Evaluation Strategies for theOptimization of Line Source Arrays
Florian Straube 2, Frank Schultz 1, Michael Makarski 3,Sascha Spors 1 and Stefan Weinzierl 21Research Group Signal Processing and Virtual Acoustics,University of Rostock2Audio Communication Group, TU Berlin3Four Audio GmbH & Co. KG, Aachen
AES 59th International Conferenceon Sound Reinforcement Engineering and Technology2015-07-16 10:00-10:20 a.m. Session 11, P.2-1
Recent Trends for Line Source Array ApplicationsCurved LSA-intensity shading by curvingband-zoning/array morphing
e.g. Meyer Sound MAPP XTe.g. L-ACOUSTICSSoundVision
Curved LSA-curving and electronic beamforming (FIR/IIR)
e.g. EASE Focus FIR Makere.g. Martin Audio Displaye.g. Duran Audio AXYS withDigital Directivity Synthesis(DDS)→ now with JBL
Straight LSA-electronic beam forming (FIR)
e.g. EAW Resolution
Version: 0.00 2015-07-19
Straube et al. | Line Array Optimization | Motivation 1 / 10
Optimization MethodComplex-Directivity Point Source Model (CDPS) [Mey84, vB00, Fei09]
P (m, f ) =
i=LN∑i=1
D(i , f )︸ ︷︷ ︸FIR-Filter
·Hpost(β(m, i), f ) ·e−j
2π fc|xm−x0,i|
4π |xm − x0,i|·Λy ,LSAL︸ ︷︷ ︸
G(m,i,f )
.
Least Squares Optimization / Pressure Matching (DDS like) [vB00, Col14] with-Tikhonov regularization /energy constraint on the loudspeaker weights [Bet12]
mind(f )‖G(f )d(f )− pdes(f )‖22 subject to: ‖d(f )‖22 ≤ D2max
Other Optimization Approaches-active noise control, personal audio, multi-zone sound field synthesis [Cho02, Bai14, Col14]-find minimum of constrained nonlinear multivariable function, Matlab: fmincon() [Tho09]-solve multiobjective goal attainment problems, Matlab: fgoalattain() [Tho11, Fei13]-near field & far field beam forming (DDC like) [vB00, Bai13]Straube et al. | Line Array Optimization | Method 2 / 10
Evaluation SetupLSA Cabinet Model [Mey84]height 0.372m, ideal circular / line pistons, ideal X-Over 400 Hz & 1.5 kHz
# falias/Hz diameter/length in inch ∆y in inch dBSPL@1W,1mLF 1 461 12 (circ) 14.65 96MF 4 1844 3 (circ) 3.66 86HF 10 4610 1.2 (line) 1.46 112
LSA Model16 cabinets, length≈6m, tilt angle +3◦, splay angles top to down: 5x2◦, 3◦, 2◦, 6x3◦, 2x4◦
0 20 40 60−15
−10
−5
0
5
10
15
29525 16325
15124
13062
12063
1000114011
x / m
y / m
Straube et al. | Line Array Optimization | Method 3 / 10
SPL Distribution in xy-Planef = 200 Hz f = 500 Hz f = 1 kHz
f = 5 kHz f = 10 kHz f = 16 kHz
Straube et al. | Line Array Optimization | Results 4 / 10
Frequency Responses & DirectivityPosition Index Plot [Tho09, Fei13] Frequency Responses in Audience Zone
200 1k 10k 20k30
40
50
60
70
80
90
100
110
f / Hz
L p /
dBS
PL
Venue Setup
0 20 40 60−15
−10
−5
0
5
10
15
29525 16325
15124
13062
12063
1000114011
x / m
