Round-Off Noise of Multiplicative FIR Filters Implemented on an FPGA Platform

Transcription

1 Appl. Sc. 4, 4, 99-7; do:.339/app499 Artcl OPEN ACCESS appld sccs ISSN Roud-Off Nos of Multplcatv FIR Fltrs Implmtd o a FPGA Platform Ja-Jacqus Vadbussch, *, Ptr L ad Joa Putma Dpartmt ESAT, KU Luv Kulab, Zdj Ostd 84, Blgum; E-Mal: joa.putma@uluv.b School of Egrg ad Dgtal Arts, Uvrsty of Kt, Catrbury Kt CT-7NT, UK; E-Mal: P.L@t.ac.u * Author to whom corrspodc should b addrssd; E-Mal: jajacqus.vadbussch@uluv.b; Tl.: ; Fax: Rcvd: 8 Novmbr 3; rvsd form: 7 Fbruary 4 / Accptd: 9 Fbruary 4 / Publshd: 5 March 4 Abstract: Th papr aalyzs th ffcts of roud-off os o Multplcatv Ft Impuls Rspos (MFIR) fltrs usd to approxmat th bhavor of pol fltrs. Gral xprssos to calculat th sgal to roud-off os rato of a cascad structur of Ft Impuls Rspos (FIR) fltrs ar obtad ad appld o th spcal cas of MFIR fltrs. Th aalyss s basd o fxd-pot mplmtatos, whch ar most commo dgtal sgal procssg algorthms mplmtd Fld-Programmabl Gat-Array (FPGA) tchology. Thr wll ow scalg mthods,.., L boud; fty boud ad absolut boud scalg ar cosdrd ad compard. Th papr shows that th ordrg of th MFIR stags, combato wth th scalg mthods, hav a mportat mpact o th roud-off os. A optmal ordrg of th stags for a chos scalg mthod ca mprov th roud-off os prformac by db. Kywords: MFIR; FIR-fltrs; lar phas fltrs; FPGA; fxd pot dgtal sgal procssg DSP; roud-off os; fltr cascad structur. Itroducto Multplcatv Ft Impuls Rspos (MFIR) fltrs ar a class of fltr structurs that wr orgally troducd by Fam th arly 98s []. It was show that MFIR fltrs ca b usd to

2 Appl. Sc. 4, 4 rplac rcursv Ift Impuls Rspos (IIR) fltrs wth FIR quvalts rqurg sgfcatly lss hardwar tha classcal FIR archtcturs that fulfll th sam spcfcatos []. Th rplacmt of a pol that s mplmtd a rcursv structur, by a o-rcursv FIR, has th advatag that t wll always b stabl. Ths s partcularly trstg wh th orgal pol s stuatd clos to th ut crcl. Th MFIR fltrs ar abl to ralz low-pass, hgh-pass, bad-pass, ad otch fltrs. Although th MFIR structurs rqur approxmatly th sam umbr of dlay lmts as th classcal FIR mplmtatos, thy rqur, logarthmcally, fwr addrs ad multplrs [,]. MFIR fltrs ar basd o th dtty [3]: P P = = ( x ) x = +. () Ths dtty ca b usd to approxmat both ral pol ad cojugat pol par fltrs. I cas H r (z) s th trasfr fucto of a stabl IIR fltr wth a ral pol, th cascad MFIR fltr approxmato usg Equato () ylds: Hr ( z) = = ( λz ) λ < λz = P P P H r ( z) ( λz ) ( λz = + ) = M( z) = M ( z). = = = wth th obvous dftos for M (z) ad M(z). I ths corrspodc, vry sgl M (z) wll b calld a stag of th MFIR fltr. It has b show [,] that v for th approxmato of pols xtrmly clos to th ut crcl, maxmum stags (P = ) ar rqurd wh a dvato smallr tha. db th magtud rspos btw th IIR fltr ad th MFIR approxmato s allowd. Th ffccy of th MFIR approxmato s mmdatly clar from Equato (). Oly P multplrs ad addrs ar rqurd for th MFIR fltr whl th drct form rqurs P multplrs ad addrs. A IIR fltr wth a trasfr fucto H c (z) havg a cojugat pol par λ = r +jθ ad λ * = r jθ wth r <, ca b approxmatd wth a cascad MFIR structur [,]: Hc ( z) = * λz λ z P * ( λz ) ( λ z ) P + + = = P + cos + = = + P ( rz ) ( θ ) ( rz ) M ( z) M ( z) = = j MFIR fltrs wth a lar phas ad a dsrd magtud rspos H( ω ) () (3) (Hr ω s th ormalzd agular frqucy: ω = ωt S, ad T S s th samplg prod.) ca also b ralzd usg th followg procdur [,4]. j. Dsg a IIR fltr that approxmats H ω.. Approxmat th pols of th IIR fltr wth th MFIR structur. 3. Cascad to vry zro th rsultg MFIR fltr ts rcprocal wth rspct to th ut crcl.

3 Appl. Sc. 4, 4 Cosqutly, th lar phas approxmato of a stabl IIR fltr H r (z) wth a ral pol at λ, ca b ' dsgd by dtrmg ( j ω ) H = H ad approxmatg th ral pol λ' of H r '(z) wth: r r = + ' + = λ ' z P ( λ z ) M z P + M ( z) = + ( λ ') + z + z. = ( λ ') (4) Th lar phas approxmato of a stabl IIR fltr H c (z) wth a complx-cojugat pol par λ = r +jθ ad λ * = r jθ ', ca b dsgd by dtrmg ( j ω ) λ' = r' +jθ' ad λ'* = r' jθ' of H c '(z) wth: M z H = H ad approxmatg th pols c ( ( ) ( θ )) ( θ ) (( r' ) ( r' ) ) cos( θ ') z z c... P r' r' cos ' z r' 4cos ' r' z =. (5) = + ( + ) + Dspt th advatag of th logarthmcally mor ffct us of multplrs ad addrs, MFIR fltrs hav ot b popular. Idd, th larg umbr of dlay lmts rqurd to approxmat th bhavor of a IIR fltr was cosdrd prohbtvly xpsv. Ths mad thm mpractcal for mplmtato o stadard DSP platforms wth fxd mmory maps. Advacs Vry Larg Scal Itgrato VLSI tchology gral, ad Fld-Programmabl Gat-Array (FPGA) archtcturs partcular, ma t cssary to r-valuat MFIR fltrs ad th tchcal barrrs to thr wdsprad us. Svral applcatos ad mplmtatos of MFIR fltrs modr FPGA fabrcs [4 6], hav show that FPGAs ar a dal targt platform for mplmtg ffct MFIR fltrs that ar compttv to stadard FIR ad IIR quvalts mplmtd o th sam fabrc. Th ffcts of coffct-quatzato hav b studd ad hav show that MFIR fltr structurs ar lss coffct-quatzato suscptbl tha th IIR fltr thy approxmat [7]. It s clar from Equatos ( 5) that MFIR fltrs ar bascally a cascad of smpl spars FIR fltrs. A xampl of a cascad of thr MFIR stags ralzg a MFIR fltr approxmatg a ral pol s gv Fgur. Fgur. Gral archtctur of a MFIR fltr approxmatg a ral pol (thr stags). λ λ 4 λ z z z 4 Evry stag a MFIR fltr has th sam structur as th othr stags but th umbr of dlay lmts dffrs. Thrfor, t s obvous to dsg a optmzd compot that mplmts a gral MFIR stag pr MFIR fltr approxmato,.., a ral pol approxmato, a complx-cojugat pol par approxmato ad thr rspctv lar phas typs.

