Chapter 2

Soundfield Modelling

2.1 Introduction

Acoustic signal processing techniques have been studied now for many years. These

techniques perform well under ideal conditions but still face difficulty in real envi-

ronments, where captured signals are corrupted by reverberation. In the interest

of developing algorithms more robust to reverberation, it is important to have a

simple analytical technique to quantify algorithmic performance. Researchers often

resort to computer simulation and real-world experiment for testing of algorithms.

However, the results may not promote deep insights or generality. In this chapter,

we describe general soundfield models useful for analyzing the performance of sig-

nal processing algorithms in reverberant rooms. These models provide the tools

that are used throughout the rest of the thesis.

Acoustic waves are well described by the wave equation. Any soundfield model

of reverberation must be consistent with it. One way of representing the wave

equation solutions in a source free region is with the single layer potential [18,51].

The single layer potential has proven useful for exploring the statistical properties

of wireless communication channels in different scattering environments [51, 100].

In a wide range of room environments, the soundfield can be described with

statistical room acoustics [83,84]. Here the sound pressure at each point is thought

of as a random variable, with predictable statistical properties. An important

model for describing these statistical properties of sound pressure is the diffuse

field model [57], where sound is evenly diffused throughout the room. A diffuse

soundfield can be thought of as the superposition of a large number of plane waves,

arriving evenly from all directions. With this understanding, one can derive the cor-

relation of sound pressure between different positions (and different time instances)

in a diffuse field [79].

In other disciplines, wave field models have benefitted by recognizing that plane

waves do not always arrive evenly from all directions. Often they are physically

constrained to a limited range of directions. Examples are ambient sea noise [20,22]

25

z

j

φ

θx

r

yx

k

i

(a)

xφ

r

i

j

xy

(b)

Figure 2.1: Representation of a position vector x in (a) the 3-D spherical coordi-nates (r, θ, φ) and (b) the 2-D polar coordinates (r, φ).

and the multipath of telecommunications systems [76, 100]. Accounting for such

directionality here has resulted in more accurate system models.

In this chapter, we derive a statistical description of the field pressure based

on a generalization of the diffuse field model. This so called generalized diffuse

field model, extends the classical diffuse field model to soundfields of a directional

nature. We also explore some general representations of the reverberation: in

particular, the wave equation solutions describing the soundfield in a source-free

region, and the geometric interpretation of such a soundfield as a single layer po-

tential. Relationships between these representations are established. The models

are used in subsequent chapters to quantify the performance of acoustic systems

and to motivate new system designs.

Previewing the chapter’s content, Section 2.2 establishes basic notation used in

this thesis. Section 2.3 provides general solution to the Wave equation. Section

2.4 develops the generalized diffuse field model of a reverberant field. Section 2.5

presents the geometric interpretation of a reverberant field. Section 2.6 describes

sample geometric configurations. Finally, Section 2.7 explores the 2-D case.

2.2 Coordinate Systems and Notation

Before describing the soundfield models, we define the notation for representation

of geometric features: positions, directions, surfaces and volumes in space.

In 3-D space, it is convenient to represent coordinates in the spherical coordinate

system. As shown in Figure 2.1(a), the spherical coordinates (r, θ, φ) of a position

vector x ∈ R3 are related to the Cartesian coordinates (x, y, z) through:

(x, y, z) = (r sin θ cos φ, r sin θ sin φ, r cos θ),

where r , ‖x‖ is the distance from the origin, θ is the polar angle, φ is the azimuthal

angle and the operator ‖·‖ denotes Euclidean distance. Direction is specified either

through a pair of polar and azimuthal angles (θ, φ) or through the unit vector φ =

[sin θ cosφ, sin θ sinφ, cos θ]T pointing in the direction of interest. Representations

of position and direction are used interchangeably in parameterizations of functions.

That is, f(r, θ, φ) ≡ f(x) and g(θ, φ) ≡ g(φ).

In 2-D space, it is convenient to represent coordinates with the polar coordinate

system shown in Figure 2.1(b). The polar coordinates (r, φ) of a position vector

x ∈ R2 are related to Cartesian coordinates (x, y) through:

(x, y) = (r cos φ, r sin φ),

where r is the distance from the origin and φ is the polar angle. Direction is

specified through either the angle φ or the unit vector φ , [cos φ, sinφ]T .

We utilize notation which covers the 2-D and 3-D cases simultaneously. Position

is represented with a vector x ∈ R` and direction with a unit vector φ ∈ R

` where

` = 2 for 2-D and ` = 3 for 3-D. We refer to the unit surface S`−1 , {x ∈ R

` :

‖x‖ = 1}, which represents a unit circle S = {(r, φ) : r ≡ 1, 0 ≤ φ < 2π} for ` = 2

and unit spherical shell S2 = {(r, θ, φ) : r ≡ 1, 0 ≤ θ ≤ π, 0 ≤ φ < 2π} for ` = 3.

The integral:∫

S`−1

f(φ)ds(φ),

for ` = 2 represents a line integral over a circle where ds(φ) = dφ is the differential

curve element at φ. For ` = 3, it represents a surface integral over a spherical shell

where ds(φ) = sin θ dθ dφ is the differential surface element at φ.

2.3 Wave Equation Solution

Following [114], we derive the sets of solutions to the Helmholtz wave equation.

These solutions describe the sound pressure of any soundfield inside a source-free

region. They provide an efficient parametrization of the sound pressure in terms

of orthogonal functions, useful for describing the reverberation in signal processing

problems.

We first summarize the frequency domain approach of solving the wave equation

(Section 2.3.1). We derive the general 3-D solution (Section 2.3.2). We then restrict

attention to the soundfield inside a source-free spherical region (Section 2.3.3).

2.3.1 Helmholtz Equation

In this thesis, we seek to study the propagation of sound waves, desiring to know

the pressure p(x, t) at any point x and time t. For an homogeneous medium un-

dergoing inviscid fluid flow, one can linearize the equations governing the dynamic

behavior to the fluid, namely the Euler’s equation, the continuity equation, the

state equation and the adiabatic hypothesis [18], to obtain the wave equation,

∇2p =1

c2∂2p

∂t2, (2.1)

where ∇2 is the Laplacian operator and c is the speed of sound. The wave equation

provides a good description of the propagation of sound waves of small amplitude in

air. It accurately describes the pressure in the soundfield provided |p(x, t)| � ρoc2

where ρ0 is the density of the propagation medium at equilibrium [49, p. 11].

