Parameterization of the three-dimensional room transfer function in

Bu et al.: JASA Express Letters

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Parameterization of the three-dimensional room transfer function in horizontal plane Bing Bua) Speech and Audio Signal Processing Laboratory, School of Electronic Information and Control Engineering, Beijing University of Technology, Beijing 100124, China [email protected]

Thushara D. Abhayapala Applied Signal Processing Group, Research School of Engineering, College of Engineering and Computer Science, The Australian National University, Canberra, Australian Capital Territory 0200, Australia [email protected]

Chang-chun Baob) Speech and Audio Signal Processing Laboratory, School of Electronic Information and Control Engineering, Beijing University of Technology, Beijing 100124, China [email protected]

Wen Zhang Applied Signal Processing Group, Research School of Engineering, College of Engineering and Computer Science, The Australian National University, Canberra, Australian Capital Territory 0200, Australia [email protected]

Abstract: This letter proposes an efficient parameterization of the threedimensional room transfer function (RTF) which is robust for the position variations of source and receiver in respective horizontal planes. Based on azimuth harmonic analysis, the proposed method exploits the underlying properties of the associated Legendre functions to remove a portion of the spherical harmonic coefficients of RTF which have no contribution in the horizontal plane. This reduction leads to a flexible measuring-point structure consisting of practical concentric circular arrays to extract horizontal plane RTF coefficients. The accuracy of the above parameterization is verified through numerical simulations. C 2015 Acoustical Society of America V

[NX] Date Received: June 5, 2015

Date Accepted: August 13, 2015

1. Introduction Modeling a room transfer function (RTF) plays a vital role in many acoustic applications, such as room equalization, echo cancellation, and sound field reproduction. The techniques employed to RTF modeling can be broadly grouped into three categories, namely, geometric model methods, physics-based methods, and measurement-based methods. In general, the geometric model methods1,2 are efficient to only calculate the high-frequency component of RTF due to lack of wave characteristics (e.g., diffraction effects). In contrast, the physics-based methods,3,4 which are able to inherently model the wave-related characteristics, are usually applied to the low-frequency component of RTF. Recent works5 have attempted to build hybrid models via the combination of geometric model methods and physics-based methods in order to synthesize accurate RTF over the entire audible bandwidth and improve the computational efficiency. However, the attainable levels of model accuracy of the above methods excessively depend on the a priori knowledge about room geometry and reflection coefficients of wall covering. The measurement-based method is the only way to avoid using the room geometrical properties as a priori knowledge. In practice, the RTF measured at particular positions are usually modeled as a pole/zero system implemented with a finite impulse response filter.6 An alternative method has been used for room equalization in the multichannel reproduction which measures the RTF over multiple points to design an

a)

Also at: Applied Signal Processing Group, Research School of Engineering, College of Engineering and Computer Science, The Australian National University, Canberra ACT 0200, Australia. b) Author to whom correspondence should be addressed. EL280 J. Acoust. Soc. Am. 138 (3), September 2015

