ESTIMATING EAR CANAL GEOMETRY AND EARDRUM REFLECTION COEFFICIENT FROM EAR CANAL INPUT IMPEDANCE

ESTIMATING EAR CANAL GEOMETRY AND EARDRUM REFLECTION COEFFICIENT FROM EAR CANAL INPUT IMPEDANCE Huiqun Deng and Jun Yang Institute of Acoustics, Chinese Acadey of Sciences, Beijing, China 9 ABSTRACT Based on the signal odel of ear canals, a novel ethod for solving the inverse proble of estiating the unique solution of the ear canal area function and the eardru reflection coefficient given the acoustic input ipedance at the entrance of an ear canal is presented. Up-sapling techniques to iprove the accuracy of the estiates are also presented. The perforance of this ethod and factors affecting the accuracy of the estiates are investigated via siulations. It is found that the accuracy of the estiates is liited by the easureent bandwidth of the given ear canal input ipedance. In the audio frequency range, the estiates obtained approxiate well to the true ones. To obtain ore accurate estiates, a wider easureent bandwidth of the ear canal input ipedance is required. Index Ters Ear canal area function, eardru reflection coefficient, acoustic ipedance, inverse ethod. INTRODUCTION Ear canal cross-sectional areas and eardru reflection coefficients are iportant factors deterining the sounds received at huan ears, and their easureents are iportant in any applications such as iddle-ear pathology, psychoacoustic easureents, hearing aid design, and sound reproduction via headphones, to ention a few. It is difficult and invasive to easure ear canal area functions and eardru reflection coefficients inside huan ear canals [-4]. Soe non-acoustical ethods such as CT scan and laser easureents of the ear canals and eardru vibrations cannot reveal the acoustic transforation properties of the ears. Therefore, the developent of non-invasive acoustical ethods to easure ear canal area functions and eardru reflection coefficients has been a research topic. Estiating ear-canal area functions and eardru reflection coefficients fro acoustic easureents at the entrances of ear canals are considered to be non-invasive. Siilar inverse probles have been encountered in speech signal processing to estiate vocal-tract area functions fro acoustic easureents at lips. Based on the Webster s wave equation, it is known that the spatial Fourier coefficients of a vocal-tract area function are related to the forant frequencies of the vocal tract assuing that the glottal boundary is rigid [5-6]. However, the assuption about glottal boundary is unrealistic. To exclude the reflections fro unknown glottal ipedance, short-tie sound pressure signals at the lip opening in response to a unit ipulse of volue velocity are used to derive the area functions of vocal tracts based on the Webster s wave equation [7-8]. This ethod has been adopted to estiate ear canal area functions and the eardru ipedances at reference planes [9]. However, direct easureents of the required shorttie sound pressure ipulse responses are difficult. A gradient ethod for estiating the ear canal area function fro the phase response of the reflection coefficient of an ear canal above 3 k is presented []. This ethod suffers fro the probles of slow convergence. In [], the tiedoain reflection coefficient at the entrance of an ear canal is obtained via inverse Fourier transfor of the frequencydoain reflection coefficient of the ear canal, and is used to estiate the ear canal area function according to the Webster s wave equation. However, the solution to the eardru reflection coefficient is not provided. A ethod for jointly estiating the ear-canal area function and the paraeters of a siplified iddle-ear ipedance odel is proposed via nonlinear optiization given the easured reflection coefficient of an ear canal []. However, the resulting estiates ay be degraded by the siplified iddle-ear odel and initial values of the optiization. The present work odels an ear canal as a ultisectional tube with a varying cross-sectional area function, and derives the relationship between the input ipedance of the ear canal and the ear-canal area function and the eardru reflection coefficient at the eardru reference plane, based on the signal odel of ear canals. Fro this relationship, the ear canal area function and the reflection coefficient at the reference plane are estiated, without iposing any odel of iddle ear ipedances, and hence the potential degradation to the estiates caused by initial values and inaccurate assuptions about the iddle ear ipedance is avoided. In Sec., the signal odel of ear canals is presented, and the relationship between the input ipedance of the ear canal and the ear canal geoetry and eardru reflection coefficient is derived. In Sec. 3, the siulations to validate the ethod and investigate the factors affecting the estiation accuracy are presented. 978--4799-9988-/6/$3. 6 IEEE 5 ICASSP 6

