Spatially characterized pseudo-perfect diffuseness via finite-degree spherical harmonic diffuseness

Tanaka, Tatsuhiro; Otani, Makoto

doi:10.1121/10.0026466

Perfectly diffuse sound fields play an important role in architectural acoustics and there are established theoretical characterizations of perfect diffuseness. Although sound fields in real rooms are diffuse to some extent, they are not perfectly diffuse, and therefore theories are required to describe pseudo-perfectly diffuse sound fields. Here, we aim to spatially characterize pseudo-perfect diffuseness via directional characterization of that, finite-degree spherical harmonic diffuseness. Our results show that finite-degree diffuse sound fields yield local spatial diffuseness, suggesting that spatial pseudo-perfect diffuseness is characterized using the effective radius of diffuseness.

1. Introduction

Perfectly diffuse sound fields are theoretically characterized by using their directional and spatial properties. For example, a random-wave model^1,2 composed of an infinite number of isotropic and incoherent plane waves simulates perfectly diffuse sound fields based on the directionally random properties. The random-wave model can also be defined by using random coefficients of spherical wave functions.^3,4 The random-wave model also enables spatial characterization of the perfect diffuseness. The spatial perfect diffuseness is defined by exploiting the correspondence of the two-point covariance of sound pressures at a pair of positions with the imaginary part of the Green's function of the field between the positions⁵ or with the sinc function² (the imaginary part of the Green's function in a free sound field). The above models and characterizations establish perfect diffuseness theoretically and provide the foundations for architectural acoustics such as the measurements of absorption coefficients using reverberation rooms.⁶

In contrast, sound fields in real environments never achieve perfect diffuseness; therefore, previous studies in room acoustics have attempted to evaluate the extent of diffuseness in a room.^7–10 Although they are not perfectly diffuse, real sound fields are assumed to be diffuse to some extent. This therefore requires another theory describing pseudo-perfect diffuseness.

We have recently characterized pseudo-perfect diffuseness by discretizing or truncating the directional properties of perfect diffuseness.^11,12 In these studies, discrete random-wave models composed of a finite number of random plane waves simulate pseudo-perfectly diffuse sound fields, and finite-degree spherical harmonic diffuseness, which is defined using random coefficients of truncated spherical harmonics, characterizes pseudo-perfect diffuseness. Nonetheless, spatial characterization of pseudo-perfect diffuseness remain unclear.

Here, we show that pseudo-perfect diffuseness can be defined spatially as local diffuseness, spatial diffuseness valid for a restricted area, via finite-degree spherical harmonic diffuseness.

Section 2 preliminarily introduces models of perfectly diffuse sound fields, defines spatially characterized perfect diffuseness, and introduces finite-degree diffuseness, directionally characterized pseudo-perfect diffuseness. Section 3 then shows that spatial diffuseness is locally valid in sound fields with finite-degree diffuseness and suggests defining spatial pseudo-perfect diffuseness as local diffuseness with an effective radius.

2. Preliminaries

2.1 Models of perfectly diffuse sound fields

There are two established methods of modeling perfectly diffuse sound fields, as shown below.

A (continuous) random-wave model simulating perfectly diffuse sound fields consists of an infinite number of isotropic and incoherent plane waves,¹ which is rigorously formulated as follows:⁵^,

P (k, r) = \int_{0}^{π} \int_{- π}^{π} A (k, θ, ϕ) e^{i k n \cdot r} sin θ d ϕ d θ,

(1)

where

P (k, r)

denotes sound pressure for a wavenumber k [rad/m] and a position

r

⁠.

A (k, θ, ϕ)

is a complex-valued function representing plane wave amplitudes arriving from a direction indicated by a unit vector,

n (θ, ϕ) = {(cos ϕ sin θ, sin ϕ sin θ, cos θ)}^{⊤} .

