Audio Engineering Society Convention Paper Presented at the 111th Convention 2001 September New York, NY, USA

Transcription

1 Audio Enginring Socity Convntion Papr Prsntd at th th Convntion 200 Sptmbr 2 24 Nw York, NY, USA This convntion papr has bn rproducd from th author's advanc manuscript, without diting, corrctions, or considration by th Rviw Board. Th AES taks no rsponsibility for th contnts. Additional paprs may b obtaind by snding rqust and rmittanc to Audio Enginring Socity, 60 East 42 nd Strt, Nw York, Nw York , USA; also s All rights rsrvd. Rproduction of this papr, or any portion throf, is not prmittd without dirct prmission from th Journal of th Audio Enginring Socity. Loudspakr Transfr Function Avraging and Intrpolation David W. Gunnss Eastrn Acoustic Works, Inc. Main St., Whitinsvill, MA 0588 USA David.Gunnss@aw.com ABSTRACT Transfr functions of acoustical systms usually includ significant phas lag du to propagation dlay. Whn this dlay varis from on transfr function to anothr, basic mathmatical oprations such as avraging and intrpolation produc unusabl rsults. A calculation mthod is prsntd which producs much bttr rsults, using wll-known mathmatical oprations. Applications of th tchniqu includ loudspakr complx dirctional rspons charactrization, complx avraging, and DSP filtr dsign for loudspakr string. 0 INTRODUCTION Th phas rspons of a loudspakr typically consists of a narly minimum-phas charactristic plus xcss phas lag du to th propagation tim from th sourc to th microphon. Put anothr way, loudspakrs xhibit a rlativly short impuls rspons that dos not bgin at zro tim. This rsults in rapidly wrapping phas plots at high frquncis. Phas wrap can b minimizd during th masurmnt procss by adjusting th offst dlay of th masurmnt systm; but if th objctiv of a sris of masurmnts is to charactriz variations of phas rspons, th rlativ diffrncs in arrival tim must b prsrvd. Invitably, som of th masurmnts will xhibit significant wrapping. Phas wrapping is not ncssarily dtrimntal, in and of itslf. Howvr, it can lad to som mislading rsults whn complx transfr functions ar subjctd to mathmatical oprations such as intrpolation, smoothing, and avraging. Thr ar svral tasks that call for acoustical transfr functions to b intrpolatd, smoothd or avragd. Ths tasks includ complx dirctional rspons charactrization, ral-tim masurmnt and tuning of loudspakr systms, convrsion of data for us in loudspakr modling programs, and th application of complx smoothing for th purpos of frquncy-scald tim windowing. Th algorithm prsntd in this papr was dvlopd as part of a projct to masur and rprsnt th thr-dimnsional complx rspons of loudspakrs. That projct is th topic of a papr by William Hoy and Charls McGrgor []. 0. Complx Polar Rspons A sourc s dirctionality is dirctly rlatd to th physical shap and siz of th wav front it mits. A point sourc has no dirctionality. If an acoustical sourc has siz, it is not possibl to rmain quidistant from it whil swping a masurmnt microphon through a circular arc to masur its polar rspons (xcpt in th cas of a sphrical sourc). Consquntly, th propagation tim from ral sourcs will vary with dirction. In fact, th particular way in which propagation tim varis with dirction is th most critical pic of information providd by complx polars. This information is rquird for accurat intrfrnc calculations, by which th prformanc of arrays of loudspakrs can b prdictd.

