WASPAA 2017 Angular spectrum decomposition-based 2.5D ... · 2.5D SDM for exterior sound field...
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Simulation condition Original sound field: modeled by 81 random numbers up to 64 loudspeakers with and Intermediator: Plane wave decomposition of 3D sound field (Herglotz integral) Spherical harmonic expansion of 3D sound field 3D cylindrical harmonic expansion of 3D sound field Analytical conversion from spherical to 3D cylindrical harmonic spectrums with Analytical conversion into angular spectrums Sound pressure synthesized by a linear sound source Analytical driving function for 2.5D SDM Comparison with focused point source synthesis by 2.5D SDM 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 Angular spectrum decomposition-based 2.5D higher-order spherical harmonic sound field synthesis with a linear loudspeaker array Takuma OKAMOTO, National Institute of Information and Communications Technology, Japan 1. Introduction 2. Proposed formulation 3. Analytical driving function for 2.5D SDM 4. Computer simulations This study was partly supported by JSPS KAKENHI Grant Number JP15K21674. ˇ S m n ˜ S(R, φ,k x ) Data-based synthesis Model-based synthesis D(x 0 ) R φ y 0 x x x y y y z z z Appropriate rotation ˜ S(y syn , 0,k x ) y syn Plane wave decomposition ˜ S m (k x ) 0 0.5 1 1.5 2 2.5 3 3.5 Temporal frequency [kHz] -30 -25 -20 -15 -10 -5 0 Averaged synthesis error [dB] -1 0 1 2 3 -2 -1 0 1 2 -1 -0.5 0 0.5 1 -1 0 1 2 3 -2 -1 0 1 2 -1 -0.5 0 0.5 1 -1 0 1 2 3 -2 -1 0 1 2 -1 -0.5 0 0.5 1 -1 0 1 2 3 -2 -1 0 1 2 -25 -20 -15 -10 -5 0 -1 0 1 2 3 -2 -1 0 1 2 -25 -20 -15 -10 -5 0 Original f = 2 kHz y 0 =0.5m y 0 = -0.5m E(x) = 10 log 10 |S org (x) - S syn (x)| 2 |S org (x)| 2 y syn =2.0m S(x)= Z 1 -1 D(x 0 )G 3D (x, x 0 )dx 0 ˜ S(y syn , 0,k x )= ˜ D(k x ) ˜ G 3D (k x ,y,y 0 )= ˜ D(k x ) j 4 H 0 (k r (y syn - y 0 )) ˜ P ps = - j p 4⇡ ˇ S 0 0 k x s = y s =0 m =0 ˜ D(k x )= -4j ⇡ kH 0 (k r (y syn - y 0 )) · 1 X m=-1 j m H m (k r y syn ) 1 X n=|m| j -n ˇ S m n Y m n (k x ) ˜ D(k x ) ps = ˜ P ps e -jkxxs H 0 (k r (y syn - y s )) H 0 (k r (y syn - y 0 )) Synthesized Error S(r, ✓, φ)= 1 4⇡ Z 2⇡ 0 Z ⇡ 0 S(✓ 0 , φ 0 )e jk T x sin ✓ 0 d✓ 0 dφ 0 S(r, ✓, φ)= 1 X n=0 n X m=-n ˇ S m n h n (kr)Y m n (✓, φ) e jk T x =4⇡ 1 X n=0 n X m=-n j n h n (kr)Y m n (✓, φ)Y m n (✓ 0 , φ 0 ) ⇤ S(✓ 0 , φ 0 )= 1 X n=0 n X m=-n j -n ˇ S m n Y m n (✓ 0 , φ 0 )= 1 X m=-1 1 X n=|m| j -n ˇ S m n Y m n (✓ 0 , 0)e jmφ0 Y m n (✓, φ)= s (2n + 1) 4⇡ (n - |m|)! (n + |m|)! P |m| n (cos ✓)e jmφ ˇ S m n = 1 h n (kr) Z 2⇡ 0 Z ⇡ 0 S(r, ✓, φ)Y m n (✓, φ) ⇤ sin ✓d✓dφ ˇ S m n = j n Z 2⇡ 0 Z ⇡ 0 S(✓ 0 , φ 0 )Y m n (✓ 0 , φ 0 ) ⇤ sin ✓ 0 d✓ 0 dφ 0 e jk T x = 1 X m=-1 e jm(φ-φ0) 1 2⇡ Z 1 -1 2⇡j m H m (k r R)δ(k x - k cos ✓ 0 )e jkxx dk x S(R, φ,x)= 1 X m=-1 e jmφ 1 2⇡ Z 1 -1 ˜ S m (k x )H m (k r R)e jkxx dk x ˜ S m (k x )= j m 2 Z 2⇡ 0 Z ⇡ 0 S(✓ 0 , φ 0 )e -jmφ0 δ(k x - k cos ✓ 0 ) sin ✓ 0 d✓ 0 dφ 0 k x = k cos ✓ 0 p k 2 - k 2 x = k r = k sin ✓ 0 ˜ S m (k x )= j m 2 Z 2⇡ 0 Z -k k S(✓ 0 , φ 0 )e -jmφ0 δ(k x - k cos ✓ 0 ) k r k d dk x ⇢ cos -1 ✓ k x k ◆ dk x dφ 0 = j m ⇡ k 1 X n=|m| j -n ˇ S m n Y m n (k x ) d dk x ⇢ cos -1 ✓ k x k ◆ = -1 p k 2 - k 2 x = - 1 k r 1 2⇡ Z 2⇡ 0 S(✓ 0 , φ 0 )e -jmφ0 dφ 0 = 1 X n=|m| j -n ˇ S m n Y m n (k x ) Y m n (k x )= s (2n + 1) 4⇡ (n - |m|)! (n + |m|)! P |m| n ✓ k x k ◆ y x z φ ✓ r (a) kx k (b) ky kz ✓0 φ0 d✓ 0 ! dk x Flowchart of proposed approach WASPAA 2017 Equivalent to proposal when n =8 Δx =0.05 m F ˜ S(R, φ,k x )= 1 X m=-1 ˜ S m (k x )H m (k r R)e jmφ = ⇡ k 1 X m=-1 j m H m (k r R)e jmφ 1 X n=|m| j -n ˇ S m n Y m n (k x ) 17th Oct. 2017
Transcript of WASPAA 2017 Angular spectrum decomposition-based 2.5D ... · 2.5D SDM for exterior sound field...
Simulation condition Original sound field: modeled by 81 random numbers up to 64 loudspeakers with and
Intermediator: Plane wave decomposition of 3D sound field (Herglotz integral)
Spherical harmonic expansion of 3D sound field
3D cylindrical harmonic expansion of 3D sound field
Analytical conversion from spherical to 3D cylindrical harmonic spectrums with
Analytical conversion into angular spectrums
Sound pressure synthesized by a linear sound source
Analytical driving function for 2.5D SDM
Comparison with focused point source synthesis by 2.5D SDM
Analytical approaches to sound field recording and reproduction Wave field synthesis (WFS), Spectral division method (SDM): Planer and linear microphone / loudspeaker arraysHigher-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 arrayApplications: listening location informed situations, such as cinema
Angular spectrum decomposition-based 2.5D higher-order spherical harmonic sound field synthesis with a linear loudspeaker array
Takuma OKAMOTO, National Institute of Information and Communications Technology, Japan
1. Introduction
2. Proposed formulation
3. Analytical driving function for 2.5D SDM
4. Computer simulations
This study was partly supported by JSPS KAKENHI Grant Number JP15K21674.
