Sound reproduction in personal audio systems using the leastsquares approach with acoustic contrast control constraint Yefeng Cai, Ming Wu, and Jun Yanga) Key Laboratory of Noise and Vibration Research, Institute of Acoustics, Chinese Academy of Sciences, Beijing 100190, China

(Received 14 August 2013; revised 3 December 2013; accepted 17 December 2013) This paper describes a method for focusing the reproduced sound in the bright zone without disturbing other people in the dark zone in personal audio systems. The proposed method combines the leastsquares and acoustic contrast criteria. A constrained parameter is introduced to tune the balance between two performance indices, namely, the acoustic contrast and the spatial average error. An efficient implementation of this method using convex optimization is presented. Offline simulations and real-time experiments using a linear loudspeaker array are conducted to evaluate the performance of the presented method. Results show that compared with the traditional acoustic contrast control method, the proposed method can improve the flatness of response in the bright zone by sacrificing C 2014 Acoustical Society of America. the level of acoustic contrast. V [http://dx.doi.org/10.1121/1.4861341] PACS number(s): 43.60.Fg, 43.60.Dh, 43.38.Hz [MRB]

I. INTRODUCTION

This study investigates the problem of local sound reproduction in personal audio systems. Personal audio systems focus sound around the user by utilizing a set of loudspeakers. These systems can provide individuals with a private listening space without disturbing other people. One common method for implementing personal audio systems, namely, acoustic contrast control (ACC),1–4 aims to maximize the ratio of acoustic potential energy density between the bright and the dark zones. Shin et al.5 proposed an ACC based method to maximize the acoustic energy difference between two selected acoustic zones and eliminate the ill-conditioning problem caused by matrix inversion. The robustness of ACC is also improved by using a regularization approach.6,7 However, only acoustic contrast is considered in these ACC-based methods. By contrast, the spatial average error between the desired and reproduced field is usually considered in sound field reproduction (SFR) systems, where wave field synthesis (WFS),8,9 Ambisonics,10–13 and least-squares (LS)14–19 approaches are usually used. The WFS method is based on the Huygens–Fresnel principle, which states that sound pressure inside a region is determined by the sound pressure and its gradient on the boundary of the region.9 The Ambisonics approach represents the desired and reproduced sound fields in terms of spherical or cylindrical harmonic functions.11 The mode-matching approach is then applied to compute the coefficients of harmonic functions. The LS approach directly optimizes loudspeaker weight to minimize the spatial average error.14 In contrast to the first two analytical solutions the LS method does not restrict the reproduction setup. Most of these SFR methods only take into account the spatial average error in the bright zone, while they ignore the acoustic contrast in the SFR analysis. However, in practice, spatial

a)

Pages: 734–741

average error should be considered together with acoustic contrast in the design of a local sound field that produces high-quality sound around the user. Recently, a multi-zone reproduction technique has been proposed to reproduce sound in the bright zone while limiting acoustic potential energy in the dark zone, where both of the two performance indices are considered.13–16 Chang and Jacobson20,21 proposed the acoustic contrast control pressure matching (ACC-PM) method and introduced a weighting factor parameter to tune the tradeoff between the two performance indices. However, the relationship between the weighting factor and the acoustic contrast remains unclear. This study proposes an approach that integrates LS-based SFR with the ACC constraint and introduces a constrained parameter that denotes the minimum maintained acoustic contrast. Spatial average error is minimized only when the acoustic contrast is larger than the value of the constrained parameter. The remainder of this paper is organized as follows. The LS-based SFR, ACC, and ACC-PM methods are outlined in Sec. II. The mathematical formulation of the SFR-ACC method is given and its solution is derived in Sec. III. The experimental results are provided in Sec. IV, and the conclusions are drawn in Sec. V. II. PRELIMINARIES AND PROBLEM STATEMENTS A. LS approach for the SFR problem

