2017 IEEE Workshop on Applications of Signal Processing to Audio and Acoustics

October 15-18, 2017, New Paltz, NY

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
3-5, Hikaridai, Seika-cho, Soraku-gun, Kyoto, 619-0289, Japanokamoto@nict.go.jp
ABSTRACTThis paper derives an analytical driving function to synthesize an
exterior sound field described by spherical harmonic expansion coefficients using a linear loudspeaker array. An exterior sound field
is decomposed into both spherical and 3D cylindrical harmonic expansion coefficients. The spherical harmonic expansion coefficientsare analytically converted into 3D cylindrical ones by plane wave
decomposition. The angular spectrum coefficients at the synthesis reference line are then analytically obtained from the converted
3D cylindrical harmonic expansion coefficients and directly synthesized by the spectral division method with a linear loudspeaker array. The results of computer simulations indicate the effectivenessof the proposed analytical formulation.
Index Terms— 2.5D sound field synthesis, linear loudspeakerarray, plane wave decomposition, spherical harmonics, spectral division method
1. INTRODUCTION
Higher-order Ambisonics (HOA) [1–4] have been investigated fora sound field synthesis technique that can synthesize sound waves
coming from all directions. In HOA, the surrounding spherical andcircular loudspeaker arrays are used for synthesis. A sound fieldin HOA is described by spherical and circular harmonic expansion
coefficients [5]. In addition, an exterior sound field with a higherorder radiation pattern, such as instrumental sound propagation, isalso represented by these coefficients and synthesized by a spherical
loudspeaker array mounted on a spherical rigid baffle [6].
For synthesizing sound waves from a half space, wave field synthesis (WFS) [7–9] and a spectral division method (SDM) [4,10–14]
have been proposed. In WFS and SDM, the planar and linear loudspeaker arrays are introduced. A sound field in SDM is describedby angular spectrum coefficients [5].
Since these methods provide analytical solutions based on thespatial Fourier transform, sound field representations must haveidentical formats between the recording/modeling and synthesis
stages. In contrast, there is no representation consistency between
these stages in pressure matching-based [15–20] and plane wave
decomposition-based [21] least squares approaches. These numerical approaches, however, are quite unstable because the acoustic
inverse problem is very ill-conditioned [5, 13].
For these reasons, the flexibility of analytical approaches tosound field synthesis must be investigated. In addition, such systems are frequently simplified for synthesis in the horizontal planeThis study was partly supported by JSPS KAKENHI Grant Number
15K21674.

for practical implementations. The sound sources are then arranged
on a line or a circle. In actual implementations, 3D monopole
sources instead of 2D line sources are usually employed as soundsources. Such approaches are called 2.5D sound field synthesis [3, 8–12, 14, 19, 22–25].
An analytical approach to 2.5D WFS with truncated linear
loudspeaker arrays has been provided [22] for synthesizing interior sound fields described by spherical harmonic expansion coefficients. An analytical method of 2.5D HOA with a circular loudspeaker array for an interior sound field described by angular spectrum coefficients has also been investigated [23].
This paper focuses on the 2.5D synthesis of an exterior soundfield with a higher-order radiation pattern, described by sphericalharmonic expansion coefficients with a linear loudspeaker array.
Some approaches have already been investigated [26–29]. This
technique efficiently provides an exterior sound field in a wider
area on the horizontal plane to multiple listeners and is suited to be
combined with big screen video display systems [13]. The existing
methods, however, introduced far field approximation and cannot
effectively control the sound field near the loudspeakers.
One possible solution to this problem is extending conventional
2.5D WFS for interior field synthesis [22] into that for exterior field
synthesis. In a typical 2.5D synthesis with a linear array, the synthesis reference line must be set where precise sound pressures aresynthesized. In an interior field synthesis, the reference line is set
on the recording/modeling center [22]. However, to synthesize an
exterior field with a linear array, the reference line is not on therecording/modeling center, and the sound pressures on the referenceline must be obtained from the spherical harmonic expansion coefficients. It is therefore difficult to directly apply the conventional
2.5D WFS-based method to exterior sound field synthesis.
This paper provides an alternative analytical method for the
2.5D synthesis of an exterior sound field with a linear loudspeakerarray and analytically converts the spherical harmonic expansion
coefficients into 3D cylindrical ones by plane wave decomposition.
The angular spectrum coefficients at the synthesis reference line
are then analytically obtained from the converted 3D cylindricalharmonic expansion coefficients and directly synthesized by 2.5D
SDM with a linear loudspeaker array.
2. PROPOSED FORMULATION
2.1. 3D exterior sound field described by spherical harmonic
expansion coefficients
In the proposed formulation, spherical coordinates relative to Cartesian coordinates for both the spatial and wavenumber k(= ω/c) domains are respectively defined in Figs. 1(a) and (b). ω = 2πf is the© 2017 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including
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2017 IEEE Workshop on Applications of Signal Processing to Audio and Acoustics

x

kx

✓

✓0
r

A plane wave is represented by the spherical harmonic expansion [5] and given as
∞
n
X
X

T

k

z

October 15-18, 2017, New Paltz, NY

ej k x = 4πkz

j n hn (kr)Ynm (θ, φ)Ynm (θ0 , φ0 )∗ ,(5)

n=0 m=−n
0

y

ky

(a)

where j =(b)

Figure 1: Definition of spherical coordinates relative to Cartesian
coordinates: (a) spatial domain and (b) wavenumber domain.

