INTERPOLATION OF LOW-ORDER HRTF FILTERS USING A ZERO DISPLACEMENT MEASURE

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1 INTERPOLATION OF LOW-ORDER RTF FILTERS USING A ZERO DISPLACEMENT MEASURE PACS: P hbboglu, usey So Arts Reserh Cetre (SARC) Shool of Computer See Quee s Uversty Belfst Belfst, BT7 NN, UK Tel: ++44 () Fx: ++44 () E-ml: h.hhbboglu@qub..uk ABSTRACT e-relte trsfer futos (RTF) re geerlly mesure t srete zmuth elevto vlues. Cotuous moto of vrtul sou mges oly be obte by terpolto of RTFs for termete zmuth elevto vlues mmersve uo system. RTFs be moelle s ombto of low-orer mmum-phse fte mpulse respose (FIR) flters ostt ely le. A ew metho for terpolto of FIR flter zeros usg proxmty-bse vetorl splemet mesure s propose ths work. INTRODUCTION e-relte trsfer futo (RTF) efes the spetrl shpg of the sou sgl o ts wy from soure loto the free fel to the exterl er. RTFs re geerlly mesure t srete postos o spherl gr equstt from etre of the hum he (or mequ) whose mesuremets re beg tke []. These mesuremets prove set of RTFs t srete vlues of zmuth elevto. The RTFs prove frequey epeet futo of terurl testy fferee (IID) ostt terurl tme ely (ITD). The RTF t obte from these mesuremets re use for esgg low-orer gtl flters to be me vlble for use rel-tme uo sptlsto ppltos. A umber of fferet pprohes regrg the RTF flter esg proess hve bee propose [2][3][4]. A vrtul utory sply be esge usg the RTF flters. For the vrtul utory sply to be semless oe, termete postos betwee the tlly mesure postos hve to be sythesse s well. Ths sythess be osere, s terpolto rther th extrpolto s RTFs spg omplete smple spe of fferet sou soure lotos re lrey vlble. Ths terpolto tsk requres smooth trsto of the spetrl shpe of the RTF. Ths volves both the preservto of burl ues tmbrl propertes of the utory evet. Smooth trsto of flters ebles more relst moto more urte postog of the vrtul sou mges. I ths pper, bref revew of the RTF flter esg wll be gve frst. Seoly, RTF flter terpolto for FIR flters wth vetorl zero-splemet mesure wll be presete followg quk summry of other terpolto methos reporte the lterture. A soluto to geerl problem use by the rel zeros of the system wll be the gve.

2 RTF FILTER DESIGN It s essetl to smooth the mgtue spetr of the RTF t before the flter esg proess orer to obt low orer flters. Dfferet methos for spetrl smoothg lue smoothg wth gmm-toe flterbks [5], utory weghtg [6], retgulr wowg [7] smoothg wth wvelet trsforms [2]. After the smoothg proess, vrous geer methos be use to esg fte mpulse respose (IIR) or fte mpulse respose (FIR) flters from smoothe mgtue respose. The FIR flters use ths work were obte from the KEMAR RTF t [8][9] usg the pplto of two methos. The mgtue resposes of the RTFs were smoothe usg the reut wvelet trsform [2]. The, mmum-phse represetto of the resultg futo ws obte usg rel epstrum lyss [] the tme-om. The frst 64 smples of the resultg mmum-phse sgl were retly use s the FIR flter oeffets resultg 64 th orer FIR flters. Although the flter orer ws slghtly hgher th prevously reporte reserh, ths prove uform spg of flter zeros (see Fg. ) resg terpolto ury. Therefore, t ws more esrble to use 64 th orer FIR flters for ths purpose. t r P y r g m I x ) B ( e u t g M Rel Prt Normlze Frequey () (b) Fg. () The pole-zero plot of the RTF FIR flter o the horzotl ple t 45 zmuth. (b) The mgtue respose of the sme RTF flter. RTF FILTER INTERPOLATION USING A ZERO-DISPLACEMENT MEASURE The mjorty of the prevously reporte terpolto methos use the RTFs o the futol level rther th elg wth flter roots [][2][3]. These methos utlze ler [][2], sple [] or trgulr [3] terpolto to terpolte ew mgtue spetrum betwee two or three mgtue spetr esg ew flter from the terpolte mgtue respose. A more omplex pproh [4] uses ombto of RTFs to erve geerl sptl frequey respose surfes to obt terpolte RTFs. These pprohes prove y ombto of two RTFs, but leve the bure of esgg ew flters from the terpolte RTFs. Ths woul use ffultes for rel-tme system, whh woul ymlly terpolte betwee two sou soure postos. The pproh ths work tkes the RTF terpolto problem s flter-morphg problem, elg retly wth the zeros of the FIR flters. The terpolto metho propose ths pper ssumes the sme umber of omplex rel zeros of both flters to be terpolte. Exessve rel zeros re therefore ouple substtute wth equvlet omplex zeros. The followg ervto s use for ths tsk: + ( z! )( z! b) = ( z! )( z! *), = p + j!,, b, p! R, where b re postve rel zeros of the flter, * eotes the omplex ojugte operto ε s suffetly smll. Whe the equto bove s evlute t z= (.e. ω=), p s fou s:

3 p =! (!! b + b) Ths operto gurtees tht the substtute zeros re se the ut rle, ot sturbg the mmum-phse property of the RTF flter. The mxmum error the mgtue spetrum s less th.3b whe ε s selete s.. The sme operto lso be pple to ouple of zeros tht re both egtve, whh wll proue ew egtve qus-omplex zeros. The umber of rel zeros ws erese to 2 for ll of the FIR flters. After ths reuto, t s possble to represet the FIR flters s ombto of 62 omplex zeros 2 rel zeros: 3 & 2 & # #! ' = ' R = $ ( ( z ± z ( ))( z ± z ( )*) $ ( ( z ± z r ( j))! % = " % j= " The seo prt of the equto bove refers to seo orer flter, whh hs hghpss, lowpss or bpss hrtersts epeg o the sgs of the rel zeros. Ths seo orer flter mposes fferet terpolto problem, so t s trete vully. The omplex zeros of the FIR flters to be use the terpolto re sorte zero splemet vetor s lulte s: r µ ) = z ( ) z ( ), =,2,..., 62 (! 2 where!( z ( )) >!( z ( + ))!( z2 ( )) >!( z2 ( + )) (.e. the omplex zeros re sorte org to ther gles wth moulo π). Lotos of vul omplex zeros be lulte usg ths zero splemet vetor s: z ( ) = z ( )! k( ) µ ( ) = (! k( )) z ( ) + k( ) z 2 where k s ether ostt betwee (whh results ler terpolto of the zeros) or eve futo of. The seleto of k epes o the level of smlrty of the terpolte flter to oe of the flters use for tht terpolto. ee, the omplex porto of the terpolte RTF s forme s: ( z) = 3! = ( ) ( z " ( " k( )) z ( ) " k( ) z2 ( ))( z " (( " k( )) z ( ) " k( ) z2 ( ))*) Fgure 2 shows the terpolto proess for the omplex prts of the two fferet RTFs obte by vryg ostt k lerly from to wth tervls of ) B ( e u t g M Normlze Frequey Fg. 2 () Movemet of some of the zeros of the terpolte FIR flter for vryg k, (b) Iterpolte mgtue spetr of the omplex zeros of 45 zmuth (sol le o the top) 55 zmuth (sol le o the bottom) the horzotl ple (wth 4B seprto)