y / m
Farfield Radiation Pattern
Straube et al. | Line Array Optimization | Results 5 / 10
Technical Quality: ErrorsAbsolute errorfrequency dependent
�abs(f ) =‖G(f )d(f )− pdes(f )‖22
Relative errorfrequency & position dependent
�rel(m, f ) =∣∣Pdes(m, f )− P (m, f )
Pdes(m, f )
∣∣2
200 1k 10k 20k−10
0
10
20
30
40
50
f / Hz
10 lg
|εab
s(f )
/ 1
Pa2
| / d
B
Straube et al. | Line Array Optimization | Results 6 / 10
Technical Quality: Acoustic Contrast & Error Distribution[Cho02, Bai14, Col14]
Acoustic contrast (bright vs. dark zone)audience vs. non-audience zone
Lp,a,na(f ) =
10 log10
(1Ma‖pm∈Ma(f )‖22
1Mna‖pm∈Mna(f )‖22
)Distribution measureq ={0.05, 0.25, 0.5, 0.75, 0.95} quantiles
Lp,des,opt,q(f ) =
Qqm
[10 log10
(|Pdes(m, f )|2
|P (m, f )|2
)]
200 1k 10k 20k0
5
10
15
20
f / Hz
L p,a
,na(
f ) /
dBre
l
desiredno dist compdist comp
200 1k 10k 20k−10
−5
0
5
10
f / Hz
L p,d
es,o
pt,q
(f )
/ dB
rel
Straube et al. | Line Array Optimization | Results 7 / 10
FIR Filters→Driving Function Index Plots [Tho08]Magnitude in dB
LF
f / Hz
i
200 300 400
1
4
8
12
16
20 lg|D(i, f )| / dB
−20
−14−8
−2410
1622
283440
MF
f / Hz
i400 1k 1.5k
1
10
20
30
40
50
6064
20 lg|D(i, f )| / dB
−20
−14−8
−2410
1622
283440
HF
f / Hz
i
1.5k 5k 10k 20k
1
20
40
60
80
100
120
140
160
20 lg|D(i, f )| / dB
−50
−44−38
−32−26−20
−14−8
−2410
Group Delay in ms
LF
f / Hz
i
200 300 400
1
4
8
12
16
τ / ms
6
14.6
23.2
31.8
40.4
49
MF
f / Hz
i
400 1k 1.5k
1
10
20
30
40
50
6064
τ / ms
0
5.6
11.2
16.8
22.4
28
HF
Straube et al. | Line Array Optimization | Results 8 / 10
Technical Quality: Array Effort [Cho02, Bai14, Col14]
LB1(f ) =
Qqi
[|D(i , f )|2
]maxi[|D(i , f )|2]
200 1k 10k 20k0
20
40
60
80
100
f / Hz
LB1(
f ) /
%
LB2(i) =
Qqf
[|D(i , f )|2
]maxf[|D(i , f )|2]
LF
1 2 4 6 8 10 12 14 160
20
40
60
80
100
i
LB2(
i) / %
MF
1 10 20 30 40 50 60 640
20
40
60
80
100
i
LB2(
i) / %
HF
1 20 40 60 80 100 120 140 1600
20
40
60
80
100
i
LB2(
i) / %
q ={0.05, 0.25, 0.5, 0.75, 0.95} quantiles
Straube et al. | Line Array Optimization | Results 9 / 10
ConclusionLS optimization above spatial aliasing frequency?
technical error measures give further hints on R&D and optimization algorithmrequirements
usage of smaller waveguides→ spatial aliasing shifted to higher frequenciesphase of optimized sound field?
what sound fields are needed in terms of perception?
slides @ http://spatialaudio.net
Straube et al. | Line Array Optimization | Conclusion 10 / 10
http://spatialaudio.net
References [Bai13, Bai14, Bet12, Cho02, Col14, Fei09, Fei13, Mey84, Tho08, Tho09, Tho11, Tho13, vB00][Bai13] Bai, M.R.; Chen, C.C. (2013): “Application of convex optimization to acoustical array signal processing.” In: J. of Snd.
