2.5D HIGHER ORDER AMBISONICS FOR A SOUND FIELDDESCRIBED BY ANGULAR SPECTRUM COEFFICIENTSTakuma 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 solution to convert sound field representation from the angular spectrum to the circular harmonics expansion. A sound field is decomposed to plane waves by the spatial
Fourier transform and represented by the angular spectrum. A plane
wave is also described by the circular harmonics expansion. In the
proposed formulation, these two representations are integrated and asound field can be represented by the circular harmonics expansion
with the angular spectrum coefficients. For actual implementations,the driving function of a circular sound source for 2.5D higher order
Ambisonics is analytically derived from a sound field described by
the angular spectrum. The results of the computer simulations showthat the proposed method with a circular loudspeaker array can reproduce a sound field recorded by a linear microphone array withappropriate accuracy around the center of the circular array.
Index Terms— Angular spectrum, circular harmonics, sphericalharmonics, sound field reproduction, Ambisonics
1. INTRODUCTION
Sound field synthesis is an important acoustic communication technique and several methods have been investigated in the last decades.
Higher order Ambisonics (HOA) [1–7] has been investigated fora sound field synthesis technique which can synthesize sound waves
coming from all directions. In HOA, spherical and circular microphone arrays [8–10] are introduced to record sound fields and spherical and circular loudspeaker arrays are used in reproduction stage.
A sound field in HOA is described by the spherical and cylindrical harmonics expansions which are derived from the spatial Fouriertransform in spherical and cylindrical coordinates [11].
For synthesizing sound waves from a half space, wave field synthesis (WFS) [12–15] and spectral division method (SDM) [6, 16–18] have been proposed. In WFS and SDM, planar and linear arrays of microphones and loudspeakers [15,17,18] are introduced forsound field recording and reproduction, respectively. A sound fieldin SDM is described by the angular spectrum which is derived fromthe spatial Fourier transform in Cartesian coordinates [11].
Since these methods provide analytical solutions based on thespatial Fourier transform, sound field representations should be
the same formats between recording and reproduction stages. For
this reason, in these methods, the configurations of receivers andsecondary sources should be spherical/circular and planar/linear
arrangements. In HOA, several decoding methods for hemispherThis study was partly supported by JSPS KAKENHI Grant Number
15K21674.

ical [19], multiple circular [20] and irregular [21, 22] loudspeakerlayouts have been investigated.
In contrast, there is no representation consistency between
recording and reproduction stages in pressure-matching based leastsquares (LS) [23–26], lasso [27] and matching pursuit [28] approaches. An HOA encoding method with LS based plane wave
decomposition for irregular microphone arrays also has been proposed [29]. These numerical approaches, however, are quite unstablebecause the acoustic inverse problem is very ill-conditioned [11,30].
For these reasons, it is highly important to improve the flexibilityof analytical approaches for sound field recording and reproduction.
An analytical approach for WFS with truncated linear loudspeakerarrays to reproduce a sound field described by the spherical harmonics expansion coefficients has been provided [31]. For sound fieldrecording by using spherical and circular microphone arrays, an analytical method of converting between cylindrical and spherical harmonics representations of sound fields has been proposed [32].
This paper provides an analytical solution for converting soundfield representation from the angular spectrum to the circular harmonics expansion. The proposed formulation enables a sound field
captured by a linear microphone array to be reproduced by a circular
loudspeaker array based on HOA.
For actual implementations, sound field reproduction systems
are frequently simplified to be reproduced in the horizontal plane.
Then the secondary sources are arranged on a line or a circle rather
than a plane or a sphere. In actual implementations, monopole
sources instead of line sources are usually employed for secondary
sources. Such approaches are called 2.5D sound field synthesis [3–5, 7, 16, 33]. The driving functions for both 2D and 2.5D
HOA [3–5, 7], therefore, are analytically derived from a sound field
described by the angular spectrum.
2. CONVERTING SOUND FIELD REPRESENTATIONFROM ANGULAR SPECTRUM TO CIRCULARHARMONICS EXPANSION
Spherical coordinates relative to Cartesian coordinates for both thespatial and wavenumber k(= ω/c) domains are defined in Fig. 1(a)and (b), respectively. ω = 2πf is the angular frequency, f denotes
the temporal frequency and c is the speed of sound.
Assuming that all sound waves are come from a half space y < 0on the x-y plane (z = kz = 0 and θ = θk = π/2).
In this case, the√trace wavenumber in the y direction is ky = k2 − kx2 ≥ 0 and asound field P (x, y, ω) is then represented as [11, 16]
Z ∞
√ 2 2
1
P̃ (kx , ω)e−j k −kx y e−jkx x dkx , (1)P (x, y, ω) =2π −∞© 2016 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
reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, orreuse of any copyrighted component of this work in other works.