It is convenient to consider (2.1) in the frequency domain [114, p. 18]. The

Fourier transform operator, represented by calligraphic letter F , is defined with

the analysis equation

F{p(x, t)}(ω) =

∫ ∞

−∞

p(x, t)e−iωtdt. (2.2)

Writing the Fourier transform of (2.1) and utilizing the derivative property,

F{

∂p(x, t)

∂t

}

(ω) = −iωP (x;ω),

where P (x;ω) , F{p(x, t)}(ω), (2.1) implies

∇2P (x;ω) + k2P (x;ω) = 0, (2.3)

where k is called the wave number, and is related to angular frequency ω and

wavelength λ through

k =ω

c=

2π

λ.

Equation 2.3 is known as the Helmholtz equation and underlies the soundfield

models of this thesis.

In this thesis we present a frequency domain formulation where the angular

frequency ω is arbitrary but fixed. This formulation applies directly to narrow-band

problems but extends naturally to broadband problems. Because the Helmholtz

equation is linear and time invariant, the steady-state dependence of pressure on

time is obtained through the inverse Fourier transform of P (x;ω),

p(x, t) =1

2π

∫ ∞

−∞

P (x;ω)eiωtdω. (2.4)

Quantities in the frequency domain are written with capital letters and an explicit

dependence on ω.

2.3.2 General Solution

The Helmholtz equation can be solved for sound pressure P (x;ω) in spherical

coordinates using the following separation of variables:

P (x;ω) = X(r;ω)Θ(θ;ω)Φ(φ;ω), (2.5)

where θ, φ and r are the polar angle, azimuthal angle and length of x respectively.

Substituting (2.5) and the expression for the Laplacian operator in spherical coor-

dinates,

∇2(·) =1

r2

∂

∂r

[

r2 ∂

∂r(·)

]

+1

r2 sin θ

∂

∂θ

[

sin θ∂

∂θ(·)

]

+1

r2 sin2 θ

∂2

∂φ2(·), (2.6)

into (2.3) leads to separation into three ordinary differential equations [87, pp. 379-

380]:d2Φ

dφ2+m2Φ = 0, (2.7)

d

dθ

(

sin θdΘ

dθ

)

+

[

n(n + 1) − m2

sin2 θ

]

Θ = 0, (2.8)

1

r2

d

dr

(

r2dX

dr

)

+ k2X − n(n + 1)

r2X = 0, (2.9)

where m and n are integers. The solutions to (2.7) are Φ+(φ) = eimφ and Φ−(φ) =

e−imφ. The finite solutions of (2.8) are Θ(θ) = Pmn (cos θ), the associated Legendre

functions of degree n and order m. Now Pmn (cos θ) = 0 for m > n so m can only

take values −n, . . . , n. From (2.9), the solutions for the radial component X(r) are

the spherical Bessel functions of the first and second kinds of order n, jn(kr) and

yn(kr), with r = ‖x‖. These functions are related to the ordinary Bessel functions

of first and second kind1 respectively through:

jn(x) =

√

π

2xJn+1/2(x),

yn(x) =

√

π

2xNn+1/2(x).

The solutions of the angular functions are conveniently expressed in the spher-

ical harmonic functions Y mn (·) [114]:

Y mn (φ) , Λm

n Pmn (cos θ)eimφ,

where Λmn ,

√

2n+14π

(n−m)!(n+m)!

is a normalization term. To avoid confusion, please

1The functions of second kinds Nn(·) and yn(·) are also known as the Neumann and sphericalNeumann functions respectively.

note that this definition differs from that mentioned in several sources of literature

[18, 49]2. Spherical harmonic functions satisfy the orthogonality property,

∫

S2

Y mn (φ)[Y p

q (φ)]∗ ds(φ) = δmpδnq, (2.10)

where [·]∗ is the complex conjugation operator and δnm is the Kronecker delta

function

δnm ,

{

1, m = n

0, m 6= n.

Spherical harmonic functions form an orthonormal set spanning the unit spherical

shell S2.

Noting that jn(k‖x‖)Y mn (x) and yn(k‖x‖)Y m

n (x) are both solutions to (2.3),

the complete solution is written as [114, p. 186],

P (x;ω) =

∞∑

n=0

n∑

m=−n

[

β(j)nm(ω)jn(k‖x‖) + β(y)

nm(ω)yn(k‖x‖)]

Y mn (x). (2.11)

The sound pressure in any soundfield is fully represented by the set of coefficients

{β(j)nm(ω)}n∈Z∗

⋃

{β(y)nm(ω)}n∈Z∗ where Z

∗ = {0, 1, 2, . . .} is the set of non-negative

integers. Conversely any function P (x;ω) that cannot be written in this form does

not satisfy the Helmholtz equation and is not a valid pressure field3.

In the next section, we present a set of solutions for the case where no sound

sources lie in the region of interest. We shall see that the spherical Neumann

function coefficients are all zero for this case.

2.3.3 Interior Field Solution

The soundfield inside a source free spherical region is known as the interior field.

Define a spherical ball centered about the origin of radius R, B` = {x ∈ R

` : ‖x‖ ≤R}, as shown in Figure 2.2, excludes all sound sources. In this section we write the

general solution in the interior of B` for the case ` = 3.

The interior field solution consists of the solutions of the Helmholtz equation

(2.11) that are finite at the origin. Since the spherical Neumann function yn(k‖x‖)becomes infinite at the origin (limx→0+ yn(x) = −∞), each β

(y)nm(ω) must be zero4.

The general interior field solution is hence:

P (x;ω) =∞

∑

n=0

n∑

m=−n

βnm(ω)jn(k‖x‖)Y mn (x). (2.12)

2The definition in this thesis has the property that Y −mn (φ) = (−1)m[Y m

n (φ)]∗.3One such example of an invalid pressure field is P (x; ω) ≡ 1 for x ∈ R

3.4The spherical Bessel function jn(‖x‖) remains finite for x ∈ R

3.

O

Sources

Region B`

R

Sources

x

Figure 2.2: Illustration of the interior field problem, where the pressure at a pointx in the source-free ball B

`, with ` = 2, 3, is determined.

The coefficients βnm(ω) shall be called the soundfield coefficients and functions

jn(k‖x‖)Y mn (x) the modes of the soundfield5. The radial dependence of the modes

is illustrated in Figure 2.3, where several of the low order Bessel functions are

shown. The soundfield coefficients characterize the soundfield completely. All

information about the reverberant field within B3 is specified through coefficients

βnm(ω).

The soundfield coefficients can be determined from the sound pressure P (x;ω).

If the soundfield pressure function P (x;ω) is known, the associated soundfield

coefficients can be determined from P (x;ω). Multiplying both sides of (2.12) by

[Y pq (x)]∗, integrating over S

2 and applying the orthogonality property (2.10) yields:

βqp(ω) =1

jq(k‖x‖)

∫

S2

P (x;ω)[Y pq (x)]∗ds(x), (2.13)

provided jq(k‖x‖) 6= 0.

Some of the zeros of the spherical Bessel functions are shown in Figure 2.3.

For each spherical Bessel function the zeros are spaced approximately 2π apart

[114, p. 118]. For frequencies at which jq(k‖x‖) = 0, the associated soundfield

coefficients {βqp(ω)}qp=−q can be found with an equation derived by performing the

integration over B3 rather than S

2 [51]. Alternative is to perform integration over

a pair of concentric spherical shells, and is presented in Chapter 5.

The usefulness of the modal expansion (2.12) originates from the following

properties:

(i) An efficient parametrization of a soundfield over B3. As is exploited in Chap-

5The term mode contrasts with use of the term in the audio field [66], where it refers tofrequency resonances. In this thesis, each mode represents a particular standing wave.

0 2 4 6 8 10 12 14 16 18 20−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x

j n(x)

j0(x)

j1(x)

j2(x)

j5(x)

j10

(x)

Figure 2.3: Plot of the spherical Bessel function jn(x) for n ∈ {0, 1, 2, 5, 10}.

ter 5, the modal expansion allows representation of the soundfield in B3 with

a small number of terms. In fact, (2.12) can be truncated to n ≤ N = dkRewhere operator d·e is the integer ceiling function, creating an expansion with

(N + 1)2 terms [48, 106].

(ii) The orthogonality of the spherical harmonic functions. It ensures simple ex-

pressions when integrating linear and certain quadratic functions of (2.12)

over the spherical shell S2.

In the next section, we describe the principal model of a reverberant field used

in this thesis.

2.4 Diffuse Field Model of Reverberation

A general model of soundfield pressure, largely free from the geometric details of

the room, is an important part of this thesis. We describe such a model in this

section.

We differentiate in the model between direct and reverberant components. The

direct component Pd(x;ω) is the part of the sound that arrives from a sound source

directly, without reflection or diffraction (shown in blue in Figure 1.3). The rever-

berant component Pr(x;ω) is what remains after subtraction of the direct compo-

O Reverb

ReverbSouce

Reverb −φ2

B3

R

−φ1

−φ3

x

Figure 2.4: Representation of a soundfield resulting from a component comingdirectly from the sound source and generalized diffuse component consisting ofplane waves originating from directions φn for n = 1, . . . , 3. Each plane wave hasamplitude ξr(φn;ω).

nent6. Complex sound pressure P (x;ω) can then be expressed

P (x;ω) = Pd(x;ω) + Pr(x;ω). (2.14)

The direct component Pd(x;ω) is given by free-field expressions for sound pressure

(1.3) and (1.4). We shall model the reverberant component of sound pressure with

a generalized diffuse field, described below. In this model we use statistical room

acoustics, where the pressure Pr(x;ω) at each position x modelled as a random

variable.

The relative size of direct and reverberant components at any point x in a

soundfield is described by the direct-to-reverberant ratio γP (x;ω) [57, 67]. The

direct-to-reverberant ratio is the ratio of the energy density of the direct component

to the energy density of the reverberant component. We can write:

γP (x;ω) ,|Pd(x;ω)|2

E{|Pr(x;ω)|2} , (2.15)

where E{·} is the expectation operator. The expectation is used to obtain an

ensemble average of reverberant energy density.

6The utility of the distinction arises from the ease of predicting the direct component. Thedirect component is a function of only the distance to the sound source and source strength.Further, for many signal processing algorithms to be practical, the direct signal must comprisea significant proportion of the reverberated signal energy [13, 99, 108]. For example, data inde-pendent beamforming, which operates by aligning direct components, is ineffective if the directenergy is negligible.

2.4.1 Basic Description

We start with a concise mathematical description of the reverberant sound pressure

in the diffuse field, and extend the definition to the generalization of the diffuse

field. The key to this description is in modelling the sound pressure at each point

in a room as a random variable.

In a diffuse field, the sound pressure at any point results from superposition of

planar waves. We can represent this fact in equation form as [18, p. 53]:

Pr(x;ω) =

∫

S`−1

ξr(φ;ω)e−ikx·φds(φ), (2.16)

where the reverberation geometry function ξr(φ;ω) describes the complex ampli-

tude of the planar wave propagating from each direction φ and e−ikx·φ describes

the spatial variation of the plane wave with position. The basic idea is illustrated

in Figure 2.4. |ξr(φ;ω)| represents the amplitude and ∠ξr(φ;ω) the phase of each

plane wave. ξr(φ;ω) is written as a spatial random process in the variable φ. This

equation is applicable to both 2-D (` = 2) and 3-D (` = 3) spaces.

The diffuse field is usually defined in terms of plane waves. To quote some

textbook definitions of a diffuse soundfield: “plane waves are incident from all

directions with equal probability and random phase” [28, p. 157] and “the sound

energy arriving at any point inside the enclosure is uniformly distributed over all

possible directions of incidence” [57, p. 1110].

Such textbook definitions are substantiated by the literature where these prop-

erties are more precisely defined [19, 57, 59, 102]. The field is only diffuse if the

amplitudes of plane waves from different angles of incidence are statistically inde-

pendent [59]. The relative phases of the plane waves are uniformly distributed [60]

and the amplitudes often modelled with equal intensity [57,59].

These properties of the plane waves comprising the diffuse field are neatly sum-

marized into the following definition.

Definition 2.4.1 (The Diffuse Field) In the diffuse field, the pressure at each

position x is described by (2.16) where the reverberant geometry function ξr(θ;ω) =

|ξr(θ;ω)|ei∠ξr(φ;ω) is a random variable possessing the following properties:

(i) The phase ∠ξr(φ;ω) of each plane wave is uniformly distributed over [0, 2π]

and independent of the plane wave amplitude |ξr(φ;ω)|.

