A new adaptive strategy to improve online secondary path modeling in active noise control systems using fractional signal processing approach

Signal Processing ] (]]]]) ]]]–]]]

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A new adaptive strategy to improve online secondary path modeling in active noise control systems using fractional signal processing approach Muhammad Saeed Aslam a, Muhammad Asif Zahoor Raja b,n a b

Pakistan Institute of Engineering and Applied Sciences, Islamabad, Pakistan COMSATS Institute of Information Technology, Attock Campus, Attock, Pakistan

a r t i c l e i n f o

abstract

Article history: Received 31 December 2013 Received in revised form 6 April 2014 Accepted 7 April 2014

In this paper, a new adaptive strategy based on fractional signal processing is proposed using multi-directional step size fractional least mean square algorithm for online secondary path modeling, which is a fundamental problem in practical active noise control systems, as opposed to the generally-employed increasing step size strategy that compromises model accuracy for faster convergence. The design approach presents step size strategy in relation with disturbance signal in the desired response of modeling filter which is not available directly so an indirect approach is used to track its variations. Comparative results for narrowband and broadband noise signals show that the proposed technique outperforms other state-of-the-art methods in terms of model accuracy and convergence rate. & 2014 Elsevier B.V. All rights reserved.

Keywords: Modified filtered-x least mean square (FxLMS) Active noise control Fractional least mean square (LMS) Online secondary path modeling

1. Introduction Application of fractional calculus to signal processing has been found in a diverse field of applied science and engineering. For example, fractional Brownian motion [1], description of damping involving fractional operators [2], fractional system identification for lead acid battery state charge estimation [3], continuous-time fractional linear systems [4], transfer function identification from frequency response [5], parameter estimation of input nonlinear control autoregressive system [6] and so on. Introductory material on subject term of fractional signals processing is given in [7,8]. Beside this, the dedicated special issues of reputed journals on fractional signal processing and its applications are also published [9,10]. Moreover, recently, many authors are attracted to the applications of fractional dynamics to signal process such as

n

Corresponding author. E-mail addresses: [email protected] (M.S. Aslam), [email protected] (M.A.Z. Raja).

fractional time integral approach to image structure denoising [11] and Design of adjustable fractional order differentiator [12]. It motivates the author to investigate in applications of fractional signal processing specially in the field of active noise control system. Acoustic noise becomes a serious problem with the extensive use of industrial equipments and this issue can be addressed by active and passive methods. Active methods outperform passive methods by being more effective at low frequencies and able to block noise selectively. After emergence of successful applications of active noise control (ANC) systems in medical instruments and consumer electronics in the recent years [13–15], interest in this field has grown rapidly. ANC system is applied for mitigation of unwanted signals in scenarios involving small axial flow fans in [16,17], privacy-phone handsets [18], neonatal intensive care units [19], magnetic resonance imaging [20], motorcycle helmets [21] and compressor of an actual heating, ventilation, and air-conditioning system [22]. Development, theoretical performance analysis and realtime experimentation of more effective ANC algorithms are

http://dx.doi.org/10.1016/j.sigpro.2014.04.012 0165-1684/& 2014 Elsevier B.V. All rights reserved.

Please cite this article as: M.S. Aslam, M.A.Z. Raja, A new adaptive strategy to improve online secondary path modeling in active noise control systems using fractional signal processing approach, Signal Processing (2014), http://dx.doi.org/ 10.1016/j.sigpro.2014.04.012i

M.S. Aslam, M.A.Z. Raja / Signal Processing ] (]]]]) ]]]–]]]

2

open for further research and point of interest in present studies. It would be interesting to investigate, whether we can get better performance by employing fractional least mean square (LMS) based ANC system with a step size parameter that increases from small value to ensure fast stable convergence and then decreases from a certain high value to account for minor adjustments. 2. An overview of active noise control system and problem statement ANC systems work on the principle of destructive interference between sound fields for reducing unwanted sound (primary noise) [23]. Essentially, primary noise is cancelled around the location of error microphone by generating canceling noise [24]. A single-channel feedforward ANC system is shown in Fig. 1. In this system, one reference microphone is used to pick up reference signal x (n), one error microphone to pick up the residual noise e (n) and a speaker to propagate canceling signal y(n) generated by adaptive noise control filter W(n). A common adaptation algorithm for ANC systems is filtered-x least mean square (FxLMS) algorithm [13], which is a modified version of LMS algorithm [25]. Here x(n) is filtered through an estimated secondary path Ŝ(n) before being supplied to LMS algorithm for weight updation. This filtering is performed to compensate for the effect of secondary path. The FxLMS algorithm is fairly robust to the modeling errors between the secondary path and the modeling filter [26]. Secondary path may be estimated offline (when primary noise is absent) prior to the operation of ANC system, but presence of fixed modeling errors (as in case of offline modeling), FxLMS converges to a biased solution [27]. Online identification of the secondary path characteristics is preferred to ensure the stability of FxLMS and maintain the noise reduction performance while tracking the variations of secondary path due to the aging of components, thermal variations and environmental modifications [28,29]. Normally, two different approaches are adopted for the secondary path modeling. A first approach involves the injection of additional random noise into the ANC system, thus utilizing a system identification method to model the

