Nonlinear active noise control using spline adaptive filters

Applied Acoustics 93 (2015) 38–43

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Nonlinear active noise control using spline adaptive filters Vinal Patel, Nithin V. George ⇑ Department of Electrical Engineering, Indian Institute of Technology Gandhinagar, Gujarat 382424, India

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 31 August 2014 Received in revised form 11 January 2015 Accepted 12 January 2015

A spline adaptive filter (SAF) based nonlinear active noise control (ANC) system is proposed in this paper. The SAF consists of a linear network of adaptive weights in cascade with an adaptive nonlinear network. The nonlinear network, in-turn consists of an adaptive look-up table followed by a spline interpolation network and forms an adaptive activation function. An update rule has been derived for the proposed ANC system, which not only updates the weights of the linear network, but also updates the nature of the activation function. An extensive simulation study has been conducted to evaluate the noise mitigation performance of the proposed scheme and the new method has been shown to provide improved noise cancellation efficiency with a lesser computational load in comparison with other popular ANC systems. Ó 2015 Elsevier Ltd. All rights reserved.

Keywords: Nonlinear active noise control Spline adaptive filter Functional link artificial neural network Volterra filter

1. Introduction

where

Active noise control (ANC), which is based on the destructive superposition of sound waves has found enhanced interest amongst the research community in the recent past owing to the advancements made in control theory as well as in semiconductor technology. ANC systems can be feed-forward or feedback in nature. A basic feed-forward ANC system contains a reference microphone to measure the reference signal xðnÞ, an active loudspeaker to generate the necessary anti-noise and an error microphone to sense the level of noise mitigation obtained. The input to the active loudspeaker is governed by an adaptive controller, which is updated using an adaptive algorithm [1]. Fig. 1 shows the block diagram of a filtered-x least mean square (FxLMS) algorithm based ANC system, where PN ðzÞ is the transfer function of the primary path (path from reference microphone to the error microphone), SN ðzÞ is the transfer function of the secondary path (electro-acoustic path from the output of the controller to S N ðzÞ is the transfer function the output of the error microphone), b of the model of the secondary path, WðzÞ is the transfer function of the adaptive controller, dðnÞ is the output of the primary path and ys ðnÞ is the output of the secondary path. The weights wðnÞ of the controller are updated using FxLMS algorithm [1] as

wðn þ 1Þ ¼ wðnÞ þ 2leðnÞx0 ðnÞ

ð1Þ

⇑ Corresponding author. Tel.: +91 9429207079; fax: +91 7923972622. E-mail addresses: (N.V. George).

[email protected]

(V.

http://dx.doi.org/10.1016/j.apacoust.2015.01.009 0003-682X/Ó 2015 Elsevier Ltd. All rights reserved.

Patel),

[email protected]

l is the learning rate, eðnÞ is the residual noise and x0 ðnÞ is

the primary noise xðnÞ filtered through b S N ðzÞ. It has been reported that the FxLMS algorithm based ANC schemes fail to effectively mitigate noise in the presence of nonlinearities in the ANC system. Several nonlinear ANC schemes have been developed in the recent past to improve the noise cancellation efficiency in such scenarios. An adaptive Volterra filter based ANC scheme which employs a Volterra filtered-x least mean square (VFxLMS) algorithm for weight update has been proposed in [2,3]. A nonlinear ANC system, which uses a functional link artificial neural network (FLANN) as the adaptive controller and employs a filtered-s least mean square (FsLMS) algorithm has been reported in [4]. Many attempts have been made in the recent past to improve the noise mitigation in nonlinear ANC systems [5–10]. Scarpiniti et al. has recently proposed a nonlinear spline filter [11]. The nonlinear spline filter consists of an adaptive linear network followed by an adaptive activation function and have been shown to effectively identify parameters in a nonlinear system identification problem. The authors have also shown improved performance in comparison with adaptive Volterra filters, which is a common filter used in ANC applications. An extension of the work has also been lately reported for nonlinear system identification [12]. In an endeavour to improve the noise cancellation achieved in a nonlinear ANC system, this paper proposes a spline adaptive filter based nonlinear ANC system. The adaptive nature of the activation function, makes the nonlinear spline filter a suitable candidate for effective noise cancellation in the presence of varying degrees of nonlinearities, without the need of increasing the computational load in terms of multipliers.