. INVERSION METHOD The goal of this section is to solve for the ear canal area function and the eardru reflectance given the input ipedance of the ear canal. It is known that below 5 k, sound waves in ear canals can be assued as planar waves, and an ear canal can be odeled as an acoustic tube with a varying cross-sectional area function, and the effect of the eardru can be odeled as a concentrated ipedance connected to the ear canal at the ubo point [3, 4]. The portion of the ear canal fro its entrance to the eardru reference plane, which is the plane of wave front at the ubo position, is odeled as an M-sectional tube with equal sectional length L, as shown in Fig., where the first section starts fro the entrance of the ear canal. The terinal ipedance of the M th section is fored by the parallel ipedance of the eardru ipedance and the input ipedance of the residual ear canal beyond the reference plane. Let the reflection coefficient fro the end of the M th section be r T, which is deterined by and the M th crosssectional area as shown in [4]. It is noted that r T corresponds to the eardru reflection coefficient easured at the reference plane in huan ears, and contains the effect of both eardru ipedance and the residual ear canal. Let u + (t) and u - (t) be the going-in and going-out volue velocities at the beginning of the th section, respectively, =,, M. Let U + (f) and U - (f) be the Fourier transfors of u + (t) and u - (t), respectively. In the frequency doain, the continuity of sound pressure and the continuity of volue velocity at the boundary of the th and (+) th sections lead to the following equation [4], jk ( ) L U f ( ) e r U f jk L jk L () U ( f ) r r e e U ( f ) where k f / c j.58 f / D c, () r ( S S ) / ( S S ) (3) D is the diaeter of the th section, c is the sound speed and S is the cross-sectional area of the th section. Define G ( f ) U ( f )/ U ( f ) (4) Then, fro Eqs. () and (4) the following equation holds: ( ) ( jkl (f)) / ( ( ) jkl G f r e G r G f e ) (5) G (f) is related to the input ipedance of the ear canal Z (f): P( f ) U ( f ) U ( f ) ( ) c G f c Z ( f) (6) U ( f ) S ( ) S U ( f ) U ( f ) G f where P (f) and U (f) are the Fourier transfors of the total sound pressure and total volue velocity at the entrance of the ear canal, respectively, S is the area of the entrance of the ear canal, and ρ is the air density. Eq. (6) leads to G ( f ) ( Z ( f )S / c) / ( Z ( f )S / c). (7) Assue that the tube attenuation can be ignored, i.e., Fig. The tube odel of an ear canal. k =πf/c, that the sound signals are sapled at a rate F s, and that the sectional length L of the tube odel is related to F s and the sound speed c as F s =c/l. (8) Then, the discrete-tie signals in the tube odel (Fig.) can be represented using their Z transfors as shown in Fig. [5], where r T (z) is the Z transfor of the reflection coefficient fro the end of the M th section. In the Z doain, Eq. (5) becoes G (z) ( r z G (z)) / ( r G (z) z ) (9) where G (z) is the Z transfor corresponding to G (f). According to the signal flow graph shown in Fig., G (z) is the transfer function of an IIR (Infinite Ipulse Response) filter, and can be expressed as G ( z) r z ( r ) r z... () g () z g () z... where g (n) is the ipulse response of G (z). Eq. () eans that g (n)=, n<=, () and that g () r () Given g (n), we derive g + (n), the ipulse response of G + (z), as follows. Inserting the second line of Eq. () into Eq. (9) leads to the following equation: 3 ( r g ()) z g () z g (3) z... G ( z) (3) 3 ( r g () ) z r ( g () z g (3) z...) Inserting Eq. () to Eq. (3), then g + (n) and g (n) are related as: (g () ) g ( n ) g ()( g () g ( n ) g (3) g ( n 3)...) g () ( n ) g (3) ( n 3)... g (n) Replacing n with n+ in Eq. (4) leads to the following equation: g u + (t) u - (t) n u M + (t) u M - (t) i ( n) g () (4) g ( n ) g () g ( i ) g ( n i) (5) Thus, for =, M-, g + (n) can be derived fro g (n) according to Eqs. (5), r, can then be obtained fro g () according to Eq. (), S + can be derived fro S according to Eq. (3), G + (f) can be derived fro G (f) according to Eq. (5). g (n), which is the ipulse response of the volue velocity reflection coefficient fro the entrance 5