secondary path S(n) [30–35]. Second approach models it directly from the output y(n) of the control filter W(n). A detailed comparison of these two approaches is given in [36], which concludes that first approach is superior to second approach on convergence rate, updating duration, speed of response to changes of primary noise and computational complexities. However, the injected auxiliary noise damages the noise cancellation performance by contributing to the residual noise and it is preferable to keep it as low as possible. Eriksson et al. [30] proposed the basic additive random noise technique for online secondary path modeling. As shown in Fig. 2, there are two adaptive filters in this ANC system: FxLMS algorithm-based noise control filter W(n), and the LMS algorithm-based secondary path modeling filter S(n). The system of [30] suffers from slow convergence and the low estimation accuracy of these filters. Signal at the error microphone of this system has two components: the auxiliary noise filtered by the secondary path and the residual noise of the ANC system. First component disturbs the control filter adaptation, while the second component disturbs the secondary path modeling. Third adaptive filter is used to address this problem in [31–33]. In these improved methods, a third adaptive filter is employed to improve the convergence speed and accuracy of the secondary path modeling by removing the residual noise from the error signal of the secondary path modeling filter, thus acting as a noise suppressor. In [24], a cross-update strategy is also employed for removing the auxiliary noise from the error signals of the control filter and noise suppressor. More recently, improved performances were obtained with the ANC system proposed in [35]. This ANC system uses only two adaptive filters, one for adapting the control filter and other secondary path modeling, but an improved convergence speed of the control filter is obtained by introducing the delay compensation scheme of [37]. In FxLMS algorithm, delay introduced by the secondary path slows down the convergence and reduces the upper bound for the step size [23]. Two fixed filters are introduced in modified FxLMS

Noise Source

e(n) S(n)

d(n)

P(n)

Noise Source

d(n)

P(n)

y’(n)-v’(n)

e(n) x(n)

S(n)

W(n)

Ŝ(n)

x(n)

y’(n)

y(n)

W(n)

y(n)

Ŝ(n)

LMS

LMS

f(n)

v(n) White Noise

x’(n)

x’(n)

v’(n)

Ŝ(n)

LMS

FxLMS Algorithm

FxLMS Algorithm

Fig. 1. Feedforward ANC system.

Fig. 2. ANC system of Fig. 1 with online secondary path modeling (Eriksson's method).

Please cite this article as: M.S. Aslam, M.A.Z. Raja, A new adaptive strategy to improve online secondary path modeling in active noise control systems using fractional signal processing approach, Signal Processing (2014), http://dx.doi.org/ 10.1016/j.sigpro.2014.04.012i

M.S. Aslam, M.A.Z. Raja / Signal Processing ] (]]]]) ]]]–]]]

d(n)

P(n)

Noise Source

e(n) S(n)

x(n)

(Actual)

v’’(n)

y(n)

W(n)

Ŝ(n)

Ŝ(n)

W(n)

Ŝ(n)

y’’(n)

v(n) x’(n)

d’’(n) LMS

y’(n)-v’(n)

VSS LMS

f(n)

White Noise

g(n)

from a small value to an upper bound for fast convergence and later it starts decreasing for minor adjustments. Vigorous simulations are performed to determine the effectiveness of proposed method. We tested system with three types of reference signals: Single tone signal corrupted with Gaussian noise, multi-tone signal corrupted with Gaussian noise and a broadband signal. Its performance is also tested for varying acoustic paths. The performance evaluation is carried out on the basis of the residual error reduction parameter R (dB) and the relative modeling error, ΔS (dB). Expressions for these parameters can be given as follows: n

∑ eðiÞ2

RðnÞ ¼ i ¼n 0 ∑ dðiÞ2 and

d’(n)

Modified FxLMS Algorithm

Fig. 3. Proposed ANC system with online secondary path modeling.