39

V. Patel, N.V. George / Applied Acoustics 93 (2015) 38–43 T

where u ¼ ½u3 ; u2 ; u; 1 ; C is the B-spline basis matrix given by

2

C¼

1

3

3 1

16 6 3 6 6 6 4 3 0 1 4

3 3 1

3

07 7 7 05

ð6Þ

0

and qi ¼ ½qi ; qiþ1 ; qiþ2 ; qiþ4 T is the control point vector. 3. Proposed nonlinear ANC system In the proposed spline adaptive filter based ANC scheme, the adaptive controller WðzÞ in Fig. 1 is replaced with the adaptive spline filter discussed in the previous section. The residual noise measured by the error sensor is given by Fig. 1. Block diagram of an FxLMS algorithm based ANC system.

eðnÞ ¼ dðnÞ  yðnÞ  sN ðnÞ

The rest of the paper is organized as follows. A brief introduction to a spline adaptive filter is made in Section 2. An adaptive spline filter based nonlinear ANC system is designed in Section 3. An adaptive learning algorithm for updating the weights as well as the activation functions is also derived and a convergence analysis of the update rule has also been presented in the section. A set of simulation exercises have been carried out in Section 4 to evaluate the performance of the proposed nonlinear ANC scheme. Concluding remarks are made in Section 5.

nðnÞ ¼ E½e2 ðnÞ

ð8Þ

where E½ is the expectation operator and nðnÞ is dependent on both aðnÞ and qi ðnÞ. Assuming E½e2 ðnÞ  e2 ðnÞ, the update rule for the weight vector aðnÞ is given by

2. Spline adaptive filter

aðn þ 1Þ ¼ aðnÞ 

The spline adaptive filter shown in Fig. 2 is essentially a linear non-linear network, where in the linear network is a finite impulse response (FIR) filter and the nonlinear network consists of an adaptive look-up table followed by a spline interpolation network. In the figure, xðnÞ is the input to the controller, aðnÞ is the output of the linear network and yðnÞ is the output of the spline filter. The output of the linear network is given by

where

aðnÞ ¼ aT ðnÞxðnÞ

ð2Þ

where aðnÞ ¼ ½a0 ; a1 ; . . . ; aN1 T is the adaptive weight vector of the linear network and xðnÞ ¼ ½xðnÞ; xðn  1Þ; . . . ; xðn  N þ 1ÞT is the tap delayed input signal vector with N as the length of the tap delay line. The output of the linear network, aðnÞ and the output of the spline adaptive network, yðnÞ are related using a nonlinear activation function q, which is determined using the span index i and the local parameter u. The local parameter is computed as

u¼

aðnÞ Dx

 

aðnÞ



Dx

ð3Þ

and the span index is obtained by



 aðnÞ Q  1 þ i¼ 2 Dx

ð4Þ

where Q is the total number of control points in the activation function, Dx is the gap between the control points and bc is the floor operator. The output of the spline adaptive filter is given by

yðnÞ ¼ qi ðuÞ ¼ uT Cqi

ð5Þ

ð7Þ

where sN ðnÞ is the impulse response of the transfer function SN ðzÞ and  is the convolution operator. In the proposed ANC system, the weights aðnÞ of the linear network as well as the control parameters of the non-linear network are updated using a gradient descent update rule [1] which minimizes the cost function

la @nðnÞ 2 @aðnÞ

@nðnÞ @½yðnÞ  sN ðnÞ ¼ 2eðnÞ @aðnÞ @aðnÞ   @y @u @ a  sN ðnÞ ¼ 2eðnÞ @u @ a @aðnÞ   1 xðnÞ  sN ðnÞ ¼ 2eðnÞ u_ T Cqi Dx