4 Appl. Sc. 4, 4 As floatg-pot arthmtc s oly rctly avalabl FPGAs, oly fxd-pot arthmtc s ta to cosdrato. Ufortuatly, vry fxd-pot addto or multplcato rqurs a wdg of th bus wdth, whch has to b avodd; ths mpls that roudg must b appld to p th data path wdths maagabl ad mplmtabl. I ths txt, roudg s dfd as th procss whrby th wdth of th data path aftr a multplcato or a addto s rducd to th orgal wdth of th data path bfor th multplcato or addto. Ths s do by tag th most sgfcat bts, covtoal roudg of th rsult, ad usg saturato f cssary. Scalg s dfd as th procss that chags th fltr coffcts ordr to cras th basd o th fulfllmt of a spcfd crtro (as wll b dfd Scto 3) at th output of th fltr. I ordr to crat gral MFIR stag compots, th data path bt-wdth at th put ad at th output of ach stag ar pt costat. Isd th stag, th bt wdth s appropratly crmtd to avod accumulato of roud-off rrors. Cosqutly, practc at th d of ach stag, a roudg bloc wll brg th output bt wdth bac to th orgal put bt-wdth. It s, howvr, ot xcludd that a chag bus wdth btw th stags would yld bttr rsults. Howvr, thr ar so may possbl combatos that t s almost mpossbl to vstgat th bhavor of all ths possbl mplmtatos. I ths corrspodc, th Sgal to Nos Rato dgradato ffct of th coscutv roudgs ad scalgs th cascadd MFIR structurs s aalyzd ad coclusos o optmal ordrg of th stags ar draw. I Scto, th roud-off modl ad th dffrt trasfr fuctos ar dfd. Scto 3 dscusss th scalg mthods as dfd [8]. Scto 4 troducs a gral thory for roud-off os dtrmato fltr cascad structurs, mplyg th thory dvlopd ths scto s gral ad s applcabl o ay fltr havg a cascad structur. I th ffth scto, th dvlopd gral thory s appld o th MFIR structur. Th papr ds wth a cocluso ad suggstos for futur wor. Although t wll ot always b xplctly mtod, ths corrspodc, rfrs to th Sgal to roud-off Nos Rato,.., os du to roudg rrors s cosdrd ad othr os sourcs ar ot ta to accout.. Th Roud-Off Nos Modl.. Itroducto I ordr to avod ovrflow of th sgal data du to th succssv multplcatos of th sgal wth th fltr coffcts, roudg (as dfd abov) wll hav to b prformd. Roudg ca b s as a quatzato acto o th sgal. Each roudg acto s tratd as a radom procss wth uform probablty dsty fucto, producg wht os that s ucorrlatd wth th sgal ad othr quatzato sourcs th fltr. As ths corrspodc th mplmtato of th MFIR fltrs s vstgatd for fxd-pot mplmtato, th roudg procss s of vtal mportac. A. Fam dvlopd, [], th pur multplcty proprty whch forms th bass of th vstgato of th rlatoshp btw th os varac at th output (rlatv to th roud-off

5 Appl. Sc. 4, 4 3 os varac) ad th ordrg of th dffrt stags of th MFIR fltr. Ths s do for a mplmtato wthout scalg (whch s ot vry ralstc for fxd pot mplmtatos) ad a mplmtato wth L scalg. Th aalyss [] s prformd for MFIR structurs that approxmat a ral pol or a cojugat pol par usg th forward lattc structur. Th approxmato of a cojugat pol par cascad s ot aalyzd as t dos ot hav th pur multplcty proprty []. Howvr, th prst rsarch, th forward lattc MFIR mplmtato s ot cosdrd bcaus of ts (upractcal) larg hardwar mpact []. Cosqutly, th study of th os bhavor of th complx-cojugat pol par cascad MFIR mplmtato ad th lar phas MFIR mplmtatos ar compltly w. Morovr, t s suggstd [] that th aalyss of th roud-off os for th MFIR structurs that do ot hav th pur multplcty proprty should b do th styl of [8] or [9]. I ths txt, th mthod of [8] wll b usd for th ral pol approxmato as wll as for th complx-cojugat pol par cascad approxmato. All thr wll ow scalg mthods (L boud, fty boud ad absolut boud) [8] wll b cosdrd. Th os prformac s valuatd by frst calculatg th os varac at th output du to th os varac of th roud-off rror sourcs. Howvr, a good valuato of th roud-off os prformac ca oly b mad wh th actual sgal to roud-off os rato () s cosdrd []. Wth scalg, th output sgal s also scald, mplyg th actual sgal to os rato at th output ad ot oly th os varac s cosdrd for all possbl scalg cass. Th objctv of th papr s to dtrm how much th of a sgal s dtroratd by th os (du to roud-off rrors) th ovrall MFIR structur. Although th roud-off os prformac of FIR ad IIR cascad structurs has b studd tsvly ovr th past dcads, [8 8] o drctly applcabl xprsso of th dgradato cascad structurs could b foud. Thrfor, th thory wll frst b dvlopd a gral mar. Mor prcsly, th stags th cascad wll ot b cosdrd as MFIR stags wth trasfr fucto M ( z ), but as gral fltr stags wth trasfr H z. Th dx wll b usd as a dx for th gral stags th cascad. Not that fucto s ot th sam as th dx that s usd to dcat th stags of th MFIR structur ad th powr of th multplr coffcts of th MFIR stags (as Equatos () ad (3)). It wll b show that th ordrg has a larg mpact o th. Howvr, P stags of a MFIR structur ca b ordrd P! possbl ordrgs, mag a xhaustv sarch for th optmal ordrg upractcal. Ul typcal cascad structurs studd [5 8], th trasfr fuctos of th stags of th MFIR structur ar fxd by th xprssos gv Equatos ( 5),.., vry stag th groupg of th zros s fxd by th MFIR approach. Ths mpls that th optmzato of th prformac ca oly focus o th ordrg of th stags... Th Roud-Off Nos Modl Th study s basd o th followg assumptos. Each multply ad accumulat acto a stag s modld as a ft prcso multplr, followd by a summato od. Aftr th summato od, roudg s prformd ad cosqutly roud-off os s addd to th systm. I th prst papr, t s assumd that th roudg procss uss covtoal roudg ad saturato. For th ral pol