(MFxLMS) algorithm [37]. As shown in Fig. 3, error signal for the control filter is generated using the extra secondary-path-modeling filter and FxLMS algorithm based adaptation is avoided by using extra control filter. Upper bound for the step size parameter is large owing to the adaptation of control filter using simple LMS algorithm and thus, fast convergence can be achieved [35]. Variable step size (VSS) algorithm for modeling filter in [35] uses power of the disturbance signal in the desired response of the modeling filter. In this method, fast convergence is achieved by varying step size from lower bound to an upper bound, but after that step size and modeling filer reaches steady state. Different VSS algorithms are proposed in [38–42], where initially large step is used for fast convergence and then it is decreased to a small value for minor adjustments. In this paper, a new scheme is developed for active noise control system with online secondary path modeling and its main features are in summarized paragraphs. A recently developed Fractional LMS algorithm [43] is proposed for the weight adaptation of modeling filter. We can see from overview of previous methods that simple modifications and extensions of the existing algorithms, improve the robustness of ANC systems. It would be interesting to investigate, whether we can get better performance by employing proposed ANC system on standard theoretical testing problems. New variable step size algorithm is proposed for modeling filter adaptation. Step size is varied in relation with disturbance signal power of the modeling filter. This multidirectional step size (MSS) algorithm is different from algorithms proposed in [28–32] where a large step size is selected at start for fast convergence and finally a small value is used for misadjustments. The proposed MSS algorithm is supported by the fact that the disturbance signal for the modeling filter, is decreasing in nature. Infact, initially this interference may be large enough to slow down the online secondary path modeling as compared with offline modeling. This issue is solved by using step size parameter whose value increases

3

ð1Þ

i¼0

( ) 2 ^ jjSðnÞ  SðnÞjj ΔS ¼ 10 log10 : jjSðnÞjj2

ð2Þ

Large value of R depicts better noise reduction capability of the system and small value of ΔS stands for accurate modeling of secondary path. The performance of the proposed method is compared with established methods presented in [30,33–35]. 3. Proposed methodology We can see the block diagram of the proposed ANC system in Fig. 3. Let control filter W(n) be an finite impulse response (FIR) filter of length L and x(n) is the reference signal, then control filter output signal y(n) is computed as yðnÞ ¼ WðnÞT xL ðnÞ;

ð3Þ

where WðnÞ ¼ ½w0 ðnÞw1 ðnÞ⋯wL  1 ðnÞT ;

ð4Þ

xL ðnÞ ¼ ½xðnÞxðn  1Þ⋯xðn  ðL 1ÞÞT :

ð5Þ

A zero mean white Gaussian noise signal, v(n), which is uncorrelated with x(n), is injected at the output of the W (n). Thus, the residual error signal e(n) is given as eðnÞ ¼ dðnÞ y0 ðnÞ þ v0 ðnÞ;

ð6Þ

where d(n) is primary disturbance signal, y'(n) is the canceling signal and v'(n) is the modeling signal. These signals can be represented by following expressions: dðnÞ ¼ pðnÞnxðnÞ; 0

ð7Þ

y ðnÞ ¼ sðnÞnyðnÞ;

ð8Þ

v0 ðnÞ ¼ sðnÞnvðnÞ;

ð9Þ

where s(n) and p(n) are the impulse responses of secondary and primary paths respectively. Also, n denotes linear convolution. Assuming that secondary path modeling filter Ŝ(n) is an FIR filter of length M, its output is given as ^ T vM ðnÞ; v″ðnÞ ¼ SðnÞ

ð10Þ

where ^ SðnÞ ¼ ½s^ 0 ðnÞs^ 1 ðnÞ⋯s^ M  1 ðnÞT ;

ð11Þ

Please cite this article as: M.S. Aslam, M.A.Z. Raja, A new adaptive strategy to improve online secondary path modeling in active noise control systems using fractional signal processing approach, Signal Processing (2014), http://dx.doi.org/ 10.1016/j.sigpro.2014.04.012i

M.S. Aslam, M.A.Z. Raja / Signal Processing ] (]]]]) ]]]–]]]

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vM ðnÞ ¼ ½vðnÞvðn  1Þ⋯vðn ðM  1ÞÞT :

