Development of Volterra FXRLS Algorithm for Active Control of Nonlinear Noise Processes

1DEBI PRASAD DAS,2GANAPATI PANDA,2PRIYABRATA NAYAK 
1.Department of Electrical Engineering
Indian Institute of  Technology
Kharagpur,West Bengal
INDIA
2.Department of Electronics and Instrumentation Engineering
National Institute of Technology
Rourkela,Orissa
INDIA
debi_das_debi@yahoo.com, rkl_ganapat@sancharnet.in, jinu_nayak@yahoo.com

Abstract: -When the noise that travels as a plane wave in the duct is a low frequency broadband noise which may be periodic, tonal or random in nature, linear techniques can be successfully applied to actively mitigate the noise. But in many applications the noise that generates from a dynamic system is nonlinear and deterministic or colored nongaussian in nature. Another issue is that the actuators are generally nonminimum phase type. Issue is that under this condition linear controllers perform suboptimaly. So to get better performance under nonlinear condition nonlinear controllers have been implemented in ANC system. This paper derives a Volterra filtered-x recursive least square algorithm which tries to solve the problem under nonlinear situation. Computer simulations verify that the developed algorithm has achieved performance improvement for nonlinear reference noise with nonminimum phase secondary path transfer function or when the primary path between the reference sensor and the loud speaker itself is nonlinear.

 Key-Words: - Active noise control, Adaptive Volterra filter, chaotic noise, Volterra series, nonminimum phase type, chaotic noise.

  1 Introduction       

WEIGHT UPDATE

MEMORY

A(z)

P(z)

FILTERING

e(n)

d(n)

out(n)

 y(n)

(n)

 x(n)

x1N

w1M

Fig. 1 Block diagram of ANC 

With the increase of large industrial equipments, high density traffics and housings noise pollution is increasing in a alarming rate. Traditional approach to control the noise level is to use passive elements such as barriers or enclousers. These passive barriers are suitable for broadband noise sources. For low frequencies these passive absorbers tend to be bulky and costly. To overcome these problems active noise control systems are used. ANC is an electro acoustic device which electronically generates a antinoise which is equal in magnitude and 180o  out of phase to that of reference noise source thus cancels each other. The conventional ANC system is essentially linear in nature and has been successfully applied in controlling both broadband and narrow band noise source. The application areas are heating, ventilating and air conditioning (HVAC) systems [2],[4], headsets [2], [6] and air planes[2].
A typical ANC system in a duct  is shown in the figure-1. In this system two sensors one close to the reference noise source known as reference microphone, other a error sensor and other a 

loud speaker which acts as a secondary noise source. The reference microphone senses the reference noise and convert the noise signal into corresponding electrical signal. The electrical signal acts as the input to the active noise controller which generates the required anti noise that drives the secondary loud speaker to produce the canceling sound that attenuates the acoustic noise in the duct .The error microphone is used to measure the residual error signal which is used to adjust the filter coefficients to minimize the error while spacing the microphone and the loud speaker care should be taken so that the principle of causality and high coherence must be satisfied .Actual implementations require additional considerations to handle acoustic effects in the duct.

WEIGHT UPDATE

MEMORY

A(z)

P(z)

FILTERING

e(n)

d(n)

out(n)

 y(n)

(n)

 x(n)

FUNCTIONAL EXPANSION

x1N

s1M

w1M

 	To ensure the optimal performance under the changing environment conditions,degradation of system components and alternation of noise source[2] the filter should be adaptive in nature .Most of the conventional ANCs are linear in nature and they employ some adaptive algorithms to update their filter coefficients to achieve optimal performance under varying conditions.The common practice is to use filtered-x algorithm[1],[2] as it involves less computational complexity for broad band and narrow band noise .Under practical applications , several non linear distortion may degrade the performance of the linear adaptive filter used by the system . so a linear ANC                  shows performance degradation under the following situations .
(1)- The components of ANC system such as analog to digital converter, loud speaker, anti aliasing filter, pre amplifier might exhibit non linearities themselves. The loud speaker for an example can excite both the frequency of interest and its associated harmonics [2], [6].The noise from a dynamic system may be non linear and deterministic (chaotic rather than stochastic, white or tonal) noise process. For example far noise often shows chaotic behavior [4].
(2)- The primary noise at the canceling point may exhibit non linear distortion due to the non linear transfer function of the primary path.
(3)- Secondary transfer function between the secondary speaker and the error microphone may exhibit non minimum phase.

 Non linear filters may be required for the applications with the above said case. There is no unique theory for modeling   and characterizing non linear systems. In the recent literatures[2-5], several methods have been proposed for the development of active noise control of non linear noise process. They have demonstrated that FXLMS algorithm is not capable of compensating non linear noise process as this algorithm exploits   the linearity of the controller. Normalized Gaussian radical basis neural network is used for compensating for secondary path transfer function with non minimum phase to control the nonlinear   effects in the secondary path transfer function  [2],[3] .In [2] adaptive volterra filters   using filtered-x LMS algorithm has been introduced for feed forward active noise control .

It is observed that the recursive least square (RLS) algorithm are superior than LMS algorithms in terms of faster convergence. Hence attempt has been made to develop the RLS counter point of volterra filtered-x RLS algorithm. The algorithm is derived for single channel case. 

