Active noise control using wavelet function and network approach

Abstract

Active noise control (ANC) is based on the principle of superposition of waves. It means that an algorithm is used to tune a secondary source to make an anti-noise with equal amplitude but opposite phase with the primary source. In this paper, a wavelet function and network (WAVENET) approach is designed for ANC. The algorithm is used to train parameters of an anti-noise filter for omitting the undesired noise. FXLMS and NLMS are the conventional methods of ANC that need complex acoustic plant models and these necessities make the methods complex and inaccurate. In the WAVENET approach, this complexity can be accounted for. Numerical simulations for a WAVENET approach are presented to demonstrate the performance of the WAVENET approach scheme.

Keywords

Active noise control ANC FXLMS WAVENET

Introduction

The active noise control (ANC) system, which uses anti-noise signals to cancel undesired noise, has received considerable attention during the past decades.¹ In the recent years, improvements in electronics have enabled researchers to apply different algorithms for ANC. An alternative approach of noise reduction is passive noise control (PNC) which is mostly used for high frequency acoustic noise reduction.¹

It has been verified by many researchers that PNC can decrease high frequency noise more than the low frequency noise.^2–5 However, in industrial applications, the power spectrum of the acoustic noise is in the low frequency band. Therefore, PNC methods will not be cost-effective in such applications. Moreover, PNC methods need bulky material in such applications and this constitutes another disadvantage of these methods.

In comparison with PNC method, ANC approach has the ability of reducing noise of low frequency.^6–9 Another advantage of this approach is the decrease in the weight, volume and cost of the noise control system.¹

The first idea of employing a microphone and a loudspeaker for ANC is a patent of Paul Lueg in 1936. Due to lack of essential equipment in those years, the implementation of the idea was only realised in 1975.^10,11 One of the essential necessities for implementation of the ANC system using conventional methods is identification of acoustic paths and transducers, because, designing the ANC system filter is based on the transfer function of the acoustic paths. One should have an accurate transfer function to generate anti-noise signal correctly.

Many researchers have applied the filtered-X algorithm in the ANC system because of its simplicity in computation and implementation. The filtered-X algorithm is an adaptive filter which uses the least mean square (LMS) algorithm to tune the weighting parameters. This adaptive algorithm is referred to as FXLMS. In contrast to PNC, this method can decrease the acoustic noises in low frequencies in an effective manner. This advantage of approach has allowed its realisation in different applications such as reduction of compressor noise, engine acoustic noise and fan noise.¹² Besides the advantages of this approach, there are some disadvantages. The approach is just applicable for ANC systems with simple acoustic paths and that nonlinearities of the system and the loudspeaker are not accounted for. One of the most important issues in FXLMS is the low convergence speed in adapting the parameters. Small step in the FXLMS algorithm causes the algorithm to have the low convergence speed. A small step is used to guarantee that the system remains stable, because, the big changes in the step can make the system unstable.¹²

Because of these disadvantages, many researchers have tried to find new methods to overcome such difficulties. To overcome the nonlinearity property of the ANC system, neural network is applied to ANC systems,¹³ because, neural networks are one of the most powerful structures to deal with nonlinear behaviours. On the other hand, neural networks have nonlinear properties and these properties help them to handle nonlinear processes.^14–16 In Salmasi and Pourghassem’s¹⁷ study, it is shown that neural networks can perform more effective than the adaptive filter in the presence of nonlinearities.

Fuzzy logic has extensively been applied in ANC systems in recent years. Most of these have used conventional methods such as finite impulse response (FIR) filters or proportional- derivative (PD) controllers with fuzzy logic to tune the parameters of the ANC system.^1,18–20 These approaches are found to be complex.

WAVENET can benefit from the nonlinear properties of a neural network and proficiency of identification and prediction of wavelets. In this paper, a novel approach using WAVENET for ANC system is proposed. In comparison to other ANC approaches, it is easy to implement and has less complexity. The ease of implementation has led to its use in various applications with neural networks^21–24 such as transportation noise control, appliances noise and industrial noises.

