Functional link artificial neural network applied to active noise control of a mixture of tonal and chaotic noise

Applied Soft Computing 23 (2014) 51–60

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Applied Soft Computing journal homepage: www.elsevier.com/locate/asoc

Functional link artificial neural network applied to active noise control of a mixture of tonal and chaotic noise Santosh Kumar Behera a,b,∗ , Debi Prasad Das a , Bidyadhar Subudhi b a b

Process Engineering and Instrumentation Cell, CSIR-Institute of Minerals and Materials Technology, Bhubaneswar 751013, India Department of Electrical Engineering, National Institute of Technology, Rourkela, India

a r t i c l e

i n f o

Article history: Received 8 February 2013 Received in revised form 13 January 2014 Accepted 6 June 2014 Available online 13 June 2014 Keywords: Active noise control (ANC) Narrowband ANC Chaotic noise Hybrid ANC Functional link artificial neural network (FLANN)

a b s t r a c t Many practical noises emanating from rotating machines with blades generate a mixture of tonal and the chaotic noise. The tonal component is related to the rotational speed of the machine and the chaotic component is related to the interaction of the blades with air. An active noise controller (ANC) with either linear algorithm like filtered-X least mean square (FXLMS) or nonlinear control algorithm like functional link artificial neural network (FLANN) or Volterra filtered-X LMS (VFXLMS) algorithm shows sub-optimal performance when the complete noise is used as reference signal to a single controller. However, if the tonal and the chaotic noise components are separated and separately sent to individual controller with tonal to a linear controller and chaotic to a nonlinear controller, the noise canceling performance is improved. This type of controller is termed as hybrid controller. In this paper, the separation of tonal and the chaotic signal is done by an adaptive waveform synthesis method and the antinoise of tonal component is produced by another waveform synthesizer. The adaptively separated chaotic signal is fed to a nonlinear controller using FLANN or Volterra filter to generate the antinoise of the chaotic part of the noise. Since chaotic noise is a nonlinear deterministic noise, the proposed hybrid algorithm with FLANN based controller shows better performance compared to the recently proposed linear hybrid controller. A number of computer simulation results with single and multitone frequencies and different types of chaotic noise such as logistic and Henon map are presented in the paper. The proposed FLANN based hybrid algorithm was shown to be performing the best among many previously proposed algorithms for all these noise cases including recorded noise signal. © 2014 Elsevier B.V. All rights reserved.

Introduction In the modern day society, noise pollution is an important concern due to increased usage of noise polluting machineries. Excessive noise issue in industry, public places, transportation, and medical devices has motivated researchers for developing noise control devices. Passive enclosures and barriers are not effective for low frequency noises. Continuous exposure to high dB low frequency noise can lead to hearing loss, headache and other physiological diseases. Active noise control (ANC) is a technique which uses the principle of destructive interference to control the noise [1,2]. The ANC device is an electro-acoustic system which

∗ Corresponding author at: Process Engineering and Instrumentation Cell, CSIRInstitute of Minerals and Materials Technology, Bhubaneswar 751013, India. Tel.: +91 9438182457. E-mail addresses: [email protected], santosh behera [email protected] (S.K. Behera), debi das [email protected] (D.P. Das), [email protected] (B. Subudhi). http://dx.doi.org/10.1016/j.asoc.2014.06.007 1568-4946/© 2014 Elsevier B.V. All rights reserved.

can generate antinoise by a loudspeaker which is of same amplitude and opposite phase of the offending noise. This principle of noise control has been a topic of research for quite sometimes where novel algorithms are being developed to combat various implementational issues and also to improve the noise controlling performance. The ANC system generates the antinoise by a controller which drives an electro-acoustic device to control the noise. The controller can be adaptive or fixed. The adaptive controllers are effective for dynamic environment and noise characteristics. The ANC is expected to be used as a tool to control noise pollution in industry and residential areas due to rapid urbanization and growing demand of manufacturing. Noise is generally produced by motors, compressors, blowers, machines, transformers, etc. Based on the frequency content, the noise is classified as tonal, narrowband or broadband. Noise is also classified as uniform or Gaussian based on its statistical property. Another type of noise which has gained research interest these days is the chaotic noise [3]. The chaotic noise is a nonlinear deterministic noise, whereas, the tonal is a linearly predictable one. The chaotic noise can be nonlinearly

