PROCEEDINGS of the23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Feedforward Control of Fan Noise in Ducts using Multichannel
Order-reduced Inverse Filters
Mingsian R. Bai; Shin-Cheng Huang
Department of Power Mechanical Engineering, National Tsing Hua University, TaiwanABSTRACT
Single-channel feedforward control is a commonly used approach for active noise control (ANC). In thispaper, a time-domain underdetermined multichannel inverse filtering technique is proposed for thefeedforward active control (ANC) of duct noise. In the commonly used filtered-x least-mean-square(FXLMS) algorithm, the feedforward control problem is formulated as an overdetermined inverse filtering
problem which generally results in non-zero residual noise. By introducing multiple secondary sources, theproblem can be reformulated into an underdetermined system, which admits infinite number of exact
solutions with zero residual noise. Linearly constrained minimum variance (LCMV) method is employed in
the controller design. However, as a major shortcoming of the time-domain approach, the length of the filters
tends to be too long to admit digital signal processor (DSP) implementation. To address the problem, the least
absolute shrinkage and selection operator (LASSO) algorithm, is exploited to effectively reduce thecontroller orders. A two-channel system is implemented to suppress the fan noise of an air-cleaner.
Simulation and experiment results have demonstrated that the proposed approach has achieved significantly
higher noise reduction than the conventional FXLMS algorithm.
Keywords: Active noise control, Multiple channels, Inverse filtering

1. INTRODUCTION
Active noise control (ANC) has been a field of active research for several decades. ANC is amethod for attenuating undesired noise by secondary sources, which is particularly effective in low
frequency range (1, 2). There are three control structures of ANC system: the feedforward structure (3,4), the feedback structure (5,ġ 6), and the hybrid structure (7, 8, 9) which combines the former twostructures. This paper is focused on the widely used feedforward structure.
The feedforward active noise control problem is an overdetermined inverse filtering problem
which typically leads to non-zeros residual noise. A method of multiple-input/output inverse theorem(MINT) was suggested (10) to convert the overdetermined problem to a square one by adding
additional channels of secondary sources. In this paper, the MINT is extended to a time-domain
underdetermined inverse filtering (TUMIF) approach (11), where a regulated underdetermined
problem is posed in a model-matching framework.
Traditionally, feedforward control for ducts is implemented with adaptive filters such as thefiltered-x least mean square (FXLMS) algorithm (12, 13). In contrast, the TUMIF controllers are
implemented as fixed filters obtained using optimizations methods, e.g., the least squares (LS) withan L2-norm regularizer, also known as the Tikhonov regularization (TIKR) (14, 15). However, theresulting FIR filters are prohibitively long. To overcome this difficulty, a sparse coding (SC)technique (16) named the least absolutely shrinkage and selection operator (LASSO) algorithm (17)is employed to reduce the controller’s complexity.
To further enhance the TUMIF controllers, the linearly constrained minimum variance (LCMV) (18)
approach is exploited by considering the noise covariance matrix and multiple linear constraints posedin the TUMIF. The LCMV algorithm seeks to minimize the array output power subject to linear
constraints. The major contribution is to implement the LCMV-TUMIF approach on an active fannoise control problem of duct. In addition, the orders of controllers are reduced by using sparse
coding techniques to enable real-time implementation. Experimental results are given and discussed.

2. THE TIME-DOMAIN UNDERDETERMINED MULTICHANNEL INVERSE

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FILTERS (TUMIF)
2.1 Two-channel TUMIF system
The idea of TUMIF is to incorporate N-channels (N > 1) of secondary sources into the ANC
system. This section focuses on a two-channel system. For a general N-channel TUMIF-based
feedforward system, we refer the interested readers to a previous paper (11).

Figure 1 – The block diagram of a two-channel feedforward ANC system.

