# Active Control of Broadband Noise Inside a Car Using a Causal Optimal Controller
## 4 Algorithm Verification and Discussion  
The proposed causal optimal controller in Equation (15) is verified in this section using the data measured from a car driving at 60 km/h in third gear with a sampling frequency of 1600 Hz. The positions of the loudspeaker and error microphone are shown in Figure 2. Vibrational signals of the car are adopted as reference signals, and they are acquired by installing eight triaxial accelerometers on the chassis of the car. The positions of accelerometers are shown in Figure 5, where ACC1--ACC8 represent eight triaxial accelerometers, and Mic and Sp represent the error microphone and door loudspeaker. Due to the distance limitation between the error microphone and ears, only the noise below 200 Hz is expected to be cancelled.  
Six reference signals are selected from twenty-four vibrational signals according to the multiple coherence results, and they are ACC2-X, ACC2-Z, ACC4-X, ACC6-X, ACC6-Y, and ACC7-Y. The multiple coherence between the selected reference signals and the undesired noise is shown in Figure 6. It can be observed that the average of the multiple coherence between 30 and 200 Hz is over 0.6. Therefore, the reference signals are acceptable.  
Based on above analysis, a multichannel control system is designed as is shown in Figure 7, where \(x_{i}\) and \(W_{i}\), \(i=1,\cdots,6\) denote six reference signals and controllers, respectively. Each controller \(W_{i}\) has the expression as Equation (15). If only the broadband noise is going to be cancelled, using six proposed causal optimal controllers can achieve a satisfactory result. If the broadband as well as engine harmonic noises are expected to be reduced, it is appropriate to add an adaptive controller \(W_{h}\) into the control system to cancel the harmonic component. The design of \(W_{h}\) is based on FXLMS algorithm, and the adopted reference signal should include the engine harmonics [26].  
Figure 5: The installation positions of accelerometers and error microphone on the car.  
Figure 6: Multiple coherence between reference signals and the undesired noise.  
A six-channel FXLMS algorithm is also adopted here for performance comparison. The resultant error signal of the algorithm is derived as follows:  
\[e(n)=d(n)-s(n)*\sum_{i=1}^{6}y_{i}(n), \tag{17}\]  
where * denotes convolution and \(s(n)\) is the impulse response of the secondary path \(S\), which is identified offline in advance [26]. By taking the square of the instantaneous error \(e(n)\) as a cost function, the updating equation for each adaptive filter of FXLMS algorithm is obtained [26] as follows:  
\[W_{i}(n+1)=W_{i}(n)+\mu/[\delta+norm(x_{if})]e(n)x_{if}(n),\ i=1,\ 2,\ \cdots,\ 6, \tag{18}\]  
where \(\mu\) is the step size for updating the control filter; \(\delta\) is a normalization constant; and \(x_{if}(n)\) is the \(i\)th filtered reference signal vector, which is the convolution result of the \(i\)th reference signal and the secondary path \(s(n)\).  
The noise reduction is evaluated using the parameter \(Rd\), which is defined as follows:  
\[Rd=10\lg(\frac{e^{2}(n)}{d^{2}(n)}). \tag{19}\]