Active noise cancellation with a fuzzy adaptive filtered-X algorithm C.-Y. Chang and K.-K. Shyu Abstract: The authors present a fuzzy adaptive filtered-X algorithm for active noise cancellation (ANC) in ducts, using fuzzy rules to construct an anti-noise filter to cancel out the undesired noise. Complex acoustic plant models, such as those used in conventional ANC systems, can be disregarded in the proposed fuzzy technique. As a new method for ANC, the proposed system tunes the free parameters automatically and changes the IF–THEN rules adaptively to minimise the residual noise as new information becomes available. Hence, the proposed algorithm can be easily constructed at the beginning, and the complexity of building an ANC system can be reduced. Direct numerical simulations demonstrate the effectiveness of the proposed design.

1

Introduction

The technique of active noise cancellation (ANC), which uses artificial signals to cancel undesired noise [1], has received much attention during the past decade because of recent advances in electronics and microcomputers. Conventional methods, called passive noise control (PNC), have the ability to suppress the higher frequency acoustic noise rather than the lower frequency noise, as proven by several researchers in many papers. However, industrial acoustic noise often has its main power on lower frequencies, where the wavelength of sound is so long that passive techniques are no longer cost-effective because they require material that is too bulky and heavy [2], such as the silencer of a car. In contrast to passive methods, active methods not only permit the cancellation of lower frequency noise, but also reduce the weight, volume and cost of the overall noise control system. To put in place an ANC system, one has to identify some transfer functions of acoustic plants and transducers, such as microphone, speaker and duct plant, to generate the correct anti-noise signal. Many researchers have employed the filtered-X [3] algorithm as an active noise controller because of its simplicity. The filtered-X algorithm is also an adaptive filter and its weighting parameters can be automatically updated by the least mean square (LMS) algorithm. This approach is effective at attenuating lower frequency noise, such as that from a fan, compressor, or engine noise in an acoustic duct [4]. However, there are still several problems with the filtered-X strategy. One of the most critical disadvantages of the filtered-X LMS algorithm is the low convergence speed. This is because the filtered-X algorithm needs a small step gain to update the weighting parameters in order to maintain stable performance of the

r IEE, 2003 IEE Proceedings online no. 20030406 doi:10.1049/ip-cds:20030406 Paper first received 16th July and in revised form 23rd December 2002. Online publishing date: 19 August 2003 C.-Y. Chang is with the Department of Electronic Engineering, Ching Yun Institute of Technology, Chung-Li, Taiwan 320, Republic of China K.-K. Shyu is with the Department of Electrical Engineering, National Central University, Chung-Li, Taiwan 320, Republic of China

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system [3, 4]. A concurrent difficulty is that small step gains cannot update the weights in time to keep up with the change of residual noise, which plays a leading role in the filtered-X algorithm. Hence, the tracking speed is very slow and an accurate anti-signal cannot be derived to cancel the undesired noise. These decrease the performance of broadband noise reduction. A few ANC systems based on fuzzy logic have been proposed, most of which use FIR filters or PD controllers to suppress noise, and then use a fuzzy method to adapt the system parameters. However, these approaches still require mathematical information about the duct plant, and they are still very complex. Rather than another fuzzy ANC system, this paper proposes a fuzzy adaptive filtered-X algorithm to enhance the performance of ANC systems. The fuzzy filtered-X algorithm is mainly composed of linguistic information from human experts, and only a little numerical information is needed. Moreover, this method can minimise the residual noise of a fuzzy adaptive ANC system by properly setting the initial weighting parameters. Hence, the critical problem of low convergence speed is overcome and the residual noise can be minimised. Compared with other ANC schemes, the proposed fuzzy approach provides a very easy way to develop an active noise controller. In addition, the proposed fuzzy adaptive system can be used for many other applications [5–9]. 2 Active noise cancellation with filtered-X algorithm The usual ANC system, which utilises the filtered-X algorithm, is shown in Fig. 1. In the plant model, Hp(z) represents the primary speaker and P(z) denotes the mathematical transfer function of the acoustic duct, the input signal xk represents the kth undesired noise sample, and the residual noise ek is measured by an error microphone. The output signal of the filtered-X algorithm uk drives an anti-noise speaker with transfer function Hs(z), in an attempt to cancel the undesired noise in the duct. The acoustic transfer function from the location of the anti-noise speaker to the error microphone Mp(z) is referred to as the error path and is represented by He(z). The (N+1)th order adaptive active noise controller with weighting parameters IEE Proc.-Circuits Devices Syst., Vol. 150, No. 5, October 2003

primary speaker

duct

microphone He(z) Mp(z)

