Multi-tone active noise control (multi-tone ANC) technology has been widely studied to reduce the tonal noise with low frequencies. In the case of an ANC system with a time-varying secondary path, an on-line modeling method is necessary. In this work, a multi-tone ANC system with a simplified local on-line secondary-path modeling method is presented. A two-tone signal with small amplitude is added at the sideband of each noise tone to conduct the local modeling. The least mean square algorithm is used as the modeling algorithm which provides tracking for a time-varying secondary path. Notch filters and bandpass filters are adopted to separate the primary tones and the additive tones. Simulations and experimental results show that the proposed algorithm has good performance.

## 1. Introduction

Active noise control (ANC) technology has good performance in the lower frequency range.^{1–8} In the case of a system with harmonic noise, a multi-tone parallel ANC algorithm using the filtered-x least mean square (FxLMS) algorithm is often adopted. In these multi-tone ANC systems based on the FxLMS algorithm, secondary-path modeling plays an important role. This modeling of transfer function is often performed in the off-line method when the secondary path is assumed as a time-invariant one. However, in the applications where the secondary path changes over time, the performance of a multi-tone ANC system with the off-line modeling method becomes poor and an on-line secondary-path modeling has to be implemented. In the literature,^{9,10} an overall on-line secondary-path modeling algorithm is presented. This method suffers from robustness especially in the cases where both the primary path and the secondary change at the same time. More often, the on-line secondary-path modeling is conducted in the form of broadband modeling^{11–16} by adding a broadband signal. This way suffers from two drawbacks in the multi-tone ANC system: (I) The broadband modeling requires a large filter order which means high computational complexity. (II) The additive signal is generated as a broadband one. If the variance of this additive signal is small, it will lead to slow convergence and a large modeling error. On the other hand, if the variance is large, the additive noise itself might turn to another noise.

Several improved strategies have been presented in the literature^{17,18} to deal with problems (I) and (II). However, these methods still suffer from computational burden as the modeling is conducted in the form of a broadband signal. By considering that only the narrowband identification is implemented in the multi-tone ANC applications, it is unnecessary to conduct a broadband identification. Two narrowband on-line secondary-path strategies are presented in Refs. 19 and 20 to reduce the computational complexity. In the literature,^{19} a local modeling is conducted in each adjacent frequency sub-band by using narrowband white additive signals filtered by bandpass filters. As the modeling is conducted in the narrowband, the order of the secondary-path model can be decreased and the computational complexity is simplified. Another similar method is given in the literature,^{20} where the sub-band modeling is replaced by local modeling just near the frequency of the primary noise. These narrowband modeling technologies have improved performance. However, there is still a conflict involving the bandwidth, the modeling order, and the convergence rate. If the modeling order is chosen too low (e.g., 2), the zero-pole parameters must be set close to 1 (e.g., 0.9999) to provide a small modeling bandwidth which leads to slow convergence (otherwise it is hard to conduct an accurate secondary-path modeling with a low-order filter as the bandwidth of the modeling signal is large). If the modeling order is set as a high one, the drawbacks (I), (II) of the broadband on-line modeling will arise again.

In this work, a simplified local secondary-path modeling is given. On the basis of steady-state sinusoidal response, the secondary path of each frequency is simplified as a complex parameter.^{3,8} In this way, the local modeling is conducted by replacing the additive narrowband white signal with tonal signals. More precisely, a two-tone additive signal with small amplitude is generated around the modeling center frequency of the primary noise at each sub-band. These two tones have similar frequencies, one of which is slightly lower than the modeling center frequency while the other is slightly higher. Notch filters and bandpass filters are used to separate the additive tones. Benefiting from the periodic response of sinusoidal signals, the modeling process has fast convergence and small additive noise and the computation complexity is always at the level of 2-order modeling.

## 2. A multi-tone ANC system with a simplified local on-line secondary-path modeling

In this section, a multi-tone ANC system with a simplified local secondary-path modeling is introduced. As shown in Fig. 1, the structure of the proposed system [Fig. 1(b)] is similar to the typical multi-tone ANC system^{3,4,6,8,12} [Fig. 1(a)], and differs only in the on-line secondary-path modeling. In this work, the primary noise signal with *N* tones is assumed as

where $pi(n)$ is the *i*th term of the noise with frequency $fi$, and $v(n)$ is the system noise which is assumed as a white noise with zero mean and $\sigma 2$ variance in this work. Each term of the primary signal is assumed as

where the angular frequency $\omega i$ is defined as

$Api$ represents the amplitude of the tonal signal, $\varphi di$ is the primary phase, and $fs$ is the sampling frequency in the system. Based on the steady-state response of the periodic signal,^{3,8} the secondary path can be assumed as a simplified frequency-domain one which has a form of

where $Asi$ is the amplitude and $\varphi si$ represents the phase. The secondary-path modeling can be conducted in the on-line or off-line way and the result is given in the form of

where $S\u0302i$, $A\u0302si$, and $\varphi \u0302si$ are the estimation of the real parameters. As the primary signal is a harmonic one, the reference signal of the multi-tone ANC system can be used in the form of two complex signals with unity amplitude and $\pi /2$ shift in the phase

Then the multi-tone ANC method is conducted using a parallel FxLMS algorithm^{3}

where $ai(n)$ and $bi(n)$ are the weights of the adaptive filter, $\mu $ is the step size, and $e(n)$ is a real error signal obtained by the error sensor. The output signal can be generated as

where “$\u2022*$” is the complex conjugate of term “$\u2022$.” So, the final secondary signal at the error point can be given as

where $yi\u2032(n)$ is the output $yi(n)$ affected by the secondary path transfer function $Si$.

In order to conduct the local on-line secondary-path modeling, the frequency of each primary signal $fi$ is chosen as the modeling center and an additive signal with two tones $uIi(n)$ and $uIIi(n)$ is generated in the form

where the amplitudes $AIi$ and $AIIi$ are much smaller than the amplitude $Api$ of the primary signal and $\Delta \omega $ is defined as

where $\Delta f$ is a small frequency gap (e.g., 10 Hz). The secondary path of these two additive tones are assumed as $SIi(z)$ and $SIIi(z)$. Then, the secondary-path modeling is conducted by using the least mean square (LMS) algorithm at both $fi\u2212\Delta f$ and $fi+\Delta f$. In this work, the secondary path is assumed as a continuous function of frequency around the center frequency, and the modeling of $fi$ can be estimated by the sideband modeling results provided a small frequency gap $\Delta f$. To separate the primary and the additive sinusoids, notch filters and bandpass filters are adopted. First, the error signal $e(n)$ is filtered by a set of cascaded notch filters to remove the primary tones

where the notch filter block $Hi(z)$ is a second-order infinite impulse response (IIR) filter^{21,22}

where $\omega i$ represents the center angular frequency of the local modeling, and $\alpha $ is a zero-pole parameter with a value between 0 and 1. Then bandpass filters are used to separate the additive tones, and the bandpass filters can be defined in the expression^{18,21}

where $\rho $ is the zero-pole parameter of the bandpass filter which is positive but smaller than unity. Further, the results of the bandpass filters $HIi(z)$ and $HIIi(z)$ are defined as $eIi(n)$ and $eIIi(n)$ which are used to obtain the estimation $S\u0302Ii(z)$ and $S\u0302Iii(z)$ of the secondary path $SIi(z)$ and $SIIi(z)$ by using the LMS algorithm. Finally, the estimation of $Si(z)$ can be given by averaging $S\u0302Ii(z)$ and $S\u0302Iii(z)$ in both real and imaginary parts

where Re[·] represents the real component of the term and Im[.] is the imaginary term. On the basis of assumption that the secondary path is a continuous one around the modeling center frequency, the linear average method in Eq. (15) can give an accurate estimation of the secondary path $S\u0302i(z)$ at the modeling center frequency provided small $\Delta f$.

## 3. Simulations

In this section, simulations are conducted to show the performance of the proposed multi-tone ANC system with a simplified local on-line secondary-path modeling and compare it with some excellent algorithms. The sampling frequency $fs$ is set as 5000 Hz in all simulations.

