Effect of transducer mismatch on the performance of spherical microphone arrays

Transcription

1 Effect of tranducer mmatch on the performance of pherca mcrophone array Dan Rao 2 Acoutc ab., Phyc Dept., South Chna Unverty of Technoogy, Guangzhou, Chna, 2 State Key aboratory of Subtropca Budng Scence, South Chna Unverty of Technoogy,Guangzhou, Chna ABSTRACT Spherca mcrophone array(sma were wdey ued n ound fed recordng, beamformng,etc. for t advantage of hgh geometrca ymmetry. Many factor coud affect the performance of SMA, ncudng tranducer error. In th paper, effect of tranducer mmatch n amptude and phae on the performance of SMA wa tuded through muaton cacuaton. A SMA wth 64 neary unformy dtrbuted tranducer wa ued n the muaton. Beam pattern wth and wthout tranducer mmatch error were cacuated, and then the dfference between thoe pattern were compared ung beam pattern correaton coeffcent a the metrc. The muaton and anya reut how that under the condton that the pherca harmonc expanon order not arger than ka(the product of wave number and array radu, the correaton coeffcent not e than.97 when the max random tranducer amptude error e than 3dB or the maxmum random tranducer phae error e than degree. Th reut ndcate that the performance of SMA ha a good toerance to tranducer mmatch error when pherca harmonc truncaton order not arger than ka. Whe ka e than truncaton order, great error occur due to ow-frequency boot nature of SMA gna proceng. Keyword: Spherca mcrophone array, mmatch, Beamformng, I-INCE Number(: 74.7 Cafcaton of Subject. INTRODUCTION Spherca mcrophone array (SMA are uefu for many appcaton, uch a ound fed recordng, beamformng and ound ource ocazaton. Meyer( tuded SMA for beamformng. Durawam et.a.(2 utzed SMA to the ound fed recordng and bnaura ynthe. Nen et.a(3. propoed pherca harmonc-baed ESPRIT ocazaton method ung SMA. Due to the hgh geometrca ymmetry, SMA have the advantage of unchanged beam pattern for dfferent teerng drecton. Two type of SMA are ued n genera, the open-phere SMA and the rgd-phere SMA. Becaue of nu outon n ome frequence, the open-phere SMA how more numerca ntabty than the rgd-phere SMA, o the rgd-phere SMA are more popuar n the practce. Ony the rgd-phere SMA w be dcued n the foowng. Many error can affect the performance of SMA, ncudng tranducer poton error, meaurement noe and tranducer mmatch. Rafeay(4 comprehenvey tuded the effect of poton error, pata aang, meaurement noe on the performance of SMA, n whch the array output gan n teerng drecton wa taken a performance metrc. The tudy of the effect of tranducer mmatch error wa edom. In th paper, ung beam pattern correaton coeffcent a metrc, the effect of the tranducer mmatch n amptude and phae on SMA performance are examned. phdrao@cut.edu.cn Inter-noe 24 Page of 6