6 Appl. Sc. 4, 4 4 approxmato, th cojugat pol par cascad approxmato ad thr rspctv lar phas approxmatos, ach stag has o sgl os sourc at th output of th stag. It s assumd that vry sampl of th os sourc s ucorrlatd wth th prvous sampl, all os sourcs ar ucorrlatd, th os sourcs ar ucorrlatd wth th put sgal, vry os sourc s a tm dscrt statoary zro ma wht radom procss wth output varac q /. Hr, q s th smallst quatzato stp (q = b whr b s th umbr of bts (wthout th sg bt) usd to quatz th sgal)..3. Symbol Covtos ad Trasfr Fuctos Th followg dftos ar usd throughout th txt. Evry stag wthout roudg or scalg s dcatd by H (z) or H ( jω ) or short form H. Th total fltr trasfr fucto s wrtt as H(z) or H( jω ). A (gral) stag wth roudg ad scalg s dcatd by ), ) or short form. Th rspctv scalg factors pr stag ar dcatd by S. Th tm sampls of th roud-off os sourc ar dcatd by () (whr s th dscrt tm dx) or short form. Th os varac of a os sourc s gv by. F z s th trasfr fucto from th fltr A umbr of trasfr fuctos ar dfd Fgur. put to th output of th stag wth trasfr fucto H (z) (wthout roudg). G ( z) s th trasfr fucto from th output of th stag wth trasfr fucto H (z) to th output of th fltr (wthout roudg). Fgur. Th fltr cascad wthout roudg or scalg ad ts trasfr fuctos. G G GP H H H H H + H P F F F F For th roud-off os aalyss, t s assumd thr s a roud-off os sourc at th output of ach stag. Th trasfr fucto from th fltr put to th output of th stag wth trasfr fucto ) (os sourc of stag ot cludd, as show Fgurs 3 ad 4) s dcatd by ). Th trasfr fucto from th output of th stag wth trasfr fucto ) (os sourc of stag cludd, as show Fgur 4) to th output of th total scald fltr s dcatd by ).

7 Appl. Sc. 4, 4 5 Fgur 3. Th roudd ad scald fltr cascad ad ts trasfr fuctos. Fgur 4. Th word out roudd ad scald fltr cascad ad ts trasfr fuctos. H S + H S + H P S P + P Cosqutly, th followg holds: P ( m) G z = H z, (6) m= + P P P G ( z) = Hm( z) = SmHm z = Sm G z m= + m= + m= +, (7) F z = Hm z, (8) m= F ( z) = Hm( z) = SmHm z = Sm F z m= m= m=. (9)

8 Appl. Sc. 4, Th Gral Scalg Mthods I aalogy to [] ad [8], th cosdrd scalg mthods, ar dfd ths scto. A scalg factor s usd to multply th fltr stag coffcts ordr to obta a spcfc crtro at th output of th stag as dfd [8]. For L boud scalg, th scalg factors ar dtrmd by: m= j Sm F ω =, () F ω s dfd as th L orm th frqucy doma, gv by: j whr + π F ( ) = F ( ) dω f = π π. () Hr, f () ar th mpuls rspos sampls of th fltr gv by th trasfr fucto F z. Th rcursv vrso of Equato () ca b usd to calculat th scal factor pr stag. It s gv by: S jω ( ) H = = jω ( S m ) H m ( ) =... P. m = m = Th followg holds wh L boud scalg s usd: f th RMS valu (ovr ω ) of th put sgal s boudd by uty, th RMS valu (ovr ω ) of th sgal at ach stag output wll b boudd by uty. For fty boud scalg, L, th scalg factors ar dtrmd by: () m m= π< ω <+ π j ( ) S sup F ω = (3) Th rcursv vrso of Equato (3) ca b usd to calculat th scal factor pr stag. It s gv by: S jω sup H ( ) = π < ω < + π = jω ( S m ) sup H m ( ) =... P. m = π < ω <+ π m = (4) Th L boud scalg sts th maxmum of th frqucy rsposs of all rspctv F ( ) at db. For absolut boud scalg, th scalg factors ar dtrmd by: p S m f ( = ) m= = (5)

9 Appl. Sc. 4, 4 7 whr f () ar th mpuls rspos sampls of th fltr gv by th trasfr fucto F (z). p s th lgth of th mpuls rspos. Equato (5) rcursv form ylds: S p = f = = p ( S m) f ( ) =... P. m= = (6) Absolut boud scalg s basd o th rasog that f th pa valu of th put sgal s boudd by uty, th pa absolut valu of th sgal at ach stag output wll b boudd by uty wh absolut boud scalg s usd. Th absolut boud scalg crtro s avodg ovrflow all cass. Cotrary to th absolut boud scalg approach, wh usg fty boud scalg or L boud scalg, ovrflow s stll possbl. Although all scalg mthods prvt ovrflow accordg to a crta crtro, o of thm wll forc all multplr coffcts to b smallr tha (or qual to) o. Ths mpls that a practcal mplmtato t ca happ that th bts usd to rprst a multplr coffct ar ot suffct. Th problm ca b solvd svral ways [9]. Th prst papr uss a mthod that has mmum mpact o th ovrall accuracy. Mor prcsly, th multplr valus of th stags whr th coffcts ar largr tha ar dvdd by a powr of (usg shftg) to forc all coffcts of ths stag to b smallr tha. Ths dvso by th powr of s udo th output sgal of th stag,.., by shftg th rsult. 4. Th Ordrg of th Stags As th squtal ordrg of th stags has a larg mpact o th scalg factors ad th os prformac of th fltr, th roud-off output os varac as a fucto of th ordrg must b calculatd ad a mthod to dtrm th optmal squtal ordrg must b foud. 4.. Th Roud-Off Output Nos Varac Th roud-off output os varac s th summato of th os sourcs sd th fltr cascad, wth sutabl wghtgs ad fltrg. Ths varac s affctd by th ordrg of th stags of th fltr structur. Th varac of th os of ach roudg oprato quals. Th put sgal of th fltr has a ampltud th trval (, +). I cas b + bts (b bts + a sg bt) ar usd to rprst th sgal two s complmt, th varac of th os gratd by o roud-off os sourc s (udr th assumptos of Scto.) gv by: q =. (7) Hr, q = b. I gral, th varac of th roud-off os sourc s also gv by: + π = P d m π ω ω (8) π whr: P (ω ) s th Powr Spctral Dsty (PSD) of th os sourc.

