## 2 Development of Novel Algorithm for Control of Amplitude of Order Sound
### Novel Optimized Weights-LMS Algorithm: OW-FxLMS  
We propose a new algorithm to improve the performance of the FxLMS algorithm to control an amplitude- and frequency-modulated signal such as an engine harmonic order signal. The main objective the proposal algorithm is to control the amplitude of time varying engine order sounds with fast frequency change. In order to achieve this purpose, the design principles of the new ANC algorithm are that the frequency bandwidth of the adaptive filter should be wide and include the frequency of the engine order sound signal to be controlled by the adaptive filter. However, the limitation of this adaptive filter cannot control each order sound separately if frequencies of order sounds are too close, because the frequency bandwidth is wide. This is the limitation of the proposed algorithm. In order to overcome this limitation, the weighting of the adaptive filter should optimized. Therefore, in this section, the OW-FxLMS is proposed. In order to trace a fast frequency change of the order sound signal, the optimized sufficient frequency bandwidth is required. The frequency bandwidth of an adaptive filter for tracking the frequency modulated signal is determined by the length of the weight vector [8; 9; 14]. The proposed algorithm is an FxLMS-algorithm-based adaptive filter with an optimal length of the weight vector \(\mathbf{w}_{k}\) (\(n\)). We determined the optimal length of the weight vector in order to trace a frequency-modulated signal such as an engine order signal. The frequency bandwidth between orders for rotating machinery is given by  
\[f_{OB}(RPM,k)=\frac{RPM\times\Delta k}{60} \tag{9}\]  
where \(\Delta k\) is the difference between the orders to be controlled. According to the theory of adaptive signal processing [9], the frequency bandwidth of an adaptive filter is given by \(f_{WB}=2f_{s}/L\) where \(f_{s}\) is the sampling frequency. The optimal frequency bandwidth \(f_{\text{WB}}\) of an adaptive filter to be used by a controller should be less than the frequency bandwidth \(f_{OB}(RPM,k)\) between the orders to be controlled by a controller. The adaptive filter is a moving bandpass filter with different center frequencies. The frequency bandwidth of the frequency-modulated signal to be cancelled should be less than the bandwidth of the adaptive filter in the proposed OW-FxLMS algorithm. Therefore, in the ANC system shown in Figure 2, \(y_{k}\) (\(n\)) is given by  
\[y(n)=\mathbf{w}^{T}(n)\mathbf{x}(n)=\sum_{l=0}^{L_{\text{o}}-1}w_{l}(n)x(n-l) \tag{10}\]  
And the proposed OW-FxLMS algorithm is given by  
\[\mathbf{w}_{k}^{Lo}(n+1)=\mathbf{w}_{k}^{Lo}(n)+\mu e_{k}(n)\mathbf{x}_{k}^{ \prime}(n) \tag{11}\]  
where the length of weight vector is the optimal length \(L_{0}\). It is given by  
\[L_{\text{o}}=\frac{2\times f_{s}}{f_{OB}},\text{or }L_{\text{o}}=\frac{2 \times 60\times f_{s}}{RPM\times\Delta k} \tag{12}\]  
The step size parameter \(\mu\) should be carefully selected to ensure that the LMS algorithm converges rapidly and is stable. [PERSON] (9) proposed the condition to select the appropriate step size parameter as follows:  
\[0<\mu<\frac{2}{L_{\text{o}}P_{x}} \tag{13}\]  
where \(P_{x}\) denotes the power of \(x(n)\). From Equations (12) and (13), when the power of the reference signal changes, if \(L_{\text{o}}\) is changed, the step size \(\mu\) should also be changed according to the convergence theory of the LMS algorithm. In practice [7], the relationship between step size and filter length for convergence is as follows:  
\[\frac{0.01}{L_{\text{o}}P_{x}}<\mu<\frac{0.1}{L_{\text{o}}P_{x}} \tag{14}\]  
When the input signal has varying frequency, the NLMS algorithm with variable step size should be used for further improvements. Therefore, the estimated power \(\hat{P}_{x}(n)\) of \(x(n)\) at time \(n\) should be computed dynamically and thus, the variable step size is changed corresponding to Equation (6) because the optimal filter length \(L_{\text{o}}\) is constant.