takn to nur propr opration of th adaptiv cacadd tructur. h updat xprion uing th NLMS algorithm, according to th xprion obtaind in Sction 4, ar α i( n+ ) i + W x (44) W x +ψ α w( n+ ) w + I x. (45) I x +ψ hn, th corrponding implifid xprion ar givn by α i( n+ ) i + x w (46) x w +ψ α w( n+ ) w +. (47) +ψ h implifid xprion (4)-(43) (46)-(47) for th LMS NLMS algorithm, rpctivly, ar prntd in [9]-[]. In [9], uch xprion ar obtaind by auming that ach filtr i tim-invariant. hi i alo th ca in th drivation givn in [] [], whr th NLMS affin projction (AP) algorithm ar alo conidrd. Howvr, th impact of th adoptd approximation on th algorithm bhavior i not highlightd in th papr. h u of F filtr for rplacing th tard filtr i only intrting if th implifid tructur i adoptd. h computational burdn rquird by th tard filtr i much mallr than that rquird by F on without any implification. Howvr, it i vry important to notic that, if implifid xprion ar ud, pcial car with th ud tp-iz valu mut b takn, avoiding an impropr algorithm opration. h imulation rult prntd in th nxt ction provid mor inight on thi fact. 6. SIMULAION RESULS In thi ction, conidring a ytm idntification problm, om xampl ar prntd aiming to vrify th propod algorithm. h curv of th F tructur (without with implification) ar compard with tho of th on, for both th LMS NLMS algorithm. Hr, th curv of th tructur ar hown only for prformanc comparion purpo. Such a tructur do not conidr any tratgy for rducing th rquird computational burdn. For all imulation th input ignal i whit with unit varianc. W alo add 6 to dn ( ) a maurmnt noi with a varianc σ v (SNR 6dB). Simulation with colord input ignal ar not hown hr inc th obtaind rult ar imilar to tho by uing whit input ignal. Exampl : In thi xampl, th adaptiv tructur ar ud for o modling a plant givn by th lngth- vctor w [..7.5.37.6.9.4..7.5.4]. h paramtr ud for th F ar L N 6. Hr, th adaptiv algorithm i th LMS on th tp iz ud i μ μ /5 (for all filtr), whr μ i th imum tp iz for algorithm convrgnc (xprimntally dtrmind). h obtaind valu for μ ar.7 for th,.5 for th,.6 for th F,.3 for th implifid F tructur. In th ca of th F tructur, th am tp iz i lctd for both algorithm to nur tability. h curv ar obtaind from Mont Carlo imulation (avrag of indpndnt run), which ar hown in Fig. 4. From thi figur, w not that th F tructur prnt a bttr tady-tat prformanc than that obtaind with th on. Obrv that th convrgnc rat of th F tructur i largr than that of th implifid on whn attaining th am tady-tat valu. - - -3-4 -5-6 F Simplifid F -7 3 4 5 6 7 Itration Figur 4 Exampl. Plant with cofficint. Svral filtr tructur uing th LMS algorithm. curv (avrag of run). Exampl : For thi xampl, th plant ud i th lngth-5 vctor o w [..3.7..6.4.8.37.39.49.46.54.49.54.46.48.38.37.8.4.6..7.3.]. Again th adaptiv algorithm i th LMS on th choic of th 7 EURASIP 373
5th Europan Signal Procing Confrnc (EUSIPCO 7), Poznan, Pol, Sptmbr 3-7, 7, copyright by EURASIP tp iz i dtrmind a in Exampl. h valu obtaind for μ ar.3 for th filtr,. for th,. for th implifid F tructur. h curv ar hown in Fig. 5, in which again w vrify that th F prformanc i bttr than th on xhibitd by th tructur. By uing th am tp iz, th F prformanc i imilar to that of it implifid vrion, corroborating th commnt mad in Sction 3. - - -3-4 -5-6 F Simplifid F -7 3 4 5 6 7 Itration Figur 5 Exampl. Plant with 5 