ð12Þ

The error signal for Ŝ(n), f(n) is obtained by subtracting v″ (n) from e(n) and can be written as f ðnÞ ¼ ½dðnÞ  y0 ðnÞ þ ½v0 ðnÞ  v″ðnÞ:

ð13Þ

The tap-weights of Ŝ(n) are updated using MSS fractional LMS algorithm, which will be explained later. Here y(n) is filtered through another modeling filter Ŝ(n) to get the desired response for the dummy W(n) ^ T y ðnÞ; d ðnÞ ¼ f ðnÞ þ SðnÞ M 0

ð14Þ

where yM ðnÞ ¼ ½yðnÞ yðn  1Þ⋯yðn  ðM  1ÞÞT :

ð15Þ

The input to the dummy W(n) is obtained by filtering the reference signal x(n) through Ŝ(n) ^ T xM ðnÞ; x0 ðnÞ ¼ SðnÞ

ð16Þ

where xM ðnÞ ¼ ½xðnÞ xðn 1Þ⋯xðn  ðM 1ÞÞT :

ð17Þ

This ANC system exploits the delay compensation scheme of [25] to improve the convergence properties. In the delay compensation scheme, an estimated primary noise disturbance d'(n) is used for the adaptation of W(n) as can be seen in Fig. 3. The signal g(n) is obtained by adding estimated secondary path signal (y″(n)  d″(n)) to the modeling filter error signal f(n). The error signal for W(n) is obtained by 0

gðnÞ ¼ d ðnÞ d″ðnÞ ¼ f ðnÞ þ y″ðnÞ  d″ðnÞ:

ð18Þ

The estimate of y″(n) is obtained by filtering y(n) through Ŝ(n). The estimate of d″(n) is obtained by filtering x'(n) through W(n). So delay compensation technique requires two additional filters: filter for estimating d″(n) from the filtered reference signal x'(n), and the delay compensation filter used for computing y″(n). The extra control filter is used to avoid FxLMS algorithm based adaption because upper bound for step size is small for FxLMS. The simple LMS algorithm is used for adapting W (n) and hence upper bound for the step size parameter is larger than that for FxLMS algorithm. Since larger step size can be selected, fast convergence can be achieved. Modified FxLMS algorithm is used for adaptation of WðnÞ and it adapts with following rule: Wðn þ 1Þ ¼ WðnÞ þ μw gðnÞx'ðnÞ;

ð19Þ

where μw is a fixed step size parameter. The secondary path is estimated by injecting zero mean white Gaussian noise in the secondary path. Ŝ(n) is adapted with the MSS fractional LMS algorithm which is explained in next subsection. 3.1. Fractional least mean square algorithm Research community has shown great interest in design and application of variants of LMS algorithms by introducing the fundamental concepts and theories of fractional calculus. It is utterly wrong to categorize the fractional calculus as a young scientific field. In fact, the birth of fractional calculus takes place almost at the same

time as that of classical calculus itself. However, most of the extensive applications of the fractional order calculus are found to be in the last three decades such as in viscoelasticity and damping, diffusion and wave propagation, electromagnetism, chaos and fractals, heat transfer, biology, electronics, signal processing, robotics, system identification, traffic systems, genetic algorithms, percolation, modeling and identification, telecommunications, chemistry, irreversibility, physics, control systems, economy and finance [43–46]. Fractional LMS algorithm is designed on the basis of generalization of the idea behind LMS algorithm to find filter weights by minimizing a cost function by using steepest descent, i.e., taking partial derivatives with respect to the individual members of the filter coefficient vector.   n JðnÞ ¼ E f ðnÞf ðnÞ ¼ E½jf ðnÞj2 ; ð20Þ where f(n) is given by (13) and E{.} denotes the expected value. This cost function J(n) is the mean square error and the adaptive weight for kth entry in fractional LMS is given by [45] sk ðn þ 1Þ ¼ sk ðnÞ þ μs

∂J ∂f r J þ μf f r ; ∂sk ∂s

ð21Þ

k

where μf is the fractional step-size, fr is order of fractional derivative and is a real number. The fractional integral and derivative have been expressed in the literature in a variety of ways, including Riemann–Liouville, Caputo, Erdélyi–Kober, Hadamard, Grünwald–Letnikov, Riesz type etc. Equivalence of these definitions for some function has been given in standard fractional calculus reference books [43–45]. Grünwald–Letnikov (GL) definition of fractional derivative a Dft r of order fr with lower terminal at a for the function f(t) is given as [46]   1 fr fr m fr D f ðtÞ ¼ lim h ∑ ð 1Þ f ðt þ ðf r  mÞhÞ a t h-0 m m¼0 Using above relation then fractional derivative of order fr for the function f(t)¼tν is given as fr ν a Dt t