(2)

To simulate an isotropic and incoherent mix of plane waves, the distributional function

A (k, n)

is assumed to provide a spherical Gaussian random field with the following statistical properties⁵ for

k \neq 0

⁠:

E [A (k, n)] = 0, E [A (k, n) A (k, n^{'})] = 0, E [A (k, n) \bar{A (k, n^{'})}] = δ_{S^{2}} (n - n^{'}),

(3)

where

E [\cdot]

represents an ensemble average (i.e., the expected value for repeatedly synthesized sound fields),

\bar{(\cdot)}

denotes taking a complex conjugate, and

δ_{S^{2}} (\cdot)

denotes the Dirac delta on the unit sphere in a 3-dimensional space. The last two equations respectively represent the pseudo covariance and the covariance of the distribution of plane wave amplitudes, although the Dirac-delta covariance may be difficult to physically understand as also indicated in an optical literature.¹³ Notably, the above (integral) formulation is more rigorous than a formulation including the infinite sum of plane waves,¹⁴ where the arrival directions of the plane waves are provided by words rather than equations. The latter formulation is not employed here also because the question remains whether it is equivalent to the following spherical harmonic formulation of perfectly diffuse sound fields. We hereafter assume

k \neq 0

as well and omit the argument k without notice.

The random-wave model can also be constructed using random coefficients

a_{l, m}

of spherical wave functions^3,4

P (k, r) = \sum_{l, m} 4 π i^{l} a_{l, m} (k) j_{l} (k r) Y_{l, m} (θ, ϕ),

(4)

where

j_{l} (\cdot)

denotes the lth-order spherical Bessel function of the first kind,

Y_{l, m} (\cdot)

denotes the real-valued spherical harmonics of the lth degree and the mth order, with

\sum_{l, m} : = \sum_{l = 0}^{\infty} \sum_{m = - l}^{l}

⁠. The coefficients

a_{l, m}

are assumed to have Gaussian distributions with the following statistical properties:^3,4,11

(5)

E [a_{l, m} (k)] = 0,

(5a)

E [a_{l, m} (k) a_{l^{'}, m^{'}} (k)] = 0,

(5b)

\begin{matrix} E [a_{l, m} (k) \bar{a_{l^{'}, m^{'}} (k)}] = {\begin{matrix} 1, & (l, m) = (l^{'}, m^{'}), \\ 0, & (l, m) \neq (l^{'}, m^{'}) . \end{matrix} \end{matrix}

(5c)

In short, the real and the imaginary parts of each coefficient are zero means and independent, and have the same variance of

1 / 2

⁠. The above statistical properties of

a_{l, m}

define infinite-degree spherical harmonic diffuseness, which represents directionally characterized perfect diffuseness.

2.2 Spatially characterized perfect diffuseness

Spatial diffuseness is defined by the statistical properties of sound pressure yielded using the random-wave model. First, sound pressure in Eq. , given by the linear sum of Gaussian random coefficients, has a Gaussian distribution. Second, the expected value and the two-point pseudo covariance of sound pressures are both zeros,

E [P (k, r)] = \sum_{l, m} 4 π i^{l} j_{l} (k r) E [a_{l, m}] Y_{l, m} (θ, ϕ) \overset{(5a)}{=} 0,

(6a)

E [P (k, r) P (k, r^{'})] = \sum_{l, m} \sum_{l^{'}, m^{'}} {(4 π)}^{2} i^{l + l^{'}} j_{l} (k r) j_{l^{'}} (k r^{'}) E [a_{l, m} a_{l^{'}, m^{'}}] Y_{l, m} (θ, ϕ) Y_{l^{'}, m^{'}} (θ^{'}, ϕ^{'}) \overset{(5b)}{=} 0.