(ii) The random variables ξr(φ;ω) and ξ

r(ϕ;ω) are independent for all φ 6= ϕ.

(iii) The distribution of ξr(φ;ω) is invariant with φ.

We extend and generalize the diffuse field to encompass fields with a directional

character, so that distribution of plane waves in such a field can be non-uniform.

We make this generalization by relaxing the property (c) of Definition 2.4.1:

Definition 2.4.2 (The Generalized Diffuse Field) In the generalized diffuse

field, the pressure at each position x is described by (2.16) where the reverberant

geometry function ξr(θ;ω) possesses the following properties:

(i) The phase ∠ξr(φ;ω) of each plane wave is uniformly distributed over [0, 2π]

and independent of the plane wave amplitude |ξr(φ;ω)|.

(ii) The random variables ξr(φ;ω) and ξ

r(ϕ;ω) are independent for all φ 6= ϕ.

The distribution of ξr(φ;ω) is no longer invariant with φ. In particular the variance

σ2ξ (φ;ω) , E{|ξr(φ;ω)|2} can be an arbitrary function of φ. In the next section

we derive the statistical properties of ξr(φ;ω) common to both definitions.

2.4.2 Statistical Properties

We now statistically describe the relationship of the sound pressure in a general-

ized diffuse soundfield. These statistical properties are utilized in the performance

analyses of Chapters 3 and 4. We start by establishing the underlying statistical

properties of the reverberant geometry function.

The Reverberant Geometry Function

Theorem 2.4.1 below summarizes some properties implied for the reverberant ge-

ometry function ξr(φ;ω) by the common property (i) of Definitions 2.4.1 and 2.4.2.

Theorem 2.4.1 For a random variable ξr= |ξ

r|ei∠ξr whose phase ∠ξ

ris uniformly

distributed over [0, 2π] and independent of amplitude |ξr|:

(i) ξris zero mean.

(ii) The real and imaginary parts of ξrare uncorrelated.

(iii) The variances of the real and imaginary parts of ξrare identical.

The proof is in the appendix at the end of this chapter.

We next write the covariance of the reverberant geometry functions between

two directions φ and ϕ. Before we do so, we define the following Dirac delta

functions. Let the delta functions δ`−1(φ − ϕ), ` = 2, 3 satisfy the sifting property

of the surface integral:

∫

S`−1

f(φ)δ`−1(φ − ϕ)ds(φ) = f(ϕ), ` = 2, 3. (2.17)

Delta functions that satisfy the sifting property (2.17) are, for ` = 2 (φ, ϕ ∈ R2):

δ(φ − ϕ) , δ(φ− ϕ), (2.18)

where φ and ϕ represent polar angles φ and ϕ respectively, and for ` = 3 (φ, ϕ ∈R

3):

δ2(φ − ϕ) , δ(φ− ϕ)δ(cos θ − cos ϑ), (2.19)

where φ and ϕ represent the polar and azimuthal angle pairs (θ, φ) and (ϑ, ϕ)

respectively.

From property (b) of Definitions 2.4.1 and 2.4.2, functions ξr(φ;ω) and ξr(ϕ;ω),

φ 6= ϕ, are independent. The implication for the covariance is summarized as

follows.

Observation 2.4.1 For the random process ξr(φ;ω) of the generalized diffuse field

defined in Definition 2.4.2, the covariance can be written

E{ξ∗r(φ;ω)ξ

r(ϕ;ω)} = σ2

ξ (φ;ω)δ`−1(φ − ϕ), (2.20)

where σ2ξ (φ;ω) = E{|ξ

r(φ;ω)|2} and the Dirac delta function δ`−1(φ− ϕ), ` = 2, 3

satisfies sifting property (2.17).

This statistical relation is used to derive the covariance of the reverberant field

pressure Pr(x;ω) between two points.

Generalized Diffuse Field Pressure

We now identify statistical properties of the sound pressure in the generalized

diffuse soundfield model.

It can readily be proved that complex sound pressure in a diffuse field has a

complex Gaussian distribution [110]. Provided certain conditions on the reverber-

ant geometry function are upheld, sound pressure in the generalized diffuse field

can also be seen to be complex Gaussian:

Theorem 2.4.2 (Gaussianity of Complex Pressure) The sound pressure Pr

(x;ω) in a generalized diffuse field, defined by (2.16), is a complex Gaussian random

variable provided:

(i) both real and imaginary parts of the integrand ξr(φ;ω)e−ikx·φ in (2.16) satisfy

the Lindeberg condition and

(ii) the domain of angles over which ξr(φ;ω) is nonzero is a region with nonzero

measure.

The Gaussianity property follows from application of the Lindeberg-Feller Central

Limit Theorem [21, p. 58]. The Lindeberg condition is a prerequisite for application

of this theorem. It requires the variances of individual random variables are small

compared to their integral sum so that no single angle φ has a variance σ2ξ (φ;ω)

that dominates. Condition (ii) ensures the integral (2.16) represents an infinite

sum of random variables, another prerequisite for application7.

The mean and covariance of sound pressure, are summarized as follows.

Theorem 2.4.3 (Statistics of Complex Pressure) The sound pressure Pr(x;

ω) in a generalized diffuse field possesses a zero mean E{Pr(x;ω)} = 0, a covari-

ance of:

E{P ∗r(x;ω)P

r(x + r;ω)} =

∫

S`−1

σ2ξ (φ;ω)e−ikr·φds(φ), (2.21)

and the circularity property:

E{Pr(x;ω)P

r(x + r;ω)} = 0, (2.22)

where σ2ξ (φ;ω) , E{|ξ

r(φ;ω)|2} and the reverberant geometry function ξ

r(φ;ω)

obeys the properties of Definition 2.4.2.

