## 2 Development of Novel Algorithm for Control of Amplitude of Order Sound  
### Conventional Method for ANC of Engine Noise  
The FxLMS-algorithm-based adaptive notch filter has been used for the active noise cancellation of engine noise [4; 5; 22]. It is called the \"conventional method\" in the paper. The scheme of this ANC system is shown in Figure 1. This system uses two single sinusoidal reference signals, \(x_{0}\) and \(x_{1}\), and two single weights, \(w_{0}\) and \(w_{1}\), to control the engine noise. \(S\)(z) is the secondary path of the transfer function, and \(\hat{S}(z)\) is the estimation of the secondary path. A suitable reference signal can be generated synthetically by measuring the current rotation speed \(f_{k}{}^{i}\) of the engine and utilizing a discrete-time oscillator [23]. The primary disturbance \(d(n)\) comprises a multi-sinusoidal signal, which is the engine noise inside the cabin of a car. The multi-sinusoidal signal is correlated to the reference signals. In order to control the multi-sinusoidal signal, this scheme could be expanded to a multi-input/multi-output FxLMS control algorithm [6; 7]. In the simplified single-frequency ANC system as shown in Figure 1, \(y_{k}(n)\) is given by  
\[y^{k}(n)=w_{0,k}(n)x_{0,k}(n)+w_{1,k}(n)x_{1,k}(n) \tag{1}\]  
The two weights \(w(n)\) of the adaptive filter are updated as  
\[ \begin{array}{l}w_{0,k}(n+1)=w_{0,k}(n)+\mu e_{k}(n)x^{\prime}_{0,k}(n)\\ w_{1,k}(n+1)=w_{1,k}(n)+\mu e_{k}(n)x^{\prime}_{1,k}(n)\end{array}  \tag{2}\]  
where \(\mu\) is the step size controlling the stable convergence of the controller and \(x^{\prime}(n)\) is the filtered version of \(x(n)\) and is given by  
\[x^{\prime}_{i,k}(n)=\sum_{m=1}^{M}\hat{s}_{m}x_{i,k}(n-m),\ i=0,\ 1 \tag{3}\]  
where \(\hat{s}_{m}\) is an \(M\)th-order estimate of the secondary path impulse response. The adaptive notch filter removes the primary spectral components within a narrow band centered at the frequency \(f_{k}{}^{i}\) (\(t\)) of the reference signal. The performance of the adaptive notch filter to cancel a sinusoidal signal depends heavily on the accuracy of the frequency of the reference signal [16] since the center frequency of the notch filter depends on the sinusoidal reference signal, whose frequency is equal to the frequency of the sinusoidal noise to be cancelled. The increment of step size \(\mu\) results in a stability problem because, in an adaptive notch filter, the convergence step size is proportional to the bandwidth of the filter [9; 15] and the higher the step size, the higher the bandwidth. Considering stability, the optimal step size is proposed [7] as follows:  
\[0<\mu<\frac{4}{LA^{2}} \tag{4}\]  
where \(L\) is filter length of adaptive filter and \(A\) is the amplitude of the sinusoidal reference. The 3 dB bandwidth \(B\) of the notch filter can be estimated as  
\[B\approx\frac{\mu LA^{2}}{4\pi\Omega} \tag{5}\]  
Figure 1: Narrowband FxLMS algorithm based adaptive notch filter. where \(\Delta\) is sampling time. From Equation (8), when the instantaneous frequency of the sinusoidal reference varies rapidly, the bandwidth \(B\) adaptive filter should be very narrow in order to track the time varying frequency and thus, the step size \(\mu\) decreases. The small step size \(\mu\) yields a slow convergence. The slow convergence cannot track the subsequent rapid varying frequency of the sinusoidal reference. Therefore, except for a quasi-stationary sweep of the instantaneous frequency of the sinusoidal reference, this system is not appropriate for controlling a frequency-modulated signal with a quickly variable bandwidth. Moreover, the performance of this ANC system is reduced when the run-up speed of the engine is changed rapidly [10; 12]. Through further improvements, a new LMS algorithm based on the variable step size is proposed. Several variable step size strategies have been suggested to improve the performance of the LMS algorithm.