Chapter 5 Theory and Design of Soundﬂeld Reproduction · Theory and Design of Soundﬂeld Reproduction 5.1 Introduction A problem relevant to emerging surround sound technology

Chapter 5

Theory and Design of Soundfield

Reproduction

5.1 Introduction

A problem relevant to emerging surround sound technology is the accurate repro-

duction of a soundfield over a region of space. Using a set of loudspeakers, it is

possible for listeners to spatialize sound and fully experience what it is actually

like to be in the original sound environment. Soundfield reproduction has been

discussed since the 1960s. However much of the work does not directly address

soundfield reproduction in reverberant environments. In this chapter, using an

efficient parametrization of the room transfer function we extend soundfield repro-

duction to reverberant enclosures.

Early work in soundfield reproduction was performed by Gerzon [32]. With

his ambisonics system, Gerzon reproduced the first order spherical harmonics of a

plane wave soundfield around a point in space. Ambisonics has since been unified

with holography [74, 75], both of which rely on the Kirchoff-Helmholtz equation.

Here, soundfield reproduction inside a control region is achieved by controlling the

sound pressure and its normal derivative over the boundary of the control region

[8]. In similar work, global soundfield reproduction techniques [46, 97] have been

proposed which control sound pressure over the boundary. By controlling sound

at additional points inside the control region, these techniques obviate the need

for velocity microphones. Unfortunately, such techniques require a large number

of loudspeakers. For lesser numbers of loudspeakers, least squares techniques have

been suggested by Kirkeby et al. [52, 54]. Recently, using a spherical harmonic

analysis, the theoretical minimum number of loudspeakers required for accurate

reproduction of a plane wave has been established [106].

The reverberant case is made difficult by the rapid variation of the acoustic

transfer functions over the room [68]. The standard approach has been to equalize

the transfer functions over multiple points using least squares techniques [53, 72].

103

104 Theory and Design of Soundfield Reproduction

However equalization away from the design points is poor. In contrast soundfield

reproduction would require the equalization to extend over the whole control region.

Alternatively the acoustic transfer functions can be measured and incorporated

into the soundfield reproduction algorithm directly. Methods for estimating the

acoustic transfer functions over a region have been established by Mourjopou-

los [69] and Bharitkar and Kyriakakis [9], which sample the field at a number

of points and use a spatial equalization library. However these techniques do not

determine transfer functions with the accuracy required for soundfield reproduction

in a reverberant room.

In this chapter, we present a method of performing soundfield reproduction in

a reverberant room. This method is based on an efficient parametrization of the

acoustic transfer function. We show that the acoustic transfer function can be

written as a weighted sum of the modes of the control region using a small num-

ber of terms. Using this parametrization, we reconstruct a soundfield accurately

over the whole control region. This approach exploits the standing wave structure

of the reverberant field generated by each loudspeaker to reproduce the desired

soundfield. We also describe a practical method for determining the active modes

of the acoustic transfer function between each loudspeaker and the control region,

by sampling sound pressure at a small number of points.

This chapter is structured as follows. Section 5.2 casts soundfield reproduction

into a least squares framework, using the modal approach to gain insight into the

fundamental parameters of the problem. In particular we use the number of modes

active in the control region to estimate the required number of speakers. In Section

5.3, we describe a method for measuring the acoustic transfer functions from each

speaker to any point within the control region. This method hinges on a modal

parametrization of the acoustic transfer function. In this section we show how to

determine the modal parameters of the acoustic transfer function. We also analyze

the effect of noisy pressure samples on determining the modal parameters (Section

5.3.3). In Section 5.4 we extend the algorithm to the three dimensional case.

Section 5.5 demonstrates the performance of our soundfield reproduction technique

with several examples. Finally Section 5.6 discusses practical implementation.

5.2 Sound Field Reproduction

In this section, we devise a scheme of performing 2-D soundfield reproduction within

a reverberant enclosure. This 2-D technique ensures good reproduction in the

plane of the loudspeakers, provided each loudspeaker possesses a sizeable vertical

dimension. It is applicable to enclosures with highly sound-absorbing floors and

5.2 Sound Field Reproduction 105

G2(ω)

G`(ω)

B2

x

G1(ω)

GL(ω)

H`(x; ω)

Figure 5.1: Use of L loudspeakers to reproduce a desired field in a control regionB2 with loudspeaker filters G`(ω) and acoustical transfer functions H`(x, ω) fromthe `th loudspeaker to a point x ∈ B2.

ceilings1.

The theory we develop here is extended to 3-D space in Section 5.4. This

extension entails replacing the 2-D modal functions described below with the 3-D

equivalent functions. However for accurate reproduction much larger numbers of

speakers are required [106]. We focus most attention on the 2-D case as it is more

practical.

Below we formulate the problem in the frequency domain. The objective is to

determine the loudspeaker filter weights required to reproduce a desired soundfield

in a reverberant room.

5.2.1 Problem Definition

We aim to reproduce the pressure Pd(x; ω) of a desired soundfield2 at each point

x and angular frequency ω in the source-free region of interest B2 using an array

of L loudspeakers. The desired soundfield could be a plane wave, a field resulting

from a monopole, a field measured in a real-life scenario or the field of a surround

sound system. For purposes of simplifying analysis in this chapter, we choose the

control region B2 to be the circle of radius R centered about origin:

B2 = {x ∈ R2 : ‖x‖ ≤ R}.

As shown in Figure 5.1, each loudspeaker ` transmits an output signal G`(ω).

1Due to the common use of carpet on the floors and foam spacers on the ceilings, both strongabsorbers of sound, such conditions hold in a large number of rooms.

2In contrast to all other chapters of this thesis, here a subscripted or superscripted d refer tothe desired soundfield and not the direct field component.


This signal encapsulates both the input signal applied to loudspeaker ` as well as

any filtering of it. To characterize the acoustic properties of the enclosure, define

the acoustic transfer function H`(x; ω). Each acoustic transfer function is the

frequency response between loudspeaker ` and point x. It describes the soundfield

due to a source and its reverberant reflections from the surface of the enclosure,

when a unit input signal G`(ω) ≡ 1 is applied to the source. The sound pressure

at any point x due to loudspeaker ` is equal to:

P`(x; ω) = G`(ω)H`(x; ω). (5.1)

From Figure 5.1, the sound pressure in the reproduced field resulting from the L

loudspeakers is then equal to

P (x; ω) =L∑

`=1

P`(x; ω) =L∑

`=1

G`(ω)H`(x; ω). (5.2)

The design task of soundfield reproduction is to choose filter weights G`(ω) to

minimize the normalized reproduction error J over B2,

J =1

E∫

B2

|P (x; ω)− Pd(x; ω)|2da(x) (5.3)

where the normalizing factor E is the energy of the desired soundfield over B2:

E =

∫

B2

|Pd(x; ω)|2da(x), (5.4)

da(x) = x dx dφx is the differential area element at x, x = ‖x‖ and φx is the polar

angle of x.

The popular approach to solving this problem is to write the least squares

solution over a set of uniformly-spaced points in B2 [52, 54]. A better approach is

to perform the design over the whole region. This approach is proposed by Asano

and Swason [5] for the related problem of equalization. Yet by discretizing, these

authors end up implementing a multi-point method. Below we outline a modal space

approach, which utilizes an efficient parametrization of acoustic transfer functions,

to perform the design over the whole region3. More insight is gained into the filter

design procedure through the modal space approach than through multi-point least

squares techniques.

3The modal space approach contrasts with the modal approaches of Asano and Swason [5]and Santillan [81], which only investigate equalization for rectangular room geometry.