C 2015 Acoustical Society of America V

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inverse filter using least squares techniques. In such cases, good equalization performance can be shown at the set of design points, while large degradations may be caused elsewhere within the control region. This is due to the fact that even a small-scale variation in source/receiver positions can cause a drastic variation in the RTF.8 In the two-dimensional (2D) reverberant room, an RTF parameterization method9 was first proposed by decomposing the measured RTF into a finite set of cylindrical harmonic coefficients, which is sufficient to represent the RTF between a fixed source point and a receiver circular region. Later works10,11 have attempted to use higher order loudspeakers and a calibration microphone array to measure the RTF coefficients and actively compensate the 2D sound fields. However, these approaches do not consider reflections from floors and ceilings. To address this problem, a novel three-dimensional (3D) RTF parameterization,12,13 which is available for variations in the source and receiver spherical regions, was given in terms of a modal expansion of 3D basis functions. As the listening positions are usually bounded to the horizontal plane, it seems to be redundant to parameterize RTF between spherical regions. In the 3D reverberant room, this letter investigates the 3D RTF parameterization in the horizontal plane, which can be applied to cancel unintended effects from reflections for horizontal implementation of room equalization. The problem is different with 2D RTF parameterization12 that only considers height-invariant reflections. Our goal is to seek and extract a set of efficient RTF coefficients to completely represent the horizontal plane RTF between two continuous circular regions. According to the underlying properties of the associated Legendre functions, the RTF coefficients are described in terms of azimuth harmonic decomposition for both source and receiver regions. Correspondingly, we also design a novel measuring-point structure consisting of concentric circular arrays with flexible loudspeaker/microphone placements. The structure is practical to derive the loudspeaker weights to generate a unit-amplitude outgoing mode in the source region, and to extract the RTF coefficients from the measured RTF in the receiver region. 2. Problem formulation As shown in Fig. 1, we assume the receiver region Xsph to be a sphere of radius Rr centered at the origin O, and the source region Ksph to be another sphere of radius Rs centered at the origin Os. In a spherical coordinate system, the receiver point within Xsph is denoted by x  (rx, hx, /x), and the source point within Ksph is denoted by y where ðsÞ ðsÞ y ¼ y(s) þ Rsr with yðsÞ  ðry ; hðsÞ y ; /y Þ representing the same source location with respect to Os and Rsr representing the vector connecting O to Os. In a reverberant environment, the RTF can be decomposed into spherical harmonics, which form an orthogonal basis set of the solution to the wave equation. Specifically, the RTF between x and y can be given in terms of the direct path Hdir(x, y, k) and the reverberant path Hrvb(x, y, k) as follows:13,14 H ðx; y; kÞ ¼ Hdir ðx; y; kÞ þ Hrvb ðx; y; kÞ 1 X n X 1 X v X eikkxyk þ ik anm ¼ vl 4pkx  yk n¼0 m¼n v¼0 l¼v       jn krðysÞ jv ðkrx ÞYnm hðysÞ ; /ðysÞ Yvl ðhx ; /x Þ;

(1)

where k ¼ 2pf/c is the wavenumber (with f the frequency and c the speed of wave propagation), jn ðÞ is the first kind spherical Bessel function of order n; Ynm ðÞ denotes the

Fig. 1. The illustration of 3D RTF in horizontal plane: the gray areas are the source and receiver regions of interest, and the black dots show the loudspeaker and microphone array configuration. J. Acoust. Soc. Am. 138 (3), September 2015

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spherical harmonic of order n and degree m, and ðÞ denotes the complex conjugate operation. anm vl are the RTF coefficients, which do not depend on the source/receiver positions. Also, anm vl represents the sound field coefficients of the reverberant path at Xsph caused by a unit-amplitude nth-order and mth-mode outgoing sound field from Ksph . In the horizontal plane, the source/receiver spherical regions are bounded to be circular regions, namely, Kc and Xc, respectively. Both the co-latitude angles with respect to O and Os are equal to p/2. In this case, we first investigate the contribution of spherical harmonic basis for different orders n and modes m. The spherical harmonics at the co-latitude angle of p/2 are termed as azimuth harmonics, defined as Ynm ðp=2; /Þ ¼ P nm ð0ÞEm ð/Þ;

(2) pffiffiffiffiffiffi where P nm ðÞ is the normalized associated Legendre function and Em ð/Þ ¼ e = 2p. According to Ref. 15, P nm (0) ¼ 0 if n þ jmj is odd. By applying the underlying property and Eq. (2) into Eq. (1), the azimuth harmonic decomposition of the RTF is given by im/

Hhor ðx; y; k Þ ¼

Ns X Nr X n v X X eikkxyk a nm þ ik vl 4pkx  yk n¼0 m¼n v¼0 l¼v |fflfflfflffl{zfflfflfflffl} |fflfflfflffl{zfflfflfflffl}





jn krðysÞ

nþjmj¼even vþjlj¼even

   jv ðkrx ÞP nm ð0ÞP vl ð0ÞEm /ðysÞ El ð/x Þ;

(3)