U + (z) U - (z) z -/ z -/ r +r -r U + +(z) Fig. The Z-doain signal odel of the ear canal. of the ear canal, is the key to the inverse solution. g (n) can be obtained fro the inverse Fourier transfor of G (f). G (f) can be deterined fro the input ipedance Z (f) of the ear canal according to Eq. (7). Z (f) and S can be easured at the entrance of the ear canal. The perforance of this ethod is investigated via siulation in the next section. 3. SIMULATION 3.. Synthesis of the input ipedance of the ear canal The focus of the present paper is on the new ethod for obtaining the ear canal area function and the eardru reflection coefficient at the eardru reference plane given the input ipedance of the ear canal, rather than the easureents of the ear canal ipedances and area functions of huan ears, which are nontrivial tasks. Therefore, synthetic input ipedances of a odel ear canal are used to investigate the perforance of the proposed ethod. The cross-sectional radius of the ear canal odel in Fig. is deterined according to the odel ear canal specified in [6] at positions [.5,,.5,, 7] fro the entrance of the ear canal, as shown using the thin line in the iddle-right panel of Fig. 3. The sectional length of the ear canal odel is L=.5. The total length of the ear canal is 7, and the ubo position is at 4 fro the end of the ear canal. The eardru ipedance values at frequencies f=[5,, 5,, 5] [4] are used here, and the agnitude and phase responses of are plotted using the thick dotted lines in the top-left and the top-right panels of Fig.3, respectively. Given the ear canal crosssectional area and, the input ipedance of the ear canal Z is calculated iteratively according to [7] c c c (6) Z ( Z j tan( k L)) / ( jz tan( k L)) S S S r T(z) where Z, =M, M-,, is the ipedance looking fro the beginning of th section into the end of the ear canal, and Z M+ =. is calculated given and the ear canal area function as shown in [4]. The agnitude and phase responses of and Z are plotted using dash and solid lines, respectively, in the two top panels of Fig.3. -r U - + (z) z -/ z -/ U + M (z) U - M (z) z -/ z -/ 3.. Inverse solution Given Z and the area of the entrance of the ear canal S, then the frequency response of G (f) is calculated according to Eq. (7). The agnitude and phase responses of G (f) are plotted using dash-dot lines in the botto-left and bottoright panels of Fig.3, respectively. The upper frequency liit of and G (f) is 5 k. If the inverse Fourier transfor of G (f) is used as g (n), then the sapling rate for the inversion is F s =3 k, and the sectional length for the inversion of the ear canal geoetry is L inv =c/f s =5.9, which is low in the spatial resolution considering that the average length of ear canals is about 7. The spatial resolution can be iproved by obtaining an up-sapled g (n) as follows. First, according to the signal flow shown in Fig., one can odel G(z) as an IIR filter with N poles and N zeros. The optial G(z) that atches G (f) is obtained via the Matlab function invfreqz(h,w,n,), given the values of G (f) at frequencies f=[5,, 5,, 5], the sapling rate F s =3. k, and G(z=e jπ )=G (f=5k). In this work, the optial N value is deterined such that the axiu difference between G (f) and G(z=e j π f/fs ) is iniu copared to that given by other N values in the range of 3 and 3: * N G G e (7) j fi/ Fs argmin(ax (f ) ( ) ) i 3N 3 i where f i f. Second, obtain the ipulse response of G(z σ ), where σ=, followed by a low-pass filter with a cut-off frequency 5 k at the sapling rate F sinv =σf s =6 k. Let g (n) be sapled at F sinv and truncated to length M=6. The frequency response of G(z σ ) at frequencies f inv =[, F sinv /M,., (M-)F sinv /M] is calculated. To apply the low-pass filter to G(z σ ), set the frequency response of G(z σ ) to zeros for 5<f inv <F sinv -5. The inverse Fourier transfor of the low-pass filtered frequency response of G(z σ ) yields g (n) at the sapling rate F sinv, as shown in the iddle-left panel of Fig. 3. The sectional length for the estiate of the ear canal area function is L inv,=c/f sinv =.9 in this case. Given g (n), then g (n), r, =,, is derived according to Eq. (5) and Eq. (), respectively. Given G (f), S and r, then G (f) and S, =,, are derived according to Eq.(5) and Eq. (3), respectively. For this siulation, the first axiu negative peak of g (n) is located at about.398 s (iddle-left panel of Fig.3), which corresponds to a reflection plane at a distance L =4.6 fro the entrance of the ear canal, i.e., about L =.6 beyond the eardru reference plane. Thus, S and G can only be estiated up to <=(L -L )/L inv,. The radius of S is shown using the thick line in the iddle-right panel of Fig.3. The agnitude and phase responses of G at different distances to the reference plane are shown using different lines in the 53