Šnm = j n

∞
n
X
X

Z 2πZ π0

S(θ0 , φ0 )Ynm (θ0 , φ0 )∗ sin θ0 dθ0 dφ0 ,(6)0

and the inverse is obtained as∞n
X
X

S(θ0 , φ0 ) =angular frequency, f is the temporal frequency, and c is the speedof sound.
The exterior expansion of a 3D sound field for regions exterior
to any sound sources is given as
S(r, θ, φ) =√−1. From (1), (4), and (5),j −n Šnm Ynm (θ0 , φ0 )n=0 m=−n
∞
X

=

∞
X

j −n Šnm Ynm (θ0 , 0)ejmφ0 .

(7)

m=−∞ n=|m|Šnm hn (kr)Ynm (θ, φ),(1)n=0 m=−nwhere Šnm and hn are the exterior sound field expansion coefficientsand the n-th order spherical Hankel function of the first kind, ands(2n + 1) (n − |m|)! |m|m
Pn (cos θ)ejmφ (2)Yn (θ, φ) =4π(n + |m|)!

2.3. Plane wave decomposition of exterior sound field describedby 3D cylindrical harmonic expansion coefficients
An exterior sound field is also decomposed into 3D cylindrical harmonic expansion coefficients S̃m (kx ) [5] and represented as
S(R, φ, x) =In data-based synthesis, Šnm are obtained from the discretized
representation of (3) where S(r, θ, φ) are recorded by a surrounding
spherical microphone array [31] (Fig. 2(a)). In contrast, Šnm are
directly given in model-based synthesis (Fig. 2(b)).
In typical spherical harmonic expansion coefficients, spherical coordinates relative to Cartesian coordinates are defined as[x, y, z]T = [r sin θ cos φ, r sin θ sin φ, r cos θ]T [5]. In thiscase, spherical harmonic expansion rotation techniques [32] are introduced. Rotation schemes are also useful for synthesizing the arbitrary directions of the exterior sound field (Fig. 2(b)).

ejmφ

m=−∞

|m|

is the spherical harmonics, and Pn is the associated Legendre
function [30]. The exterior field expansion coefficients are obtained
from the sound pressures on a sphere with radius r [5] and given as
Z 2πZ π1
Šnm =
S(r, θ, φ)Ynm (θ, φ)∗ sin θdθdφ. (3)hn (kr) 0 0

∞
X

1
2π

Z ∞

S̃m (kx )Hm (kr R)ejkx x dkx ,

−∞

(8)

p

√
where R = y 2 + z 2 , kr = k2 − kx2 , and Hm is the m-th order
Hankel function of the first kind [5]. A plane wave is also describedby 3D cylindrical harmonic expansion [34] and represented asT

ej k x =
1
·
2π

∞
X

ejm(φ−φ0 )

m=−∞
Z ∞
−∞

2πj m Hm (kr R)δ(kx − k cos θ0 )ejkx x dkx , (9)where δ is a Dirac delta function [5]. From (4), (8), and (9),Z Z
j m 2π πS(θ0 , φ0 )e−jmφ0
S̃m (kx ) =2 0 0· δ(kx − k cos θ0 ) sin θ0 dθ0 dφ0 .

(10)2.2. Plane wave decomposition of exterior sound field describedby spherical harmonic expansion coefficients

2.4. Analytically converting sound field representation fromspherical to 3D cylindrical harmonic expansion

As in the conventional interior field approach [22], a sound field
is decomposed into plane waves and described by the 3D Herglotz
integral [21, 30, 33]:
Z 2πZ πT
1
S(r, θ, φ) =S(θ0 , φ0 )ej k x sin θ0 dθ0 dφ0 , (4)4π 0 0From (7) and (10), spherical harmonic expansion coefficientsŠnm are analytically converted into 3D cylindrical ones S̃m (kx )(Fig. 2(b)) and simply represented aswhere S(θ0 , φ0 ) is the Herglotz weight coefficient of a plane wave
propagating from [θ0 , φ0 ]T andk = [kx , ky , kz ]T = [k cos θ0 , k sin θ0 cos φ0 , k sin θ0 sin φ0 ]T ,x = [x, y, z]T = [r cos θ, r sin θ cos φ, r sin θ sin φ]T .