where a nm vl represents the sound field coefficients of the reverberant path within Xc caused by a unit-amplitude nth-order and mth-mode outgoing sound field from Kc. For the rest of the letter, we denote a nm vl as horizontal plane RTF coefficients. In order to extract horizontal plane RTF coefficients, the outgoing modes from Kc need to be produced to obtain corresponding reverberant field. In Sec. 3.1, the outgoing modes can be synthesized by a weighted sum of the loudspeaker signals, which avoids the physical production of each outgoing mode. According to the natural truncation property,16 the outgoing field from Kc can be truncated to Ns ¼ dkeRs/2e. Similarly, the interior field within Xc can be truncated to Nr ¼ dkeRr/2e. In terms of the horizontal plane, the above parameterization shows that a set of ½ðNs þ 1ÞðNs þ 2Þ=2  ½ðNr þ 1ÞðNr þ 2Þ=2 coefficients can completely represent the RTF between the source region of radius Rs and the receiver region of radius Rr. 3. Extraction of horizontal plane RTF coefficients using multiple concentric circular arrays Since we are interested in the RTF that is restricted to the horizontal plane, the geometry of the open spherical shell13 is not the most efficient strategy for extracting horizontal plane RTF coefficients. This section develops a practical and flexible geometry consisting of multiple concentric circular arrays of loudspeakers and microphones to generate a unit-amplitude outgoing mode from Kc, and then to extract the sound field coefficients of the corresponding room response within Xc. 3.1 Production of a unit-amplitude outgoing mode We propose the mode-matching on circles17 to derive the weight vector of the loudspeaker array for generating a specific unit-amplitude outgoing wave of order n0 and mode m0 . The unit-amplitude outgoing wave can be expressed as ðsÞ ðsÞ Sout ðzðsÞ ; kÞ ¼ hn0 ðkrðsÞ z ÞYn0 m0 ðhz ; /z Þ; ðsÞ

(4)

ðsÞ ðsÞ ðrðsÞ z ; hz ; /z Þ

denotes the observation point outside Kc, and hn0 ðÞ is the where z ¼ first kind spherical Hankel function of order n0 . The sound field coefficients of the outgoing wave are given by  1; n ¼ n0 and m ¼ m0 ; (5) anm ¼ 0; otherwise: Suppose there is a set of Q concentric circles located at radius rq with respect to Os, where q ¼ 1,…,Q. First, consider a continuous loudspeaker distribution on the qth circle, the loudspeaker weights are described by a spatial function called a circular continuous aperture function qq(/, k),18 which is a periodic function of azimuth angle /. We can expand the aperture function by a Fourier series as qq ð/; kÞ ¼

1 X

bðqÞ m ðkÞEm ð/Þ;

(6)

m¼1

where bðqÞ m ðkÞ are the Fourier coefficients of the aperture function. EL282 J. Acoust. Soc. Am. 138 (3), September 2015

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Based on the mode-matching on circles, each mode of the outgoing wave in Eq. (4) is matched to the corresponding mode of the aperture function. As a result, we obtain a set of simulation equations in matrix form, as Am ¼ Tm Bm ; for m ¼ Ns ; …; Ns ;

(7)

where 2

jjmj ðkr1 ÞP jmjm ð0Þ

 .. . .. . 

6 6j ðkr ÞP ð0Þ 6 Tm ¼ ik 6 jmjþ2 1 ðjmjþ2Þm .. 6 4 . jNe ðkr1 ÞP Ne m ð0Þ

jjmj ðkrQ ÞP jmjm ð0Þ

3

7 jjmjþ2 ðkrQ ÞP ðjmjþ2Þ m ð0Þ 7 7 7; .. 7 5 . jNe ðkrQ ÞP Ne m ð0Þ

(8)

T ð2Þ ðQÞ T Bm ¼ ½bð1Þ m ; bm ; …; bm  , and Am ¼ ½ajmjm ; aðjmjþ2Þm ; …; aNe m  . Ne is equal to Ns if jmj þ Ns is even; otherwise it is equal to Ns  1. The vector of the Fourier coefficients Bm can be solved using the least squares method as Bm ¼ T†m Am , where ðÞ† denotes the Moore-Penrose inverse. Because Tm is only related to the radii of circles, the mode-matching on circles method provides a way to choose the radii of circles appropriately according to the underlying structure of Tm. In practical measurements, the circular aperture functions are discretized to derive the loudspeaker weights. According to the Shannon sampling theorem, if we place Lq  2Nq þ 1 loudspeakers equiangularly on the qth circle, we can exactly reproduce qq(/, k).18 Thus, the weights of the lth loudspeaker on the qth circle are given by 0

Ns X

0

wn m ð/ql ; kÞ ¼

bðqÞ m ðkÞEm ð/ql ÞDq ;

(9)

m¼Ns

where Dq ¼ 2p/Lq is the angular space of the loudspeakers and /ql ¼ lDq results from uniform sampling. Nonuniform sampling could also be recommended as long as the maximum angular space is not larger than Dq. 3.2 Extraction of horizontal plane RTF coefficients Using a concentric circular array of microphones, we discuss how to extract the horizontal plane RTF coefficients for a specific order n0 and mode m0 . The RTF coefficients are equivalent to the spherical harmonic coefficients of the reverberant path within Xc caused by a unit-amplitude n0 th-order and m0 th-mode outgoing wave from Kc. Considering the l0 th microphone located on the q0 th circle, the incident pressure of the reverberant path at this microphone is described as Pq0 l 0 ðn0 ; m0 ; kÞ ¼

