Oversampled Wilson-Type Cosine Modulated Filter Banks with Linear Phase

Transcription

1 Pacific Grov (C Nov 6 Ovrsampld WilsonTyp Cosin Modulatd Filtr Banks with Linar Phas Hlmut Bölcski Franz Hlawatsch INTHFT inna Univrsity of Tchnology Gusshausstrass 5/ inna ustria Phon: Fax: mail: holcsk@auroranttuwinacat stract W introduc Wilson filtr anks (WFBs as a nw typ of cosin modulatd filtr anks (CMFBs corrsponding to th discrttim Wilson xpansion WFBs allow linar phas filtrs in all channls W formulat prfct rconstruction (PR conditions for ovrsampld critically sampld WFBs show that PR WFBs corrspond to PR filtr anks with twic th ovrsampling factor Gnralizing WFBs w thn propos th nw family of vnstackd CMFBs allowing oth PR linar phas filtrs in all channls This CMFB family contains WFBs as wll as CMFBs rcntly introducd y Lin aidyanathan Finally aftr xtnding convntional ( oddstackd CMFBs to th ovrsampld cas w formulat unifid PR conditions for oth vn oddstackd ovrsampld critically sampld CMFBs W show that PR CMFBs ar always rlatd to PR filtr anks of th sam stacking typ with twic th ovrsampling factor Introduction Rcnt intrst in ovrsampld filtr anks (FBs [][5] is mainly du to thir incrasd dsign frdom nois immunity [ 3 4] Ovrsampld FBs [6][8] [3] ovrsampld cosin modulatd FBs (CMFBs [][5] [8] ar spcially attractiv as thy allow an fficint implmntation n advantag of CMFBs ovr FBs is th fact that thir su signals ar ral for ral input signals ral analysis prototyp This papr introducs studis a nw (possily ovrsampld CMFB which w call Wilson FB (WFB sinc it corrsponds to th discrttim Wilson xpansion [6] Wilson xpansions ar asd on cosin sin modulation of a prototyp function can constructd to hav good tim frquncy localization sids ing orthonormal [7][] [6] n important advantag (spcially in imag coding applications of WFBs ovr convn Funding y FWF grant P053ÖPH tional CMFBs [][5] is that thy hav linar phas filtrs in all channls if th prototyps hav linar phas Organization of papr ftr th introduction of (possily ovrsampld WFBs in ction ction 3 shows a clos rlation twn WFBs FBs with twic th ovrsampling factor ction 4 provids prfct rconstruction (PR conditions for (possily ovrsampld WFBs shows that PR WFBs can always drivd from PR FBs ction 5 prsnts a nw gnralizd framwork for linar phas CMFBs that contains WFBs a CMFB typ rcntly introducd y Lin aidyanathan [0] Wilson Filtr Banks W considr a WFB with channls dcimation factor Th WFB is critically sampld if ovrsampld if Th analysis FB consists of two partial FBs with impuls rsponss! #"$ rspctivly that ar drivd from an analysis prototyp as ' * + $$/ : :<= BCDC EGF 7 <IH "KJIL + Y CCC M N*OCPRQ UT M for vn F Hr F for odd imilarly th synthsis FB consists of th two partial FBs [ \ [ ] ^ R" drivd from a synthsis prototyp as ' I + _ `/0 4 6 a7 ` `< = c+ Y CCC daf 7 <H5 JL + _ $M $*OCP Q T d X

2 7 7 BCDC Not that thr ar channls ut only diffrnt channl frquncis 7 + BCDC < as dpictd schmatically in Fig (a In particular th + channl is cntrd at frquncy which is a diffrnc from convntional CMFBs (s ction 5 If th analysis prototyp has linar phas i with som w hav W F < < 7 < * + BDC * 7 < * BDC Thus all analysis filtrs hav linar phas as wll imilarly for a linar phas synthsis prototyp all synthsis filtrs hav linar phas This is an important advantag of WFBs ovr convntional CMFBs [][5] (a ( H N N N N N H N H H 0 H 0 H N H N N 3 H N H 0 