jm
2

Z 2πZ −kS(θ0 , φ0 )e−jmφ0

 
kr d
kx
cos−1
dkx dφ0· δ(kx − k cos θ0 )
k dkx
k
∞
j m π X −n m m=
j Šn Yn (kx ),(11)kS̃m (kx ) =

0

k

n=|m|

2017 IEEE Workshop on Applications of Signal Processing to Audio and Acoustics

October 15-18, 2017, New Paltz, NY

Model-based synthesis

z

R

z

z
D(x0 )

Appropriate rotation
y

ysyn

y
Šnm

y0

x

x

x

S̃m (kx )Data-based synthesis(a)yS̃(R, , kx )

(b)

(c)

S̃(ysyn , 0, kx )(d)Figure 2: Processing flowchart of proposed method: (a) recording exterior sound field with a surrounding spherical microphone array;(b) analytically converting sound field representation from spherical harmonic expansion coefficients Šnm to 3D cylindrical ones S̃m (kx );(c) obtaining angular spectrum coefficients S̃(R, φ, kx ) on a cylinder with radius R and angle φ; (d) synthesizing horizontal sound fieldS̃(ysyn , 0, kx ) using a linear loudspeaker array with driving function D(x0 ).
where sin θ0 = kr /k, the integral by substitution dθ0 → dkx andthe relationships

 
d
1
kx
−1
=− ,
cos−1
= √ 2
(12)
dkx
k
kr
k − kx2
Z 2π∞
X
1
S(θ0 , φ0 )e−jmφ0 dφ0 =j −n Šnm Ynm (kx ) (13)
2π 0
n=|m|

are applied, andYnm (kx ) =s(2n + 1) (n − |m|)! |m|Pn
4π(n + |m|)!



kx
k



.

(14)

As illustrated in Fig. 2(c), the exterior sound field on a cylinder
along the x-axis with radius R and angle φ is analytically obtainedby (8) where only the Fourier series expansion component is calculated and represented as
S̃(R, φ, kx ) =∞X

S̃m (kx )Hm (kr R)ejmφ ,

(15)

m=−∞

which is obtained as the angular spectrum coefficients for kx .
2.5. Analytical driving function of a linear sound source
Sound pressure S(x), which is synthesized at position x by a continuous linear sound source with an infinite length along the x-axis,
is given as
Z ∞S(x) =
D(x0 )G3D (x, x0 )dx0 ,(16)−∞where D(x0 ) is the sound source driving function at position x0 =
[x0 , y0 , 0]T , and G3D (x, x0 ) is the transfer function of the soundsource placed at x0 to point x. Under the free-field assumption,
ejk|x−x0 |G3D (x, x0 ) =4π|x − x0 |asS̃(ysyn , 0, kx ) = D̃(kx )G̃3D (kx , y, y0 )j= D̃(kx ) H0 (kr (ysyn − y0 )) ,4

where the synthesis reference line is set to y = ysyn > 0, z = 0
and y0 < ysyn (Fig. 2(d)). From (15) and (18), the driving function
for synthesizing exterior sound field S̃(ysyn , 0, kx ) in the angular
spectrum domain is then analytically derived as−4j S̃(ysyn , 0, kx )H0 (kr (ysyn − y0 ))
∞
∞
X
X
−4jπ
=
j m Hm (kr ysyn )j −n Šnm Ynm (kx ),kH0 (kr ysdm ) m=−∞D̃(kx ) =

n=|m|

(19)

where ysdm = ysyn − y0 . The driving function of the proposed
method in the temporal frequency domain is finally derived by the
inverse spatial Fourier transform:
Z k
1
D(x0 ) =D̃(kx )ejkx x0 dkx ,
(20)
2π −k
where only the propagation wave components are considered andevanescent components |kx | > |k| are discarded to calculate stable
driving functions [13, 14].
For actual implementations, a linear loudspeaker array is usedinstead of a continuous linear sound source, and (20) must be discretized and truncated.
2.6. Comparison with conventional virtual point source synthesis by SDM
The driving function for virtual point source synthesis, which is located at xs = [xs , ys , 0]T by the SDM [11, 25], is given asD̃(kx )ps = P̃ps e−jkx xs(17)

is the 3D free-field Green’s function [5]. In the SDM [10], the spatial Fourier transform of (16) along the x-axis is applied and given(18)

H0 (kr (ysyn − ys )),H0 (kr ysdm )(21)

where P̃ps is the√temporal spectrum of the virtual point source.
When P̃ps = −j 4π Š00 /k and xs = ys = 0, D̃(kx )ps is equivalent to proposed driving function D̃(kx ) in (19) for the 0-th order.