ð9Þ

ð10Þ ð11Þ ð12Þ

with u_ ¼ @u and la as the learning rate. The weight update rule for @u the linear network is given by

aðn þ 1Þ ¼ aðnÞ þ la eðnÞx0 ðnÞ 0

_T

ð13Þ

Cqi D1x xðnÞ

where x ðnÞ is u filtered through a model of the secondary path. Similarly, the control points are updated as

qi ðn þ 1Þ ¼ qi ðnÞ 

lq @nðnÞ 2 @qi ðnÞ

ð14Þ

where

@nðnÞ @½yðnÞ  sN ðnÞ ¼ 2eðnÞ @qi ðnÞ @qi ðnÞ  T  ¼ 2eðnÞ u C  sN ðnÞ h i ¼ 2eðnÞ C T u0

ð15Þ ð16Þ ð17Þ

with lq as the learning rate for updating the control points and u0 is the filtered version of u. The update rule for the control points is given by

Fig. 2. Linear non-linear network.

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V. Patel, N.V. George / Applied Acoustics 93 (2015) 38–43

h

i

qi ðn þ 1Þ ¼ qi ðnÞ þ lq eðnÞ C T u0 :

ð18Þ

The update rules in (13) and (18) together form the new filtered-c least mean square (FcLMS) algorithm. It may be noted that, the adaptive activation function is updated by updating only four control points in an iteration. The selection of the control points is achieved using the output of the linear network in the adaptive spline filter. The complete schematic diagram of the proposed scheme for an ANC system in a duct is shown in Fig. 3.

and thus

0 < la ðnÞ 6

2

ð27Þ

kx0 k2

lq ðnÞ many be computed in a similar fashion as

The bounds on given below.

eðn þ 1Þ ¼ eðnÞ þ

@eðnÞ Dq ðnÞ þ g @qTi ðnÞ i

ð28Þ

Using (7), we get

3.1. Convergence analysis The conditions under which the update rules of the proposed scheme converges are derived in this section. The Taylor series expansion of the error signal eðn þ 1Þ is given by

@eðnÞ @½yðnÞ  sN ðnÞ ¼ @qTi ðnÞ @qTi ðnÞ

ð29Þ

¼ uC T  sN ðnÞ

ð30Þ

0T

¼ Cu

ð31Þ

where g is the higher order terms in the Taylor series expansion and DaðnÞ ¼ aðn þ 1Þ  aðnÞ. Using (7), we can write

Dqi ðnÞ ¼ lq eðnÞC T u0

ð32Þ

@eðnÞ @y @u @ a ¼  sN ðnÞ @aT ðnÞ @u @ a @aT ðnÞ 1 T x ðnÞ  sN ðnÞ ¼ u_ T Cqi Dx 0T ¼ x ðnÞ

ð20Þ

h i eðn þ 1Þ ¼ 1  lq ðnÞkC T u0 k2 eðnÞ

ð21Þ

For convergence, we have the following bound on

eðn þ 1Þ ¼ eðnÞ þ

@eðnÞ DaðnÞ þ g @aT ðnÞ

ð19Þ

ð22Þ

and

and

which leads to

T

0 2

j1  lq ðnÞkC u k j 6 1

ð33Þ

lq ðnÞ. ð34Þ

or

DaðnÞ ¼ la ðnÞeðnÞx0 ðnÞ

ð23Þ 0 < lq ðnÞ 6

Using appropriate substitutions in (19), we get

h

i

eðn þ 1Þ ¼ 1  la ðnÞkx0 k2 eðnÞ

ð25Þ

which requires

j1  la ðnÞkx0 k2 j 6 1

kC u0 k2

ð35Þ

ð24Þ

The norm of eðn þ 1Þ should be less than the norm of the terms in the right hand side of (24) to ensure convergence. Therefore

jeðn þ 1Þj 6 j1  la ðnÞkx0 k2 jjeðnÞj

2 T

ð26Þ

3.2. Computational complexity The computational complexity of the proposed ANC scheme for one iteration is computed in this section, where b is the order of the spline matrix, N is the length of the FIR filter in the linear section of the adaptive spline filter and L is the length of the secondary path model. Table 1 shows the comparison of computational load with that of VFxLMS and FsLMS algorithms. It may be noted that

Fig. 3. Schematic diagram of an ANC system in a duct using a spline adaptive filter.

