Document [original]

Florian Straube, Frank Schultz, David Albanés Bonillo, Stefan

Weinzierl

An Analytical Approach for Optimizing the

Curving of Line Source Arrays

Open Access via institutional repository of Technische Universität Berlin

Document type

Journal article | Accepted version

(i. e. final author-created version that incorporates referee comments and is the version accepted for

publication; also known as: Author’s Accepted Manuscript (AAM), Final Draft, Postprint)

This version is available at

https://doi.org/10.14279/depositonce-15234

Citation details

Straube, Florian; Schultz, Frank; Bonillo, David Albanés; Weinzierl, Stefan (2018). An Analytical Approach for

Optimizing the Curving of Line Source Arrays. J. Audio Eng. Soc., vol. 66, no. 1/2, pp. 4–20.

https://doi.org/10.17743/jaes.2017.0043.

This work is protected by copyright and/or related rights. You are free to use this work in any way permitted by

the copyright and related rights legislation that applies to your usage. For other uses, you must obtain

permission from the rights-holder(s).

An Analytical Approach for Optimizing the Curving

of Line Source Arrays

FLORIAN STRAUBE,

AES Associate Member

([email protected])

, FRANK SCHULTZ,

AES Associate Member

([email protected])

DAVID ALBAN ´

ES BONILLO

([email protected])

, AND STEFAN WEINZIERL

([email protected])

Audio Communication Group, TU Berlin, DE-10587 Berlin, Germany

Line source arrays (LSAs) are used for large-scale sound reinforcement aiming at the syn-

thesis of homogeneous sound fields for the whole audio bandwidth. The deployed loudspeaker

cabinets are rigged with different tilt angles and/or are electronically controlled in order to

provide the intended coverage of the audience zones and to avoid radiation towards the ceil-

ing, reflective walls or residential areas. This contribution introduces the analytical polygonal

audience line curving (PALC) approach for finding appropriate LSA cabinet tilt angles with

respect to the geometry of the receiver area and the intended coverage. PALC can be applied

in advance of a numerical optimization of the loudspeakers’ driving functions. The method

can be used with different objectives, such as a constant interaction between adjacent cabinets

with respect to the receiver geometry or by additionally considering amplitude attenuation over

distance. PALC is compared with typical standard LSA curving schemes. The advantages of

the presented approach regarding sound field homogeneity and target-oriented radiation are

evaluated based on technical quality measures.

0 INTRODUCTION

For the optimization of the curving and the electronic

control of line source arrays (LSAs) for advanced sound re-

inforcement there is no standard procedure. In practice, both

a pure geometric and a pure electronic wavefront shaping

as well as combinations thereof are realized. Even state-

of-the-art line array systems with extensive beam steering

capabilities differ significantly [1]. Since they comprise

several individually controllable, small drivers, beam steer-

ing is feasible up to high audio frequencies. While the

cabinets of some array systems are curved in addition to

the beam steering, the cabinets of other systems are rigged

as a straight line.

Recent, mostly proprietary software such as Martin Au-

dio Display [2], EAW Resolution 2 [3], d&b ArrayCalc

[4], and AFMG FIRmaker [5] offer (numerical) optimiza-

tion schemes but the algorithms and the parametrization

are rarely publicly documented. In the literature the cal-

culation of appropriate driving signals, i.e., finite impulse

response (FIR) filters for the individual LSA loudspeak-

ers in order to generate a desired sound field by numerical

optimization techniques, was discussed in [6–12]. These

approaches yield considerable improvements with respect

to homogeneous audience coverage and/or avoidance of

high side lobe energy compared to manually adjusted se-

tups. In [8, 10, 13] also the LSA cabinet tilt angles are

determined by numerical optimization methods. The estab-

lished Wavefront Sculpture Technology (WST) criteria for

LSAs [14, p. 929] also comprise a geometrical criterion for

audience coverage considering the splay angles between

LSA cabinets and the source-to-receiver distances.

As the process for combined geometric-electronic op-

timization typically starts with the curving, this article is

only focused on finding optimal tilt angles. These could

be taken as a pre-processing stage for the optimization of

the loudspeakers’ driving functions, i.e., for the calculation

of the FIR filter coefficients, or could also be applied for

uniformly driven line arrays without further computation.

In this article we aim at introducing the analytical polyg-

onal audience line curving (PALC) approach for finding

appropriate LSA cabinet tilt angles with respect to the ge-

ometry of the receiver area and the intended coverage. The

method can be used with different objectives, such as a con-

stant interaction between adjacent cabinets with respect to

the receiver geometry or by additionally considering am-

plitude attenuation over distance, i.e., sound pressure level

(SPL) loss over distance. PALC is evaluated in comparison

with typical standard LSA curving schemes (straight, arc,

J, progressive, numerically optimized). Acoustic simula-

tions based on the complex-directivity point source (CDPS)

model [6, 15–17] including far-field radiation patterns

Table 1. List of abbreviations.

ARF active radiating factor

ATF acoustic transfer function

BEM boundary element method

CDPS complex-directivity point source

FIR finite impulse response

HF high frequency

LF low frequency

LSA line source array

MA Martin Audio

MF mid frequency

MZSFS multi-zone sound field synthesis

PAL polygonal audience line

PALC polygonal audience line curving

PIP position index plot

prog progressive array

SPL sound pressure level

str straight array

WST Wavefront Sculpture Technology

Fig. 1. Sketch of the LSA setup under discussion. A total of

N=16 LSA cabinets of the height y,LSA =0.372 m is used. See

Table 11 and Table 12 for the deployed tilt angles γn.

LSA cabinet, respectively. With the radiation angle β(m,i),

the source-receiver configuration is specified for the i-th

source and the m-th receiver position. Detailed information

on the geometric setup can be found in [19, 20].

Built from three-way cabinets in this article, the exem-

plarily chosen LSA consists of VLF =1, VMF =4 and VHF

=10 vertically stacked, individually controlled drivers per

cabinet for the low, the mid, and the high frequency band

(LF, MF, HF). 12-inch, 3-inch, and 1.2-inch speakers are

used for LF, MF, and HF, respectively. Thus, the LSA con-

sists of a total of VNsources with i=1, 2, ..., VNfor each

frequency band.

Different frequency independent loudspeaker sensitivi-

ties are assumed in order to obtain realistic sound pressure

values, SLF =96 dB, SMF =88 dB and SHF =112 dB for

vertical radiation in this case. The relation of the pistons’

dimensions to the fixed distance between adjacent piston

centers that is also known as Active Radiating Factor (ARF)

[14, Sec. 3.2], [21] amounts to approximately 0.82. For LF

and MF the circular piston model is deployed, while for

HF the line piston model is used. For the frequency band

crossover, fourth-order Linkwitz-Riley filters with the tran-

sition frequencies fLF,MF =400 Hz and fMF,HF =1500 Hz

are applied.

1.2 Venue Geometry

A multi-stand arena [10, Sec. 6.1] and an open-air am-

phitheater [18, Sec. 4.2.2] with audience and non-audience

sections, i.e., zones to be covered and zones to be avoided,

are modeled by two dimensional slice representations. The

multi-stand arena slice representation consists of four au-

dience lines with different tilt angles and typifies a rather

complex source-receiver configuration. Venue 2 resembles

the Waldbuehne in Berlin and is composed of two audience

lines with different tilt angles for the sake of simplicity.

It conforms to an extreme long-throw application. In this

article near-fills, side-fills and delayed arrays— that are

of baffled line and circular pistons provide the basis for

an evaluation of the introduced approach. One uniformly

driven LSA model is analyzed for two concert venues.

The article is organized as follows. In Sec. 1 the chosen

LSA model and the selected concert venues are presented.

Mathematical fundamentals—among them especially the

adjusted CDPS model—are shortly revisited in Sec. 2. In

Sec. 3 the established WST approach is considered and the

PALC algorithm is described. The evaluation criteria and

the results for the different LSA curving schemes are shown

in Sec. 4 and are discussed in Sec. 5. In Table 1 frequently

used abbreviations of this article and in the Appendix in

Table 2, Table 3, Table 4, Table 5 as well as in Table 6

frequently used mathematical variables of this article are

arranged with regard to their application and formalism.

