IEEE/ACM TRANSACTIONS ON AUDIO, SPEECH, AND LANGUAGE PROCESSING 1
Active Noise Control Over Space: A Wave Domain
Approach
Jihui Zhang, Student Member, IEEE, Thushara D. Abhayapala, Senior Member, IEEE,
Wen Zhang, Member, IEEE, Prasanga N. Samarasinghe, Member, IEEE, and Shouda Jiang
Abstract—Noise control and cancellation over a spatial region
is a fundamental problem in acoustic signal processing. In this
paper, we utilize wave-domain adaptive algorithms to iteratively
calculate the secondary source driving signals and to cancel the
primary noise field over the control region. We propose wave-
domain active noise control algorithms based on two minimiza-
tion problems, (i) minimizing the wave-domain residual signal
coefficients and (ii) minimizing the acoustic potential energy
over the region, and derive the update equations with respect to
two variables, (a) the loudspeaker weights and (b) wave-domain
secondary source coefficients. Simulation results demonstrate the
effectiveness of the proposed algorithms, more specifically the
convergence speed and the noise cancellation performance in
terms of the noise reduction level and acoustic potential energy
reduction level over the entire spatial region.
Index Terms—Active noise control (ANC), wave domain, mul-
tichannel, spatial noise, reverberant room.
I. INTRODUCTION
A. Motivation and Background
Active noise control (ANC), or noise cancellation, em-
ploys secondary sound sources to generate secondary signals,
which collectively cancel the primary sound field [1], [2].
In applications, such as noise cancellation in aircraft [3] and
automobiles [4]–[7], the control zone is large, which requires
noise cancellation to be performed over the entire region,
instead of at some spatial points. Furthermore, real noise
fields are often time-varying and unknown, which requires an
adaptive algorithm to iteratively calculate the secondary source
driving functions and to produce the secondary sound field.
ANC over space is often approached via multichannel ANC
systems with multiple sensors and multiple secondary sources
[8]–[10], employing either feed-forward [11] or feedback
control systems [12], [13]. Conventional multichannel ANC
algorithms in the frequency domain [14]–[16] perform noise
cancellation directly on a set of multiple observation points
J. Zhang is with the Research School of Engineering, College of Engineer-
ing and Computer Science, The Australian National University, Canberra,
ACT 2601, Australia. She is also with School of Electrical Engineering and
Automation, Harbin Institute of Technology, Harbin Heilongjiang 150001,
China (e-mail: jihui.zhang@anu.edu.au).
T.D. Abhayapala and P.N. Samarasinghe are with the Research School
of Engineering, College of Engineering and Computer Science, The
Australian National University, Canberra, ACT 2601, Australia (e-mail:
thushara.abhayapala@anu.edu.au; prasanga.samarasinghe@anu.edu.au).
W. Zhang is with Center of Intelligent Acoustics and Immersive Commu-
nications, School of Marine Science and Technology, Northwestern Polytech-
nical University, Shaanxi 710072, China (e-mail: wen.zhang@nwpu.edu.cn).
S. Jiang is with School of Electrical Engineering and Automation, Harbin
Institute of Technology, Harbin Heilongjiang 150001, China (e-mail: js-
d@hit.edu.cn).
(MP) in the control region, which are fairly straightforward
and are widely used in practice [17]. These control systems
minimize the sum of the squared pressures, which is equal
to minimizing the potential energy density at the microphone
locations. Although these approaches lead to significant noise
reduction at the target points, the consistency over a continuous
spatial region is low.
To overcome this problem and to enlarge the control re-
gion, some researchers have proposed ANC systems based
on energy density. By utilizing the acoustic energy density
sensors [18], Parkins captured more global information in the
enclosure and minimized the acoustic energy density (AED)
[19]. Similar methods have been proposed such as minimizing
the acoustic potential energy (APE) [20], and minimizing the
generalized acoustic energy density (GED) [21]. Montazeri
described the acoustic potential energy in terms of room
modes, which depends greatly on the room geometry [22].