2 Thr has bn much spiritd dbat within th Audio Enginring Socity in th last dcad concrning th angular rsolution rquird for complx polar data. Various conclusions hav bn rachd, ranging from 0 dgrs to 0.5 dgrs pr data point. Evn without consulting transcripts of ths dbats, on can rasonably assum that proponnts of th finr rsolutions hav no faith in th ffctivnss of intrpolation. Such mistrust is not unfoundd, as th following xampl illustrats: Figur 2: Linar Intrpolation of Complx Transfr Function AN ALGORITHM FOR INTERPOLATING TRANSFER FUNCTIONS Figur : Magnitud and Impuls Rspons at 5 & 20 dgrs Off Axis Figur prsnts two rspons masurmnts of a highly dirctional high frquncy horn, takn at 5 dgrs and 20 dgrs off axis. This particular dvic and ths off-axis angls wr slctd bcaus thy rprsnt a practical worst cas, as far as th angular rsolution rquird. Th impuls rspons chart shows clarly that th propagation tim varid by about 0.07 ms btwn th two masurmnts. If w apply simpl linar intrpolation to stimat th complx transfr function at th midpoint btwn th two (thir avrag) w gt an impuls rspons which lis btwn th impuls rsponss of th two masurmnts at vry point, as shown in Figur 2. Th Fourir transform of that impuls rspons (th frquncy rspons), on th othr hand, dos not li btwn th original frquncy rsponss. In intuitiv trms, adding th two transfr functions and dividing by two shows th intrfrnc of th two functions, rathr than stimating th midpoint of a transition from on to th othr. Th intrpolatd impuls rspons consists of ½-amplitud arrivals at th tims of th original impulss. Howvr, what is rquird is a singl impuls, similar in shap to th original impulss, but occurring at a tim halfway btwn th two original impulss. Furthrmor, th intrpolatd magnitud rspons should li btwn th two original magnitud rsponss. Concptually, th intrpolation of loudspakr transfr functions can b dividd into two sparat problms: intrpolating th propagation dlay, and intrpolating th charactristic complx frquncy rspons (which is to say, th shap of th impuls). If w can liminat th diffrnc in arrival tim, th phas rspons of th two transfr functions will bcom similar nough that intrpolation is ffctiv.. Using th Cross-corrlation Function to Eliminat Excss Dlay Th cross-corrlation of two impuls rspons functions givs an indication of thir similarity for various offst tims. If th shaps of th impuls rsponss of th two functions ar similar nough, thn th cross-corrlation function will hav a wll-dfind pak that indicats th rlativ arrival tim, or xcss dlay. Cross-corrlation is dfind as th convolution of on squnc with a rvrs-ordr scond squnc. In th prsnt cas, th squncs ar impuls rsponss. Th cross-corrlation can b fficintly calculatd in th frquncy domain by multiplying th Fourir transform of th first impuls rspons by th complx conjugat of th Fourir transform of th scond rspons: thn taking th invrs Fourir transform (IFT) of th rsult. In psudocod, this looks lik: Cross-Corrlation = IFT(FFT(Impuls ) Conj(FFT(Impuls 2 ) ) ) For th particular cas of intrpolating complx polar rspons, adjacnt transfr functions should b xtrmly similar. So, th cross-corrlation function would b xpctd to hav a wlldfind pak. Figur 3 shows th cross corrlation of th two transfr functions illustratd in Figur, indicating an xcss dlay of.075 ms. () AES TH CONVENTION, NEW YORK, NY, USA, 200 SEPTEMBER

3 .2 Gomtric Intrpolation Th rquirmnts for an ffctiv intrpolation mthod ar that: Figur 3: Cross-corrlation of 5-dgr and 20-dgr Transfr Functions Th propagation tim diffrnc should b backd out of th lat transfr function. This is accomplishd by synthsizing a transfr function rprsnting th xcss dlay. This transfr function will hav unity magnitud and a phas rspons corrsponding to th dlay. Th complx valu at ach frquncy is dfind by, ( ωt ) + j ( ωt ) F( ω, t ) = cos sin, (2) whr ω is 2πf, and t is th xcss dlay. Th lat transfr function is dividd by th xcss dlay transfr function. Th rsulting transfr function is no longr lat, but is approximatly synchronous with th arly transfr function. In fact, its phas rspons would b xpctd to b as similar as possibl to that of th arly transfr function. Figur 4 shows th two xampl impuls rsponss, as wll as th phas rspons (unwrappd for radability) aftr rmoving th xcss dlay. a) Th intrpolatd magnitud rspons should vary logarithmically from on magnitud rspons to th othr. Effctivly, w would lik to linarly