2017 IEEE Workshop on Applications of Signal Processing to Audio and Acousticsz = 0 [m]

0

1

1

0.5

0

0

-1

-0.5

-2
-1

Averaged synthesis error [dB]

x [m]

2

-1

0

1

2

3

y [m]

(a) Original exterior sound fieldz = 0 [m]z = 0 [m]221October 15-18, 2017, New Paltz, NY

1

Proposed (only 0th order) with y0 = 0.5 m
Proposed (only 0th order) with y0 = −0.5 m
Numerical with y0 = 0.5 m
Proposed with y0 = 0.5 m
Numerical with y0 = −0.5 m
Proposed with y0 = −0.5 m

-5
-10
-15
-20
-25
-30

0

0.5

1

1.5

2

2.5

3

3.5

1

0.5

0

0

-1

-2
-1

0

1

2

1

0.5

0

0

-0.5

-1

-0.5

-1

-2
-1

x [m]

x [m]

Temporal frequency [kHz]

3

-1

0

1

y [m]

2

3

y [m]

(b) Synthesized sound fieldz = 0 [m]z = 0 [m]

2

2

0

-5

0
-15

-1

-2
-1

-20
-25

0

1

2

y [m]

3

[dB]

x [m]

x [m]

1
-10

-10

E(x) = 10 log10|Sorg (x) − Ssyn (x)|2,|Sorg (x)|2

(22)

0
-15

-1

-2
-1

-20
-25

0

1

2

y [m]

3

[dB]

(c) Synthesis error
y0 = 0.5 m

merically calculated by the discrete spatial Fourier transform where∆x = 0.05 m and −6.4 m ≤ x ≤ 6.35 m.
To estimate the synthesized sound field, the synthesis error at
position x was defined as

0

-5

1

Figure 4: Results of averaged synthesis error for numerical and proposed methods with a linear array of 64 loudspeakers

y0 = −0.5 m

Figure 3: Results: (a) original exterior sound field modeled by Šnm
up to n = 8-th order with 81 random numbers; (b) synthesizedsound field; and (c) synthesis error for proposed method using alinear array of 64 loudspeakers at temporal frequency f = 2 kHzwith ysyn = 2.0 m. Black circles in (b) and (c) are 64 loudspeakers.

3. COMPUTER SIMULATIONS
Computer simulations evaluated the proposed analytical approachand compared it with the conventional numerical SDM [12]. In all
the simulations, a three-dimensional free field was assumed. The
speed of sound c was 343.36 m/s.
The original exterior sound field was modeled by Šnm up to then = 8-th order with (n + 1)2 = 81 random numbers and calculated by (1). A linear array of loudspeakers was set on y0 = 0.5
or −0.5 m and centered around x = 0. The number of loudspeakers was L = 64, and the distance between adjacent loudspeakers
was ∆x = 0.05 m. Its spatial Nyquist frequency was then about
3.4 kHz. The synthesis reference line was set to ysyn = 2.0 m. dkx
in (20) was discretized into ∆kx = 2π/4L∆x = 0.4909.
In the numerical SDM, the sound pressures on the synthesis
reference line were obtained by (1), and S̃(ysyn , 0, kx ) were nu-

where Sorg (x) and Ssyn (x) are respectively the original and synthesized sound pressures at position x.
Figure 3 shows the results of the original exterior sound field,
the synthesized sound field, and the synthesis error in the horizontal plane for the proposed method at a temporal frequency off = 2 kHz with y0 = 0.5 and −0.5 m, respectively. The resultsof the averaged synthesis error calculated on a plane with z = 0 m,
−1.5 m ≤ x ≤ 1.5 m, and 1.0 m ≤ y ≤ 3.0 m for both the numerical and proposed methods up to the spatial Nyquist frequency
were plotted in Fig. 4. Those of the virtual point source synthesis discussed in Section 2.6 were also plotted as references. These
results indicated that the proposed method achieved 2.5D exteriorsound field synthesis where the original higher-order sound source
is located both in front of and behind the linear loudspeaker arraywith identical accuracy as the numerical approach.
Consequently, based on angular spectrum decomposition, theproposed analytical driving function can effectively synthesize an
exterior sound field described by the spherical harmonic expansion
coefficients using a linear loudspeaker array.
4. CONCLUSIONSThis paper provided an analytical approach to the 2.5D synthesis ofan exterior sound field described by spherical harmonic expansion
coefficients using a linear loudspeaker array. The spherical harmonic expansion coefficients were analytically converted into 3D
cylindrical ones by plane wave decomposition. The angular spectrum coefficients at the synthesis reference line were then analytically obtained from the converted 3D cylindrical harmonic expansion coefficients and directly synthesized by 2.5D SDM with a linear loudspeaker array. The results of computer simulations showed
the effectiveness of the proposed analytical driving function.

2017 IEEE Workshop on Applications of Signal Processing to Audio and Acoustics

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