z
θ

θk
r

k

y

ky

φk

φ
x

introduced as

kz

kx

(a)

(b)

where P̃ (kx , ω) is the angular spectrum. (1) indicates that a soundfield is decomposed to each plane wave propagating to φk with eachweight coefficient P̃ (kx , ω). When a far field is assumed, P (x, y, ω)
only contains the propagation wave components [11] and (1) can be
represented as
Z k−jP̃ (kx , ω)e

√ 2

=

e

dkx .

(2)

−k

1
2π

Z 0

P̃ (φk , ω)e−jkr(sin(φ) sin(φk )+cos(φ) cos(φk ))

k
2π

Z π

P̃ (φk , ω)e−jkr cos(φ−φk ) sin(φk )dφk ,π· ((k cos(φk ))0 dφk(3)0where x = r cos(φ), y = r sin(φ), kx = k cos(φk ) and the integration by substitution dkx → dφk is applied.
At the same time, a plane wave propagating to the direction φk isalso represented by the circular harmonics expansion [11] and givenase−jkr cos(φ−φk ) =∞X(−j)m Jm (kr)ejm(φ−φk ) ,(4)

where Jm is the m-th order Bessel function. From (3) and (4), thecircular harmonics representation of P (r, φ, ω) is analytically obtained as=k(−j)m Jm (kr)
2π

Z π

(6)

the m-th order spherical harmonics of the n-th degree [11], respectively. From (3) and (6), the 2.5D spherical harmonics representationof P (r, φ, ω) is also analytically derived as
Z πkP̃ (φk , ω)Ψ̊m (r, φk , ω) sin(φk )dφk ,P̊2.5D,m (r, ω) =
2π 0
(7)
where
Ψ̊m (r, φk , ω)∞X(n − m)! m 2 −jmφk=(−j)n jn (kr)(2n + 1)
Pn (0) e, (8)(n + m)!

and Pnm is the m-th order associated Legendre polynomial of then-th degree [11].
As a result, a sound field described by the angular spectrum canbe represented by the circular harmonics expansion with the angular
spectrum coefficients as (5) and (7).
3. 2.5D HOA FOR A SOUND FIELD DESCRIBED BY
ANGULAR SPECTRUM COEFFICIENTS
3.1. Analytical formulation
By using the 2D circular and 2.5D spherical harmonics expansion
coefficients of a sound field derived in (5) and (7), the driving functions of 2D and 2.5D HOA are analytically derived in this section.
The angular spectrum representation of a sound field can beobtained by a continuous linear receiver. When the receiver is located along with the x-axis, the recorded sound pressure P (x, 0, ω)
is transformed to the angular spectrum coefficients by the spatial
Fourier transform [11] along with x and given as
Z ∞P̃ (φk , ω) = P̃ (kx , ω) =P (x, 0, ω)ejkx x dx−∞
Z ∞=P (x, 0, ω)ejk cos(φk )x dx.
(9)
−∞

m=−∞

P̊2D,m (r, ω)m (φ ,ω)
P̌n
k

n=|m|

2 y −jk x
k −kxx

From Fig. 1, P (x, y, ω) is expressed as a sum of plane waves propagating to the direction φk from π to 0 with the weight coefficientsP̃ (kx , ω).
For converting the representation of (2) from the angular spectrum to the circular harmonics expansion, the coordinate system of(2) is transformed from Cartesian coordinates to cylindrical coordinates, and represented as
P (r, φ, ω) =n=|m|where jn and Ynm are the n-th order spherical Bessel function and