Recently, ANC over space has been approached via Wave
Field Synthesis (WFS) based adaptive algorithms [23]–[25]
and cylindrical/spherical harmonics expansion based wave
domain adaptive algorithms [26], [27]. Please note that in
this manuscript, we use the terminology “wave-domain signal
processing” to refer to harmonics (cylindrical/spherical) based
sound field processing. Harmonics based wave-domain signal
processing is a technique commonly used for spatial sound
field recording/reproduction over spatial regions using discrete
transducer arrays. The principle of harmonic representation of
sound fields is to use fundamental solutions of the Helmholtz
wave-equation as basis functions to express sound over a
spatial region. Thus, the sound field can be thought of as
superimposed set of orthogonal and continuous basis fields
(cylindrical/spherical harmonics) with corresponding knobs to
control relative weights (coefficients) of each basis wave field.
Since wave-domain signal processing controls propagating
sound fields in whole rather than a distributed set of target
points, it naturally provides a more insightful and efficient
method for ANC over space.
Initial work on harmonics based solutions to ANC appeared
in [26], [27]. The authors use cylindrical/spherical harmonics
as basis functions and their respective coefficients to represent
the noise field and secondary field over the desired spatial
region. Instead of minimizing the sum of the squared error
signals [28], wave-domain ANC tends to minimize the har-
monic coefficients, which in turn control the entire spatial
region directly. Authors of [26], [27] showed that wave-domain
ANC can achieve significant noise cancellation over the entire
region of interest with faster convergence speeds.
IEEE/ACM TRANSACTIONS ON AUDIO, SPEECH, AND LANGUAGE PROCESSING 2
B. Approach and Novel Contributions
In this paper, for the first time, we present a comprehensive
analysis on adaptive wave-domain ANC based on a feedback
control system, while studying multiple cost functions and
multiple update algorithms. We use a a microphone array
to measure the residual signals and a loudspeaker array to
generate the secondary sound signals. We utilize the harmonic
coefficients to characterize the noise field and calculate the
acoustic potential energy. In existing spatial ANC work, the
adaptive algorithms are based on minimizing residual sound
field coefficients and updating the secondary sound field
coefficients to update the secondary source weights. In this
paper, we develop normalized wave-domain ANC algorithms
in two different ways: (i) minimizing the residual sound field
coefficients and (ii) minimizing the acoustic potential energy
of the residual sound field. We also derive the update equations
with respect to two variables: (a) the loudspeaker weights
and (b) secondary sound field coefficients. Thus, resulting
four different method of implementing harmonics based wave-
domain ANC systems. We compare these four methods respect
to each other as well as with the the conventional multi-point
method. We show that there are trade-offs in selecting each
one of the four algorithm over the other. To the best of our
knowledge, the detailed analysis of these four wave-domain
adaptive algorithms have not been reported in the literature.
The proposed algorithms are shown to give better convergence
results and improve noise reduction performance within the
control region when compared to the conventional multi-point
method.
The rest of the paper is organized as follows. In Section
II we formulate the spatial noise cancellation problem and
the ANC system in the wave domain. The four variants of
wave-domain multichannel ANC algorithms are proposed in
Section III. We demonstrate the simulation results to compare
the ANC performance of the proposed wave-domain methods
and the conventional multi-point method in Section IV, and
draw some conclusions in Section V.
II. PROBLEM FORMULATION
A. System Model
In this section, we address the problem of ANC to cancel
the noise over a spatial region. Let the interested control zone
be a circular region (S) with a radius R
1
. We assume that the
noise sources are located outside the control region.
We consider an ANC system in two-dimensional space
using (i) a single microphone array on the boundary of the
control region to measure the residual signals and (ii) a single
loudspeaker array outside the region to generate the secondary
sound field [26], as shown in Fig.1. The theory we develop in
this paper can be extended to 3-D space.
Any arbitrary observation point within the control region
is denoted as x ≡ {r, φ
x
}. In the ANC system, the residual
signal at this point is given by
e(x, k) = v(x, k) + s(x, k), (1)
where k = 2πf/c is the wave number, f is the frequency, c is
the speed of sound propagation, v(x, k) is the noise signal
Microphone array
Loudspeaker array
Noise
sources
x
R
1
R
2
φ
x
Fig. 1: A spatial ANC region (black) consists of a circular
microphone array of radius R
1
and a circular loudspeaker
array of radius R
2
.
and s(x, k) is the secondary sound field generated by the
loudspeakers.
The secondary sound field generated by the loudspeaker
array can be represented by
s(x, k) =
Q
X
q=1
d
q
(k)G(x|y
q
, k), (2)
where d
q
(k) is the driving signal of the q
th
loudspeaker,
and G(x|y
q
, k) denotes the acoustic transfer function (ATF)
between the q
th
loudspeaker and the observation point x. For
example, for sound propagation in free field, G(x|y
q
, k) =
i
4
H
(2)
0
(kky
q
− xk) , where H
(2)
0
(·) is the zero
th
-order Hankel
function of the second kind.
Instead of using the measurements on the microphone points
directly, the wave domain approach employs the wave equation
solutions as basis functions to express the sound field over
the entire spatial region of interest, and designs the secondary
signals accordingly. Below we transform each component in
(1) and (2) into the wave domain.
B. Primary noise field
The cylindrical harmonic based wave equation solution
decomposes any homogeneous incident wave field v(x, k)
observed at x into
v(x, k) =
∞
X
m=−∞
β
m
(k)J
m
(kr) exp(imφ
x
), (3)
where J
m
(·) is the Bessel function of order m and exp(·)
denotes the exponential function [29]. The decomposition
coefficients β
m
(k) represent the primary noise field in the
wave domain. Within the circular region r ≤ R
1
, we can
use a finite number of modes to approximate
1
the noise field
[30], thus (3) reduces to
v(x, k) ≈
M
X
m=−M
β
m
(k)J
m
(kr) exp(imφ
x
), (4)
where M = dekr/2e [30], [31].
1
The infinite summation in (3) can be truncated at M = dekr/2e [30],
[31] due to inherent properties of Bessel functions
IEEE/ACM TRANSACTIONS ON AUDIO, SPEECH, AND LANGUAGE PROCESSING 3
C. Secondary sound field
Using the cylindrical harmonic expansion, the generated
secondary sound field inside the control region can also be
represented by
s(x, k) ≈
M
X
m=−M
γ
m
(k)J
m
(kr) exp(imφ
x
), (5)
where coefficients γ
m
(k) represent the secondary sound field
in the wave domain.
The ATF in (2) can be parameterized [32] as
G(x|y
q
, k) ≈
M
X
m=−M
T
m,q
(k)J
m
(kr) exp(imφ
x
), (6)
where T
m,q
(k) are the ATF coefficients in the wave do-
main and assumed to be prior knowledge obtained from pre-
calibration.
By substituting (5) and (6) into (2), we can get
γ
m
(k) =
Q
X
q=1
d
q
(k)T
m,q
(k), for m = −M, · · · , M. (7)
Therefore, in matrix form, the relationship between the sec-
ondary source decomposition coefficients and the loudspeaker
weights are given by
γ(k) = T d(k), (8)
where T is a (2M + 1) × Q matrix with the (i, j) element
given by T
i−M−1,j
, γ is the secondary coefficients vector with
the (i) element given by γ
i−M−1
(k), and d is the vector of
loudspeaker driving signals.
D. Residual signals
Substituting (4) and (5) into (1), the residual signals can be
represented by
e(x, k) ≈
M
X
m=−M
(β
m
(k) + γ
m
(k)
|
{z }
α
m
(k)
)J
m
(kr) exp(imφ
x
), (9)
where α
m
(k) is the residual signal decomposition coefficients.
In our ANC system, e(x, k) are the frequency domain sound
pressure measured by the error microphones. From (9), we can
obtain the wave domain α
m
(k), which is a good indicator of
the residual sound field over the entire region.
The objective of wave-domain adaptive ANC is to design
the loudspeaker driving signals d(k) based on the wave-
domain residual signal α
m
(k) and the acoustic transfer func-
tion T (k), so that the noise field v(x, k) is canceled by the
generated secondary sound field S(x, k) over the control re-
gion of interest. The proposed ANC algorithms are introduced
in the following section.
III. MULTI-CHANNEL WAVE-DOMAIN ACTIVE NOISE
CONTROL
We adopt a block-wise operation and transform the micro-
phone measurements into time-frequency domain, and decom-
pose the noise field into the wave-domain coefficients using
(9).
In the wave-domain adaptive algorithm, the residual signals
in each iteration (the n
th
time block) can be expressed as
α(n, k) = β(n, k) + γ(n, k), (10)
where α(n, k) = [α
−M
(n, k), . . . , α
M
(n, k)]
T
,
the superscript (·)
T
denotes the transpose of a
vector, β(n) = [β
−M
(n, k), . . . , β
M
(n, k)]
T
and
γ(n, k) = [γ
−M
(n, k), . . . , γ
M
(n, k)]
T
.
Here onwards, we omit the dependency k in each vector for
notational simplicity, thus have
α(n) = β(n) + γ(n). (11)
Below we derive different wave-domain adaptive algorithms
by solving two minimization problems, (a) squared residual
signal coefficients, and (b) acoustic potential energy.
A. Minimization of squared residual signal coefficients
Minimizing the sum of the squared residual signal coeffi-
cients, the cost function becomes
ξ
1
(n) =
M
X
m=−M
|α
m
(n)|
2
= α
H
(n)α(n), (12)
where the superscript (·)
H
denotes the conjugate transpose.
Using the steepest descent algorithm, the adaptive algorithm
follows the update equation
w(n + 1) = w(n) −
µ
2
∇ξ
1
(n), (13)
where w is the update variable, µ is the step size. Below
we derive the wave-domain update function for two cases, (1)
loudspeaker weights are updated directly, and (2) secondary
sound field coefficients are updated directly.
Case 1: Update the loudspeaker weights directly.
If we perform the adaptive process on the loudspeaker
weights directly, we can obtain the loudspeaker weights for
each iteration from the update equation. In this case, the update
variable in (13) can be replaced by d, that is w = d =
[d
1
, . . . , d
Q
]
T
.
By the complex LMS algorithm [33], taking a derivative
of ξ
1
(n) with respect to d, the gradient of the cost function
(Theorem 1) can be written by
∇ξ
1
(n) = 2T
H
α(n). (14)
The proof is given in Appendix A.
Substituting (14) into (13), the final adaptive equation in wave
domain can be written as
d(n + 1) = d(n) − µT
H
α(n). (15)
The block diagram of the algorithm is shown in Fig. 2.
By replacing the LMS filter by the normalized LMS filter,
the final update equation of the normalized wave-domain
IEEE/ACM TRANSACTIONS ON AUDIO, SPEECH, AND LANGUAGE PROCESSING 4
Σ
ATF
Adaptive
algorithm
WD
transform
+
+
d
ν
s
e
α
P
Q
Fig. 2: The block diagram of wave-domain ANC system,
when updating the loudspeaker driving signals. Block of
WD transform represents the wave-domain transform for the
residual signals.
algorithm updating driving signals (NWD-D) can be written
as
d(n + 1) = d(n) −
µ
0
kT
H
k
2
2
T
H
α(n), (16)
where k·k
2
denotes the Euclidean norm for a vector or matrix,
and µ
0
∈ [0, 1] denotes the normalized step size.
Case 2: Update the secondary sound field coefficients.
If we update the wave-domain secondary sound field coef-
ficients (γ) first, and calculate the loudspeaker driving signals
(d) later, the update variable in (13) can be replaced by γ,
then we have w = γ = [γ
−M
, . . . , γ
M
]
T
. Taking a derivative
of ξ
1
(n) with respect to γ, the gradient of the cost function
(Theorem 2) can be written by
∇ξ
1
(n) = 2α(n). (17)
The proof is given in Appendix B.
Substituting (17) into (13), the adaptive equation in wave-
domain coefficients can be written as
γ(n + 1) = γ(n) − µα(n). (18)
Thus, the final update equation of the normalized wave-
domain algorithm updating mode coefficients (NWD-M) is as
follows,
γ(n + 1) = γ(n) − µ
0
α(n). (19)
From (8), we obtain the loudspeaker weights d(n) by d =
T
+
γ, where the superscript (·)
+
denotes the pseudoinverse
of a matrix. The block diagram for updating the wave-domain
coefficients is shown in Fig 3.
B. The minimization of acoustic potential energy
Minimizing the total acoustic potential energy (APE) in an
enclosed noise field can achieve global reduction in sound
pressure throughout the enclosure [34], [35]. Here we derive
the acoustic potential energy in terms of the wave-domain
coefficients to obtain global reduction over the control region.
By definition, acoustic potential energy is
E
p
(k) =
1
2ρ
0
c
2
P (k), (20)
Σ
ATF
T
+
d
Adaptive
algorithm
WD
transform
+
+
γ
ν
s
e
α
P
Q
Fig. 3: The block diagram of wave-domain ANC system,
when updating the wave-domain coefficients. Block of WD
transform represents wave domain transform for the residual
signals.
where ρ
0
denotes the density of the media and P (k) is the
average energy of the residual signal given by
P (k) =
Z
S
e
∗
(x, k)e(x, k)dS
=
Z
2π
0
Z
R
1
0
e
∗
(x, k)e(x, k)rdrdφ
x
, (21)
with superscript (·)
∗
denoting the complex conjugate. Since
the potential energy is a scalar multiple of the average spatial
energy, by defining P (k) to be the cost function, we can
effectively minimize the potential energy.
We represent P (k) in the spherical harmonics domain by
substituting (9) into (21) as,
P (k) =
M
X
m=−M
α
∗
m
(k)α
m
(k)(2π
Z
R
1
0
(J
m
(kr))
2
rdr
| {z }
u
m
(k)
), (22)
where the integral in (22) is estimated by numerically evaluat-
ing the integral between 0 and R
1
, which is the integral over
the interested region.
Then, (22) can be written in matrix form as
P (k) = α
H
Uα, (23)
where α = [α
−M
, . . . , α
M
]
T
, U = diag(u
−M
, . . . , u
−M
),
and u
m
= 2π
R
R
1
0
(J
m
(kr))
2
rdr .
Therefore, the new cost function becomes
ξ
p
(n) = P (n) = α
H
(n)Uα(n). (24)
where the frequency dependency k is omitted for notational
simplicity.
We derive the update equation for the new cost function in
two cases.
Case 1: Update the loudspeaker weights directly.
The gradient of the cost function (Theorem 3) can be written
by
∇ξ
p
(n) = 2T
H
Uα(n). (25)
The proof is given in Appendix C.
Substituting (25) into (13), the final adaptive equation in wave
domain can be written as
d(n + 1) = d(n) − µT
H
Uα(n). (26)
IEEE/ACM TRANSACTIONS ON AUDIO, SPEECH, AND LANGUAGE PROCESSING 5
Similar to (16), the update equation of the normalized
energy-based wave domain algorithm updating driving signals
(NEWD-D) can be written as
d(n + 1) = d(n) −
µ
0
kT
H
Uk
2
2
T
H
Uα(n). (27)
Case 2: Update the secondary sound field coefficients.
The gradient of the cost function (Theorem 4) can be written
as
∇ξ
p
(n) = 2Uα(n). (28)
The proof is given in Appendix D.
Substituting (28) into (13), the adaptive equation in the wave-
domain coefficients can be written as
γ(n + 1) = γ(n) − µUα(n). (29)
The final update equation of the normalized energy-
based wave domain algorithm updating the mode coefficients
(NEWD-M) can be written as
γ(n + 1) = γ(n) −
µ
0
kU k
2
2
Uα(n). (30)
Then the loudspeaker weights d(n) can be calculated by d =
T
+
γ in each iteration.
IV. SIMULATION RESULTS
A. Simulation setup
In this section, performance of the proposed four wave
domain algorithms (i) normalized wave domain algorithm up-
dating driving signals (NWD-D), (ii) normalized wave domain
algorithm updating mode coefficients (NWD-M), (iii) normal-
ized energy-based wave domain algorithm updating driving
signals (NEWD-D), and (iv) normalized energy-based wave
domain algorithm updating the mode coefficients (NEWD-
M) are compared with the conventional normalized multi-
point (NMP) algorithm
2
, in both free-field and reverberant
environments. We assume the desired control zone to be a
circular region of a radius of 1 m (black area in Fig. 1), and
the noise field to be generated by point sources, which are
outside of the control region.
We utilize a feedback ANC system for control on a 2D
plane, where the circular microphone array of radius 1 m is
placed on the boundary of the control region and the circular
loudspeaker array of radius 2 m is placed outside the control
region. The speed of sound is c = 343 m/s and the density of
the air is ρ
0
= 1.225 kg/m
3
. The simulation of the reverberant
environment is modeled as a rectangular room of size 6 m×6
m with perfectly absorbing ceiling and floor, and all the side
walls have a reflection coefficient of 0.75. The reverberation
is simulated using the image-source method [36].
The simulation starts in the time domain. We adopt a block-
wise operation and transform the microphone recordings into
the time-frequency domain. Based on (9), we further transfer
the signal into wave-domain coefficients. A sampling rate of
8 kHz and a window length of 3200 samples are employed.
White Gaussian noise with signal-to-noise (SNR) ratio of 40
2
Here, NMP algorithm is the normalized version of the MC algorithm in
[10].
dB is added to each microphone recordings to model the
internal thermal noise of microphones.
To evaluate the primary noise reduction performance, we
study the (i) instantaneous noise reduction on the microphones
N
b
r
(n), (ii) noise reduction inside the interest region N
in
r
(n),
and (iii) acoustic potential energy over region E
p
(n).
The instantaneous noise reduction on the microphones can
be defined as
N
b
r
(n) , 10 log
10
P
z
E{|e
z
(n)|
2
}
P
z
E{|e
z
(0)|
2
}
, (31)
where e
z
(n) represents the sound pressure received on the
z
th
microphones at the n
th
iteration, and e
z
(0) represents the
sound pressure received on the z
th
microphones before the
ANC process.
To evaluate the noise reduction performance inside the
control region, sound pressure at L = 1296 points uniformly
placed inside the regions e
in
are examined. We define the
instantaneous noise reduction inside the interest region N
in
r
(n)
as follows,
N
in
r
(n) , 10 log
10
P
l
E{|e
in l
(n)|
2
}
P
l
E{|e
in l
(0)|
2
}
, (32)
where e
in l
(n) denotes the residual signals at the l
th
point
inside the region at the n
th
iteration, and e
in l
(0) represents
the primary noise field at the l
th
point in the region.
As mentioned above, acoustic potential energy is another
measure of evaluating the noise reduction over the entire
spatial region [37], which can be considered as a more
insightful measure in practice. From (20) and (23), the acoustic
potential energy over the control region for each iteration can
be calculated by
E
p
(n) =
1
2ρ
0
c
2
α
H
(n)Uα(n), (33)
where α(n) can be conveniently captured by circular micro-
phone arrays, and calculated based on (9).
In addition to the noise reduction measures mentioned
above, we analyze two more performance measures, (i) the
residual noise field in the control region, and (ii) the conver-
gence speed. We simulate the ANC algorithms to deal with a
single frequency noise field and a multi-frequency noise field
as shown in the following two subsections.
B. Single frequency
First, we investigate the narrowband performance of differ-
ent algorithms. Three noise sources are located at (2.2, 0
◦
),
(2.5, 45
◦
), and (3, 240
◦
) with magnitude of 10, 15 and 5,
which are marked as pink
0
+
0
in Fig. 4 and Fig. 5. The
frequency of the noise field is 200 Hz. The control region (R
1
= 1 m) in such a noise field can be represented by m[−5, 5]
modes, thus, we place 2N + 1 = 11 microphones on the
boundary to capture the information of the residual noise field
for each modes. We select the same normalized step values
for different algorithms, µ
0
= 0.8 in free-field and µ
0
= 0.5
in reverberant environments.