10 Appl. Sc. 4, 4 8 m s th ma valu of th os sourc. Udr th assumptos gv Scto., th ma m = ad th PSD of th roud-off os sourc s dpdt of th frqucy, mplyg: ( ω ) = P. (9) Th PSD of th os gratd at th output of th total fltr structur, by th os sourc of a (scald ad roudd) stag s gv by: Th varac of ths os s gv by: j ω P = G ω () vv π v = Pvv d mv π π ω ω. () As th os sourc has a zro ma valu ad th stags ar lar, m v =. Th PSD of th os gratd at th output of th fltr structur, by all roud-off os sourcs s gv by: P j P ( ) G ( ω ω = + ). () = Th output os varac du to th roud-off os sourcs of all stags s thus gv by (usg Equato ()): j ( ω ) (3) P = + G = As ca b s Equato (3), th cotrbuto of a os sourc of ay stag to th output os varac oly, dpds o th trasfr fucto from ths os sourc to th output. As ths s vald for a gral stag, t s vald for all stags. From ths rasog, t s obvous to ordr th stags from j hgh powr amplfcato to low powr amplfcato ordr to p all G ω (for ay ) as small as possbl. Ths approach mmzs th output roud-off os varac. Howvr, Equato (3), th trasfr fucto from a os sourc to th output s scald, mplyg that th optmal ordrg must b drvd from th scald stag quatos, whch s rathr covt. Thrfor, Equato (3) wll b furthr word out. It s clar from Fgur 3 ad Equato (7) that Equato () ca b wrtt as: P P j P ( ω ) = + SmHm ( ω ). = m= + (4) Idpdt of th stag ordrg, th followg holds for fty boud ad L boud scalg: P m= + S P ω ( ) m m = m = = j m = S F S m H α α (5) whr, α = or α = rspctvly. Usg Equatos (7) ad (5) Equato (4) ylds:

11 Appl. Sc. 4, 4 9 P P ( ω ) = F ( ) G ( + ) = α H ( ) α (6) ad by applyg Equato (): P = F ( ) G ( + ). = α H( ) α (7) I Equatos (6) ad (7), th scalg s ta to accout ad th u-scald stag quatos ca ow b usd F ( ) ad G ( ) to dtrm th optmal ordrg for mmal output roud-off os varac. It s clar that vry F ( ) ad G ( ) cotrbut to th output os. For a gv P fltr, th output os varac s mmal wh F ( ) G ( ) s mmal,.., a = α optmal ordrg must b foud to mmz ths sum. j I cas of L boud scalg (α = ), F ω j ad G ω vry sum trm wll hav th sam cotrbuto, mplyg that th ordrg of th stags has o mportac. (I.., cas of ordrg from = to = P ad usg L boud scalg, Equato (7) ylds: = + ( H H HH H H H H ). L H I cas of ordrg from = P to = ad usg L boud scalg, Equato (7) ylds: = + ( HHH 3... H + HH 3... HH +... ). L H Two dtcal quatos ar obtad.) If fty boud scalg, L, s cosdrd (α = ), th optmal ordrg wll b dtrmd by th rato: for vry {,,..., P }. F G ( ) ( ) I cas ths rato s sgfcatly largr tha for vry valu, t s bst to ordr th stags from small pa ga to larg pa ga. I cas ths rato s ot sgfcatly largr tha, th optmal ordrg should b dtrmd xhaustvly. (8)

12 Appl. Sc. 4, 4 I cas of absolut boud scalg th quvalt of Equato (5) s gv by: P m= + S P p m m= = m = = p m m= = S f S h. (9) Hr h() ar th mpuls rspos (havg a lgth p) sampls of th fltr gv by th trasfr H z. Applyg Equato (9) o Equato () ad usg Equato (7) ylds: fucto P p P ( ω ) = + f ( ) G ( ) p = = h( ) = ad by applyg Equato ():. h = P p = + f ( ) G ( ) p = = (3) (3) As for fty boud scalg, th optmal ordrg to obta a mmal output roud-off os varac s dtrmd by th rato: for vry {,,..., P }. p j ( ω ) =, (3) G f I cas ths rato s sgfcatly largr tha for vry valu, t s bst to ordr th stags crasg coffct magtud,.., th stag wth th largst coffct(s) at th d. I cas th rato s ot sgfcatly largr tha, th optmal ordrg should b dtrmd xhaustvly. 4.. Th Sgal to Roud-off Nos Rato Th roud-off os prformac of a fltr may ot b corrctly valuatd by oly aalyzg th roud-off os. A mor rlabl rsult s obtad by calculatg th Sgal to roud-off os rato (). Th sgal to os ratos (wh usg th prvously dscussd scalg mthods) ar vstgatd ths scto. Th dscrt put sgal of th fltr s dcatd by x() ad th dscrt output sgal by y(). Th varac of th dscrt put sgal x() s dcatd by. Th varac of th dscrt output sgal y() of th fltr wll b dcatd by. All calculatos prstd ths scto ar basd o th out codtos that th put sgal x() s zro ma ad has a costat, frqucy dpdt, Probablty Dsty Fucto (PDF).