5.2.2 Modal Space Approach

In the modal space approach, we express the sound pressure variables Pd(x; ω),

P (x; ω) and the acoustic transfer functions H`(x; ω) in terms of the modes of

the soundfield. Provided all sound sources (including image-sources produced by

reflection) lie outside of B2, at any point inside B2 the above variables can be

written in modal form using the interior field solution (2.44) of Chapter 2. We

write the desired sound pressure Pd(x; ω) as:

Pd(x; ω) =∞∑

n=−∞β(d)

n (ω)Jn(kx)einφx , (5.5)

where β(d)n (ω) is the nth order modal coefficient of the desired soundfield at fre-

quency ω. Reviewing, the functions {Jn(kx)einφx}n∈Z are called the modes of the

soundfield. Appropriate choice of modal coefficients generate any valid soundfield

inside B2.

Similarly, we write the reproduced sound pressure P (x; ω) as:

P (x; ω) =∞∑

n=−∞βn(ω)Jn(kx)einφx , (5.6)

where βn(ω) are the modal coefficients of the reproduced soundfield. Reproduction

of the sound pressure Pd(x; ω) over B2 with P (x; ω) is equivalent to reproduction

of the modal coefficients {β(d)n (ω)}n∈Z with {βn(ω)}n∈Z.

Because H`(x; ω) is equal to the soundfield pressure when the loudspeaker is

excited by a unit impulse, we can also write it in modal form as:

H`(x; ω) =∞∑

n=−∞αn(`, ω)Jn(kx)einφx , (5.7)

where αn(`, ω) are the modal coefficients of the room responses for loudspeaker `.

These modal coefficients completely characterize the reverberant soundfield gener-

ated by each loudspeaker within B2:

Observation 5.2.1 When the modal coefficients αn(`, ω) for each loudspeaker are

known for a given room environment, the room response H`(x; ω) between each

loudspeaker and any position x inside B2 is also known, and is given by (5.7).

Substituting (5.5) and (5.7) directly into (5.2), the modal coefficients of the

reproduced soundfield are related to αn(`, ω) through

βn(ω) =L∑

`=1

G`(ω)αn(`, ω). (5.8)


The sequences of coefficients (β(d)n (ω))n, (βn(ω))n and (αn(`, ω))n associated

with any wave field in a source-free region are shown to be bounded in Jones et

al. [48].

We now derive an expression for the energy E and normalized error J of the

reproduced soundfield over B2 as a function of the modal coefficients. Starting with

the field energy, we substitute (5.5) into (5.4) to yield:

E =

∫

B2

∣∣∣∣∣∞∑

n=−∞β(d)

n (ω)Jn(kx)einφx

∣∣∣∣∣

2

da(x).

It follows that:

E =∞∑

m=−∞

∞∑n=−∞

[β(d)m (ω)]∗β(d)

n (ω)

∫ 2π

0

e−imφxeinφxdφx

×∫ R

0

Jm(kx)Jn(kx)x dx,

where we have applied da(x) = x dx dφx. Applying the orthogonality property

(2.42), the field energy reduces to:

E = 2π∞∑

n=−∞|β(d)

n (ω)|2∫ R

0

[Jn(kx)]2x dx

=2π

k2

∞∑n=−∞

wn(kR)|β(d)n (ω)|2, (5.9)

where

wn(kR) , k2

∫ R

0

[Jn(kx)]2x dx

=

∫ kR

0

[Jn(x)]2xdx. (5.10)

The second step was performed with the variable substitution x′ = kx. Similarly

substituting (5.5) and (5.6) into (5.3), the normalized error becomes:

J =1

E∫

B2

∣∣∣∣∣∞∑

n=−∞[βn(ω)− β(d)

n (ω)]Jn(kx)einφx

∣∣∣∣∣

2

da(x).

Utilizing the orthogonality property, the normalized error reduces to:

J =2π

k2E∞∑

n=−∞wn(kR)|β(d)

n (ω)− βn(ω)|2. (5.11)

We shall call wn(kR) in (5.11) the modal coefficient weighting function.

Since the summations in (5.5), (5.6) and (5.7) have an infinite numbers of terms,


it may seem that the soundfield or room response parametrization needs an infinite

number of coefficients. However in the next section, we show that for any finite

control region, the above parametrization needs only a finite number of coefficients

to accurately represent a soundfield or a room response.

5.2.3 Active Modes

Because of the high-pass character of Bessel functions, not all of the modes make

a significant energy contribution to the soundfield inside B2. Studying (5.11),

because the sequences of modal coefficients (β(d)n (ω))n and (βn(ω))n are bounded,

the energy contribution of each modal term to reproduction error is controlled by

wn(kR). The reason for domination by wn(kR) is that for fixed kR, wn(kR) drops

rapidly to zero as n is increased. To see this fact, consider the following upper

bound on Jn(x) derived by Jones et al. [48]:

|Jn(x)| ≤ 1√2πn

(ex

2n

)n

.

Substituting this upper bound into (5.42) and then performing the integration:

wn(kR) ≤∫ kR

0

1

2πn

(ex

2n

)2n

xdx

≤ 2n

2n + 1

(ekR

2n

)2n+2

.

Hence wn(kR) drops exponentially to zero past |n| > dekR/2e. The modes for

which |n| > ekR/2 make increasingly negligible contribution to the soundfield

energy inside B2.

Previous work [48, 106] asserts that only the modes of modal index up to

N = dkRe contribute significant energy to the soundfield inside B2. This result is

supported by Figure 5.2, where wn(kR) is plotted4 against n. The weighting is seen

to be small past |n| > N . The 2N + 1 modes, J−N(kx)e−iNφx , . . . , JN(kx) eiNφx

are referred to as the active modes of B2. The remaining modes are referred to as

inactive in B2. These modes make their contribution to the soundfield outside of

B2.

Accurate soundfield reproduction requires reproduction of these active modes.

Also, the room response between each loudspeaker and any points inside B2, as

mentioned in Observation 1, can be accurately determined just by measuring the

modal coefficients {αn(`, ω)}Nn=−N of the active modes.

4Note the weighting functions of the negative modal index are a mirror of those of the positivemodal index. This can be seen by applying the Bessel function property Jn(x) = (−1)nJ−n(x).


0 10 20 30 40 50 60 700

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

|n|

w|n

|(kR

)

2λ 4λ 6λ 8λ 10λ 1λ

Figure 5.2: Plot of modal coefficient weighting function w|n|(kR) versus |n| forseveral radii R = [1λ, 2λ, . . . , 10λ]. Note that wn(kR) = w−n(kR).

5.2.4 Least Squares Solution

We now derive the least squares solution for the speaker filter weights that minimize

the reproduction error in (5.11). This solution is expressed in terms of the modal

coefficients.

The previous section established that due to the presence of wn(kR) in (5.11),

weighting of modal terms in the normalized error rapidly diminishes for n ≥ N .

Consequently the normalized error expression (5.11) can be truncated to n ≤ NT

for NT ≥ N :

JNT=

2π

k2ENT∑

n=−NT

wn(kR)|βn(ω)− β(d)n (ω)|2. (5.12)

This truncated reproduction error JNTcan be written in matrix form, as follows.

Defining the vector of loudspeaker filter weights,

g = [G1(ω), G2(ω), . . . , GL(ω)]T ,

the vector of the modal coefficients of the reproduced soundfield,

β = [β−NT(ω), β−NT +1(ω), . . . , βNT

(ω)]T ,


and the matrix of the modal coefficients of the room responses of all loudspeakers,

A =

α−NT(1, ω) α−NT

(2, ω) . . . α−NT(L, ω)

α1−NT(1, ω) α1−NT

(2, ω) . . . α1−NT(L, ω)

......

. . ....

αNT(1, ω) αNT

(2, ω) . . . αNT(L, ω)

, (5.13)

equation 5.8 can be rewritten as β = Ag. Additionally, define the vectors of the

modal coefficients of the desired soundfield,

βd = [β(d)−NT

(ω), β(d)−NT +1(ω), . . . , β

(d)NT

(ω)]T , (5.14)

and the diagonal weighting matrix,

W =

w−NT(kR) 0 . . . 0

0 w−NT +1(kR) . . . 0...