The proof is in the appendix at the end of this chapter. An important observation

is that the covariance is independent of the reference position x. The covariance is

dependent only on the displacement r between points. The diffuse property of the

diffuse soundfield [85], that mean square pressure E{|Pr(x;ω)|2} is independent of

position, is seen from (2.21):

E{|Pr(x;ω)|2} =

∫

S`−1

σ2ξ (φ;ω)ds(φ). (2.23)

We are able to reveal more insight into (2.21) by considering the spherical

harmonic expansion of σ2ξ (φ;ω). In the 3-D case:

σ2ξ (φ;ω) =

∞∑

n=0

n∑

m=−n

χnm(ω)Y mn (φ), (2.24a)

χnm(ω) =

∫

S2

σ2ξ (φ;ω)[Y m

n (φ)]∗ds(φ), (2.24b)

where χnm(ω) shall be called the reverberant geometry variance coefficients. Since

σ2ξ (φ;ω) is real, χ∗

nm(ω) = (−1)mχn(−m)(ω). The covariance can be expressed in

terms of these coefficients. Substituting (2.24a) and the Jacobi-Anger expansion

[18],

e−ikr·φ = 4π

∞∑

n=0

n∑

m=−n

(−i)njn(k‖r‖)Y mn (r)[Y m

n (φ)]∗, (2.25)

where r , r/‖r‖, into (2.21) and applying the orthogonality property (2.10), the

7For example, the soundfield in Figure 2.4 does not possess Gaussian pressure, since here thedomain over which ξr(φ; ω) is nonzero, namely {φ1, φ2, φ3}, is a region with zero measure.

covariance reduces to

E{P ∗r (x;ω)Pr(x + r;ω)} = 4π

∞∑

n=0

n∑

m=−n

(−i)nχnm(ω)jn(k‖r‖)Y mn (r). (2.26)

Note the similarity of this covariance relation to the interior field solution (2.44) for

sound pressure. The covariance is a linear combination of the modes {jn(k‖r‖)Y mn

(r) : n ∈ Z∗, m = −n, . . . , n} of the soundfield.

The generalized diffuse field has application to room acoustics modelling. It

preserves the diffuse property of a diffuse soundfield, while extending it to describe

fields with a directional character. Fields possessing a directional character are

quite common. In concert halls, sound absorbers are often strategically placed,

and the room geometry selected to limit range of the directions of propagation of

reverberant reflections. In an office with a heavy curtain covering one wall, the

curtain, acting as a sound absorber, modifies the directional distribution of the

reverberation. The generalized diffuse field could provide a first order model for

such room inhomogeneities.

Summarizing the results of this section, sound pressure in a generalized dif-

fuse field is circularly complex Gaussian, with a covariance given by (2.21). The

main use of these results is in Chapters 3 and 4, where we study the reverber-

ant performance of combined beamforming and equalization, and combined source

localization and tracking.

2.5 Geometric Representation of Reverberation

One way to represent a reverberant field is geometrically, by specifying the positions

of a set of sound sources that can generate it. This was already done in the

generalized diffuse field model, by defining the field in terms of plane waves, which

can be viewed as originating from farfield sound sources. Actually, by appropriately

arranging sound sources over a sphere, any soundfield can be constructed. We

present such a geometric representation of a soundfield here.

Consider the soundfield created by distributing sources, which we call reverber-

ant sources, over a spherical shell of fixed radius R′ > R concentric with the region

of interest B3. We describe the complex amplitude of the reverberant source at

each point R′φ on the shell with the reverberant geometry function ξr(φ;ω). The

sound pressure Pr(x;ω) can then be calculated as:

Pr(x;ω) =

∫

S2

ξr(φ;ω)e−ik‖R′φ−x‖

4π‖R′φ − x‖ds(φ). (2.27)

For ξr(φ;ω) appropriately chosen, any reverberant soundfield can be represented

by (2.27). This fact is shown by demonstrating the equivalence of (2.27) to the

general modal form (2.12), as will now be shown.

Perform a spherical harmonic expansion on the reverberant geometry function:

ξr(φ;ω) =∞

∑

n=0

n∑

m=−n

ψnm(ω)Y mn (φ), (2.28a)

ψnm(ω) =

∫

S2

ξr(φ;ω)[Y mn (φ)]∗ ds(φ), (2.28b)

where ψnm(ω) are the spherical harmonic coefficients of ξr(φ;ω). These coefficients

shall be called the reverberant geometry coefficients. These coefficients are closely

related to the reverberant field coefficients of modal form (2.12). Substituting

(2.28a) and the spherical harmonic expansion [18, p. 30],

e−ik‖y−x‖

4π‖y − x‖ = −ik∞

∑

n=0

n∑

m=−n

h(2)n (k‖y‖)[Y m

n (y)]∗ jn(k‖x‖)Y mn (x), ‖y‖ > ‖x‖,

(2.29)

where y = y/‖y‖ into (2.27), and applying the orthogonality property (2.10) yields:

Pr(x;ω) = −ik∞

∑

n=0

n∑

m=−n

ψnm(ω)h(2)n (kR′)jn(k‖x‖)Y m

n (x). (2.30)

Equation 2.30 is of the form of (2.12) with the reverberant soundfield coefficients

given by:

β(r)nm(ω) = −ikh(2)

n (kR′)ψnm(ω). (2.31)

Since spherical Hankel functions h(2)n (x) have no zeros, this equation can always

be solved for ψnm(ω). For any given field inside B3 specified through reverberant

soundfield coefficients β(r)nm(ω), we can always choose a reverberant source geometry

in (2.27) that will generate that field:

ξr(φ;ω) =∞

∑

n=0

n∑

m=−n

i

k

β(r)nm(ω)

h(2)n (kR′)

Y mn (φ). (2.32)

The relationships between sound pressure Pr(x;ω), geometric parameter ξr(φ;ω)

and modal parameters β(r)nm(ω) and ψnm(ω) are summarized in Figure 2.5.

In summary, the spatial variation of sound pressure in a reverberant soundfield

can equivalently be interpreted modally (as the sum of orthogonal functions) or

geometrically (as reverberant sources arranged over a sphere). For the interested

reader, Appendix A at the end of this thesis explores the modal interpretation for

the generalized diffuse field. There we determine the statistical properties of the

modal coefficients implied by the generalized diffuse field model.