13 Appl. Sc. 4, 4 I cas th put sgal s a wd ss statoary radom sgal wth uform PDF ad varac, th output sgal varac of th fltr s gv by: out + π = H( ) dω π (33) π whr h fltr. j H ω out = (34) out = h (35) s th sum of th squard mpuls rspos sampls of th (scald ad roudd) I cas of L or L boud scalg, Equato (34) ca b wrtt as (usg Equato () or Equato (3) as approprat):, out H = (36) H α whr α = or α =, rspctvly. Combg Equatos (36) ad (7) ylds. H( ) ( ) P ( ) out = =. H F G + α = α (37) I cas of absolut boud scalg, th s gv by: out H p P h + f = = = G j = = p ω I Equatos (37) ad (38) s dpdt o th typ of roudg usd ad th umbr of bts that ar usd to quatz th sgal th fltr structur. dpds o th put sgal. Quatzato ad put sgal dpdt factors ar gv by:. = P jω j L ω F ( ) G ( ) = + (39) H( ). (38)

14 Appl. Sc. 4, 4. =. P j j L jω ω ω F ( ) G ( H ) = + H( ) H( ). =. p p P abs j f ( ω h ) G ( ) = = = + jω H( ) H( ) (4) (4) I gral for a arbtrary A(ω) [8]: whch mpls that: A A... A a, (4) p H H h =. (43) Not that Equato (43) s dpdt of th ordrg of th stags. P As mtod Scto 4., cas of L boud scalg, F ( ) G ( ) = Equato (39) s dpdt of th stag ordrg, mplyg that th stag ordrg has o mpact o th. For a gv ordrg of th stags, t s clar from Equatos (39 4) ad (43) that.... L L abs Ev cas of a optmzd ordrg for fty boud scalg or absolut boud scalg, L boud scalg wll always hav th bst. Not that ths txt s sgal to roud-off os rato ad ot th ovrall of th fltr. Howvr, a optmzd ordrg for absolut boud scalg ca hav a bttr tha a o-optmzd ordrg for fty boud scalg ad vc vrsa. I cas of L boud scalg, th optmal must b dtrmd by fdg a ordrg that mmzs (44) P F G = (45). I cas of absolut boud scalg th optmal must b dtrmd by fdg a ordrg that mmzs p P = = j f G ω. (46) Comparg ths rqurmts wth thos of th output os varac (formulatd Scto 4.), t ca b cocludd that th optmzato of th, by fdg th optmal ordrg, uss th sam crtra as th mmzato of th output roud-off os as dscussd Scto 4..

15 Appl. Sc. 4, 4 3 Notc that th quatos drvd ths scto ar grally vald for ay fltr cascad (wth o os sourc at th output of ach fltr stag) ad ot oly for MFIR structurs. 5. Th of MFIR Fltrs I ths scto, th gral thory dvlopd Scto 4 wll b appld to th MFIR fltr structurs. 5.. Th Trasfr Fuctos I cas MFIR fltrs ar cosdrd, th stags ar dcatd by M (z) or M ( ) or short form M. Th total fltr trasfr fucto s wrtt as M (z), M ( ) or M. A scald stag s dcatd by ), ) or. Th ordrg whr th frst stag s stag =, th scod s = ad so o, wll b calld th forward (squtal) ordrg. Cosqutly, cas of MFIR structurs ad forward ordrg: P m G = M, (47) m=+ P P P = = =, (48) G Mm SmMm Sm G m=+ m=+ m=+ m F = M, (49) m= = = =. (5) F Mm SmMm Sm F m= m= m= Notc that F P ( jω ) = M ( jω ) ad G P ( jω ) = (s Fgurs ad 3). Fgur 5. Th uscald ad scald MFIR fltr cascad trasfr fuctos for rvrs ordrg.

16 Appl. Sc. 4, 4 4 Th ordrg whr stag = P s th frst stag, = P s th scod stag ad so o, s calld th rvrs (squtal) ordrg. I cas of rvrs ordrg of MFIR stags, th trasfr fuctos ar show Fgur 5 ad ar dfd by: m G = M, > (5) m= = = =, > (5) G Mm SmMm Sm G m= m= m= P m F = M, (53) m= P P P = = =. (54) F Mm SmMm Sm F m= m= m= Notc that cas of rvrs ordrg F ( ) = M( ) ad G ( ) =. Although thr ar P! Possbl ordrg combatos of th MFIR stags, xprmts hav show that choosg th bst opto btw forward ad rvrs squtal ordrg s usually satsfactory. 5.. Th Scalg Factors Th quatos drvd Scto 3 adaptd to MFIR structurs ar gv Tabl. Tabl. Scalg factors for Multplcatv Ft Impuls Rspos (MFIR) structurs. Forward ordrg j S F ω = Rvrs ordrg P j S F ω = L boud m m m= j Ifty boud Sm = sup F( ω ) m= π< ω <+ π p m m= = Absolut boud S = f m= j ( F( )) ω P Sm = sup m= π< ω <+ π p P S m f( = ) m= = Tabl. Lgth of th mpuls rsposs of th partal MFIR trasfr fuctos. Forward ordrg f ( ) or f ( ) Ral pol p = Ral pol lar phas p = Complx-cojugat pol par p = Complx-cojugat pol par lar phas 3 p = 4 Rvrs ordrg f ( ) or f ( ) Ral pol P p = Ral pol lar phas P+ p = + Complx-cojugat pol par P+ p = + Complx-cojugat pol par lar phas P+ p = +

17 Appl. Sc. 4, 4 5 I cas of absolut boud scalg, th lgth p of th mpuls rspos f () must b ow. As ths lgth s fltr dpdt, Tabl gvs a ovrvw of th mpuls rspos lgths for forward ad rvrs ordrg of MFIR stags [,9]. It s show [9] that cas of a ral pol approxmato, th scalg factors oly dpd o th stag o whch thy ar appld,.., ot o ay of th prvous or xt stags Th ad Optmal Ordrg for MFIR Fltrs Gral MFIR Exprssos I cas of forward ordrg, Equatos (39 4) bcom:. =, P jω j L ω F( ) G( ) = + M ( ). =, P j j L jω ω ω F ( ) G ( M ) = + M ( ) M ( ) (55) (56) j Hr, G ( ω ) ad F ( j ω ). =. p p P abs j f ( ω m ) G ( ) = = = + jω M ( ) M ( ) (57) ar gv by Equatos (47) ad (49) rspctvly. Th valu of p s gv Tabl for forward ordrg. m() ar th mpuls rspos sampls of th total MFIR fltr. p Th valu p m( ) Equato (57) s th lgth of th total MFIR fltr mpuls rspos ad = ca b foud by sttg = P Tabl for forward ordrg. I cas of rvrs ordrg, Equatos (39 4) bcom:. =, P jω jω L F ( ) G ( ) = + jω M ( ). =, P j jω L ω M ( ) F ( ) G ( ) = + M ( ) M ( ) (58) (59)

