# A Review of Sound Field Control
## 4 Active Noise Control
### ANC System Model  
An ANC system mainly consists of reference microphones, controllers, and error microphones. The reference signal vector \(\mathbf{x}(n)=[x(n),x(n-1),\ldots,x(n-L+1)]\) is  
Figure 5: (**a**) Microphone array for sound field recording, (**b**) linear loudspeaker array for personal audio systems [51] (**c,d**) circular loudspeaker arrays for personal audio systems.  
processed by the controller \(\mathbf{w}(n)=[w_{0}(n),w_{1}(n),\cdots,w_{L-1}(n)]\) to produce an output signal:  
\[y(n)=\mathbf{w}^{T}(n)\mathbf{x}(n), \tag{6}\]  
where \(L\) is the length of the control filter. The error signal \(e(n)\) measured by the error sensor represents the acoustical combination of the desired signal \(d(n)\) and the output signal [13; 14; 15; 53; 54]. The control filter is updated as:  
\[\mathbf{w}(n+1)=\mathbf{w}(n)+\mu\mathbf{x}(n)e(n), \tag{7}\]  
where \(\mathbf{x}^{\prime}(n)\) is filtered-x signal obtained by convolving the reference signal and the secondary path \(S(z)\).  
ANC systems can be divided into narrowband systems and wideband systems according to the noise bandwidth to be eliminated. Narrowband systems are mainly used to handle periodic (harmonic) noise. In a narrowband system, the output of the control filter is the harmonic frequency signal  
\[\begin{array}{c}y(n)=w_{1}x_{a}(n)+w_{2}x_{b}(n),\\ x_{a}(n)=\cos(\omega n),x_{b}(n)=\sin(\omega n),\end{array} \tag{8}\]  
where \(\omega\) is the frequency of narrowband signal, and \(w_{1}\) and \(w_{2}\) are the control filter coefficients. Each periodic noise requires independently a two-order control filter. Frequency mismatch refers to the inconsistency of the frequency of the reference signal and the noise signal, which destroys the noise reduction performance. Solving the frequency mismatch is an important issue for narrowband ANC systems [55; 56; 57; 58]. By contrast, broadband systems use a high-order filter to deal with broadband noise and do not need to control each frequency point. In the presence of wideband and narrowband noise components, it is necessary to separate the narrowband noise first and then process the narrowband noise and the wideband noise independently.  
Alternatively, ANC systems can be divided into feedforward and feedback systems depending on the control structure [59; 60; 61; 62]. Feedforward systems are a classic type of ANC systems, which obtain a reference noise signal through a reference sensor and then performs noise reduction at the error point. Feedforward systems have good stability and have thus been widely used. The block diagram of a single channel feedforward ANC system is shown in Figure 6. However, this system requires high coherence between the reference signal and the primary noise signal, which often requires experienced selection of the reference point. Feedback systems, on the other hand, require no reference microphone and are suitable for occasions where the reference signal is not easy to obtain. Therefore, feedback systems perform well in consumer electronics with low cost and complexity such as a noise-canceling headset. However, the feedback algorithm needs to make a trade-off among water bed effect, stability, and noise amplification. Moreover, hybrid feedforward and feedback systems can improve the noise reduction performance with the poor coherence between the feedforward reference signal and the noise signal.  
Figure 6: Block diagram of a single channel feedforward ANC system.