2 Page 2 of 6 Inter-noe PRINCIPE 2. Beamformng n SMA Suppong a unt amptude pane wave arrve from (θ,φ, n the pherca coordnate ytem, the preure on a rgd phere can be wrtten a p( a, θ, ϕ θ, ϕ = Y, ϕ b Y, ϕ ( = m= Where (a,θ,φ denote poton on the phere, a the radu of the phere, θ eevaton ange ranged from o to 8 o, φ azmuth ange ranged from o to 36 o, k wave number, Y are pherca harmonc, whch are defned by (2 + ( m! Y, ϕ 4π ( + m! jmϕ = P (coθ e (2 Where =,,.., m=-,-+,,,... In Eq.(, b (ka gven by j h j h b = 4π j [ ] (3 h Where j (ka and h (kaare pherca Bee and Hanke functon, repectvey, j (ka,h (ka are ther dervate. For SMA, becaue mcrophone tranducer arranged on the phere urface fnte, meaurng oud preure ung SMA equvaent to ampng preure on the phere urface. Mutpyng preure amped by mcrophone on the phere wth weght w (θ,φ and ummng them up wth weght α, t gve the array output M = A( θ, ϕ = α p( a, θ, ϕ θ, ϕ w, ϕ = p w (4 = m= Where the weght α depend on the ampng cheme, the truncaton order of pherca harmonc, p are the pherca harmonc decompoton of preure p, M the mcrophone number. When the weght w are taken a w = Y, ϕ / b, the array output A n Eq.(4 regarded a the beam pattern wth the teerng drecton (θ,φ A( θ, ϕ θ, ϕ = = M = = m= = m= Y α p( a, θ, ϕ θ, ϕ Y p, ϕ / b, ϕ Y, ϕ / b Beam pattern A repreent the array repone for pane wave from dfferent ncdence drecton whe teerng at drecton (θ,φ. Subttutng Eq.( n Eq.(5, the beam pattern wthout tranducer error acheved., ϕ = Y, ϕ Y, = m= A( θ, ϕ θ ϕ (6 2.2 Effect of Tranducer Mmatch In Eq.(6, the tranducer are aumed dentca. When the tranducer mmatch condered, Eq.(6 w be reved. Tranducer mmatch can be condered a unformy-dtrbuted random error n tranducer gan and phae. Takng the mmatch error nto account, the mcrophone output become o p = p ( + Δg exp( jδφ (7 Where Δg and ΔΦ are amptude and phae mmatch error repreented by unformy -dtrbuted random noe. Subttutng Eq.( and (7 n Eq.(5 yed the beam pattern wth tranducer mmatch (5 Page 2 of 6 Inter-noe 24

3 Inter-noe 24 Page 3 of 6 A( θ, ϕ θ, ϕ = M α( + Δg exp( jδφ ( Y, ϕ b Y = = m= = m = = b / b Y(, ϕ Y( θ, ϕ e mm = m= = m = Where e mm repreent the effect of mmatch e M m = ( + Δg exp( jδφ Y m, ϕ Y, ϕ =, ϕ Y, ϕ Y, ϕ / b θ (8 α (9 Many tude ued the man gan error of A for pane wave from teerng drecton,.e. (θ,φ =(θ,φ, a the error metrc. However, th metrc coud not evauate the beam pattern entrey. Here, a new metrc caed beam pattern correaton coeffcent (BPCC propoed, that a cc = N = o A, ϕ A, ϕ / A, ϕ A( θ, ϕ ( = o Where N ampng number n beam pattern, A the actua beam wth error, A o the orgna beam. BPCC evauate the marty between the beam pattern nfuenced by mmatch error and the orgna one. For mot tuaton, keepng the hape of beam pattern more mportant n practce. The vaue of BPCC from to, arger BPCC mean more mar between thee two pattern. 3. Smuaton 3. Smuaton condton A SMA wth 64 tranducer neary unformy arranged on a rgd phere urface wa ued n the muaton. The arrangement cheme hown n Fg., whch wa propoed by Fege(5. Fgure 64 tranducer ampng cheme on a phere urface. Accordng to the tranducer number, truncaton number =6 wa ued. When frequence ka are hgher than, the aang error occur. In order to avod thee error, ony frequence ka are condered n the foowng muaton. A to mmatch error, maxmum amptude error Δg max =3dB, db and maxmum phae error ΔΦ max = o,5 o are adopted n muaton. BPCC w be cacuated a an average over 5 reazaton of random error. Prenary muaton found that there wa no gnfcant effect of teerng drecton on error Inter-noe 24 Page 3 of 6

4 Page 4 of 6 Inter-noe 24 reut, o ony one drecton((9 o, o cacuated n th paper. 3.2 Smuaton reut 3.2. Cae for ka==6 Fg.2 how magntude curve of actua beam pattern A and the orgna beam pattern A o n the horzonta pane (θ=9 o under poar coordnate for amptude mmatch. It can be een that, the maxmum magntude occur at (9 o, o concdent wth the teerng drecton a expected. For the cae of Δg max =db, two curve are aot concdent, and the BPCC.99. For the cae of Δg max =3dB, ony ght dfference are oberved, and the BPCC equa.97. Thee reut how, when ka= the hape of beam pattern keep very we even f the tranducer have reatvey arge mmatch error. (a Δg max =db,bpcc=.99 (b Δg max =3dB,BPCC=.97 Fgure 2 Beam pattern for amptude mmatch n the horzonta pane (θ=9 o, ka==6; red dot ne for A o, dark od ne for A For phae mmatch, magntude curve hown n fg.3. Smar to amptude mmatch cae, 5 o error, even o error ony ntroduce ght mpact on the beam pattern. Thee reut how that the gna proceng of SMA ha a good toerance to tranducer mmatch error. (a ΔΦ max =5 o,bpcc=.99 (b ΔΦ max = o,bpcc=.99 Fgure 3 Beam pattern for phae mmatch n the horzonta pane (θ=9 o, ka==6; red dot ne for A o, dark od ne for A Page 4 of 6 Inter-noe 24

