Acoustic contrast sensitivity to transfer function errors in the design of a personal audio system

Transcription

1 Acoutic contrat enitivity to tranfer function error in the deign of a peronal audio yte Jin-Young Park, Jung-Woo Choi, a) and Yang-Hann Ki Center for Noie and Vibration Control, Departent of Mechanical Engineering, Korea Advanced Intitute of Science and Technology, 291 Daehak-ro, Yueong-Gu, Daejeon, , Republic of Korea jy_park@kait.ac.kr, khepera@kait.ac.kr, yanghannki@kait.ac.kr Abtract: An analytic ean to evaluate the error enitivity of a peronal audio yte i propoed. The peronal audio yte, which focue acoutic energy into a zone of interet uing ultiple loudpeaker, i ubject to variou error when ipleented. The perforance of a peronal audio yte, defined a an energy ratio between the zone of interet and the ret, i inevitably influenced by error. Thu the ability to predict perforance change at the deign tage i crucial when building a robut peronal audio yte. The dependence of the energy ratio change on variou type of error i forulated. VC 2013 Acoutical Society of Aerica PACS nuber: Hz, Fg [DC] Date Received: February 25, 2013 Date Accepted: May 23, Introduction Peronal audio yte 1,2 are ued to produce private or iolated ound zone without diturbing other litener. To achieve thi goal, a peronal audio yte utilize an array of loudpeaker and control the radiation of ound fro each loudpeaker uch that the ditribution of acoutic potential energy i concentrated over a elected zone through the contructive and detructive interference of ound wave. Therefore the function that decribe ound radiation fro each loudpeaker, often denoted a the tranfer function, play a crucial role in deterining the control ignal of loudpeaker. In practice, tranfer function of loudpeaker are either odeled or eaured; hence, odeling or eaureent error alway exit. When a peronal audio yte i driven by control ignal that are calculated fro the odeled tranfer function, perforance change can occur due to the dicrepancy between the odeled and actual tranfer function. The perforance degradation 3 ha been invetigated in cae of the ditortion of the calculated control ignal due to the randoly ditributed tranfer function error. A nuber of robut control ethod 4 have been propoed to reduce the perforance degradation due to the tranfer function error. Although thee tudie can infor the nature of the yte and iprove the robutne of a yte againt the arbitrary or rando error ditribution, the fundaental behavior of a yte ubjected to the pecific type of error ha not been addreed well. For exaple, there can be variou ource of error, uch a the loudpeaker poition iatch, the eaureent icrophone iatch, or even the gain and phae of each loudpeaker can be different. In thi regard, the perforance enitivity of a peronal audio yte to variou type of error i invetigated. The perforance of a peronal audio yte can be repreented by the ratio of acoutic energie fro two different zone. It i noteworthy that there are alway two different type of zone involved with the concept of the peronal audio; one i the bright zone in which a louder ound i deirable than in the urrounding area. The other zone i the dark (quiet) zone in which the ound preure level need to be a) Author to who correpondence hould be addreed. EL112 J. Acout. Soc. A. 134 (1), July 2013 VC 2013 Acoutical Society of Aerica Downloaded 23 Sep 2013 to Reditribution ubject to ASA licene or copyright; ee

2 iniized. For intance, Choi and Ki 5 propoed the ue of the acoutic energy ratio of bright and dark zone, which i tered a acoutic contrat, a a eaure of perforance and a an objective function. In view of all thee, the enitivity of the acoutic contrat againt the error can be a ueful indicator of yte robutne. The eaured tranfer function include both the electro-acoutic repone of the tranducer and the characteritic of acoutic wave propagation between the loudpeaker and icrophone poition. For intance, the acoutic propagation repone between loudpeaker and icrophone ay vary if there i the phyical preence/ oveent of a litener or the change of the reflection boundary. The effect of uch