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predictable. The chaotic noise is neither tonal nor random. The chaotic noise is generated from a dynamic systems like rotating machines, fans and airfoils [3–5]. The ANC algorithms are applied to varieties of noises such as tonal, broadband and chaotic type. The tonal noise, which consists of discrete frequency components are easier to control using adaptive technique and shows greater degradation of noise intensity. However, a practical noise consists of tonal components and broadband components. For example, a water pump, fan or grinder has a motor rotating at a particular speed generates the tonal components and the interaction of the rotating part with the axis and air generates broadband noise. Therefore, the noise generated is a mixture of tonal and broadband noises. The broadband noise can be either random or chaotic. As discussed earlier, the chaotic noise is a nonlinearly deterministic noise and a nonlinear predictive controller is essential for this. The linear controller such as the filtered-X last mean square (FXLMS) algorithm is sufficient for the random broadband noise and is not effective for nonlinear noise such as chaotic noise [1]. The FXLMS algorithm based ANC for controlling stochastic broadband noise does not show great performance due to band limited hardware systems. However, due to deterministic property of chaotic broadband noise, many nonlinear ANC algorithms have been proposed in recent past which show better performance over the FXLMS algorithm for chaotic noises [5–11]. Nonlinear active noise controllers are mostly based on the soft-computing tools like different versions of neural network for noise and vibration control. Neural network [12–17] and radial basis function based neural network [5] have been used for active control of nonlinear noise/vibration. Volterra series expansion based ANC algorithms [6,8,11] are also popular due to its low complexity. Functional link neural network (FLANN) [18] based filtered-s LMS algorithm (FSLMS) was initially proposed in Ref. [7]. This was shown to be very simple and computationally and performance wise efficient. Extension of this FLANN based FSLMS algorithm can be seen for multichannel ANC [9], frequency domain implementation [19], virtual ANC in [3,20–22]. Recently, works have been carried out to develop better nonlinear ANC algorithms using the Volterra filter and FSLMS algorithm [23–25]. Combination of both Volterra filter and FLANN is proposed in [25] for effective cancelation of nonlinear noise processes. Other competitive nonlinear control algorithms are based on the usage of polynomial Nonlinear AutoRegressive models with eXogenous variables (NARX) models [26], adaptive bilinear filter [27] etc. In addition to these algorithms, other nonlinear prediction and control algorithms using softcomputing tools [28–31] can be used for controlling chaotic noise. All these papers attempted to control only the nonlinear noise processes with primary path nonlinearity, secondary path nonlinearity or the chaotic noise input with non-minimum phase secondary path. They have not tried their algorithms for a mixture of tonal and chaotic noises. The previously proposed nonlinear adaptive ANC controllers show degraded convergence performance when a mixture of tonal and chaotic noise is to be controlled. Recently, a feedforward hybrid ANC algorithm is proposed where a mixture of tonal and broadband noise is considered [32]. This has been the basis of our present study. In this paper, the authors have proposed a new feedforward hybrid ANC algorithm which consists of several subsystems such as a sinusoidal noise canceller and two separate ANC controllers (for broadband and narrowband). This new ANC system has been shown to be effectively canceling the mixture of narrowband and broadband noise. The broadband controller used in this paper uses a finite impulse response (FIR) filter as the controller. The FIR controller is adapted using the FXLMS algorithm. This controller is a linear controller and does not have capability to model and predict the nonlinear noise process such as chaotic noise and hence its control performance is