1 SIMULATION SETUP

An LSA setup is examined for two concert venues fol-

lowing practical examples presented in [10, Sec. 6.1] and

in [18, Sec. 4.2.2] with audience and non-audience sec-

tions given within the vertical LSA radiation plane, i.e.,

the xy-plane in this case. While the first location is a typi-

cal multi-stand arena, the second one resembles a common

open-air amphitheater geometry.

1.1 Line Source Array Setup

The LSA setup and the geometry under discussion are

schematically depicted in Fig. 1 for calculating the sound

pressure P(m, ω)atthem-th receiver position characterized

by the vector xm at the angular frequency ω. A total of

N = 16 LSA cabinets with n = 1, 2, ..., N is deployed.

The front grille’s height y,LSA of a single LSA cabinet is

set to 0.372 m resulting in an overall LSA length of ca.

5.96 m. γn denotes the individual tilt angle of the n-th LSA

cabinet and x0,i denotes the vector of the front grille center

position of the i-th LSA driver. The vectors xt,n, xc,n and

xb,n define the top, center, and bottom position of the n-th

0 20 40 60

-15

-10

-5

297 165

152

131

122

101

151

x / m

y / m

(a) Venue 1: multi-stand arena

0 20 40 60 80 100

-10

466 246

231

112

x / m

y / m

(b) Venue 2: amphitheater

Fig. 2. Venue slices within the xy-plane with audience (black)

and non-audience/avoid (gray) zones and selected index numbers

(change of audience/avoid zone and/or polygonal line’s section

angle) from Mreceiver positions.

routine in practical realizations—are not considered. Only

the xy-plane is considered for vertical radiation, cf., Fig. 2.

This is a common approach for optimization schemes of the

loudspeakers’ driving functions as the horizontal radiation

is assumed to be convenient anyway, cf., [6–13].

The number of receiver positions taken into account is

Mv1 =297 for venue 1 and Mv2 =466 for venue 2. They

each comprise the receiver positions mwith m=1, 2, ...,

M. This corresponds to a distance of 0.5 m between the re-

ceiver positions. The receiver positions are composed of Ma

audience positions from the set Maand Mna non-audience

positions from the set Mna with M=Ma+Mna.They

are characterized by the position vectors xm=(xm,ym,0)T

and are numbered counterclockwise starting from the po-

sition under the LSA that is closest to the LSA (index 1,

cf., Fig. 2). The venue slice coordinates are documented

in Table 7 for venue 1 and in Table 8 for venue 2 in the

Appendix.

Note that the terms bright zone and dark zone used in the

field of multi-zone sound field synthesis (MZSFS) [22–24]

correspond to the audience zone and the non-audience zone

used in the field of sound reinforcement.

2 CALCULATION MODEL

The calculation model presented in this section is used

for sound field prediction and for the evaluation of the

different curving schemes. However, please note that the

introduced analytical approach for finding LSA cabinet tilt

angles (PALC) does not depend on a specific sound field

prediction model.

Modeling multi-way cabinets, the total sound pressure

at the receiver position mat the angular frequency ωis

composed of the complex sound pressures, i.e., magnitudes

and phases, of the different frequency bands as

P(m,ω)=PLF(m,ω)+PMF(m,ω)+PHF(m,ω).(1)

Since the calculations are performed separately for each

frequency band with a subsequent summation, the fre-

quency band indices (LF, MF, HF) are omitted for gener-

alization in the article. The sound field prediction is based

on a complex-directivity point source (CDPS) model of

baffled piston far-field radiation patterns. Its fundamental

equation [16, Eq. (5)], [6, Eqs. (3–5)], [17, Sec. 1.1], [15,

Eq. (11)] reads

P(m,ω)=

i=VN



i=1

G(m,i,ω)D(i,ω)(2)

considering the sources iwith a total of NLSA cabinets

each equipped with Vloudspeakers in a specified frequency

band.

P(m,ω) denotes the sound pressure at the receiver po-

sition xmat the angular frequency ω.G(m,i,ω)termsthe

acoustic transfer function (ATF) from the i-th source to

the m-th receiver position. The complex driving function

D(i,ω)ofthei-th source at the angular frequency ωis

directly proportional to the source’s velocity spectrum.

Utilizing Eq. (2) for the sound field prediction, the cal-

culated sound fields result from the superposition of the

impact of each individual source i. The impact of each

source is characterized by the source-receiver propagation

characteristics—described by the ATF G(m,i,ω)—and by

the signal characteristics, i.e., the signal input as well as

the electronic filters affecting the input of each source—

described by the driving function D(i,ω).

Eq. (2) is modified including a loudspeaker sensitivity

standardization in order to obtain absolute SPLs. Therefore

G(m,i,ω) is considered as a scaled ATF for unit transfor-

mation

G(m,i,ω)=p010 S(i,ω)

20 R(β(m,i),ω)e−jω

c|xm−x0,i|

|xm−x0,i|(3)

with the distance |xm−x0,i|in meter, the reference sound

pressure p0in Pascal and the sensitivity S(i,ω)indB

SPL

at 1 meter per Watt. The scaled ATF is composed of a

specific far-field radiation pattern R(β(m,i), ω) for the ra-

diation angle β(m,i) at the angular frequency ω, the ideal

point source wave propagation e−jω

c|xm−x0,i|

|xm−x0,i|with the velocity

of sound c, the reference sound pressure p0that commonly

amounts to 2 ·10−5Pa in air and the loudspeaker sensitivity

S(i,ω) specifying the SPL in 1 m distance for 1 W electrical

input power. For all drivers and all frequencies per fre-

quency band, the sensitivity is assumed to be constant, i.e.,

S(i, ω) = const.

The complex driving function D(i, ω)ofthei-th source

at the angular frequency ω consists of the signal input

Din(i, ω), the complex optimized filter Dopt(i, ω) and the

complex frequency band crossover as well as high-/lowpass

filter Dxo(ω), thus

D(i,ω)=Din(i,ω)Dopt(i,ω)Dxo(ω).(4)

Gain and delay are mathematically considered as the

amplitude and phase of these complex functions. As this

article is exclusively focused on the curving of the LSA

cabinets, only uniformly driven sources are considered, i.e.,

Dopt(i,ω)=1∀iand ∀ω.(5)

The far-field radiation pattern of the baffled circular pis-

ton with radius with a constant surface velocity is the

normalized so-called jinc function [25, Eq. (26.42)]

Rc(β,ω)=2J1ω

csin β

csin β,(6)

denoting the cylindrical Bessel function of first kind of first

order as J1(·) [26, Eq. (10.2.2)]. The line piston models

an ideal waveguide of height yfor the HF band and its

sinc-function far-field radiation pattern can be written as

[25, Eq. (26.44)]

Rl(β,ω)=

sin ω

y

2sin β

y

2sin β

.(7)

Note that these patterns exhibit main lobe unity gain (i.e.,

0dBforβ=0) in order to control the energy radiated by

the pistons via the assumed sensitivities.

Accounting for all receiver positions Mfor a single an-

gular frequency ω, Eq. (2) is rewritten in matrix notation,

p(ω)=G(ω)d(ω)(8)

1.2. In [8, 10, 13] this is executed by a numerical multi-

objective optimization method that is used for determining

the electronic drive as well. From [14, p. 929] the Wavefront

Sculpture Technology (WST) criteria for LSAs are known.

The practical consequences of criterion number four for

determining the tilt angles are revisited in Sec. 3.1.

In Sec. 3.2 and Sec. 3.3 of this contribution, a purely

analytical approach for finding practical LSA cabinet tilt

angles with respect to the geometry of the receiver area and

the intended coverage is presented: the polygonal audience

line curving (PALC). PALC was originally developed to be

applied beforehand to a numerical optimization of the loud-

speakers’ driving functions. The method can be used with

different objectives, such as a constant interaction between

adjacent cabinets with respect to the receiver geometry or

by additionally considering SPL loss over distance.

