Realiation of Narrowband /$ltistage ;ilters Tehni!o Re"ontni Ja(od KCa!aK Dr. Dragise /iso(i!a 17<' 59 Ca!a' erbia Using Low->ass )ran!h of Co"ple"entar% Tehni!o Re"ontni Ja(od KCa!aK Dr. Dragise /iso(i!a 17<' 59IIR Ca!a' erbia ;a!$lt% of Te!hni!al !ien!es Ca!a' Uni(ersit% of rag$e(a!' (etog a(e t. 7' 59 Ca!a' erbia ;ilter >airs with >ipelining-Interlea(ing >ipelining-Interlea(in g Ca!a' erbia Tehni!o Re"ontni Ja(od KCa!aK Dr. Dragise /iso(i!a 17<' 59 /ileno >. ?iri@a ' Foan / Radoni! b ' Radoa R rneta!' Nenad tefano(i!d
a
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Abstract — This paper concentrates on the efficient reali realizat zation ionss of low-pas low-passs narrowb narrowband and digital digital filters filters using using Xilinx field program programmable mable gate array array (FPG! chips" The narrow bandwidth of these filters (less than a #$%& of the samp sampli ling ng fre' fre'ue uenc ncy! y! is acco accomp mpli lish shed ed by mult multir irat atee multistage multistage filtration and using a low-pass branch of Class III complem complementa entary ry infinite infinite impulse impulse respon response se ()! ()! filter filter pairs for Kernel for Kernel filter* filter* instead of the usual realization with finite impulse response response (F)! filters" +y choosing a low-pass bran branch ch of Class Class III complem complementa entary ry ) filter pairs we achie,e impro,ements in the sense of reducing the stop-band and transition zone errors errors which are caused by a significant increas increasee in the complex complexity ity of the filter filter and fixed-point fixed-point realization" dditional efficiency enhancements and reduced consumptions consumptions of hardware hardware resourc resources es ha,e been enabled by the simultaneous application of PI (Pipelining-nterlea,ing! (Pipelining-nterlea,ing! techni'ues what allows time-sharing of hardware resources resources on the chip" This paper shows one such implementation implementation and presents the ad,antages of such an application" Keywords Keywords Class Class III complem complementa entary ry filter filter pairs* pairs* Field prog program ramma mabl blee gate gate array array . FPG FPG chips chips** /ultis /ultistag tagee mult multir irat atee filt filter ers* s* Para Parall llel el allall-pa pass ss filt filter er real realiz izat atio ion* n* Pipelining-nterlea,ing (P! techni'ue* Tapped cascaded allpass filters realization"
1. I NTRODUCTION
H
narrow narrowba band nd digita digitall filte filters rs with with a sharp sharp transition one are be!o"ing a !o""on re#$ire"ent in "an% appli!ations. appli!ations. <ho$gh designers in re!ent re!ent %ears %ears !an $se $se "ore "ore and and "ore "ore hard hardwa ware re reso reso$r $r!!es' the the narrowband filter re#$ire"ents also si"$ltaneo$sl% grow. Howe(er' stri!t filter re#$ire"ents lead to a signifi!ant in!re in!reas asee in filt filter er orde order. r. )esi )eside dess the prob proble le" " of the !ons$"ption of hardware reso$r!es' s$!h !o"ple* filters also e*hibit e*pressed errors in stop band and transition one. These errors are the res$lt of an in!reased n$"ber of arith"eti! operations +!a$sed b% filter !o"ple*it%, and the fi*ed point i"ple"entation. This effe!t is the "ost sign signif ifi! i!an antt li"i li"iti ting ng fa!t fa!tor or in the the desi design gn of high highl% l% narrowband filters. One of the sol$ti sol$tions ons for narrowband narrowband digital digital filter filterss wo$l wo$ldd be to appl% appl% so"e so"e of the the "$lti "$ltirat ratee te!h te!hni# ni#$e $ess +"$lt +"$ltis ista tage ge filte filterin ring' g' fre# fre#$e $en! n!%%-re resp spons onsee "as "asing ing te!h te!hni# ni#$e $e'' et!. et!.,. ,. )% appl appl%%ing so"e so"e of the "$lti "$ltirat ratee te!hni#$es' a filter order !an be red$!ed to a (al$e that is realiable in pra!ti!e' and the i"ple"entation of the filter "a% be possible. /$ltistage filters are s$itable for the realiation of filters where the bandwidth is less than a tent tenthh of the the sa"p sa"pli ling ng fre# fre#$e $en! n!%% 01' 01' and and in so"e so"e IGHLY