(6b)

Finally, the two-point covariance results in a sinc function^2,5

\begin{matrix} E [P (k, r) \bar{P (k, r^{'})}] = {(4 π)}^{2} \sum_{l, m} \sum_{l^{'}, m^{'}} i^{l - l^{'}} j_{l} (k r) j_{l^{'}} (k r^{'}) E [a_{l, m} \bar{a_{l^{'}, m^{'}}}] Y_{l, m} (θ, ϕ) Y_{l^{'}, m^{'}} (θ^{'}, ϕ^{'}) \\ \overset{(5c)}{=} {(4 π)}^{2} \sum_{l, m} j_{l} (k r) j_{l} (k r^{'}) Y_{l, m} (θ, ϕ) Y_{l, m} (θ^{'}, ϕ^{'}) \overset{add .}{=} \frac{4 π sin (k | r - r^{'} |)}{k | r - r^{'} |}, \end{matrix}

(6c)

where the addition theorems¹⁵ of spherical harmonics and spherical Bessel functions allow the last transformation. Notably, the sinc function corresponds to the imaginary part of a Green's function in free sound fields. In fact, Weaver and Lobkis have shown that the two-point covariance in diffuse sound fields includes the imaginary part of the Green's function between the two positions and suggested defining the spatial diffuseness by the correspondence between the covariance and the Green's function.⁵ The sinc covariance derived above thus represents a special case of the diffuseness in a free sound field.

2.3 Directionally characterized pseudo-perfect diffuseness: Its definition and sound field model

Finite-degree spherical harmonic diffuseness, directionally characterized pseudo-perfect diffuseness, is defined by simply truncating the infinite-degree spherical harmonic diffuseness.^11,12 Namely, Gaussian random coefficients $α_{l, m}$ of $0 \leq l \leq L$ and $| m | \leq l$ with the statistical properties in Eqs. (5a)–(5c) define the Lth-degree spherical harmonic diffuseness.

The Lth-degree spherical harmonic diffuseness is achieved by using a discrete random-wave model of the Lth-degree. The model is composed of a finite number of Gaussian random plane waves with the following statistical properties:^11,12

E [A_{q} (k)] = 0,

(7a)

E [A_{q} (k) A_{q^{'}} (k)] = 0,

(7b)

E [A_{q} (k) \bar{A_{q^{'}} (k)}] = {\begin{matrix} w_{q}, & q = q^{'}, \\ 0, & q \neq q^{'}, \end{matrix}

(7c)

for

q, q^{'} = 1, \dots, Q

⁠, where

A_{q} (k)

denotes the amplitude of the qth plane wave, Q denotes the number of plane waves, and

w_{q}

denotes the weight of a 2Lth-degree spherical quadrature rule.

A spherical quadrature rule of the 2Lth exactitude degree is a set of points and weights

{n_{q} (θ_{q}, ϕ_{q}); w_{q}}_{q = 1}^{Q}

on the sphere accurately calculating the integrals of spherical functions, yielding the following equality:¹⁶^,

\sum_{q = 1}^{Q} w_{q} F_{2 L} (θ_{q}, ϕ_{q}) = \int_{0}^{π} \int_{- π}^{π} F_{2 L} (θ, ϕ) sin θ d ϕ d θ,

(8)

where

F_{2 L} (θ, ϕ)

is a function written by the linear sum of spherical harmonics up to the 2Lth degree,

F_{2 L} (θ, ϕ) = \sum_{l = 0}^{2 L} \sum_{m = - l}^{l} f_{l, m} Y_{l, m} (θ, ϕ),

(9)

which can represent any polynomial function on the sphere of degree at most 2L.¹⁷ In addition, the weights of the quadrature rule are all positive:

w_{q} > 0

⁠, applied

4 π

-normalization:

\sum_{q} w_{q} = 4 π,

and the sampling points of the quadrature rule

{n_{q} (θ_{q}, ϕ_{q})}_{q = 1}^{Q}

determine the arrival directions of the plane waves in the discrete random-wave model. Importantly, the following Lth-degree discrete orthonormality of spherical harmonics is enabled by the 2Lth-degree quadrature rules¹⁶

\begin{matrix} \begin{matrix} \sum_{q = 1}^{Q} w_{q} Y_{l, m} (θ_{q}, ϕ_{q}) Y_{l^{'}, m^{'}} (θ_{q}, ϕ_{q}) \overset{(8)}{=} \int_{0}^{π} \int_{- π}^{π} Y_{l, m} (θ, ϕ) Y_{l^{'}, m^{'}} (θ, ϕ) sin θ d ϕ d θ = {\begin{matrix} 1, & (l, m) = (l^{'}, m^{'}), \\ 0, & (l, m) \neq (l^{'}, m^{'}), \end{matrix} l + l^{'} \leq 2 L, \end{matrix} \end{matrix}