P (z)

xk

3

error

Hp(z)

Hs(z) secondary speaker

ek

antinoise uk a

xk

Hp(z)

P (z)

silent zone + He(z)

Mp(z)

ek

Hs(z)

acoustic path

uk

artificial path [Wk ] Hs(z)*He(z)*Mp(z) filtered-X LMS b

Fig. 1

ANC for duct

a Duct system b Block diagram of filtered-X LMS ANC

Wk is updated by the filtered-X LMS algorithm. Hence, the residual noise ek becomes ek ¼ xk Hp PHe Mp þ Wkt Xk Hs He Mp

ð1Þ

where the weighting parameters and the input noise sequence are

Fuzzy adaptive filtered-X ANC system

Since the introduction of fuzzy sets by Zadeh in 1965 [10], many researchers have applied this theory to diverse engineering topics. In general, the knowledge base and rule base are designed from expert experience, and then an inference engine is selected to make up a fuzzy system [11–13]. Traditionally, active noise controllers are designed on the basis of a mathematical description and its linearised model. However, it is difficult to control undesired noise in a nonlinear duct plant. Hence, the performance of broadband noise reduction in a conventional ANC system is worse than the performance of narrowband noise reduction. Furthermore, existing ANC systems use complicated mathematical transfer functions to design the controllers. For instance, the conventional filtered-X based ANC systems often use hundreds of weighting functions to control undesired noise. Therefore, resulting numerical errors, such as the round-off and quantisation are inevitable. In contrast, the proposed self-tuning fuzzy-based ANC system can process both the numerical data and linguistic information to adapt the ANC system, as shown in Fig. 3. The advantages of the proposed algorithm include the reduction of system complexity and the property of nonlinear compensation. Furthermore, due to its generality, the fuzzy filtered-X scheme can also be applied to many engineering applications, such as power stabilisers, shock testing experiments, hearing aids, and vibration control systems [5–9]. The following describes the procedures for designing a fuzzy filtered-X ANC system. xk

Wk ¼ ½w0k ; w1k ;    ; wNk t

ð2Þ

Xk ¼ ½xk ; xk1 ;    ; xkN t

ð3Þ

acoustic path

The corresponding gradient estimation of the adaptive system at time sample k is given by

artificial path

2 ^k ¼ @ek ¼ 2ek ðXk Hs He Mp Þ r @Wk

P (z)

ek He(z)

+

Mp(z)

Hs(z)

uk fuzzy controller

ð4Þ

where the weighting parameters are updated with the LMS algorithm: Wkþ1

silent zone Hp(z)

^k ¼ Wk  2mek ðXk Hs He Mp Þ ¼ Wk  mr

Fig. 3

Architecture of a fuzzy-based active noise controller

ð5Þ

This differs from the traditional LMS algorithm: Wkþ1 ¼ Wk  2mek Xk

fuzzy filtered-X

ð6Þ

It is seen that the difference between (5) and (6) is in the correction term. The last term of (5) is similar to the output of noise sequence Xk that came into the composite filter HsHeMp. This condition is shown in Fig. 2. Therefore, (5) is referred to as the filtered-X LMS algorithm. In addition, there is a stepping parameter m for each wlk, l ¼ 0, ?, N.

Step 1: Generally speaking, the output of a (N+1)th order FIR filter is composed of the moving averages of the past (N+1) input noise samples. Hence, one uses the past (N+1) noise samples to design the fuzzy filter. For a (N+1)th order fuzzy ANC system, one can define M fuzzy sets Fil for each input noise sample xki with Gaussian membership functions 2 !2 3 l xki  xi;k 1 5 mFil ðxki Þ ¼ exp4 ð7Þ 2 slxi

Correction term of filtered-X LMS algorithm

where l ¼ 1,2, ?M, i ¼ 0,1,2,?, N and k and denotes the time samples. In (7), xil are the centres and slxi are standard deviations of the Gaussian functions, respectively. In this study, we set M ¼ 7, N ¼ 20 and the initial standard deviations equal to 0.2. Hence, each input sample xki, has seven linguistic terms which initially are equally distributed in the input signal range [1 1]; NB (negative big), NM

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Xk

Fig. 2

HsHeMp

XkHsHeMp

step. The filter is represented as f ðXk Þ:

1.0 NB

NM

NS

AZ

PS

PM

PB

M P

membership degree

0.8

ylk  minðmF l ðx k Þ; mF l ðx k1 Þ;  ; mF l ðx kN ÞÞ

f ðXk Þ ¼ l¼1M P l¼1

0.6

0

N

1

0

N

1

ð9Þ The authors use uk to control the undesired noise, such as shown in Fig. 3. Step 4: To minimise the power of residual noise, the fuzzy filtered-X LMS algorithm uses the following [3, 14] to l update the free parameters ylk ; xi;k and slxi;k at each sample time k:

0.4

0.2

0 –1.0

–0.5

0

0.5

ylkþ1 ¼ ylk þ l  ek 

1.0

interval for input variables

Fig. 4 Membership functions of input and output linguistic variables

(negative medium), NS (negative small), AZ (almost zero), PS (positive small), PM (positive medium) and PB (positive big). Figure 4 shows the results of this step in constructing a twenty-first order fuzzy FIR filter. Step 2: Define a set of M fuzzy IF–THEN rules of the following type: l

Rule : IF xk is

F0l

and    and

xkN is FNl THEN yk is U l

Table 1: Rule map of fuzzy ANC system Antecedents

slxi;kþ1 ¼ slxi;k þ l  ek glk where

xk1

Rule 1

NB

NB

NB

NB

NB

Rule 2

NM

NM

NM

NM

NM

Rule 3

NS

NS

NS

NS

NS

Rule 4

AZ

AZ

AZ

AZ

AZ

Rule 5

PS

PS

PS

PS

PS

Rule 6

PM

PM

PM

PM

PM

Rule 7

PB

PB

PB

PB

PB

yyy xkN

yk

Step 3: This step illustrates the processes of the inference engine and defuzzifier. In a fuzzy inference engine, fuzzy logic principles combine fuzzy rules into a mapping from an input fuzzy set to an output fuzzy set. Suppose that the given input noise of the fuzzy filter at sample time k is Xk ¼ ½x k ; x k1 ;    ; x kN . The proposed filter, using a minimum the inference engine and centroid defuzzifier, is obtained by combining the M rules defined in the previous

l 2 ðxki  xi;k Þ

2

qk ¼

7 X

ð10Þ

ðslxi;k Þ2

ðslxi;k Þ3

l 1 xki  xi;k pkl ¼ min@exp4 i 2 slxi;k

!2 31 5A

pkl

ð11Þ

ð12Þ

ð13Þ

ð14Þ

l¼1

glk ¼

ylk þ uk l pk qk

ð15Þ

and where l is a small positive constant. The last terms of (10)–(12) are similar to the input noise xk sent into the fuzzy filters. Thus, (10)–(12) are referred to as the fuzzy filtered-X LMS algorithm, which differs from the conventional filtered-X LMS algorithm in the correction term and in the fuzzy algorithm. In this study, one can set the following initial parameters as: l l xi;0 ¼ xki and xi;0 ¼ 6 xki for l ¼ 6 1

Consequents

xk

0

pkl qk

l xki  xi;k

l l xi;kþ1 ¼ xi;k þ l  ek glk

ð8Þ

l ¼ 1, ?, M. In (8), Fil is defined in the previous step and the set Ul is a fuzzy singleton at ylk . At first the crisp value yl0 can be arbitrarily chosen in the dynamic range [1 1], and the adaptive law of fuzzy rules is shown in next step. The seven rules form a fuzzy FIR filter, which acts as an antinoise filter to cancel undesired noise. The rule base of the proposed system is shown in Table 1.

418

9uk

min ðmF l ðx k Þ; mF l ðx k1 Þ;    ; mF l ðx kN ÞÞ

ð16Þ

and choose y10 ¼ xk 

Hp P Hs

ð17Þ

Moreover, the other yl0 ðl ¼ 6 1Þ can be chosen to be zero. Thus, by (9) and (16)–(17), the anti-noise uk will equal y10 . After driving the secondary speaker Hs, the artificial noise will become xkHpP, which compensates the undesired noise in the silent zone. In addition, (17) can be achieved by applying the transfer functions HeMp at the rear end of (17), hence Hp PHe Mp ð18Þ y10 ¼ xk  Hs He Mp The numerator becomes HpPHeMp and the denominator becomes HsHeMp. Both the numerator and the denominator are transfer functions between electrical signals. One can use broadband white noise as the input signal of the primary speaker Hp and then apply the microphone Mp at the rear end of the duct to measure the numerator. The input/output response is the numerator of (18). Moreover, one can apply the broadband noise as the input of Hs and IEE Proc.-Circuits Devices Syst., Vol. 150, No. 5, October 2003

