# A New Effective Narrowband Active Noise Control System for Accommodating Frequency Mismatch
## 3 Proposed Algorithm
### Stability Analysis in the Mean Sense  
The stability analysis of the proposed algorithm is undertaken by the following. Using (1)-(4), (5), (7)-(9) and (14)-(21), the error signal (4) can be represented as:  
\[ \begin{split}& e(n)=\sum\limits_{i=1}^{q}\left\{\left[a_{p,i}- \left(\alpha_{i}\overset{\wedge}{a}_{i}(n)-\beta_{i}\overset{\wedge}{b}_{i}( n)+\alpha_{k_{i}}\overset{\wedge}{a}_{k_{i}}(n)-\beta_{k_{i}}\overset{\wedge}{b}_{k_{i}}( n)\right)\right]x_{ai}(n)\right.\\ &+\left[b_{p,i}-\left(\beta_{i}\overset{\wedge}{a}_{i}(n)+\alpha _{i}\overset{\wedge}{b}_{i}(n)+\beta_{k_{i}}\overset{\wedge}{a}_{k_{i}}(n)+ \alpha_{k_{i}}\overset{\wedge}{b}_{k_{i}}(n)\right)\right]x_{bi}(n)\right\}+v _{p}(n)\end{split}  \tag{26}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameters** & **K = 8** & **K = 14** & **K = 20** & **K = 32** \\ \hline FM = 10\% & \(-\)8.4999 & \(-\)9.43 & \(-\)9.012 & \(-\)9.424 \\ FM = 5\% & \(-\)9.041 & \(-\)9.428 & \(-\)8.677 & \(-\)9.007 \\ Without FM & \(-\)8.796 & \(-\)9.159 & \(-\)8.212 & \(-\)8.972 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The MSE with Different k under Different FMs(\(dB\)). Figure 4: MSE of different k under different FM. (**a**) Without FM (**b**) Under 5%FM. (**c**) Under 10%FM (**d**) Under 15%FM. Obviously, from the above error signal expression,  
\[\left( \begin{array}{c}a_{p,i}\\ b_{p,i}\end{array} \right)=\left( \begin{array}{c}\alpha_{i}\overset{\wedge}{a}_{ i}(n)-\beta_{i}\overset{\wedge}{b}_{i}(n)+\alpha_{k_{i}}\overset{\wedge}{a}_{ k_{i}}(n)-\beta_{k_{i}}\overset{\wedge}{b}_{k_{i}}(n)\\ \beta_{i}\overset{\wedge}{a}_{i}(n)+\alpha_{i}\overset{\wedge}{b}_{i}(n)+\beta _{k_{i}}\overset{\wedge}{a}_{k_{i}}(n)+\alpha_{k_{i}}\overset{\wedge}{b}_{k_{i }}(n)\end{array} \right)=\left( \begin{array}{c}\alpha_{i}&-\beta_{i}\\ \beta_{i}&\alpha_{i}\end{array} \right)\left( \begin{array}{c}\overset{\wedge}{ a}_{i}(n)\\ \overset{\wedge}{b}_{i}(n)\end{array} \right)+\left( \begin{array}{c}\alpha_{k_{i }}&-\beta_{k_{i}}\\ \beta_{k_{i}}&\alpha_{k_{i}}\end{array} \right)\left( \begin{array}{c}\overset{ \wedge}{a}_{k_{i}}(n)\\ \overset{\wedge}{b}_{k_{i}}(n)\end{array} \right) \tag{27}\]  
We assume that, it is worth emphasizing that m is an arbitrary constant  
\[\left( \begin{array}{cc}\alpha_{i}&-\beta_{i}\\ \beta_{i}&\alpha_{i}\end{array} \right)\left( \begin{array}{c}\overset{\wedge }{a}_{i,opt}\\ \overset{\wedge}{b}_{i,opt}\end{array} \right)=m\left( \begin{array}{c}a_{p,i} \\ b_{p,i}\end{array} \right)\left(0<m<1\right) \tag{28}\]  
\[\left( \begin{array}{cc}\alpha_{k_{i}}&-\beta_{k_{i}}\\ \beta_{k_{i}}&\alpha_{k_{i}}\end{array} \right)\left( \begin{array}{c}\overset{ \wedge}{a}_{k_{i}}(n)\\ \overset{\wedge}{b}_{k_{i}}(n)\end{array} \right)=\left(1-m\right)\left( \begin{array}{c}a_{p,i}\\ b_{p,i}\end{array} \right) \tag{29}\]  
Therefore, the optimal DFCs for a perfect performance of targeted noises are given by:  
\[\left( \begin{array}{c}\overset{\wedge}{a}_{i,opt}\\ \overset{\wedge}{b}_{i,opt}\end{array} \right)=m\left( \begin{array}{cc}\alpha _{i}&-\beta_{i}\\ \beta_{i}&\alpha_{i}\end{array} \right)^{-1}\left( \begin{array}{c}a_{p,i}\\ b_{p,i}\end{array} \right) \tag{30}\]  
\[\left( \begin{array}{c}\overset{\wedge}{a}_{k_{i},opt}\\ \overset{\wedge}{b}_{k_{i},opt}\end{array} \right)=\left(1-m\right)\left( \begin{array}{cc}\alpha_{k_{i}}&\alpha_{k_{i}}\\ \beta_{k_{i}}&\alpha_{k_{i}}\end{array} \right)^{-1}\left( \begin{array}{c}a_{p, i}\\ b_{p,i}\end{array} \right) \tag{31}\]  
Define the estimation errors of DFCs as:  
\[\left( \begin{array}{cc}\varepsilon_{a_{i}}(n)\;\varepsilon_{a_{k_{i}}}(n)\\ \varepsilon_{b_{i}}(n)\;\varepsilon_{b_{k_{i}}}(n)\end{array} \right)=\left( \begin{array}{cc}\overset{\wedge}{a}_{i,opt}-\overset{\wedge}{a}_{i}(n)\; \overset{\wedge}{a}_{k_{i},opt}-\overset{\wedge}{a}_{k_{i}}(n)\\ \overset{\wedge}{b}_{i,opt}-b_{i}(n)\;b_  
Combining (34)-(37), we obtain the transformation matrix:  
\[A_{i}=\left( \begin{array}{cccc}p_{1}&p_{2}&0&0\\ p_{2}&p_{1}&0&0\\ 0&0&p_{k1}&p_{k2}\\ 0&0&p_{k2}&p_{k1}\\ \end{array}