degraded when it is applied to control a mixture of chaotic and tonal noise. Therefore, the present paper proposes a nonlinear hybrid ANC algorithm which provides superior noise canceling performance for a mixture of tonal and chaotic noise compared to either the single nonlinear controller [7] or the hybrid controller [32]. The organization of the rest of the paper is as follows. Section “Chaotic noise” details about the characteristics of various noises with emphasis on chaotic noise. The proposed nonlinear hybrid algorithm is presented in Section “Proposed nonlinear hybrid ANC algorithm”. The computational complexity analysis is presented in Section “Computational complexity analysis”. The simulation results are presented in Section “Simulation study”. Section “Conclusions” concludes the paper. Chaotic noise The noise generated from different noise polluters is not simple sinusoidal waves and they belong to nonlinear acoustics [33]. The nonlinear acoustic waves may not show a simple combination of discrete frequencies but have a complicated dynamic characteristic resembling the stochastic behavior. This type of acoustic noise is also called deterministic chaos or acoustic chaos as defined in [33]. In this paper, we call these types of noise as the chaotic noise which is different from both tonal and random noises. The chaotic noise is mostly broadband and nonlinearly deterministic. There are many methods to know whether a noise is chaotic or not from discrete samples [34]. To get an initial idea about chaotic nature of the noise, a set of noise samples are plotted with respect to its delayed samples [3]. The chaotic noise shows dense periodic orbits in such a plot. The density of periodic orbits means that every point in the space is approached arbitrarily closely by periodic orbits. There are many types of chaotic noises such as logistic, Lorenz, Duffing and quadratic [35]. The noise generating from motors and fans has both tonal and chaotic components. The tonal component is related to the rotating speed of the fans or motors and the chaotic components may be due to the airfoil noise. The frequency spectrum of noise polluters such as a vacuum cleaner and a grinder used in household work is shown in Fig. 1. Both the spectrum clearly shows that these noises consist of discrete tonal components and other broadband noises. These broad band noises are assumed to be chaotic in nature. Many synthetic chaotic time series can be generated to simulate the proposed algorithm. The logistic chaotic noise can be generated by the following c(n) = c(n − 1)[1 − c(n − 1)]

(1)

where  = 4 and c(0) = 0.9, n is the time/sample index. The Lorentz chaotic map can be generated by solving the following equations dx = s(y − x) dt

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

dy = rx − y − xz dt ⎪ ⎪

dz = xy − bz dt

(2)

⎪ ⎪ ⎪ ⎭

where x, y and z are the chaotic time series with s = 16, r = 45.92 and b = 4; The initial values of x, y and z are chosen as 0.1 each. This can be generated using Matlab software [35]. The Henon map is simulated by solving

⎫

dx ⎬ = 1 − ax2 + y ⎪ dt dy = bx dt

⎪ ⎭

(3)

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Fig. 2. Random stochastic, tonal and chaotic noise in frequency domain showing chaotic noise as a broadband noise.

Fig. 1. Frequency spectrum of noise polluters (a) vacuum cleaner and (b) grinder.

where x and y are the chaotic time series. The parameters a = 1.4 and b = 0.3. The initial values of x and y are chosen as 0.1 each. This can be generated using Matlab software [35]. The composite signal which is a mixture of tonal/periodic noise and the broadband chaotic noise is generated as follows. xm (n) =

H−1  h=0

Ah sin

 2f n  h fs

+ ch(n).

(4)

where fh is the hth frequency, fs is the sampling frequency, Ah is the amplitude of hth component, n is the sample index and ch(n) is the chaotic noise. In this paper, both logistic and Henon chaotic maps are used to evaluate the performance of the proposed algorithm and some previously proposed algorithms. The phase plots of tonal, chaotic and random noises are presented in [3]. Fig. 2 shows the frequency spectrum of a tonal, logistic chaotic and random stochastic signal. This shows that the logistic chaotic noises are broadband in nature. Proposed nonlinear hybrid ANC algorithm To improve the noise canceling performance of the ANC for a mixture of chaotic and tonal noise, a new nonlinear hybrid ANC algorithm is proposed in this section. In this ANC, there are

three subsystems i.e. sinusoidal synthesizer, a nonlinear and a narrowband controller. Sinusoidal synthesizer estimates the narrowband component which is then subtracted from the reference signal to estimate the chaotic component in the noise. The tonal noise is controlled by a narrowband ANC and the chaotic noise is controlled by a nonlinear ANC which works parallel to each other. The proposed algorithm assumes that the tonal and chaotic components are separated by a method. Separation of the narrowband noise from the broadband can be done by either filtering or by the time domain subtraction; out of which the later is a better method. The proposed nonlinear ANC uses the adaptive method to estimate the tonal noise by waveform synthesis method. This assumes that the frequency of the tonal noise is known and the method will adjust the phase and the amplitude adaptively. Let us define the input signal xm (n) which is a mixture of both tonal and chaotic noise xm (n) = xp (n) + xc (n),

(5)

where xp (n) is the periodic/tonal component and xc (n) is the chaotic component. The first objective of this algorithm is to remove the sinusoidal/periodic/tonal component of the noise from xm (n). The adaptively estimated periodic component xˆ p (n) is subtracted from the received reference signal xm (n) to estimate the chaotic component of the noise xˆ c (n) as follows xˆ c (n) = xm (n) − xˆ p (n).