.... . .

...

0 0 . . . wNT(kR)

.

Writing the numerator of (5.12) in matrix form:

NT∑n=−NT

wn(kR)|βn(ω)− β(d)n (ω)|2 = (β − βd)

HW (β − βd),

the truncated reproduction error can be written as:

JNT=

(β − βd)HW (β − βd)

βHd Wβd

.

Since β = Ag, we expand the truncated reproduction error as a quadratic form in

the vector of loudspeaker filter weights:

JNT(g) =

1

d(gHBg − bHg − gHb + d),

where B = AHWA, b = AHWβd, d = βHd Wβd. From [5], this quadratic form

possesses it’s global minimum at:

g = B−1b = (AHWA)−1AHWβd, (5.15)

with the associated minimum in truncated reproduction error:

JNT(g) = 1− 1

dbHB−1b.

This modal space approach is superior to the conventional least squares approach


in that it ensures reproduction over the whole control region. It is also superior

to many previous techniques because it allows reproduction of any soundfield, not

just a plane wave. Further, once (AHWA)−1AHW is calculated for the acoustical

environment, the reproduced soundfield can be changed easily by modifying βd in

(5.15).

5.2.5 Mode-Matching Solution

As shown in Section 5.2.3, accurate soundfield reproduction is obtained by sim-

ply reproducing exactly the active modes of B2. In this reproduction strategy,

assuming non-degenerate speaker placement, only one loudspeaker is required to

reproduce each active modal coefficient (L = 2N + 1). Each active mode controls

the soundfield over a limited region of the space in B2.

For odd L, appropriate filter weights are chosen in order to set βn(ω) = β(d)n (ω)

for n = −(L− 1)/2,−(L− 1)/2 + 1, . . . , (L− 1)/2. Similar to the mode-matching

approach of [75], it requires solving the L× L linear system:

Ag = βd,

where A and βd are defined in (5.13) and (5.14) respectively here with NT =

(L − 1)/2. This solution is equivalent to a least squares solution with an equal

weighting matrix of low order modes, that is W = I2NT +1 where In is the n × n

identity matrix. Due to weighting the importance of different modes, the least

squares approach yields a slightly better reproduction performance. However, the

mode-matching approach is simpler.

The next section describes a method for measuring modal coefficients for the

acoustic transfer function in matrix A.

5.3 Estimation of Soundfield Coefficients

In this section we describe how to fully determine the soundfield inside a control

region B2 through measurement of the modal coefficients. This task is important

as it is required to calculate {αn(`, ω)}n∈Z that characterizes the reverberant field

generated by each loudspeaker.

We write the sound pressure P (x; ω) inside B2 generated by a loudspeaker

outside B2 in a reverberant enclosure as the modal expansion:

P (x; ω) =∞∑

n=−∞βn(ω)Jn(kx)einφx , (5.16)

where βn(ω) is the modal coefficient of order n. To determine the field pressure in-

5.3 Estimation of Soundfield Coefficients 113

βn

DFT(n) 1/Jn(kR)

R

(a)

R1

anDFT(n)

DFT(n) bn

R2

βn

(b)

Figure 5.3: Methods for measuring the modal coefficients in (a) narrow-band caseand (b) wide-band case.

side B2, we describe a simple means of measuring βn(ω). Modal coefficients αn(`, ω)

can then be found in the following way. Excite a loudspeaker with an appropriate

source output function G`(ω) (whether that be an impulse, sweeping sine wave or

pseudo random training sequence [14]) and calculate the modal coefficients βn(ω)

for each loudspeaker position. Coefficients αn(`, ω) are then obtained by dividing

βn(ω) by G`(ω).

The method used to determine the modal coefficients βn(ω) varies depending

on whether they are required in a narrow range of frequencies (Section 5.3.1) or a

wide range of frequencies (Section 5.3.2).

5.3.1 Narrow-band Method

In the case that soundfield reproduction is performed in a narrow frequency range

for a choice of R, away from the zeros of Jn(kR), good modal coefficient estimates

are obtained by sampling pressure on a circle of radius R.

Computation of Modal Coefficients

The modal coefficients are obtained from the modal analysis equation (2.13),

βn(ω) =1

2πJn(kx)

∫ 2π

0

P (x; ω)e−inφxdφx, (5.17)

provided x is not a zero of Jn(kR). Interpreting this equation, the modal coefficients

and hence soundfield can be known over the whole of B2 just by measuring sound

pressure on a circle of radius x.

In this chapter we sample pressure at x = R, on the boundary of B2. Now

at a radius x, only modes up to order dkxe are active. Over the boundary all

of the active modes of B2 are active, while the higher order modes are inactive.

Heuristically this choice of sample radius makes sense.

Approximate modal coefficients βn(ω) are obtained by sampling sound pressure


at M evenly-spaced points (R, φm) where φm = 2πm/M for m = 0, 1, . . . , M −1. Equation 5.17 shows that βn(ω)Jn(kR) are the Fourier series coefficients of

P (R, φ, ω) in variable φ. Consequently it can be approximated with the discrete

Fourier transform (DFT) relationship:

βn(ω) =1

Jn(kR)DFT{P (R, φm; ω)}(n), (5.18)

where DFT{f(m)}(n) is the M -point DFT defined by:

DFT{f(m)}(n) =1

M

M−1∑m=0

f(m)e−i 2πmnM . (5.19)

Coefficients βn(ω) are recognized as the DFT of the sampled field pressure around

the circle {P (R, 2πm/M ; ω), m = 0, 1, . . . , M−1}, weighted by the Bessel function

term 1/Jn(kR) (Figure 5.3(a)).

An appropriate choice for the number of sound pressure samples M can be

deduced from the number of active modes of B2. Since only 2N + 1 modes are

active over B2, (5.16) can be truncated on the boundary to:

P (R, φm; ω) ≈N∑

n=−N

βn(ω)Jn(kR)einφm , (5.20)

where N = dkRe. Since solving an equation is required for each unknown, we

should sample pressure at M = 2N + 1 points. One point is required for each

active mode. This rule for choosing M is consistent with Nyquist sampling theory.

Next the error in the approximated modal coefficients is analyzed. This analysis

reveals that due to the presence of 1/Jn(kR) in (5.18), if kR is near one of the Bessel

zeros, the coefficient error is amplified.

Approximation Error

We commence the analysis by writing an expression for the error caused by esti-

mating the modal coefficients βn(ω) with βn(ω).

Theorem 5.3.1 The error in the approximated modal coefficients βn(ω) is given

by:

βn(ω)− βn(ω) =1

Jn(kR)

∞∑q=−∞6=0

Jn+qM(kR)βn+qM(ω). (5.21)


Proof

Substituting (5.19) into (5.18) yields:

βn(ω) =1

Jn(kR)

1

M

M−1∑m=0

P (R, φm; ω)e−i 2πmnM , (5.22)

where φm = 2πm/M . Evaluating the modal expansion of the soundfield (5.16) at

the M points (R, φm):

P(R, 2πm

M; ω

)=

∞∑q=−∞

βq(ω)Jq(kR)ei 2πmqM . (5.23)

Substituting (5.23) into (5.22) and interchanging summations:

βn(ω) =1

Jn(kR)

1

M

∞∑q=−∞

βq(ω)Jq(kR)M−1∑m=0

ei2πm(q−n)

M .

Now the summation of the complex exponential is given by:

M−1∑m=0

ei2πm(q−n)

M =

{M, if q − n|M0, otherwise.

Hence:

βn(ω) =1

Jn(kR)

∞∑s=−∞

βn+sM(ω)Jn+sM(kR).

Rearranging:


Jn(kR)

∞∑s=−∞6=0

βn+sM(ω)Jn+sM(kR).