ANC systems can be classified into two categories: feedforward control and feedback control. In feedforward ANC system, sensor and error sensor are placed in separate locations. In feedback ANC systems, only one sensor is used, which acts as both reference and error sensor.¹⁴

Figures 1 and 2 show the feedforward ANC system and feedback ANC system, respectively.

The main idea of this paper was firstly proposed in Yamada et al.²² Feedback ANC system and its block diagram are studied in ‘Feedback ANC system’ section. ‘WAVENET theory’ section presents the WAVENET theory needed for the proposed method and ‘WAVENET ANC system’ section presents the proposed WAVENET ANC system. The ability of the proposed approach in ANC is shown in simulation results in ‘A simulation study’ section. Finally, conclusions are drawn in ‘Conclusion’ section.

Feedback ANC system

In this paper, the proposed approach is based on the feedback ANC. Figure 2 shows a feedback ANC system, where the primary noise signal $d (n)$ is not accessible. Hence, the main point of an adaptive feedback ANC system is to measure the error signal and regenerate the reference signal $d (n)$ from the measurements.¹⁴ One can see the basic block diagram of the feedback ANC system in Figure 3, where $e (n)$ is the residual noise, $d (n)$ is the primary noise, $y (n)$ is the anti-noise signal. Moreover, $W (z)$ is the transfer function of the controller and $S (z)$ is the transfer function of the secondary path from cancelling loudspeaker to the error microphone.

From Figure 3, one can express the primary noise in the z-domain as,

D (z) = E (z) + S (z) Y (z)

(1)

\overset{\land}{S} (z)

is the estimation of the secondary path transfer function

S (z) .

Therefore, one can estimate the primary noise

d (n)

with notation

\overset{\land}{d} (n)

. Using

\overset{\land}{d} (n)

as a synthesized reference signal

x (n)

as,

X (z) \equiv \overset{\land}{D} (z) = E (z) + \overset{\land}{S} (z) Y (z)

(2)

Figure 4 shows a complete block diagram of the feedback ANC system.

From Figure 3, one can see that the reference signal $x (n)$ and the secondary signal $y (n)$ can be expressed as,

x (n) \equiv \overset{\land}{d} (n) = e (n) + \sum_{m = 0}^{M - 1} {\overset{\land}{s}}_{m} y (n - m)

(3)

y (n) = \sum_{l = 0}^{L - 1} w_{l} (n) x (n - l)

(4)

where

\overset{\land}{s} (m), m = 0, 1, \dots, M - 1

is the

M th

order FIR filter used to estimate the secondary path transfer function.

w_{l} (n), l = 0, 1, \dots, L - 1

are the coefficients of the Lth order adaptive FIR filter

W (z)

at time n. These coefficients are updated by the FXLMS algorithm as,

w_{l} (n + 1) = w_{l} (n) + μ x' (n - l) e (l)

(5)

where μ is the step size and

x' (n)

is the filtered reference signal. From equations (1) and (2) it is obvious that if

\overset{\land}{S} (z) = S (z)

, then

x (n) = d (n)

If this condition holds, which is assumed so, the adaptive feedback ANC system will be equivalent to the feedforward ANC system. If the LMS algorithm converges with a slow pace, i.e. the step size μ is small, then the adaptive filter $W (z)$ transforms with the secondary path transfer function $S (z)$ . Moreover, if it is assumed that the secondary path $S (z)$ can be modelled with a pure delay, i.e. $S (z) = z^{- Δ}$ , then the feedback ANC system is equivalent to the standard adaptive predictor.¹⁴ Figure 5 shows a block diagram of the standard adaptive predictor.