3.1 Wavefront Sculpture Technology Approach

In [14, 27, 28], several criteria based on the Fresnel ap-

proach and analytical derivations of the diffraction theory

are denoted as Wavefront Sculpture Technology. WST con-

sists of five criteria how to create a homogeneous wavefront

based on geometric LSA shaping. These criteria include

the spatial sampling condition specifying the maximum al-

lowed distance between adjacent sources and the Active

Radiating Factor (ARF) theorem specifying the maximum

relation of the pistons’ dimensions to the fixed distance

between the acoustic centers of adjacent sources. A further

criterion defines the maximum allowed wavefront curvature

at the waveguide’s exit for high frequency radiation. These

criteria are aimed for minimized grating lobe radiation that

generally enables homogeneous wavefront shaping. The re-

maining criteria define the maximum allowed splay angle

between adjacent LSA cabinets and an optimal array curva-

ture to provide a homogeneous and frequency independent

SPL loss over the audience using an appropriate wavefront

shape, cf., [21, 29, 30].

Considering the splay angles, criterion number four

states that the product of the splay angle αn, i.e., the differ-

ence of the tilt angles of the (n+ 1)-th and the n-th LSA

cabinet, and the distance nfrom the center of the respec-

tive boxes to the receivers should be constant for an SPL

attenuation of 3 dB per distance doubling.

Three different LSA curvatures derived from WST num-

ber four are introduced in [31]: the constant curvature, the

J-shape curvature, and the progressive curvature as visual-

ized in Fig. 3. The progressive curvature corresponds to the

SPL attenuation of 3 dB per distance doubling directly re-

sulting from WST number four. For an SPL attenuation of 6

dB per distance doubling the constant curvature with αn=

const. is recommended. If a constant sound pressure level

over the audience zone is demanded, the J-shape curvature

with αn·2

n=const. is used.

3.2 Polygonal Audience Line

The positions of the audience zones in the vertical radia-

tion plane can be mathematically interpreted as polygonal

line with the total length 0. Fig. 4 represents an exemplary

with p(ω) denoting the (M × 1) vector of sound pressures

for all considered receiver positions xm, G(ω) denoting the

(M × VN) scaled ATF matrix for all sources i and for all

receiver positions m and d(ω) denoting the (VN × 1) vector

of the complex driving functions for all sources i per angular

frequency ω.

In line with this modeling, air absorption is neglected, a

constant velocity of sound (c = 343 m/s) and for the mod-

eled sources infinite, straight baffles and a constant piston’s

surface velocity are assumed. The sound field predictions

are performed for a logarithmically spaced frequency vec-

tor with fstart = 200 Hz, fstop = 20 kHz and 1/36 octave

resolution.

3 CURVING OPTIMIZATION

Since it is a common approach to restrict the optimization

schemes for the LSA driving functions to the vertical plane,

it may also be advantageous to seek appropriate tilt angles

of the LSA cabinets based on the venue slices from Sec.

Δ3

Δ5

Δ3

Δ5

Δ3

Δ5

n=const.

n·Δ2

n=const.

n·Δn=const.

-6 dB 0dB -3 dB

per distance doubling

Fig. 3. Constant curvature, J-shape curvature, and progressive

curvature according to WST for obtaining the denoted target

SPL characteristics per distance doubling. The prerequisites for

the splay angles αnand the distances nto the receivers are

denoted.

5=0

3=0

xPAL,5

xPAL,4

xPAL,3

xPAL,2

xPAL,1

xPAL,0

Fig. 4. Polygonal audience line (PAL) with Ksections (in this

case: K=5). The start position of the k-th line section is

specified by the vector xPAL,k−1and the stop position is given

by the vector xPAL,k.εkdenotes the tilt angle of the k-th line

section.

polygonal audience line (PAL) that is similar to venue 1

with Ksections [k=0, 1, 2, ..., K] with the tilt angles εk.

The k-th line section is specified by the vectors xPAL,k−1

for the start position and xPAL,kfor the stop position. With

these vectors, the total PAL length is

0=

k=K



k=1|xPAL,k−xPAL,k−1|.(9)

The KPAL sections are covered by NLSA cabinets with

n=1, 2, ..., N. The polygonal audience line is therefore

divided into Nsegments that represent the main radiation

area of the LSA cabinets. ndenotes the length of the n-th

segment with the distance n,1from the top to the center

xa,b,n−xc,n

xa,t,n−xc,n

xa,c,n−xc,n

Ξn,1

Ξn,2

Fig. 5. Sketch of one section of the polygonal audience line with

the n-th segment including only one section. The line section is

not changed.

position and n,2from the center to the bottom position of

the segment, i.e., n=n,1+n,2.From

=

n=N



n=1

n(10)

the total length of the covered audience line sections can

be concluded. The different audience zones are indexed

from the lowest to the highest audience positions. In order

to calculate the position and the tilt angle γnof each LSA

cabinet, it is necessary to start with the uppermost cabinet

and compute iteratively from top to bottom.

3.3 PALC Algorithm

Starting with n=1 and k=K, i.e., the topmost LSA

cabinet and the topmost audience positions, nis iteratively

increased and kis decreased. See Fig. 5 for detailed geomet-

ric information. Either the coverage angles ψnof all LSA

cabinets are specified resulting in Nrequired LSA cabinets

that may differ from the intended original number of cab-

inets or the number Nof LSA cabinets is fixed. The latter

means that the angles ψnare found iteratively depending

on the length of the polygonal audience line and start-

ing from the initial coverage angles ψinit for all LSA boxes.

Note that only discrete values can be typically set for the tilt

angles of practical LSAs. By rounding the tilt angles γnin

Eq. (11) after each calculation, the algorithm can be easily

adapted concerning this matter. The algorithm is designed

as follows:

I) Compute the tilt angle γn of the n-th LSA cabinet from

the slope

tan(−γn+ψn)=ya,t,n−yc,n

xa,t,n−xc,n

(11)

with the vector xa,t,n=(xa,t,n,ya,t,n)Tof the top position of

the n-th polygonal audience line segment. The vector xc,n

of the n-th LSA cabinet center position is given as

xc,n=xc,n

yc,n=xt,n

yt,n−Λy,LSA

2sin γn

cos γn(12)

with the height Λy,LSA of a single LSA cabinet and the vec-

tor xt,n=(xt,n,yt,n)Tof the n-th LSA cabinet top position.

The initial values are set to

xt,1

yt,1=xH

yH(13)

and

xa,t,1=xPAL,K.(14)

xHand yHare the coordinates of the top position of the

uppermost LSA cabinet and the vector xPAL,Kpoints at the

top position of the K-th polygonal audience line section.

II) Calculate the center position vector xc,nof every LSA

cabinet with Eq. (12), i.e., for n=1, 2, ..., N.

III) Compute the required distance n,1from the top

to the center position of the n-th polygonal audience line

segment with

n,1=|xa,t,n−xc,n|sin ψn

cos(εk−γn).(15)

including the n-th coverage angle ψnand the tilt angle εk

of the k-th polygonal audience line section.

We have to consider two cases: for case (i) equation

n,1≤|xa,t,n−xPAL,k−1|(16)

holds, i.e., the calculated distance n,1is equal or smaller

than the distance from the start position of the k-th polyg-

onal audience line section to the top position of the n-th

polygonal audience line segment, so that the section of the

polygonal audience line is not changed. The center position

vector of the current polygonal audience line segment can

be calculated with

xa,c,n=xa,t,n+n,1cos εk

sin εk(17)

and hence the segment’s upper partial length n,1is

n,1=n,1.(18)

If for case (ii) equation

n,1>|xa,t,n−xPAL,k−1|(19)

audience line is changed. The n-th partial segment angle

ψnhas to be calculated with

|xa,t,n−xPAL,k−1|

|xa,t,n−xc,n|=sin(ψn−˜

ψn)

cos(εk−γn+˜

ψn).(20)

The partial distance ˜

n,1from the top to the center posi-

tion of the n-th polygonal audience line segment is thus

n,1=|xPAL,k−1−xc,n|sin ˜

ψn

cos(εk−1−γn)(21)

and the center position vector of the current polygonal au-

dience line segment can be written as

xa,c,n=xPAL,k−1+˜

n,1cos εk−1

sin εk−1(22)

and hence

n,1=|xa,t,n−xPAL,k−1|+˜

n,1(23)

is the segment’s upper partial length.