2 DEVLOPMENT OF VOLTERRA F-X RLS ALGORITHM        
s 
 Block diagram of single channel broad band feed forward ANC system is shown in the figure-2. 

Fig. 2 Block diagram of ANC with VFXRLS  algorithm

We have , p(z) the transfer function of primary path from the reference microphone  to the error 
microphone, d(n) the primary noise signal at the canceling point, e(n) the residual noise sensed by the error microphone which is expressed as  
e(n)=d(n)+a(n)*y(n)                                       (1)
Where a(n) is the impulse response of the secondary path transfer function A(Z) and * denotes convolution operation.  is the estimate of secondary path which is present within the filtering block of figure(2) and v(n) is the filtered vector in which each signal element is obtained by filtering the corresponding signal  elements of s(n) using the secondary path estimate . The output of the adaptive second order volterra can be expressed as 

                                                                        (2)

Where w1(i)   is the time varying coefficient of the linear filter and  w2(i,j) represents the coefficient of the non linear adaptive filter. In the vector form w can be written as 

and the input signal is defined as

Hence equation(1) can be written as
                      (3)
In the z-domain equation (3) can be written as 
 E(z)=D(z)+A(z)W(z)S(z)
where E(z), D(z), A(z), W(z) and S(z) are the z- transform of the residual noise e(n), desired noise to be cancelled d(n), secondary path impulse response a(n), weight vector w(n) and the functionally enhanced signal element vector s(n). Let the output of the Volterra filter be 

y(n)=wT(n)s(n) as the dimension of s(n) and w(n) are identical and is equal to   . To derive Volterra FXRLS algorithm we rewrite equation (1) as
e(n)=d(n) + a(n)*y(n)
=d(n)+a(n)*{w(n)s(n)}
Let us defined cost function as the residual noise power i.e
 =E[e2(n)]
The weight vector can be adjusted according to the update equation which minimizes the mean square error    .
 
 W(n+1)=W(n)-                                  (4)
Where=instantaneous estimate of the gradient of with respect to the weight vector W(n).
using RLS algorithm in place of LMS algorithm the update equation of (4) becomes
 
wi(n+1)=wi(n)-e(n)ki(n)
where i =1,2,… 
 The individual kalman gain vector,

 
And the inverse of auto correlation matrix 
 
Where  is the forgetting factor,which weights the recent data more heavily in order to accommodate nonstationary signals.This algorithmis named here as Volterra F-x RLS(VFXRLS) algorithm.

3   Simulation Results

To verify the effectiveness of FSRLS algorithm over the recently proposed FSLMS algorithm computer simulation study has been under taken with the well known nonlinear noise processes. The simulation considers both nonminimum phase and minimum phase transfer function for the secondary path and its estimate [5]. 
3.1	Experiment I
	The primary noise signal is chosen to be a unit power logistic chaotic type [5]. The primary path transfer function is considered to be . The secondarypath transfer function is a minimum phase type. The VFXLMS and VFXRLS algorithms 

Fig. 3 Experiment 1

with M=10 are considered. Fig. 3 shows a comparative plot of the normalized mean-square error (NMSE) defined in [5]. The convergence characteristics of different ANC controller with VFXLMS, and VFXRLS are shown in Fig. 3. From Fig. 3 it is observed that the proposed VFXRLS algorithm outperforms the conventional VFXLMS algorithm in terms of faster convergence performance. 
3.2	Experiment II
In this experiment, the primary noise signal 
and the primary path transfer function used are same as those of Experiment I. However, the secondary path transfer function is chosen to be the nonminimum-phase type. Fig. 4 shows the plot between NMSE achieved by each ANC 
controller versus the number of iterations. From this plot it is evident that the proposed VFXRLS algorithm exhibits faster convergence 

Fig.4 Experiment 2

performance compared to that of the conventional VFXRLS algorithm. 

3.3 Experiment III
In this case, the example of nonlinear primary path of [5] is used in the simulation. The primary noise at the canceling point is generated based on the following third-order polynomial model given in [7],

where 	 and * denotes convolution operation and . The reference signal is a sinusoidal wave of 500 Hz sampled at the rate of 8000 samples/s and then Gaussian noise of 40 dB SNR is added. The secondary path transfer function is chosen to be minimum-phase type. Fig. 5 depicts the comparative plot of NMSEs for all the algorithms. It is found that there is a significant improvement of performance VFXRLS algorithm.

Fig. 5 Experiment 3

3.4	Experiment IV
In this case, the same nonlinear primary path is taken for simulation study. The primary noise at the canceling point is identical to that given in Experiment III. The secondary path transfer function in this case is chosen to be nonminimum phase. Fig. 6 depicts the results of NMSEs for the two algorithms.In each of these experiments the NMSEs are averaged after many independent trials to obtain the smoother convergence

Fig. 6 Experiment 4
(a) VFXLMS (b) VFXRLS

4   Conclusion

This paper proposes a faster converging algorithm called VFXRLS algorithm for nonlinear noise processes. Its performance in terms of mean square erroe has been compared to that of recently proposed VFXLMS algorithm a.It is shown that in terms of NMSE , the proposed algorithm performs better than the VFXLMS algorithm.The VFXRLS algorithm can also be used as an alternative when the standard filtered-x LMS performs well.
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