# Active Control of Broadband Noise Inside a Car Using a Causal Optimal Controller
## 4 Algorithm Verification and Discussion
### Robustness and Computational Complexity Analysis  
In practice, the real primary path \(P_{k,r}\Big{(}e^{j2\pi f}\Big{)}\) of a physical system may vary from the identified result \(P_{k}\Big{(}e^{j2\pi f}\Big{)}\). Thus, the robustness of the proposed control algorithm has to be examined by defining the uncertainty of the primary path \(P_{k,r}\Big{(}e^{j2\pi f}\Big{)}\) as follows:  
\[\Delta=\left\|\frac{P_{k,r}(e^{j2\pi f})-P_{k}(e^{j2\pi f})}{P_{k}(e^{j2\pi f} )}\right\|_{\infty}, \tag{20}\]  
where \(\|\ \|_{\infty}\) denotes the infinite norm of the expression.  
The control performances are evaluated under different uncertainties. In the examination, six primary paths \(P_{1},\cdots,P_{6}\) are identified using the method proposed in Section 3, and six broadband random noise signals are applied to the real primary paths \(P_{1,r},\cdots,P_{6,r}\) as excitation sources to produce a noise signal, where the real primary paths \(P_{k,r}\Big{(}e^{j2\pi f}\Big{)}\), \(k=1,\cdots,6\) are derived as \(P_{k,r}\Big{(}e^{j2\pi f}\Big{)}=\ (1+\Delta V)P_{k}\Big{(}e^{j2\pi f}\Big{)}\) according to different uncertainties \(\Delta\), and \(V\) is a normalized random signal vector with maximal amplitude of one. Meanwhile, six causal optimal controllers derived using the identified primary paths \(P_{1},\cdots,P_{6}\) are adopted as controllers. Since the coefficients of the proposed COC controller approximate zero when its length is larger than 768, each filter of proposed COC algorithm and FXLMS algorithm is selected with the same memory length of 768. For the FXLMS  
Figure 7: The diagram of a multichannel ANC system.  
algorithm, the step size and normalization constant are \(\mu=0.1\), \(\delta=0.1\). Under a specific uncertainty \(\Delta\), a random signal vector \(V\) is generated to obtain the different real primary paths \(P_{k,r}\big{(}e^{j2\pi f}\big{)}\), \(k=1,\cdots,6\). The simulation results of averaging 50 independent runs for each specific uncertainty \(\Delta\) are displayed in Figure 8. FXLMS algorithm is adaptive, and it can adjust its coefficients to different primary paths. Therefore, the control results of FXLMS algorithm for different primary paths are similar, and only the result with uncertainty \(\Delta=0\) is shown in Figure 8.  
It can be observed from Figure 8 that the nominal noise reduction of proposed COC (\(\Delta=0\)) is about 8 dB, and by increasing uncertainty \(\Delta\) to 300%, the corresponding noise reduction decreases to 3 dB, which is at the same level as the result of FXLMS algorithm. Therefore, the proposed method can achieve better noise-reduction performance if the modelling accuracy of primary paths is within 300%. In most of cases, the modelling uncertainty can be limited within this range. Thus, the modelling errors of primary paths do not significantly deteriorate the noise-reduction performance, and the results of proposed algorithm are better than that of FXLMS algorithm. It should be noted that even though all the undesired noise is caused by the reference signals, only a small reduction is obtained. This is because the results are deteriorated by the nonminimum-phase characteristics of the secondary path.  
The computational complexity of the proposed COC algorithm and FXLMS algorithm is shown in Table 1, where \(L\), \(N\), \(M\) denote the length of the control filter, the secondary path, and the number of channels; \(\pm\) and \(\times\) denote the manipulations of addition and subtraction, and multiplication, respectively. Compared with the FXLMS algorithm, the proposed COC algorithm requires less computation irrespective of cancelling broadband noise or broadband as well as harmonic noises. Its advantage becomes more significant when the number of channels \(M\) is increased.