IV) Update k:kis not changed if the section of the

polygonal audience line was not changed (step III, case (i)).

khas to be decreased by 1 if the section of the polygonal

audience line was changed (step III, case (ii)).

V) Calculate the required distance n,2from the center

to the bottom position of the n-th polygonal audience line

segment with

n,2=|xa,c,n−xc,n|sin ψn

cos(εk−γn−ψn)(24)

including the n-th coverage angle ψnand the tilt angle εk

of the k-th polygonal audience line section.

We again have to consider two cases: for case (i) equation

n,2>|xa,c,n−xPAL,k−1|(25)

holds, i.e., the calculated distance n,2is greater than the

distance from the start position of the k-th polygonal audi-

ence line section to the center position of the n-th polygonal

audience line segment, so that the section of the polygonal

audience line is changed. The n-th partial segment angle

ψnthen has to be calculated with

|xa,c,n−xPAL,k−1|

|xa,c,n−xc,n|=sin ˜

ψn

cos(εk−γn−˜

ψn).(26)

Therefore, the partial distance ˜

n,2from the center to the

bottom position of the n-th polygonal audience line segment

n,2=|xPAL,k−1−xc,n|sin(ψn−˜

ψn)

cos(εk−1−γn−ψn)(27)

and the bottom position vector of the current polygonal

audience line segment can be written as

xa,b,n=xPAL,k−1+˜

n,2cos εk−1

sin εk−1.(28)

Hence the segment’s lower partial length is

n,2=|xa,c,n−xPAL,k−1|+˜

n,2.(29)

If for case (ii) equation

n,2≤|xa,c,n−xPAL,k−1|(30)

holds, i.e., the calculated distance n,1 is greater than the

distance from the start position of the k-th polygonal audi-

ence line section to the top position of the n-th polygonal

audience line segment, so that the section of the polygonal

init

∀n

|Γ0−Γ|>Γe

Fig. 6. Overview of the iterative PALC process with the initial

coverage angle ψinit,then-th coverage angle ψn, the tilt angle γnof

the n-th LSA cabinet, the total polygonal audience line length 0,

the total length of the covered polygonal audience line sections 

and the termination condition e.

is valid, i.e., the calculated distance n,2is equal or smaller

than the distance from the start position of the k-th polygo-

nal audience line section to the center position of the n-th

polygonal audience line segment, so that the section of the

polygonal audience line is not changed, the bottom position

vector of the current polygonal audience line segment can

be calculated with

xa,b,n=xa,c,n+n,2cos εk

sin εk(31)

and eventually

n,2=n,2.(32)

is the segment’s lower partial length.

VI) Update k:kis not changed if the section of the

polygonal audience line was not changed (step V, case (ii)).

khas to be decreased by 1 if the section of the polygonal

audience line was changed (step V, case (i)).

The steps I) – VI) have to be repeated until n=Nand k

=0. If the number Nof LSA cabinets is fixed and the cov-

erage angles ψnare to be determined, the values of the total

polygonal audience line length 0and of the total length

of the covered polygonal audience line sections are com-

pared after each complete iteration. All ψnare decreased

if the total length of the covered polygonal audience line

sections is greater than the total polygonal audience line

length 0, and all ψnare increased if is smaller than 0.

This termination condition is denoted as e.InFig.6,the

iteration process is visualized.

4EVALUATION

Acoustic simulations based on the CDPS model includ-

ing far-field radiation patterns of baffled line and circu-

lar pistons provide the data for an evaluation of the intro-

duced approach. The evaluation is performed for two condi-

tions and in comparison with typical standard LSA curving

schemes such as straight, arc, J, and progressive [30] as

well as two numerically optimized versions resulting from

the Martin Audio Display prediction and optimization soft-

ware [8, 13]. Note that we deliberately distinguish between

the WST curvatures from Sec. 3.1 and the specifications

in [30]. The nomenclature from the latter is used for the

curvings in this article.

Fig. 7. Sketch of the LSA setup following the PALC1 and PALC2

approach. It is exemplarily shown for the second and the third LSA

box. γndenotes the tilt angle of the n-th LSA cabinet, ψndenotes

the n-th coverage angle, |xa,c,n−xc,n|denotes the distance from

the center positions of the n-th polygonal audience line segment

and of the n-th LSA cabinet and ndenotes the length of the n-th

polygonal audience line segment.

PALC1 incorporates the goal of an invariant interaction

between adjacent cabinets with respect to the receiver ge-

ometry in order that the radiated sound of the different

sources overlap at a constant coverage angle ψin the far-

field of the individual sources. This constraint simply reads

PALC1: ψ1=ψ2=ψ3=... =const.(33)

PALC1 is similar to an arc array but the goal does not re-

fer to the array itself, i.e., constant splay angles between all

cabinets, but it refers to the shape of the receiver geometry.

The distances of the different positions from the sources

and the desired sound field are considered in PALC2. It

demands a constant product of the coverage angle ψand

the distance from the source to the receiver positions, i.e.,

the distance from the center positions of the n-th polygonal

audience line segment and of the n-th LSA cabinet,

PALC2: ψ1·|xa,c,1−xc,1|

=ψ2·|xa,c,2−xc,2|=...=const.,(34)

cf., Fig. 5. This results from an approximation of

tan ψ1·|xa,c,1−xc,1|

=tan ψ2·|xa,c,2−xc,2|=...=const. (35)

for small ψn. Eq. (35) arises from a simplification of attain-

ing a constant length nfor all npolygonal audience line

segments. In Fig. 7 the geometric variables being relevant

for PALC1 and PALC2 are exemplarily shown for the sec-

ond and the third LSA box. The PALC2 constraint should

not be confused with the Wavefront Sculpture Technology

criterion number four, cf., Sec. 3.1, Table 9, and Table 10.

For the evaluation cases MA1 and MA2, the tilt angles

were extracted from the commercially available predic-

tion and optimization software Martin Audio Display (ver-

sion 2.1.10) that provides suitable tilt angles and also—

if desired—the electronic control by means of a numeri-

cal multi-objective optimization scheme [8, 10, 13]. The

(a) PIP PALC1 (b) PIP PALC2

0.2 0.5 1 2 5 10 20

f / kHz

Lp,a,na(f ) / dBrel

(e) Lp,a,na(f)

0.2 0.5 1 2 5 10 20

f / kHz

H1(f ) / dBrel

(f) H1(f)

str arc J prog PALC1 PALC2 MA1 MA2

Fig. 8. Position index plots (PIPs) depending on the frequency fand the position index mfor PALC1, PALC2, MA1 and MA2, the

acoustic contrast Lp,a,na(f) and the homogeneity measure H1(f) depending on the frequency ffor all analyzed curvings for venue 1

from Fig. 2a. The dashed horizontal lines in the PIPs refer to the selected index numbers in Fig. 2a which represent changes of the

audience/avoid zone and/or of the polygonal line’s section angle.

ferent curving approaches. The position index plots (PIPs)

show the resulting SPL spectra at all receiver positions xm,

i.e., the sound pressure levels Lp(ω,m) depending on the

angular frequency ωand the position index m.Theyare

depicted in Fig. 8 for PALC1, PALC2, MA1, and MA2 for

venue 1 and in Fig. 9 for venue 2. Also known as positional

map, the PIPs were used in [7, 8, 10, 11, 13] as well. The

"ideal" PIP depends on the application. In general, it is

weighting parameters for target and leakage each are set to

1 for MA1 and are set to 10 and 4 for MA 2 allowing a

SPL attenuation of 10 dB from the first to the last audience

position. Note that the optimization parameters target and

leakage are quantitatively specified as the absolute error

Eq. (36) and the acoustic contrast Eq. (39) in our case.