(10)

which is necessary for the coefficients to satisfy the spherical harmonic diffuseness up to the Lth degree.^11,12

Sound pressure in this model can then be written as follows:

P (k, r) = \sum_{q = 1}^{Q} A_{q} e^{i k n_{q} \cdot r}, n_{q} = {(cos ϕ_{q} sin θ_{q}, sin ϕ_{q} sin θ_{q}, cos θ_{q})}^{⊤} .

(11)

The sound pressure can also be represented by the linear sum of spherical wave functions

P (k, r) = \sum_{l, m} 4 π i^{l} α_{l, m} j_{l} (k r) Y_{l, m} (θ, ϕ),

(12)

with

α_{l, m} = \sum_{q = 1}^{Q} A_{q} Y_{l, m} (θ_{q}, ϕ_{q}) .

(13)

Note that the above models of diffuse sound fields simulate sound fields generated by multiple sound sources emitting Gaussian noise⁵ rather than room impulse responses.

3. Results and discussion: Spatially characterized pseudo-perfect diffuseness

This section shows that the statistical properties of sound pressure yielded using discrete random-wave models can characterize pseudo-perfect diffuseness spatially. First, we theoretically show that the expected value and the two-point pseudo covariance of sound pressures always result in zeros regardless of the diffuseness degree, L. Following the definition of spatial diffuseness suggested by Weaver and Lobkis,⁵ we then show numerically that the two-point covariance locally corresponds with the imaginary part of the Green's function between the two positions and suggest defining local diffuseness, which represents spatially characterized pseudo-perfect diffuseness, by exploiting such correspondence.

3.1 The expected value and the two-point pseudo covariance

The amplitudes of the plane waves in the discrete model have Gaussian distributions, and thus sound pressure , written as the linear sum of the amplitudes, also has a Gaussian distribution. The expected value and the two-point pseudo covariance of

P (k, r)

are zeros,

E [P (k, r)] \overset{(11)}{=} E [\sum_{q = 1}^{Q} A_{q} e^{i k n_{q} \cdot r}] = \sum_{q = 1}^{Q} E [A_{q}] e^{i k n_{q} \cdot r} \overset{(7a)}{=} 0,

(14)

E [P (k, r) P (k, r^{'})] = \sum_{q = 1}^{Q} \sum_{q^{'} = 1}^{Q} E [A_{q} A_{q^{'}}] e^{i k n_{q} \cdot r} e^{i k n_{q^{'}} \cdot r^{'}} \overset{(7b)}{=} 0.

(15)

These results show that only the zero statistical properties derived above cannot characterize spatial pseudo-perfect diffuseness because they hold for any pair of positions, $r$ and $r^{'},$ regardless of the diffuseness degree, L. In addition, these zero properties are identical to those in the (continuous) random-wave model [see Eqs. (6a) and (6b)].

3.2 The two-point covariance

First, we calculated the two-point covariance in pseudo-perfectly diffuse sound fields synthesized using the Lth-degree discrete random-wave models under free conditions without scatterers or reflectors present in the field, as follows:

E [P (k, r) \bar{P (k, O)}] \approx \frac{1}{T_{s} - 1} \sum_{τ = 1}^{T_{s}} (\sum_{q = 1}^{Q} A_{q}^{(τ)} e^{i k n_{q} \cdot r} \bar{\sum_{q^{'} = 1}^{Q} A_{q^{'}}^{(τ)}}),

(16)

where

O

denotes the origin and

τ

is the index indicating the number of syntheses of sound fields with T_s = 50 000 denoting the total number of syntheses. The random amplitudes

A_{q}

with the statistical properties as in Eqs. to and the arrival directions

n_{q}

of the plane waves were generated by using a numerical quadrature rule of the 2Lth degree.¹⁸ The frequency and the sound speed were respectively 880 Hz and 345 m/s, yielding

k \approx 16.0

rad/m. The positions

r

were on 64 straight lines through the origin, whose directions were selected using random points on a unit hemisphere.