H H 0 0 N N N 0 N 3 N H N N N H N H H H H H N N N H N H N N Figur Transfr functions of th channl filtrs in (a 3 a WFB or mor gnrally an vnstackd CMFB with channls ( an channl oddstackd CMFB 3 Rlation to Filtr Banks W shall nxt show that thr is a clos rlationship twn WFBs FBs with twic th ovrsampling factor Th asic ida comining positiv ngativ frquncis in a FB to otain a CMFB has n introducd y Dauchis t al [7] in a signal xpansion contxt has also n usd in filtr ank thory for many yars [4] (in filtr ank thory howvr mphasis has n placd on th narpr cas Th inputoutput rlation in a WFB with channls dcimation factor is #"$ ^ 3 ^3 "$ B "$* ( whr B3 B3 " 3 3 " B B * + * + G* + G* + BCD BCD _ CC Y CC Not that ( dscris an xpansion of th rconstructd signal into th synthsis functions this xpansion can altrnativly writtn as R" ^3 B3d ^ ^ B 3 ( imilarly th 3 inputoutput rlation in an vnstackd FB [6] with channls dcimation factor using analysis prototyp $ synthsis prototyp is R" [ = with = " H 3 = with = " H! (3 Hr whr "! #" It can now shown that th WFB s synthsis functions can drivd from th FB s synthsis functions as %$& < (' B3 d BCCD (4 (5 "J (6 whr $ $ " + F has n dfind in ction Indd for th righth sid of (4 is '& 8 R" "*'/ For $: : $ E "0 H5 " " 0 H5 " " N/03 Q UTd 3 " "*(' L L Eq (4 can vrifid in a similar way vrification of (5 (6 is straightforward X

3 + imilarly th WFB s analysis functions can drivd from th FB s analysis functions as 3 d $ < ' 7 Y CCC (7 (8 "J ( " 3 " ' Insrting th rlations (4(6 (7( in ( arranging trms lads to th following important rsult Thorm [] For a WFB th rconstructd signal can dcomposd as 7 < 7 < $ (0 whr th oprators ar dfind as #"$ 7 < ^ 3 I 7 < #"$ < ^ 3 #" 3 7 Not that th FB inputoutput rlation in (3 can writtn as 7 < thus th oprator corrsponds to an vnstackd FB with channls dcimation factor i with twic th ovrsampling factor of th CMFB 4 Prfct Rconstruction Conditions With (0 it can shown [] that a ncssary sufficint condition for PR with zro dlay is ( whr dnot th idntity zro oprator rspctivly For th WFB s inputoutput rlation (0 rducs to 7 < i th inputoutput rlation of an vnstackd FB with channls dcimation factor Thus PR WFBs corrspond to vnstackd PR FBs with twic th ovrsampling factor In particular it can shown [] that paraunitary [4] WFBs corrspond to paraunitary FBs (hr R Th oprators can xprssd in th tim domain as 7 < $ Rd 7 < 7 < $ 7 $ ( < RW 7 < (3 With ( th first PR condition in ( if only if E $ R is satisfid whr is th unit sampl This is quivalnt to th PR condition for a FB with channls dcimation factor i with twic th CMFB s ovrsampling factor In gnral (3 dos not lad to a similarly simpl condition for th scond PR condition In th cas of in tgr ovrsampling howvr i with IN it can shown [] that is satisfid if only if 7 < M 7 < 7 for BDCC sampling is a spcial cas with Th oprators Not that critical can also xprssd in th frquncy in th polyphas domains this lads to corrsponding formulations of th PR conditions in ( [] 5 Unifid Framwork for CMFBs In a WFB all channl filtrs hav linar phas if th prototyps hav linar phas W now introduc a gnralizd framwork for linarphas CMFBs In analogy to vnstackd FBs w call this nw CMFB class vnstackd W show that th class of vnstackd CMFBs contains WFBs th linar phas CMFBs rcntly introducd y Lin aidyanathan [0] usquntly w xtnd th convntional oddstackd CMFBs to th ovrsampld cas finally w prsnt unifid PR conditions that ar valid for oth vn oddstackd CMFBs Evnstackd CMFBs Th analysis FB in an vnstackd CMFB with channls dcimation fac tor consists of two partial FBs [ \B[ ^ #"$ drivd from an analysis prototyp ' $E * + _ $$/ d = BDC $EGF 7 <H "KJIL + : $ ODP T 7 : <! CC imilarly th synthsis FB consists of \B[ B[ ^ #"$ dfind ' " * + $`/ M! = BCD af 7 <H JIL + partial FBs in trms of a synthsis prototyp $ as