Pressure{

Pr(x) : x ∈ S2}

Sound FieldCoefficients

{

β(r)nm : n = 0, 1, . . . ,

∞, m = n, . . . , n}

{

ξr(φ) : φ ∈ S2}

∫

S2

ξr(φ)e−ik‖x−Rφ‖

4π‖x −Rφ‖ds(φ)

∑

n,m

ψnmYmn (φ)

Reverberant Geometry

∑

n,m

β(r)nmjn(k‖x‖)Y m

n (x)

∫

S2

ξr(φ)[Y mn (φ)]∗ds(φ)

1

jn(k‖x‖)

∫

S2

Pr(x)[Y mn (x)]∗ds(x)

{

ψnm : n = 0, 1, . . . ,

∞, m = −n, . . . , n}

Coefficients

Sound Field Reverberant GeometryFunction

iβ(r)nm

kh(2)n (kR)

−ikh(2)n (kR)ψnm

[*]

Figure 2.5: Transformation map of parameters specifying a soundfield ina source-free region. Dependence on angular frequency ω is suppressedfor brevity. The sum

∑

n,m is short for∑∞

n=0

∑nm=−n. Equation [*] is

−ik∞

∑

n=0

n∑

m=−n

[h(2)n (kR′)/jn(k‖x‖)]

∫

S2

Pr(x)[Y mn (x)]∗ds(x)Y m

n (φ).

2.6 Geometric Configurations of Reverberation

In this section we describe several geometric configurations of reverberant sources,

appropriate for studying the relationship between the geometric parameters ξr(φ;ω)

and σ2ξ (φ;ω) and properties of the resultant soundfields. Although these config-

urations do not describe real reverberant environments exactly, they are useful

idealizations.

We investigate the reverberant source configurations shown in Figure 2.6. These

configurations have in common the geometric parameter set equal to 1 over the

subset D2 of the unit sphere:

U(φ) =

{

1, φ ∈ D2

0, otherwise,(2.33)

where U(φ) = σ2ξ (φ;ω) for studying the generalized diffuse field and U(φ) =

ξr(φ;ω) for studying general soundfield properties. The configurations are de-

scribed below, along with acoustic conditions under which each geometry can be

applied.

(e)(a) (b) (c) (d)

θc

Figure 2.6: Configurations of reverberant sources around a sphere. (a) Isotropicshell. (b) Conical sector with half cone angle θc. (c) Spherical slice. (d) Circularring (e) Circular arc.

By Theorem 2.4.2, provided D2 is a region of nonzero measure, the sound pres-

sure in this generalized diffuse field is complex Gaussian. Since the variance of

ξr(φ;ω) is constant over D2, the set of random variables {ξr(φ;ω)e−ikx·φ : φ ∈ D

2}obeys the Lindeberg condition8.

We now calculate the associated soundfield and geometry variance coefficients

for these configurations. Substituting (2.33) into (2.24a) or (2.28b), the equation

for the spherical harmonic coefficient in either case reduces to:

υnm =

∫

D2

[Y mn (φ)]∗ds(φ), (2.34)

where υnm = ψnm(ω) or υnm = χnm(ω). The resulting expressions for the coef-

ficients are summarized in Table 2.1. Using these coefficient expressions we can

study:

(i) the sound pressure (2.27) in an arbitrary soundfield and

(ii) the sound pressure covariance (2.26) in a generalized diffuse field,

as a function of the reverberant source configuration. The configurations themselves

are now described.

Spherical Shell

In this configuration we evenly distribute a continuum of reverberant sources over

a spherical shell (Figure 2.6(a)). The spherical shell yields the simplest field prop-

erties of all the source geometries. Setting D2 = S

2 in (2.34), we see the soundfield

given by (2.27) is composed of only one mode, j0(kx)Y00 (x). The generalized diffuse

field reduces to the classical diffuse field.

Spherical Sector

Here, we uniformly distribute a continuum of reverberant sources over the sector

of the sphere. Two spherical sectors considered are the conical sector D2cone =

8The Lindeberg condition in general holds when no random variable has a dominating variance.Here they all have the variance σ2

ξ (φ; ω).

{(θ, φ) : 0 < θ < θc, 0 < φ < 2π} (Fig. 2.6(b)) and the spherical slice D2slice =

{(θ, φ) : 0 < θ < π, φ0 − ∆φ < φ < φ0 + ∆φ}. (Fig. 2.6(c)). The equations

for the soundfield parameters are summarized in Table 2.1.9 The spherical sector

is useful for describing non-spherically symmetrical fields where the reverberation

comes only from a certain range of directions.

Circular Ring and Circular Arc

For this configuration, we uniformly distribute a continuum of reverberant sources

over the circular ring D2ring = {(θ, φ) : θ = π

2, 0 < φ < 2π} (Fig. 2.6(d)) and circular

arc D2arc = {(θ, φ) : θ = π

2, φ0 − ∆φ < φ < φ0 + ∆φ}, both centered about origin

in the x-y plane. These geometries are the 2-D analogs of the conical sector and

spherical slice respectively. The circular ring generates a diffuse-like field in the x-y

plane. We expect these geometries to describe the reverberant field best in rooms

with a highly sound-absorbing floor and ceiling.

As is the case when reverberant source geometries are axisymmetric about the

z-axis (including the conical sector geometry), we see from Table 2.1 that the

coefficients χnm(ω) and ψnm(ω) for which m 6= 0 are zero.

Image-Source Model

It is noteworthy to contrast the above-mentioned reverberant source geometries

with the image-source model [4] described in Chapter 1. In the image-source model,

the reverberant field is written as the sum of the soundfield produced by each

image-source,

Pr(x;ω) =∞

∑

`=1

ζ`(ω)e−ik‖y`−x‖

4π‖y` − x‖ , (2.35)

where ζ`(ω) and y` denote the accumulated reflection coefficient and position for

the `th image-source, respectively.

Here the reverberant geometry coefficients ψnm(ω) are found from β(r)nm(ω) using

(2.31), where β(r)nm(ω) are obtained by substituting (2.29) into (2.35) and comparing

with (2.44). The result is:

ψnm(ω) =

∞∑

`=1

ζ`(ω)h

(2)n (k‖y`‖)h

(2)n (kR′)

[Y mn (y`)]

∗. (2.36)

This spherical harmonic expansion is utilized in Chapter 3.

9The following expression was used to derive the coefficients for the spherical slice case:

∫ φ0+∆φ

φ0−∆φ

eimφdφ = 2∆φeimφ0j0(m∆φ).

Configuration χnm(ω) or ψnm(ω)

Spherical shell√

4πδm0δn0

Conical sector 2πΛ0n

∫ 1

cos θc

Pn(u)du δm0

Spherical slice 2∆φΛmn e

−imφ0j0(m∆φ)

∫ 1

−1

Pmn (u)du

Circular ring 2πΛ0nPn(0) δm0

Circular arc 2∆φΛmn e

−imφ0j0(m∆φ)Pn(0)

Arbitrary D2

∫

D2

[Y mn (φ)]∗ds(φ)

Table 2.1: Reverberant geometry variance coefficients for various geometric config-urations in a generalized diffuse field.