cofficint. Svral filtr tructur uing th LMS algorithm. curv (avrag of run). Exampl 3: For thi xampl, th plant i th am a in Exampl. Now, th algorithm ud i th NLMS on, with α.5 ψ ψ 6. h obtaind rult ar hown in Fig. 6. A xpctd from Exampl, th F tructur prnt a bttr bhavior than th on. Now, comparing th prformanc btwn th F it implifid vrion, th diffrnc btwn thm i rmarkabl. In th lattr, w not a highr tady-tat a wll a th prnc of om pik both du to th valu ud for th paramtr α α. - - -3-4 -5-6 F Simplifid F -7 5 5 Itration Figur 6 Exampl 3. Plant with cofficint. Svral filtr tructur uing th NLMS algorithm with α.5. curv (avrag of run). Exampl 4: In thi xampl, w hav ud th am data of Exampl 3, but now w hav rducd th valu of th paramtr to α.. h curv ar hown in Fig. 7. Now, w obrv almot th am prformanc for both th F tructur it implifid vrion. hi fact dmontrat how critical i th choic of th tp-iz control paramtr valu for th NLMS algorithm if a atifactory prformanc i rquird. - - -3-4 -5-6 Simplifid F F -7 5 5 Itration Figur 7 Exampl 4. Plant with cofficint. Svral filtr tructur uing th NLMS algorithm with α.. curv (avrag of run). 7. CONCLUDING REMARKS In thi work, gnral xprion for th LMS NLMS adaptation of cacadd FIR tructur ar prntd. h application of uch xprion to fully adaptiv intrpolatd FIR filtr i alo dicud, highlighting it advantag, limitation a wll a om implification aiming to rduc th rquird computational burdn. h prntd rult bring nw inight on adaptiv intrpolatd tructur, xping thu it rang of application. REFERENCES [] Y. Nuvo, C. Y. Dong, S. K. Mitra, Intrpolatd finit impul rpon digital filtr, IEEE ran. Acout., Spch, Signal Proc., vol. 3, no. 3, pp. 563-57, Jun. 984. [] A. Abouaada,. Aboulnar, W. Stnaart, An cho tail cancllr bad on adaptiv intrpolatd FIR filtring, IEEE ran. Circuit Syt. II, Analog Digit. Signal Proc., vol. 39, no. 7, pp. 49-46, Jul. 99. [3] O. J. obia, R. Sara Jr., R. Sara, Echo cancllr bad on adaptiv intrpolatd FIR filtr, in Proc. IEEE Int. lcom. Symp., Natal, Brazil, Spt., pp. -5. [4] S. S. Lin W. R. Wu, A low-complxity adaptiv cho cancllr for xdsl application, IEEE ran. Signal Proc., vol. 5, no. 5, pp. 46-465, May 4. [5] S. Kuo D. R. Morgan, Activ Noi Control Sytm, John Wily & Son, 996. [6] L. S. Niln J. Sparo, Digning aynchronou circuit for low powr: An IFIR filtr bank for a digital haring aid, Procding of th IEEE, vol. 87, no., pp. 68-8, Fb. 999. [7] O. J. obia R. Sara, Analytical modl for th firt cond momnt of an adaptiv intrpolatd FIR filtr uing th contraind filtrd-x LMS algorithm, IEE Procding Viion, Imag, Signal Proc., vol. 48, no. 5, pp. 337-347, Oct.. [8] R. Sara, J. C. M. Brmudz, E. Bck, A nw tchniqu for th implmntation of adaptiv IFIR filtr, in Proc. Int. Symp. Signal, Sytm, Elctronic (ISSSE), Pari, Franc, vol., Spt. 99, pp. 644-647. [9] M. D. Gron, Nw FIR tructur for fixd adaptiv digital filtr, Ph.D. Dirtation, Univrity of California, Santa Barbara, CA, Unitd Stat, 987. [] R. C. Bilcu, P. Kuomann, K. Egiazarian, On adaptiv intrpolatd FIR filtr, in Proc. IEEE Int. Conf. Acoutic, Spch, Signal Procing (ICASSP), Montral, Canada, vol., May 4, pp. 665-668. [] R. C. d Lamar R. Sampaio-Nto, Adaptiv rducd-rank M filtring with intrpolatd FIR filtr adaptiv intrpolator, IEEE Signal Proc. Lttr, vol., no. 3, pp. 77-8, Mar. 5. [] S. Haykin, Adaptiv Filtr hory, 4 d., Prntic-Hall,. 7 EURASIP 374