Equation 5.21 reveals a type of aliasing, since the higher order coefficients

{βn+qM(ω)}∞q=−∞,6=0 are mapped onto each βn(ω). It also shows that the magnitude

of the approximation error is controlled by the size of 1/Jn(kR). We shall refer

to the summation term in (5.21) as the aliasing error and the preceding 1/Jn(kR)

term as the error scaling. If Jn(kR) is small, the error scaling is large.

Concepts of aliasing error and error scaling will be put aside for the mo-

ment, as we explore the error contribution made by each term in the summa-

tion of (5.21). In (5.21), each higher order mode makes a termwise contribu-

tion of [Jn+qM(kR)/Jn(kR)]βn+qM(ω) for q 6= 0 to the approximation error in

βn(ω). We aim to identify a choice of M that ensures the termwise scaling factor

Jn+qM(kR)/Jn(kR) arbitrarily small. This choice will make the termwise contri-

bution small. We start by writing an upper bound on the termwise scaling factor.


Theorem 5.3.2 Define κn as the largest termwise scaling factor in βn(ω):

κn , maxq=−∞,...,∞6=0

∣∣∣∣Jn+qM(kR)

Jn(kR)

∣∣∣∣ , (5.24)

for n = −N,−N + 1, . . . , N . Then:

maxn=−N,...,N

κn(ω) ≤ maxn=0,...,N

1/|Jn(kR)| × 1√2π(M −N)

[ekR

2(M −N)

]M−N

. (5.25)

Proof

For the immediate discussion, we view Jn(kR) as a function of its order n. For

n ≥ N = dkRe, Jn(kR) is observed to be a monotone decreasing function in n,

decaying exponentially toward zero. (This property can be observed in Figure 5.5

for n up to 20.) Similarly for n ≥ N , |J−n(kR)| is also monotone decreasing in n.

From (5.24), the largest termwise scaling factor for coefficient βn(ω) is:

κn(ω) = 1/Jn(kR)× maxq=−∞,...,∞6=0

|Jn+qM(kR)| .

Now since M > 2N , for n = −N,−N + 1, . . . , N and q = ±1,±2, . . ., we have

|n + qM | > N . Each of these Bessel functions Jn+qM(kR) is hence sampled over

the above-mentioned monotone decreasing interval. Consequently κn is maximized

when |n + qM | is minimized:

κn =

{J−(M−n)(kR)/Jn(kR), n ≥ 0,

JM+n(kR)/Jn(kR), n < 0.

Calculating now the maximum κn over the active modes of positive order, the

termwise scaling factor is bounded by:

maxn=0,...,N

κn = maxn=0,...,N

∣∣∣∣J−(M−n)(kR)

Jn(kR)

∣∣∣∣≤ max

n=0,...,N1/|Jn(kR)| × max

n=0,...,N

∣∣J−(M−n)(kR)∣∣ .

Again due to the monotone decreasing property of the Bessel function:

maxn=0,...,N

κn = maxn=0,...,N

1/|Jn(kR)| ×∣∣J−(M−N)(kR)

∣∣

Similarly, due to the Bessel function property J−n(x) = (−1)nJn(x), the bound is

the same for maxn=−N,...,−1

κn(ω). Then applying the Bessel function bound from [48]:

maxn=−N,...,N

κn(ω) = maxn=0,...,N

1/|Jn(kR)| × 1√2π(M −N)

[ekR

2(M −N)

]M−N

.


The first term of the upper bound in (5.25), maxn=0,...,N 1/|Jn(kR)|, is the maximum

error scaling and shall be denoted as κes(ω). The second term is an upper bound

on the Bessel function JM−N(kR) obtained from [48]. We note in (5.25) that the

largest termwise scaling factor decays exponentially to zero as M is increased past

N + dekR/2e. This observation suggests choosing M ≈ N + dekR/2e. Further the

bound on the termwise scaling factor motivates use of the following procedure for

the choice of M .

Conservative Estimate of M

We now describe a procedure that allows a more accurate choice of M than above:

a) Choose the desired bound ε on the termwise scaling factor; i.e. choose a bound

for which maxn=−N,...,N

κn(ω) < ε.

b) Calculate the maximum error scaling:

κes(ω) = maxn=0,1,...,N

|1/Jn(kR)|.

c) Determine ∆N = M −N through the relationship:

1√2π∆N

[ekR

2∆N

]∆N

= ε′(ω) (5.26)

where ε′(ω) = ε/κes(ω). This relationship has been plotted in Figure 5.4 for

several values of ε′(ω).

d) The required number of samples M = N + d∆Ne.

A judicious choice of radius R will ensure that κes and hence M is minimal.

Interestingly Figure 5.4 shows a linear relationship between ∆N and kR for

large kR. Rearranging (5.26):

∆N =e

2[√

2π∆Nε′(ω)]−1

∆N kR.

As ∆N → ∞, the term [√

2π∆Nε′(ω)]−1/∆N → 1, causing this expression to

reduce to ∆N ≈ ekR/2. This relationship explains the linear section of the curves

in Figure 5.4 and is consistent with the N + dekR/2e rule.

In summary, we require at least M = 2N + 1 pressure samples to measure the

active modal coefficients where N = dkRe. Analysis of the error in approximated

modal coefficients shows that for accurate measurement of modal coefficients, more

pressure samples may be required. The larger M is required to negate the effects

of error scaling.


1 2 3 4 5 6 7 8 9 10

10

20

30

40

50

60

70

80

90

100

r / λ

∆N

−0 dB

−10 dB

−20 dB

−30 dB

−40 dB

−50 dB

Figure 5.4: Sample measurement parameter ∆N required for several values of ε′(ω)(in dB).

5.3.2 Wide-band Method

In frequency ranges and sizes of B2 of interest to practical problems, the Bessel

term Jn(kR) is guaranteed to be zero at a number of frequencies. These zeros cause

problems when the above narrow-band method is applied over a wide frequency

range, since at each zero, one or two modes becomes unmeasurable. Further due

to error scaling in the vicinity of a zero, robustness to measurement noise about

these zeros is also poor (c.f. Section 5.3.3).

To illustrate the magnitude of the problem, consider the asymptotic behavior

(that is, the behavior for large kR) of the Bessel function [114]:

Jn(kR) ∼√

2

πkRcos(kR− nπ/2− π/4). (5.27)

One can see from this equation that each Bessel functions has zeros spaced ap-

proximately kR = π or f = c/2R apart. In a region B2 with radius R = 0.3m

and speed of sound c = 342m/s, each Jn(kR) has zeros spaced 570Hz apart. In a

0-4kHz frequency range, for the 47 active modes there are 58 zeros present (Figure

5.5), ranging from 7 zeros for J0(kR) to 1 zero for J16(kR). On average one zero

occurs every 69Hz, with the larger concentration of zeros at higher frequencies.

To combat this problem, we propose an alternative method. Instead of sampling

over a single radius R we sample over two concentric circles of radii R1 = R − δR

and R2 = R (Figure 5.3(b)).


0

5

10

15

20

0

5

10

15

20

25−0.5

0

0.5

1

nkr

J n(kr)

Figure 5.5: Bessel functions Jn(kR) for kR < 22 corresponding to a control regionof radius 0.3m and frequencies up to 4kHz. Bessel zeros are marked with a dot.


Multiplying both sides of (5.16) by modal function Jn′(kx)e−in′φx and integrating

over the thin shell of thickness δR = R2 − R1, {x ∈ R2 : R1 ≤ ‖x‖ ≤ R2}, the

orthogonality property (2.42) is used to show that:

βn(ω) =1

2π∫ R2

R1[Jn(kx)]2x dx

∫ R2

R1

∫ 2π

0

P (x; ω)Jn(kx)e−inφx x dx dφx.

For small δR, we can approximate the integral in x with the zeroth order approxi-

mation: ∫ R2

R1

f(x)dx =1

2[f(R1) + f(R2)]δR. (5.28)

Using (5.28), we express βn(ω) as a sum of two weighted Fourier Series equations.