Equation (2.36) describes the geometric mapping of the image-sources onto a

sphere of radius R′. In contrast to the simple geometries mentioned above, ψnm(ω)

and hence ξr(φ;ω) depend nonlinearly10 on the angular frequency ω.

2.7 Two Dimensional Case

A simpler reverberant performance analysis is offered with a two dimensional sound-

field model. For that reason we sometimes make use of them in this thesis. We

shall see that the relationships and properties closely correspond to those of the

3-D case.

We start this section by outlining the derivation of the interior field solution

to the wave equation (Section 2.7.1). We describe the differences from the 3-D

case in the generalized diffuse model (Section 2.7.2). We then apply the geometric

interpretation to the 2-D case and define reverberant source geometries (Sections

2.7.3 and 2.7.4).

10If all image-sources lie in the farfield however, the dependence on angular frequency is signif-icantly simpler. Rewriting (2.35) for farfield image-sources:

Pr(x; ω) =

∞∑

`=1

ζ`(ω)e−ikx·y` .

Using the sifting property (2.17), this equation is rewritten:

Pr(x; ω) =

∫

S2

∞∑

`=1

ζ`(ω)δ2(φ − y`)e−ikx·φds(φ). (2.37)

Comparing (2.37) with (2.16), we can identify the reverberant geometry function as ξr(φ; ω) =∑

∞

`=1ζ`(ω)δ2(φ − y`). In cases where the wall reflection coefficients ζ`(ω) are independent of

angular frequency, so is ξr(φ; ω).

2.7.1 Wave Equation Solution

Following [114], we solve the Helmholtz equation (2.3) for sound pressure P (x;ω)

using the separation of variables

P (x;ω) = X(r;ω)Φ(φ;ω). (2.38)

Substituting (2.38) and the expression for the Laplacian operator in polar coordi-

nates:

∇2(·) =∂2

∂r2+

1

r

∂

∂r+

1

r2

∂2

∂φ2(2.39)

into (2.3) leads to separation into two ordinary differential equations [114, pp. 117]:

d2Φ

dφ2+ n2Φ = 0, (2.40)

∂2X

∂r2+

1

r

∂X

∂r+

(

k2 − n2

r2

)

X = 0, (2.41)

where φ and r are the polar angle and length of x respectively. Solutions to (2.40)

are again Φ+(φ) = einφ and Φ−(φ) = e−inφ with n integer. Solutions to (2.41) are

the Bessel functions of the first and second kinds with order n, Jn(kr) and Nn(kr).

The exponential functions summarize the angular dependence of the solution. They

possess the orthogonality property,

∫ 2π

0

e−inφeimφdφ = 2πδnm. (2.42)

Writing the complete solution to the Helmholtz equation in R2,

P (x;ω) =∞

∑

n=−∞

[

β(J)n (ω)Jn(k‖x‖) + β(N)

n (ω)Nn(k‖x‖)]

einφ. (2.43)

Sound pressure is fully represented by the set of coefficients {β(J)n (ω)}n∈Z

⋃

{β(N)n (ω)}n∈Z

where Z is the set of integers. For the interior field problem, the field in a region

B2 = {x ∈ R

2 : x ≤ R} absent of any sources reduces to:

P (x;ω) =∞

∑

n=−∞

βn(ω)Jn(k‖x‖)einφ, (2.44)

where {βn(ω)}n∈Z are the 2-D soundfield coefficients. Here sound pressure is ex-

pressed in terms of the 2-D modes {Jn(k‖x‖)einφ}n∈Z of the soundfield. Similar to

the 3-D case in (2.13), each βn(ω) can be determined with the relation:

βn(ω) =1

2πJn(kx)

∫ 2π

0

P (x;ω)e−inφdφ, (2.45)

provided Jn(kx) 6= 0. The advantage of the modal expansion (2.44) lies in its

efficient parametrization of the soundfield (c.f. Chapter 5) and the functional de-

pendence of terms on orthogonal functions.

2.7.2 Generalized Diffuse Field

The generalized diffuse field, as defined in Definition 2.4.2 encompasses both 2-D

and 3-D cases. Field properties are still governed by Theorems 2.4.2 and 2.4.3 but

the Fourier expansion of the spatial covariance is different from the 3-D case.

For 2-D plane waves, (2.16) reduces to:

Pr(x;ω) =

∫ 2π

0

ξr(φ;ω)e−ikx·φdφ. (2.46)

The spatial covariance property (2.21) becomes:

E{P ∗r (x;ω)Pr(x + r;ω)} =

∫ 2π

0

σ2ξ (φ;ω)e−ikr·φdφ, (2.47)

where σ2ξ (φ;ω) , E{|ξr(φ;ω)|2}. The Fourier expansion of σ2

ξ (φ;ω) is written as:

σ2ξ (φ;ω) =

∞∑

n=−∞

χn(ω)einφ, (2.48a)

χn(ω) =1

2π

∫ 2π

0

σ2ξ (φ;ω)e−inφdφ. (2.48b)

Substituting the Jacobi-Anger expression [18],

e−ikx·y =∞

∑

n=−∞

(−i)ne−inφyJn(kx)einφx , (2.49)

into (2.47) and applying (2.48b) we show that:

E{P ∗r (x;ω)Pr(x + r;ω)} = 2π

∞∑

n=−∞

(−i)nχn(ω)Jn(kr)einφr , (2.50)

where φr is the polar angle of r.

2.7.3 Geometric Representation of Reverberation

We now present the geometric interpretation of the reverberant field in the 2-D

case. Any soundfield in the circular region B2 can be interpreted as resulting from

a continuum of reverberant sources arranged on a circle of radius R′ > R concentric

with B2:

Pr(x;ω) =

∫ 2π

0

ξr(φ;ω)H(2)0 (k‖R′φ − x‖) dφ, (2.51)

where H(2)n (·) is the Hankel function of the second kind and order n. The term

H(2)0 (k‖R′φ − x‖) is the fundamental solution of the Helmholtz equation in R

2.