4 @ BCC furthrmor!f for ` *OCP T 7 N < Hr w dfin th phass as 3 +! with F for vn F odd Thus vnstackd CMFBs ar paramtrizd For all analysis filtrs hav in trms of th two paramtrs any choic of linar phas if 7 < _ with som imilarly all synthsis filtrs hav linar phas if $ 7 CMFBs hav frquncis cntrd at frquncy < _ Not that vnstackd channls ut only diffrnt channl (+ BDCC with th + channl (s Fig (a spcial vnstackd CMFB is th WFB introducd in ction it is otaind y choosing th paramtrs as = = nothr important spcial cas is th CMFB rcntly introducd (for critical sampling y Lin aidyanathan [0] its gnralization to aritrary ovr sampling is otaind for Oddstackd CMFBs Oddstackd CMFBs ar th traditional CMFB typ prviously dfind for critical sampling [][5] In th gnral cas of an oddstackd CMFB with channls dcimation factor (not that th CMFB is ovrsampld for w dfin th analysis synthsis filtrs rspctivly as o o BCCD K/0 T 7 + $/0 T < o < E! o Hr dnot th analysis synthsis prototyp rspctivly th phass ar dfind as o 3 T + " with this xtnds th phas dfinition givn in [] for th spcial cas of critical sampling Not that th channl frquncis ar in particular th + channl is cntrd at frquncy (s Fig ( disadvantag of oddstackd CMFBs is that th channl filtrs do not hav linar phas vn if th prototyps hav linar phas [][5] Rlation to FBs For oth vn oddstackd CMFBs th following dcomposition of th rconstructd signal (gnralizing (0 can shown [] 7 < 7 < whr th oprators ar dfind as #"$ 7 < ^ 3 3 #"$ 7 < ' ^ 3 #" 3 I (Th oprators in ctions 3 4 ar a spcial cas In th vnstackd cas w hav 3 $ * " 0 H " L * 0 L " H 7 < In th oddstackd cas ar otaind y formally rplacing + with + in th aov xprssions furthrmor o For an vnstackd (oddstackd CMFB th oprator corrsponds 3 to an vnstackd (oddstackd FB [6] with channls dcimation factor this FB has thus th sam stacking typ twic th ovrsampling factor of th CMFB PR conditions W nxt provid unifid PR conditions for ovrsampld critically sampld vn oddstackd CMFBs Ths PR conditions gnraliz th PR conditions for WFBs givn in ction 4 For PR with zro dlay it is ncssary sufficint that [] (4 (Not that ( is a spcial cas For = th CMFB s 7 < which is th inputoutput rlation of th corrsponding FB Hnc PR CMFBs corrspond to PR FBs of th sam stacking typ with twic th ovrsampling factor In particular paraunitary CMFBs can shown to corrspond to paraunitary FBs Th oprators can xprssd in th tim domain as inputoutput rlation rducs to 7 < R (5 7 < 7 < 3 ]` $ (6 with Y 7 < in th vnstackd cas 7 < Y in th oddstackd cas With (5 _ th first PR condition in (4 if $E $ RW is satisfid if only