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

1
P (x, y, ω) '2π

e−jkr cos(φ−φk )
∞
∞
X
X
=
jn (kr) 4π(−j)n Ynm (π/2, φk )∗ Ynm (π/2, φ),
|
{z
}
m=−∞

P̃ (φk , ω) sin(φk )e−jmφk dφk . (5)0To derive the driving function of 2.5D HOA in the next session,the 2.5D spherical harmonics expansion of a plane wave [3,4] is also

A sound field synthesized by a continuous circular sound source
distribution with radius r0 centered at the origin on the x-y plane isgiven as
Z 2π
P (r, φ, ω) =D(r 0 , ω)G(r, r 0 , ω)r0 dφ0 ,(10)0where G(r, r 0 , ω) is the transfer function from a sound source position r 0 to a receiver position r. Under the free-field assumption,G(r, r 0 , ω) is the two-dimensional free-field Green’s function for
2D HOA and the three-dimensional free-field Green’s function givenase−jk|r −r 0 |G3D (r, r 0 , ω) =,(11)
4π|r − r 0 |z = 0 [m]z = 0 [m]z = 0 [m]

2

2

1

2

1.5

0.8

1.5

z = 0 [m]
0

0

2

0.8

1.5
0.6

-0.5

-0.2

1
0.5

0.2
0

0

-0.2

-0.5

-10

y [m]

0

-5

1
0.5

0.4

0.5

y [m]

0.2

0

-5

0.6

1

0.4

0.5

y [m]

y [m]

1

1.5

0
-15

-0.5

-10

0
-15

-0.5

-0.4

-0.4

-1

-1

-1

-1

-0.6

-0.6

-1.5

-1.5

-20

-20

-1.5

-0.8

-1.5

-0.8

-2
-2

-1

0

1

-2
-2

2

-1

-1

x [m]

0

1

-2
-2

2

0

1

2

x [m]

(a) original sound fieldz = 0 [m]
2

2

1

1.5

0.8

0.6

1

0.6

1

0.4
0.2

0

0

-0.5

-0.2

0.4

0.5

y [m]

0.5

0.2

2

[dB]

f = 3 kHz

Fig. 3. Synthesis error E(r) defined in (18) of ideal point source
synthesis by conventional 2.5D HOA using circular array of 64 loudspeakers.
z = 0 [m]-0.4z = 0 [m]
0

2

-1

-1.5

-1.5

1.5

-0.8

1.5

-0.8

-5

-2
-2

2

-1

-1

x [m]

0

1

z = 0 [m]y [m](b) reproduced sound field

1
0.5

-10

0
-15

-0.5

z = 0 [m]
0

2

-1

-15

-0.5
-1

-20

1.5
-5

-5

1

-10

0

0

2

1.5

-5

1
0.5

2

x [m]

y [m]

1

0

2

-0.6

-0.6

-1.5

-20

-1.5

1

0.5
0

-15

-0.5
-1

-2
-2

0.5

-10

y [m]

y [m]

1

-0.2

-0.5

-1

0

0

0

0

-0.4

-1

-25

-1

x [m]

f = 500 Hz

0.8

-2
-2

-2
-2

[dB]

z = 0 [m]

1.5

y [m]

-25

-1

x [m]

-25

-1

0
-15

-0.5

0

1

x [m]

-10

f = 500 Hz

2

[dB]

-2
-2

-25

-1

0

1

x [m]

2

[dB]

f = 3 kHz

-1
-20

-1.5

-20

-1.5
-25

-2
-2

-1

0

1

2

x [m]

[dB]

-2
-2

-25

-1

0

1

x [m]

2

[dB]