intrpolat dcibl valus. b) Th phas rspons should intrpolat corrctly rgardlss of wrapping. Corrct rsults will b vidncd by an impuls shap that morphs smoothly from on impuls shap to th othr. Ths rquirmnts ar mt if w adopt an intrpolation mthod that corrsponds to th gomtric man, rathr than th arithmtic man. For purposs of illustration, lt s compar th linar man of two quantitis (intrpolation of th midpoint) to th gomtric man: ( Valu ) _ + Valu Linar Man = 2 (3) 2 Gomtric _ Man = Valu Valu (4) * Considr two complx quantitis which diffr by narly 80 dgrs in phas: + 0j, and j. Th linar man of ths two valus is ( j) / 2, or j. Th magnitud of this rsult is much smallr than th magnitud of ithr of th original valus, so it obviously fails to satisfy rquirmnt (a). Th gomtric man is j, or j. This rsult falls nicly btwn th original valus in both magnitud and phas. Working a fw mor xampls will dmonstrat that th gomtric man satisfis both rquirmnts, as long as th phas diffrnc is no gratr than π. 2 In ordr to xploit th dsirabl bhavior of th gomtric man, w must xpand th avraging quation to an intrpolation quation: Valu Gomtric _ Intrpolation Valu Valu = * 2, (5) whr ratio is th fraction of intrpolation (if th ratio is 0.0, th function rturns Valu ; if th ratio is.0, th function rturns Valu 2 ). This xprssion should not b rducd to Valu (- ratio) Valu 2 ratio, bcaus th rducd xprssion only provids valid rsults for phas diffrncs of lss than π/2 and absolut phas of +/- π/2; whras, quation (5) is valid for any absolut phas valu, and phas diffrncs of +/- π. Whn gomtric intrpolation is applid to th xampl transfr functions, th rsult is as shown in Figur 5. ratio Figur 4: Transfr Functions with Excss Dlay Rmovd W ar now rady to intrpolat th impuls rspons. AES TH CONVENTION, NEW YORK, NY, USA, 200 SEPTEMBER

4 It has bn found that for marginally matchd transfr functions, idntification of th xcss dlay is accomplishd mor rliably if th nvlop of th cross-corrlation function is usd, rathr than th cross-corrlation itslf. Th nvlop may b calculatd by trating th cross-corrlation as an impuls rspons, and calculating its corrsponding analytic impuls [2]. Th nvlop of th cross-corrlation is th magnitud of its analytic impuls. Th cross-corrlation rsult could b thought of as bing wightd by th rspons magnituds of diffrnt frquncis. Th frquncis at which th rspons magnitud is rlativly larg hav a gratr ffct on th cross-corrlation rsult than do th frquncis at which th rspons magnitud is rlativly small. For most loudspakr applications this is a dsirabl ffct, bcaus it naturally wights mor havily th frquncy rang ovr which th loudspakr oprats ffctivly. Howvr, in som applications it may b dsirabl to qualiz th lvls of th various frquncis bfor calculating th cross-corrlation. In th fild of computr imaging, th function so obtaind is rfrrd to as th phas corrlation [3], bcaus only th phas of th input functions affcts th rsult - th magnitud rspons is ffctivly ignord..5 Limitations of th Algorithm Figur 5: Alignd and Intrpolatd Transfr Functions.3 Intrpolating th Propagation Dlay Intrpolating th propagation dlay is a simpl mattr. W simply work th mthod dscribd in sction., in rvrs. For th dlay w us th xcss dlay multiplid by th intrpolation ratio. For th xampl, w multiply th intrpolatd rsult by th transfr function quating to ½ of.075 ms, or.0375 ms.. Th intrpolatd rspons is shown in Figur 6, ovrlaid with th actual masurd rspons it stimats. In this xampl, th stimatd and masurd rsults ar narly idntical. Th mthod dscribd in this papr is an algorithm, not a transform. It falls far short of th lvl of robustnss xpctd of a mathmatical transform. It only rsults in good stimats if: a) Th two transfr functions to b intrpolatd ar similar to ach othr. b) Each of th transfr functions can b charactrizd as having a particular arrival tim (th impuls rspons is sufficintly focusd). A particular condition that violats th scond rquirmnt is whn an impuls rspons contains two discrt arrivals, narly qual in lvl. Gnrally, th polar rspons of a singl loudspakr satisfis both constraints. But, if a loudspakr consists of multipl sourcs, thy should b masurd sparatly and mathmatically summd to produc th nt acoustical rspons. A complt tratmnt of this procss is givn in Hoy/McGrgor []. 2 OTHER APPLICATIONS In addition to loudspakr polar rspons, thr ar svral othr audio-rlatd applications of this tchniqu. 