The first simulation is given as an overall effect of the simplified local on-line modeling method. The data of the secondary path are obtained in a real air pipe ANC system which is the same as our previous work.^{5,8} The broadband secondary path is acquired in the form of a finite impulse response filter with 100 order length (*L* = 100). Then the simplified local on-line secondary-path modeling is conducted at each frequency point (from 20 to 500 Hz by using the obtained transfer function) in the method of internal model simulation and results are shown in the same figure. The obtained broadband secondary path and the result of an off-line local simplified modeling process are compared with the on-line method. In Figs. 2(a) and 2(c), the modeling performance is compared in the form of amplitude (which is normalized on the basis of the system response range), and the comparison of the phase is given in Figs. 2(b) and 2(d). To show the influence of the broadband noise in the system, two zero-mean white noise signals with different variance is added. In Figs. 2(a) and 2(b), the system noise is set to make the signal-to-noise ratio (SNR) $\sigma 2/AKi2=10$ (*K* = I or II) and this ratio changes into 1 in Figs. 2(c) and 2(d). The off-line and on-line secondary-path modeling are conducted in the proposed algorithm with $\Delta f$ = 10 Hz. Although the variance of the modeling error increases slightly in the poor SNR conditions, the proposed algorithm shows good performance.

The second simulation is conducted to show the performance of the multi-tone ANC system with a simplified local on-line secondary-path modeling algorithm. The conventional on-line secondary-path modeling and other two wonderful algorithms (Ma's algorithm^{18} and Chang's algorithm^{20}) are compared with the proposed algorithm. The primary noise is a multi-tone signal with three frequencies 100, 200, and 300 Hz. The amplitudes of all three tones are set as unity. A zero-mean white noise with $\sigma 2=0.1$ is added in Fig. 2(e), and this variance of the noise changes into $\sigma 2=0.01$ in Fig. 2(f). The frequency gap of the additive modeling signal is set as 20 Hz in the proposed algorithm, and the zero-pole parameters are set as $\alpha =0.99$ and $\rho =0.995$. The additive modeling signals are set at the same amplitude and all the parameters are chosen carefully to keep the same noise reduction value in the steady state in each algorithm to ensure the fair comparisons in convergence rate. All of the simulation results are given by averaging over 100 independent runs. Figures 2(e) and 2(f) show the excess mean-square error performance of all the algorithms. To show the robustness, the amplitude of the secondary path transfer function is suddenly changed by multiplying 0.8 at the middle of the iterations. As shown in Figs. 2(e) and 2(f), the abrupt change has little impact on all the algorithms, and the proposed algorithm still has a faster convergence rate. Figures 2(g)–2(i) show the tracking curve of the secondary-path modeling in the form of a real component and an imaginary component. To be more specific, Fig. 2(g) shows the tracking curve of $S\u0302100\u2009Hz(z)$ and Figs. 2(h) and 2(i) give the secondary-path modeling at 200 and 300 Hz. As shown in Figs. 2(g)–2(i), the proposed algorithm shows a fast convergence rate as well as a small error. Although the secondary path changes at the middle of the iterations, all three estimation curves of the secondary path keep good robustness.

## 4. Experiments

In this section, the results of the experiment are given with discussion. The experiment is carried out in a pipe ANC system with an active-passive muffler which is the same as our previous work.^{5,8} The signal processing is conducted by a controller with a development board on the basis of TMS320c 6747. The sampling frequency $fs$ is chosen as 5 KHz in the experiment. As shown in Fig. 3, the primary noise is a signal constituted by three tones and a broadband noise signal, and the sound pressure level (SPL) of the periodic signal is 97 dB (100 Hz), 91 dB (200 Hz), and 91 dB (300 Hz) while the broadband background signal has a SPL of about 84 dB. The reference signal is generated in the form of sinusoids and the additive signal is given with large distance ($\Delta f$ = 20 Hz) and small amplitude (SPL = 67 dB) at each modeling center frequency. The proposed algorithm shows good performance, and the noise reduction is about 40 dB at each primary frequency.

## 5. Analysis and discussion

In this work, the additive signals of the proposed simplified on-line secondary-path modeling method are tonal signals. The periodicity of the sinusoidal signal simplifies the modeling complexity, and the secondary path can be estimated as a complex parameter. As the modeling is conducted online using the LMS algorithm, the proposed algorithm has good tracking performance and robustness in the system with a time-varying secondary path. Notch filters and bandpass filters are employed to separate tones instead of generate narrowband signals. So, the value of zero-pole parameters can be set smaller in the proposed algorithm than in the methods of Delegà *et al*.^{19} and Chang *et al*.^{20} (e.g., 0.99 instead of 0.9999) as the additive signal has been a narrowband one, and the convergence rate is improved. Further, the proposed algorithm has a high local SNR at each modeling frequency because the additive signal is a tonal one and notch filters and bandpass filters remove most of the noise power (involving the tonal noise and broadband noise). So the amplitude of the additive modeling signal can be set as a very small value while the modeling accuracy is high.

