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