A reasonable selection of the evaluation criteria that were

suggested in [20] is utilized to assess the quality of the dif-

(a) PIP PALC1 (b) PIP PALC2

0.2 0.5 1 2 5 10 20

f / kHz

Lp,a,na(f ) / dBrel

(e) Lp,a,na(f)

0.2 0.5 1 2 5 10 20

f / kHz

H1(f ) / dBrel

(f) H1(f)

str arc J prog PALC1 PALC2 MA1 MA2

Fig. 9. Position index plots (PIPs) depending on the frequency fand the position index mfor PALC1, PALC2, MA1 and MA2, the

acoustic contrast Lp,a,na(f) and the homogeneity measure H1(f) depending on the frequency ffor all analyzed curvings for venue 2

from Fig. 2b. The dashed horizontal lines in the PIPs refer to the selected index numbers in Fig. 2b which represent changes of the

audience/avoid zone and/or of the polygonal line’s section angle.

desired that only little energy is radiated into the non-

audience zones compared to the audience zones, i.e., for the

position indices mthat belong to the non-audience zones,

low SPLs are expected all frequencies f. For the receiver

positions mthat belong to the audience zones, maximum

SPLs, a predefined SPL or a constant SPL at all frequencies

may be desired.

The quantitative evaluation is based on three technical

quality measures. Denoting the squared Euclidean norm

·2

2[26, Eq. (3.2.13)], the frequency dependent absolute

amplitude error [20, Eq. (16)]

E(ω)=



p(ω)

m∈Ma−pdes(ω)

m∈Ma



(36)

between the obtained sound field p(ω) and the desired

sound field pdes(ω) in the audience zone, i.e., m∈Ma,

is additionally smoothed in third-octave bands. The desired

sound field pdes(ω) could in principle be set arbitrarily.

However, the used array geometry restricts the choice to

physically realizable wave fronts. Typically a desired level

decay over the audience zone and a level offset for the avoid

zone can be defined in practical realizations [8].

We have chosen

Pdes(m,ω)∝e−jω

c|xm−xS|

√|xm−xS|(37)

at the receiver position mand the angular frequency ωas the

basis for the comparison. This desired sound field complies

with a sound field generated by a virtual line source at the

position xSdeploying the large argument-approximation of

the 2D Green’s function, i.e., for high frequencies and/or in

the far-field [32, Eq. (26)]. The source position xSis indi-

vidually calculated for each LSA configuration depending

on the top position vector xt,1of the first (topmost) LSA

cabinet and the bottom position vector xb,16 of the last (bot-

tommost) LSA cabinet, i.e.,

xS=1

2xt,1

yt,1 +xb,16

yb,16 .(38)

A target sound pressure level of 100 dBSPL at the first

receiver position within the audience zone is expected.

Alternatively, we could have also chosen other desired

sound fields, such as constant SPLs or maximum SPLs

at all receiver positions mor a 6 dB SPL loss per dis-

tance doubling. Since this article is focused on modeled

loudspeaker data, we selected an ideal 3 dB SPL loss per

distance doubling which is provided by an infinite, con-

tinuous line source and can be unambiguously expressed

by Eq. (37). Using measured loudspeaker data, other tar-

get sound fields may be more advantageous, such as sound

fields that are directly based on the sound fields generated

by uniformly driven LSAs.

Moreover, the frequency dependent relation of the ob-

tained average SPLs of the audience and the non-audience

zone

Lp,a,na(ω)=10 log10 ⎛

⎜

⎝

Ma



p(ω)

m∈Ma



Mna 



p(ω)

m∈Mna



⎞

⎟

⎠

(39)

[20, Eq. (18)] is evaluated, i.e., considering the sound pres-

sures for the receiver positions of the set of the audience

positions in the numerator and of the set of the non-audience

positions in the denominator. This measure is depicted in

Fig. 8e and Fig. 9e and corresponds to the acoustic con-

trast [22, Eq. (16)], [23, Eq. (2)] established in MZSFS.

Quantifying the homogeneity of the generated sound field

based on the magnitudes, neglecting the phases in this case,

the frequency dependent standard deviation of the distance

compensated SPLs of all audience positions

H1(ω)=σ

m∈Ma20 log10 |P(m,ω)|

p0|xm−xS|

|xmin(m)−xS|,

(40)

cf., [11, e.g. Fig. 6, Fig. 8], is analyzed. σdenotes the

standard deviation and xmin(m)is the vector for the first

audience position in this case. Note that the root term re-

sults from the distance compensation referring to the de-

sired 3 dB SPL loss per distance doubling of a virtual line

source. The homogeneity measure is visualized in Fig. 8f

and Fig. 9f.

5 DISCUSSION

For two concert venues, the proposed algorithm for op-

timizing the tilt angles of line array cabinets was evaluated

with respect to different quality measures. The position

index plots (PIPs) for venue 1 (Fig. 2a) shown in Fig. 8 re-

veal that only little energy is radiated into the non-audience

zones compared to the audience zones for all of the al-

gorithmic curving schemes. Note that the dashed horizon-

tal lines in the PIPs refer to the selected index numbers

in Fig. 2 which represent changes of the audience/avoid

zone and/or of the polygonal line’s section angle. For

both venues, the audience zone is located between the

second uppermost and the bottommost dashed horizontal

line.

The intended behavior is confirmed by Lp,a,na(ω)from

Eq. (39) in Fig. 8e for the optimized curvings as well as for

the standard curving schemes (straight, arc, J, and progres-

sive) that were manually adjusted to the receiver geometry.

It can be seen in the PIPs and by means of Lp,a,na(ω) that

the relation of the energy radiated into the audience and the

non-audience zones is very similar for all tilt angle sets and

increases with increasing frequency. The latter results from

the radiation characteristics of the sources: the radiation is

more directed, the higher the frequency.

Lp,a,na(ω) features acceptable values larger than 12 dB

for frequencies above 1 kHz and below ca. 8.5 kHz. For fre-

quencies above ca. 8.5 kHz spatial aliasing effects are vis-

ible leading to more energy in the non-audience zones and

significantly reduced acoustic contrast values. The grating

lobes that are causal for that appear at rather high frequen-

cies compared to conventional LSA designs due to the small

distances between the HF sources. Choosing equal weights

for target and leakage, the final angle of the numerically

optimized MA1 is rather small. Therefore, the first audi-

ence rows are hardly reinforced as it can be deduced from

the MA1 PIP. Increasing the target weight in relation to

the leakage weight, the final angle of MA2 approximately

corresponds to those of the other curving methods. Com-

paring MA1 and MA2, the effect of the reduced focus on

leakage can be clearly observed by means of Lp,a,na(ω)for

frequencies above ca. 700 Hz.

Significant performance differences can be found with

the help of the homogeneity measure H1(ω) from Eq. (40).

The straight array does not cover the whole audience zone

so that there are large deviations considering the front, the

middle, and the back audience positions. For the front po-

sitions, MA1 shows a similar performance due to the small

final angle. The arc and J array suffer from limited adjust-

ment capabilities to the given receiver geometry. PALC2

and MA2 provide the best homogeneity values with MA2

being more homogeneous than PALC2 for frequencies be-

low ca. 2 kHz. For frequencies above ca. 2 kHz, H1(ω)of

PALC2 is smaller than the one of MA2 for up to (1...1.5)

dB. The PIP of MA2, however, reveals some coverage gaps

for the middle positions around m ≈ 110 due to some large

splay angles. An invariant interaction between adjacent cab-

inets that was intended with PALC1 does not seem to be

practical considering homogeneity. As expected, the pro-

gressive curving yields solid results without paying atten-

tion to the specific composition of the receiver geometry.