A two-channel feedforward TUMIF system is shown in Figure 1. This can be viewed as amodel-matching problem, as described in the following equation:
c1[k ] g1[k ]  c2 [k ] g2 [k ] m[k ],(1)where k denotes the discrete time index and “ ” denotes the convolution. The impulse response m[k]represents the primary path, whereas g 1[k] and g 2 [k] denote the secondary paths. The aim is tominimize the residual noise e[k] by using the feedforward controllers c 1[k] and c 2 [k]. Assume thatthe primary path, two secondary paths, and two feedforward controllers are modeled by L m, L g, andL c -tapped finite impulse response (FIR) filters. In matrix notations, the model -matching problems
can be written asGT cT  m 0,(2)whereGTGm

Lm u2 Lc,0ª g m [0]º« g [1]»g m [0]m»««»
g m [1]«»« g m [ Lg  1]
»«g m [ Lg  1]
g m [0] »«»g m [1] »««»«»0g m [ Lg  1]¼»¬«>cn [0] cn [1]

cncT

m

32

ª¬G1 G 2 @ 

Lm u Lc, m 1, 2;cn [ Lc  1]@ , n 1, 2;TTªc1T cT2 º ¼¬>m[0] m[1]

(3)

2 Lc

(4)

(5)
(6)

,m[ Lm  1]@ ,T(7)LmLc  Lg  1  2Lc ,(8)Lc ! Lg  1,(9)where “ T ” denotes matrix transpose.

Figure 2 - Vector subspace for the full-rank underdetermined problem. There exist infinite number of exactsolutions. cr is the projection of cT in N (GTT ) and cn is the projection of cT in N (GT ) .

The TUMIF system described by Equation 2 is an underdetermined system if the inequalities ofEquations 8 and 9 are fulfilled by choosing appropriate filter lengths, L g, Lc and Lm . Assuming that
GT has full-rank, the vector subspaces (19) associated with GT is illustrated in Figure 2. N (GTT )shrinks to 0, so m  R(GTT ) , suggesting solutions always exist. Furthermore, N (GT ) is not {0}, soinfinite number of exact solutions exist. The minimum-norm solution isc LS GTT (GT GTT )1 m.
(10)
The controller gain can be regularized by the TIKR method
cTIKR (GTT GT  E 2 I)1 GTT m,(11)where ȡ ġ is the regularization parameter to trade-off the controller gain and noise reduction
performance.
To enhance the feedforward ANC system, the LCMV design is employed to minimize the controleffort.
22E ª« c1T x º»  E ª« cT2 x º»¬¼¬
¼cTT R xx cT ,(12)where E >@ denotes the expected value, R xx consist of R ss , and x is the input noise vector.
x> x[k ] x[k  1]x[k  Lc  1]@T(13)ª R ss 0 º(14)« 0 R »,ss ¼¬
is the noise covariance matrix. That is, the LCMV optimization problem can be written as
and R xx

where R ss

min cTT R xx cTcst. GT cTm.

The optimal controllers of the preceding constrained optimization problem arec LCMV (R xx  H I)1 GTT (GT ((R xx  H I)1 )GTT  P I)1 m,
where H and P denote the regularization terms in TIKR approach.

(15)

(16)

2.2 Order Reduction of Feedforward Controllers
The preceding controllers tend to be too long to be implemented in real-time. To overcome thedifficulty, a sparse coding technique will be exploited to reduce the order of controller. The33

TUMIF-based ANC problem can be posed as a constrained optimization problem:
min c 0 st. GT c  m 0,cwhere(17)c 0 denotes the zero-norm, or the cardinality of the solution vector, c . To accommodate noise

robustness, inequality constraint is used instead.
min c 0 st. GT c  m 2  K ,cwhere η is a threshold and(18) 2 donates the L2-norm. This problem can be solved by using relaxation

methods or greedy methods. This paper adopts one of relaxation methods, the LASSO algorithm.
Instead of the zero-norm, the LASSO approach uses the L1-norm in the cost function to obtain aconvex optimization problem (20, 21, 22) as shown
min c 1 st. GT c  m 2  K ,cwhere(19) 1 denotes the L1-norm. Besides, Equation 19 can also be converted into an equivalent

unconstrained convex optimization problem.
minc12GT c  m 2  O c 1 ,2(20)where O denotes the regularization parameter that serves as a weighting for sparsity.
To extend this to the LCMV controllers, we rewrite Equation 15 into a modified LASSO form:
minc

where12
G T cT  m 2  O cT2rxx 1 ,(21)denotes the Hadamard product and vector rxx is derived from the noise covariance matrix
dvec(R xx )rxx,(22)dvec(R xx ) f

where dvec() denotesġthe operation to convert diagonal entries of a matrix into a column vector, and f denotes the infinity norm.ġ The convex optimization problems can be solved by the package CVX(23) or the alternating direction method of multipliers (ADMM) (24) method.