As a conclusion, one can use an adaptive predictor of the primary noise as feedback ANC system to minimize the residual error noise. In Figure 5, $H (z)$ is the overall transfer function of the feedback ANC system from $d (n)$ to $e (n)$ and is given by,

H (z) = \frac{E (z)}{D (z)} = 1 - S (z) W (z)

(6)

WAVENET theory

Identification using WAVENET

In the last few years, several approaches using artificial neural network (ANN) have been developed for identification of linear and nonlinear systems. To start the identification, one should collect a set of input–output plant data through experiments. Hence, a sequence of $(x^{n}, y_{m}^{n})$ as training data can be established, where $x^{n} = {x_{1}^{n}, x_{2}^{n}, \dots, x_{Ni}^{n}}$ is input and $y_{m}^{n}$ is output vector.²⁵ The main step in the identification procedure is to find a relation between previous data input $[x^{t - 1}, y^{t - 1}]$ and the current output of the system $y (t)$ , which can be described as

y (t) = f (x^{t - 1}, y^{t - 1}) + e (t)

(7)

where

e (t)

is noise signal which denotes that the output

y (t)

is not equal to the previous output exactly. Consider that the function f can be written as

f (x^{t - 1}, y^{t - 1}, θ) = f (ϕ (t), θ)

(8)

where

ϕ (t) = [y (t - 1) \dots y (t - n_{a}) x (t - 1) \dots x (t - n_{b})]^{T}

(9)

with

d = n_{a} + n_{b}

and

ϕ (t) \in ℜ^{d}

. Using a sequence of measured data as training data, shown as

Z^{N} = {y (t), ϕ (t), k = 1, 2, \dots, N}

(10)

then the following relation can be presented

\overset{\land}{y} (t) = {\overset{\land}{f}}_{N} (ϕ (t), θ)

(11)

where

{\overset{\land}{f}}_{N}

is an unknown nonlinear function. In fact, the model

{\overset{\land}{f}}_{N} (ϕ (t))

can be specified using different approaches. One can use the basis function to approximate the nonlinear function

{\overset{\land}{f}}_{N}

as follows

{\overset{\land}{f}}_{N} (ϕ (t), θ) = \sum_{k = 1}^{K} w_{k} h_{k} (ϕ)

(12)

where h_k is a basis function. One of the local basis functions is a wavelet basis function. In this paper, the nonlinear function

{\overset{\land}{f}}_{N}

will be specified using WAVENET (wavelet network) approach, where θ is the WAVENET parameter.

Wavelet function and network

The term wavelet means “a little wave”. A little wave in the positive direction and also negative direction, of its amplitude, has a minimum oscillation and a fast decay to zero. This property is similar to a required condition of a function that is used for the wavelet transform.

As studied by many researchers,^26,27 WAVENET can be used to approximate a signal. To do this, the goal is to find daughter wavelets. The daughter wavelet is created by a dilation (scale or compression) and translation (shift) of the original wavelets or mother wavelets. The wavelet theory has created a transformation called wavelet transform. The transform operates on a function by incorporating it with an adjusted form of some kernel functions called mother wavelet and also some kernel functions called daughter wavelet.

A function should have a condition to be known as mother wavelet. For a given function $h (t)$ , the permissibility condition for wavelet function is

\int_{- \infty}^{+ \infty} h (t) d t = 0

(13)

The wavelet transform of a function $f \in L^{2} (R)$ with respect to a given permissible mother wavelet $h (t)$ is defined as²³

W_{f} (a, b) = \int_{- \infty}^{+ \infty} f (t) h_{a, b}^{*} (t) d t

(14)

where * means complex conjugate, even though furthermost wavelets are real. The daughter wavelets are built from single mother wavelet

h (t)

by dilation and translation, or

h_{a, b} (t) = \frac{1}{\sqrt{a}} h (\frac{t - b}{a})

(15)

where a and b are the dilation and translation factors, respectively.²⁵

Wavelet function can be categorized into two classes: orthogonal wavelet and wavelet frame.²⁵ When related wavelet transform is orthogonal, it results to an orthogonal wavelet. Furthermore, with operating the simple operations of translation and dilation on mother wavelet function, the wavelet frame will be produced. The wavelet frame is a powerful tool for nonlinear identification.²⁸

WAVENET algorithm

Approximation of a nonlinear function is an important task for many applications. To approximate a nonlinear function using WAVENET, first one should construct a network, and then, error minimization should be performed by the algorithm. A WAVENET is classified in feedforward neural networks with benefits from wavelet as activation function. There is a relation between the inverse of discrete wavelet transformation with a single hidden layer of the neural network.²⁵ If a nonlinear auto-regressive with exogenous inputs (NARX) model structure is used as approximation, then the proposed network configuration for system identification will have WAVENET serial-parallel model. The proposed structure is shown in Figure 6.