In addition, the center modeling frequency is assumed as the known knowledge of the modeling process in this work. In the system with time-varying frequencies, the additive signal and the parameters of the notch/bandpass filters should be adjusted all the time (otherwise the system might work poorly). An adaptive notch filter in the cascaded or parallel structure^{8} can be used to track the frequency and provide the previous knowledge of the modeling algorithm. Then the center frequency and other parameters can be adjusted continuously by using the results of the frequency tracking.

However, the proposed algorithm still suffers from one certain problem. If noise signal has close frequencies, the interval has to be set as a very small value and the bandpass filters have to be set as a very small window which means slow convergence. This problem, however, can only be alleviated rather than completely solved. In the cases where the primary noise signal has a sufficient frequency gap (e.g., over 5 Hz) or the secondary-path phase changes gently in the frequency domain, the proposed algorithm has good performance. In the opposite cases [with a small frequency gap (e.g., smaller than 3 Hz) and abrupt changing secondary-path phase in the frequency domain] the convergence rate and the modeling accuracy have to be compromised.

Then, the computational complexity has to be analyzed. In Table 1, detailed computational complexity of the conventional algorithm (the classical broadband on-line secondary-path modeling^{11}) and the proposed algorithm is given, where *L* is the length of the secondary-path model, and *N* represents the number of noise frequencies. Although the computational burden increases a little because of the additional notch filters and bandpass filters, the complexity of the secondary path in the proposed algorithm is maintained at the 2-order modeling level. So, the complexity of the modeling process and FxLMS algorithm can be reduced, and the overall calculation burden of the ANC system is much lower than the conventional algorithm.

Algorithm . | FxLMS algorithm . | On-line modeling . | Total . | Example: L = 200 (Total)
. | ||||
---|---|---|---|---|---|---|---|---|

Multiplication . | Addition . | Multiplication . | Addition . | Multiplication . | Addition . | Multiplication . | Addition . | |

Conventional | (2L + 6)N | (2L + 2)N − 1 | 3L + 1 | 2L | (2L + 6)N + 3L + 1 | (2L + 2)N + 2L − 1 | 406N + 601 | 402N + 399 |

Proposed | 10N | 6N − 1 | 32N | 27N − 1 | 42N | 33N − 2 | 42N | 33N − 2 |

Algorithm . | FxLMS algorithm . | On-line modeling . | Total . | Example: L = 200 (Total)
. | ||||
---|---|---|---|---|---|---|---|---|

Multiplication . | Addition . | Multiplication . | Addition . | Multiplication . | Addition . | Multiplication . | Addition . | |

Conventional | (2L + 6)N | (2L + 2)N − 1 | 3L + 1 | 2L | (2L + 6)N + 3L + 1 | (2L + 2)N + 2L − 1 | 406N + 601 | 402N + 399 |

Proposed | 10N | 6N − 1 | 32N | 27N − 1 | 42N | 33N − 2 | 42N | 33N − 2 |

## 6. Conclusions

A multi-tone ANC system with a simplified on-line modeling algorithm is presented in this work. The proposed algorithm conducts the local secondary-path modeling by generating a tonal additive signal. Notch filters and bandpass filters are used to separate each tone and the adaptive iteration process is on the basis of the LMS algorithm. As the additive signal is in the form of a sinusoid, the secondary path can be modeled in the form of a complex parameter and this proposed algorithm has three improvements: fast convergence rate, small amplitude of the additive modeling signal, and 2-order modeling computational burden. Simulations and experiment results show that the proposed algorithm has good performance in both steady state and time-varying state. In the future, the simplified on-line modeling algorithm will be equipped with a frequency tracking technology to provide the frequencies of the noise and modify the center modeling frequencies and the notch/bandpass filters. Then the on-line secondary-path modeling can be conducted in real time. In this way, the primary noise signal with time-varying frequencies can also be effectively reduced.

## Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant Nos. 11474306 and 11472289) and Youth Innovation Promotion Association, the First Action Plan Program of Institute of Acoustics, Chinese Academy of Sciences.