Figure 1 shows the overview of the LS-based SFR problem. The loudspeaker elements are located at rsj , j ¼ 1;…; L, where L is the number of loudspeaker elements. Coordinates of the control points in the bright zone are rbi , i ¼ 1;…; M, where M is the number of control points. The sound pressure at the ith control point can be expressed as pðrbi ; xÞ ¼ gðrbi j rs ; xÞT wðxÞ;

(1)

gðrbi j rs ; xÞ ¼ ½gðrbi j rs1 ; xÞ; gðrbi j rs2 ; xÞ; Author to whom correspondence should be addressed. Electronic mail: [email protected]

734

J. Acoust. Soc. Am. 135 (2), February 2014

0001-4966/2014/135(2)/734/8/$30.00

:::; gðrbi j rsL ; xÞT ;

(2)

C 2014 Acoustical Society of America V

eq ¼

max where x is the angular frequency; gðrbi jrsj ; xÞ denotes the transfer function between the jth loudspeaker element and the ith control point; and wðxÞ ¼ ½w1 ðxÞ; w2 ðxÞ;…; wL ðxÞT is the weight vector of the loudspeaker array, which should be carefully designed. The LS-based SFR problem amounts to solving the following optimization problem: min wðxÞ

EðxÞ ¼

kdðxÞk22

;

(3)

where kk2 represents the ‘2 norm, GðxÞ ¼ ½gðrb1 jrs ; xÞ; gðrb2 jrs ; xÞ;…; gðrbM jrs ; xÞT , and dðxÞ ¼ ½pd ðrb1 ; xÞ; pd ðrb2 ; xÞ;…; pd ðrbM ; xÞT is the desired sound field. EðxÞ denotes the normalized spatial average error and its level is defined as 10 log10 EðxÞ.The solution to problem (3) is given by w ¼ ðGH GÞ1 GH d;

(6)

where N is the number of control points and pðrqi Þ is the sound pressure at the ith control point rqi in the dark zone. Q is defined as the transfer function matrix between the loudspeaker array and the control points in the dark zone. The idea behind the ACC approach is to maximize the ratio of acoustic potential energy density between the bright and dark zones,1 which leads to the following maximization problem:

FIG. 1. LS-based SFR problem.

kGðxÞwðxÞ  dðxÞk22

N 1X 1 pðrqi ÞH pðrqi Þ ¼ wH QH Qw; N i¼1 N

(4)

w

C¼

eb N wH GH Gw ; ¼ eq M wH QH Qw

(7)

where C denotes the acoustic contrast and its level is defined as 10 log10 C. The optimal weight vector that maximizes C is given as ðQH QÞ1 GH Gwmax ¼ Cmax wmax :

(8)

wmax is equal to a unit eigenvector, which corresponds to the largest eigenvalue of the matrix ðQH Q þ dIÞ1 GH G.1,7 The non-negative d is a preselected regularization parameter for each frequency and can improve the robustness of the system. Additionally, C remains unchanged when w undergoes arbitrary phase rotation and amplitude scaling. Therefore, a more general optimal solution is expressed as w ¼ b expðjhÞwmax ;

(9)

where b is the amplitude scaling factor larger than zero, and h is the arbitrary angle.

where x is no longer marked for convenience in the following text.

C. ACC-PM

B. ACC

The ACC-PM (Ref. 20) method considers the acoustic contrast and the spatial average error simultaneously. The optimization problem can be expressed as

The acoustic potential energy density in the bright zone (Fig. 2) is defined as

min jkQwk22 þ ð1  jÞkGw  dk22 ; w

M 1X 1 pðrbi ÞH pðrbi Þ ¼ wH GH Gw: eb ¼ M i¼1 M

(5)

Similarly, the acoustic potential energy density in the dark zone is

(10)

where j is a weighting factor that brings a tradeoff between the potential energy in the dark zone and the spatial average error in the bright zone. The optimal solution can be derived as w ¼ ½jQH Q þ ð1  jÞGH G1 ð1  jÞGH d:

(11)

III. SFR-ACC

The SFR-ACC method considers both performance indices from a different point of view. It minimizes the spatial average error on the condition that the level of acoustic contrast is no less than a predefined level. The proposed SFR-ACC method can then be described as the following optimization problem: FIG. 2. ACC problem. J. Acoust. Soc. Am., Vol. 135, No. 2, February 2014

min w

kGw  dk22 ;

Cai et al.: Sound reproduction in personal audio systems

(12a) 735

s:t: 10 log10 C  10 log10 Ccon ;

(12b)

where Ccon is the constrained parameter and denotes the minimum allowable acoustic contrast in the sound reproduction system. Equation (8) shows that the acoustic contrast has a maximum value of Cmax . If Ccon is larger than Cmax , then the optimum value of problem (12) does not exist. To guarantee that the solution w is always available, the value of Ccon should be set not to exceed Cmax . The solution to problem (12) for different values of Ccon is discussed in the following paragraphs. A. Ccon 5Cmax

In this case, the spatial average error is optimized without any loss in acoustic contrast. Problem (12) can be rewritten as min w

kGw  dk22 ;

(13a)

(

N wH GH Gw s:t: arg max w M wH QH Qw

) :

(13b)

b; h

H b2 wH max ðG GÞwmax H H ; 2bRefexpðjhÞwH max G dg þ d d

(14)

where Refg stands for the real part of the element. The optimal values can be found by setting the partial derivative of problem (14) with respect to b and h to zero: H absðwH max G dÞ ; H wH max ðG GÞwmax

(15)

where /ðÞ and absðÞ represent the argument and modulus of the element, respectively. Substituting Eq. (15) into Eq. (9), the solution to problem (13) can be obtained as follows: w¼

H absðwH max G dÞ H wH max ðG GÞwmax

   H exp j/ wH max G d wmax :

nðkÞ ¼ min 1ðx; kÞ x ( yT ðH1 þ kH2 Þ1 y; H1 þ kH2  0 ¼ 1 otherwise;

(16)

and

where  implies that the matrix is symmetric semi-definite. The Lagrangian dual problem is max

yT ðH1 þ kH2 Þ1 y

s:t:

k0 H1 þ kH2  0

k

y ¼ ðH1 þ kH2 Þx: 736

min w

s: t:

kGw  dk22 ;   H M H H Ccon Q Q  G G w  0: w N

S¼

M Ccon QH Q  GH G; N " # RefSg ImfSg ImfSg

:

(18)

RefSg

By substituting Eq. (18) into problem (17) and discarding the constant term dH d, the following optimization problem can be expressed with real variables: min x

xT H1 x  2yT x;

J. Acoust. Soc. Am., Vol. 135, No. 2, February 2014

(19)

Problem (19) is a non-convex problem and thus cannot be directly solved using conventional convex optimization techniques. The Lagrangian of problem (19) can be written as 1ðx; kÞ ¼ xT H1 x  2yT x þ kxT H2 x;

(20)

where k  0 is the Lagrange multiplier. Its dual function is expressed as

y ¼ ðH1 þ kH2 Þx

(21)

Defining yT ðH1 þ kH2 Þ1 y  c and using a Schur complement,22 the dual problem can be reformulated as c

max k

" s:t: (22)

(17)

ðM=NÞCcon QH Q  GH G is not a semi-definite matrix, thus the optimization problem (17) is a non-convex problem. We assume that " # " # Refwg RefGH dg ; x¼ ; y¼ Imfwg ImfGH dg " # RefGH Gg ImfGH Gg H1 ¼ ; ImfGH Gg RefGH Gg

s: t: xT H2 x  0:

H h ¼ /ðwH max G dÞ;

b¼

In this case, the spatial average error can be further reduced at the cost of degrading acoustic contrast as follows:

H2 ¼

Using Eq. (9), problem (13) can be expressed as min

B. Ccon