18 Appl. Sc. 4, 4 6 j Hr, G ( ω ) ad F ( j ω ). =. p p P abs j f( ω m ) G( ) = = = + jω M ( ) M ( ) gv Tabl for rvrs ordrg. Th valu p m (6) ar gv by Equatos (5) ad (53) rspctvly. Th valu of p s p = Equato (6) s th lgth of th total MFIR fltr mpuls rspos ad ca b foud by sttg = Tabl for rvrs ordrg. I ordr to p th txt mor radabl, th fgurs of th svral calculato rsults of Equatos (55 6) ar groupd togthr at th d of th scto. Tabl 3 gvs a ovrvw of th rsults. Tabl 3. Ovrvw of th fgurs of th calculatos. Approxmato of Scalg typ Ordrg Rag Fgur Ral pol L Forward ad Rvrs [., ) Fgur 6 Ral Pol Abs ad If. Forward ad Rvrs [., ) Fgur 7 Ral Pol Lar Phas L Forward ad Rvrs [., ) Fgur 8 Ral Pol Lar Phas Abs ad If. Forward ad Rvrs [., ) Fgur 9 Compl. Coj. r =.9 L Forward ad Rvrs Pol par [, ) Fgur Compl. Coj. r =.9 If Forward ad Rvrs Pol par [, ) Fgur Compl. Coj. Pol par Abs Forward ad Rvrs r =.9 [, ) Fgur Compl. Coj. Pol par If Rvrs r =.8;.85;.9;.95 [, ) Fgur 3 Compl. Coj. Pol par L, If, Abs Forward ad Rvrs r =.9 [, ) Fgur 4 Compl. Coj. Pol par, L Phas Compl. Coj. Pol par, L Phas Compl. Coj. Pol par, L Phas L, If, Abs L, If, Abs If Forward Rvrs Rvrs r =.9 [, ) r =.9 [, ) r =.8;.85;.9;.95 [, ) Fgur 5 Fgur 6 Fgur 7

19 Appl. Sc. 4, Prformac of a MFIR Fltr Approxmatg a Ral Pol Fltr Fgurs 6 ad 7 show th valus for ral pols λ th trval [., ). I cas of th MFIR approxmato, for ach λ valu, th rqurd umbr of stags, P, s dtrmd to obta a maxmum dffrc of. db btw th MFIR magtud rspos ad th magtud rspos of th approxmatd IIR fltr. Th dgs th curvs dcat whr a xtra MFIR stag s addd to fulfll ths rqurmt. Th IIR fltr rsults ar basd o ([], Equato.48): Hr, S s th scalg factor of th IIR fltr. Fgur 6. S = for ral pols λ th trval [., ) cas of L boud scalg. (Th MFIR forward ad MFIR rvrs rsults ar suprmposd). (6) Fgur 7. for ral pols λ th trval [., ) cas of absolut or fty boud scalg.

20 Appl. Sc. 4, 4 8 Fgur 8. of MFIR fltrs approxmatg th squard magtud rspos of ral pols λ th trval [., ) cas of L boud scalg. (Th MFIR forward ad MFIR rvrs rsults ar suprmposd). Fgur 9. of MFIR fltrs approxmatg th squard magtud rspos of ral pols λ th trval [., ) cas of absolut ad/ or fty boud scalg.

21 Appl. Sc. 4, 4 9 Fgur. for complx-cojugat pol pars wth magtud.9 ad L boud scalg; Th MFIR approxmato uss P = 7 stags. (Th MFIR forward ad MFIR rvrs rsults ar suprmposd). Fgur. for complx-cojugat pol pars wth magtud.9 ad fty boud scalg; Th MFIR approxmato uss P = 7 stags.

22 Appl. Sc. 4, 4 Fgur. for complx-cojugat pol pars wth magtud.9 ad absolut boud scalg; Th MFIR approxmato uss P = 7 stags. Fgur 3. fucto of pol magtud r ad pol agl θ, cas of fty boud scalg ad rvrs ordrg. (surfac = MFIR; dscrt curvs = IIR fltr).

23 Appl. Sc. 4, 4 Fgur 4. of a MFIR fltr approxmatg a complx-cojugat pol par wth magtud.9, ad P = 7. Fgur 5. of a lar phas MFIR fltr approxmatg th squard magtud rspos of a complx-cojugat pol par wth magtud.9 ad P = 7 forward ordrg.

24 Appl. Sc. 4, 4 Fgur 6. of a lar phas MFIR fltr approxmatg th squard magtud rspos of a complx-cojugat pol par wth magtud.9 ad P = 7 rvrs ordrg. Fgur 7. of a lar phas MFIR fltr fucto of th pol agl ad magtud cas of fty boud scalg ad rvrs ordrg.

25 Appl. Sc. 4, 4 3 As xplad Scto 4., th ordrg has o mpact o th os prformac wh L boud scalg s usd. Absolut boud scalg ad fty boud scalg yld th sam os prformac ad rvrs ordrg s bttr (max. 4 db) tha forward ordrg. L boud scalg always ylds a bttr os prformac tha absolut ad fty boud scalg. I cas of absolut or fty boud scalg, th MFIR fltr has a bttr sgal to os prformac tha th approxmatd ral pol fltr for λ.86. For L boud scalg, oly for λ.95 th MFIR fltr has a bttr prformac tha th approxmatd IIR fltr. I cas λ <.9, th worst-cas dffrcs btw th MFIR fltr ad th IIR fltr ar 3.6 db for absolut (ad fty) boud scalg ad 5 db for L boud scalg. Ths s, howvr, ot dramatc bcaus th rag of trst ( λ >.9), gral th MFIR fltr has a bttr sgal to os prformac tha ts corrspodg IIR fltr (xcpt for th small rgo btw λ =.9 ad λ =.95 for L boud scalg whr th dffrc s maxmum. db favor of th IIR fltr). Th mor λ approachs th ut crcl, th bttr th os prformac of th MFIR fltr compard to th IIR fltr. For xampl, for λ =.999, th MFIR approxmato s 7 db bttr for absolut (ad fty) boud scalg, for L boud scalg th MFIR fltr s 7 db bttr tha th approxmatd IIR fltr Prformac of a Lar Phas MFIR Fltr Approxmatg th Squard Magtud Rspos of a Ral Pol Fltr Fgurs 8 ad 9 show th prformac of lar phas MFIR fltrs approxmatg th squard magtud rspos of ral pol fltrs wth λ th trval [., ). For ach λ valu, th umbr of stags, P, s pt th sam as Scto Thr s o objctv comparso possbl btw ths MFIR approxmato ad th IIR fltr, sc th IIR fltr s ot a lar phas fltr ad th MFIR fltr approxmats th squard magtud rspos of th pol dcatd o th horzotal axs. Howvr, t s clar that th roud-off os prformac s comparabl wth th ral pol o-lar phas MFIR approxmatos. Th ordrg has o mpact wh L boud scalg s usd ad th os prformac wth L boud scalg s always bttr tha absolut ad fty boud scalg. Ifty boud scalg ad absolut boud scalg hav th sam prformac. Rvrs ordrg ylds aga bttr rsults tha forward ordrg. Howvr, th maxmum dffrc btw th two ordrgs s rathr small (3.3 db). Compard to th o-lar phas approxmato, th dffrc btw L boud scalg ad th othr scalg mthods ar somwhat largr Prformac of a MFIR Fltr Approxmatg a Complx-Cojugat Pol Par Fltr Cascad Fgur shows th valus wh ralzg pol pars wth a magtud r =.9 ad agls θ th trval [, π) cas of L boud scalg. Fgurs ad show th rsults cas of valus fty boud scalg ad absolut boud scalg rspctvly. I Fgur 3 th ar calculatd for th pol magtuds:.8,.85,.9 ad.95 approxmatd wth P = 5, 6, 7, ad 7