appl appli! i!at atio ions ns the% the% are are a!t$ a!t$al all% l% the the on onl% l% sol$ sol$ti tion on enfor!eable in pra!ti!e 09' 5. Unfo Unfort$ rt$nat natel el%%' s$!h s$!h "$lti "$ltista stage ge narrow narrowba band nd filt filter er realiations also ha(e li"its. Aith an in!reasing downsa"ple +$p-sa"ple, fa!tor a "$ltistage digital filter will be!o"e narrower. na rrower. )$t in addition to this effe!t' effe!t' with an e*!essi(e down-sa"ple +$p-sa"ple, fa!tor in!rease' errors in transition one and stop-band will again begin to grow' depen depending ding on the (al$es (al$es of dow down-s n-sa"p a"ple le +$p-sa" +$p-sa"ple ple,, fa!to fa!torr and the initial initial filter filter order. order. Th$s' Th$s' in "$ltist "$ltistage age i"pl i"ple" e"en enta tati tion onss with with ;IR ;IR Kernel filte ilterr 0B' 0B' ' ' the the infl$en!e of a high initial filter order is e*pressed. The ad(a ad(anta ntage gess of s$!h s$!h reali realiat atio ions ns are are a linea linearr ph phas asee !hara!teristi! !hara!teristi! and si"plified si"plified !riti!al-loop !riti!al-loop !al!$lations !al!$lations'' d$e to $sed ;IR str$!t$re. Ae ha( ha(ee $sed $sed "$ltista "$ltistage ge filter filtering ing as a "$ltirat "$ltiratee te!hni#$e in order to red$!e the filter order. Ae ha(e a!hie a!hie(ed (ed f$rther f$rther filter filter narrowing narrowing b% $sing a low-pa low-pass ss bran!h of Class III !o"ple"entar% !o"ple"entar% IIR filter pairs 07 for initial + Kernel) Kernel) filter realiation. & "a*i"all% low order propert% of this !lass of filters allows f$rther filter narrowing and pre(ents growing of stop-band errors. Ae ha(e shown that the appli!ation of a "$ltistage filter with a Kernel filter realied as a low-pass bran!h of !o"ple"entar% IIR filter pairs enables the realiation of a digital filter narrower than an e#$i(alent "$ltistage "$ltistage filter with an IIR +e.g. ellipti!, Kernel filter. $!h narrowband filters "a% be $sed for !onstr$!ting digital !rosso(ers for lo$dspeaer appli!ations 0<' for the non-$nifor" !hannel separation in a "$lti-band a$dio s%ste" 06 and also for !o"p$tationall% effi!ient b$ilding blo!s in "$lti-!hannel filter bans 01. &ddi &dditio tional nal effi effi!i !ien en!% !% enh enhan! an!e" e"ent entss and red$ red$!e !edd !ons$"ptions of hardware reso$r!es ha(e been a!hie(ed b% appl%ing Pipelining-Interl eaving te!hni# te!hni#$es $es for filtering filtering of se(eral se(eral identi!al !hannels with the sa"e filter 08. 9. /ULTIT&GE /ULTIR&TE ;ILTER This se!tion presents the properties and i"ple"entation i"ple"entation str$!t$res str$!t$res of "$ltistage "$ltistage filters when the% are i"ple"ented b% PI te!hni#$e. te!hni#$e. The general str$!t$re of these filters is shown in ;ig$re 1.
2
Corresponding a$thor. Tel.3 456 171 7859966: ;a*3 4561595<9<15.
E-mail address: address: !!iro=bl$net.rs +/.>. ?iri@,.
;ig$re 1 The general str$!t$re of the "$ltistage filter
&s we see fro" ;ig$re 1' a "$ltistage filter !onsists of a blo! for red$!ing the sa"pling fre#$en!%' Kernel filter' and a blo! for in!reasing the sa"pling fre#$en!%. D$e to de!i"ation and interpolation operations' a "$ltistage filter !ontains additional filters to s$ppress aliasing H D ( z ) and to re"o(e i"aging !o"ponents H I ( z ) . a"pling fre#$en!% is first red$!ed to a lower (al$e' so the filtering with Kernel filter is perfor"ed at a lower fre#$en!%. Hen!e' this "ethod is s$itable for narrowband filter design 0B' . $ppose now that we need to realie "ore identi!al narrowband "$ltistage filters. One of the "ost effe!ti(e sol$tions wo$ld be the $se of PI pro!ed$re 09 whi!h enables $sing onl% one digital filter instead of $sing digital filters for ea!h !hannel. )% $sing the PI te!hni#$e' signals are pro!essed in parallel and thereb% the realiation with se#$ential pro!essing is a(oided.
&nalogo$sl% to this' an algorith" for "$ltistage interpolation part of the filter +we will na"e it EI' i.e.' the e#$i(alent interpolator, wo$ld be3 -
a(e two !onse!$ti(e sa"ples' Insert 2 ∗ ( M K − 1 ) eros' Repeat the pro!ess $ntil the last sa"ple e*pires.
These and si"ilar str$!t$res are ideal !andidates for the realiation of progra""able hardware. The entire str$!t$re !an be realied as a single blo!' and then $sed se(eral ti"es for ea!h stage +e#$i(alent de!i"ator blo! and e#$i(alent interpolator blo!,.
;ig$re B The final str$!t$re of the filter H+ 9, ;ig$re 9 /$ltistage de!i"ation
;ig$re B shows a three-stage realiation' with one and three e#$i(alent de!i"ators +ED, and e#$i(alent interpolators +EI, blo!s. Ea!h de!i"ator re#$ires antialiasing + H Di + z , , and ea!h interpolator
Kernel filter
Consider now the $se of PI te!hni#$e on a "$lti!hannel "$ltistage filter. &s we said' the de!i"ation part of s$!h a filter will !ontain se(eral downsa"pler blo!s and se(eral de!i"ations +as shown in ;ig$re 9,' re#$ires antii"aging filtering + H Ii + z , ,. then the Kernel filter and finall% an interpolation part whi!h is analogo$s to the de!i"ation part. )oth !hannels sho$ld be identi!al. 5. /ULTIT&GE ;ILTER AITH ;>G& CO/>ONENT O$r filter !an be realied b% "erging the pairs of identi!al filters with PI te!hni#$e of both !hannels +the oftware tools for designing str$!t$res that digital first de!i"ation filter of the first !hannel and the first progra""able hardware "an$fa!t$res pro(ide for $sers de!i"ation filter of the se!ond !hannel' the se!ond in re!ent %ears be!o"e "ore !o"plete and offer "ore and de!i"ation filter of the first !hannel and the se!ond "ore freedo" in design. The $ser has the option to !hoose de!i"ation filter of the se!ond !hannel' et!.,. whether to $se one of the sol$tions fro" a wide range of Obser(ing the str$!t$re between two ada!ent filters for"s for a spe!ifi! digital str$!t$re' or design will start res$lting fro" operations related to the PI pro!ess and fro" the beginning' $sing the original "an$fa!t$rer tool' operation of red$!ing the fre#$en!% +;ig$re 5,' we noti!e or $se so"e of the standard tools' in!l$ding Matlab 01' that the algorith" that des!ribes the de!i"ation part of 11 and then again (ia software "an$fa!t$rers "ae a "$ltistage filter +we will na"e it ED' i.e.' the e#$i(alent !o"pilation in FHDL' Ferilog HDL' or the Ferilog !ode. de!i"ator, is3 Ae ha(e !onsidered realiation of "$lti!hannel narrowband low-pass filter with the Xilin !P"# $irte% - a(e two !onse!$ti(e sa"ples' series $sing "$ltistage and PI realiation as shown in 9 + 1 , ∗ M − ;ig$re . The filter is i"ple"ented $sing Milin*s %ste" - Re"o(e the following sa"ples' K Generator 019. Ae defined blo!s of e#$i(alent - Repeat the pro!ess $ntil the last sa"ple e*pires. de!i"ators and e#$i(alent interpolators as s$bs%ste"s' and "ade "$ltiple $se of the". ;ig$re 7 shows the pro!ess of the Kernel filter sele!tion' $sing the !D# t&&l . i"$lation "odel of the entire filter fro" ;ig$re will be $sed for all i"ple"entations' !hanging the Kernel filter' de!i"ationinterpolation filters realiation and the (al$es for the fa!tor M . ;or the beginning' Kernel filter will be i"ple"ented as an IIR ellipti! filter with !oeffi!ients3 ;ig$re 5 One degree of "$ltistage filter +E#$i(alent De!i"ator,
0N'wnPellipord+ .915
"ethods +in!reasing of fa!tor M or $sing a "ore !o"ple* Kernel filter, will res$lt in in!reasing the !o"ple*it% of the o(erall "$ltistage filter. It will be a se(enth-order filter' with fo$r se!ond-order $!h in!rease in the filter !o"ple*it% !a$ses in!reasing se!tions' realied as Dire!t-;or" II Transposed se!tions. of errors related to a fi*ed-point i"ple"entation. This is The e*a"ple fro" ;ig$re shows a "$ltistage realiation !onfir"ed b% ;ig$re <' whi!h shows the re!orded in three stages +for larger (al$es of fa!tor M ' "$ltistage a"plit$de !hara!teristi!s of "$ltistage filter' with the filtering is re!o""ended,. Therefore' the operations of "entioned ellipti! IIR Kernel filter and for fa!tor M = C de!i"ation and interpolation for this e*a"ple are also . perfor"ed in three stages +blo!s De!i"ation1' Ae !an a(oid s$!h an effe!t and i"pro(e filter De!i"ation9Q..QInterpolation5,. ;or ea!h de!i"ation !hara!teristi!s b% sele!ting a Kernel filter with enhan!ed +interpolation, operation we need additional filters to !hara!teristi!s in ter"s of low stop-band sensiti(it%' and a s$ppress aliasing +blo!s De!i"ation1' De!i"ation9 and Kernel filter whi!h at the sa"e ti"e retains good De!i"ation5, and to re"o(e i"aging !o"ponents +blo!s !hara!teristi!s in ter"s of bandwidth and transition one. Interpolation1' Interpolation9 and Interpolation5,. In &s we now' the "ini"$" stop-band