¼

Γðν þ1Þ ν  f r t ; Γðν  f r þ 1Þ

ν4 1

After taking first order and fractional order derivative of cost function based on GL procedure and carrying out necessary simplifications, we obtain the final update relation for the weight vectors [47,48] ! jsk ðnÞj1  f r sk ðn þ 1Þ ¼ sk ðnÞ þ μs þμf f ðnÞvðnÞ; ð22Þ Γð2  f rÞ where |  | is used to denote absolute value as fractional power of a negative entry can cause problems and Γ(.) denotes the gamma function given as Z 1 xt  1 e  x dx: ΓðtÞ ¼ 0

Adaptation rule of (21) is used for updating the modeling filter coefficients in the method proposed in this paper.

Please cite this article as: M.S. Aslam, M.A.Z. Raja, A new adaptive strategy to improve online secondary path modeling in active noise control systems using fractional signal processing approach, Signal Processing (2014), http://dx.doi.org/ 10.1016/j.sigpro.2014.04.012i

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Table 1 Computational complexity comparison of proposed method with existing methods. Number of computations per iteration

Phase (degrees)

Magnitude (dB)

EM ZM-a ZM-b AM PM

Additions

Multiplications

Additions (L¼ M¼ N)

Multiplications (L¼ M¼ N)

2L þ3M  1 2L þ3M þ 2N þ1 3L þ4M þ 3N þ3 3L þ4M þ 5 3L þ5M þ 6

2L þ3M þ 2 2L þ3M þ 2N þ3 3L þ4M þ 3N þ12 3L þ4M þ 10 3L þ5M þ 13

5L  1 7L þ 1 10L þ1 7L þ 5 8L þ 6

5L þ 2 7L þ 3 10L þ3 7L þ 10 8L þ 13

Table 2 Parameter values used in simulations.

20 0

Parameter values used in simulations

−20 Primary Path 1 Primary Path 2

−40 −60

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1000 Primary Path 1 Primary Path 2

0 −1000 −2000 −3000

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Phase (degrees)

Magnitude (dB)

Normalized Frequency ( ×π rad/sample) 20 0 −20 −40

Secondary Path 1 Secondary Path 2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

500 0 −500 Secondary Path 1 Secondary Path 2

−1000 −1500

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Normalized Frequency ( ×π rad/sample) Fig. 4. Frequency response of the acoustic paths.

