WASPAA 2017

Angular spectrum decomposition-based 2.5D higher-order
spherical harmonic sound field synthesis with a linear loudspeaker array

17th Oct. 2017

Takuma OKAMOTO, National Institute of Information and Communications Technology, Japan

1. Introduction

Model-based synthesis

Analytical approaches to sound field recording and reproduction
Wave field synthesis (WFS), Spectral division method (SDM):
Planer and linear microphone / loudspeaker arrays
Higher-order Ambisonics (HOA): Spherical and circular arrays
Analytical methods for interior sound field recording and synthesis with flexible
arrangements of arrays
2.5D WFS for spherical harmonic spectrums: Ahrens et al. JASA 2012.
2.5D HOA for angular spectrums: Okamoto ICASSP 2016.
Proposal: analytical approach for exterior sound field
2.5D SDM for exterior sound field described by spherical harmonic spectrums
Recording with a surrounding spherical array, and synthesis with a linear array
Applications: listening location informed situations, such as cinema

Sound pressure synthesized by a linear sound source

00S(r, ✓, ) =1
Šnm =hn (kr)Šnm hn (kr)Ynm (✓, )n=0 m= nYnm (✓, ) =Šnm = j nZ 2⇡Z ⇡0sS(✓0 ,00)

⇤sin ✓0 d✓0 d 0ej k x = 4⇡

Z 1

S̃m (kx )Hm (kr R)e1

S̃m (kx ) =m Z 2⇡Z ⇡j20jkx x(b)S(r, ✓, )Ynm (✓, )⇤ sin ✓d✓dD̃(kx )ps = P̃ps e jkx xs

j n hn (kr)Ynm (✓, )Ynm (✓0 ,0)S(✓0 ,0) =1 X
nX

jn

Šnm Ynm (✓0 ,0) =dkxejk

Tx=

1Xe0 )ejm 0(kx1X

jjm(10)2⇡

Z 11k cos ✓0 ) sin ✓0 d✓0 d 00nŠnm Ynm (✓0 , 0)ejm 0m2⇡j Hm (kr R) (kxk cos ✓0 )ekx = k cos ✓0pk 2 kx2 = kr = k sin ✓0

Analytical conversion from spherical to 3D cylindrical harmonic spectrums⇢✓ ◆Z Zj m 2⇡ k
kr dkxwith d✓0 ! dkx
S̃m (kx ) =S(✓0 , 0 )e jm (kx k cos ✓0 )cos 1
dkx d 0
20

k dkx

k1

⇢1
X

m= 1

S̃m (kx )Hm (kr R)ejm =◆dkx
1
1cos 1=p=dkxkkr
k 2 kx2Z 2⇡
1X1S(✓0 , 0 )e jm 0 d 0 =j n Šnm Ynm (kx )2⇡ 0Analytical conversion into angular spectrumsS̃(R, , kx ) =k✓1⇡ X m
j Hm (kr R)ejm
k m= 1

n=|m|

1
X

n=|m|

j n Šnm Ynm (kx )m= 1

j n Šnm Ynm (kx )n=|m|ys ))y0 ))Equivalent to proposal whenp
j 4⇡ Š00
kx s = ys = 0P̃ps =

m=0

jkx x

Simulation condition
Original sound field: modeled by 81 random numbers up to n = 864 loudspeakers with x = 0.05 m and ysyn = 2.0 m

dkx
2

110.5

00-1-0.5-2-1-10jm⇡ X=j n Šnm Ynm (kx )kn=|m|s✓ ◆(2n + 1) (n |m|)! |m| kx
mYn (kx ) =P
4⇡(n + |m|)! n
k1Xj m Hm (kr ysyn )4. Computer simulations

m= 1 n=|m|m= 1

S(✓0 ,1X

1X

H0 (kr (ysynH0 (kr (ysyn⇤

n=0 m= n3D cylindrical harmonic expansion of 3D sound field
1
S(R, , x) =ejm
2⇡
m= 1ky(a)1 XnXy0 ))·Comparison with focused point source synthesis by 2.5D SDM

0n=0 m= n1Xkzy0T(2n + 1) (n |m|)! |m|P (cos ✓)ejm
4⇡(n + |m|)! nm0 )Yn (✓0 ,04j⇡
kH0 (kr (ysynkz

Spherical harmonic expansion of 3D sound fieldZ 2⇡Z ⇡D̃(kx ) =✓0ry0 ))

Analytical driving function for 2.5D SDM

kx✓jS̃(ysyn , 0, kx ) = D̃(kx )G̃3D (kx , y, y0 ) = D̃(kx ) H0 (kr (ysyn4F

D(x0 )G3D (x, x0 )dx0

101 XnXZ 1012Originalf = 2 kHz

321210.5

1000-1-0.5-1-2-1-1-2-1

120E(x) = 10 log10-5-100

1232
10.5-200123-250-5

1-10000-15-1-0.5-5-10-15-20-25-30-1-20|Sorg (x) Ssyn (x)|2|Sorg (x)|20-15

Averaged synthesis error [dB]0S̃(ysyn , 0, kx )S̃(R, , kx )3. Analytical driving function for 2.5D SDM

y0 = 0.5 mx sin ✓ d✓ dx

Flowchart of proposed approach

0.5 m0 )eTxS̃m (kx )y0 =0S(✓0 ,jkyy0x

Data-based synthesis

ysynyPlane wave decomposition

Intermediator: Plane wave decomposition of 3D sound field (Herglotz integral)
1S(r, ✓, ) =
4⇡

z

Šnm

yS(x) =xRz

D(x0 )

2. Proposed formulation
Z 2⇡Z ⇡

Appropriate rotation

z00.5

1

1.5

2

2.5

Temporal frequency [kHz]-2-10123

Synthesized-1-2-101

23-25Error

This study was partly supported by JSPS KAKENHI Grant Number JP15K21674.

33.5