The upper panel of Fig. 1 shows the real part of the two-point covariance computed using discrete random-wave models of select degrees, (a) $L = 8$ ⁠, (b) $L = 16$ ⁠, (c) $L = 24$ ⁠, and (d) $L = 32.$ Colored lines denote the two-point covariance on randomly selected straight lines and circles denote the sinc function, $sin (k r) / k r$ ⁠. The two-point covariance was scaled by $1 / 4 π$ ⁠. The lower panel of Fig. 1 shows the absolute error between the sinc function and the two-point covariance above.

Fig. 1.

View large Download slide

(Upper panel) Colored lines show the real part of two-point covariance computed between the origin and the positions on randomly selected 64 straight lines thorough the origin in sound fields repeatedly synthesized using discrete random-wave models of select diffuseness degrees, (a) $L = 8$ ⁠, (b) $L = 16$ ⁠, (c) $L = 24$ ⁠, and (d) $L = 32$ ⁠. The number of sound field syntheses was 50 000. The two-point covariance was scaled by $1 / 4 π$ ⁠. Circles denote the sinc function, $sin (k r) / k r,$ with $k \approx 16.0$ rad/m. (Lower panel) Absolute errors between the sinc function and the two-point covariance above. Colors of the lines therein correspond to those in the above figures.

As shown in Figs. 1(e)–1(h), the two-point covariance remarkably corresponds to the sinc function for

r < 1

⁠,

r < 2

⁠,

r < 3

⁠, and

r < 4

⁠, respectively. This implies that the radius of the area of such correspondence can be connected with the diffuseness degrees L and the wavenumber k as follows:

R = 2 L / k .

(17)

Next, we calculated the two-point covariance in pseudo-perfectly diffuse sound fields with a sound-hard spherical scatterer of radius,

r_{sc} = 0.6

m, at the origin. (Unless otherwise specified, the calculation conditions are the same as in the above experiment.) The synthesized sound fields can be written as follows:

P_{sc} {(k, r)}^{(τ)} = \sum_{l = 0}^{L_{tr}} \sum_{m = - l}^{l} 4 π i^{l} α_{l, m}^{(τ)} [j_{l} (k r) - \frac{j_{l}^{'} (k r_{sc})}{h_{l}^{'} (k r_{sc})} h_{l} (k r)] Y_{l, m} (θ, ϕ), r > r_{sc},

(18)

where

j_{l} (\cdot)

is the spherical Bessel functions of the first kind;

h_{l} (\cdot)

is the spherical Hankel functions of the first kind; and their first derivatives are represented by

j_{l}^{'} (\cdot)

and

h_{l}^{'} (\cdot)

⁠, respectively, and

α_{l, m}^{(τ)}

is written as

α_{l, m}^{(τ)} = \sum_{q = 1}^{Q} A_{q}^{(τ)} Y_{l, m} (θ_{q}, ϕ_{q}),

(19)

where again,

τ

is the index for the number of syntheses of sound fields. As the truncation degree in Eq. , we let

L_{tr} = 49

to accurately calculate sound fields for the positions such that

r < 3

⁠. We then computed the two-point covariance between

r_{c} = {(- 1, 1, 0)}^{⊤}

and the positions

r

on

z = 0

plane,

E [P_{sc} (k, r) \bar{P_{sc} (k, r_{c})}] \approx \frac{1}{T_{s} - 1} \sum_{τ = 1}^{T_{s}} P_{sc} {(k, r)}^{(τ)} \bar{P_{sc} {(k, r_{c})}^{(τ)}} .