Case 1

Case 2

Case 3

Case 4

PM μ1min μ1max η1 η2 A μsmax C1 C2 μfmax μw

7.5E  03 2.5E  02 0.9995 2.0E  04 7.0000 2.4E  02 12.000 0.4000 2.4E  02 9.0E  04

7.5E  03 2.5E  02 0.9995 6.0E  06 0.7500 2.4E  02 6.0000 0.8000 2.4E  02 5.0E  04

7.5E  03 2.5E  02 0.9995 5.0E  06 12.00 1.2E  02 7.000 1.000 1.2E  02 5.0E  04

7.5E  3 2.5E  2 0.9995 2.0E  04 7.0000 2.4E  02 12.0000 0.4000 2.4E  02 9.0E  04

EM μw μs

5.0E  04 3.0E  03

5.0E  04 3.0E  03

5.0E  04 3.0E  03

5.0E  04 3.0E  03

ZM-a μw μs μh

5.0E  04 1.0E  02 1.0E  02

5.0E  04 1.0E  02 2.5E  02

5.0E  04 7.5E  03 1.0E  02

5.0E  04 1.0E  02 1.0E  02

ZM-b μw μs μh C Δ Λ Tw Ts Th

5.0E  04 1.3E  02 3.0E  04 9.0000 30.0000 0.9900 5.0000 8.0000 3.0000

5.0E  04 1.0E  02 1.0E  03 9.0000 8.0000 0.9900 6.0000 8.0000 4.0000

5.0E  04 7.5E  03 1.0E  02 3.5000 30.0000 0.9900 5.0000 8.0000 3.0000

5.0E  04 2.0E  02 3.0E  04 7.0000 30.0000 0.9900 5.0000 8.0000 3.0000

AM μsmin μsmax μw

7.5E  03 2.5E  02 5.0E  04

7.5E  03 2.5E  02 5.0E  04

7.5E  03 2.5E  02 5.0E  04

7.5E  03 2.5E  02 5.0E  04

3.2. Multi-directional step size parameter From Fig. 3, we can see that modeling filter error signal f ðnÞ comprises of two parts; (d(n) y'(n)) and (v'(n) v″(n)). First part is a disturbance for the modeling process and initially this disturbance is large which degrades performance of modeling filter (in worst case filter may become unstable). As ‘n’ increases, this disturbance decreases and ideally would converge to zero. Thus, initially step size parameter μs, should be a small value and later when disturbance signal decreases, parameter value should increase accordingly. The disturbance signal (d(n)y'(n)) is not available directly but e(n) is available as it is being received by error microphone. On the basis of e(n) and f(n), a candidate approach for variation of μs can be given as

follows: μ1 ðnÞ ¼ ρðnÞμ1 min þð1  ρðnÞÞμ1 max ;

ð23Þ

where ρ(n) is the ratio between the estimation error signal power Pf(n) and the error microphone signal power Pe(n). The expressions for Pf(n) and Pe(n) are given below: 2

P f ðnÞ ¼ λP f ðn  1Þ þð1  λÞf ðnÞ;

ð24Þ

P e ðnÞ ¼ λP e ðn 1Þ þ ð1 λÞe2 ðnÞ;

ð25Þ

where λ is a forgetting factor (close to 1). The parameter μ1 of (23) varies between a minimum value μ1min and a maximum value μ1max. The choice of (23) as μs is supported

Please cite this article as: M.S. Aslam, M.A.Z. Raja, A new adaptive strategy to improve online secondary path modeling in active noise control systems using fractional signal processing approach, Signal Processing (2014), http://dx.doi.org/ 10.1016/j.sigpro.2014.04.012i

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by the fact that during the early phases of adaptation of ANC system, the convergence of the secondary path model is degraded by the large disturbance at the error microphone. Thus, a small step-size is recommended. With the convergence of the noise control filter, the disturbance reduces and a large value for μs can be used in the adaptation of Ŝ(n). So selection of μ1min is helped by the limit that it should not be very small as it will slow down the adaptation process and μ1max should not be too large to make system unstable so suitable zone for parameter selection can be obtained with less effort. We can see in [35] that ρ(n)E 0 in the early stages of adaptation, while ρ (n)E 1 when the ANC system is converged and this scheme shall contribute in fast convergence. But this is not the only expression used for step size parameter. Another mechanism that is based on decreasing nature of modeling error signal f(n), can be given as 3

μ2 ðnÞ ¼ η1 μ2 ðn  1Þ þ η2 f ðnÞ;

ð26Þ

where η1 and η2 are constants used to articulate a decreasing function. η1 is the forgetting factor and is usually

greater than 0.9 and less than 1. η2 is for updation and is usually very small (of the order of 1E 3). This decreasing nature of this function is similar to the disturbance signal and thus can trigger minor adjustments for improved modeling accuracy. Step size parameters μs and μf vary by using following rules: 9 8 if μ1 o μs max > > = < μ1 μ if μ1 4 μs max and μ2 oμs max ð27Þ μs ¼ A2 > > ; :μ otherwise s max and 8 C μ > < 1 1 μf ¼ C 2 μ2 > :μ

f max

if μ1 o μf

max

if μ1 4 μf

max

and μ2 o μf

otherwise

9 > = max

> ;

ð28Þ

where μsmax and μfmax are the upper bounds on step size parameter μs and μf for modeling filter. A, C1 and C2 are used as scaling factor to use same increasing and decreasing functions for both fractional and non-fractional terms. These rules make sure that step size parameters change from small

x 104

x 104 Fig. 5. Performance comparison of proposed method with previous methods for Case 1.

Please cite this article as: M.S. Aslam, M.A.Z. Raja, A new adaptive strategy to improve online secondary path modeling in active noise control systems using fractional signal processing approach, Signal Processing (2014), http://dx.doi.org/ 10.1016/j.sigpro.2014.04.012i

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to large values for achieving fast convergence and then decrease again for improving the modeling accuracy.