Vibr., 332(25):6596–6616.[Bai14] Bai, M.R.; Wen, J.C.; Hsu, H.; Hua, Y.H.; Hsieh, Y.H. (2014): “Investigation on the reproduction performance versus
acoustic contrast control in sound field synthesis.” In: J. Acoust. Soc. Am., 136(4):1591–1600.[Bet12] Betlehem, T.; Withers, C. (2012): “Sound field reproduction with energy constraint on loudspeaker weights.” In: IEEE
Trans. Audio Speech Language Process., 20(8):2388–2392.[Cho02] Choi, J.W.; Kim, Y.H. (2002): “Generation of an acoustically bright zone with an illuminated region using multiple
sources.” In: J. Acoust. Soc. Am., 111(4):1695–1700.[Col14] Coleman, P.; Jackson, P.J.B.; Olik, M. (2014): “Personal audio with a planar bright zone.” In: J. Acoust. Soc. Am.,
136(4):1725–1735.[Fei09] Feistel, S.; Thompson, A.; Ahnert, W. (2009): “Methods and limitations of line source simulation.” In: J. Audio Eng.
Soc., 57(6):379–402.[Fei13] Feistel, S.; Sempf, M.; Köhler, K.; Schmalle, H. (2013): “Adapting loudspeaker array radiation to the venue using
numerical optimization of FIR filters.” In: Proc. of 135th Audio Eng. Soc. Convention, New York, #8937.[Mey84] Meyer, D.G. (1984): “Computer simulation of loudspeaker directivity.” In: J. Audio Eng. Soc., 32(5):294–315.[Tho08] Thompson, A. (2008): “Real world line array optimisation.” In: Proc. of the Institute of Acoustics, 30(6).[Tho09] Thompson, A. (2009): “Improved methods for controlling touring loudspeaker arrays.” In: Proc. of 127th Audio Eng.
Soc. Convention, New York, #7828.[Tho11] Thompson, A.; Baird, J.; Webb, B. (2011): “Numerically optimised touring loudspeaker arrays - Practical
applications.” In: Proc. of 131st Audio Eng. Soc. Convention, New York, #8511.[Tho13] Thompson, A.; Luzarraga, J. (2013): “Drive granularity for straight and curved loudspeaker arrays.” In: Proc. of the
Institute of Acoustics, 35(2).[vB00] van Beuningen, G.W.J.; Start, E.W. (2000): “Optimizing directivity properties of DSP controlled loudspeaker arrays.”
In: Proc. of the Institute of Acoustics, 22(6).Straube et al. | Line Array Optimization | References 10 / 10
Optimization Parameters
D2max
200 1k 10k 20k10
20
30
40
50
f / Hz
Dm
ax /
dB
regularization value λreg
200 1k 10k 20k0
50
100
150
200
f / Hz
λ reg
condition number κ
200 1k 10k 20k1
10
100
400
f / Hz
κ
d(f , λreg) = [G(f )HG(f ) + λregILN ]
−1G(f )Hpdes(f )
Straube et al. | Line Array Optimization | Appendix A-1 / 10
Evaluation Setup LSA1LSA1 Cabinet Model [Mey84]height 0.372m, ideal circular / line pistons, ideal X-Over 400 Hz & 1.5 kHz
# falias/Hz diameter/length in inch ∆y in inch dBSPL@1W,1mLF 1 461 12 (circ) 14.65 96MF 2 922 6 (circ) 7.32 94HF 1 461 12 (line) 14.65 112
LSA Model16 cabinets, length≈6m, tilt angle +3◦, splay angles top/down: 5x2◦, 3◦, 2◦, 6x3◦, 2x4◦
0 20 40 60−15
−10
−5
0
5
10
15
29525 16325
15124
13062
12063
1000114011
x / m
y / m
Straube et al. | Line Array Optimization | Appendix | Results LSA1 A-2 / 10