Sampling the field with M evenly-spaced sensor pairs positioned at (R1, φm) and

(R2, φm), the modal coefficients βn(ω) are estimated with:

βn(ω) = an(ω)DFT{P (R1, φm; ω)}(n) + bn(ω)DFT{P (R2, φm; ω)}(n). (5.29)

where for R1 ≈ R2, the DFTs are weighted by:

an(ω) =Jn(kR1)

[Jn(kR1)]2 + [Jn(kR2)]2, (5.30)

bn(ω) =Jn(kR2)

[Jn(kR1)]2 + [Jn(kR2)]2. (5.31)

The approximated modal coefficients can hence be obtained through taking the

DFT of the pressure samples around each circle and calculating a weighted average.


Next we analyze the error in the approximated modal coefficients for the wide-

band case.

Approximation Error

For the wide-band method, the error in the approximated modal coefficients is:

βn(ω)− βn(ω) = an(ω) maxq=−∞,...,∞

q 6=0

Jn+qM(kR1)βn+qM(ω)

+ bn(ω)∞∑

q=−∞6=0

Jn+qM(kR2)βn+qM(ω). (5.32)

This expression is proven by substituting (5.19) into (5.29) and simplifying the

resulting expression in a manner similar to (5.21). In contrast to the narrow-band

case in (5.21), the wide-band case possesses two error scaling factors an(ω) and

bn(ω).

The presence of two Bessel functions Jn(kR1) and Jn(kR2) in the error scaling

factors (5.30) and (5.31) improves robustness at their zeros. They achieve the

improvement by weighting greater the pressure measurements from the radius at

which Jn(kR) is larger.

The critical parameter in the wide-band technique is δR. δR controls the maxi-

mum value of the error scaling terms an(ω) and bn(ω), as we will now show. When

either kR1 or kR2 is a zero of the Bessel function, approximation error simplifies to

that of the narrow-band case (5.21). For example, if Jn(kR1) = 0, the error scaling

terms reduce to an(ω) = 0 and bn(ω) = 1/|Jn(kR2)|. For δR small, Jn(kR2) is also

small and the linear approximation Jn(kR2) = kδRJ ′n(kR1) can be made. By the

derivative property of the Bessel function xJ ′n(x) = nJn(x)− xJn+1(x) [64, p. 24],

we see that J ′n(kR1) = Jn+1(kR1) so the nonzero error scaling term is

bn(ω) ≈ 1

kδR|Jn+1(kR1)| .

From this equation, it seems advantageous to choose δR large, as a larger δR

implies a smaller error scaling. However if δR is too large, error scaling will start

to increase. In the worse case Jn(kR2) coincides with another zero of the same

Bessel function. As these Bessel functions are regularly spaced, we can choose a

δR to avoid this case. From (5.27) the Bessel zeros of Jn(kR) are spaced π apart.

Because the extrema of the Bessel functions are approximately half way between

the zeros, set kδR < π/2 or δR < λ/4. An appropriate choice of δR is hence 1/4

of the smallest acoustic wavelength of interest.


5.3.3 Impact of Measurement Noise

This section studies how noise impacts the measurement of the modal coefficients.

In real rooms with background noise and sensor noise, it is non-trivial to obtain

clean measurements of the acoustical transfer functions. It is hence important to

ensure that such noise does not significantly impact the determination of modal

coefficients.

Firstly we model the measurement noise η(x; ω) at each sensor position x as

additive white noise of zero mean and variance σ2(ω). The noisy pressure is:

P (x; ω) = P (x; ω) + η(x; ω). (5.33)

Calculating the DFT of both sides of (5.33) and comparing with (5.18), the noisy

modal coefficient estimates βn(ω) are shown to be related to the noiseless estimates

βn(ω) by:

βn(ω) = βn(ω) +1

Jn(kR)DFT{η(R, φm; ω)}(n).

Inserting the definition of the DFT in (5.19) and rearranging:


Jn(kR)

1

M

M−1∑m=0

η(R, φm; ω)e−i 2πmnM . (5.34)

Equation 5.34 is used to derive the mean and variance of the noisy modal coefficient

estimates. Taking the expectation of both sides of (5.34), the zero mean property

of η(R, φm; ω) implies that:

E{βn(ω)− βn(ω)} = 0,

or E{βn(ω)} = βn(ω). Measurement of the modal coefficients remains unbiased by

noise with zero mean. Multiplying (5.34) by its complex conjugate and taking the

expectation, the variance is given by:

E{|βn(ω)− βn(ω)|2} =1

[Jn(kR)]21

M2

M−1∑m1=0

M−1∑m2=0

E{η∗(R, φm1 ; ω)

× η(R, φm2 ; ω)} exp{i2π(m1 −m2)n/M}.

In the case that noise is spatially uncorrelated, E{η∗(R, φm1 ; ω)η(R, φm2 ; ω)} =

σ2(ω)δm1m2 and the variance reduces to:

E{|βn(ω)− βn(ω)|2} =1

M

σ2(ω)

Jn(kR)2 .

The variance is influenced by error scaling factor 1/|Jn(kR)|. In the wide-band


case, we can use a similar derivation to show that the modal coefficient estimates

are also unbiased and have a variance given by:

E{|βn(ω)− βn(ω)|2} =1

M

σ2(ω)

[Jn(kR1)]2 + [Jn(kR2)]2. (5.35)

The Bessel functions in the denominators of (5.35) show that similar error scaling

occurs in the noise error of the wide-band case. This error scaling has the potential

to greatly amplify the noise in measured modal coefficients if the Bessel functions

are small.

This error scaling of the noise also has implications on the measurability of the

inactive modal coefficients. For the inactive modes, the Bessel terms Jn(kR) are so

small as to be effectively zero. Thus error scaling in the inactive modes is so large

that the modal coefficients are unmeasurable.

5.4 Three Dimensional Case

We now extend the above theory to the reproduction of a soundfield in a volume

of space, by replacing the modal functions of the 2-D wave equation with the

analogous 3-D modal functions. A notable strength of this method is the ability

to reproduce the soundfield over the entire region of interest.

In Section 5.4.1 we define the soundfield reproduction problem in 3-D space and

in Section 5.4.2 we describe a method for determining the 3-D modal coefficients.

5.4.1 Sound Field Reproduction

We now show how to perform soundfield reproduction over a 3-D control region of

spherical shape. As for the 2-D case, our approach is to identify and reproduce the

active modes of the control region.

Problem Definition

The aim is to reproduce the sound pressure Pd(x; ω) of a desired field inside the

source-free region B3 = {x ∈ R3 : ‖x‖ ≤ R}. Each loudspeaker ` transmits the

output signal G`(ω) and possesses the acoustic transfer function H`(x; ω) between

the loudspeaker and each point in B3. The sound pressure P (x; ω) in the repro-

duced soundfield is again related to G`(ω) and H`(x; ω) for ` = 1, 2, . . . , L through

(5.1) and (5.2).

The design task of our 3-D soundfield reproduction is to choose filter weights

G`(ω) to minimize the normalized reproduction error J ,

J =1

E∫

B3

|P (x; ω)− Pd(x; ω)|2dv(x), (5.36)

5.4 Three Dimensional Case 123

where the energy of the desired soundfield E over B3 is:

E =

∫

B3

|Pd(x; ω)|2dv(x),

and dv(x) = x2 sin θx dx dθx dφx is the differential volume element at x and θx and

φx are the polar and azimuthal angles of x respectively. The next section describes

the modal approach of soundfield reproduction.