(20)

The real parts of the two-point covariance computed using discrete random-wave models of diffuseness degrees,

L = 16

⁠,

L = 24,

and

L = 32

⁠, are shown in the upper panel of Fig. 2. The two-point covariance was scaled by

k / {(4 π)}^{2}

for easier comparison with the imaginary part of the Green's function for a point source at

r_{c}

shown in Fig. 2(d). The Green's function was computed as follows:

G (k; r, r_{c}) \approx \frac{e^{i k | r_{c} - r |}}{4 π | r_{c} - r |} + \sum_{l = 0}^{L_{tr}} \sum_{m = - l}^{l} β_{l, m} (k) h_{l} (k r) Y_{l, m} (θ, ϕ), r > r_{sc},

(21)

with

β_{l, m} (k) = - i k h_{l} (k r_{c}) Y_{l, m} (θ_{c}, ϕ_{c}) \frac{j_{l}^{'} (k r_{sc})}{h_{l}^{'} (k r_{sc})} .

(22)

The lower panel of Fig. 2 shows the absolute error between the imaginary part of the Green's function and the two-point covariance above.

As shown in the lower panel of Fig. 2, the area of correspondence between the Green's function and the two-point covariance extends around the reference position of covariance,

r_{c}

⁠, as the diffuseness degree increases. Here, again, the radius of such area can be written as in Eq. . Thus, in the Lth-degree directionally diffuse sound fields, spatial diffuseness can hold for any position,

r

⁠, such that

| r - r_{c} | < 2 L / k .

(23)

That is, we can define spatial pseudo-perfect diffuseness as local diffuseness with the effective radius,

2 L / k,

via the Lth-degree directional diffuseness. Notably, in plane wave sound fields, the effective radius of local diffuseness is constant regardless of the reference position,

r_{c}

⁠; however, in sound fields not composed only of plane waves, the effective radius will depend on the diffuseness degree at the reference position.

Fig. 2.

View large Download slide

(Upper panel) Real part of the two-point covariance between $r_{c} = {(- 1, 1, 0)}^{⊤}$ and the positions on $z = 0$ plane in sound fields repeatedly synthesized using discrete random-wave models of select diffuseness degrees, (a) $L = 16$ ⁠, (b) $L = 24$ ⁠, and (c) $L = 32$ ⁠. The two-point covariance was scaled by $k / {(4 π)}^{2}$ ⁠. (d) Imaginary part of the Green's function for a point source at $r_{c}$ ⁠. (Lower panel) Absolute errors between the imaginary part of the Green's function and the two-point covariance above.

3.3 Supplementary remarks

In addition to characterizing pseudo-perfect diffuseness spatially, the present results provide insight into the sufficient number of isotropic and incoherent sound sources for generating locally diffuse sound fields, which is not very trivial.⁵ Namely, the number of sampling points in the 2Lth-degree quadrature rule, which composes the Lth-degree discrete random-wave model, can be regarded as the sufficient number of random sound sources. For example, we can numerically construct the 2Lth-degree spherical quadrature rules with

Q \approx ⌈ \frac{{(2 L + 1)}^{2}}{3} ⌉ + 1, L = 0, 1, \dots, 50,

(24)

sampling points¹⁹ and positive weights,¹⁸ where

⌈ \cdot ⌉

denotes a ceiling function.

Note that emerging the imaginary part of the Green's function from the two-point covariance of sound pressures in diffuse sound fields is different from reproducing a sound field based on theories such as higher-order ambisonics²⁰ (HOA). For example, HOA usually deals only with interior problems (i.e., sound fields without sound sources or scatterers). Our results should therefore not be interpreted based on HOA.

4. Conclusion

Aiming to spatially characterize pseudo-perfect diffuseness, we presented locally diffuse sound fields with the effective radius, $R = 2 L / k$ ⁠, via the Lth-degree spherical harmonic diffuseness. Our findings connect the directional and the spatial characterizations of pseudo-perfect diffuseness.