For venue 2 (Fig. 2b) resembling a typical open-air am-

phitheater, the results are very similar to those for venue

1. Only the expected difference of the acoustic contrast

Lp,a,na(ω) between MA1 and MA2 is not observable for

venue 2. Note that the absolute error E(ω) from Eq. (36) is

not visualized as it provides no meaningful results for the

uniformly driven LSAs.

The tilt angles calculated with the PALC2 approach were

compared with the established WST criterion number four

for venue 1 and venue 2. It can be found in Table 9 and

Table 10 in the Appendix that PALC2 provides results that

are in accordance with the demand for a constant product

of the splay angles and the respective source-to-receiver

distances. The absolute values of the products are constant

depending on the polygonal audience line section that the

involved LSA cabinets are pointing at.

Using the PALC approach, it is assumed that all deployed

LSA cabinets have similar radiation patterns. Frequency

dependent behavior and individual characteristics of the

LSA radiation are not considered. All audio frequencies are

treated equally, there is no preference for selected frequen-

cies or frequency bands. This enables a low-computational

and straight-forward calculation without the need of nu-

merical algorithms requiring high computational effort, but

this may reduce the achievable accuracy. However, PALC

completely incorporates the present receiver geometry. In

addition, the comparison of the PALC2 results and the re-

spective WST criterion reveals that the assumptions do not

turn away from the assumptions made for the established

WST approach.

The evaluation is not based on measurements in this arti-

cle but only on simulations. Therefore, the results may dif-

fer from measured and perceived sound fields of practical

LSAs. Since typical trial-and-error-approaches by sound

engineers, i.e., manual adjustment of the LSA cabinet tilt

angles, follow similar principles as PALC and due to the

PALC tilt angles being in accordance with WST, similar

results can be expected for measurements. The precision

of the sound field prediction may be improved by incor-

porating data from balloon measurements that also include

the influence of adjacent cabinets or by using boundary

element method (BEM) data [17]. Measured loudspeaker

data can be included as far-field radiation patterns R(β,

ω) in the deployed CDPS calculation model. Comparing

the sound fields generated with modeled and these mea-

sured loudspeaker data, we concluded that the design and

development of optimization algorithms can be performed

independent of the data [33]. Future verifications based on

hands-on measurements are, however, essential.

6 CONCLUSION

Sound reinforcement in different venues requires adapted

curving of line source arrays (LSAs). A purely analytical

approach for finding appropriate LSA cabinet tilt angles

was presented in this article. The polygonal audience line

curving (PALC) is based on the geometry of the receiver

area and the intended coverage. In comparison with typical

standard LSA curving schemes, we conclude that the PALC

is superior due to its flexible adaptability with respect to

the receiver geometry.

This algorithm is faster than numerical methods. Since

identical specifications for the presented analytical and the

evaluated numerical optimization approach cannot be com-

pletely ensured, a final comparative statement regarding

accuracy does not seem to be advisable. These specifi-

cations especially comprise the exact conversion from the

selectable goal parameters to the desired sound field as well

as the considered frequency ranges, possibly also weighted,

of the numerical algorithms used as a reference and the fact

that these are based on proprietary, not extractable loud-

speaker directivity data. The effort and the computing time

of the numerical approach are however significantly higher

than for the analytical approach.

The PALC algorithm can be easily extended so that it

only seeks from a discrete set of tilt angle values as it is

required for practical realizations. It can be easily used in

combination with subsequent electronic wavefront shaping.

Since the LSA setup is restricted to fundamentals in this

article, neglecting several aspects such as changes and un-

certainties of the source configuration and of the frequency-

dependent behavior, different LSA setups and their charac-

teristics along with several curving optimization schemes

should be discussed in the future. Also the human percep-

tion of phase effects in sound fields generated by LSAs

should be examined. By listening tests, it may be possible

to find quality criteria for phase position index plots (PIPs)

analogue to the magnitude PIPs.

7 REFERENCES

[1] D. Scheirman, “Large-Scale Loudspeaker Arrays:

Past, Present and Future (Part Two—Electroacoustic Con-

siderations),” presented at the AES 59th International Con-

ference: Sound Reinforcement Engineering and Technology

(2015 July), conference paper 1-3.

[2] Martin Audio Ltd., “User Guide. Display 2.2,

version 3.2,” Tech. rep., https://martin-audio.com/

downloads/software/Display-2.2-User-Guide-v3.2.pdf,

last seen on 2017-08-21 (2017).

[3] Eastern Acoustic Works Inc., “EAW Resolu-

tion 2. Help File,” Tech. rep., http://eaw.com/docs/

3_Manuals/EAW last seen on 2017-08-21 (2017).

[4] d&b Audiotechnik GmbH, “ArrayCalc Simulation

Software V8. ArrayProcessing Feature, Technical White

Paper,” Tech. rep., http://www.dbaudio.com/en/support/

downloads/category/download/4195.html, last seen on

2017-08-21 (2017).

[20] F. Straube, F. Schultz, M. Makarski, S. Spors, and

S. Weinzierl, “Evaluation Strategies for the Optimization

of Line Source Arrays,” presented at the AES 59th Interna-

tional Conference: Sound Reinforcement Engineering and

Technology (2015 July), conference paper 2-1.

[21] F. Schultz, F. Straube, and S. Spors, “Discussion of

the Wavefront Sculpture Technology Criteria for Straight

Line Arrays,” presented at the 138th Convention of the

Audio Engineering Society (2015 May), convention paper

9323.

[22] J.-W. Choi and Y.-H. Kim, “Generation of

an Acoustically Bright Zone with an Illuminated

Region Using Multiple Sources,” J. Acoust. Soc.

Am., vol. 111, no. 4, pp. 1695–1700 (2002 Apr.),

doi:https://doi.org/10.1121/1.1456926.

[23] P. Coleman, P. J. Jackson, M. Olik, M. Møller,

M. Olsen, and J. A. Pedersen, “Acoustic Contrast, Pla-

narity and Robustness of Sound Zone Methods Us-

ing a Circular Loudspeaker Array,” J. Acoust. Soc.

Am., vol. 135, no. 4, pp. 1929–1940 (2014 Apr.),

doi:https://doi.org/10.1121/1.4866442.

[24] M. R. Bai, J.-C. Wen, H. Hsu, Y.-H. Hua, and Y.-

H. Hsieh, “Investigation on the Reproduction Performance

versus Acoustic Contrast Control in Sound Field Synthe-

sis,” J. Acoust. Soc. Am., vol. 136, no. 4, pp. 1591–1600

(2014 Oct.), doi:https://doi.org/10.1121/1.4894693.

[25] E. Skudrzyk, The Foundations of Acoustics

(Springer, New York, 1971).

[26] F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C.

W. Clark, NIST Handbook of Mathematical Functions, 1st

ed. (Cambridge University Press, Cambridge, 2010).

[27] C. Heil and M. Urban, “Sound Fields Radiated by

Multiple Sound Sources Arrays,” presented at the 92nd

Convention of the Audio Engineering Society (1992 Mar.),

convention paper 3269.

[28] M. Urban, C. Heil, and P. Bauman, “Wavefront

Sculpture Technology,” presented at the 111th Convention

of the Audio Engineering Society (2001 Nov.), convention

paper 5488.

[29] M. S. Ureda, “Line Arrays: Theory and Applica-

tions,” presented at the 110th Convention of the Audio En-

gineering Society (2001 May), convention paper 5304.

[30] M. S. Ureda, “Analysis of Loudspeaker Line Ar-

rays,” J. Audio Eng. Soc., vol. 52, pp. 467–495 (2004 May).

[31] L’Acoustics, “Training Module Variable Curvature

Line Source” (2016).

[32] J. Ahrens and S. Spors, “Sound Field Reproduction

Using Planar and Linear Arrays of Loudspeakers,” IEEE

Trans. on Audio Speech Language Process., vol. 18, no. 8,

pp. 2038–2050 (2010 Nov.).