atten$ation in the ;ig$re ' two inp$t signals are !o"bined b% $sing >I obser(ed range of the o(erall filter is deter"ined fro" the te!hni#$e' for filtering two signals with the sa"e filter. stop-band atten$ations of the Kernel filter' interpolation &!!ording to this' at the last stage of the filter' signal and de!i"ation filters 01. The "ain role of separation is perfor"ed with down sa"ple blo!s and de!i"ationinterpolation filters is not to ens$re the stopdela% ele"ents. band atten$ation of the o(erall filter b$t to s$ppress Aith an in!reasing downsa"ple$psa"ple fa!tor M ' aliasing in de!i"ation' and to re"o(e i"aging in or with narrowing of the Kernel filter we !an de!rease the interpolation. Therefore' we will perfor" a "odifi!ation filter bandwidth of "$ltistage filter to the li"its. On the of Kernel filter whi!h has the !r$!ial infl$en!e on stopother hand' with the in!rease in fa!tor M ' the n$"ber band atten$ation. of degrees to whi!h we "$st di(ide the filter grows. )oth
;ig$re ;>G& realiation of "$ltistage filter
design of lower order filters. Ae will perfor" the "odifi!ation grad$all%' going fro" si"pler sol$tions whi!h prod$!e slight i"pro(e"ents to the "odern realisations that pro(ide signifi!ant i"pro(e"ents in ter"s of red$!ing errors in stop-band and transition one. B. >&R&LLEL &LL->& RE&LIJ&TION O; K E'(E ;ILTER Ae ha(e started Kernel filter "odifi!ation b% $sing parallel str$!t$res with all-pass se!tions as shown in ;ig$re 6. &t the beginning' for realiation of all-pass se!tions' we $sed a sol$tion with a !as!aded latti!e realiation +;ig$re 8,' with filter spe!ifi!ations identi!al to IIR filter sol$tion3 0N'wnPellipord+ .915
;ig$re 8 Cas!aded latti!e realiations of all-pass se!tions It will be a se(enth-order filter' with three se!ond-order and one first-order all-pass se!tion. Co"pared with other realiations +for e*a"ple the e#$iripple ;IR filter wo$ld be a <-th order and the )$tterworth IIR filter a 97-th order, we ha(e a!hie(ed so"e i"pro(e"ents in ter"s of red$!ing the !o"ple*it% of the filter. &t the sa"e ti"e' ;ig$re < Chara!teristi!s of "$ltistage filters with ellipti! o$r filter has a sharp transition one and the !$t-off IIR Kernel filter fre#$en!% + * = L'9Cπ . o' we !an !hoose this filter as a Kernel and for lower (al$es of the fa!tor M tr% to &fter these "odifi!ations and after $sing s$!h a Kernel a!hie(e a filter with "ore stringent !hara!teristi!s. filter in a "$ltirate str$!t$re fro" ;ig$re ' we e*pe!t If we $se this filter as a Kernel in the "$ltistage PI "ini"al errors d$e to a fi*ed point-i"ple"entation str$!t$re whi!h is shown in ;ig$re for a fa!tor M , = C Instead of $sing read%-"ade sol$tions offered b% a software tool' we ha(e to design this filter fro" the ' and perfor" re!ording of a"plit$de !hara!teristi!s' we beginning' $sing so"e of the te!hni#$es that enable will get the !hara!teristi! ill$strated in ;ig$re 1. &s we !an see' we obtained so"e i"pro(e"ents +still not
s$ffi!ient, in ter"s of red$!ing stop-band and transition tapped !as!aded inter!onne!tion of identi!al all-pass s$bone errors. It "eans that we are on the right wa% and filters' and espe!iall% with filters whi!h are denoted as that we sho$ld !ontin$e with "odifi!ations of all-pass Class II and Class III !o"ple"entar% filter pairs 01' 17. parallel str$!t$red filters. Ae ha(e to find an ade#$ate Aith this propert%' we !an realie a Kernel filter as a lowrealiation' and "ini"ie the n$"ber of arith"eti! pass bran!h of these filters +we will $se Class III operations. Onl% in this wa% we will a!hie(e f$rther !o"ple"entar% filter pairs,' and tr% to red$!e stop-band i"pro(e"ents of the filter !o"ple*it% and "ini"ie and transition one errors. errors in the stop-band and transition one. & Class III filter pair [ H P + z ,] [ H HP + z ,] satisfies both the all-pass !o"ple"entar% and the "agnit$de-!o"ple"entar% properties. The "agnit$de response of [ H HP + z ,] os!illates in an e#$iripple "anner in the range [ 1 1 - δ ] in the pass-band' and in the range [.- δ ] in the stop-band' δ 0 1. - # s / 2. is the "ini"al stop-band atten$ation. The low-pass transfer f$n!tions is gi(en b%
# s
H P ( z ) =
∑ a P ( n )[ #. ( z )]
n
[ #1 ( z )] −n '
n =.
