# Active Control of Broadband Noise Inside a Car Using a Causal Optimal Controller
## 3 Formulation of the Causal Optimal Controller  
Since broadband road noise inside a car is produced by the interaction between road surface and tires, it is complicated and at least six reference signals are required to achieve a satisfactory result [5]. Therefore, the combination of reference signals needs to be chosen according to coherence function between input signal \(x(n)\) and output undesired noise \(d(n)\)[24]. Since the vibrational signals measured in a vehicle are correlative to each other, the multiple coherence cannot be calculated directly, and the partial coherence function is required, which is defined as follows:  
\[\gamma_{x_{k}x_{1}\cdots x_{k-1}}^{2}(e^{j2 nf})=\big{|}S_{x_{k}d\cdot x_{1} \cdots x_{k-1}}(e^{j2 nf})\big{|}^{2}/\big{[}S_{x_{k}x_{k}\cdot x_{1}\cdots x_{k- 1}}(e^{j2 nf})S_{dd\cdot x_{1}\cdots x_{k-1}}(e^{j2 nf})\big{]}, \tag{8}\]  
where \(S_{x_{k}x_{k}x_{1}\cdots x_{k-1}}(e^{j2 nf})\), \(S_{dd\cdot x_{1}\cdots x_{k-1}}(e^{j2 nf})\), and \(S_{x_{k}d\cdot x_{1}\cdots x_{k-1}}(e^{j2 nf})\) are the residual power spectral density functions and the residual cross-spectral density function of signals \(x_{k}\), \(d\) after removing the components relating to the signals \(x_{1}\cdots x_{k-1}\) at frequency \(f\). When \(k=1\), they are simplified as \(S_{x_{1}x_{1}}\big{(}e^{j2 nf}\big{)}\), \(S_{dd}\big{(}e^{j2 nf}\big{)}\), and \(S_{dd\cdot x_{1}}\big{(}e^{j2 nf}\big{)}\), which can be derived from the autocorrelation and cross-correlation functions using discrete Fourier transformation [23]. The residual spectral density function can be derived using the following expression:  
\[S_{ij\cdot x_{1}\cdots x_{k-1}}(e^{j2 nf})=S_{ij\cdot x_{1}\cdots x_{k-2}}(e^{ j2 nf})-S_{ix_{k-1}\cdot x_{1}\cdots x_{k-2}}(e^{j2 nf})S_{x_{k-1}\cdot x_{1} \cdots x_{k-2}}(e^{j2 nf})/S_{x_{k-1}\cdot x_{1}\cdots x_{k-2}}(e^{j2 nf}), \tag{9}\]  
where \(i\) and \(j\) denote the signals \(x_{k}\) or \(d\). The partial coherence function \(\gamma_{x_{k}dx_{1}\cdots x_{k-1}}^{2}\big{(}e^{j2 nf}\big{)}\) of Equation (8) represents the contribution of signal \(x_{k}\) to the power of signal \(d\) after removing signals \(x_{1}\cdots x_{k-1}\) from them. Once they are acquired, the multiple coherence between reference signals \(x_{1}\cdots x_{K}\) and the undesired noise \(d\) can be derived as follows [24]:  
\[\gamma_{x_{1}\cdots x_{K}d}^{2}(e^{j2 nf})=1-(1-\gamma_{x_{1}d}^{2}(e^{j2 nf})) (1-\gamma_{x_{2}d\cdot x_{1}}^{2}(e^{j2 nf}))\cdots(1-\gamma_{x_{K}d\cdot x_{1} \cdots x_{K-1}}^{2}(e^{j2 nf})). \tag{10}\]  
The result of Equation (10) represents the multiple coherence between the undesired noise and the set of reference signals. As we know, a widely accepted expression to estimate the maximal noise reduction is [4; 11; 21], as follows:  
\[NR(e^{j2 nf})=-10\lg(1-\gamma_{x_{1}\cdots x_{K}d}^{2}(e^{j2 nf})). \tag{11}\]  
Figure 4: The impulse response of the noncausal inversion for the secondary path. The above equation shows that the larger the result of Equation (10), the more potential noise reduction can be achieved. Thus, the most appropriate reference signals can be obtained according to the calculation results of all the different combinations of \(K\) signals from all the measured vibrational signals. The combination of \(K\) signals with the largest multiple coherence value will be used as reference signals, and the corresponding primary paths of vibration and noise propagation are required to be identified. Since the disturbances in the input and output signals cannot be acquired, it is appropriate to assume that their power ratio equals one and identify the primary system paths using the following equation [25]:  
\[P(e^{j2\pi f})=\bigg{\{}S_{dd}(e^{j2\pi f})-S_{xx}(e^{j2\pi f})+\sqrt{\big{[} S_{dd}(e^{j2\pi f})-S_{xx}(e^{j2\pi f})\big{]}^{2}+4\big{|}S_{xd}(e^{j2\pi f}) \big{|}^{2}}\bigg{\}}/\big{[}2S_{dx}(e^{j2\pi f})\big{]}. \tag{12}\]  
The first primary path \(P\big{(}e^{j2\pi f}\big{)}\) can be identified directly using Equation (12).