To approximate a signal $y (t)$ with a WAVENET, a set of daughter wavelets which made of the mother wavelet with dilation and translation should form a linear combination with each other.

Supposing that the output function of the network fulfils the permissibility condition and the network could approximate the desired signal, and, time-frequency region which is efficiently covered by their K windows, then the approximated signal of the network $\overset{\land}{y} (t)$ can be described as

\overset{\land}{y} (t) = u (t) \sum_{k = 1}^{K} w_{k} h_{a_{k}, b_{k}} (t)

(16)

where K is the number of wavelets and w_k is the weight coefficient.²⁵ Figure 7 shows the configuration of adaptive wavelet network.²⁵

One should define the cost function to tune the neural network parameters $w_{k}, b_{k}, a_{k}$ using LMS. The error is as follows:

e (t) = y (t) - \overset{\land}{y} (t)

(17)

where y(t) is the expected output/target. The cost function is as follows:

E = \frac{1}{2} \sum_{t = 1}^{T} e^{2} (t)

(18)

One should apply the steepest descent approach to minimize E. To calculate the steepest descent, one needs the gradient $\frac{\partial E}{\partial w_{k}}$ , $\frac{\partial E}{\partial a_{k}}, \frac{\partial E}{\partial b_{k}}$ to tune the parameters $w_{k}, b_{k}, a_{k}$ for each wavelet. The gradient of the cost function E to each network parameter is given as:

\frac{\partial E}{\partial w_{k}} = - \sum_{t = 1}^{T} e (t) h (τ)

(19)

\frac{\partial E}{\partial b_{k}} = - \sum_{t = 1}^{T} e (t) w_{k} \frac{\partial h (τ)}{\partial b_{k}}

(20)

\frac{\partial E}{\partial a_{k}} = - \sum_{t = 1}^{T} e (t) w_{k} τ \frac{\partial h (τ)}{\partial b_{k}} = τ \frac{\partial E}{\partial b_{k}}

(21)

Each of the network parameters w, b, a of the WAVENET is updated as:

\underline{w} (n + 1) = \underline{w} (n) + μ_{w} Δ \underline{w}

(22)

\underline{b} (n + 1) = \underline{b} (n) + μ_{b} Δ \underline{b}

(23)

\underline{a} (n + 1) = \underline{a} (n) + μ_{a} Δ \underline{a}

(24)

where μ is the learning rate.²⁵

WAVENET ANC system

In this paper, a WAVENET is used as a predictor of the primary noise. Figure 8 shows a block diagram of the proposed approach. In the simulation section, the secondary path S(z) is supposed to be a pure delay $S (z) = z^{- 1}$ .

The input of the WAVENET is N samples which are passed through the pure delay. The WAVENET uses these N samples to predict the $(N + 1) th$ sample. The $(N + 1) th$ sample is the output of the WAVENET and is applied to the anti-noise speaker to generate an anti-noise signal. The loudspeaker generates a sampled signal which has the same amplitude as the $(N + 1) th$ sample but with 180° phase difference with the $(N + 1) th$ sample.