Modal Space Approach

In the modal space approach, the sound pressure variables Pd(x; ω) and P (x; ω)

and each acoustic transfer function H`(x; ω) are expressed in terms of the modes

of the soundfield. Applying the 3-D interior field solution (2.12) from Chapter 2,

within a source-free region these quantities are written as:

Pd(x; ω) =∞∑

n=0

n∑m=−n

β(d)nm(ω)jn(kx)Y m

n (x), (5.37)

P (x; ω) =∞∑

n=0

n∑m=−n

βnm(ω)jn(kx)Y mn (x), (5.38)

H`(x; ω) =∞∑

n=0

n∑m=−n

αnm(`, ω)jn(kx)Y mn (x), (5.39)

where β(d)nm(ω), βnm(ω) and αnm(`, ω) are the modal coefficients of the desired sound-

field, reproduced soundfield and acoustical transfer function for loudspeaker ` re-

spectively. Like in the 2-D case, soundfield reproduction can be performed by

reproduction of modal coefficients {β(d)nm(ω)} with {βnm(ω)}.

Similar to Observation 5.2.1, the modal coefficients αnm(`, ω) completely char-

acterize the acoustic transfer functions H`(x; ω) between each loudspeaker and any

position inside B3. Substituting (5.37) and (5.39) for H`(x; ω) into (5.2), the modal

coefficients of the reproduced soundfield are related to αnm(`, ω) through

βnm(ω) =L∑

`=1

G`(ω)αnm(`, ω). (5.40)

An expression for the energy E and normalized error J of the reproduced soundfield

over B3 as a function of the modal coefficients can be derived in a manner similar

to the 2-D case. The result is:

J =1

k3E∞∑

n=0

n∑m=−n

wn(kR)|βnm(ω)− β(d)nm(ω)|2, (5.41)


E =1

k3

∞∑n=0

n∑m=−n

wn(kR)|β(d)nm(ω)|2,

where the modal coefficient weighting function is given by:

wn(kR) =

∫ kR

0

[jn(x)]2x2dx. (5.42)

Next the activity properties of the modes in the 3-D case, are summarized.

Active Modes

Similar to the 2-D case, not all modes make a significant contribution to the field

inside the control region. Due to the low-pass property of the spherical Bessel

functions, some modes are active within B3 while others are inactive. In fact, Ward

and Abhayapala show that [106] as a rule of thumb only the modes {jn(kx)Y mn (x)}

for which n ≤ N = dkRe are active in the control region (just as in the 2-D case).

Once the coefficients of these modes are measured, the soundfield in B3 is accurately

known. This property is observed in the plot of wn(kR) versus n in Figure 5.6.

The wn(kR) curves drop rapidly to zero past n = kR in a manner almost identical

that in the 2-D case (Figure 5.2).

The major difference in the 3-D case is that the number of active modes has

an approximate second order dependence on the frequency-radius product kR.

There are 2n + 1 active modes for each value of index n, creating∑N

n=0(2n + 1) =

(N+1)2 active modes instead of 2N+1. This dependence causes a large loudspeaker

requirement for accurate reproduction in moderately-sized control regions. For

example to reproduce the soundfield in a 1m diameter sphere we are required

to reconstruct 202 = 400 modes at 2kHz. For a practical limit on the number

of speakers, we must restrict 3-D reproduction to small control regions and low

frequencies.

Least Squares Solution

We now derive the least squares solution for the speaker weights that minimizes

(5.41). Because wn(kR) decays rapidly to zero for n > N = dkRe, while modal

coefficients {βnm(ω)} and {β(d)nm(ω)} are bounded in a similar sense to the 2-D case,

(5.41) is accurately truncated to NT for NT ≥ N :

JNT=

1

k3ENT∑n=0

n∑m=−n

wn(kR)|βnm(ω)− β(d)nm(ω)|2. (5.43)


0 10 20 30 40 50 60 700

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

n

wn(k

R)

2λ 4λ 6λ 8λ 10λ 1λ

Figure 5.6: Plot of modal coefficient weighting function wn(kR) versus n for 3-Dsoundfield reproduction at several radii R = [1λ, 2λ, . . . , 10λ].

Similar to the 2-D case, (5.43) is written as a quadratic form in the vector g of

loudspeaker filter weights:

JNT(g) =

1

d(gHBg − bHg − gHb + d), (5.44)

where B = AHWA, b = AHWβd and d = βHd Wβd. In the 3-D case the diagonal

weighting matrix W is:

W =

w0(kR)I1 01(3) . . . 01(2NT +1)

03(1) w1(kR)I3 . . . 03(2NT +1)

......

. . ....

0(2NT +1)(1) 0(2NT +1)(2) . . . wNT(kR)I2NT +1

,

where 0nm is the n×m zero matrix, the vector of desired modal coefficients βd ∈C(N+1)2 is:

βd = [β(0)Td ,β

(1)Td , . . . , β

(NT )Td ]T , (5.45)

where

β(n)d = [βn(−n), βn(−n+1), . . . , βnn]T ,


and the matrix of the modal coefficients of the room responses of all loudspeakers,

A =

α0(1, ω) α0(2, ω) . . . α0(L, ω)

α1(1, ω) α1(2, ω) . . . α1(L, ω)...

.... . .

...

αNT(1, ω) αNT

(2, ω) . . . αNT(L, ω)

,

where

αn(`, ω) = [αn(−n)(`, ω), αn(−n+1)(`, ω), . . . , αnn(`, ω)]T .

The least squares solution to (5.44) is again g = B−1b. This solution reproduces

the desired soundfield over the whole of B3.

Modal-Matching Solution

The mode-matching solution is identical in principle to the 2-D case. However at

least L = (N + 1)2 loudspeakers are required to match all the active modes of B3.

5.4.2 Estimation of 3-D Soundfield Coefficients

In this section, we devise a scheme for determining the modal coefficients of the

soundfield in the reproduction region. This task is extremely important to sound-

field reproduction in reverberant rooms. However, as will be seen below, it is of

limited practical interest because of the very large number of measurements needed.

Hence we limit treatment to the narrow-band case and only present a basic analy-

sis of the measurement error in the modal coefficients. Writing the sound pressure

P (x; ω) inside B3 generated by a loudspeaker outside as the modal expansion,

P (x; ω) =∞∑

n=0

n∑m=−n

βnm(ω)jn(kx)Y mn (x). (5.46)

where βnm(ω) are the modal coefficients of the soundfield that we desire to measure.


The modal coefficients are obtained from the modal analysis equation (2.13) of

Chapter 2,

βnm(ω) =1

jn(kx)

∫

S2P (x; ω)[Y m

n (x)]∗ds(x), (5.47)

provided jn(kx) 6= 0. Sampling the pressure at M points Rφ1, Rφ2, . . . , RφM over

the boundary of B3, one can approximate the integral in (5.47) as:

βnm(ω) =1

jn(kR)

M∑t=1

P (Rφt; ω)[Y mn (φt)]

∗∆st, (5.48)


Figure 5.7: Arrangement of 49 points over a sphere, with a minimal point-to-pointdistance [88]. This arrangement is used to illustrate the error analysis of coefficientestimation.

where ∆st is a spatial weighting term used to account for the spacing of pressure

samples.

The simplest way to evaluate (5.48) is to evenly space pressure samples over B3.

In such a case, the spatial weighting terms are the same for all samples ∆st = 4πM

.

However the problem of evenly spacing sensors over the sphere is nontrivial. It

has been solved by Sloan et al. as the so-called sphere packing problem, “Place M

points on a sphere so as to maximize the minimal distance between them” [88].

Sloan et al. have collated the coordinates for packings of up to 130 points [88].

Now except for a limited set of geometries (eg. tetrahedral or octahedral) it is not

theoretically possible to space the sensors perfectly uniformly over the sphere. The

distributions are close to uniform however.

The number of pressure samples required for measurement of the modal coeffi-

cients can be deduced from (5.46). Now the inactive modal terms make a negligible

contribution to this equation, allowing truncation of (5.46) to n ≤ N . To measure

the remaining (N + 1)2 active modes of B3, the truncated equation that implies at

least M = (N + 1)2 pressure samples are required around the boundary. We again

see a second order dependence on kR. In reality because the inactive modes still

contribute some energy to the soundfield, it is wise to perform some oversampling.