variation in acoutic repone, however, depend on the characteritic of the catterer or the reflection boundary uch that it hould be tudied for cae by cae. Thu two different type of error have been conidered here: The agnitude and phae error due to the electro-acoutic repone iatch and the error that arie due to the poition iatch. Thoe iatche in tranducer ay be iportant iue when there are odel iatche between the actual product and experiental prototype or anufacturing variance. Herein, the enitivitie of acoutic contrat in repone to the error that arie fro electro-acoutic and poition iatche are forulated, repectively. An exeplary analyi, which ue a linear array of loudpeaker, explain which paraeter ha to be preciely controlled and which can be regarded a le iportant, in the deign of a peronal audio yte. 2. Proble definition 2.1 Sound field controlled by ultiple loudpeaker Conider a peronal audio yte coniting of L loudpeaker and driven by a ingle frequency x. Control ignal q (l ¼ 1; 2; ; L,: Nuber of loudpeaker) are fed into each loudpeaker located at a poition * r. Microphone located at * ðnþ r (n ¼ 1; ; N, N: Nuber of eaureent point) eaure the generated ound preure pðr * ðnþ Þ. The preure field that i generated by the control ignal and eaured at variou icrophone poition can be expreed a p ¼ Hq; (1) where hðr * ðnþ jr* p ¼½pðr * ð1þ Þ ðnþ pðr* Þ Š T, ½HŠ n;l ¼ hðr * ðnþ jr* Þ, and q ¼½q ð1þ q ðlþ Š T. Here Þ i a tranfer function between the lth loudpeaker and the nth icrophone. 2.2 Preure perturbation due to the tranfer function error When the eaured (or odeled) tranfer function differ fro the actual tranfer function of the real loudpeaker, the difference between the two can be conidered a the tranfer function error. In general, a tranfer function that include error can be expreed a ~H ¼ H þ dh; (2) where dh i a atrix repreenting the tranfer function error [½dHŠ n;l ¼ dhðr * ðnþ jr* Þ], and ~H denote the tranfer function atrix with error. The preure perturbation dp that arie fro the tranfer function error then can be expreed a follow: dp ¼ dhq: (3) The atheatical expreion of dp depend on the ource of error Electro-acoutic iatch In the cae of electro-acoutic iatch, the tranfer function error incur a the agnitude or phae variation of frequency repone, which can be forulated 6 a J. Acout. Soc. A. 134 (1), July 2013 Park et al.: Acoutic contrat enitivity to error EL113 Downloaded 23 Sep 2013 to Reditribution ubject to ASA licene or copyright; ee

3 ~hðr * ðnþ jr* Þ¼ð1þa Þe j/ ð1 þ a ðnþ Þej/ðnÞ hðr * ðnþ jr* Þ; (4) where a and / are the agnitude and phae error of a tranfer function for the lth loudpeaker, and a ðnþ and / ðnþ denote the error for the nth icrophone, repectively. The error are all real valued variable (a and a ðnþ 1). Auing all agnitude and phae error, uch that ð1 þ aþe j/ 1 þ a þ j/, the perturbed tranfer function can be approxiated a ~hðr * ðnþ jr* Þhðr * ðnþ jr* Þþ½ða þ j/ Þþða ðnþ þ j/ðnþ ðnþ ÞŠhðr* jr* Þ: (5) Equation (5) yield the tranfer function error in atrix for a dh ¼ HE þ E H; (6) where the error in loudpeaker and icrophone are expreed by diagonal atrice: E ¼ diag½ða ð1þ þj/ ð1þ Þ; ;ða ðlþ þj/ ðlþ ÞŠ and E ¼ diag½ða ð1þ þ j/ ð1þ Þ; ; ðaðnþ þ j/ ðnþ ÞŠ. With agnitude and phae error only in the lth loudpeaker, the error atrix E can be written by E ¼ diag½d l Šða þ j/ Þ, where d l denote the Kronecker delta vector the ole nonzero eleent of which equal 1 in the lth entry. Then the preure perturbation dp can be expreed a dp ¼ HE q ¼ h ða þ j/ Þq ¼ða þ j/ Þp ; (7) where h repreent the lth colun of H, and p ¼ h q denote the preure field of the lth loudpeaker. It i noteworthy that the preure perturbation due to the electro-acoutic iatch of a loudpeaker i proportional to the loudpeaker tranfer function, h. When the electro-acoutic iatch exit only in the nth icrophone uch that E ¼ða ðnþ þ j/ ðnþ Þdiag½d nš, the preure perturbation i expreed a dp ¼ E Hq ¼ E p ¼ða ðnþ Poition iatch When there are poition iatche at * r the tranfer function can be expreed a ~hðr * ðnþ jr* Þ¼hðr * ðnþ j and jdr* If it i aued that jdr * ðnþ Eq. (9) can be approxiated 6 a ~hðr * ðnþ Þhðr * ðnþ Þþr hðr * ðnþ jr* jr* þ j/ ðnþ Þpð~rðnÞ Þd n: (8) and r * ðnþ with dr* and dr * ðnþ, repectively, þ ðnþ dr* jr* þ dr * Þ: (9) jr* j are all copared to the wavelength, Þdr * þr hðr * ðnþ jr* Þdr * ðnþ ; (10) where r and r are gradient operator with repect to the ource poition r * and the icrophone poition r *. The tranfer function error of Eq. (10) can be written in atrix for a dh ¼r H T * þ T * r H; (11) where T * ¼ diag½dr * ð1þ ; ; dr * ðlþ Š and T * ¼ diag½dr * ð1þ ; ; ðnþ dr* Š denote the tranlation of the loudpeaker and the icrophone, repectively. If the poition iatch i preent only at the lth loudpeaker (T * ¼ dr * diag½d l Š), the preure perturbation can be expreed a dp ¼r p dr * : (12) Fro Euler equation, the gradient of a preure field rp i related to the velocity field u * by rp ¼ jq 0 x u *, where q 0 i the ediu denity. Becaue the gradient i EL114 J. Acout. Soc. A. 134 (1), July 2013 Park et al.: Acoutic contrat enitivity to error Downloaded 23 Sep 2013 to Reditribution ubject to ASA licene or copyright; ee

4 uually evaluated with repect to the oberver poition, the reciprocal velocity field * u, generated by the lth loudpeaker, can be defined (r p ¼ jq 0 x * u ). Hence, we have dp ¼ jq 0 xu * dr * : (13) On the other hand, if poition iatch i preent only at the nth icrophone, the preure perturbation i written a dp ¼½dr * ðnþ r pðr * ðnþ ÞŠ d n ¼ jxq 0 ½dr * ðnþ ðr * ðnþ u* ÞŠ d n; (14) where r pðr * ðnþ Þ¼jxq 0u * ðr * ðnþ Þ fro Euler equation. 2.3 Perturbation of acoutic contrat due to preure perturbation The acoutic contrat 6 i defined a the pace-averaged acoutic energy ratio between the bright and dark zone. That i, b ¼ðp H b p b=n b Þ=ðp H d p d=n d Þ; (15) where the ubcript b and d in Eq. (15), repectively, denote the bright zone (V b ) and the dark zone (V d ). Fro the firt order approxiation of dp, the relative contrat perturbation can be expreed a db=b 2ðRe½ðp H b =jjp bjjþðdp b =jjp b jjþš Re½ðp H d =jjp djjþðdp d =jjp d jjþšþ: (16) The operator Re½ Šdenote the real part of a variable. The ter inide of ½Šin Eq. (16) repreent the relative energy perturbation in a zone. Therefore the acoutic contrat perturbation i deterined by the difference of energy perturbation in the bright and dark zone, each of which i given by the real part of the inner product of two vector dp=jjpjj and p=jjpjj. 3. Acoutic contrat enitivity with repect to tranfer function error 3.1 Electro-acoutic iatch of a loudpeaker The contrat perturbation can be forulated in ter of the acoutic contrat enitivity by ubtituting Eq. (7) into Eq. (16). Specifically, db=b ¼ K;ag a þ K ;ph / ; (17) K;ag ¼ 2ðRe½ðpH b =jjp bjjþðp b =jjp bjjþš Re½ðp H d =jjp djjþðp d =jjp djjþšþ; (18) K ;ph ¼ð 2ÞðI½ðpH b =jjp bjjþðp b =jjp bjjþš I½ðp H d =jjp djjþðp d =jjp djjþšþ; (19) where K;ag and K ;ph repreent contrat enitivitie due to the agnitude and phae error in the lth loudpeaker, repectively. The contrat enitivity depend on the inner product of the total preure p and the preure of the lth loudpeaker p. Therefore in general, the loudpeaker that contribute ore to the total preure field ha greater contrat enitivity. However, the final contrat enitivity i deterined by the real part of the inner product for the agnitude iatch, and the iaginary part for the phae iatch. 3.2 Electro-acoutic iatch of a icrophone In the cae of electro-acoutic iatch of nth icrophone, unlike the cae of the loudpeaker, the preure perturbation only occur at the poition of the nth icrophone. Fro Eq. (8) and (16), the contrat enitivity can be decribed a K ðnþ ;ag ¼ 8 < : 2jpðr*ðnÞ Þj2 =jjp b jj 2 2jpðr * ðnþ Þj2 =jjp d jj 2 ðr * ðnþ ðr * ðnþ 2 V bþ 2 V dþ; (20) J. Acout. Soc. A. 134 (1), July 2013 Park et al.: Acoutic contrat enitivity to error EL115 Downloaded 23 Sep 2013 to Reditribution ubject to ASA licene or copyright; ee