A simulation study

In this section, results of simulations are presented to show the effectiveness of the proposed method. One of the common methods of evaluating the ANC system performance is to use Gaussian white noise. However, this noise is not completely suitable for this work, because the acoustic noise does not cover all the ranges of the frequency. Furthermore, in acoustic noise despite the white noise the samples are dependent on each other. Therefore, for the simulation exercises, the acoustic noise is produced from 2000 samples of white noise and passed through a type-2 Chebychev low pass filter. The degree of the filter, attenuation range and cut-off frequency used were 9, 50 dB and 6 kHz, respectively. Using the obtained coloured noise, the training of the WAVENET can be done. For the training process the first 1000 samples were used. The data was fed into the WAVENET using the tapped delay line. Since the transfer function $S (z)$ is assumed to be $z^{- 1}$ , the objective is to find the next sample, i.e. according to $d (n - 1), d (n - 2), \dots, d (n - 20)$ , the $d (n)$ sample will be anticipated. In Figure 9, the compatibility of the WAVENET is shown in the training process. According to Figure 9, the WAVENET could fit with the 1000 data with 86.35% precision. To test the trained WAVENET the 1000 data samples were used. The expectation from the trained WAVENET is to produce the current sample using the former data. Figures 10 –12 show the second 1000 data samples of the coloured noise, 1000 data samples produced using the WAVENET (which is negated for use as the anti-noise signal) and the error data (which is the residual noise). In the test procedure, performance of the trained WAVENET in noise attenuation is calculated. Noise attenuation is evaluated using:

Noiseattenuation = 10 \times {log}_{10} \frac{Inputnoiseenergy}{Residualnoiseenergy}

The noise attenuation for this case was calculated as 29.9165 dB. To continue, with the trained WAVENET in the frequency range 0–5 kHz, the attenuation will be studied in different frequency ranges to check the performance of the system in new situations. Table 1 shows three experiments which were performed using the WAVENET trained in the 0–5 kHz frequency range. It is noted that the WAVENET performance was acceptable in frequency ranges where the system was trained, however, in the 0–6 kHz frequency range, the trained WAVENET performance was not acceptable.

Table 1.

Performance of the trained WAVENET with the coloured noise over 0–5 kHz frequency range.

Attenuation frequency range	Test 1	Test 2	Test 3
0–3 (kHz)	59.0210 (dB)	51.1524 (dB)	51.8115 (dB)
0–4 (kHz)	24.9165 (dB)	25.0260 (dB)	24.4195 (dB)
0–6 (kHz)	−0.8172 (dB)	−0.8058 (dB)	−0.8232 (dB)

Table 2 shows the results of the WAVENET trained in the frequency range of 0–3 kHz. As it is shown in Table 1, the performance of the WAVENE system was acceptable in 0–3 kHz, but the performance decreased as the frequency range increases.

Table 2.

Performance of the trained WAVENET with the coloured noise over 0–3 kHz frequency range.

Attenuation frequency range	Test 1	Test 2	Test 3
0–3 (kHz)	69.8912 (dB)	68.9315 (dB)	69.0121 (dB)
0–4 (kHz)	58.6324 (dB)	58.3212 (dB)	58.7318 (dB)
0–6 (kHz)	−1.2975 (dB)	−1.7136 (dB)	−1.2142 (dB)

For the next set of simulations, the input noise was periodic with frequency 250 Hz. The sampled noise passed through the pure delay and entered the WAVENET. Figure 13 shows the acoustic noise and the residual noise for the first simulation. In the second simulation, the input noise was the combination of two periodic noises with frequencies of 250 Hz and 350 Hz. Figure 14 shows the residual noise for the second simulation. In the third set of the simulation, the input noise was broadband. Figure 15 shows the input noise of the ANC system. To check the effectiveness of the proposed method, the power spectral density (PSD) of the input noise and of the residual noise is shown in Figures 16, 17 and 18 for the three simulations, respectively.

Figure 16 shows the PSD of the $250 Hz$ input noise and the PSD of the residual noise. Moreover, Figure 17 shows the PSD of the 250 Hz and 350 Hz input noise and the PSD of the residual noise. The broadband noise PSD and the residual noise PSD can be seen in Figure 18. As it is shown in Figures 16 and 17, the proposed approach has a good performance in the range of the main frequency spectrum.

Conclusion

In this paper, a WAVENET ANC system has been proposed. The conditions for converting the feedback ANC system to a signal predictor have been discussed. The proposed method was tested in simulations to show its effectiveness in producing an anti-noise signal to make a silent zone in the presence of the disturbance. The PSDs of the input noises and the residual noise in the silent zone have shown promising results for effectiveness of this method.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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