[33] F. Straube, F. Schultz, M. Makarski, and S.

Weinzierl, “Optimized Driving Functions for Curved Line

Source Arrays Using Modeled and Measured Loudspeaker

Data,” Fortschritte der Akustik: Tagungsband d. 42. DAGA,

Aachen, pp. 1136–1139 (2016).

[5] AFMG Technologies GmbH, “AFMG FIRmaker.

AFMG White Paper,” Tech. rep., http://www.afmg-

support.eu/SoftwareDownloadBase/FIRmaker/FIRmaker_

White_Paper.pdf, last seen on 2017-08-21 (2017).

[6] G. W. van Beuningen and E. W. Start, “Optimizing

Directivity Properties of DSP Controlled Loudspeaker Ar-

rays,” Proc. of the Inst. of Acoustics: Reproduced Sound,

vol. 22, no. 6, pp. 17–37 (2000).

[7] A. Thompson, “Real World Line Array Optimisa-

tion,” Proc. of the Inst. of Acoustics, vol. 30, no. 6 (2008).

[8] A. Thompson, “Improved Methods for Controlling

Touring Loudspeaker Arrays,” presented at the 127th Con-

vention of the Audio Engineering Society (2009 Oct.), con-

vention paper 7828.

[9] M. Terrell and M. Sandler, “Optimising the Controls

of a Homogenous Loudspeaker Array,” presented at the

129th Convention of the Audio Engineering Society (2010

Nov.) convention paper 8159.

[10] A. Thompson, J. Baird, and B. Webb, “Numer-

ically Optimised Touring Loudspeaker Arrays - Practi-

cal Applications,” presented at the 131st Convention of

the Audio Engineering Society (2011 Oct.), convention

paper 8511.

[11] S. Feistel, M. Sempf, K. K¨ohler, and H. Schmalle,

“Adapting Loudspeaker Array Radiation to the Venue Us-

ing Numerical Optimization of FIR Filters,” presented at

the 135th Convention of the Audio Engineering Society

(2013 Oct.), convention paper 8937.

[12] A. Thompson and J. Luzarraga, “Drive Granular-

ity for Straight and Curved Loudspeaker Arrays,” Proc.

of the Inst. of Acoustics, vol. 35, no. 2, pp. 210–218

(2013).

[13] A. Thompson, “Line Array Splay Angle Optimi-

sation,” Proc. of the Inst. of Acoustics, vol. 28, no. 8, pp.

135–148 (2006).

[14] M. Urban, C. Heil, and P. Bauman, “Wavefront

Sculpture Technology,” J. Audio Eng. Soc., vol. 51, pp.

912–932 (2003 Oct.).

[15] S. Feistel, A. Thompson and W. Ahnert, “Methods

and Limitations of Line Source Simulation,” J. Audio Eng.

Soc., vol. 57, pp. 379–402 (2009 Jun.).

[16] D. G. Meyer, “Computer Simulation of Loud-

speaker Directivity,” J. Audio Eng. Soc., vol. 32, pp. 294–

315 (1984 May).

[17] P. Meyer and R. Schwenke, “Comparison of the Di-

rectional Point Source Model and BEM Model for Arrayed

Loudspeakers,” Proc. of the Inst. of Acoustics, vol. 25, no.

4 (2003).

[18] F. Schultz, Sound Field Synthesis for Line Source

Array Applications in Large-Scale Sound Reinforcement,

Ph.D. thesis, University of Rostock (2016).

[19] F. Straube, F. Schultz, and S. Weinzierl, “On the

Effect of Spatial Discretization of Curved Line Source Ar-

rays,” Fortschritte der Akustik: Tagungsband d. 41. DAGA,

Nuremberg, pp. 459–462 (2015).

APPENDIX

Table 2. List of variables—sound field prediction and evaluation

cvelocity of sound

D(i,ω) complex driving function of the i-th source at the angular frequency ω

d(ω)(VN×1) vector of the complex driving functions for all sources iper angular frequency ω

Din(i,ω) (complex) signal input of the i-th source at the angular frequency ω

Dopt(i,ω) complex optimized filter of the i-th source at the angular frequency ω

Dxo(ω) complex frequency band crossover at the angular frequency ω

E(ω) frequency dependent absolute error between the obtained and the desired sound field in the audience zone

ffrequency

fLF,MF low/mid crossover frequency

fMF,HF mid/high crossover frequency

fstart start frequency for the calculations

fstop stop frequency for the calculations

G(m,i,ω) scaledATFfromthei-th source to the m-th receiver position at the angular frequency ω

G(ω)(M×VN) scaled ATF matrix for all sources iand for all receiver positions mat the angular frequency ω

H1(ω) frequency dependent standard deviation of the distance compensated SPLs of all audience positions

j imaginary unit, j2=−1

Lp(ω,m) frequency and position index dependent SPL

Lp,a,na(ω) frequency dependent relation of the obtained average SPLs of the audience and the non-audience zone

P(m,ω) sound pressure at the m-th receiver position at the angular frequency ω

Pdes(m,ω) desired sound pressure at the m-th receiver position at the angular frequency ω

p(ω)(M×1) vector of sound pressures at all receiver positions mat the angular frequency ω

pdes(ω)(M×1) vector of desired sound pressures at all receiver positions mat the angular frequency ω

p0reference sound pressure, p0=2·10−5Pa

R(β,ω) specific far-field radiation pattern for the radiation angle βat the angular frequency ω

S(i,ω)i-th loudspeaker sensitivity specifying the SPL in 1 m distance for 1 W electrical input power at the angular frequency ω

σstandard deviation

ωangular frequency

Table 3. List of variables—LSA setup and venue characteristics

iindex of the LSA loudspeakers, with i=1, 2, ..., VN

Knumber of sections of the PAL

kindex of the PAL section, with k=0, 1, 2, ..., K

Mnumber of receiver positions in the vertical radiation plane, with M=Ma+Mna

Manumber of audience positions in the vertical radiation plane

Maset of audience positions in the vertical radiation plane

Mna number of non-audience positions in the vertical radiation plane

Mna set of non-audience positions in the vertical radiation plane

Mv1 number of receiver positions in the vertical radiation plane of venue 1

Mv2 number of receiver positions in the vertical radiation plane of venue 2

mindex of the receiver positions, with m=1, 2, ..., M

Nnumber of individual LSA cabinets and therefore also number of PAL segments

nindex of the LSA cabinet, with n=1, 2, ..., N

Vnumber of vertically stacked loudspeakers per LSA cabinet (different for each frequency band)

Table 4. List of variables—lengths

total length of the covered PAL sections

0total length of the PAL

etolerated difference of the total PAL length and the total length of the covered PAL sections

nlength of the n-th PAL segment, with n=n,1+n,2

n,1distance from the top to the center position of the n-th PAL segment, i.e., the segment’s upper partial length

n,2distance from the center to the bottom position of the n-th PAL segment, i.e., the segment’s lower partial length

ndistance from the center of the (n+ 1)-th and the n-th LSA cabinet to the receiver positions (for WST)

radius of a baffled circular piston

yheight of a line piston

y,LSA front grille’s height of a single LSA cabinet

n,1required distance from the top to the center position of the n-th PAL segment

n,2required distance from the center to the bottom position of the n-th PAL segment

n,1partial distance from the top to the center position of the n-th PAL segment

n,2partial distance from the center to the bottom position of the n-th PAL segment