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V. Patel, N.V. George / Applied Acoustics 93 (2015) 38–43 Table 1 Comparison of computational complexity: (a) VFxLMS algorithm based ANC scheme, (b) FsLMS algorithm based ANC scheme, and (c) FcLMS algorithm based ANC scheme. VFxLMS

FsLMS

FcLMS

+

*

+

*

+

*

Controller output

NðN þ 3Þ=2  1

NðN þ 3Þ=2

Nð2P þ 1Þ  1

Nð2P þ 1Þ

Filtered signal

NðN þ 3Þ ðL  1Þ 2 NðN þ 3Þ=2  1

NðN þ 3ÞL=2

Nð2P þ 1ÞðL  1Þ

Nð2P þ 1ÞL

N þ b2 þ 2 ðN þ bÞðL  1Þ

N þ b þ b2 þ 4 ðN þ bÞL

NðN þ 3Þ=2 þ 1

Nð2P þ 1Þ

Nð2P þ 1Þ þ 1

N þ 2b þ bðb  1Þ

N þ b2 þ 3b þ 2

NðN þ 3Þ ðL þ 1Þ  1 2

NðN þ 3Þ ðL þ 2Þ þ 1 2

Nð2P þ 1Þ ðL þ 1Þ  1

Nð2P þ 1ÞðL þ 2Þ þ 1

2N þ b2 þ bðb  1Þ þ2b þ ðL  1ÞðN þ bÞ þ 2

2N þ LðN þ bÞ þ 2b2 þ 4b þ 6

Weight update Total

the computational load of an FcLMS based ANC scheme is independent of Q, the number of knots in a spline filter.

computational load of the proposed ANC scheme is evident from the comparison.

3.2.1. Computing controller output

4. Simulation study A set of simulations experiments have been conducted in this section to evaluate the performance of the proposed ANC scheme. The noise mitigation capability of the new scheme has been compared with that obtained by FsLMS as well as VFxLMS algorithms in terms of the mean square error (MSE), which is defined as

Multiplications: N þ b þ b2 þ 4. Additions: N þ b2 þ 2. 3.2.2. Obtaining filtered signal

   MSE ¼ 10log10 E e2 ðnÞ

Multiplications: ðN þ bÞL. Additions: ðN þ bÞðL  1Þ.

ð36Þ

The reference signal xðnÞ employed is a random noise uniformly distributed in the range ½0:5; 0:5 and all the secondary path are assumed to be perfectly modelled through an offline system identification process. An additive white Gaussian measurement noise of signal to noise ratio of 30 dB is considered for all experiments.

3.2.3. Weight update Multiplications: N þ b2 þ 3b þ 2. Additions: N þ 2b þ bðb  1Þ.

4.1. Experiment 1

3.2.4. Complete ANC operation using FcLMS

In this experiment, the primary noise sensed by the error microphone is given by

2

Multiplications: 2N þ LðN þ bÞ þ 2b þ 4b þ 6. Additions: 2N þ b2 þ bðb  1Þ þ 2b þ ðL  1ÞðN þ bÞ þ 2.

dðnÞ ¼ uðn  2Þ þ 0:8u2 ðn  2Þ  0:4u3 ðn  1Þ

Fig. 4 shows the variation of multiplication and addition count with respect to memory size (N) for FsLMS, VFxLMS and the proposed FcLMS, considering L ¼ 12; b ¼ 4 (FcLMS), P ¼ 1 (FsLMS) and a second order Volterra filter (VFxLMS). The reduced

where uðnÞ ¼ xðnÞ  qðnÞ with qðnÞ as the impulse response of the transfer function

QðzÞ ¼ z3  0:3z4 þ 0:2z5

ð38Þ

4

No. of Additions

2

x 10

Proposed Scheme VFxLMS FsLMS

1.5 1 0.5

(a) 0

0

5

10

15

20

25

30

35

40

45

50

25

30

35

40

45

50

4

No. of Multiplications

2

x 10

Proposed Scheme VFxLMS FsLMS

1.5 1 0.5

(b) 0

0

5

10

15

20

ð37Þ

Memory (N) Fig. 4. Variation of (a) addition count and (b) multiplication count with respect to memory size (N) for FsLMS, VFxLMS and FcLMS.