Q X L X

0

0

wn m ð/ql ; kÞHrvb ðxq0 l0 ; yql ; kÞ;

(10)

q¼1 l¼1

where Hrvb ðxq0 l0 ; yql ; kÞ denotes the reverberant path of RTF between the microphone at ðmicÞ

ðsÞ

ðsÞ

xq0 l 0  ðrq0 ; p=2; uq0 l 0 Þ and the loudspeaker at yql ¼ yql þ Rsr with respect to O, where yql  ðrq ; p=2; /ql Þ represents the lth loudspeaker location on the qth circle with respect to Os. Due to the orthonormality constraint of spherical harmonics, horizontal plane RTF coefficients cannot be directly extracted by a sum of the incident pressures at every microphone for a concentric circular array. Therefore, we adopt the conversion relationship between azimuth harmonic coefficients and spherical harmonic coefficients15 to derive horizontal plane RTF coefficients. Based on the azimuth harmonic 0 0 ;n ;m0 Þ decomposition, we can calculate the azimuth harmonic coefficients cðq ðkÞ of the l reverberant path on the q0 th circle (q0 ¼ 1, 2,…, Q0 ) using the following approximation: L

0 0 0 cðlq ;n ;m Þ ðkÞ

q 2p X Pq0 l 0 ðn0 ; m0 ; k ÞEl ðuq0 l0 Þ; Lq0 l0 ¼1 0

(11)

where Lq0 > 2Nq0 þ 1 is the number of microphones on the q0 th circle, and Nq0 ðmicÞ ¼ dkerq0 =2e is the truncation order. The azimuth harmonic coefficients can be expressed as a weighted sum of RTF coefficients, which can be written in matrix form 0

0

0

0

Jl anl m ¼ cnl m ; J. Acoust. Soc. Am. 138 (3), September 2015

for l ¼ Nr ; …; Nr ;

(12) Bu et al. EL283

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where 0

0

0

0

0

0

0

0

m m anl m ¼ ½a njljl ; a nðjljþ2Þl ; …; a nNm T ; e0 l

2 6 6 Jl ¼ 6 6 4 0

0

ðmicÞ

jjlj ðkr1

ðmicÞ

ÞP jljl ð0Þ jjljþ2 ðkr1

ðmicÞ

ÞP ðjljþ2Þl ð0Þ    jNe0 ðkr1 ..

.. .

.. .

ðmicÞ jjlj ðkrQ0 ÞP jljl ð0Þ

ðmicÞ jjljþ2 ðkrQ0 ÞP ðjljþ2Þl ð0Þ

0

0

0

0

0

0

ÞP Ne0 l ð0Þ

.. .

.

ðmicÞ

   jNe0 ðkrQ0 ÞP Ne0 l ð0Þ

3 7 7 7; (13) 7 5

0

and cnl m ¼ ½clð1;n ;m Þ ; clð2;n ;m Þ ; …; clðQ ;n ;m Þ T . Ne0 is equal to Nr if jlj þ Nr is even; otherwise it is equal to Nr  1. Then, we can solve Eq. (12) in the least square sense as 0 0 0 0 anl m ¼ J†l cnl m . In Sec. 3.3, we choose the radii of circles appropriately for avoiding the ill-conditioning problem of Jl, which guarantees that horizontal plane RTF coefficients are extracted efficiently using concentric circular array. So far, we present the procedure of extracting the RTF coefficients for a specific order n0 and mode m0 . Considering all outgoing modes from Kc, we need to repeat the above procedure to extract all the RTF coefficients. 3.3 Array placement The array placement impacts the validity of Moore-Penrose inverse T†m and J†l . To avoid the ill-conditioning problem of Tm, we adopt the guideline15 to make Tm become close to an upper triangular matrix within the broadband frequency k 2 ½kl ; ku . Specifically, we choose the radii of circles as rq ¼ 0;

4 8 2Nsmax ; ; …; ; ku e ku e ku e

for q¼ 1; 2; 3; …; Q;