26 Appl. Sc. 4, 4 4 stags rspctvly cas of fty boud scalg ad rvrs ordrg. Th surfac plots ar th valus ( ) valus for th MFIR fltrs. Th half crcl shapd curvs ar th for th corrspodg IIR fltrs. Fgur 4 compars th os prformacs of th approxmato of th complx-cojugat pol par wth a magtud r =.9 ad ay agl btw ad π, for svral scalg mthods ad ordrgs. All pol magtuds btw.8 ad.99 stps of. hav b calculatd but ar ot all show hr. Th rsults show th fgurs ar howvr rprstatv for all combatos of scalg mthods, pol magtuds ad agls that wr calculatd. Aftr xtsv aalyss of th data, th followg coclusos ca b draw for th MFIR approxmato of a complx-cojugat pol par fltr ralzd usg th cascad structur. Th roud-off os prformac of a MFIR fltr approxmatg a complx-cojugat pol par fltr s sgfcatly bttr (up to db) tha th os prformac of ts corrspodg IIR fltr wh th approxmatd pols ar stuatd th ghborhood of th ral axs; s far lss pol agl θ dpdt comparso wth th corrspodg IIR fltr; s up to.5 db bttr for fty boud scalg tha for absolut boud scalg (usg rvrs ordrg); s always bttr for L boud scalg tha for th othr scalg mthods (obys Equato (44)); s pol magtud dpdt, but ot that much as th corrspodg IIR fltr; s farly sstv to a xtra MFIR fltr stag (typcally db); s vry sstv to th stag ordrg for absolut ad fty boud scalg; s ordrg dpdt cas of L boud scalg; s gral bttr rvrs ordrg tha forward ordrg, xcpt for pol agls th ghborhood of π/; s for pol agls th ghborhood of π/, for absolut ad fty boud scalg, bttr forward ordrg tha rvrs ordrg (It was alrady rmard Scto 4., from a thortcal pot of vw, that ths stuato could occur.); ca b up to 6 db wors tha th corrspodg IIR fltr for pol agls th ghborhood of π/. Th wdth of ths rgo ad th magtud of th dffrc dcrass howvr wth crasg pol magtud ; I th ormal rag of pol magtuds that ar cosdrd for MFIR approxmatos ( r >.9), th os prformac s gral (dpdg o th pol agl) bttr tha for th corrspodg IIR fltr Prformac of a Lar Phas MFIR fltr Approxmatg th Squard Magtud Rspos of a Complx-Cojugat Pol Par Fltr Fgur 5 shows th valus for pol pars wth magtud r =.9 ad agls θ th trval [, π) for forward ordrg. Fgur 6 shows th valus for rvrs ordrg. I Fgur 7, th cas of rvrs ordrg ad fty boud scalg for th pol magtuds.8,.85,.9 ad.95 approxmatd wth P = 5, 6, 7 ad 7 stags rspctvly, s show.

27 Appl. Sc. 4, 4 5 Compard wth th o-lar phas approxmato, th s mor pol agl dpdt. Th dps at th pol agls π/4, π/ ad 3π/4 ar also dpr. Th fgurs show that th ordrg of th stags has o ffct wh L boud scalg s usd. I cas of absolut or fty boud scalg, rvrs ordrg prforms bttr tha forward ordrg xcpt th rgos whr θ = π/4, π/, ad 3π/4. Howvr, th wdth of ths rgos s pol magtud dpdt. Cosqutly, practc both ordrgs wll hav to b cosdrd wh pols wth agls ths rgos ar to b approxmatd. Ufortuatly th ghborhood of th pol agls θ = π/4, π/ ad 3π/4 th lar phas approxmato clarly prforms wors tha th o-lar phas approxmato. Th o-optmal prformac for pol agls θ = π/4, π/ ad 3π/4 ca b xplad by usg a xampl. I cas r =.9, θ = π/ ad P = 9, M (z) s gv by (usg Equato (5)): ad M 8 (z) s gv by: 4 M z = +.45z + z (6) M8 z = +.35* z +.678* z +.35* z + z. (63) I Scto 4. t s show that cas of fty boud scalg, th stags wth th largst pa gas ( th frqucy doma), should fall most oft G (z) to rduc th output roud-off os varac. I cas of absolut boud scalg, th stags wth th largst coffcts should fall most oft G (z) to rduc th output roud-off os varac. It s clar from Equatos (6) ad (63) that forward ordrg (M 8 (z) most oft G (z) ) s ths cas much bttr tha rvrs ordrg. Thr s a combd ffct volvd th lar phas approxmato of complx-cojugat pol pars wth agls th ghborhood of π/4, π/ ad 3π/4: / r factors Equato (5) ca hav vry larg valus wh s larg, cos( θ ) factors Equato (5) ar for most stags clos to uty mplyg th coffcts ar ot rducd by th cos fuctos. Ev cas of forward ordrg, th prformac for ths agls s ot good. Idd th stags wth largr valus, stll hav vry larg coffcts mplyg that Equato (45) or Equato (46) wll vr b vry small. 6. Coclusos A approach to modl roud-off os gral cascad fltr structurs has b studd. Ths roud-off os dpds o th usd scalg mthod ad o th ordrg of th stags. Ths gral rsults ar usd to study ad optmz th roud-off os bhavor of MFIR fltrs. It has b show that th roud-off os prformacs of th MFIR pol approxmatos dd dpd o th usd scalg mthod. L boud scalg rsults th bst prformac, followd by fty ad absolut boud scalg. I gral, t ca b cocludd that th rgo of trst (approxmatg pol bhavors wth magtuds r >.9) th MFIR approxmatos prform bttr tha th approxmatd IIR fltrs. Ev