Acknowledgments

This work was supported by JSPS KAKENHI (Grant Nos. JP19H04153, JP22H00523) and Grant-in-Aid for JSPS Fellows (Grant No. 22KJ1941).

Author Declarations

Conflict of Interest

The authors have no conflicts to disclose.

Data Availability

The data are available from the corresponding author upon reasonable request.

References

1.

C. G.

Balachandran

and

D. W.

Robinson

, “

Diffusion of the decaying sound field

,”

Acustica

19

(

5

),

245

–

257

(

1967

).

Google Scholar

2.

R. K.

Cook

,

R. V.

Waterhouse

,

R. D.

Berendt

,

S.

Edelman

, and

M. C.

Thompson

, Jr. , “

Measurement of correlation coefficients in reverberant sound fields

,”

J. Acoust. Soc. Am.

27

(

6

),

1072

–

1077

(

1955

).

https://doi.org/10.1121/1.1908122

Google Scholar

Crossref

3.

S.

Yan

,

H.

Sun

,

U. P.

Svensson

,

X.

Ma

, and

J. M.

Hovem

, “

Optimal modal beamforming for spherical microphone arrays

,”

IEEE Trans. Audio. Speech. Lang. Process.

19

(

2

),

361

–

371

(

2011

).

https://doi.org/10.1109/TASL.2010.2047815

Google Scholar

Crossref

4.

B.

Rafaely

,

Fundamentals of Spherical Array Processing

, 2nd ed. (

Springer

,

Cham, Switzerland

,

2019

), Sec. 7.1.

Google Scholar

Crossref

5.

R. L.

Weaver

and

O. I.

Lobkis

, “

Diffuse fields in open systems and the emergence of the Green's function (L)

,”

J. Acoust. Soc. Am.

116

(

5

),

2731

–

2734

(

2004

).

https://doi.org/10.1121/1.1810232

Google Scholar

Crossref

6.

H.

Kuttruff

,

Room Acoustics

, 6th ed. (

CRC Press

,

Boca Raton, FL

,

2016

), Sec. 8.7.

Google Scholar

Crossref

7.

E.

Meyer

and

R.

Thiele

, “

Raumakustische Untersuchungen in zahlreichen Konzertsälen und Rundfunkstudios unter Anwendung neuerer Messverfahren

” (“Room acoustic studies in numerous concert halls and radio studios using newer measurement methods”),

Acustica

6

,

425

–

444

(

1956

) (in German with English summary).

Google Scholar

8.

M.

Koyasu

and

M.

Yamashita

, “

Evaluation of the degree of diffuseness in reverberation chambers by spatial correlation techniques

,”

J. Acoust. Soc. Jpn.

27

(

3

),

132

–

143

(

1971

) (in Japanese with English summary).

https://doi.org/10.20697/jasj.27.3_132

Google Scholar

9.

K. J.

Ebeling

, “

Statistical properties of random wave fields

,” in

Physical Acoustics

, edited by

W. P.

Mason

and

R. N.

Thurston

(

Academic Press

,

Orlando, FL

,

1984

), Vol.

17

, Secs. 5.3 and 5.4, pp.

233

–

310

.

Google Scholar

10.

M.

Nolan

,

E.

Fernadez-Grande

,

J.

Brunskog

,

A.

Richard

, and

C. H.

Jeong

, “

A wavenumber approach to characterizing the diffuse field conditions in reverberation rooms

,” in

Proceedings of the 22nd International Congress on Acoustics

, paper ICA2016-578, Buenos Aires, Argentina (

2016

).

Google Scholar

11.

T.

Tanaka

and

M.

Otani

, “

A diffuseness of a sound field in the domains of spherical harmonics and plane waves

,”

Inter-Noise

268

(

6

),

2164

–

2171

(

2023

).

https://doi.org/10.3397/IN_2023_0319

Google Scholar

Crossref

12.