4. Computational complexity The computational complexity of an algorithm is determined by the number of computations performed per iteration [31]. Comparison of proposed and existing methods on the basis of computational complexity is given in Table 1. Here, EM stands for Eriksson's method of [30], ZM-a stands for Zhang's method of [33], ZM-b stands for Zhang's method of [34] and AM stands for Akhtar's method of [35]. Here L is tap-weight length of control filter W'(n) and M is tap-weight length of modeling filter Ŝ(n). For Zhang's method, tapweight length of third filter H(n) is N. The proposed algorithm (PM) is similar to AM, but a new variable step size fractional LMS algorithm is used for adaptation of modeling filter. Computational complexity of proposed method is slightly larger than that of previous methods.

7

5. Simulations and results In this section, we present the simulation results that verify the effectiveness of the proposed method. The performance of the proposed method is compared with established methods reported in [30,33–35]. Primary path P(n), secondary path S(n), control filter W(n) and modeling filter Ŝ(n) are FIR filters of tap-weight length 48, 16, 32 and 16 respectively. Primary path and secondary path are obtained by truncating impulse response of acoustic path models provided in the companion disk of [13]. The frequency responses of P(n) and S (n) are shown in Fig. 4. The control filter is initialized by a null vector and modeling filter is initialized with vector which gives relative modeling error of  5 dB (as described in [35]). A sampling frequency of 2 kHz is used and step size parameters are obtained by experimentation for fast and stable convergence. A zero-mean white Gaussian noise of variance 0.05 is used in the modeling process. Simulations are performed multiple times for different values of fractional order fr (between 0.1 and 1) and it was observed that on average, fr

Fig. 6. Performance comparison of proposed method with previous methods for Case 2

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values from 0.6 to 0.7 provide better results, so parameter value of 0.7 is used for results presented in this paper. Parameters are adjusted, by trial-and-error, for achieving fast, stable and better performance and are summarized in Table 2. Effectiveness of the proposed method is shown by performing extensive simulations. All the results given below are obtained by averaging over ten experiments.

5.1. Case 1 Here a tonal signal of 300 Hz is used as reference and variance of this signal is adjusted to 2.0. This signal is corrupted with a zero-mean white Gaussian noise until a 30-dB SNR ratio is obtained. We can see from modeling error curves in Fig. 5(a) that EM and ZM-b converge at a very slow rate while AM and ZM-a reach steady state before n¼20,000. But proposed method manages to reduce the modeling error at a faster rate than the previous methods. Among previous methods, best modeling error reaches

steady state value of 32 dB at about n¼4400 while for proposed method, same accuracy is achieved at about n¼2900 which represents an improvement of 35 percent in convergence rate. Steady state value of ΔS for proposed method is  41.5 dB which is an improvement in steady state accuracy by 30 percent. The curves for the noise reduction and step size parameters are shown in Fig. 5(b) and (c) respectively. The noise reduction performance of all the methods is quite similar as can be seen in Fig. 5(b). We can see in Fig. 5(c) that during the early stages, values of μs and μf increase from a small value to accelerate convergence and later their values decrease to account for misadjustments. Thus, overall performance of system is improved.

5.2. Case 2 In this case, reference signal is multi-tonal with frequencies of 100, 200, 300, and 400 Hz. The resulting signal's variance is adjusted to 2.0 and a zero-mean white

Fig. 7. Performance comparison of proposed method with previous methods for Case 3

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Gaussian noise is added until SNR of 30 dB is achieved. The simulation results are shown in Fig. 6. Again, we can see in Fig. 6(a) that proposed method performs better than previous methods just like in Case 1. Best modeling error, for previous methods, reaches steady state value of 35.5 dB at about n ¼4750 while for proposed method, modeling error achieves same accuracy at about n ¼2760 which means that an improvement of 42 percent is achieved in convergence rate. Modeling error steady state value for proposed method is  43 dB which is an improvement in steady state accuracy by 21 percent. The curves for the noise reduction and step size parameters are shown in Fig. 6(b) and (c) respectively and their behavior is similar in nature to that of case 1.

5.3. Case 3 A broad-band signal is obtained by filtering a zero mean white Gaussian noise of unit variance through a bandpass

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filter with the passband 100–400 Hz. Variance of resulting signal is adjusted to 2 and is used as reference signal. The simulation results are given in Fig. 7. We can see from Fig. 7 (a) that all the methods reach same steady state accuracy before n¼20,000. But among the previous methods, EM reaches steady state in a slow fashion but its response is very smooth. Response of ZM-a, ZM-b and AM is very much similar to each other. Best modeling error, for previous methods, reaches steady state value of  20 dB at about n¼5200 while for proposed method, modeling error achieves same accuracy at about n¼3250 which means that an improvement of 37 percent is achieved in convergence speed. Modeling error steady state value for proposed method is  28.5 dB which is better than previous methods by 40 percent. Also, modeling error reduces in a smoother way for our method while its behavior is very jerky for AM and ZM. The curves for the noise reduction parameter and step size parameters in the proposed method are shown in Fig. 7(b) and (c) respectively and their behavior in a manner similar to that in cases 1 and 2.