.k.s. acoustic oh.k.s. acoustic oh 8, and Z Phase responses of, and Z 5, and Z Phase responses of, and Z 5 7-5 -5 Z 6. Z 3 4 Estiated g at F sinv =6 k Z 3 4 Estiated and true sectional radius 5 5. Z 4 Estiated g at F sinv =6 k 4 Estiated and true sectional radius 5. 4. 4 3 3 -. -. -. -.3.5 s estiated true distance fro the entrance () -. -.3.5 s estiated true distance fro the entrance () r T and G at distances fro ubo Phase responses of r T and G r T and G at distances fro ubo Phase responses of r T and G.8.6.4. G 3.35 8.384 3.44.499 3 4 - -4-6 G 3.35 8.384 3.44.499 3 4.8.6.4. G 3.3 8.47 3.33.4 4 - -4-6 G 3.3 8.47 3.33.4 4 Fig. 3 Siulation given realistic ear canal ipedance Z. botto-left and botto-right panels of Fig. 3. In practice, the ubo position is unknown. It can be estiated that the section closest to the eardru reference plane is nubered as <=(L -L )/L inv, where L can be estiated fro the peak index of g (n), and L is about.6. 3.3. Factors affecting the estiates It is found fro iddle-right panel of Fig. 3 that there are soe differences between the estiate of the ear canal radius function and the original one used for synthesizing Z, and that further up-sapling of g (n) cannot reduce the differences. This is explained by the liitation in the bandwidth of the given input ipedance Z. It is known that the th spatial Fourier coefficient of the area function of a tube with a varying sectional area is related to the th resonance frequency of the tube [5-6]. Since the given input ipedance of the ear canal is liited to audio frequencies, only the first a few resonance frequencies of the ear canal are available, and hence only the first a few spatial Fourier coefficients of the ear canal area function can be obtained, resulting in an incoplete set of the spatial Fourier coefficients to represent the area function of the ear canal. To verify this cause, another siulation is perfored assuing that the frequency response of the given Z are specified at f=[5,,, 5], which is synthesized according to Eq. (6) with the values of and hence being specified at f=[5,,, 5]. The agnitude and phase responses of the superbandwidth, and Z are shown in the top panels of Fig. Fig. 4. Siulation given Z up to 5 k 4 using the dotted green line, the dash red line and the solid blue line, respectively. For the cases with F s =3. k and σ=, the sae procedure as described in Sections 3.-3. is applied. It is shown in iddle-right panel of Fig.4 that the estiate of the ear canal radius function is nearly equal to the true one. Siilar results are obtained for σ=, 3, 4, confiring the effect of bandwidth on the estiate of ear canal geoetry. It is noted that for both cases shown in Figs. 3 and 4, when th section does not contain the eardru reference plane, as the distance between the beginning of the th section and the eardru reference plane decreases, G (f) approxiates r T ore and ore. In the inversion, the tube attenuation is assued according to Eq. () as given in [8]. For estiation on real huan ears, a realistic assuption about the tube attenuation of the ear canal is required to obtain accurate estiate of eardru reflection coefficients. It is also found (not shown) that the estiate of the ear canal area function is not affected by assued tube attenuation. 4. CONCLUSION The present ethod shows that the unique solution of the ear canal area function and the eardru reflection coefficient at the reference plane given the input ipedance of an ear canal can be derived based on the signal odel of ear canals, without using iddle-ear odels, and the probles of such odels and non-linear optiization are avoided. The ethod is expected to have applications in hearing aid and headphone syste design, iddle ear pathology, auditory odel, psycho-acoustic easureents, etc. 54

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