Smn
S(R,�, kx
)Data-based synthesis
Model-based synthesis
D(x0)
R�
y0
x x x
y y y
z z zAppropriate rotation
S(ysyn, 0, kx)
ysyn
Plane wave decomposition
Sm
(kx
)
0 0.5 1 1.5 2 2.5 3 3.5Temporal frequency [kHz]
-30
-25
-20
-15
-10
-5
0
Ave
rage
d sy
nthe
sis e
rror [
dB]
-1 0 1 2 3-2
-1
0
1
2
-1
-0.5
0
0.5
1
-1 0 1 2 3-2
-1
0
1
2
-1
-0.5
0
0.5
1
-1 0 1 2 3-2
-1
0
1
2
-1
-0.5
0
0.5
1
-1 0 1 2 3-2
-1
0
1
2
-25
-20
-15
-10
-5
0
-1 0 1 2 3-2
-1
0
1
2
-25
-20
-15
-10
-5
0
Originalf = 2 kHz
y 0=
0.5m
y 0=
�0.5m
E(x) = 10 log
10
|Sorg
(x)� Ssyn
(x)|2
|Sorg
(x)|2
ysyn = 2.0 m
S(x) =
Z 1
�1D(x0)G3D(x,x0)dx0 S(ysyn, 0, kx) = D(k
x
)G3D(kx, y, y0) = D(kx
)j
4H0 (kr(ysyn � y0))
Pps = �jp4⇡S0
0
kxs = ys = 0
m = 0
D(kx
) =�4j⇡
kH0 (kr(ysyn � y0))·
1X
m=�1jmH
m
(kr
ysyn)1X
n=|m|
j�nSm
n
Y m
n
(kx
)
D(kx
)ps = Ppse�jk
x
xsH0 (kr(ysyn � ys))
H0 (kr(ysyn � y0))
Synthesized Error
S(r, ✓,�) =1
4⇡
Z 2⇡
0
Z ⇡
0S(✓0,�0)e
jkTx sin ✓0d✓0d�0
S(r, ✓,�) =1X
n=0
nX
m=�n
Smn hn(kr)Y
mn (✓,�)
ejkTx = 4⇡
1X
n=0
nX
m=�n
jnhn(kr)Ymn (✓,�)Y m
n (✓0,�0)⇤
S(✓0,�0) =1X
n=0
nX
m=�n
j�nSmn Y m
n (✓0,�0) =1X
m=�1
1X
n=|m|
j�nSmn Y m
n (✓0, 0)ejm�0
Y mn (✓,�) =
s(2n+ 1)
4⇡
(n� |m|)!(n+ |m|)!P
|m|n (cos ✓)ejm�
Smn =
1
hn(kr)
Z 2⇡
0
Z ⇡
0S(r, ✓,�)Y m
n (✓,�)⇤ sin ✓d✓d�
Smn = jn
Z 2⇡
0
Z ⇡
0S(✓0,�0)Y
mn (✓0,�0)
⇤ sin ✓0d✓0d�0
ejkTx
=
1X
m=�1ejm(���0) 1
2⇡
Z 1
�12⇡jmH
m
(kr
R)�(kx
� k cos ✓0)ejk
x
xdkx
S(R,�, x) =1X
m=�1ejm�
1
2⇡
Z 1
�1Sm
(kx
)Hm
(kr
R)ejkx
xdkx
˜Sm
(kx
) =
jm
2
Z 2⇡
0
Z⇡
0S(✓0,�0)e
�jm�0�(kx
� k cos ✓0) sin ✓0d✓0d�0
kx
= k cos ✓0p
k2 � k2x
= kr
= k sin ✓0
˜Sm
(kx
) =
jm
2
Z 2⇡
0
Z �k
k
S(✓0,�0)e�jm�0�(k
x
� k cos ✓0)kr
k
d
dkx
⇢cos
�1
✓kx
k
◆�dk
x
d�0
=
jm⇡
k
1X
n=|m|
j�n
ˇSm
n
Y m
n
(kx
) d
dkx
⇢cos
�1
✓kx
k
◆�=
�1pk2 � k2
x
= � 1
kr
1
2⇡
Z 2⇡
0S(✓0,�0)e
�jm�0d�0 =
1X
n=|m|
j�n
ˇSm
n
Y m
n
(kx
)
Y m
n
(kx
) =
s(2n+ 1)
4⇡
(n� |m|)!(n+ |m|)!P
|m|n
✓kx
k
◆
y
x
z
�
✓
r
(a)
kx
k
(b)ky
kz
✓0
�0
d✓0 ! dkx
Flowchart of proposed approach
WASPAA 2017
Equivalent to proposal when
n = 8
�x = 0.05 m
F
S(R,�, kx
) =1X
m=�1Sm
(kx
)Hm
(kr
R)ejm� =⇡
k
1X
m=�1jmH
m
(kr
R)ejm�
1X
n=|m|
j�nSm
n
Y m
n
(kx
)
17th Oct. 2017