(c) reproduction error E(r) defined in (18)f = 500 Hz

f = 3 kHz

(2)

Fig. 2. Results of proposed 2.5D HOA reproduction of sound fieldrecorded by linear array of 64 microphones. White circles along with
x-axis are 64 microphones and white crosses are 64 loudspeakers.
for 2.5D HOA, respectively [11].
Just as in [3, 4], when the spatial Fourier series expansion [11]is applied to (10), the circular convolution theorem holds and thedriving function of a circular sound source is directly derived asP̊m (r, ω)D̊m (r0 , r, ω) =.
2πr0 G̊m (r, r0 , ω)=k(−j)m+1(2)

π 2 r0 Hm (kr0 )Z π
0P̃ (φk , ω) sin(φk )e−jmφk dφk ,D̊2.5D,m (r = 0, r0 , ω) =

m
P̌|m|
(ω)
1
·
,
(2)
2πr0 −jkh (kr0 )Y m (π/2, 0)∗
|m|

|m|

(15)

(2)

(13)

derived in [11] and the driving function of 2D HOA is analyticallyderived from (5), (12) and (13) asD̊2D,m (r0 , ω)

where Hm is the m-th order Hankel function of the second kind.
In (14), Jm (kr) is cancelled and the driving function of 2D HOA is
independent of r.
Compared to 2D HOA, on the other hand, the driving functionof 2.5D HOA is dependent on r [3, 4]. In [3, 4], for avoiding forbidden frequencies where jn (kr) = 0 [34], r is set to 0 by using thede l’Hospital’s rule and the driving function of 2.5D HOA for r = 0
is analytically derived as(12)

In this case, the reproduction area is assumed to be inside a circular source and r < r0 is considered.
In 2D HOA,
j
(2)
G̊2D (r < r0 , ω) = − Jm (kr)Hm(kr0 ),
4

Fig. 4. Reproduction error E(r) defined in (18) of conventional
LS based reproduction of sound field recorded by linear array of 64
microphones using semicircular array of 32 loudspeakers.

(14)

where hn denotes the n-th order spherical Hankel function of thesecond kind [11].
Finally, by integrating (3), (6) and (15), the driving function of
the proposed 2.5D HOA is analytically derived asD̊2.5D,m (r0 , ω)1
1=·
2πr0 −jkh(2) (kr0 )Y m (π/2, 0)∗
|m|
|m|
Z π
k
m
·
P̃ (φk , ω)4π(−j)|m| Y|m|
(π/2, φk )∗ sin(φk )dφk2π 0Z π(−j)|m|−1=P̃ (φk , ω) sin(φk )e−jmφk dφk .
(16)(2)
πr0 h|m| (kr0 ) 0

0For an actual implementation, a circular loudspeaker array insteadof a continuous circular source is used and (16) must be discretized.
When the number of loudspeakers is Lsp , the order m of the spatial
Fourier series in (16) can be calculated up to M = b(Lsp − 1)/2c,
where b·c is the floor function. Finally, the driving signal of each
loudspeaker at φl in the temporal frequency domain is obtained as
D(r0 , φl , ω) =MX

D̊m (r0 , ω)ejmφl ,m=−Ml = 1, 2, · · · Lsp .
(17)

4. COMPUTER SIMULATIONS|Porg (r, ω) − Psyn (r, ω)|2,
|Porg (r, ω)|2(18)

where Porg (r, ω) and Psyn (r, ω) were the original and reproduced
sound pressures at position r, respectively.
The results of the proposed method were compared with thatof ideal point source synthesis by the conventional 2.5D HOA [3,4] as a reference and the conventional pressure-matching based LS
method [23, 24].
In ideal point source synthesis, the driving signals for synthesizing a sound field radiated from a point source located at r s =[rs , θs , φs ]T were calculated from (15) where
(2)

m
m
P̌|m|
(r s , ω) = −jkh|m| (krs )Y|m|(θs , φs )∗ ,

-5
-10
-15

(19)

derived in [4] was introduced.
When a sound field recorded by a linear microphone array is reproduced by the LS method using a circular loudspeaker array inside
the circle, it is obvious that only the semicircular portion is available
for not only matching the sound pressures at the control points but
also reproducing the wave front in the circle. In the simulations,a semicircular array of 32 loudspeakers was then used as shown in
Fig. 4.