(6)

The periodic component is estimated as follows

xˆ p (n) =

H−1 

[ach (n) cos(ωh n) + bch (n) sin(ωh n)],

(7)

h=0

where H is the total number of harmonics or individual frequency components present in the periodic noise. ωh = 2fh /fs , where fh is the frequency of the hth frequency component and fs is the sampling frequency. ach (n) and bch (n) are adaptive coefficients which are adapted as follows ach (n + 1) = ach (n) + c cos(ωh n)ˆxc (n),

(8)

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bch (n + 1) = bch (n) + c sin(ωh n)ˆxc (n),

(9)

where c is the step-size of the adaptive sinusoidal cancelation subsystem which adaptively subtract the sinusoidal components from the mixture xm (n) to get broadband chaotic signal estimate xˆ c (n). The ANC control output y(n) is a combination of two separate controllers: broadband and narrowband ANC controllers and are represented as follows y(n) = yc (n) + yp (n).

(10)

The estimated chaotic signal xˆ c (n), is passed through a filterbank structure of FLANN based ANC [9,11] to generate the broadband control signal yc (n) and the frequency synthesis method [32] using the synchronization signal is used to generate the narrowband control signal yp (n). The broadband control signal, yc (n), is the sum of the output of all the adaptive sub-filters, which is computed as



2P+1

yc (n) =

yc,i (n),

(11)

i=1

where the output of ith adaptive sub-filter of FLANN filter bank is yc,i (n) = sTi (n)wi (n),

(12) T

and wi (n) = [wi,0 (n) wi,1 (n). . .wi,N−1 (n)] , i = 1, . . ., 2P + 1, is an Npoint coefficient vector of the ith controller at time n. Here, N represents the memory size, P is the order of the functional expansion. The input signal vector of the adaptive filter is represented as si (n) = [si (n)si (n − 1). . .si (n − N + 1)]T ,

(13)

which contains N recent samples of the functionally expanded reference input xˆ c (n). Here for 1 ≤ l ≤ P,

si (n) =

⎧ xˆ (n), ⎪ ⎨ c ⎪ ⎩

if i = 1

sin[lˆxc (n)],

if i is even .

cos[lˆxc (n)],

if i is odd

(14)

(15)

where i denotes the step-size which controls the convergence speed of the FSLMS algorithm. The filtered reference signal vector, si (n) = [si (n)si (n − 1), . . ., si (n − N + 1)]T , is formed by filtering ˆ as si (n) through the secondary path estimate transfer function S(z) follows ˆ si (n) = S(z)s i (n).

(16)

The narrowband/tonal control signal yp (n) is generated as follows, where it is assumed that the frequencies ωh ’s are known. H represents total number of sinusoidal components in the narrowband signal. The narrowband signal sample are online computed as follows yp (n) =

H−1 

[ah (n) cos(ωh n) + bh (n) sin(ωh n)].

(17)

h=0

The adaptive coefficients, ah (n) and bh (n) are adapted as follows 

ah (n + 1) = ah (n) + p xhc (n)e(n), s

bh (n + 1) = bh (n) + p xh (n)e(n),

 c ˆ xhc (n) = S(z)X (z), h 

s ˆ xhs (n) = S(z)X (z), h

(18) (19)

(20) (21)