Table 5. List of variables—vectors and room coordinates

xroom coordinate

xtwo-dimensional vector with x=(x,y)T

x0,ivector of the i-th LSA source’s front grille center position

xa,b,nvector of the bottom position of the n-th PAL segment

xa,c,nvector of the center position of the n-th PAL segment

xa,t,nvector of the top position of the n-th PAL segment

xb,nvector of the bottom position of the n-th LSA cabinet

xc,nvector of the (front grille) center position of the n-th LSA cabinet

xHx-coordinate of the top position of the uppermost LSA cabinet

xmvector of the m-th receiver position

xPAL,kvector of the stop position of the k-th PAL section

xPAL,k−1vector of the start position of the k-th PAL section

xSvector of the source position of the virtual line source

xt,nvector of the top position of the n-th LSA cabinet

yroom coordinate

yHy-coordinate of the top position of the uppermost LSA cabinet

zroom coordinate

Table 6. List of variables—angles

αnsplay angle between the (n+ 1)-th and the n-th LSA cabinet

β(m,i) radiation angle for the m-th receiver position from the i-th source

γntilt angle of the n-th LSA cabinet

εktilt angle of the k-th PAL section

ψncoverage angle of the n-th LSA cabinet (for PALC)

ψnn-th partial coverage segment angle of the PAL

ψinit initial coverage angle of the LSA cabinets

Table 7. Selected venue slice coordinates according to venue 1 from Fig. 2a.

m/m ym/m

10 −11

15 7 −11

101 50 −11

122 58.1492 −4.3788

131 58.1492 0.1212

152 66.2984 6.7424

165 66.2984 13.2424

297 0.2984 13.2424

Table 8 Selected venue slice coordinates according to venue 2 from Fig. 2b.

m/m ym/m

10 −10

21 10 −10

62 30.4426 −8.4668

231 110.0423 19.8906

246 110.0423 27.3906

466 0.0423 27.3906

Table 9 Splay angles αn, source-to-receiver distances nand their product following WST criterion

number four for venue 1 using the PALC2 tilt angles.

LSA cabinets αn/deg n/m αn·n/ (rad ·m)

1...2 1.662.12 1.73

2...3 1.67 59.48 1.73

3...4 1.72 58.21.75

4...5 1.74 58.33 1.77

5...6 1.75 57.62 1.76

6...7 1.79 55.55 1.74

7...8 1.86 53.61 1.74

8...9 1.93 51.81.74

9...10 2.13 44.13 1.64

10...11 2.55 36.11.61

11...12 3.13 29.41.61

12...13 3.84 23.87 1.6

13...14 4.73 19.37 1.6

14...15 5.82 15.77 1.6

15...16 7.12 12.94 1.61

Table 10 Splay angles αn, source-to-receiver distances nand their product following WST criterion

number four for venue 2 using the PALC2 tilt angles.

LSA cabinets αn/deg n/m αn·n/ (rad ·m)

1...2 1 95.55 1.67

2...3 1.11 86.23 1.67

3...4 1.24 77.71 1.68

4...5 1.37 69.95 1.67

5...6 1.53 62.88 1.68

6...7 1.756.47 1.68

7...8 1.89 50.68 1.67

8...9 2.11 45.45 1.67

9...10 2.35 40.75 1.67

10...11 2.62 36.55 1.67

11...12 2.92 32.81.67

12...13 3.39 27.61.63

13...14 4.12 22.07 1.59

14...15 5.15 17.64 1.59

15...16 6.43 14.16 1.59

Table 11 Tilt angles of the LSA cabinets for the geometry used in Fig. 1 and for venue 1 from Fig. 2a for

the different curvings (arc, J, progressive, PALC1, PALC2, MA1, and MA2). Every cabinet of the straight

array is tilted by 7 deg.

LSA γn/deg γn/deg γn/deg γn/deg γn/deg γn/deg γn/deg

cabinet arc J prog PALC1 PALC2 MA1 MA2

1−2−1−2−1.53 −2.45 −3−3

21−1−1.62 1.84 −0.85 −2.5−2.5

34−1−0.85 5.20.82 −2−0.5

47−10.38.55 2.54 −10

510−11.83 11.88 4.29 0 4

613−13.75 15.19 6.04 1 6

716−16.05 18.39 7.83 2 10

81948.73 21.48 9.69 3 10.5

922911.824.45 11.62 4 14.5

10 25 14 15.25 27.31 13.75 5 16.5

11 28 19 19.130.05 16.3718.5

12 31 24 23.32 32.67 19.43 9 22.5

13 34 29 27.92 35.18 23.27 11 26.5

14 37 34 32.937.57 28.01 12 30.5

15 40 39 38.27 39.84 33.83 18 38

16 43 44 44 42 40.95 25.545.5

Table 12 Tilt angles of the LSA cabinets for the geometry used in Fig. 1 and for venue 2 from Fig. 2a for

the different curvings (arc, J, progressive, PALC1, PALC2, MA1, and MA2). Every cabinet of the straight

array is tilted by –4 deg.

LSA γn/deg γn/deg γn/deg γn/deg γn/deg γn/deg γn/deg

cabinet arc J prog PALC1 PALC2 MA1 MA2

1−7.14 −7−7.1–7.14 −8.24 −9.3−8.8

2−4.54 −7−6.78 −4.03 −7.24 −8.8−8.3

3−1.94 −7−6.13 −0.98 −6.12 −8.3−7.8

40.66 −7−5.15 2.01 −4.89 −7.8−7.3

53.26 −7−3.85 4.94 −3.51 −7.3−5.3

65.86 −7−2.23 7.81 −1.99 −6.8−4.8

78.46 −7−0.28 10.63 −0.29 −6.30.2

811.06 −7213.39 1.6−5.83.2

913.66 −2.13 4.616.09 3.71 −5.36.2

10 16.26 2.75 7.53 18.69 6.06 −3.39.2

11 18.86 7.63 10.78 21.18 8.68 −1.312.2

12 21.46 12.514.35 23.56 11.60.715.2

13 24.06 17.38 18.25 25.83 14.98 2.720.2

14 26.66 22.25 22.48 27.97 19.14.723.2

15 29.26 27.13 27.03 30 24.25 6.727.2

16 31.86 32 31.931.91 30.68 14.230.2

THE AUTHORS

Florian Straube Frank Schultz David Alban´

es Bonillo Stefan Weinzierl

Florian Straube received the Dipl.-Ing. degree in

electrical engineering/communications and information

technology from Technische Universit¨

at Dresden in co-

operation with Klippel GmbH in 2013. Since 2014 he

has been working as a research associate at Audio Com-

munication Group at TU Berlin focusing on sound field

synthesis and line source array applications for sound

reinforcement.

•

Frank Schultz received the M.Sc. in audio communi-

cation and technology from Technische Universit¨

at Berlin

and the Dr.-Ing. degree with distinction from Universit¨

Rostock, in 2011 and 2016, respectively. From 2003–2007

he worked at EVI Audio GmbH/Bosch Communications

Systems, Straubing, as an audio DSP engineer. Since 2016

he has been working at sonible GmbH, Graz, as senior

R&D engineer for 3D audio. Recent research interests are

sound field synthesis applications and acoustic signal pro-

cessing for loudspeaker arrays. He is currently a visiting

postdoc at the Audio Communication Group at Technische

Universit¨

at Berlin. He is a member of the AES and reviews

for the AES and the IEEE.

•

David Alban´

es Bonillo received the M.Sc. degree in

telecommunications engineering with a focus on digital

signal processing from the Universidad Europea de Madrid

in 2011. Since 2013 he has been working towards the M.Sc.

in audio communication and technology from Technische

Universit¨

at Berlin and is a research assistant at the TU

Berlin. His fields of interest are electroacoustics, loud-

speaker development, and digital signal processing.

•

Stefan Weinzierl is head of the Audio Communication

Group at the Technische Universit¨

at Berlin. His activities in

research are focused on audio technology, virtual acoustics,

room acoustics, and musical acoustics. He is coordinating

a master program in Audio Communication and Technol-

ogy at TU Berlin and teaching Tonmeister students at the

University of the Arts (UdK). With a diploma in physics

and sound engineering and a two-year study in musicology

at UC Berkeley, he received his Ph.D. from TU Berlin.

He is coordinating research consortia in the field of vir-

tual acoustics (SEACEN) and music information retrieval

(ABC DJ).