+1, where is an integer that is two ti"es an odd integer' and #. ( z ) and #1 ( z ) are the all-pass filters. The tap ;ig$re 1 Chara!teristi!s of "$ltistage filters with Kernel filter as !as!aded latti!e all-pass se!tions . LOA->& ;ILTER RE&LIJ&TION AITH T&>>ED C&C&DED INTERCONNECTION O; TAO IDENTIC&L &LL->& ;ILTER
This se!tion is a re(iew of a tapped !as!aded inter!onne!tion of two identi!al all-pass filters realiations. Class III tapped !as!aded all-pass filters design allows the transfor"ation of one fre#$en!% a*is into another' or generall%' the transfor"ation of one !o"ple* plane into another. This approa!h is $sed in order to find an opti"al Kernel filter for a "$ltistage realiation. .1. >roperties of a low-pass digital filter i"ple"ented as a tapped !as!aded inter!onne!tion of two identi!al all pass filters This approa!h was first introd$!ed in filter design b% ara"i and Renfors 015' and later e*tended to the !o"ple"entar% filter pairs b% ara"i and /ili@ 01B. The o(erall filtering tas is shared between two filters3 a (er% low order ;IR protot%pe ! P ( ω ) ' and a (er% low order ellipti! "ini"al S-fa!tors +E/S;, IIR protot%pe " P ( z ) . The role of ;IR protot%pe is to pro(ide the re#$ested passbandstopband ripple' whereas the IIR protot%pe pro(ides the pass-bandstop-band edge fre#$en!ies. &s we !on!l$ded' for a "$ltistage realiation we need a Kernel filter with i"pro(ed !hara!teristi!s in ter"s of low order and low stop-band sensiti(it% +when the fi*ed-point arith"eti! is $sed,. The sensiti(it% is !onsiderabl% red$!ed when $sing the i"ple"entation str$!t$res based on a
(al$es a P [ n] are !onstants of a fa!torable +separable, Lth-order linear-phase half-band filter ! P ( ) whose ero-phase fre#$en!% response is a nonnegati(e f$n!tion of fre#$en!%. The ;IR protot%pe ! e*hibits the sa"e P ( ) "ini"al stop-band atten$ation and !o"ple"entar% properties as the o(erall filter H P ( z ) . &ll-pass filters #. ( z ) and #1 ( z ) are the sol$tions for the all-pass
s$b-filters in the !ase of Class I low-passhigh-pass IIR filter pair [ " )P ( z ) " HP ( z ) ] . ;ilter " P ( z ) defines the IIR protot%pe filter' whi!h has the sa"e transition band as the o(erall filter H P ( z ) . ' &ll-pass filters !onstants α ' α 1 ' β l : l = 9'5...+ + 1, O 9 as well as the orders of
all-pass se!tions will ha(e a de!isi(e infl$en!e on the filter narrowing pro!ess. ;ig$re 11 shows the i"ple"entation of a low-pass bran!h of Class III filter pair' based on for"$la +1, and !onditions related to the following all-pass !o"ple"entar% and the "agnit$de-!o"ple"entar% properties. a P 0 − n = a P 0 n'
a HP 0 n = − a P 0n '
n = L'9'... n = L'9'...
9
9
−1
−1
+9, a HP 0 − n = a HP 0 n'
n = L'9'...
9
−1
Gi(en the "$ltiple $se of IIR identi!al se!tions + " P + z , and " HP + z , ,' and a!!ording to the res$lts of sensiti(it% anal%sis presented in 07' b% red$!ing the orders of all-pass se!tions we will enable a narrowband realiation with red$!ed stop-band errors.
lo!ation of the !$t-off +!rosso(er, fre#$en!% ω * and the% !an be !o"p$ted dire!tl% b% $sing for"$lae α = − !os+ω * ,
+,
.9. I"ple"entation of Class III digital filter pairs Now we will realie a Kernel filter as a low-pass Class filter. This t%pe of filter is not offered as a !o"plete sol$tion to a software "an$fa!t$rer +lie !I' *&mpiler K ,' and we will b$ild s$!h a filter fro" the beginning' ele"ent b% ele"ent' and after that' we will in!l$de this filter as the Kernel filter in the o(erall str$!t$re fro" ;ig$re . Ae start the design fro" a low order start-$p E/S; + + filter " P + z , + " HP + z , ,. In o$r e*a"ple' we !hoose se!ond-order all-pass se!tions and the desired 5-d) !$t-off fre#$en!% ω * . Note that the 5d)-!$toff is the !rosso(er fre#$en!% for the !o"ple"entar% filter pair. Then' we will shift ω * to a lower and lower (al$e in order to e*a"ine possibilities for a low-pass realiation. The algorith" for !o"p$ting !onstants α ' α 1 ' β l : l = 9'5...+ + 1, O 9 III digital
of the new E/S; filter " )P + z , + " HP + z , , follows fro" the propert% of the E/S; transfer f$n!tion that its -plane poles are pla!ed on the !ir!le orthogonal to the $nit !ir!le and !entred on the real a*is 01<'16'18. Ahen the !enter of the !ir!le approa!hes infinit%' the !ir!le degenerates into the i"aginar% a*is' and a half band filter with the poles on the i"aginar% a*is is obtained. This propert% !an be $sed to !on(ert a start-$p + + E/S; filter " P + z , + " HP + z , , with the !rosso(er fre#$en!%
+
ω *
" HP + z , whose
to a new E/S; filter
" )P + z , +
!rosso(er fre#$en!% ω * !an be !hosen
arbitraril% in the range L ≤ ω * ≤ π . The de(elop"ent of t$ning for"$lae is based on the low-pass to low-pass transfor"ation 09. Therefore' the + + $nit dela% z −1 in " P + z , + " HP + z , sho$ld be repla!ed with the all-pass f$n!tion 3 + z , =
z −1 − g 1 − gz −1
3 + z ,
gi(en b%
1
=
+1 − 1 − α 9 ,' α ≠ L .
+7,
α
In order to deri(e an e*pression for !o"p$ting !onstants β l : l = 9'5...+ + 1, O 9 in the se!ond-order all-pass se!tions' we introd$!e +5, and +B, in the o(erall representations of #L + z , and #1 + z , + +1, O 9
∏
#L + z , =
β l + α +1 + β l , z −1 + z −9 1 + α +1 + β l , z −1 + β l z − 9
l = 9' B..
+
α 1 + z −1 −
1 + α 1 z 1
+ +1, O 9
∏ l = 5'C..
β l + α +1 + β l , z −1 + z −9 −
−9
1 + α +1 + β l , z 1 + β l z
+
β l + + λ
β l =
+ l
β λ + 1
' l = 9'5...