where Xhc (z) and Xhs (z) are the Z transform of xhc (n) and xhs (n) with xhc (n) = cos(ωh n) and xhs (n) = sin(ωh n). The right hand side of (20) and (21) represent the filtering of xhc = cos(ωh n) and xhc = sin(ωh n) respectively through secondary path estimate ˆ S(z). The complete algorithm is shown in Fig. 3(a). The block diagram of the complete system is presented in Fig. 3(b) with the proposed algorithm in the shaded portion. The algorithm takes the reference and error microphone signals as input and generates antinoise through control loudspeaker. In comparison to the work published in Ref. [32] the present algorithm has an extra step (Step-5) where the reference chaotic signal is functionally expanded and Step 6 is an FIR filter operation with xˆ c (n) as its input. Similarly, in conventional FLANN [7] operations in Steps2, 3, 4, 7, 8 and the tonal part of algorithm in Step-10 was not involved. The reference signal xm (n) is directly used in Step-5 in place of xˆ c (n). Power series expansion or truncated Volterra series expansion In Volterra series expansion [11], the input signal is expanded as power series and product of present and delayed inputs. To use the Volterra series expansion in the above proposed algorithm, one can choose truncated version where the cross-terms (product of dissimilar delayed inputs) are removed from the Volterra series making it a power series expansion. The power series expansion of the estimated broadband chaotic reference signal xˆ c (n) replaces the sin–cos blocks of Fig. 3(b) and is shown in Fig. 4. In this algorithm Eq. (14) is replaced by i

The weight parameters of the nonlinear controller, wi (n), are updated using the error microphone signal e(n). The weight update equation is as follows: wi (n + 1) = wi (n) − i e(n)si (n),

where p is the step-size of the adaptive narrowband controller.   xhc (n) and xhs (n) are the filtered cosine and sine components of each harmonics, h and are computed as follows

si = (ˆxc ) ,

(22)

where the power i = 0, 1, 2, . . ., P. si (n) is nothing but the ith power of the xˆ c (n). P is the order of the truncated Volterra filter controller.

Computational complexity analysis The computational complexity of the proposed algorithm is slightly more than the hybrid ANC algorithm proposed in Ref. [4]. The complexity of the sinusoidal synthesizer and the narrowband controller are same in both these algorithms but the computational complexity of the broadband controller using FLANN is proportional to the total number of expansion. In Pth order FLANN with trigonometric expansion, there are M = 2P + 1 number of functional expansions of the broadband reference signal xˆ c (n). If N is the number of the filter coefficients in each filters of the filter bank, then there are NM multiplications to generate the broadband output component. In addition to this, each of these functionally expanded signals is filtered by the secondary path estimate. If the length of the secondary path estimate is L, then the total multiplication requirement for filtering through secondary path estimate is ML. If there are H sinusoidal components present in the noise then the requirement of total number of multiplication is 2H(L + 4)(2P + 1)(2N + L),

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Fig. 3. Proposed hybrid ANC algorithm with FLANN (trigonometric expansion) (a) algorithm steps and (b) block diagram.

total number of addition is 2H(2H − 1)(L + 2) + (2P + 1)(2N + L − 1) and total number of sin/cos computation is 4H + 2P for the proposed algorithm. Simulation study Simulation experiments are conducted to evaluate the performance of the proposed nonlinear ANC algorithm compared to the recently proposed hybrid controller [32] along with the standard FXLMS [1] and FSLMS algorithms [7] to control the noise which consists of both tonal and chaotic noises. In all these experiments, the primary and secondary paths are taken as P(z) = z−4 + 0.5z−5 − 0.3z−6 + 0.2z−7 and S(z) = z−2 + 1.5z−3 − z−4 . The secondary path is a non minimum phase system. Experiment 1: Absolutely separated tonal and chaotic noise This experiment was performed to evaluate the performance of the hybrid controllers (with absolutely separated tonal and broadband noise inputs) compared to the solo controllers (with noise mixture input) for controlling noise mixture consisting of both tonal and broadband components. Therefore, the FXLMS algorithm is first used to control the mixture of tonal (with one tone of frequency 500 Hz) and the logistic chaotic noise as given in (1). This is referred as FIR-FXLMS in Fig. 5 as the FIR filter is the main control

architecture. It was found to show very poor performance. Next, the FSLMS algorithm [7] is used with the noise mixture as the reference signal. This is shown as FLANN-FSLMS in Fig. 5 as FLANN is the control architecture. The performance of FLANN-FSLMS was found to be better than the FIR-FXLMS. Then the periodic signal is fed to the FIR controller and chaotic signal is fed to the Volterra or FLANN controller separately. The results for these two proposed algorithms named as FIR(P)-FLANN(C) and FIR(P)-Volterra(C) are also shown in compassion to FIR-FXLMS and FLANN-FSLMS algorithms in Fig. 5. In this experiment, the separation of periodic and the chaotic signal are assumed to be 100%. It was found that the convergence performance of FIR(P)-FLANN(C) is much better than other controllers which is about 27 dB and 10 dB down compared to the linear and nonlinear controllers respectively (in Fig. 5). This experiment shows that instead of the mixture signal, if the separated signals are fed to two separate controllers, the noise cancelation performance is improved. Such controllers are named as hybrid controllers. In addition to this, for the chaotic noise case, the nonlinear controller such as FLANN or Volterra controllers work better than the linear controller like FIR filter. Experimental conditions are provided in Table 1. Experiment 2: Adaptively separated tonal and chaotic noise In this experiment, the narrow band adaptive signal synthesizer is used to separate the chaotic and the periodic signal (one tone).