Fig. 8. Performance comparison of proposed method with previous methods for Case 4

Please cite this article as: M.S. Aslam, M.A.Z. Raja, A new adaptive strategy to improve online secondary path modeling in active noise control systems using fractional signal processing approach, Signal Processing (2014), http://dx.doi.org/ 10.1016/j.sigpro.2014.04.012i

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5.4. Case 4 Here we consider a situatio of varying acoustic paths. The reference signal and system conditions are same as described in Case 1. At n ¼10,000, acoustic paths change to as shown by dark curves in Fig. 4. The simulation results are shown in Fig. 8. We can see from Fig. 8(a) that response of EM fails to improve after change of acoustic path while ZM-b, AM and PM respond to the change by adapting to the similar accuracy at similar rate as before the change. PM gives better modeling accuracy at a fast convergence rate before and after the change of acoustic paths. The curves for the noise reduction and step size parameters are shown in Fig. 8(b) and (c) respectively. The noise reduction performance of all the methods is quite similar as can be seen in Fig. 8(b). 6. Conclusions A new adaptive procedure is designed using the multiple directional step size strategy in fractional LMS algorithm by exploiting the strength of fractional signal processing and applied effectively for online secondary path modeling in active noise control systems. The step size is adjusted according to status of the active noise control system as shown in simulations and compared with existing methods; it is found that the proposed scheme improves the convergence speed and modeling accuracy. For tonal reference signal, convergence speed is improved by  35 percent and modeling accuracy is improved by  20 percent. While for broadband noise signal, convergence speed and modeling accuracy are improved by  35 percent. This improvement in performance is achieved without degrading the noise control performance, which is the ultimate goal of ANC system. The computational complexity of the proposed method is comparable to that of previous methods. References [1] B. Mandelbrot, J.W. van Ness, Fractional Brownian motions, fractional noises and applications, SIAM Rev. 10 (4) (1968) 422–437. [2] L. Gaul, P. Klein, S. Kemple, Damping description involving fractional operators, Mech. Syst. Signal Process. 5 (2) (1991) 81–88. [3] J. Sabatier, M. Aoun, A. Oustaloup, G. Gregoire, F. Ragot, P. Roy, Fractional system identification for lead acid battery state charge estimation, Signal Process. 86 (10) (2006) 2645–2657. [4] M.D. Ortigueira, On the initial conditions in continuous-time fractional linear systems, Signal Process. 83 (11) (2003) 2301–2309. [5] D. Valério, M.D. Ortigueira, J. Sá da Costa, Identifying a transfer function from a frequency response, ASME J. Comput. Nonlinear Dyn. Spec. Issue Discontin. Fract. Dyn. Syst. 3 (2) (2008) 021207. [6] Naveed Ishtiaq Chaudhary, Muhammad Asif Zahoor Raja, Junaid Ali Khan, Muhammad Saeed Aslam, Identification of input nonlinear control autoregressive systems using fractional signal processing approach, Sci. World J. 2013 (2013), http://dx.doi.org/10.1155/2013/ 467276. (Article ID 467276, 13 pp.). [7] M.D. Ortigueira, Introduction to fractional signal processing. Part 1: continuous-time systems, IEE Proc. Vis. Image Signal Process. 147 (1) (2000) 62–70. [8] M.D. Ortigueira, Introduction to fractional signal processing. Part 2: discrete-time systems, IEE Proc. Vis. Image Signal Process. 147 (1) (2000) 71–78. [9] M.D. Ortigueira, J.A.T. Machado (Eds.), Special issue on fractional signal processing and applications, Signal Process. 83 (11) (2003). [10] M.D. Ortigueira, J.A.T. Machado (Eds.), Special section: fractional calculus applications in signals and systems, Signal Process. 86 (10) (2006).

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Please cite this article as: M.S. Aslam, M.A.Z. Raja, A new adaptive strategy to improve online secondary path modeling in active noise control systems using fractional signal processing approach, Signal Processing (2014), http://dx.doi.org/ 10.1016/j.sigpro.2014.04.012i