5 K ðnþ ;ph ¼ 0; (21) where K;ag ðnþ and K ðnþ ;ph are the contrat enitivitie for agnitude and phae error at the nth icrophone, repectively. Fro Eq. (20), it can be een that the contrat enitivity to the agnitude error i given by the ratio between the energy eaured by the nth icrophone and the total ound energy in a zone. Conequently, the icrophone located at the poition with higher preure level i ore enitive to the agnitude error than other icrophone. An additional intereting apect of the icrophone phae error i that the contrat enitivity i alway zero [Eq. (21)] becaue the eaured preure i quared during the energy calculation. Therefore it can be concluded that a far a the acoutic contrat i concerned, the phae calibration i unneceary. 3.3 Poition iatch of a loudpeaker or icrophone When there i a iatch only at the lth loudpeaker poition, fro Eq. (13) and (16), the contrat perturbation can be calculated a db b ¼ K * ;po dr* ; (22) K * ;po ¼ 2xq 0ðI½ðp H b =jjp bjjþðu * b =jjp bjjþš I½ðp H d =jjp djjþðu * d =jjp djjþšþ; (23) where K * ;po i the contrat enitivity vector due to the lth loudpeaker poition iatch. The contrat enitivity vector for loudpeaker poition iatch depend on the inner product 7 between the total preure field p and velocity field of the lth loudpeaker * u and the iaginary part given by Eq. (23) i proportional to the reactive intenity generated by the total preure field and the velocity field. For the poition iatch of the nth icrophone, the contrat enitivity vector K * ðnþ ;po can be derived by ubtituting Eq. (14) into Eq. (16). Specifically, ( K * ðnþ ;po ¼ 2xq 0I½pðr *ðnþ ðr * ðnþ Þu* Þ =jjp b jj 2 Š ðr * ðnþ 2 V bþ 2xq 0 I½pðr * ðnþ ðr * ðnþ Þu* Þ =jjp d jj 2 Š ðr * ðnþ 2 V dþ: The contrat enitivity for the icrophone poition iatch can be evaluated fro the reactive intenity at the icrophone location. Thi iplie that the contrat enitivitie for both the loudpeaker and icrophone poition iatche can be expreed in ter of the reactive intenity. 4. Analyi of acoutic contrat enitivity 4.1 Exeplary deign of a peronal audio yte We now evaluate the contrat enitivity of a peronal audio yte and deontrate how thi ethod of analyi can be ued practically. A an exaple of the peronal audio yte, a linear array with nine equally paced loudpeaker the aperture ize L of which i 0.34 i conidered 2 [ee Fig. 1(a)]. The bright and dark zone are apled with equal pacing (D ¼ 0.02 ) in x and y direction by 1000 (¼ðB t D t BDÞ=D 2 ) icrophone. For thi iulation, under the free field (or anechoic) condition, the loudpeaker are conidered to be onopole ource the control ignal of which are deterined by the acoutic contrat axiization. 5 The preure field at 3 khz, generated by the yte, i depicted in Fig. 1(b). 4.2 Contrat enitivity ap with repect to tranfer function error Figure 2 and 3 depict the contrat enitivitie with repect to the electro-acoutic/poition iatche of icrophone/loudpeaker, which are tered the contrat enitivity ap. (24) EL116 J. Acout. Soc. A. 134 (1), July 2013 Park et al.: Acoutic contrat enitivity to error Downloaded 23 Sep 2013 to Reditribution ubject to ASA licene or copyright; ee

6 Fig. 1. (a) Configuration of a peronal audio yte; the bright zone (V b ) i the region that include the aued uer location, and the dark zone (V d ) i configured near the bright zone (B t ¼ 0:8, B ¼ 0:4, D t ¼ 0:6, D ¼ 0:2, and D ¼ 0:02 ); nine onopole are aued a control loudpeaker (L ¼ 0:32, D ¼ 0:04 ). (b) Preure field (agnitude) generated by the contrat axiization [noralized by the preure at ðx; yþ ¼ð0; 0:4Þ]. The contrat enitivity ap reveal which loudpeaker or icrophone i enitive to the error when the peronal audio yte i driven by the pre-deterined control ignal. In Fig. 2, it can be oberved that the icrophone with higher preure level [ee Fig. 1(b)] i ore enitive (either in poitive or negative direction) than the other. Therefore both in the bright and dark zone, the agnitude calibration of a icrophone in the high aplitude region hould be perfored ore carefully. The contrat enitivity againt the phae error of a icrophone i alway zero, and hence i not hown. Meanwhile, the contrat enitivity of the loudpeaker attain the value of zero for all loudpeaker (not hown). Zero contrat enitivity i the peculiar feature of the peronal audio yte controlled by the acoutic contrat axiization, i.e., the contrat enitivity ay not be zero when loudpeaker are controlled by other control chee. The acoutic contrat