# A New Effective Narrowband Active Noise Control System for Accommodating Frequency Mismatch
## 2 Materials and Methods  
### The Conventional NANC System  
Figure 1 depicts the structural block diagram of the conventional parallel narrowband ANC system [21]:  
It is assumed that the primary noise that is removed has multiple fundamental frequency noises, and such noise signals can be expressed as:  
\[P(n)=\sum_{i=1}^{i=q}\left[a_{p,i}\cos(w_{p,i}n)+b_{p,i}\sin(w_{p,i}n)\right]+v _{p}(n) \tag{1}\]  
In this formula, \(q\) is the number of frequency components contained in the primary noise, \(\omega_{p,i}(i=1,2,3)\) is the frequency of the i-th component, \(\left\{a_{p,i},b_{p,i}\right\}_{i=1}^{i=q}\) correspond to the DFCs of the different components of the primary noise. \(V_{p}(n)\) represents the additive background noise, which is a zero-mean additive white Gaussian noise with variance \(\delta_{p}^{2}\).  
The reference signals are the software synthesized sinusoidal signal according to the information of the estimated frequencies. The \(i\)-th input channel can be, respectively, modeled as:  
\[x_{ai}(n)=a_{i}\cos(w_{r,i}n)\ \ x_{bi}(n)=b_{i}\sin(w_{r,i}n) \tag{2}\]  
where \(\left\{a_{i},b_{i}\right\}_{i=1}^{i=q}\) represent the weights of the controller filter, \(\omega_{r,i}\) represents the frequency captured by a non-acoustic sensor.  
The sum of the synthesized reference signals in all channels are:  
\[y_{i}(n)=\sum_{i=1}^{i=q}\left[\overset{\wedge}{a_{i}}(n)x_{ai}(n)+\overset{ \wedge}{b_{i}}(n)x_{bi}(n)\right] \tag{3}\]  
where \(\left\{\overset{\wedge}{a_{i}},\overset{\wedge}{b_{i}}\right\}_{i=1}^{i=q}\) denote the estimations of the controller filter weights, \(\sum_{m=1}^{m=M}\hat{s}_{m}\) is the coefficients of the estimated secondary path. Generally, the block \(S_{(z)}\) is assumed to be known, which corresponds to the secondary-path and is considered as a FIR low pass filter with length \(M,\sum_{m=1}^{m=M}S_{m}\) represents the weights of the filter.  
Figure 1: Block diagram of conventional FXLMS algorithm.  
The error signal that consists of q residual noise components in the error microphone is expressed as:  
\[e(n)=P(n)-\sum_{m=1}^{m=M}\overset{\wedge}{s}_{m}y(n-m) \tag{4}\]  
Then, the error signal is fed to the DFC estimation by FXLMS:  
\[\overset{\wedge}{a}_{i}(n+1)=\overset{\wedge}{a}_{i}(n)+\mu_{a,i}e(n)\overset{ \wedge}{x}_{ai}(n) \tag{5}\]  
\[\overset{\wedge}{b}_{i}(n+1)=\overset{\wedge}{b}_{i}(n)+\mu_{b,i}e(n)\overset{ \wedge}{x}_{bi}(n) \tag{6}\]  
where \(\left\{u_{a,i},\,u_{b,i}\right\}_{i=1}^{i=q}\) is the step size of the DFCs coefficients, \(\left\{\hat{x}_{ai}(n),\,\hat{x}_{bi}(n)\right\}_{i=1}^{i=q}\) are the input signals filtered by \(\hat{S}_{(z)}\), which are as follows:  
\[\overset{\wedge}{x}_{ai}(n)=\sum_{m=1}^{m=M}\overset{\wedge}{s}_{m}x_{ai}(n-m )=\overset{\wedge}{a}_{i}x_{ai}(n)+\overset{\wedge}{\beta}_{i}x_{bi}(n) \tag{7}\]  
\[\overset{\wedge}{x}_{bi}(n)=\sum_{m=1}^{m=M}\overset{\wedge}{s}_{m}x_{bi}(n-m )=-\overset{\wedge}{\beta}_{i}x_{ai}(n)+\overset{\wedge}{\alpha}_{i}x_{bi}(n) \tag{8}\]  
where  
\[\overset{\wedge}{\alpha}_{i}=\sum_{m=1}^{m=M}\overset{\wedge}{s}_{m}\cos(m \omega_{r,i})\overset{\wedge}{\beta}_{i}=\sum_{m=1}^{m=M}\overset{\wedge}{s}_ {m}\sin(m\omega_{r,i}) \tag{9}\]