# Active Control of Broadband Noise Inside a Car Using a Causal Optimal Controller
## 2 Secondary Path Analysis
### Noncausal Inversion  
In this part, a practical secondary path shown in Figure 2 is adopted, where Sp and Mic denote the door loudspeaker and error microphone, respectively. The secondary path from the output of the controller to the measured signal of Mic is identified with a 13 th order infinite impulse response (IIR) filter and its impulse response, zeros, and poles distribution are shown in Figure 3. It can be seen from Figure 3b that there are five zeros outside the unit circle. Thus, the secondary path is a nonminimum-phase system, and its inversion will contain both causal and noncausal parts.  
Figure 2: The positions of the loudspeaker and error microphone.  
From Figure 3b, we know that there are no repeated zeros in the secondary path. So, its inversion \(S^{-1}\)(z) can be expressed using partial fraction decomposition as [23], as follows:  
\[S^{-1}(z)=\sum_{q=1}^{Q}\frac{A_{q}}{1-p_{q}z^{-1}}, \tag{4}\]  
where \(p_{q}\) is the \(q\)th pole of \(S^{-1}\)(z) and also the \(q\)th zero of \(S\)(z) and \(A_{q}\) is the related coefficient. Because all the coefficients of the filter \(S\)(z) are real, its poles and zeros must be either purely real or appear in complex conjugate pairs. By including the unit circle into the convergence region, the term \(A_{q}/(1-p_{q}z^{-1})\) of Equation (4) corresponding to purely real \(p_{q}\) can be expanded as follows:  
\[X_{q}^{r}(z)=\left\{\begin{array}{l}A_{q}\sum\limits_{n=0}^{\infty}\left(p_{ q}z^{-1}\right)^{n}\big{|}p_{q}\big{|}\leq 1\\ -A_{q}\sum\limits_{n=1}^{\infty}\left(p_{q}^{-1}\right)^{n}\big{|}p_{q}\big{|} >1.\end{array}\right. \tag{5}\]  
For a pair of conjugate zeros \(p_{q}\) and \(p_{q}^{*}\) of \(S\)(z), their corresponding coefficients are also conjugate. Thus, the expansions can be simplified from Equation (5) into an expression of conjugate pairs. According to whether the conjugate zeros pair, \(p_{q}\) and \(p_{q}^{*}\), are inside or outside the unit circle; their expansion \(X_{q}^{c}(z)\) can be rewritten as Equations (6), as follows:  
\[X_{q}^{c}(z)=\left\{\begin{array}{l}\sum\limits_{n=0}^{\infty}2\big{|}A_{q} \big{|}r_{q}^{n}\cos(\beta_{q}n+\alpha_{q})z^{-n}\quad\big{|}p_{q}\big{|}\leq 1\\ -\sum\limits_{n=1}^{\infty}2\big{|}A_{q}\big{|}r_{q}^{-n}\cos(\beta_{q}n- \alpha_{q})z^{n}\quad\big{|}p_{q}\big{|}>1,\end{array}\right. \tag{6}\]  
where \(\alpha_{q}\), \(\beta_{q}\), and \(r_{q}\) are derived from \(A_{q}=\big{|}A_{q}\big{|}e^{i\alpha_{q}}\), \(p_{q}=r_{q}e^{j\beta_{q}}\). Equation (6) corresponds to the causal and noncausal responses of conjugate zeros pairs. By substituting Equations (5) and (6) for Equation (4), the inversion of \(S\)(z) is obtained as follows:  
\[S^{-1}(z)=\sum\left(X_{q}^{r}(z)+X_{q}^{c}(z)\right). \tag{7}\]  
The finite impulse response of \(S^{-1}\)(z) in Equation (7) corresponding to the secondary path in Figure 2 is shown in Figure 4. The noncausal part of the response is displayed in the range of \([-400,0)\), which is the summation of Equation (7) when \(|p_{q}|>1\). The causal part of the response is shown in the range of \([0,100]\), which is the summation of Equation (7) when \(|p_{q}|\leq 1\). It can be seen that it will take a  
Figure 3: The identified secondary path: (**a**) impulse response and (**b**) zeros (crosses) and poles (circles) distribution.  
long period for noncausal response to decay. Therefore, the nonminimum-phase secondary path \(S(z)\) will have a significant effect on control performance, and a large proportion of the undesired noise cannot be reduced using this secondary path.