Performing a Fourier expansion on ξr(φ;ω):

ξr(φ;ω) =

∞∑

n=−∞

ψn(ω)einφ, (2.52a)

ψn(ω) =1

2π

∫ 2π

0

ξr(φ;ω)e−inφ dφ, (2.52b)

and applying the addition property of the Hankel function [18, p. 67]:

H(2)0 (k‖x − y‖) =

∞∑

n=−∞

H(2)n (ky)e−inφyJn(kx)einφx ,

where φx and φy are the polar angles of x and y respectively, the field representation

in (2.51) is shown equivalent to the 2-D modal representation (2.44) with

β(r)n (ω) = 2πH(2)

n (kR′)ψn(ω). (2.53)

2.7.4 Geometric Configurations of Reverberation

For soundfield analysis performed in 2-D in this thesis, we only use one reverberant

source configuration - the circular arc. Here we set geometric parameters equal to

1 over the subset of angles φ ∈ [φ0 − ∆φ, φ0 + ∆φ]:

U(φ) =

{

1, φ0 − ∆φ ≤ φ < φ0 + ∆φ

0, otherwise,(2.54)

where U(φ) = σ2ξ (φ;ω) for a generalized diffuse field and U(φ) = ξr(φ;ω) for

studying general soundfield properties.

By Theorem 2.4.2, provided ∆φ 6= 0, the sound pressure in this generalized

diffuse field is complex Gaussian. Because the variance of ξr(φ;ω) is constant over

[φ0 − ∆φ, φ0 + ∆φ], the Lindeberg condition holds for the set of random variables

{ξr(φ;ω)e−ikx·φ : φ ∈ [φ0 − ∆φ, φ0 + ∆φ]}.Substituting (2.54) into (2.48b) or (2.52b), the equation for the spherical har-

monic coefficients in either case reduces to:

υn =

∫ φ0+∆φ

φ0−∆φ

e−inφdφ = 2∆φe−inφ0j0(n∆φ).

for υn = ψn(ω) or υn = χn(ω). When ∆φ = π, the reverberant source configuration

is a circular ring, in which case the generalized diffuse field reduces to the classical

2-D diffuse field described in [59].

2.8 Summary and Contribution

In this chapter we have described different models for the sound pressure in a

reverberant room. We have developed a generalized diffuse field model, and a

representation of a sound field as a single layer potential.

We now itemize the contributions made in this chapter:

i. A generalization of the diffuse field has been made to describe soundfields

where the sound propagates in as plane waves from a subset of directions.

Here the complex sound pressure is shown to be a circularly complex Gaussian

distribution and an expression for spatial covariance determined.

ii. Application of the single layer potential to create a new interpretation where

a soundfield results from nearfield reverberant sources lying on a shell. We

explored the relationship to the modal expansion of sound pressure for the

interior field problem.

iii. A classification of basic reverberant source geometries has been made, which

can be used to explore the statistical properties of the generalized diffuse field

and properties of soundfields in general.

In the coming chapters, we will utilize the soundfield models to analyze the

performance of different signal processing algorithms. In Chapters 3 and 4, these

models will be used to explore equalization and source localization algorithms re-

spectively. In Chapter 5, the interior field solution is exploited to measure the

soundfield within a portion of the room, in order to facilitate soundfield reproduc-

tion in a reverberant room.

2.9 Appendices

2.9.1 Proof of Theorem 2.4.1

We start by calculating some relevant expectations for the phase ∠ξr. Since ∠ξr

is uniformly distributed over [0, 2π], the expectation of any function f(∠ξr) of the

phase is:

E{f(∠ξr)} =1

2π

∫ 2π

0

f(∠ξr)d(∠ξr).

A simple calculation yields:

E{ei∠ξr} = 0, (2.55)

E{cos(∠ξr) sin(∠ξr)} = 0, (2.56)

E{cos2(∠ξr)} = E{sin2(∠ξr)} =1

2. (2.57)

Property (i) is seen from the independence of the amplitude and phase of ξr and

(2.55), E{ξr} = E{|ξr|}E{ei∠ξr} = 0. The real and imaginary parts of ξr can be ex-

panded as E{Re[ξr]Im[ξr]} = E{|ξr| cos(∠ξr)|ξr| sin(∠ξr)}. From the independence

property and (2.56):

E{Re[ξr]Im[ξr]} = E{|ξr|2}E{cos(∠ξr) sin(∠ξr)} = 0.

Finally, the independence property implies:

E{Re2[ξr]} = E{|ξr|2 cos2(∠ξr)} = E{|ξr|2}E{cos2(∠ξr)},

and

E{Im2[ξr]} = E{|ξr|2 sin2(∠ξr)} = E{|ξr|2}E{sin2(∠ξr)}.

Thus by (2.57), E{Re2[ξr]} = E{Im2[ξr]} = 12E{|ξr|2}. �

2.9.2 Proof of Theorem 2.4.3

The zero mean property follows from the linearity of (2.16) and zero mean property

of ξr(φ;ω) in Theorem 2.4.1. From (2.16), the covariance is calculated as:

E{P ∗r (x;ω)Pr(x + r;ω)} =

∫

S`−1

∫

S`−1

E{ξ∗r (φ;ω)ξr(ϕ;ω)}

× eikx·φe−ik(x+r)·ϕds(φ)ds(ϕ).

Applying the reverberant geometry function covariance property (2.20):

E{P ∗r (x;ω)Pr(x + r;ω)} =

∫

S`−1

∫

S`−1

σ2ξ (φ;ω)δ`−1(φ − ϕ)

× eikx·(φ−ϕ)e−ikr·ϕds(φ)ds(ϕ).

Equation 2.21 then follows from applying the sifting property (2.17).

To show (2.22), note that E{ξ2r (φ;ω)} = 0. This property can be seen by

expanding E{ξ2r (φ;ω)} into real and imaginary parts:

E{ξ2r (φ;ω)} = E{Re2[ξr(φ;ω)]} − E{Im2[ξr(φ;ω)]}

+ 2iE{Re[ξr(φ;ω)]Im[ξr(φ;ω)]}. (2.58)

Because their variances are identical, the first and second terms are equal. The

third term of (2.58) is zero by independence of real and imaginary parts of ξr(φ;ω).

Equation 2.22 is then proved in a manner identical to (2.21), by noting that

E{ξr(φ;ω)ξr(ϕ;ω)} = E{ξ2r (φ;ω)}δ`−1(φ − ϕ) = 0.