28 Appl. Sc. 4, 4 6 outsd th rgo of trst, th prformac grally dos ot dffr too much from th corrspodg IIR fltrs (max 6 db). Th aalyss prstd [] suggsts forward ordrg as a optmal ordrg cas o scalg s appld. I cas of L boud scalg th ordrg has o mpact o th roud-off os prformac. Th aalyss prstd hr, xtds ths rsults to othr practcal scalg mthods ad cocluds that ths cass, rvrs ordrg prforms bttr tha forward ordrg for most pol approxmatos. Howvr, t should b otd that spcal attto to th stag ordrg s rqurd wh approxmatg a complx-cojugat pol par havg a pol agl th ghborhood of π/ ad th cascad structur has b usd; wh approxmatg th squard magtud rspos of a complx-cojugat pol par fltr cas th pol agls ar stuatd th ghborhood of π/4, π/ ad 3π/4 ad th lar phas cascad structur has b usd. Furthr rsarch s rqurd to dtrm f altratv ordrgs ca b foud whch would yld a bttr os prformac. Not that th ordrgs cosdrd hr ar ot th oly possbl ordrgs. For bst prformac, o must dtrm th dclg amplfcato ordr for vry pol par that s approxmatd. At th momt, o systmatc approach has b foud, so tral ad rror s rqurd. Rsarch wll hav to prov f a optmal ordrg ca b calculatd. If ot, a hurstc approach as [8] for th pol zro parg or a tratv optmzato algorthm [7] or aothr ar optmal ordrg tchqu [6] could also b trstg. Coflcts of Itrst Th authors dclar o coflct of trst. Rfrcs. Fam, A.T. MFIR fltrs: Proprts ad applcatos. IEEE Tras. Acoust. Spch Sgal Procss. 98, 9, Vadbussch, J.-J.; L, P.; Putma, J. Aalyss of tm ad frqucy doma prformac of MFIR fltrs. I Procdgs of th 8 Itratoal Cofrc o Embddd Systms ad Applcatos, Las Vgas, NV, USA, 4 7 July 8; Araba, H.R., Mu, Y., Eds; CSREA Prss: Las Vgas, NV, USA, 8; pp Radmachr, H. Topcs Aalytc Numbr Thory; Sprgr Vrlag: Nw Yor, NY, USA, 973; Chaptr, pp Vadbusschm, J.-J.; Lm, P.; Putmam, J. Lar phas approxmato of ral ad complx pol IIR fltrs usg MFIR structurs. I Procdgs of th 5th Europa Cofrc o th Us of Modr Iformato ad Commucato Tchologs, Gt, Blgum, 3 March ; D Strycr, L., Ed.; Nvllad: Gt, Blgum, ; pp Vadbussch, J.-J.; L, P.; Putma, J. A FPGA basd dgtal loc- amplfr mplmtd usg MFIR rsoators. I Procdgs of th Itratoal Cofrc o Sgal Procssg, Pattr Rcogto ad Applcatos, Crt, Grc, 8 Ju ; Ptrou, M., Sappa, A.D., Tratafyllds, G.A., Eds.; Acta Prss: Calgary, AB, Caada, ; Volum , pp

29 Appl. Sc. 4, Vadbussch, J.-J.; L, P.; Putma, J. Dsg of a FPGA basd TV-Tur tst bch usg MFIR structurs. Au. J. Elctro. 3, 7, Vadbussch, J.-J.; L, P.; Putma, J. O th coffct quatzato of multplcatv FIR fltrs. Dgt. Sgal Procss. 3, 3, Jacso, L.B. O th tracto of roudoff os ad dyamc rag dgtal fltrs. Bll Syst. Tch. J. 97, 49, Cha, D.S.K.; Rabr, L.R. A algorthm for mmzg roudoff os cascad ralzatos for ft mpuls rspos dgtal fltr. Bll Syst. Tch. J. 973, 5, Mtra, S.K. Dgtal Sgal Procssg: A Computr-Basd Approach, 3rd d.; Mc Graw Hll Hghr Educato: Nw Yor, NY, USA, 6; pp Jacso, L.B. Roud-off os aalyss for fxd pot dgtal fltrs ralzd cascad or paralll form. IEEE Tras. Audo Elctroacoustcs. 97, 8, 7.. Cha, D.S.K.; Rabr, L.R. Aalyss of quatzato rrors th drct form for ft mpuls rspos dgtal fltrs. IEEE Tras. Audo Elctroacoustcs. 973,, Fttws, A. Roudoff os ad attuato sstvty dgtal fltrs wth fxd-pot arthmtc. IEEE Tras. Crcut Thory 973,, Modal, K.; Mtra, S.K. Roudoff os uppr bouds for cascadd rcursv dgtal fltr structurs. IEE Proc. Elctro. Crcut Syst. 98, 9, Lm, Y.C.; Lu, B. Dsg of cascad form FIR fltrs wth dscrt valud coffcts. IEEE Acoust. Spch Sgal Procss. 988, 36, Motgomry Smth, L.; Hdrso, M.E. Roudoff os rducto cascad ralzatos of FIR dgtal fltrs. IEEE Tras. Sgal Procss., 48, Dhr, G.F. Nos optmzd IIR dgtal fltr dsg-tutoral ad som w aspcts. Sgal Procss. 3, 83, Sh, D.; Yu, Y.J. Dsg of dscrt-valud lar phas FIR fltrs cascad form. IEEE Tras. Crcuts Syst., 58, Vadbussch, J.-J. Aalyss ad Implmtato of MFIR Fltrs FPGA Tchology. Ph.D. Thss, Uvrsty of Kt, Catrbury, UK, Sptmbr. 4 by th authors; lcs MDPI, Basl, Swtzrlad. Ths artcl s a op accss artcl dstrbutd udr th trms ad codtos of th Cratv Commos Attrbuto lcs (