# Active Control of Broadband Noise Inside a Car Using a Causal Optimal Controller
## 4 Algorithm Verification and Discussion
### Implementation and Verification  
In this subsection, the simulations are performed using the data measured from the setup of Figures 2 and 5 when the car is driving at a rough road. Since the noise inside a car contains both  
\begin{table}
\begin{tabular}{l l l l l l} \hline \hline  & **Noise** & **Output** & **Filtering** & **Updating** & **Total** \\ \hline \multirow{2}{*}{**FXLMS**} & Broadband and harmonics & \(\pm\)\(M(L-1)\) & \(M(N-1)\) & \(ML\) & \(M(2L+N-2)\) \\  & & \(\times\)\(ML\) & \(MN\) & \(M(L+1)\) & \(M(2L+N+1)\) \\ \hline \multirow{4}{*}{**COC**} & Broadband & \(\pm\)\(M(L-1)\) & 0 & 0 & \(M(L-1)\) \\  & & \(\times\)\(ML\) & 0 & 0 & \(ML\) \\ \cline{1-1} \cline{2-6}  & Broadband & \(\pm\)\((M+1)(L-1)\) & \(N-1\) & \(L\) & \((M+2)(L-1)+N\) \\ \cline{1-1}  & and harmonics & \(\times\)\((M+1)L\) & \(N\) & \(L+1\) & \((M+2)L+N+1\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Computational complexity analysis.  
Figure 8: Control results with different uncertainties.  
broadband noise and harmonics of the engine, an adaptive controller \(W_{h}\) updated using FXLMS algorithm is added into the control system to reduce the engine harmonic noise, and the proposed six-channel causal optimal controllers are used to reduce the broadband noise. The parameters for COC and FXLMS algorithms are \(L=768\), \(\mu=0.002\), and \(\delta=0.1\).  
The control results in time domain are shown in Figure 9a. It can be seen that the proposed COC algorithm outperforms FXLMS algorithm with an approximate 6 dB reduction, while FXLMS algorithm only obtains a 2.5 dB noise reduction. Comparing Figure 9a with Figure 8, we can find that the noise reductions of two algorithms are decreased. That is because all the undesired noise in Figure 8 is caused by the six reference signals, while for the noise inside a car, only part of it is induced by the reference signals. Thus, the control performance can be improved further by increasing the number of reference signals.  
The control results in frequency domain are shown in Figure 9b. The main attenuation occurs around 34 and 87 Hz. The noise peak at 34 Hz is caused by the resonance of the car. After control, FXLMS algorithm achieves about 7 dB reduction at peak, while for the proposed COC algorithm, a maximal reduction up to 12 dB is achieved. The peak at 87 Hz is the second-order harmonic of the engine, which is reduced distinctly under two algorithms. Meanwhile, for the noise between 30 and 130 Hz, the proposed COC algorithm achieves a significant reduction in the whole frequency band. However, FXLMS algorithm only reduces some peaks slightly. This is because the cost function of FXLMS algorithm is the square of the instantaneous error, which leads the FXLMS algorithm to reduce the maximal noise component first (beforehand, the other noise components cannot be attenuated effectively). Besides, FXLMS algorithm is derived by using the instantaneous error \(e^{2}(n)\) to replace its expectation \(\mathrm{E}(e^{2}(n))\); its gradient estimation deviation may be large, especially under time-varying noise. These factors will deteriorate the control result of FXLMS algorithm.