2. Masurmnt Avraging Avraging is oftn hlpful whn applid to acoustical masurmnts. Most commonly, it is usd to incras th signal to nois ratio of a masurmnt. It can also b usd with a sourcindpndnt masurmnt systm such as SIA SmaartLiv to rduc th inclusion of lat rvrbrant nrgy in th masurmnt rsults. For this to b ffctiv, th complx valus of th transfr functions must b avragd. This tchniqu is rfrrd to as vctor avraging, or complx avraging [4]. Figur 6: Intrpolatd vs Masurd Rspons.4 Additional Tchniqus Whn acoustical masurmnts ar conductd in-situ, som amount of variability of arrival tims will b obsrvd. Whn making masurmnts at long distancs outdoors (i.., 00 m or mor), arrival tim variability of 5 to 20 ms is not unusual. Evn in larg indoor vnus, th variability might b on th ordr of on or two millisconds. AES TH CONVENTION, NEW YORK, NY, USA, 200 SEPTEMBER

5 Th ffct of arrival tim variability is similar to th intrpolation illustratd in Figur 2. Succssiv masurmnts intrfr, rsulting in a mislading rduction in high frquncy lvls. Consquntly, complx avraging has only bn practicabl indoors, at distancs up to 50 mtrs or so. Complx avraging can b carrid out using th intrpolation algorithm dscribd in this papr. Using this tchniqu, th masurmnt procss bcoms tolrant of arrival-tim variability, and a bttr figur for th arrival tim itslf can b dtrmind. Th bnfits of complx avraging can thn b ralizd outdoors and in larg vnus. 2.2 DSP Filtrs for Loudspakr String [2] A. Duncan, Th Analytic Impuls, Journal of th Audio Enginring Socity, vol. 36, pp (988 May). [3] C. Kuglin and D. Hins, ``Th Phas Corrlation Imag Alignmnt Mthod,'' Proc. Int. Conf. Cybrntics and Socity, 975, pp [4] J. Andrson and C. Dayton, SmaartLiv Hlp, SIA Softwar Company, Topic Avraging and Smoothing. [5] D. W. Gunnss and J.F.S. Spck, Phasd Point Sourc Tchnology and th Rsultant KF900 Sris, Eastrn Acoustic Works, Inc., 997) Whn th intrpolation algorithm was first concivd, a numbr of rasonablnss tsts wr prformd. On of ths tsts was to intrpolat btwn som standard high-pass and low-pass filtr typs. If th magnitud, phas, and impuls rspons all appard to b rasonably similar to th input functions, it would support th fficacy of th nw mthod. A surprising rsult was obsrvd. Whn a scond-ordr Buttrworth filtr and a fourth-ordr Buttrworth filtr ar intrpolatd with a ratio of 0.5, th rsult is an xact match in both magnitud and phas to a third-ordr Buttrworth filtr. Similar rsults ar obtaind btwn ordrs of Bssl filtrs. Th intrsting aspct of this obsrvation is that th ssntial bhavior that charactrizs a particular filtr typ survivs intrpolation. An intrpolation btwn two maximally flat (Buttrworth) filtrs is anothr maximally flat filtr. By xtnsion, it stands to rason that fractional-ordr filtrs can b dfind by gomtric intrpolation; and that ths fractional ordr filtrs will also shar th ssntial charactr of th filtr typ. Exprimnts hav vrifid that this prsumption is tru. A 3.5- ordr Buttrworth filtr, for xampl, has an impuls rspons that rings slightly longr than a 3 rd -ordr filtr, and slightly shortr than a 4 th -ordr filtr. It has a slop of 2 db pr octav. Whil continuously variabl low-pass and high-pass filtrs might prov usful to dsignrs of multi-way loudspakrs, a mor intrsting valu of such a filtr st is in loudspakr string. It has bn found that th smoothst strd dirctional rspons is obtaind with a programmd-optimizr and high-prcision complx polar data [5]. Unfortunatly, th ordr of a low-pass filtr was prviously an intgr-valud function. It was not productiv to allow an optimizr to chang it, bcaus its ffct on th rsult was discontinuous. Implmnting continuously variabl filtrs could lad to bttr optimization rsults. 3 CONCLUSIONS A nw transfr function intrpolation algorithm has bn prsntd. This algorithm producs bttr stimats than linar intrpolation whn th transfr functions bing intrpolatd includ pur dlay. Th nw algorithm nabls complx polar data to b intrpolatd with sufficint accuracy for loudspakr intrfrnc calculations; and it allows complx avraging to b mployd in outdoor acoustical masurmnts, dspit th prsnc of changabl wind. Fractional ordr high-pass and low-pass filtrs wr proposd, basd on intrpolation of th familiar typs. 4 REFERENCES [] W.Hoy and C. McGrgor, Loudspakr Complx Dirctional Rspons Charactrization, to b prsntd at th th Convntion of th Audio Enginring Socity, 200. AES TH CONVENTION, NEW YORK, NY, USA, 200 SEPTEMBER