(a) point source synthesis(b) LS method(c) proposed method

-20
-25
-30

Computer simulations were performed to evaluate the proposed 2.5D
HOA for a sound field recorded by a linear microphone array.
In all the simulations, a three-dimensional free field was assumed. The speed of sound c was 343.36 m/s. A circular loudspeaker array with radius r0 = 2.0 m was centered at the origin
and located on the x-y plane. The number of loudspeakers Lsp
was 64. The sound pressures produced by a point source located at
xs = [1, −5, 0]T were recorded by a linear array of Lmic = 64 microphones centered at the origin and along with the x-axis as shownin Fig. 2(a). The distance between the adjacent microphones of thelinear array was ∆x = 0.05 m and the length of the array was 3.2 m.
The spatial Nyquist frequency of the linear array was about 3.4 kHz.
In the proposed method, the recorded sound field was transformed to the angular spectrum coefficients by the discrete Fouriertransform as the discretized and truncated representation of (9)where dx was discretized as ∆x and the infinite integral was truncated as −Lmic ∆x/2 ≤ x ≤ Lmic ∆x/2. The driving signals of thecircular loudspeaker array were also numerically calculated fromthe discretized representation of (16) where dφk was equiangularly
discretized as ∆φk = π/Lmic and (17) with M = 31. The reproduced sound field was calculated from the discretized representationof (10).
To estimate the reproduced sound field, the reproduction error at
position r was defined as
E(r, ω) = 10 log10

Averaged reproduction error [dB]3.2. Actual implementation

0.5

1

1.5

2

2.5

3

3.5

4

Temporal Frequency [kHz]Fig. 5. Results of averaged reproduction error within r ≤ 0.5 m
for (a) ideal point source synthesis using circular array of 64 loudspeakers (reference), (b) LS method using semicircular array of 32
loudspeakers, and (c) proposed method using circular array of 64
loudspeakers.
Fig. 2 shows the results of the reproduced sound field and the
reproduction error by the proposed method at f = 500 and 3000 Hz
with the original sound field. Figs. 3 and 4 also show the results of
the reproduction error by ideal point source synthesis and the conventional LS method, respectively.
In addition, the results of theRaveraged reproduction error r≤0.5 E(r, ω)dr was also calculated
up to f = 4 kHz and plotted in Fig. 5.
These results suggest that the conventional LS method can only
control sound field at low frequencies because only 32 loudspeakers were used and the spatial Nyquist frequency of the LS method
is about 1.7 kHz which corresponds to half as much as c/(2∆x) ≈3.4 kHz. The mode-matching based 2.5D HOA including the proposed method, on the other hand, can control sound field around thecenter of the array over a wide frequency range. Compared with theresults of ideal point source synthesis as a reference, the proposed
method can reproduce a sound field recorded by a linear microphonearray with appropriate accuracy. The reproduction accuracy of the
proposed method is slightly degraded because of the discretizationof (9) and (16) and the truncation of (9). Detailed analysis and discussion are required as future work.
Consequently, the proposed analytical formulation is effectivefor HOA based synthesis of a sound field described by the angular
spectrum coefficients.
5. CONCLUSION
This paper proposed an analytical method to convert sound field representation from the angular spectrum to the circular harmonics expansion. In the proposed method, these two representations are integrated and a sound field can be represented by the circular harmonics
expansion with the angular spectrum coefficients. The driving functions of a circular sound source for both 2D and 2.5D HOA were analytically derived from a sound field described by the angular spectrum. The results of the computer simulations with 2.5D HOA using
a circular loudspeaker array showed that the proposed formulation
can reproduce a sound field recorded by a linear microphone arraywith appropriate accuracy around the center of the circular array.

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