+ 1
9
+6, where λ =
+α 1+ , 9
+
α 19
+8,
1 − +α 1+ , 9 α 19
Here' the s$pers!ript T is $sed to denote the !onstants of the start-$p filter pair. In a spe!ial !ase for the start-$p filter pair being a half+ + band filter pair' i.e.' 0" P P + z ,' " HP + z , H4 H4 0" P + z ,' " HP + z , ' the !rosso(er fre#$en!% is pla!ed
'
in the "iddle of the band'
+5, where g =
α 1
+
ω *
H4
= ω *
= π 9 . In that
!ase α 1+ = α 1H4 = L and e*pression +6, si"plifies to + * + *
sin0+ω
+ ω * , O 9
sin0+ω
+ ω * , O 9
.
+B,
H4 9 l 1
β α + 1
' l = 9'5...
+ 1
9
+1,
The new E/S; filter pair " )P + z , + " HP + z , , will e*hibit the pass-bandstop-band beha(io$r of the start-$p + + filter pair " P + z , + " HP + z , . The onl% differen!e is the !rosso(er fre#$en!% "o(ed fro"
β l =
β l H4 + α 19
+
ω *
to ω * .
&ll-pass !onstants α and α 1 are deter"ined b% the
Lets loo at the res$lts of the i"ple"entation for this Class of low-pass digital filters. To do this' we start fro" a filter with spe!ifi!ations si"ilar to the spe!ifi!ations of the filter fro" the pre(io$s se!tion + ω * = .625π ' ω * =
.61π ' ω * = .6.75π
;or !hosen se!ond order all-pass se!tions' we will find a lowω * =
.6.8 π ,.
pass Class III filter with the lowest !rosso(er fre#$en!% ω * that !an be realied. Then' we will in!l$de s$!h a "a*i"all% low order filter in the "$ltirate "$ltistage str$!t$re fro" ;ig$re . The para"eters of low-pass Class III filter for ω * = .61π are shown in Table 1. The linear-phase ;IR protot%pe filter is of the order = 8 and has "ini"al stop-band atten$ation # s!I' = 8. d4 . The IIR protot%pe filter is of the order = 9 with the "ini"al stop-band atten$ation of # sII' = 1%659d4 . ;or Class III low-pass filter i"ple"entation' with !rosso(er fre#$en!% ω * = .61π ' the first-order se!tions #1 ( z ) and se!ond-order se!tions #. ( z ) are3
#. ( z ) =
'
.6:78;977:5 + 167:5.79;1. z −1 + z −2 1 + 167:5.79;1. z − + .6:78;977:5 z − 1
2
+11,
#1 ( z ) =
- .67285%252 : + z −1 1 - .67285%252: z −1
'
+19, with
tap
!oeffi!ients
a P [ . ] 6666a P [ 8 ]
and
a HP [ . ] 6666a HP [ 8 ] $sed fro" Table 1. The o(erall filter e*hibits the sa"e "ini"al stop-band atten$ation as the protot%pe ;IR filter.
;ig$re 11 I"ple"entation str$!t$res of Kernel filter
F) filter prototype (tap coefficients!
aL>0 P a H>0 P aL>07 P a H>07 P .5B<<<09 P a H>09 P aL>0B P a H>0B P .96B9<<05 P aH>05 P . aL>01 P aH>01 P aL>0 P aH>0P ) filter prototype parameters
*r&ss&ver ? * .61 @ passband edge ? p .6.%%5. @ st&pband edge ? s .6119.1 @ passband ripple # pII' .6158 d4 minimal st&pband atten,ati&n 1%659 d4 # sII' 0econd order all-pass filter constants A1 B.67285%252 A B.6;51.58518 2 .678;9775
Table 1 Kernel para"eters of filter for
ω * =
str$!t$re whi!h is shown in ;ig$re ' and again perfor" re!ording of a"plit$de !hara!teristi!s' we will get the !hara!teristi! ill$strated in ;ig$re 19. &s we !an see' we obtained signifi!ant i"pro(e"ents in ter"s of red$!ing stop-band and transition one errors !o"pared with the ellipti! Kernel filter realiation. Gi(en these res$lts' we !an !ontin$e with a Kernel filter "odifi!ation i"ple"ented in this "anner' and realie a filter with a lower !$toff fre#$en!%. The !hara!teristi!s of Class III low-pass Kernel filter for ω * = .625π ' ω * = .61π ' ω * = .6.75π and PI
.61π
;or other (al$es of ω * ' tap !oeffi!ients will eep the sa"e (al$es' while the all pass filter !onstants will !hange depending on the sele!ted fre#$en!%. If we now $se this filter as a Kernel in the "$ltistage
ω * =
.6.8 π are shown in ;ig$re 15. &s we !an see'
we ha(e the possibilities for an additional red$!tion of !$toff fre#$en!%' before the appli!ation of "$ltistage te!hni#$e. Ae ha(e to note that' d$e to errors' the !hara!teristi!s of a filter in whi!h ω * = .6.8 π and s"aller +a dashed line, "$st be ree!ted fro" the f$rther i"ple"entation of "$ltistage te!hni#$es. This also "eans that b% $sing a low-pass bran!h of Class III !o"ple"entar% filter pairs' witho$t the appli!ation of "$ltistage te!hni#$es' filters !an be narrowed down to a "a*i"$" of ω * = .6.75π for Xilin !P"# $irte% series and 59-bit pre!ision.