15

10

magnitude, dB

5

0

–5 conventional filtered-X fuzzy filtered-X after 8000 iterations

–10

–15 0

100

200

300

400

500

frequency, Hz

Fig. 5

Performance of system identification

1.5

1.0

0.5 amplitude

measure the remaining noise by microphone Mp. The denominator of (18) is also determined. Hence, (17) is achieved and no duct plant is needed. Remark 1: In (9), the fuzzy filtered-X ANC algorithm is well defined and will not become unstable during the adaptive process because the denominator of the membership function mF l ðxki Þ is always greater than zero. Furthermore, i using the definitions of initial parameters in step 4, the fuzzy filtered-X algorithm will converge faster than the conventional filtered-X algorithm. Thus, the proposed system speeds the convergence of a filtered-X system. Remark 2: Most studies have tried more rules to fulfil the fuzzy filter. However, the proposed method uses only seven rules to construct the fuzzy system. The main reason is that the LMS algorithm will update each rule. The residual error is hence converged towards zero. In addition, the antecedent part of each fuzzy rule consists of 21 input variables. Thus the proposed fuzzy filter will perform like a 21st-order FIR filter, which can control the undesired narrowband noise very well. Remark 3: From steps 1–4, it can be seen that the fuzzy algorithm uses very little numerical data to develop the ANC system because the fuzzy system can self-tune the free parameters using the proposed updating algorithm to generate the anti-noise signal. This is very helpful in compensating for the shortcomings of the usual ANC systems, such as the hardware limitations of quantisation and round-off errors and the distortion of sensors and actuators. Hence, the computational complexity of designing an ANC system is reduced.

0

–0.5

4

Simulation results –1.0

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250 Hz sine wave conventional filtered-X –1.5 0

20

40

60

80

100

time, ms a 1.5

1.0

0.5 amplitude

The duct model [15, 16] is described as follows. The acoustic plant of duct P(z) is modelled by a pure time delay of 25 samples and the error path He(z) is modelled by five samples. Both the speakers Hp(z) and Hs(z) are represented by second-order Butterworth high-pass filters with cutoff frequency of 80 Hz, and the system uses a 21st-order adaptive FIR filter with zero initial weighting parameters to perform a filtered-X algorithm. In addition, the standard deviations of Gaussian membership functions are all initially set to 0.2, and the sampling frequency of the proposed simulations is 2 kHz. Several experiments are here presented to demonstrate the utility of the fuzzy filtered-X ANC system. In the first numerical simulation, the performance of system identification of acoustic plant is illustrated. The authors use white noise to identify the duct plant. Because the duct plant is modelled by 25 pure delays, the 0 dB lines represent the frequency response of real duct plant. The solid line in Fig. 5 displays the proposed fuzzy filtered-X model and the dashed line displays the conventional 21st-order filtered-X model for companies on. One can see that the proposed fuzzy filtered-X algorithm identifies the duct well. The second simulation uses a 250 Hz periodic signal as the undesired noise. Figure 6 shows the noise cancelling performance, where the dashed lines represent the undesired noise. Figure 6a shows the results of the conventional filtered-X ANC system and Fig. 6b shows fuzzy filtered-X ANC. It is clear that the proposed fuzzy algorithm provides faster control speed and less residual noise than the conventional method. The third simulation uses a combination of two periodic signals with close frequencies, 245 Hz and 250 Hz, as the undesired noise, as shown in the dashed lines of Figs. 7a and 7b. It can be seen that the proposed fuzzy filtered-X

0

–0.5

–1.0 250 Hz sine wave fuzzy filtered-X

–1.5 0

20

40 60 time, ms

80

100

b

Fig. 6

Narrowband noise cancellation

a Conventional method b Fuzzy method

algorithm still provides excellent performance in cancelling the periodic signals. These simulation results demonstrate that the proposed algorithm provides superior performance for narrowband noise cancellation compared to conventional methods. 419