0 20 40 60 80 100

0

50

100−40

−35

−30

−25

−20

−15

−10

−5

0

n‘q‘

10lo

g 10 δ

M(n

‘,q‘)

Figure 5.8: Plot of δM(n,m, q, p) for M = 49 sensors positioned over a sphere asin Figure 5.7. Coefficient indices (n,m) and (q, p) have been re-indexed so as tobe referenced with single parameters with the bijective mappings n′ = n2 + n + mand q′ = q2 + q + p.

Approximation Error

We now derive an expression for the error in the estimated modal coefficients.

Substituting (5.46) into (5.48), estimated modal coefficients are shown to be:

βnm(ω) =1

jn(kR)

∞∑q=0

q∑p=−q

δM(n,m, q, p)jq(kR)βqp(ω), (5.49)

where

δM(n,m, q, p) ,M∑t=1

[Y mn (φt)]

∗Y pq (φt)∆st. (5.50)

Equation 5.50 is the integral approximation to the orthogonality property of spher-

ical harmonic functions (2.10) analogous to (5.48). In the limit of large number of

pressure samples M (for appropriately spaced samples):

limM→∞

δM(n, m, q, p) = δqnδpm,

and hence limM→∞ βnm(ω) = βnm(ω). As a result, δM(n, m, n, m) ≈ 1 and (5.49)

can be rearranged to show that the error in each estimated modal coefficient is

given by:

βnm(ω)− βnm(ω) ≈ 1

jn(kR)

∞∑q=0

q∑p=−q

(q,p)6=(n,m)

δM(n,m, q, p)jq(kR)βqp(ω). (5.51)

5.5 Simulation Examples 129

The key features of (5.51) are the aliasing error (the summation term) and the error

scaling (the term 1/jn(kR)). Due to the error scaling, measurement noise obscures

the measurement of modal coefficients at frequencies about which the spherical

Bessel functions jn(kR) are zero. The size of the aliasing error is dependent on

the accuracy of the integral approximation (5.50) to the orthogonality property.

The smaller the terms δM(n,m, q, p) for (n, m) 6= (q, p), the less aliasing error that

occurs.

As an example, δM(n, m, q, p) is plotted in Figure 5.8 for the 49 sensor ar-

rangement shown in Figure 5.7 for n = 0, 1, . . . , 8, m = −n . . . n, q = 0, 1, . . . , 8

and p = −q . . . q. In this plot, coefficient indices (n,m) and (q, p) have been re-

indexed so as to be referenced with single parameters with the bijective mapping

n′ = n2 + n + m and q′ = q2 + q + p. Under this mapping, n = b√n′c where b·cis the integer floor function and m = n′ − n2 − n, and similarly for q and p. The

contribution to the approximation error from off-diagonal terms is small (−20dB)

for low order coefficients, steadily rising to −5dB at around n = q = 7.

5.5 Simulation Examples

In the following examples, we illustrate sound reproduction of a plane wave and

a single monopole source at a single frequency (Section 5.5.1 and Section 5.5.2)

and at a range of frequencies (Section 5.5.4). Single frequency reproduction is per-

formed at 1kHz (ω = 2πradians / sec). Section 5.5.3 illustrates the qualitative

differences between the least squares and mode matching approaches to filter de-

sign. Section 5.5.4 demonstrates the influence of the noise in pressure samples on

the reproduction error. Finally, Section 5.5.5 illustrates the 3-D reconstruction of

a plane wave.

For the 2-D examples, the reverberant room parameters and loudspeaker place-

ment are summarized in Figure 5.9. The room is rectangular with a wall absorp-

tion coefficient of 0.3. Unless otherwise stated, the control region has radius 0.3m.

Though the soundfield reproduction design technique is applicable for any config-

uration and type of loudspeaker, we perform the soundfield reproduction by posi-

tioning omnidirectional loudspeakers in circular array concentric with the region

of interest. This setup yields an average DRR from each loudspeaker of −4.4dB at

the boundary of B2.

The loudspeaker requirements of this scenario are governed by the control re-

gion parameter N = dkRe = 6, prompting the use of 2N + 1 = 13 loudspeakers.

Following the conservative design procedure of Section 5.3.1 with ε = −20dB, the

maximum error scaling is κ(ω) = 25dB, and from Figure 5.4 the d∆Ne correspond-

ing to κ′(ω) = ε/κ(ω) is 14. We hence sample the pressure at M = N +d∆Ne = 20

points to measure the room response coefficients αn(`, ω).


2m

R = .3m

Sample Points

Speakers

6.4m

5m

Figure 5.9: Room dimensions and layout of loudspeakers and sample points forthe simulations. The loudspeakers and region of interest are centered about(3.8m, 2.4m).

The reverberation is generated with a 2-D adaptation of the image-source

method [4]. Each acoustic transfer function is given by:

H(xm; ω) = H(2)0 (ω‖xm − y‖) +

Ni∑n=1

ζnH(2)0 (ω‖xm − yn‖),

where H(2)0 (·) is the zeroth order Hankel function of the second kind5, y is source

position, and ζn and yn are the position and accumulated reflection coefficient of

the nth image-source respectively. In simulations below, all the image-sources of

up to fifth order were included (totalling Ni = 60 image-sources).

Soundfield reproduction results are illustrated in Figures 5.10 - 5.14. Here

the real and imaginary parts of the complex pressure of the reproduced field are

displayed as density plots. Details of the soundfield reproduction in each case are

described below.

5.5.1 Reproduction of a Plane Wave

Firstly the field pressure of a plane wave is reproduced. For a plane wave of unit

strength originating from direction y:

Pd(x; ω) = e−ikx·y.

5For the 2-D point source, or a cylindrical source H(2)0 (kr) gives the field at a distance r from

the source [18]; for a 3-D point source, this is equal to the more familiar expression h(2)0 (kr) =

ie−ikr/kr


Through the Jacobi-Anger expression (2.49) one sees the modal coefficients are

given by

β(d)n (ω) = (−i)ne−inφy ,

where φy is the polar angle of y. Loudspeaker filter weights are chosen using the

least squares approach of Section 5.2.4.

Figure 5.10 illustrates the reproduction of a plane wave approaching from an an-

gle of φy = π/6. We provide a free field design (Figure 5.10(a)), the same free field

design in the reverberant room described above (Figure 5.10(b)) and the reverber-

ant field design (Figure 5.10(c)). With a 307% reproduction error, the reverberant

performance of the free-field design is poor. In contrast the reverberant field design

shows accurate reproduction. A direct-to-reverberant ratio of −4.4dB (as in this

case) is common in office environments. This fact highlights the importance of re-

verberant field design techniques. Also, note that the reverberant design performs

as well in the reverberant room as the free-field design does in a free field.

5.5.2 Reproduction of a Phantom Monopole Source

The pressure field of a 2-D monopole source is now reproduced. For a monopole

source of unit strength -positioned at y, the sound pressure is:

Pd(x; ω) = H(2)0 (k‖x− y‖).

Through the addition property of the Hankel function [18]:

H(2)0 (k‖x− y‖) =

∞∑n=−∞

H(2)n (ky)e−inφyJn(kx)einφx ,

where y , ‖y‖ one sees the modal coefficients are given by

β(d)n (ω) = H(2)

n (ky)e−inφy .

Using the same design technique as for the plane wave reproduction, we simulate

reproduction of a monopole source just outside the region of interest, at (y, φy) =

(0.35m, 3π/4). Figure 5.11 shows good reproduction of this monopole source.