4 It my be observe from the fgure tht, lthough the flter shpe s strogly epeet o ts omplex zeros, the sgft fferees betwee y two oseutve flters re mly ue to ther rel zeros. INTERPOLATION OF TE REAL ZEROS The rel zeros of the flters were terpolte seprtely. The egtve rel zeros of the FIR flter eterme the flter shpe t hgh frequees, whle the postve rel zeros eterme the shpe t low frequees. Preferbly, the rel zeros oul lso be terpolte usg the metho esrbe bove, but whe the two flters ot fferet ombtos of egtve postve rel zeros, ths beomes mpossble. The terpolto metho for the rel prt of the flters els oly wth the hghest lowest gulr frequees (.e. ω= ω=π). The ervto below s for the rel prts of the two RTF flters ( R (z) 2R (z)), oe of whh ots two postve rel zeros the other ots two egtve rel zeros: R 2R ( z) = ( z! )( z! b), ( z) = ( z + )( z + ), + where, b,,! R. The m of the terpolto s to obt ew seo orer flter (.e. D (z)) suh tht: log( D ( z)) = # " (log( R ( z)) + log( 2 < # < D ( z) = ( z! m)( z + p) where λ s ostt. The equtos evlute t z= z=- yel: R ( z))), ((! )(! b)( + )( + )) ((!! )(!! b)(! + )(! + )) "! (! m)( + p) =, "! (!! m)(! + p) = Postve results for the smulteous soluto to these equtos yel m p. Seleto of the vrble λ etermes the smlrty of the terpolte seo orer flter to ether oe of the seo orer orgtor flters t hgher lower frequees. Fgure 3 shows the mgtue spetrum of the terpolte seo orer flter for λ= Fg. 3 Flters forme by the rel zeros t 4 45 zmuth gles o the horzotl ple.

5 RESULTS AND DISCUSSIONS It s possble to ombe the two seprte terpolto methos by ouplg the k prmeter of the frst metho λ prmeter of the seo metho vryg them lerly. Fgure 4 shows the splemet of the omplex zeros the mgtue spetr obte by suh terpolto betwee RTFs of zmuth the horzotl ple Fg. 4 Iterpolto betwee RTF flters for 45 zmuth (sol le o the top) 55 zmuth (sol le o the bottom) the horzotl ple for lerly resg vlues of the ouple vrbles k λ. (plotte wth 5B seprto) It my be observe from the fgure tht the terpolto metho propose ths pper s more lke morphg proess from oe RTF flter to the other. The frequey othes peks, low frequey hgh frequey etls re pture morphe orretly s well. There s suffet evee the lterture tht ler terpolto of TRFs gve suffetly urte results for zmuth gle tervls s lrge s 3 [2]. Ths mkes smoothess of the tmbrl qulty the morphe flters more mportt gol to heve for semless postog of the vrtul sou mges. CONCLUSIONS AND FUTURE WORK I ths work, ew RTF flter terpolto metho for FIR flters bse o the splemet of the flter s zeros ws propose. Ths metho be use for terpolto of the RTF flters wthout the explt formto of the shpe of the mgtue spetr of the orgtor RTFs. The opertos use the ervto gurtee the mmum-phse property of the terpolte FIR flter. Se the terpolto of ITD s rther strghtforwr tsk, t my be possble to use the metho rel-tme mmersve uo pplto. The opportutes for future work lue vestgto of pplblty of the urret metho to smple IIR flters, or ommo-pole IIR flters [3][5] whh woul prove lower flter orer reue omputtol omplexty. BIBLIOGRAPICAL REFERENCES [] F. L. Wghtm D. J. Kstler, ephoe smulto of free-fel lsteg I: Stmulus sythess, J. Aoust. So. of Am., Vol. 85, pp , 989. [2]. hbboglu, B. Guel F. Murtgh, Wvelet-bse Spetrl Smoothg for e-relte Trsfer Futo Flter Desg, Pro. of the AES 22 Itertol Coferee o Vrtul, Sythet Etertmet Auo, Espoo, Fl, Jue 22. [3] Y. e, S. Mko, Y. Ke N. Ktwk, Commo-Aoustl-Pole Zero Moelg of e-relte Trsfer Futos, IEEE Trs. Speeh Auo Pro., Vol 7, No. 2, Mrh 999.

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