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3. EXPERIMENTAL RESULTS

Figure 3 – The experimental arrangement of duct ANC system in the anechoic room.

In this section, several experiments were performed to validate the two-channel feedforward ANC
system in anechoic room, as shown in Figure 3. The noise attenuation performance of controllers iscompared with or without order reduction method. The duct is 1.75 m long, with rectangularcross-section 0.16 m×0.16 m and wall thickness 0.018 m as shown in Figure 4. The corresponding
cutoff frequency of the duct is 1078 Hz that is also regarded as the control bandwidth in which thesound field in the duct can be treated as plane waves.

Figure 4 – Dimensions of the wooden duct.

In this paper, fan noise band-limited to 4 kHz is used as the noise source. In experiment, all of thecontrollers which are offline designed under 8 kHz sampling rate are implemented on theTMS320C6416 of Texas Instrument® for two control loudspeakers. The conventional FXLMS
algorithm for control-loudspeaker 2 is adopted as a benchmarking method. However, with thelimitation of TMS320C6416’s computation power, the allowable maximum number of taps ofcontroller is 400. Therefore, the controllers in this work are selected to be 400 taps in experiment andsimulation. In addition, the step size of the FXLMS is 0.01, the regularization parameter of theTUMIF-LCMVġis 0.01, the TUMIF-TIKR is 0.1, and the other sparse coding approaches is 0.001.

35

(a)

(b)

Figure 5 – Simulation results obtained using the FXLMS, the TUMIF-TIKR, the TUMIF-LCMV, theTUMIF-LCMV-CVX, the TUMIF-LASSO-CVX and the TUMIF-LASSO-ADMM. (a) residual noise, and (b)noise reduction performance (control off – control on).

Figure 5 of simulation results show that the order-reduced controllers perform well in noise
reduction. Because of the limitation of DSP board’s computation power that mentioned previously, theperformance of the unreduced controllers will decrease severely. However, with the noise covariancematrix, the TUMIF-LCMV approach has a better and more robust performance. In experiment, theunreduced TUMIF-LCMV controllers attains 0-15dB noise suppression performance more than thebenchmarking controller in frequency range 100-770 Hz. Moreover, the order-reduced
TUMIF-LCMV achieves 0-60dB noise attenuation performance more than the benchmarking
controller in frequency range 100-1100 Hz. Therefore, Figures 6 and 7 of experimental results
indicate that the order-reduced TUMIF controllers attain larger noise attenuation than the unreducedcontrollers. In particular, the order-reduced TUMIF-LCMV approach attains 5-80 dB noise
attenuation in frequency range 100-1100 Hz, which is the best among all methods.

(a)

(b)

Figure 6 – Experimental results obtained using the FXLMS, the TUMIF-TIKR and the TUMIF-LCMV.
(a) residual noise, and (b) noise reduction performance (control off – control on).

36

(a)

(b)

Figure 7 – Experiment results obtained using the FXLMS, the TUMIF-LASSO-CVX, theTUMIF-LASSO-ADMM and the TUMIF-LCMV-CVX.
(a) residual noise, and (b) noise reduction performance (control off – control on).

4. CONCLUSIONSIn this paper, we apply the linearly constrained minimum variance (LCMV) method to reformulate
the time-domain underdetermined inverse filters for feedforward ANC. In order to simplify the FIR
controllers, we use the sparse coding technique to reduce the order of the controllers such that inreal-time implementation is possible.
Through experiments, the proposed LCMV-TUMIF controller has demonstrated superior noise
attenuation performance over other methods. However, the TUMIF-LCMV-CVX controllers will
behave like the TUMIF-LASSO controllers if regularization parameter defined in Equation 21 isslightly under-regularized.
Given a DSP board with limited computational power, the sparsely coded TUMIF controllers
significantly outperforms the FXLMS algorithm in noise attenuation. In the future, more efficient
implementation of the LCMV-TUMIF controllers will be investigated.

ACKNOWLEDGEMENTS
The work was supported by the Ministry of Science and Technology (MOST) in Taiwan, under theproject number 105-2221-E-007-030-MY3.

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