## 3 Simulation and Validation of the Proposed Algorithm  
### Synthesizing the Engine Noise Signal Inside a Car Cabin  
Engine noise is a periodic signal. From the theory of Fourier series expansion, a periodic signal (such as engine noise) can be decomposed into a sum of sinusoidal signals with the fundamental frequency and its harmonic frequencies [25]. These harmonic sinusoidal signals are called order signals. The frequency of the first-order signal matches the rotating speed of the crankshaft [3]. The frequency of a one-half order signal matches the fundamental frequency because the engine fires once whenever the crankshaft rotates 720\({}^{\circ}\). Therefore, the frequency components of engine noise are comprised of the rotating speed of the crankshaft and its harmonic. The engine noise is transferred to the inside the cabin of a car through an airborne path and a structure-borne path [2; 5; 26]. Therefore, the amplitude of each order signal is modulated according to the resonance of the structure and the cavity. When the speed of the engine changes, the rotating frequency of the crankshaft also changes. Therefore, the engine noise inside the cabin of a car is comprised of the sum of amplitude and frequency modulated order signals. This is mathematically expressed as follows:  
\[x(t)=\sum_{k=0.5}^{K}x_{k}(n)=A_{k}(t)\exp(j\theta_{k}(t)),\,k\,=\,\,0.5,\,1,\, 1.5,\,2\dots,\,K \tag{15}\]  
where \(x_{k}(t)\) is the kth order signal, \(A_{k}\) is its amplitude, and \(\theta k\) is its phase (which includes the information of the instantaneous frequency \(f_{i}\)). The instantaneous frequency of the \(k\)th order signal is given by [27]  
\[f_{k}^{i}(t)=\frac{1}{2\pi}\frac{d\theta_{k}(t)}{dt} \tag{16}\]  
For the order signal, this instantaneous frequency can be expressed in terms of the engine speed and is given by  
\[f_{k}^{i}(t)=\frac{RPM}{60}\times k \tag{17}\]  
If the engine speed is accelerated linearly, the instantaneous frequency is a linear equation and is given by \(f_{k}^{i}(t)=at+b\), where \(a\) is the sweep rate of the rotational speed and \(b\) is the initial instantaneous frequency at the starting rotational speed. If the engine speed is accelerated nonlinearly, the instantaneous frequency becomes a nonlinear equation. In this section, three signals are synthesized for the simulation. The sampling frequency used to synthesize the signals is 3000 Hz. **Signal 1** is a second-order signal with a linear instantaneous frequency, and its initial frequency \(f_{2}^{i}(t)\) is 20 Hz, which corresponds to 600 rpm. The final run-up frequency is 200 Hz, which corresponds to 6000 rpm. The run-up time from 600 rpm to 6000 rpm is 5 s. This run-up time corresponds to a short \begin{table}
\begin{tabular}{c c c} \hline _Parameters:_ & \(\mathbf{L_{\infty}}\) **= Number of Taps** & \\ \hline \multirow{3}{*}{_Initial condition:_ Data} & \(0<\overline{\mu}<2\) & \\  & **w** (0) = 0 & Use Equation (16) \\ \hline \multirow{3}{*}{(a) Given:} & \(\mathbf{x}^{\prime}(n)\); \(L_{\infty}\)-by-1 tap input vector & \(\mathbf{x}^{\prime}(n)=\mathbf{x}_{n}(n)\otimes\delta(n)\), \\  & \(\mathbf{x}_{n}(n)\): Normalized sinusoidal signal & \(\mathbf{x}_{n}(n)\): Measured impulse response of the 2 nd \\  & \(d(n)\): desired response at time \(n\) & path of the system includes electric part. \\ \cline{1-1}  & & Note: \(\otimes\) means convolution \\ \hline \multirow{3}{*}{(b) To be computed:} & \(\mathbf{w}(n+1)=\) estimate of tap-weight vector at time \(n+1\) \\  & \(n=0,1,2,\dots\) & \\ \cline{1-1}  & \(e(n)=d(n)-\mathbf{w}^{T}(n)\mathbf{x}^{\prime}(n)\) & \\ \cline{1-1}  & \(\mathbf{w}_{k}^{Lo}(n+1)=\mathbf{w}_{k}^{Lo}(n)+\frac{\mathbf{x}^{\prime}(n)} {|\mathbf{x}_{k}(n)|^{2}}\mu e_{k}(n)\) & \\ \hline \end{tabular}
\end{table} 
Table 1: Pseudo code of the OW-FxLMS algorithm. duration and the instantaneous frequency is changed quickly during this run-up time. The amplitude of the signal is unity.