Kernel filter
7. /ULTIT&GE LOA->& )R&NCH O; C # III CO/>LE/ENT&RY ;ILTER >&IR AITH ;>G& CO/>ONENT
Ahen we ha(e e*a"ined the i"ple"entation of a single Class III low-pass digital filter' we !an now in!l$de these filters in the !o"ple* str$!t$re of the "$ltistage filter realied for two !hannels in the PI te!hni#$e +;ig$re 9,. Ha(ing in "ind that we ha(e alread% realied a Kernel filter for a (er% s"all V !' and that the de"ands for the Kernel filter are alread% stri!t eno$gh' we will $se the de!i"ation-interpolation fa!tor of low (al$e + M = 2 ' M = 9 and M = % , and i"ple"ent the filter $sing onl% one or two stages. In this wa% we a(oid the errors that o!!$rred in the passband whi!h wo$ld be !a$sed b% the large (al$e of the M fa!tor for (er% narrow bandwidth filters.
;ig$re 15 Chara!teristi!s of Class III low-pass filters for ω * = .625π ' ω * = . 1π ' ω * = .6.75π and ω * =
.6.8 π
;ig$re 19 Chara!teristi!s of Kernel filter realied as low-pass bran!h of Class III low-pass filters
;ig$re 1B Chara!teristi!s of o(erall "$ltistage filters for ω *& = .625π / % ' ω *& = .61π / 2 and ω *& = .675π / 2
;ig$re 1B shows the !hara!teristi!s of3 a.
/$ltistage filter with M = 2 ' Kernel filter !$toff fre#$en!% is ω * = .6.75π +for
o(erall filter ω *& = .6 .975π ' b. /$ltistage filter with M = 2 ' Kernel filter !$toff fre#$en!% is ω * = .61π +for o(erall !.
filter ω *& = .6.5π ,' /$ltistage filter with M = 2 ' Kernel filter !$toff fre#$en!% is ω * = .625π +for o(erall
.6.825π ,. filter ω *& =
Ae !an see fro" ;ig$re 1B that we !an !onsider the !ase b + ω *& = .6.5π , as an a!!eptable sol$tion. This is so be!a$se in the !ases with stri!ter re#$ire"ents in ter"s of bandwidth +!ase !, the error in transition one is too high.
This "eans that the de!ision in !hoosing the filter sho$ld be the res$lt of a !o"pro"ise between the re#$ire"ents for the width of the transition one filter' the a"o$nt of a(ailable hardware reso$r!es' as well as the sie of the de(iations in pass-band that res$lt fro" in!reasing the n$"ber of stages. <. CONCLUION
The appli!ation of "$ltirate "$ltistage te!hni#$e !o"bined with pipelininginterlea(ing te!hni#$e pro(ides a sol$tion for the rationaliation of hardware reso$r!es for appli!ations that in(ol(e or re#$ire the appli!ation of the sa"e highl% narrow-band filters se(eral ti"es or hardware str$!t$re that !an be rearranged so that the signal pro!essing is perfor"ed in parallel. Co"bining the operations related to the i"ple"entation of PI pro!ed$re together with the operation related to the filter realiation' !an res$lt in additional i"pro(e"ents of filter str$!t$re and the rationaliation of hardware reso$r!es. In addition to the rationaliation of hardware reso$r!es' we a!hie(e a red$!tion of the n$"ber of arith"eti! operations and' a!!ordingl%' a red$!tion of errors in the stop-band and transition one for (er% narrowband filters. These errors are !a$sed b% the in!reased n$"ber of arith"eti! operations for the fi*ed-point i"ple"entation with ;>G& !hips. ;ig$re 1 Chara!teristi!s of o(erall "$ltistage filter for ;$rther i"pro(e"ents in ter"s of narrowing the filter V!P . W bandwidth !an be a!!o"plished with an additional red$!tion of the n$"ber of arith"eti! operations and b% In ;ig$re 1' a dashed line shows the !hara!teristi!s of red$!ing filter !o"ple*it%. One s$!h effe!ti(e the Class III theoreti!al filter for ω *& = .6.5π ' i"ple"entation !an be "ade with a Kernel filter obtained witho$t the fi*ed-point i"ple"entation' and a "odifi!ation as Class III !o"ple"entar% filter pairs. In solid line shows the !hara!teristi!s of the Class III real this wa% it is possible to realie a low-pass +high-pass, filter i"ple"ented with PI and "$ltistage te!hni#$e with filter with bandwidth less than a 19 of the sa"pling fre#$en!