T.

Tanaka

and

M.

Otani

, “

Directionally characterized pseudo-perfect diffuseness: A detailed comparison of its theoretical formulations

,”

Acoust. Sci. Technol.

(published online 2024).

https://doi.org/10.1250/ast.e24.16

13.

M. J.

Beran

and

G. B.

Parrent

,

Theory of Partial Coherence

(

Prentice-Hall

,

New York

,

1964

), Sec. 4.4.

Google Scholar

14.

F.

Jacobsen

and

A. R.

Molares

, “

Sound power emitted by a pure-tone source in a reverberation room

,”

J. Acoust. Soc. Am.

126

(

2

),

676

–

684

(

2009

).

https://doi.org/10.1121/1.3158918

Google Scholar

Crossref

PubMed

15.

F. W. J.

Olver

,

A. B.

Olde Daalhuis

,

D. W.

Lozier

,

B. I.

Schneider

,

R. F.

Boisvert

,

C. W.

Clark

,

B. R.

Miller

,

B. V.

Saunders

,

H. S.

Cohl

, and

M. A.

McClain

, “

NIST Digital Library of Mathematical Functions

” (2023), https://dlmf.nist.gov/ (Last viewed 2 December

2023

).

Google Scholar

16.

I. H.

Sloan

, “

Polynomial interpolation and hyperinterpolation over general regions

,”

J. Approximation Theory

83

(

2

),

238

–

254

(

1995

).

https://doi.org/10.1006/jath.1995.1119

Google Scholar

Crossref

17.

R. S.

Womersley

and

I. H.

Sloan

, “

How good can polynomial interpolation on the sphere be?

,”

Adv. Comput. Math.

14

,

195

–

226

(

2001

).

https://doi.org/10.1023/A:1016630227163

Google Scholar

Crossref

18.

T.

Tanaka

, “

Efficient spherical quadrature rules with positive weights [Dataset]

” (2023), doi: 10.5281/zenodo.10042399 (Last viewed 26 October

2023

).

Google Scholar

19.

M.

Gräf

, “

Efficient algorithms for the computation of optimal quadrature points on Riemannian manifolds

,” Ph.D. thesis,

Technische Universität Chemnitz

, Chemintz, Germany (

2013

), Chap. 6.

Google Scholar

20.

F.

Zotter

and

M.

Frank

,

Ambisonics: A Practical 3D Audio Theory for Recording, Studio Production, Sound Reinforcement, and Virtual Reality

(

Springer Nature

,

New York

,

2019

).

Google Scholar

Crossref

Spatially characterized pseudo-perfect diffuseness via finite-degree spherical harmonic diffuseness

1. Introduction

2. Preliminaries

2.1 Models of perfectly diffuse sound fields

2.2 Spatially characterized perfect diffuseness

2.3 Directionally characterized pseudo-perfect diffuseness: Its definition and sound field model

3. Results and discussion: Spatially characterized pseudo-perfect diffuseness

3.1 The expected value and the two-point pseudo covariance

3.2 The two-point covariance

3.3 Supplementary remarks

4. Conclusion

Acknowledgments

Author Declarations

Conflict of Interest

Data Availability

References

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Spatially characterized pseudo-perfect diffuseness via finite-degree spherical harmonic diffuseness

1. Introduction

2. Preliminaries

2.1 Models of perfectly diffuse sound fields

2.2 Spatially characterized perfect diffuseness

2.3 Directionally characterized pseudo-perfect diffuseness: Its definition and sound field model

3. Results and discussion: Spatially characterized pseudo-perfect diffuseness

3.1 The expected value and the two-point pseudo covariance

3.2 The two-point covariance

3.3 Supplementary remarks

4. Conclusion

Acknowledgments

Author Declarations

Conflict of Interest

Data Availability

References

Citing articles via

Submit your article

Sign up for alerts

Resources

Explore

pubs.aip.org

Connect with AIP Publishing

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