(14)

where Nsmax ¼ dkueRs/2e is the maximum truncation order. With the choice of radii, the soundfield on the qth circle is order limited to Nq ¼ dkerq/2e. Similar to Tm, we can make Jl become close to a lower triangular matrix for avoiding the ill-conditioning problem. Therefore, the microphone locations are determined similarly as the loudspeaker locations. In practice, given that the room characteristics remain unchangeable, the measured RTF can be acquired using a single loudspeaker and a single microphone moved along the respective arrays. 4. Simulation results To evaluate the performance of the proposed RTF parameterization, an example of multiple concentric circular arrays is examined in the following simulation setup. The design frequency is assumed up to fmax ¼ 1 kHz, and both the radii of source/receiver regions are limited to 15 cm, which is the suitable size of a typical human head. Specifically, we place 3 circles of loudspeakers and microphones with 1, 5, and 9 loudspeakers and microphones in respective horizontal planes. Thus, we use a total of 15 omnidirectional loudspeakers and 15 omnidirectional microphones to parameterize the RTF over a source circular region of radius Rs ¼ 15 cm and a receiver circular region of radius Rr ¼ 15 cm. We use the image source method1 to measure the actual room response in the 3D reverberant room (size 6  5  2.5 m) with its center defined as the origin O. A total of 124 image-sources up to the third order are included with wall reflection coefficients [0.9 0.9 0.9 0.9 0.7 0.7]. The vector from O to Os is considered to be Rsr  (1.5, 1, 1) m, which makes the source region and the receiver region be not in the same plane. Figures 2(a) and 2(b) show the actual and synthesized RTF between a fixed source at

Fig. 2. (Color online) (a) The actual and (b) synthesized RTF between a fixed source at y  (1.6, 1.05, 1) m and all points in the receiver circular region with Rsr  (1.5, 1, 1) m. (c) The actual and (d) synthesized RTF between a fixed receiver at x  (0.05, 0.1, 0) m and all points in the source circular region with Rsr  (1.5, 1, 1) m. EL284 J. Acoust. Soc. Am. 138 (3), September 2015

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Fig. 3. (Color online) The parameterization error curves have been averaged over 50 trial runs for several SNRs, using 60  60 ¼ 3600 combinations of the source and receiver design points.

y  (1.6, 1.05, 1) m and all points in the receiver circular region for a frequency of f ¼ 800 Hz, respectively. In contrast, as shown in Figs. 2(c) and 2(d), we also demonstrate the robustness to source variations by plotting RTF between a fixed receiver point at x  (0.05, 0.1, 0) m and all points in the source circular region. In these two examples, within the limited region of interest, both the synthesized results agree well with the actual RTF within the desired circular region and validate the accuracy of the proposed RTF parameterization. Further considering the broadband performance, we plot the parameterization error curves over a range of 200–1000 Hz. The parameterization error is defined as relative mean square error between the synthesized RTF and the actual RTF within the desired source and receiver regions. In the simulation, we calculate the parameterization error using 60 design points within Kc and 60 design points within Xc. The parameterization error curves are plotted in Fig. 3 for several signal-to-noise ratios (SNRs) averaged over 50 trial runs. The results show that the parameterization errors above 20 dB SNR always remain lower than 15 dB up to the maximum frequency of interest 1 kHz. Although the local peak values exist at a number of frequencies corresponding to the Bessel zeros, the multi-radius geometry provides a potentially practical structure for avoiding too large peaks over a wide frequency range. The problem of local peak values eventually disappears with the decrease of SNR, because the large measurement noise covers up the effect of Bessel zeros. 5. Conclusion We develop a novel measuring-point structure to parameterize the 3D RTF in a horizontal plane. The proposed parameterization not only reduces the number of RTF coefficients, but also provides a flexible array geometry with the ability to avoid Bessel zeros. We have demonstrated a practically realizable array design of three concentric circles to parameterize the RTF at frequencies below 1 kHz. Simulations suggest that the synthesized results agree well with the actual RTF between two desired circular regions for mono-frequency and broadband situations. Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant Nos. 61231015 and 61201197), Scientific Research Project of Beijing Educational Committee (Grant No. KM201310005008), the Ph.D. Programs Foundation of Ministry of Education of China (Grant No. 20121103120017), and the Australian Research Council’s Discovery Projects funding scheme (Project No. DP140103412). B.B. is a joint Ph.D. student between Beijing University of Technology and Australian National University, funded by the China Scholarship Council. References and links 1

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