%' and witho$t worsening the narrow transition the fi*ed-point i"ple"entation. Ae !o"e to a !on!l$sion that we !an realie a low- one. Aith software !apabilities offered b% a !hip pass filter +;ig$re 1, with "a*i"$"all% stri!t !hara!teristi!s gi(en in Table 9' b% appl%ing all the "an$fa!t$rer' we !an b$ild fro" this str$!t$re a general pre(io$sl% "entioned te!hni#$es' and b% $sing this sol$tion +I> CoreK,' and $se it as a tool for designing narrow-band digital filters. hardware. o we !an also !on!l$de that the proble" of the R E;ERENCE realiation of narrowband filters for "ore stringent re#$ire"ents now "o(es fro" stop-band to the transition one. 01 /ili! L. /$ltirate filtering for digital signal pro!essing3 /&TL&) ;$rther filter i"pro(e"ents !an be a!hie(ed b% appli!ations. Hershe%' >&3 Infor"ation !ien!e Referen!e: 98. in!reasing the Kernel filter order' the n$"ber of stages or 09 J. iang and &. N. Aillson. Effi!ient digital filtering ar!hite!t$res $sing pipelininginterlea(ing. IEEE Trans. Cir!$its %st.' (ol. BB' no. b% in!reasing the !odeword length. )$t all these "ethods 9' pp. 11-116' ;ebr$ar% 188B. will res$lt in a f$rther signifi!ant in!rease in spent 05 J. iang' &.N. Ailson. & pipelinedinterlea(ed IIR digital filter ar!hite!t$re. &!o$st. pee!h ignal >ro!ess. 5 +188<, 991
*r&ss&ver ? * passband edge ? p st&pband edge ? s passband ripple # p minimal st&pband atten,ati&n # s
.6.5 @ .6.%2225 @ .6 .5;15.5 @ .61 d4 8. d4
Table 9 >ara"eters of o(erall "$ltistage filter
0B Ra"stad' T.&.' ara"i' T. +1866' $ne,. /$ltistage' "$ltirate ;IR ;ilter str$!t$res for narrow transition-band filters. Pr&*6 1; IEEE Int6 >mp6 Cir*,its and >stems X IC# ' 918 X 999. 0 Ra"stad' T.&.' ara"i' T. +188' /a%,. /$ltistage' "$ltirate ;IR filter str$!t$res for narrow transition-band filters. Pr&*6 1;;. IEEE Int6 >mp6 Cir*,its and >stems X IC# ' New Orleans' Lo$isiana' 91< X 991. 07 L. /ili@ and . ?erti@. In(estigation of !o"p$tationall% effi!ient !o"ple"entar% IIR filter pairs with t$nable !rosso(er fre#$en!%. Int. . Ele!tron. Co""$n. +&EZ, 7 +911, B18XB96. 0< alo" I/' Todoro(i@ DJ' /ili@ LD. The infl$en!e of i"p$lse response length and transition bandwidth of "agnit$de !o"ple"entar% !rosso(ers on per!ei(ed so$nd #$alit%. &$dio Eng o! 9<: 738B1XB. 06 Cassid% R' "ith O. & t$nable' nons$bsa"pled' non-$nifor" filter ban for "$lti-band a$dition and le(el "odifi!ation of a$dio signals.
In3 Conferen!e of the thirt%-eight &silo"ar !onferen!e on signals' s%ste"s and !o"p$ters. 9B. p. 9996X59 08 /. Ciri! and F. Radoni!. Realiation of /$ltistage ;IR ;ilters $sing >ipelining-Interlea(ing. TEL;OR o$rnal' Fol. B' No.9' pp. 1<-11' 919. 01 ignal pro!essing toolbo* for $se with /&TL&). User[s g$ide' The /athAors In!.' 5 &pple Hill Dri(e' Nati!' /&' 97. 011 ;ilter design toolbo* for $se with /&TL&). User[s g$ide' The /athAors In!.' 5 &pple Hill Dri(e' Nati!' /&' 97. %ste" Generator for D>' release 1.1' /ar!h' 91' www.*ilin*.!o". 019 %ste" Generator for D>' release 1.1' /ar!h' 91' www.*ilin*.!o". 015 ara"i' T.' Renfors' /. +186<,. & no(el approa!h for the design IIR filters as a tapped !as!aded inter!onne!tion of identi!al allpass s$bfilters. >ro!. 186< IEEE Int. %"p. Cir!$its %st.' IC& 186<. 9' 798759. 01B /ili@ L' ara"i T. Co"ple"entar% IIR filter pairs with an ad$stable !rosso(er fre#$en!%. In3 >ro!eedings of IEEE Nordi! signal pro!essing s%"posi$". 99. p. 7. 01 /ili@' L. D.' ara"i' T. +95' b,. >ower-!o"ple"entar% IIR filter pairs with an ad$stable !rosso(er fre#$en!%. ;a!ta Uni(ersitatis' er.3 Ele!. Energ. 17+5,' 985B. 017 /ili@' L. D.' ara"i' T. +95' a,. Three !lasses of IIR !o"ple"entar% filter pairs with an ad$stable !rosso(er fre#$en!%. >ro!. IEEE Int. %"p. Cir!$its %st. IC& 95' B' 1BX1B6. 01< /ili! L' L$to(a! /. Effi!ient algorith" for the design of high-speed ellipti! IIR filters. &EZ Int Ele!tron Co""$n 95:<39X79. 016 /ili! LD' L$to(a! /D. Design of "$ltiplierless ellipti! IIR filters with a s"all #$antiation error. IEEE Trans ignal >ro!ess 1888:B<3B78X<8. 018 L$to(a! /D' /ili! LD. Design of !o"p$tationall% effi!ient ellipti! IIR filters with a red$!ed n$"ber of shift-and-add operations in "$ltipliers. IEEE Trans ignal >ro!ess 188<:B39B99X5. 09 Constantinides &C. pe!tral transfor"ations for digital filters. >ro!eed IEEE 18<:11<316X8.