42

V. Patel, N.V. George / Applied Acoustics 93 (2015) 38–43

The transfer function of the secondary path used is

4.2. Experiment 2

SN ðzÞ ¼ z2 þ 0:5z3

ð39Þ

In this experiment, the reference signal xðnÞ and the primary noise dðnÞ are related by

The various simulation parameters used in this experiment are: N ¼ 6; Q ¼ 11; Dx ¼ 0:4; la ¼ 2  102 ; lq ¼ 2  102 ; lv ¼ 2  102 ;

N 1 ¼ 27; lf ¼ 2  103 and N 2 ¼ 18 where lv is the learning rate in VFxLMS algorithm, N1 is the number of weights updated in VFxLMS algorithm, lf is the learning rate in FsLMS algorithm and N 2 is the number of weights updated in FsLMS algorithm. Fig. 5 shows the learning curves for all the three algorithms studied. The average MSEs computed using the last 1000 samples of the residual noise are 19:08; 21:11 and 21:65 dBs for FsLMS, VFxLMS and FcLMS algorithms respectively. The steady state activation functions obtained using the adaptive look-up table is shown in Fig. 6. A linear initial activation function was employed in this experiment. In an effort to understand the effect of the variation of Dx, this experiment has been repeated for Dx ¼ 0:2; 0:4 and 1:0. The MSE obtained as an average of 30 iterations are 21:57; 21:41 and 19:18 dBs for Dx ¼ 0:2; 0:4 and 1.0 respectively. As the MSE values are similar for Dx ¼ 0:2 and Dx ¼ 0:4, and the number of control points that needs update are less with Dx ¼ 0:4, we have selected Dx ¼ 0:4 for this experiment. The steady state activation function for Dx ¼ 0:2 and Dx ¼ 1:0 have also been included in Fig. 6. A similar approach has been followed in selecting Dx in other experiments.

dðnÞ ¼ xðn  5Þ þ 0:8xðn  6Þ þ 0:3xðn  7Þ þ 0:4xðn  8Þ þ 0:2xðn  5Þxðn  6Þ  0:3xðn  5Þxðn  7Þ þ 0:4xðn  5Þxðn  8Þ

The secondary path considered in this experiment, which is a Hammerstein filter, is given by

^ dðnÞ ¼ wðnÞ þ 0:2wðn  1Þ þ 0:05wðn  2Þ

wðnÞ ¼ tanh½yðnÞ

ð42Þ

The simulation parameters applied in this experiment are: N ¼ 7; Q ¼ 11; Dx ¼ 0:4; la ¼ 3  102 ; lq ¼ 2:5  102 ; lv ¼ 2:5 102 ; N 1 ¼ 35; lf ¼ 3  103 and N 2 ¼ 21. The average MSEs obtained over 100 independent trials for FsLMS, VFxLMS and FcLMS algorithms are shown in Fig. 7. The average MSEs for the last 1000 error samples are 17:4; 17:9 and 19:1 dBs for FsLMS, VFxLMS and FcLMS algorithms respectively. The initial and steady state activation functions obtained are shown in Fig. 8. The figure

Proposed Scheme VFxLMS FsLMS

Proposed Scheme VFxLMS FsLMS

−10

−15

MSE (dB)

MSE (dB)

−10

−20

ð41Þ

where

−5 −5

ð40Þ

−15

−20

−25 −25

−30 −35

0

0.5

1

1.5

−30

2

0

0.5

1

x 104

Iterations

Fig. 5. Learning curves of the ANC system in Experiment 1 using FsLMS algorithm with lf ¼ 2  103 , VFxLMS algorithm with lv ¼ 2  102 and FcLMS with la ¼ 2  102 and lq ¼ 2  102 . All observations are averaged over 100 independent trials.

1.5

2 x 104

Iterations

Fig. 7. Learning curves of the ANC system in Experiment 2 using FsLMS algorithm with lf ¼ 3  103 , VFxLMS algorithm with lv ¼ 2:5  102 and FcLMS algorithm with la ¼ 3  102 and lq ¼ 2:5  102 . All observations are averaged over 100 independent trials.

