Weighted pressure matching based on kernel interpolation for sound ﬁeld reproduction Shoichi KOYAMA and Kazuyuki ARIKAWA1

2025-05-06 0 0 2.16MB 9 页 10玖币

侵权投诉

Weighted pressure matching based on kernel interpolation

for sound ﬁeld reproduction

Shoichi KOYAMA and Kazuyuki ARIKAWA(1)

(1)Graduate School of Information Science and Technology, The University of Tokyo, Japan, koyama.shoichi@ieee.org

ABSTRACT

A sound ﬁeld reproduction method called weighted pressure matching is proposed. Sound ﬁeld reproduction is

aimed at synthesizing the desired sound ﬁeld using multiple loudspeakers inside a target region. Optimization-

based methods are derived from the minimization of errors between synthesized and desired sound ﬁelds, which

enable the use of an arbitrary array geometry in contrast with integral-equation-based methods. Pressure match-

ing is widely used in the optimization-based sound ﬁeld reproduction methods because of its simplicity of

implementation. Its cost function is deﬁned as the synthesis errors at multiple control points inside the target

region; then, the driving signals of the loudspeakers are obtained by solving a least-squares problem. However,

in pressure matching, the region between the control points is not taken into consideration. We deﬁne the cost

function as the regional integration of the synthesis error over the target region. On the basis of the kernel in-

terpolation of the sound ﬁeld, this cost function is represented as the weighted square error of the synthesized

pressures at the control points. Experimental results indicate that the proposed weighted pressure matching

outperforms conventional pressure matching.

Keywords: Sound ﬁeld reproduction, Kernel interpolation, Weighted pressure matching

1 INTRODUCTION

Sound ﬁeld reproduction is aimed at synthesizing spatial sound using multiple loudspeakers (or secondary

sources). Such a technique can be applied to virtual/augmented reality audio, generation of multiple sound

zones for personal audio, and noise cancellation in a spatial region.

Sound ﬁeld reproduction methods can be classiﬁed into two major categories: integral-equation-based and

optimization-based methods. The integral-equation-based methods are developed from the boundary integral

representations derived from the Helmholtz equation, such as wave ﬁeld synthesis and higher-order ambison-

ics [3,19,17,1,24,12]. The optimization-based methods are derived from the minimization of a certain

cost function deﬁned for synthesized and desired sound ﬁelds in a target region, such as pressure matching and

mode matching [16,9,7,17,4,21,11]. Many integral-equation-based methods require the array geometry of

loudspeakers to have a simple shape, such as a sphere, plane, circle, or line, and driving signals are obtained

from a discrete approximation of an integral equation. In optimization-based methods, the loudspeakers can be

placed arbitrarily, and driving signals are generally derived as a closed-form least-squares solution. In particular,

pressure matching is widely used among the optimization-based methods because of its simplicity of imple-

mentation. Pressure matching is based on synthesizing the desired pressures at a discrete set of control points

placed over the target region.

An issue of pressure matching is that the region between the control points is not taken into consideration

because of the discrete approximation. Therefore, its reproduction accuracy can deteriorate when the distribution

of the control points is not sufﬁciently dense. However, the smaller the number of control points, the better in

practice, because the transfer functions between the loudspeakers and control points are generally measured in

advance. We propose an optimization-based sound ﬁeld reproduction method called weighted pressure matching.

We deﬁne the cost function as the regional integration of the synthesis error over the target region. On the

basis of the kernel interpolation of the sound ﬁeld [20,22], this cost function is represented by the pressures

at the control points with the regional integration of kernel functions. When the same kernel functions are used

for interpolating primary and secondary sound ﬁelds, i.e., the desired ﬁeld and the sound ﬁeld of each loud-

speaker, respectively, the driving signal is obtained as the solution of weighted least squares problem with the

arXiv:2210.14711v1 [eess.AS] 26 Oct 2022

: Target region

Secondary source

Figure 1. The desired sound ﬁeld is synthesized in the target region Ωusing multiple secondary sources.

weighting matrix deﬁned by using the kernel functions. Experimental evaluation comparing pressure matching

and weighted pressure matching is performed.

2 PROBLEM STATEMENT AND PRIOR WORKS

2.1 Problem formulation

Suppose that Lsecondary sources (loudspeakers) are placed around a target region Ω⊂R3as shown in Fig. 1.

The sound ﬁeld usyn(r

r,ω)at the position r

r∈R3and angular frequency ω∈Rsynthesized using the secondary

sources is represented as

usyn(r

r,ω) =

∑

l=1

dl(ω)gl(r

r,ω),(1)

where dl(ω)is the driving signal of the lth secondary source (l∈ {1,...,L}), and gl(r

r,ω)is the transfer function

from the lth secondary source to the position r

r. The transfer functions gl(r

r,ω)are assumed to be known by

measuring or modeling them in advance. Hereafter, the angular frequency ωis omitted for notational simplicity.

The goal of sound ﬁeld reproduction is to obtain {dl}L

l=1of the Lsecondary sources so that usyn(r

r)coincides

with the desired sound ﬁeld, denoted by udes(r

r), inside Ω. We deﬁne the cost function to determine the driving

signal {dl}L

l=1as

J=ZΩ



∑

l=1

dlgl(r

r)−udes(r

r)



r.(2)

The optimal driving signal can be obtained by solving the minimization problem of J.

2.2 Pressure matching

Since it is difﬁcult to solve the minimization problem of Jowing to the regional integration over Ω, several

methods based on the approximation of Jhave been proposed. A simple strategy for solving it is to discretize

the target region Ωinto multiple control points, which is referred to as the pressure matching. Assume that N

control points are placed over Ωand their positions are denoted by r

rc,n(n∈ {1,...,N}). The cost function Jis

approximated as the error between the synthesized and desired pressures at the control points. The optimization

problem of pressure matching is written as

minimize

d∈CLkG

d−u

udesk2+ηkd

dk2,(3)

where d

d= [d1,...,dL]T∈CLis the vector of the driving signals, u

udes = [udes(r

rc,1),...,udes(r

rc,N)]T∈CNis the

vector of the desired sound pressures, and G

G∈CN×Lis the matrix consisting of the transfer functions gl(r

rc,n)

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

WeightedpressurematchingbasedonkernelinterpolationforsoundeldreproductionShoichiKOYAMAandKazuyukiARIKAWA(1)(1)GraduateSchoolofInformationScienceandTechnology,TheUniversityofTokyo,Japan,koyama.shoichi@ieee.orgABSTRACTAsoundeldreproductionmethodcalledweightedpressurematchingisproposed.Soundeldrepro...

展开>> 收起<<

Weighted pressure matching based on kernel interpolation for sound ﬁeld reproduction Shoichi KOYAMA and Kazuyuki ARIKAWA1.pdf

共9页,预览2页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Weighted pressure matching based on kernel interpolation for sound ﬁeld reproduction Shoichi KOYAMA and Kazuyuki ARIKAWA1

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: