PROCEEDINGS of the
23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Empirical Study of Decentralized Multi-Channel Active Noise Control
Based on the Genetic Algorithm
Guoqiang Zhang1; Jiancheng Tao2; Xiaojun Qiu1
1

University of Technology Sydney, Australia
2

Nanjing University, China

ABSTRACTIn an active noise control (ANC) system, computational complexity is one major concern when designing
practical control algorithms. One approach to reducing computational complexity is to apply a decentralized
control scheme rather than the centralized scheme. The decentralized scheme attempts to control a number of
ANC subsystems independently, where for simplicity, one subsystem consists of one loudspeaker and one
error microphone. Our recent published article has shown theoretically that a decentralized two-channel
ANC system can achieve the same noise reduction performance as the centralized one with guaranteed
convergence in the frequency domain. In this work, we attempt to extend the results from two-channel case to
N (N>1) channel case. The challenge sits in finding N complex numbers that could properly shape theeigenvalues of an N ´ N matrix for each frequency bin towards guaranteed convergence. Due to the problem
complexity, we conduct empirical study by using the genetic algorithm (GA). Simulation results on thechannel numbers of 4, 6, and 12 demonstrate that the resulting decentralized ANC controller is also able toachieve the same noise reduction performance as the centralized controller.
Keywords: ANC, decentralized control

1. INTRODUCTION
Active noise control (ANC) techniques have been successfully exploited to remove or mitigate
sound noise in many applications, such as designing ANC systems for headphone applications [1],reducing acoustic noise in magnetic resonance (MR) imaging [2], and creating a quiet zone around
listener ears [3]. The basic principle of an ANC system is to introduce a set of secondary sources tointerfere destructively with the primary sound by using multiple channels of loudspeakers and error
microphones, where each channel consists of one pair of loudspeaker and error microphone. To be able
to cancel the sound noise, a central controller is commonly employed to collect and process signals fromall the error microphones, and then make proper adjustments to the loudspeakers using the filtered-x LMS
algorithm [4], referred to as the centralized control scheme. In general, the noise reduction performanceof the system improves along with an increasing number of channels at the cost of high computational
complexity and expensive wiring, which is therefore unscalable and preventing it from broad applicability.
In recent years, it has attracted increasing attention on reducing the computational complexity of
the centralized controller for a large-scale ANC system in the sound and acoustic community. Two
schemes have been proposed in the literature to address the above issue. The first one is to introduce
the framework of distributed computing in the ANC system [5], [6], [7]. Specifically, [5] treats an
ANC system as a ring network where each secondary source is taken as one node in the network. The
computational burden is then distributed across all the secondary sources by performing incremental
computation sequentially over the ring network. At each iteration, the computation unit at eachsecondary source receives, updates and then transmits the control signal of all the secondary sources
by interacting with neighbours. The method achieves the same noise reduction performance as thecentralized controller at the cost of high transmission bandwidth and potential slow tracking speed.
The work of [6] and [7] extend [5] by utilizing a diffusion strategy in the update expressions of the
1
2{guoqiang.zhang, xiaojun.qiu}@uts.edu.au
jctao@nju.edu.cn

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control signal.
A decentralized control scheme aims to reduce both the computational complexity and expensive
wiring of the central controllers (see [8], [9]). The basic idea is to treat a large scale ANC system as a
combination of small-size ANC subsystems, where each subsystem, for example, can be taken as asingle-channel ANC. Each subsystem adjusts its own control signal only based on its associated error
microphone(s). Due to the inherent coupling between the secondary sources and error microphones, itremains challenging to design effective and robust control methods for the multiple subsystems. The
work of [8] proposed and analyzed a feedforward decentralized control method by studying matrix
eigenvalues in the frequency domain, where the matrix is determined by transfer functions of thesecondary paths of the whole system. It is found that if the inputs of the secondary sources areconstrained to be sufficiently small in magnitude, the control method is guaranteed to converge.
However, small inputs implicitly degrade the noise reduction performance, which is practicallyunattractive. The recent work [9] conducted extensive simulations and experiments for the control
method of [8] and drew a number of insightful observations.
Recently, we have reconsidered decentralized two-channel control in the frequency domain [10]. In
this situation, it is shown in [10] that properly designed decentralized control can theoretically achieve
the same noise reduce performance as the centralized controller without enforcing small inputs of thesecondary sources as in [8]. Considering each frequency bin, the basic idea of [10] is to find twocomplex values that can push the eigenvalues of a 2´2 matrix to the right complex domain (i.e., the two
eigenvalues having positive real parts). A procedure is provided in [10] for finding the two complex
values for each frequency bin. Later on, the authors in [11] attempted to extend our results todecentralized N-channel ANC (N>1, see Fig. 1). They proposed to find N complex values such that all
the eigenvalues of the corresponding N´N matrix for each frequency bin are equal to 1, which are thentransformed to solving N nonlinear equations. However, it is unclear if a solution exists or not. It also remains
open to explicitly solve the nonlinear equations to obtain the N optimal complex values.
In this paper, we reconsider decentralized N-channel ANC based on the genetic algorithm (GA)[12]. Our motivation is that the problem formulation in [11] for the N-channel case might beunnecessarily strict, making it difficult to search for the optimal solution even if it exists. In this work,
we intend to find N complex values for each frequency bin such that all the eigenvalues of theassociated N´N matrix are pushed to the right complex domain rather than being pushed to 1 as in [11]. By
relaxing the problem formulation, it would more likely lead to the existence of a solution for the
problem. Due to the problem complexity, we apply GA to compute the N complex values for each N´Nmatrix. A special nonconvex optimization problem in terms of the maximum and minimum eigenvalue
phases is constructed to facilitate the usage of the GA method. Simulation results on the channel
numbers of 4, 6, and 12 for an ANC system attached to a rectangular enclosure with a baffled opening(see the simulation setup in Section 4) demonstrate that the resulting decentralized ANC controller isalso able to achieve the same noise reduction performance as the centralized controller.

2. Multi-Channel ANC Signal Model
Consider a primary sound disturbance generated by a point monopole at a single frequency. We exploit
an ANC system to mitigate or remove the disturbance in the same environment. The considered ANC
system consists of N (N>1) secondary sources and N error microphones, where each secondary source andits collocated error microphone form a single-channel ANC subsystem (see Fig. 1). As the considered
disturbance is from one frequency, all the signals can be represented as single complex numbers. We omit
the frequency index here for concision.
Based on the above guidelines, the complex steady state signal at the 𝑖th error microphone can be
represented as
𝑒# = 𝑑# + )(1)(*+ 𝑧#( 𝑢( ,where 𝑑# represents the disturbance from the primary source at the 𝑖th error microphone, 𝑢( is thecomplex input to the 𝑗th secondary source, and 𝑧#( represents the complex response from the 𝑗th
secondary source to the 𝑖th error microphone. Eq. (1) can be further expressed into a compact vector form
as𝑧++ ⋯ 𝑧+)⋱⋮ .
𝒆 = 𝒅 + 𝒁𝒖, where 𝒁 = ⋮(2)𝑧)+ ⋯ 𝑧))The N×N complex matrix 𝒁 captures the joint response of the N secondary sources at the N error

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microphones.
Primary
SourceReference
SignalDecentralized
Controller

SecondarySources

Error
Microphones

Figure 1 – Demonstration of a decentralized multi-channel ANC system, where each channel consists of asecondary source and its collocated error microphone. The parameters 𝑧@A , s,𝑣 = 1,2 ⋯ , 𝑁, denote thefrequency response of the acoustic path from 𝑠 to 𝑣, and 𝑑A represents the noise disturbance at microphoneindex 𝑠.
The objective of a controller is to minimize the error vector 𝒆 by adjusting the input vector 𝒖 properly.
For the case that 𝒁 is nonsingular, the optimal input vector 𝒖⋆ can be computed by minimizing the sumof squared error signals directly, given by [8]+𝒖⋆ = arg min 𝒆L 𝒆 = −𝒁N+ 𝒅,
(3)
𝒖

K

where the superscripts (∙)L and (∙)N+ denote conjugate transpose and matrix inversion, respectively. It is
immediate that when 𝒖 = 𝒖⋆ , the error signal 𝒆 = 𝟎, which completely cancels out the primary noise.
When the matrix 𝒁 is singular, a common strategy for computing 𝒖⋆ is to introduce a quadratic penalty
function of 𝒖 into the objective function. One can then easily derive the optimal solution of 𝒖⋆
accordingly [8].

3. Towards Optimal Decentralized ANC Control
In this section, we first describe the optimization problem to allow for optimal decentralized control ofan N-channel ANC system. A decentralized controller is optimal if it is able to achieve the same noise
reduction performance as the centralized controller. We then explain how to solve the optimization problem
using the genetic algorithm (GA).
3.1

Problem formulation
We start with the iterative update expression for an N-channel decentralized controller. Suppose its
input to the N secondary sources at iteration 𝑘 is 𝒖 𝑘 . The input signal 𝒖 𝑘 + 1 at next iteration 𝑘 +1 can be computed by following the steepest descent algorithm as
𝒖 𝑘 + 1 = 𝒖 𝑘 − 𝜇𝑪𝒆 𝑘 ,(4)where 𝑪 =diag([𝑐+ , … , 𝑐) ]) is a diagonal matrix to be determined, and 𝜇 > 0 is the step-size of thesteepest descent algorithm. Plugging the expression (2) for 𝒆 at iteration 𝑘 into (4) yields
𝒖 𝑘 + 1 = 𝑰 − 𝑢𝑪𝒁 𝒖 𝑘 − 𝜇𝑪𝒅,5
where the matrix 𝒁 is assumed to be nonsingular. It can be shown that the iterates in (5) converge to a
fixed point if and only if all the eigenvalues of the matrix in the square bracket have modulus less than one.
By using algebra, it can be concluded that as long as the real parts of all the eigenvalues of 𝑪𝒁 are positive,a sufficient small step-size 𝑢 would allow the iterates in (5) to converge. It is not difficult to show that the
fixed point of (5) after convergence can be expressed as𝒖 ∞ = − 𝑪𝒁 N+ 𝑪𝒅 = −𝒁N+ 𝒅,(6)which coincides with the optimal solution in (3), thus reaching the optimal noise reduction performance.
The above analysis suggests that the key step to ensure convergence of (4) or (5) is to find a proper

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diagonal matrix 𝑪 such that all the eigenvalues of the matrix product 𝑪𝒁 are located in the right complex
domain. The above problem can be formulated mathematically as𝑪∗ = arg min 1
subject to 𝑪 = diag 𝑐+ , … , 𝑐) and 𝜆#,hi 𝑪𝒁 > 0 ∀𝑖 = 1, … , 𝑁,
(7)𝑪

where 1 represents a constant function, and 𝜆#,hi ∙ denotes the real part of the 𝑖th eigenvalue of amatrix. Eq. (7) defines a nonconvex optimization problem in terms of the N complex numbers {𝑐+ , 𝑐K ,…, 𝑐) } embedded in the diagonal matrix 𝑪. It is noted from (7) that there are in total N nonlinear
constraints, one for each matrix eigenvalue. That is, the number of free parameters is equal to thenumber of constraints. When N=2, it is clear from [10] that there exists 𝑪∗ such that 𝜆#,hi 𝑪∗ 𝒁 ≥ 0
for 𝑖 = 1,2. For the more general case N>2, it remains open to show the existence of a solution in (7).
We note that the optimal solution for 𝑪 in (7) is not unique if it exists. One can simply multiply apositive scalar to 𝑪𝒁, which would not change the sign of the eigenvalues. Alternatively, one can also
twist an optimal solution a bit by adding a sufficiently small noise without changing the sign of theeigenvalues. In general, it is preferable to find an optimal solution for 𝑪 such that the resulting
eigenvalues are far away from the left complex domain, which would lead to fast algorithmic
convergence.
Remark: The traditional decentralized controller in [8, 9] directly sets 𝑪 to be 𝑪 = 𝒁L
m , where 𝒁mis a diagonal matrix obtained by keeping only the diagonal elements of 𝒁. Empirical studies on an
ANC system with 2, 6, and 15 channels in [9] demonstrate that the above special setup of 𝑪 does notalways converge, where the failure cases are due to the negative real parts of the eigenvalues of 𝒁L
m 𝒁.
3.2

On shaping matrix eigenvalues using the genetic algorithm
In this subsection, we consider solving the nonconvex optimization problem of (7) using GA [12]. It is
well known that GA is quite flexible and can be used to solve highly nonlinear and nonconvex optimization
problems. Differently from the traditional optimization methods such as the steepest descent algorithm, GA
does not rely on any gradient information. Instead, it creates a population of solution candidates. At each
iteration, the candidates undergo a series of bio-inspired operations such as mutation, crossover andselection to produce new candidates. Only the candidates that produce low functional costs survive and therests are eliminated to be able to evolve towards a satisfactory solution.
For the nonconvex problem (7), it is, in general, quite difficult to compute the gradient of theeigenvalues of 𝑪𝒁 w.r.t. 𝑪. This prevents it from employing traditional optimization methods for solving(7). In principle, GA can be used for solving (7) without difficulty due to its flexibility and generality. One
property of GA is that given limited computation time, it may not reach the global optimal solution. For our
problem (7), a solution that satisfies all the N inequality constraints is sufficient.
3.2.1 On utilizing GA
To be able to employ GA for solving (7), we first need to perform problem reformulation. A constantobjective function 1 in (7) cannot be directly used in GA. To do so, we first introduce a number of notations.
Given a complex number 𝑐 = 𝑐n + 𝑗𝑐# , we use |𝑐| and ∠𝑐 ∈ −𝜋, 𝜋 to denote its amplitude and phase,respectively. To facilitate computation by GA, we set 𝑪 in (7) to be a product of two diagonal matrices
where one matrix is 𝒁L
m , which can be expressed as
𝑪 = diag 𝒂 𝒁L
8m,
where 𝒂 is an N-dimensional complex vector to be optimized. Next we define two functions 𝜃vwx (𝒂) ∈−𝜋, 𝜋 and 𝜃vyz (𝒂) ∈ −𝜋, 𝜋 to beL𝜃vwx (𝒂) = max ∠ 𝜆# diag(𝒂)𝒁L(9)m 𝒁 and 𝜃vyz (𝒂) = min ∠ 𝜆# diag(𝒂)𝒁m 𝒁 ,##where the maximization and minimization in (9) are taken over the phase values of all the N eigenvalues ofthe matrix diag(𝒂)𝒁Lm 𝒁.
We now construct a new optimization problem in terms of 𝜃vwx (𝒂) and 𝜃vyz (𝒂) as𝒂∗ = arg min 𝑔 𝒂 = 𝜃vwx (𝒂) } ×1~•€• 𝒂 ‚ƒ + 𝜃vyz (𝒂) } ×1~•„… 𝒂 †ƒ ,(10)𝒂subject to𝑏ˆ ≥ 𝑎# ≥ 𝑏Š
2𝜋 ≥ ∠𝑎# ≥ 0∀𝑖 = 1, … , 𝑁∀𝑖 = 1, … , 𝑁

1112

where 1(∙) is an indicator function. The two parameters 𝑏ˆ and 𝑏Š are the imposed upper and lower
bounds on the magnitude of elements of 𝒂. In practice, the lower bound 𝑏Š should be selected to bepositive (i.e., 𝑏Š >0) to ensure that the overall matrix diag(𝒂)𝒁Lm 𝒁 is nonsingular.
From a high level perspective, the optimization problem (8)-(12) is formulated to search for a solution6916

of 𝒂 such that the phases of all the eigenvalues of the matrix diag(𝒂)𝒁Lm 𝒁 are close to 0 as much as
possible. This is realized by penalizing the 4th order of the maximum and minimum eigenvalue phases
𝜃vwx (𝒂) and 𝜃vyz (𝒂) if necessary. If the solution space of (7) is not empty, one can alternatively solve
the problem (8)-(12) to obtain a solution to (7).
In this work, we apply GA to solve the nonconvex problem (8)-(12). The objective function 𝑔 𝒂 will
be used to evaluate the quality of each solution candidate generated by GA at each iteration, where only
those candidates that produce small functional values are kept to give birth to new child candidates. The
amplitude-phase constraints (11) -(12) are imposed on 𝒂 to facilitate the computation of GA.
3.2.2 Solution refinement after the GA procedure
We note that the global optimal solution 𝒂∗ to (8)-(12) should have the following property:
𝜃vwx 𝒂∗ + 𝜃vyz 𝒂∗ = 0.
However, in practice, the GA method may not produce the global optimal solution given limited timeand computational resources. It is highly likely that the output 𝒂‹Œ of GA would lead to 𝜃vwx 𝒂‹Œ +𝜃vyz 𝒂‹Œ ≠ 0 . That is, the maximum and minimum phases 𝜃vwx 𝒂‹Œ and 𝜃vyz 𝒂‹Œ are notsymmetric around the real axis of the complex domain.
Based on the above analysis, we can refine the solution 𝒂‹Œ by performing a rotation operation to
the eigenvalues of diag(𝒂‹Œ )𝒁L
m 𝒁 such that the resulting minimum eigenvalue phase is the negative of
the maximum eigenvalue phase. The rotation operation can be easily achieved by multiplying a scalar𝑒 (Ž to 𝒂‹Œ , where the phrase 𝜙 can be computed as
𝜙 = −(𝜃vwx 𝒂‹Œ + 𝜃vyz 𝒂‹Œ )/2.
(13)
Based on (8)-(13), we can obtain the final estimate 𝑪‹Œ for 𝑪 in (7), which can be represented as𝑪‹Œ = 𝑒 (Ž ∙ diag 𝒂‹Œ 𝒁L(14)m,where 𝒂‹Œ is computed by solving (8)-(12) using the GA method, and 𝜙 is obtained from (13).

4. Simulation Results
4.1

Simulation setup
We follow a similar simulation setup to that in [9, 10] to evaluate the proposed GA-based method fordecentralized N-channel ANC. A monopole sound source is located inside a rectangular enclosure with N
channels of loudspeakers and microphones placed near a baffled opening (see Fig. 2 for the setup of N=6,12). The enclosure dimensions are chosen as 𝐿“ = 0.432 m, 𝐿” = 0.670 m, and 𝐿• = 0.598 m. The
opening of the enclosure at 𝑧 = 𝐿• is assumed to be embedded in an infinite baffle. The primary source isplaced at 𝑟A = (0.12, 0.12, 0.16) m with a strength of 𝑞ƒ = 1×10N} 𝑚 š /𝑠.

𝑧𝑧Infinite baffle

Infinite baffle

𝐿•

𝐿•

0

𝑥

𝑞ƒ 𝛿(𝒓 − 𝒓A )𝐿”(a) 6-channel ANC

0𝑦Secondary loudspeaker
Error microphone𝑥𝑞ƒ 𝛿(𝒓 − 𝒓A )𝑦𝐿”(b) 12-channel ANC

Figure 2 - A monopole sound source inside a rectangular enclosure with a baffled opening.
The channel numbers of 𝑁 = 4, 6, 12 were studied in the simulation. For each N-channel ANC, the N
error microphones were evenly placed on the open surface at 𝑧 = 𝐿• while the N loudspeakers were
evenly placed right below the error microphones at the vertical level 𝑧 = 𝐿• − 0.1 𝑚. We consider noise

6917

reduction performance at 100 evenly sampled frequency bins from 5 Hz to 500 Hz: 𝑓 =5 Hz, 10 Hz, … , 500 Hz. As the GA method needs to be applied for each ANC setup at each frequency bin,
300 𝒁 matrices were tested in total using our proposed method. The default implementation of the GA
method in Matlab R2016a was used in our simulations. The two bounds in (11) were set to be 𝑏ˆ = 20
and 𝑏Š = 0.5, respectively.
For performance comparison, we tested both the conventional decentralized controller considered in [8,9], and our newly proposed decentralized controller. While the convergence of our proposed controller ischaracterized by the real parts of the eigenvalues of 𝑪‹Œ 𝒁 (see Section 3) for each frequency bin, theconvergence of the conventional controller in [8, 9] is determined by the eigenvalues’ real parts of 𝒁L
m 𝒁.
Therefore in the following, we mainly study the eigenvalue properties of the two matrices 𝑪‹Œ 𝒁 and 𝒁L
m𝒁
across different frequency bins over the ANC channel numbers of 4, 6, and 12.
On stability of the proposed decentralized control method
Fig. 3 shows the minima of the eigenvalues’ real parts and their signs for the two matrices 𝒁L
m 𝒁 and𝑪‹Œ 𝒁 for the channel numbers of 4, 6, 12, where it is noted that 𝒁L
𝒁
and
𝑪
𝒁
are
extracted
from
the
‹Œ
m
decentralized controller of [8, 9] and our proposed controller, respectively. It is clear from the figure thatfor each channel number, our proposed method manages to shape the eigenvalues of 𝑪‹Œ 𝒁 properly over
all the considered frequency bins, ensuring that the real parts of all the eigenvalues are positive. As a result,the proposed control method is able to achieve the same noise reduction performance as the centralized one.
This indicates that the GA method is effective for finding a satisfactory solution 𝑪‹Œ for each matrix 𝒁
measured from the considered ANC setup.
× 10 7

5
0
-5

4

100

× 10 7

200

300

400

200

300

200

300

400

100

200

300

(f): 12-channel ANC

100

200

300

Frequency (Hz)

400

500

500

400

500

-1

(e): 12-channel ANC

-6

400

0

Frequency (Hz)

-4

500

1

500

-2

400

2

Frequency (Hz)

0

0

100

(d): 6-channel ANC-4× 10 7-1(c): 6-channel ANC

-2

2

0

Frequency (Hz)

0

100

1

500

2

0

2

Frequency (Hz)sign of the minima in (c)minima of theeigenvalues' real parts(b): 4-channel ANC

100minima of theeigenvalues' real parts(a): 4-channel ANC
sign of the minima in (a)15sign of the minima in (e)minima of theeigenvalues' real parts

4.2

2
1
0
-1
100

200

300

Frequency (Hz)Figure 3 - The minima of the eigenvalues’ real part and their signs. (a)-(c)-(e): the minima of theeigenvalues’ real parts of 𝒁L
m 𝒁 and 𝑪‹Œ 𝒁. (b)-(d)-(f): the signs of the eigenvalues’ real part minima ofL
𝒁m 𝒁 and (“1” and “-1” stand for positive and negative respectively)6918

By contrast, the conventional decentralized method in [8, 9] would diverge at the frequency bins where
the minima of the eigenvalues’ real part are negative. To fix the convergence issue in this situation, thenoise reduction performance of the conventional decentralized method has to be sacrificed by imposing
constraints on the magnitude of the input to the loudspeakers (see [8, 9] for details).
4.3

Comparison of eigenvalue distributions(c): 4-channel ANC (500 Hz)
norm. eigenvalues of(d): 4-channel ANC (500 Hz)
norm. eigenvalues of

1

1

1

1

0.5

0.5

0.5

0.5

0
-0.5

0
-0.5

-1
0

1

1

-1
-1

real

0

1

-1

real

0

(h): 6-channel ANC (500 Hz)
norm. eigenvalues of

1

1

1

1

0.5

0.5

0.5

0.5

-0.5

0
-0.5

-1
0

1

0
-0.5

-1
-1

imaginary(g): 6-channel ANC (500 Hz)
norm. eigenvalues ofimaginary(f): 6-channel ANC (350 Hz)
norm. eigenvalues of

0

real

0

1

0
-0.5

-1
-1

-1
-1

real

0

1

-1

real

0

1

1

1

1

0.5

0.5

0.5

0.5

-0.5

-0.5

-1
0
real

1

0
-0.5

-1
-1

imaginary(l): 12-channel ANC (500 Hz)
norm. eigenvalues ofimaginary(k): 12-channel ANC (500 Hz)
norm. eigenvalues ofimaginary(j): 12-channel ANC (350 Hz)
norm. eigenvalues of

0

0
real

1

0
-0.5

-1
-1

1

real

(i): 12-channel ANC (350 Hz)
norm. eigenvalues of

0

1

real

(e): 6-channel ANC (350 Hz)
norm. eigenvalues ofimaginaryimaginary

0

0
-0.5

-1
-1

real

imaginary

0
-0.5

-1
-1

imaginary(b): 4-channel ANC (350 Hz)
norm. eigenvalues ofimaginary(a): 4-channel ANC (350 Hz)
norm. eigenvalues ofimaginaryimaginaryIn this subsection, we study the eigenvalue distributions of 𝒁L
m 𝒁 and 𝑪‹Œ 𝒁 in the complex domain
at two frequencies of 350 Hz and 500 Hz. Our primary motivation to do so is to reflect thecontrollability of the GA method on the resulting eigenvalues of 𝑪‹Œ 𝒁 qualitatively. The two
frequencies were chosen carefully so that the minima of the eigenvalues’ real parts of 𝒁L
m 𝒁 at 350 Hz
and 500 Hz are negative and positively, respectively, (see Fig. 3). To facilitate comparison between thetwo matrices 𝒁L
m 𝒁 and 𝑪‹Œ 𝒁, the eigenvalues of each matrix were normalized by division over its
largest eigenvalue magnitude.
Fig. 4 shows the normalized eigenvalues of the two matrices for the two considered frequencies
across the channel number of 4, 6, and 12. For the simple case of 350 Hz, it is seen that the GA methodmanaged to squeeze the phases of all eigenvalues of 𝑪‹Œ 𝒁 around 0 for all three channel numbers
while the phases of 𝒁L
m 𝒁 are scattered in the right complex domain. At 500 Hz, the eigenvalue phasesof the matrix 𝑪‹Œ 𝒁 are pushed in a relatively small phase region by the GA method. The above
properties suggest that the reformulated optimization problem (8) -(12) is effective, which makes the
GA method work as expected.

-1
-1

0
real

1

-1

0

1

real

Figure 4 – Visualization of the normalized eigenvalues of 𝒁L
m 𝒁 and 𝑪‹Œ 𝒁 at two frequencies of 350 Hz
and 500 Hz over the ANC channel numbers of 4, 6, and 12.

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5. CONCLUSIONSIn this paper, we have studied decentralized control of an N-channel ANC system in the frequency
domain. As an extension of our previous two-channel work [10], we considered shaping the
eigenvalues of an N´ N matrix for each frequency bin by using the GA method. To be able to apply the
GA method, we have built a special nonconvex optimization problem in terms of the maximum andminimum eigenvalue phases. Simulation results for the ANC channel numbers of 4, 6, and 12 over asemi-open rectangular enclosure suggest that the GA method indeed is able to effectively push all the
eigenvalues of the N´ N matrix to the right complex domain, thus ensuring the convergence of thecorresponding decentralized controller to reach the optimal noise reduction performance.
The research conducted in this paper is empirical. It still remains open to find out the existence ofa solution for an arbitrary N´ N matrix extracted from a real N-channel ANC system that can lead to
optimal decentralized ANC control.
In addition, the GA method would become increasingly heavy as the channel number N increases.
To allow for decentralized control of a large-scale ANC system (e.g., several hundred channels), it isdesired to develop advanced optimization methods other than the GA method that can properly shapethe eigenvalues of the N´ N matrix with low complexity.

ACKNOWLEDGEMENTS
This research was supported under the Australian Research Council’s Linkage Project funding scheme(LP140100987, Integrated Passive and Active Control of Humming Noise from KCGMs Haul Trucks).

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