5.5.3 Least Squares Versus Mode-Matching

The mode matching and least squares filter design techniques are now compared,

illustrating the results of designing with an insufficient number of speakers. We

reproduce a plane wave with 13 speakers, but over a region of interest of radius

0.6m. Because of the larger radius, we require more pressure samples (35 using


−0.6 −0.4 −0.2 0 0.2 0.4−0.6

−0.4

−0.2

0

0.2

0.4

0.6

x (m)

y (m

)

Real

−0.6 −0.4 −0.2 0 0.2 0.4−0.6

−0.4

−0.2

0

0.2

0.4

0.6

x (m)

y (m

)

Imaginary

(a)

−0.6 −0.4 −0.2 0 0.2 0.4−0.6

−0.4

−0.2

0

0.2

0.4

0.6

x (m)

y (m

)

Real

−0.6 −0.4 −0.2 0 0.2 0.4−0.6

−0.4

−0.2

0

0.2

0.4

0.6

x (m)

y (m

)

Imaginary

(b)

−0.6 −0.3 0 0.3 0.6−0.6

−0.3

0

0.3

0.6

x (m)

y (m

)

Real

−0.6 −0.3 0 0.3 0.6−0.6

−0.3

0

0.3

0.6

x (m)

y (m

)

Imaginary

(c)

Figure 5.10: Reproduction of a plane wave with 13 speakers and 20 pressure sam-ples in a 0.3m radius circle, for (a) a free field, (b) the same free field design inthe reverberant room, and (c) a reverberant field design in the reverberant room.Reproduction errors as defined in (5.3) are 0.87%, 307% and 0.85% respectively.

conservative procedures).

For the least squares solution, speaker weights are determined from straight

forward application of (5.15). For the modal solution, one can only reproduce 13

of the 25 active modes. We choose to reproduce the lowest order modes.

The resulting sound fields are shown in Figure 5.12. While the least squares

solution reproduces the soundfield with uniform reproduction accuracy over the

whole region of interest (Figure 5.12(a)), mode-matching reproduces accurately

within a disc of radius of 0.3m as before but not outside this disc (Figure 5.12(b)).


−0.6 −0.3 0 0.3 0.6−0.6

−0.3

0

0.3

0.6

x (m)

y (m

)

Real

−0.6 −0.3 0 0.3 0.6−0.6

−0.3

0

0.3

0.6

x (m)

y (m

)

Imaginary

Figure 5.11: Reproduction of monopole in a 0.3m radius circle of the reverberantroom with 13 speakers and 20 pressure samples. Reproduction error as defined in(5.3) is 2.12%. The position of the monopole is marked with a ’+’.

Mode-matching fails outside this disc because the 12 highest order modes, which

become active there, are not reproduced. In general, at larger radii the higher

order modes become active but the mode matching approach, unlike least squares,

makes no attempt to reproduce them.

3 3.5 4 4.5 5

1.5

2

2.5

3

3.5

x (m)

y (m

)

Real

3 3.5 4 4.5 5

1.5

2

2.5

3

3.5

x (m)

y (m

)

Imaginary

(a)

3 3.5 4 4.5 5

1.5

2

2.5

3

3.5

x (m)

y (m

)

Real

3 3.5 4 4.5 5

1.5

2

2.5

3

3.5

x (m)

y (m

)

Imaginary

(b)

Figure 5.12: Reproduction of plane wave in a 0.6m radius circle of the reverberantroom with 13 speakers and 35 pressure samples using (a) the least squares solutionand (b) the modal solution. Reproduction errors as defined in (5.3) are 26.0% and84.1% respectively.


100 200 300 400 500 600 700 800 900 1000−50

−45

−40

−35

−30

−25

−20

−15

−10

−5

0

Frequency (Hz)

Rep

rodu

ctio

n E

rror

(dB

)

40dB SNR

30dB SNR

20dB SNR

Figure 5.13: Wide-band reproduction of plane wave in a 0.3m radius circle with 13speakers and using 40 pressure samples.

5.5.4 Wide-band Reproduction with Measurement Noise

Wide-band soundfield reproduction of a plane wave is performed with noisy pres-

sure samples in the frequency range 100Hz to 1kHz, R1 = 0.3m and R2 = 0.27m.

Reproduction error is plot in Figure 5.13 for several noise SNRs averaged over

40 trial runs. This figure shows that at least 30dB SNR is required for accurate

reproduction over the whole frequency range.

The general trend in this curve is that error increases with frequency. This

trend is due to the linear increase in demand for loudspeakers and sensors with

frequency. Our design use the same number of loudspeakers and pressure samples

for all frequencies. If we desire to flatten the curve, we could use less pressure

samples and loudspeakers at lower frequencies where less modes are active.

Also observe the peaks in Figure 5.13. These peaks occur in the vicinity of the

zeros of the Bessel functions J0(kR) and J1(kR). Zeros of these Bessel functions at

460Hz and 730Hz respectively. These peaks are hence a direct result of the error

scaling mentioned in Section 5.3.3. To flatten such peaks, more pressure sampling

should be performed about these frequencies, or the sensor pairs further separated

(i.e. δR = R1 −R2 should be increased).

5.5.5 3-D Reproduction of a Plane Wave

In this 3-D example we illustrate reproduction of a plane wave in 3-space. For a 3-

D planar wave of unit strength originating from direction y, the modal coefficients

5.6 Practical Implementation 135

−0.4 −0.2 0 0.2 0.4−0.4

−0.2

0

0.2

0.4

x (m)

y (m

)

Real

−0.4 −0.2 0 0.2 0.4−0.4

−0.2

0

0.2

0.4

x (m)

y (m

)

Imaginary

Figure 5.14: 3-D reproduction of a plane wave with 25 speakers and 49 pressuresamples in a 0.2m radius circle in a reverberant room. Reproduction error asdefined in (5.41) is 3.4%.

are given by:

βnm(ω) = 4π(−i)n[Y mn (y)]∗.

This fact can be seen by examining the Jacobi-Anger expression (2.25).

The room parameters in this simulation are similar to the 2-D case, with repro-

duction taking place in a R = 0.2m radius control region centered at (3.8m, 2.4m,

1.8m) in a 6.4×5×4m room. The reverberation is generated with the image-source

model [4] with a wall absorption coefficient of 0.3. We perform the soundfield re-

production at 2kHz by positioning (dkRe+ 1)2 = 25 omnidirectional loudspeakers

about a sphere of radius 2m as per [88] and concentric with the region of inter-

est. This setup yielded an average DRR from each loudspeaker of −4.0dB to the

boundary of B3. To measure the room response coefficients αnm(`, ω), we oversam-

pled the pressure, using M = 49 pressure samples spaces as in Figure 5.7. The

reproduction of the plane wave is illustrated in Figure 5.14.

5.6 Practical Implementation

The work presented in this chapter is still preliminary. The practical implementa-

tion of the soundfield reproduction scheme and its subjective performance remain

as open questions. Such questions shall be addressed in future research.

One important issue is sensor calibration. For measuring the modal coefficients,

the sensors have been assumed identical. If sensors are not calibrated to sufficient

accuracy, of at least 1000 : 1 for the wide-band example of Section 5.5.4, modal

coefficients will be unmeasurable. Since the gain of many real microphones changes

with temperature and humidity, accurate calibration may easily be destroyed. We

thus suggest that robustness will be a major implementation issue with this tech-

nique.


5.7 Summary and Contribution

In this chapter we have developed a novel method of performing soundfield repro-

duction in reverberant acoustical environments. Key to this method is an efficient

parametrization of the acoustic transfer function over a region of space. We provide

an itemized list of our contribution:

i. The technique of sound field reproduction through modal reconstruction has

been extended to reverberant acoustic environments. Through several simu-

lation examples, it is shown to yield a reproduction accuracy in reverberation

as good as conventional free-field designs operating in a free-field.

ii. A model-based method is provided for determining the transfer function be-

tween each loudspeaker and every point in a region. This method has arisen

from a novel interpretation of the modal space parametrization of a sound-

field.

iii. A theory for choice of fundamental design parameters has been developed -

namely the numbers of loudspeakers and sensors, geometric configuration of

sensors and loudspeaker filter weights.

Chapter 5 Theory and Design of Soundﬂeld Reproduction · Theory and Design of Soundﬂeld Reproduction 5.1 Introduction A problem relevant to emerging surround sound technology

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