5 Not that this condition is indpndnt of th stacking typ it is th PR condition for a FB (vn or oddstackd with channls dcimation factor i twic th CMFB s ovrsampling factor Furthrmor for intgr ovrsampling it follows from (6 that is satisfid if only if $# 7 < $# 7 < $ for BDCC whr 7 < in th vnstackd cas 7 < in th oddstackd cas Not Y that critical sampling is a spcial cas with 6 Conclusion W introducd studid Wilson filtr anks (WFBs a nw typ of cosin modulatd filtr anks (CMFBs that corrsponds to th discrttim Wilson xpansion allows oth prfct rconstruction (PR linar phas filtrs in all channls W formulatd PR conditions for ovrsampld critically sampld WFBs w showd that PR WFBs ar associatd to PR FBs Gnralizing WFBs w thn dfind th nw class of vnstackd CMFBs that all hav th dsiral proprty of allowing oth PR linar phas filtrs in all channls Th CMFBs rcntly introducd for critical sampling y Lin aidyanathan wr xtndd to th ovrsampld cas shown to a furthr spcial cas (sids WFBs of vnstackd CMFBs Th convntional ( oddstackd CMFBs wr also xtndd to th ovrsampld cas Finally w prsntd a unifid st of PR conditions that applis to oth vn oddstackd ovrsampld critically sampld CMFBs Ths PR conditions showd that PR CMFBs ar always rlatd to PR filtr anks of th sam stacking typ with twic th ovrsampling factor W not that framthortic proprtis of ovrsampld CMFBs ar discussd in [] Rfrncs [] H Bölcski F Hlawatsch H G Fichtingr Framthortic analysis of filtr anks sumittd to IEEE Trans ignal Procssing F 6 [] Cvtkovi c M ttrli Ovrsampld filtr anks to appar in IEEE Trans ignal Procssing [3] H Bölcski F Hlawatsch H G Fichtingr Ovrsampld FIR IIR filtr anks WylHisnrg frams Proc IEEE ICP6 tlanta (G May 6 vol 3 pp 334 [4] H Bölcski F Hlawatsch H G Fichtingr Framthortic analysis dsign of ovrsampld filtr anks Proc IEEE IC6 tlanta (G May 6 vol pp 404 [5] J E M Janssn Dnsity thorms for filtr anks Nat La Rport 6858 Eindhovn (Th Nthrls pr 5 [6] R E Crochir L R Rainr Multirat Digital ignal Procssing PrnticHall 83 [7] K waminathan P P aidyanathan Thory dsign of uniform paralll quadratur mirror filtr anks IEEE Trans Circuits ystms ol C33 No Dc 86 pp 70 [8] H Bölcski F Hlawatsch Ovrsampld modulatd filtr anks to appar in Gaor nalysis: Thory lgorithms pplications H G Fichtingr T trohmr ds Birkhäusr Jun 7 [] H Malvar ignal Procssing with Lappd Transforms rtch Hous [0] R D Koilpillai P P aidyanathan Cosinmodulatd FIR filtr anks safisfying prfct rconstruction IEEE Trans ignal Procssing vol 40 no 4 pril pp [] R Gopinath C Burrus om rsults in th thory modulatd filtr anks modulatd wavlt tight frams pplid Computational Harmonic nalysis vol 5 pp [] T Q Nguyn R D Koilpillai Th thory dsign of aritrarylngth cosinmodulatd filtr anks wavlts satisfying prfct rconstruction IEEE Trans ignal Procssing vol 44 no 3 pp March 6 [3] T Ramstad J P Tanm Cosinmodulatd analysissynthsis filtr ank with critical sampling prfct rconstruction Proc IEEE ICP Toronto Canada May pp 787 [4] P P aidyanathan Multirat ystms Filtr Banks PrnticHall 3 [5] M ttrli J Kovacvi c Wavlts u Coding PrnticHall 5 [6] H Bölcski H G Fichtingr K Gröchnig F Hlawatsch Discrttim Wilson xpansions Proc IEEE P Int ympos TimFrquncy Timcal nalysis Paris (Franc Jun 6 pp 5558 [7] I Dauchis Jaffard J L Journé simpl Wilson orthonormal asis with xponntial dcay IM J Math nal vol pp [8] H G Fichtingr K Gröchnig D Walnut Wilson ass modulation spacs Math Nachrichtn vol 55 pp 77 [] P uschr Rmarks on th local Fourir ass in Wavlts: Mathmatics pplications ds J J Bndtto M W Frazir Boca Raton (FL: CRC Prss 3 pp 03 8 [0] YP Lin P P aidyanathan Linar phas cosin modulatd maximally dcimatd filtr anks with prfct rconstruction IEEE Trans ignal Procssing vol 4 no Nov 5 pp 5553 [] H Bölcski F Hlawatsch Ovrsampld cosin modulatd filtr anks to sumittd to IEEE Trans Circuits ystms