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Fig. 4. Proposed hybrid ANC algorithm with truncated Volterra filter.

Table 1 Conditions used in Experiment 1. Algorithm

Input

Control structure

Step-size

N/P

FIR-FXLMS

xm (n)

FIR

 = 0.0002

10/0

FLANN-FSLMS

xm (n)

FLANN

0 = 0.0008 1 = 0.0004

10/1

FIR(P)-FSLMS(C)

xp (n) xc (n)

FIR FLANN

p = 0.001 0 = 0.001 1 = 0.0008

10 10/1

FIR(P)-VFXLMS(C)

xp (n) xc (n)

FIR Volterra

p = 0.001 0 = 0.00081 = 0.00001

10 10/3

The periodic signal is subtracted from the noise mixture being generated by a frequency synthesis method and the chaotic signal is passed through FIR filter, FLANN and Volterra filter separately. In this manner FXLMS, FSLMS or VFXLMS algorithms are used to control the chaotic part of the noise with the control structure as FIR, FLANN or Volterra filter, respectively. The smoothed square error in dB is plotted in Fig. 6. It was found that the FLANN based hybrid controller with frequency synthesis and online noise separation outperforms the rest two algorithms. Experimental conditions for these experiments are provided in Table 2. The power spectral density of the noise when ANC is OFF and when ANC (using FXLMS, FSLMS or VFXLMS) is ON is plotted for single tone and multi tones in Fig. 7(a) and (b) respectively.

Experiment 3: Effect of change in primary path In practical situations, the primary path, which is the acoustic environment between the noise source and the point of noise cancelation, changes with respect to time. To study the tracking behavior of the proposed hybrid nonlinear algorithm to the change in primary path transfer function this experiment was conducted. In this experiment, the primary path was considered as used in the previous experiments and is abruptly changed to P(z) = z−4 − 0.5z−5 + 0.3z−6 + 0.2z−7 after 5000 iterations. It was found from Fig. 8 that the error signal in proposed hybrid algorithms converged to its minimum level after few iterations of the occurrence of the change. Out of the three algorithms, the FLANN based

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Table 2 Conditions used in Experiment 2. Algorithm FXLMS

VFXLMS

FLANN

Purpose

Input

Control structure

Step-size

N/P

Sinusoidal noise canceller

sin[ωh (n)] cos[ωh (n)]

Multiplication coefficients

c = 0.01

–

Broadband ANC

xˆ c (n)

FIR

0 = 0.001

10

Narrowband ANC

sin[ωh (n)] cos[ωh (n)]

Multiplication coefficients

p = 0.0008

–

Sinusoidal noise canceller

sin[ωh (n)] cos[ωh (n)]

Multiplication coefficients

c = 0.01

–

Broadband ANC

xˆ c (n)

Volterra

0 = 0.0011 = 0.00001

10/3

Narrowband ANC

sin[ωh (n)] cos[ωh (n)]

Multiplication coefficients

p = 0.0008

–

Sinusoidal noise canceller

sin[ωh (n)] cos[ωh (n)]

Multiplication coefficients

c = 0.01

–

Broadband ANC

xˆ c (n)

FLANN

0 = 0.001 1 = 0.0008

10/1

Narrowband ANC

sin[ωh (n)] cos[ωh (n)]

Multiplication coefficients

p = 0.01

–

Fig. 5. Convergence performance comparison of conventional and hybrid algorithms assuming separate periodic and chaotic noise sources are available.

Fig. 6. Convergence performance comparison of three hybrid algorithms where the chaotic signal is controlled by FXLMS, Volterra and FLANN algorithms. The periodic and chaotic noise are separated online.