2.5

2

2

1.5

1.5 1

1

0.5

y (n)

y (n)

0.5 0 −0.5

−0.5

−1

−1

Δx = 0.2 Δx = 0.4 Δx = 1

−1.5 −2 −2.5 −2

0

−1.5

−1

−0.5

0

0.5

1

1.5

Steady State Knots Initial Activation Function Steady State Activation Function

−1.5

2

α (n) Fig. 6. Experiment 1: Steady state activation functions for different values of Dx.

−2 −2

−1.5

−1

−0.5

0

0.5

1

1.5

α (n) Fig. 8. Experiment 2: Initial and steady state activation functions.

2

43

V. Patel, N.V. George / Applied Acoustics 93 (2015) 38–43

also shows the steady state locations of the 11 knots obtained, which are eventually used in obtaining the steady state activation function.

The primary path used in this experiment is similar to the one employed in Experiment 2. The output of the adaptive controller, yðnÞ and the controller output measured at the error microphone, ys ðnÞ [13] are related by

ys ðnÞ ¼ yðnÞ þ 0:35yðn  1Þ þ 0:09yðn  2Þ  0:5yðnÞyðn  1Þ þ 0:4yðnÞyðn  2Þ The simulation parameters employed are: N ¼ 8; Q ¼ 21; Dx ¼ 0:2; la ¼ 2  102 ; lq ¼ 1  102 ; lv ¼ 1:5  102 ; N1 ¼ 44; lf ¼ 2

103 and N 2 ¼ 24. The learning curves for the three ANC systems compared in this study, which is averaged over 100 independent trails is shown in Fig. 9. The average MSEs obtained from the last 1000 samples of the error signal for FsLMS, VFxLMS and FcLMS are 18:01; 19:84 and 22:11 dBs respectively. The nonlinear activation function obtained at steady state is plotted in Fig. 10. The figure also shows the initial linear activation function as well as the steady state location of the knots.

−5 Proposed Scheme VFxLMS FsLMS

MSE (dB)

−10

−15

−20

Number of multiplications

Number of coefficients

VFxLMS 161 FsLMS 107 FcLMS 90

139 83 74

175 95 78

190 127 116

176 106 101

221 121 106

27 18 10

35 21 11

44 24 12

The outcome of the three experiments clearly show the improved noise cancellation capability of the spline adaptive filter based nonlinear ANC system proposed in this paper. The adaptive nature of the activation function makes this filter a suitable for handling different levels of non-linearities. The computational complexity of the various algorithms studied in this paper is compared in Table 2. It is evident from the table that the improved noise mitigation achieved using FcLMS based ANC scheme is obtained at a lower computational cost in comparision with VFxLMS and FsLMS based ANC systems. 5. Conclusions In this paper, we have proposed a SAF based nonlinear ANC system. The proposed scheme employs an adaptive activation function in cascade with a linear weight network. A new FcLMS algorithm has been developed to update the weights of the linear network as well as the control points in the activation function. The dynamic nature of the activation function makes it possible for the proposed controller to effectively cancel noise in the presence of various levels of nonlinearities. As only a fraction of the control points are updated in an iteration, there is no significant increase in the computational load. Simulation results show the improved noise mitigation achieved by the new scheme. Acknowledgement This work was supported by the Department of Science and Technology, Government of India under the Fast Track Scheme for Young Scientists (SERB/ET-0018/2013).

−25

0

0.5

1

1.5

2

Fig. 9. Learning curves of the ANC system in Experiment 3 using FsLMS algorithm with lf ¼ 2  103 , VFxLMS algorithm with lv ¼ 1:5  102 and FcLMS algorithm with la ¼ 2  102 and lq ¼ 1  102 . All observations are averaged over 100 independent trials.

2 1.5 1 0.5 0 −0.5 −1 Steady State Knots Initial Activation Function Steady State Activation Function

−1.5 −1.5

−1

−0.5

0

0.5

References

x 104

Iterations

−2 −2

Number of additions

Exp. 1 Exp. 2 Exp. 3 Exp. 1 Exp. 2 Exp. 3 Exp. 1 Exp. 2 Exp. 3

4.3. Experiment 3

−30

Table 2 Comparison of computational complexity for the three ANC scenarios studied.

1

1.5

2

Fig. 10. Experiment 3: Initial and steady state activation functions.

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