1.5 20 1.0 0

magnitude, dB

amplitude

0.5

0

–0.5

–1.0

–20

–40

–60

245 Hz + 250 Hz sine wave conventional filtered-X

random noise conventional filtered-X

–1.5 0

20

40

60

80

100

–80 0

time, ms

200

400

600

800

1000

800

1000

frequency, Hz

a

a 1.5 20 1.0 0

magnitude, dB

amplitude

0.5

0

–0.5

–1.0

–20

–40

–60 245 Hz + 250 Hz sine wave fuzzy filtered-X

random noise fuzzy filtered-X

–1.5 0

20

40

60

80

100

–80 0

time, ms

200

Fig. 7

Composite narrowband noise cancellation

a Conventional method b Fuzzy method

In addition, broadband noise is also difficult to cancel by a conventional filtered-X algorithm, and the following examples demonstrate the ability of the proposed fuzzy filtered-X algorithm to cancel broadband noise. In the fourth simulation, broadband random noise is used as the undesired noise in the dynamic range of [1 1]. Figures 8a and 8b show the random noise and residual noise after noise cancellation in the frequency domain, respectively. The dashed line in Fig. 8a shows the undesired random noise and the solid line represents the noise cancellation results of the traditional filtered-X algorithm. It is clear that random noise almost cannot be cancelled by the traditional filtered-X method. Figure 8b illustrates the noise cancelling result using the fuzzy filtered-X algorithm, showing that the proposed method cancels undesired broadband noise at about 20–40 dB. In addition, the high-pass Butterworth filter, which acts as the speakers in ANC system, affects the performance of lower frequencies. When showing the noise cancelling performance in time domain, it is obvious that the proposed fuzzy filtered-X algorithm controls broadband noise very well. These are shown in Figs. 9a and 9b. The final example uses a combination of broadband and narrowband noise as the undesired noise, specifically a 420

400

600

frequency, Hz

b

b

Fig. 8

Broadband random noise cancellation, frequency domain

a Conventional method b Fuzzy method

periodic narrowband noise with frequency of 250 Hz combined with random noise. Figures 10a and 10b show these results in the frequency domain. The dashed lines in Figs. 10a and 10b show the undesired composite noise. The solid lines in Figs. 10a and 10b represent the residual noise after the conventional and fuzzy scheme controls. The performance can be divided into two parts. The narrowband noise can be cancelled by either the conventional or the fuzzy filtered-X method. However, the broadband random noise can be cancelled only by the proposed fuzzy filtered-X algorithm. It is also obvious that the proposed fuzzy method provides a better performance than the conventional scheme. Figures 11a and 11b show the noisecancelling performance of conventional and fuzzy schemes in time domain. These simulations confirm the performance of the proposed fuzzy filtered-X ANC system. From above simulation results, one can see that the conventional filtered-X algorithm can control the narrowband noise, but reduces the broadband noise only by about 3–5 dB. However, the proposed results still show that applying the fuzzy filtered-X method provides better noise reduction. The reduction of narrowband noise by the proposed method is over 70 dB and the reduction of broadband noise is up to 40 dB. It can be seen that not only IEE Proc.-Circuits Devices Syst., Vol. 150, No. 5, October 2003

2.0

1.5

1.5

1.0

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0.5 amplitude

amplitude

2.0

0 –0.5

0 –0.5

–1.0

–1.0

–1.5

–1.5 –2.0

–2.0 0

200

400

600

800

0

1000

200

400

Fig. 9

600

800

1000

time, ms

time, ms a

b

Broadband noise cancellation, time domain

80

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40

40

20

20

magnitude, dB

magnitude, dB

a Conventional method b Fuzzy method

0 –20

–20 –40

–40 –60

–60

250 Hz sine wave+random noise conventional filtered-X

0

200

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800

250 Hz sine wave+random noise fuzzy filtered-X

–80 0

–80

Fig. 10

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1000

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600

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frequency, Hz

a

b

800

1000

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2.0

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amplitude

a Conventional method b Fuzzy method

0 –0.5

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–1.0

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–2.0

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Fig. 11

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time, ms

a

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in the interests of convergence speed but also in the residual noise signal, the proposed fuzzy-based approach performs better than traditional ANC schemes. Compared with another existing ANC system [15, 16], the proposed method does not need complex computation and duct plant, and also the noise cancellation performance is also enhanced. In addition, due to its self-tuning capacity, the fuzzy filtered-X ANC system can be constructed easily at the beginning and the residual noise can be minimised in hundreds of milliseconds. 5

Conclusions

In the past few years, a variety of ANC algorithms have been proposed for cancellation of undesired. Most of the approaches need transfer functions such as duct plants, sensors and actuators to design the anti-noise filter. The proposed method uses fuzzy linguistic information to generate the ANC filter to cancel out the undesired noise. No transfer functions of duct plants, sensors and other actuators are needed to fulfil the proposed algorithm. The modified LMS algorithm is used to tune the variables of the 21st-order adaptive FIR filter, and hence only seven fuzzy rules are necessary. In addition, the initial variables can be set arbitrarily; the proposed algorithm is also convenient for other industrial applications. A method for restricting the residual noise to arbitrary accuracy is provided in this paper. The low convergence problem of the filtered-X algorithm is hence overcome. The numerical simulations show that the fuzzy adaptive filtered-X algorithm can cancel both broadband noise and narrowband noise, thus enhancing the ANC system. 6

7

References

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Acknowledgment

This work is supported by the National Science Council of the Republic of China under Grant NSC-91-2213-E-231007.

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