5 Inter-noe 24 Page 5 of Cae for ka<=6 Tabe. how BPCC for amptude mmatch. It can be oberved, BPCC decreae wth the frequency ka decreae. Epecay, when ka<-2, BPCC dramatcay decreae, whch ndcate the hape of beam pattern detroyed. Tabe BPCC for amptude mmatch ka=6 ka=5 ka=4 ka=3 ka=2 ka= Δg max =db Δg max =3dB Fg.4 gve beam pattern curve of A and A o for Δg max =db and ka=3,2 repectvey. It hown that, there are very arge dfference between A and A o. For the cae of ka=2, the hape of A coud not be dentfed a orgnatng from A o. In fact, the term /b (ka n Eq.(8 w boot when ka<. In th way, the effect of mmatch e mm w be enarged a ka decreae. (a ka=3,bpcc=.7 (b ka=2,bpcc=.2 Fgure 5 Beam pattern for amptude mmatch n the horzonta pane (θ=9 o, ΔΦ max =5 o ; red dot ne for A o, dark od ne for A Tabe.2 how BPCC for phae mmatch. Smar behavor for the cae of amptude mmatch oberved. The beam pattern curve for phae mmatch hown n Fg.5. Tabe 2 BPCC for amptude mmatch ka=6 ka=5 ka=4 ka=3 ka=2 ka= ΔΦ max =5 o ΔΦ max = o Inter-noe 24 Page 5 of 6

6 Page 6 of 6 Inter-noe 24 (a ka=3,bpcc=.8 (b ka=2,bpcc=.28 Fgure 5 Beam pattern for phae mmatch n the horzonta pane (θ=9 o, ΔΦ max =5 o ; red dot ne for A o, dark od ne for A 4. CONCUSIONS Th paper anayzed the effect of tranducer mmatch through muaton by ung beam pattern correaton coeffcent (BPCC a evauaton metrc. When ka=, reatve arge mmatch error, uch a 3dB n amptude error or o n phae error, ony ntroduce ma error n beam pattern, n whch BPCC arger than.97. Th reut ndcate SMA ha a good toerance to tranducer mmatch error when pherca harmonc truncaton order not arger than ka. However, when ka e than, arger error occur whch due to the fact /b (ka w enarge the mmatch error. So, carefuy choong the truncaton order under certan ka needed n beamformng proceng. ACKNOWEDGEMENTS Th work upported by the State Key ab of Subtropca Budng Scence, South Chna Unverty of Technoogy, grant no. 24KB23. REFERENCES. Meyer J, Eko G, Ieee. A hghy caabe pherca mcrophone array baed on an orthonorma decompoton of the oundfed. 22 Ieee Internatona Conference on Acoutc, Speech, and Sgna Proceng, Vo I-Iv, Proceedng; New York: Ieee; 22. p Durawam R, Zotkn D, Z, Gra E, Gumerov N, Dav, edtor. Hgh order pata audo capture and t bnaura head-tracked payback over headphone wth HRTF cue. AES 9th Conv; 25; New Yor NY. 3. Nen C-IC, Hafzovc I, Ho S, edtor. Robut 3D ound ource ocazaton ung pherca mcrophone array 23; Rome, Itay: Audo Engneerng Socety. 4. Rafaey B. Anay and degn of pherca mcrophone array. IEEE Tranacton on Speech and Audo Proceng 25;3(: Fege J, Maer U. The dtrbuton of pont on the phere and correpondng cubature formuae. Ima Journa of Numerca Anay 999 Apr;9(2: Page 6 of 6 Inter-noe 24