Fig. 7. PSD of the noise with and without ANC (a) single tone and (b) multi-tone.

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Fig. 8. Convergence performance with primary path change at 5000 iterations.

algorithm outperforms both FXLMS and the Volterra filter based algorithms. Experiment 4: Effect of change in the relative amplitude of tonal components In certain practical situations, the tonal components of the periodic noise vary with respect to their relative amplitudes. It is expected that the adaptive algorithm should be able to adapt its coefficients to cope with such change in relative amplitudes. In this experiment the amplitude of 500 Hz, 1000 Hz and 1500 Hz was initially taken same and equal to 1 till 5000 iterations and after that the amplitude ratio was chosen as 1:2:3. The convergence of the smoothed mean square error in dB is plotted in Fig. 9 for all the three algorithms. Out of the three algorithms, the proposed FLANN based hybrid algorithm showed the best performance. Experiment 5: Effect of error in separation of tonal component from noise mixture To study the effect of the error in separation of tonal component from the noise mixture to get the chaotic component on the overall performance of the ANC mean square error, this experiment is conducted. The adaptive waveform synthesis method as shown in Fig. 3(b) is used for separation. The step size c used in Eqs. (8) and (9) controls the speed of convergence. Two separate simulations were performed with c = 0.01 (fast separation)

Fig. 10. (a) Comparative plot of mean square error of the chaotic separation from the mixture of noise and the overall ANC performance for two different step size of adaptive separation. (b) Time domain plots of samples of noise mixture before and after separation.

and c = 0.001 (slow separation). The fast separation convergences faster with a higher steady state mean square error. Where as the slow separation has a lower steady state mean square error with relatively slower convergence. The comparative plots of separation error and the overall ANC error are shown in Fig. 10(a). Fig 10(b) shows the time domain signal samples for different acoustic noises such as mixture of chaotic and tonal noise, only chaotic noise, estimated chaotic noise (adaptively separated chaotic noise) and the error in separation. This shows that the adaptive tonal component separation based on a waveform synthesis method is capable of separating the tonal component from the mixture of chaotic and tonal noise. This experiment also shows that the slower convergence in separation leads to better ANC performance. Experiment 6: Recorded noise

Fig. 9. Convergence performance with change in amplitude of harmonics at 5000 iterations.

This experiment was conducted by taking a recorded noise from a home grinder. The sampling frequency was 8 kHz. First the frequency spectrum was plotted using fast Fourier transform (FFT) to know the discrete frequency content. As seen in Fig. 1(b), there are many discrete frequencies such as 236 Hz and 2118 Hz. These two frequencies are removed from the composite signal and the proposed algorithm is applied. The whole signal is used while simulating FLANN (Das and Panda [7]). However, the separated signals are used in hybrid algorithms of [32] and the proposed one. It is found in Fig. 11 that the proposed FLANN based hybrid controller out perform the other two algorithm in term of faster convergence and minimum mean square error. In addition to this, the frequency spectra of the noise before and after cancelation at different frequency ranges are shown in Fig. 12. In this study, either single

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Table 3 Conditions used in Experiment 7. Algorithm

Purpose

Input

Control Structure

Step-size

N/P

FIR-FXLMS

ANC

xm (n)

FIR

 = 0.006

10/0

FLANN-FSLMS

ANC

xm (n)

FLANN

0 = 0.0061 = 0.002

10/1

Sinusoidal noise canceller

sin[ωh (n)] cos[ωh (n)]

Multiplication coefficients

C = 0.003

–

Broadband ANC

xˆ c (n)

FIR

0 = 0.006

10

Narrowband ANC

sin[ωh (n)] cos[ωh (n)]

Multiplication coefficients

p = 0.001

–

Sinusoidal noise canceller

sin[ωh (n)] cos[ωh (n)]

Multiplication coefficients

c = 0.003

–

Broadband ANC

xˆ c (n)

Volterra Filter

0 = 0.0061 = 0.004

10/3

Narrowband ANC

sin[ωh (n)] cos[ωh (n)]

Multiplication coefficients

p = 0.001

–

Sinusoidal noise canceller

sin[ωh (n)] cos[ωh (n)]

Multiplication coefficients

c = 0.003

–

Broadband ANC

xˆ c (n)

FLANN

0 = 0.0061 = 0.002

10/1

Narrowband ANC

sin[ωh (n)] cos[ωh (n)]

Multiplication coefficients

p = 0.001

–

Hybrid FIR

Hybrid VFXLMS

Hybrid FLANN

Fig. 11. Mean square error with recorded noise.

Fig. 13. Mean square error with Henon chaotic noise mixture.

Experiment 7 frequency 236 Hz or double frequency 236 Hz and 2118 Hz were separated and the responses are plotted in Fig. 12 as single frequency separator (FS) and double FS. In addition, the residual noise using conventional FLANN and making ANC off was also plotted for comparison. It was seen that the proposed nonlinear hybrid controller with either single or double FS shows superior performance to the conventional FLANN controller.

In this experiment the Henon type chaotic map as shown in Eq. (3) is used to generate the chaotic noise. The tonal signal of frequency 200 Hz, amplitude 0.5 sampled at 1000 Hz and the chaotic noise with power 0.04 were mixed for generating the mixture noise for this simulation. The FXLMS [1], FSLMS [7] the hybrid FIR [32] and the two proposed nonlinear hybrid structures were simulated. The

Fig. 12. Frequency spectrum of noise before and after cancelation (a) 150–500 Hz, (b) 900–1500 Hz, (c) 1550–1800 Hz.

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input signal, corresponding control structure and step sizes used in different algorithms used in this experiment are presented in Table 3. It is shown in Fig. 13 that the for Henon type of chaotic noise also, the proposed FLANN based hybrid structure shows the best performance among all the algorithms. Conclusions In this paper, it has been shown that many practical noise sources are a mixture of tonal and chaotic noise. Many conventional ANC algorithms use this noise mixture as the reference signal to generate the antinoise, which have suboptimal performance. The present paper proposes an algorithm which first separates the chaotic signal from the noise mixture and the estimated chaotic signal was used in a FLANN/Voterra based nonlinear controller. The tonal component uses a frequency synthesis method to control the tonal part. Therefore, this algorithm uses a narrow band controller and a broadband controller. Hence it is called as hybrid controller. The proposed hybrid nonlinear controller shows greater noise reduction capability compared to many previously published algorithms such as FXLMS, FSLMS and hybrid ANC. Exhaustive computer simulation experiments with single and multitone signal mixed with synthetic chaotic signal (logistic and Henon map) were performed. In addition to this, a recorded grinder noise was used in the simulation study to show the efficacy of the proposed algorithm. Acknowledgements Authors acknowledge CSIR-IMMT for funding this research work under OLP-49 project and CSIR for funding YSP-3/2013 project to carry out part of this work. Authors also acknowledge the anonymous reviewers for their valuable suggestions which helped in improving the quality of the paper. References [1] S.M. Kuo, D.R. Morgan, Active noise control: a tutorial review, Proc. IEEE 87 (6) (1999) 943–973. [2] C.H. Hansen, S. Snyder, Q. Xiaojun, L. Brooks, D. Moreau, Active Control of Noise and Vibration, 2nd ed., CRC Press, Florida, 2013. [3] D.P. Das, D.J. Moreau, B.S. Cazzolato, A nonlinear active noise control algorithm for virtual microphones controlling chaotic noise, J. Acoust. Soc. Am. 132 (2) (2012) 779–788. [4] S.B. Behera, D.P. Das, N.K. Rout, Nonlinear feedback active noise control for broadband chaotic noise, Appl. Soft Comput. 15 (2014) 80–87. [5] P. Strauch, B. Mulgrew, Active control of non linear noise processes in a linear duct, IEEE Trans. Signal Process. 46 (1998) 2404–2412. [6] L. Tan, J. Jiang, Adaptive Volterra filters for active control of nonlinear noise processes, IEEE Trans. Signal Process. 49 (8) (2001) 667–1676. [7] D.P. Das, G. Panda, Active mitigation of nonlinear noise processes using a novel filtered-s LMS algorithm, IEEE Trans. Speech Audio Process. 12 (3) (2004) 313–322. [8] G.L. Sicuranza, A.C. Carini, Filtered-X affine projection algorithm for multichannel active noise control using second-order Volterra filters, IEEE Signal Process. Lett. 11 (11) (2004) 853–857.

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