SPL Distribution in xy-Planef = 200 Hz f = 500 Hz f = 1 kHz
f = 5 kHz f = 10 kHz f = 16 kHz
Straube et al. | Line Array Optimization | Appendix | Results LSA1 A-3 / 10
Frequency Responses & DirectivityPosition Index Plot [Tho09] Frequency Responses in Audience Zone
200 1k 10k 20k30
40
50
60
70
80
90
100
110
f / Hz
L p/ d
BS
PL
Venue Setup
0 20 40 60−15
−10
−5
0
5
10
15
29525 16325
15124
13062
12063
1000114011
x / m
y / m
Farfield Radiation Pattern
Straube et al. | Line Array Optimization | Appendix | Results LSA1 A-4 / 10
FIR Filters→Driving Function Index PlotsMagnitude in dB
LF
f / Hz
i
200 300 400
1
4
8
12
16
20 lg|D(i, f )| / dB
−20
−14−8
−2410
1622
283440
MF
f / Hz
i400 1k 1.5k
1
10
20
32
20 lg|D(i, f )| / dB
−30
−24−18
−12−60
612
182430
HF
f / Hz
i
1.5k 5k 10k 20k
1
4
8
12
16
20 lg|D(i, f )| / dB
−40
−34−28
−22−16−10
−42
81420
Group Delay in ms
LF
f / Hz
i
200 300 400
1
4
8
12
16
τ / ms
1
9.6
18.2
26.8
35.4
44
MF
f / Hz
i
400 1k 1.5k
1
10
20
32
τ / ms
0
4.8
9.6
14.4
19.2
24
HF
f / Hz
i1.5k 5k 10k 20k
1
4
8
12
16
τ / ms
13
13.8
14.6
15.4
16.2
17
Straube et al. | Line Array Optimization | Appendix | Results LSA1 A-5 / 10
Technical quality: ErrorsAbsolute errorfrequency dependent
�abs(f ) =‖G(f )d(f )− pdes(f )‖22
Relative errorfrequency & position dependent
�rel(m, f ) =∣∣Pdes(m, f )− P (m, f )
Pdes(m, f )
∣∣2
200 1k 10k 20k−10
0
10
20
30
40
50
f / Hz
10 lg
|εab
s(f )
/ 1
Pa2
| / d
B
Straube et al. | Line Array Optimization | Appendix | Results LSA1 A-6 / 10
Technical quality: Acoustic Contrast[Cho02, Bai14, Col14]Acoustic contrast (bright vs. dark zone)audience vs. non-audience zone
Lp,a,na(f ) =
10 log10
(1Ma‖pm∈Ma(f )‖22
1Mna‖pm∈Mna(f )‖22
)Distribution measureq ={0.05, 0.25, 0.5, 0.75, 0.95} quantiles
Lp,des,opt,q(f ) =
Qqm
[10 log10
(|Pdes(m, f )|2
|P (m, f )|2
)]
200 1k 10k 20k0
5
10
15
20
f / Hz
L p,a
,na(
f ) /
dBre
l
desiredno dist compdist comp
200 1k 10k 20k−10
−5
0
5
10
f / Hz
L p,d
es,o
pt,q
(f )
/ dB
rel
Straube et al. | Line Array Optimization | Appendix | Results LSA1 A-7 / 10
Technical quality: Array Effort [Cho02, Bai14, Col14]
LB1(f ) =
Qqi
[|D(i , f )|2
]maxi[|D(i , f )|2]
200 1k 10k 20k0
20
40
60
80
100
f / Hz
LB1(
f ) /
%
5 %25 %50 %75 %95 %
LB2(i) =
Qqf
[|D(i , f )|2
]maxf[|D(i , f )|2]
LF
1 2 4 6 8 10 12 14 160
20
40
60
80
100
i
LB2(
i) / %
MF
1 5 10 15 20 25 30 320
20
40
60
80
100
i
LB2(
i) / %
HF
1 2 4 6 8 10 12 14 160
20
40
60
80
100
i
LB2(
i) / %
Straube et al. | Line Array Optimization | Appendix | Results LSA1 A-8 / 10
Optimization Parameters
D2max
200 1k 10k 20k10
20
30
40
50
f / Hz
Dm
ax /
dB
regularization value λreg
200 1k 10k 20k0
20
40
60
80
100
f / Hz
λ reg
condition number κ
200 1k 10k 20k1
10
100
200
f / Hz
κ
Straube et al. | Line Array Optimization | Appendix | Results LSA1 A-9 / 10
MotivationMethodResultsConclusionReferencesAppendixResults LSA1