axiization chooe the tationary point of the acoutic contrat (@b=@q ¼ 0) a the optial olution atifying the relation ðh H b H b =N bþq ¼ bðh H d H d =N dþq, 5 which draw the zero contrat enitivity. However, thi doe not ean that the acoutic contrat doe not change irrepective of the loudpeaker error. If the all error auption of Eq. (5) and (10) i not valid, the econd-order ter of the contrat perturbation cannot be neglected; db=b ¼ ðjjp b jj2 =jjp b jj 2 jjp d jj2 = jjp d jj 2 Þfða Þ 2 þð/ Þ 2 g. The ditribution of contrat enitivity vector i arked with arrow in Fig. 3. The direction of each arrow indicate the direction that i ot enitive to the icrophone/loudpeaker poition iatch, and the length of the arrow reflect the odulu of Fig. 2. Contrat enitivity ap (frequency at 3 khz) with repect to the electro-acoutic iatch of a icrophone (K;ag). ðnþ The poitive/negative ign of enitivity repreent the increae/decreae of the contrat, and for each cae, the enitivitie are noralized by the overall axial enitivity (jk;agj ðnþ ax ¼ 6: ). J. Acout. Soc. A. 134 (1), July 2013 Park et al.: Acoutic contrat enitivity to error EL117 Downloaded 23 Sep 2013 to Reditribution ubject to ASA licene or copyright; ee

7 Fig. 3. Contrat enitivity ap (drawn for a ingle frequency of 3 khz) with repect to the poition iatch of (a) icrophone (K * ðnþ ;po ) and (b) loudpeaker (K* ;po )in 1. The length of arrow i noralized by the overall axiu odulu of enitivity vector for each cae. contrat enitivity vector. The reultant contrat perturbation i deterined by the inner product of the contrat enitivity vector and the poition iatch vector (dr * or dr * ðnþ ). 5. Concluion The change in the perforance of a peronal audio yte i invetigated for variou tranfer function error. To thi end, the enitivity of acoutic contrat i defined and atheatically expreed in ter of the degree of preure perturbation with an auption that the error or perturbation in the tranfer function i o all that it higher order ter can be neglected. The contrat enitivity i exained for the electro-acoutic iatch of icrophone and loudpeaker, a well a for their poition iatche. To viualize the ditribution of contrat enitivity for variou type of error, a contrat enitivity ap i introduced, and the reult for an exeplary cae of a line array of loudpeaker are explained. It i deontrated that the analyi on the acoutic contrat enitivity can be a ueful guide in the realization of a robut peronal audio yte. Acknowledgent Thi work wa upported in part by the Minitry of Knowledge Econoy grant funded by the Korea governent (Grant No ), and the BK21 (Brain Korea 21) project initiated by the Minitry of Education, Science and Technology, the Converging Reearch Center Progra funded by the Minitry of Education, Science and Technology (2012K001329), and Unanned Technology Reearch Center (UTRC) at Korea Advanced Intitute of Science and Technology (KAIST), originally funded by DAPA, ADD. Reference and link 1 S. J. Elliott and M. Jone, An active headret for peronal audio, J. Acout. Soc. A. 119(5), (2006). 2 J.-H. Chang, C.-H. Lee, J.-Y. Park, and Y.-H. Ki, A realization of ound focued peronal audio yte uing acoutic contrat control, J. Acout. Soc. A. 125(4), (2009). 3 J.-Y. Park, M. Song, J.-H. Chang, and Y.-H. Ki, Acoutic brightne/contrat control perforance due to tranfer function error, in Proceeding of 16th International Congre on Sound and Vibration, Krakow, Poland (July 5 9, 2009). 4 S. J. Elliott, J. Cheer, J.-W. Choi, and Y. Ki, Robutne and regularization of peronal audio yte, IEEE Tran. Audio, Speech, Lang. Proce. 20(7), (2012). 5 J.-W. Choi and Y.-H. Ki, Generation of an acoutically bright zone within an illuinated region, J. Acout. Soc. A. 111(4), (2002). 6 K.-U. Na and Y.-H. Ki, Error due to enor and poition iatch in planar acoutic holography, J. Acout. Soc. A. 106(4), (1999). 7 J.-W. Choi and Y.-H. Ki, Manipulation of ound intenity within a elected region uing ultiple ource, J. Acout. Soc. A. 116(2), (2004). EL118 J. Acout. Soc. A. 134 (1), July 2013 Park et al.: Acoutic contrat enitivity to error Downloaded 23 Sep 2013 to Reditribution ubject to ASA licene or copyright; ee