# A New Effective Narrowband Active Noise Control System for Accommodating Frequency Mismatch
## 3 Proposed Algorithm  
In this section, inspired by the convex combination structures and FE-NANC system [32; 33], a reference signal is generated by adding the weighted sum of both the cosine and sine components of each frequency present in the noise, and then the frequency compensation and the AR model work simultaneously. Excellent steady-state errors can be obtained after considering larger step sizes. We thoroughly demonstrate the proposed algorithm with the block diagram, shown in Figure 3 below. The AR model is optimized by the LMP criterion; with the coefficients of the AR model adjusted by the LMP and good performance of the AR model in the tracking frequencies, the NANC system can achieve a high standard in canceling the narrowband noises and fast convergence. In the above graph, the reference signal synthesized in each frequency channel is composed of two parts: one part is that the cosine and sine signal are adjusted by the  
Figure 3: The block of proposed NANC system. AR model, and then filtered by FXLMS; the other part is the frequency compensation combination with the delay module, which is expressed as:  
\[y_{i}(n)={y_{i}}^{(0)}(n)+{y_{i}}^{(-k)}(n) \tag{14}\]  
where  
\[{y_{i}}^{(0)}(n)=\overset{\wedge}{a_{i}}(n)x_{ai}(n)+\overset{\wedge}{b_{i}}(n )x_{bi}(n) \tag{15}\]  
\[{y_{i}}^{(-k)}(n)=\overset{\wedge}{a_{k}}(n)x_{ai}(n-k)+\overset{\wedge}{b_{k _{i}}}(n)x_{bi}(n-k) \tag{16}\]  
According to the FXLMS algorithm, we have:  
\[\overset{\wedge}{a_{k_{i}}}(n+1)=\overset{\wedge}{a_{k_{i}}}(n)+\mu_{k_{k_{i} }}e(n)\overset{\wedge}{a_{ai}}(n-k) \tag{17}\]  
\[\overset{\wedge}{b_{k_{i}}}(n+1)=\overset{\wedge}{b_{k_{i}}}(n)+\mu_{k_{k_{i} }}e(n)\overset{\wedge}{x_{bi}}(n-k) \tag{18}\]  
Where  
\[\overset{\wedge}{x_{ai}}(n-k)=\sum\
olimits_{m=1}^{m-M}\overset{\wedge}{s}_ {m}x_{ai}(n-k-m)=\overset{\wedge}{a_{k_{i}}}x_{ai}(n)+\overset{\wedge}{b_{k_{ i}}}x_{bi}(n) \tag{19}\]  
\[\overset{\wedge}{x_{bi}}(n-k)=\sum\
olimits_{m=1}^{m=M}\overset{\wedge}{s}_ {m}x_{bi}(n-k-m)=-\overset{\wedge}{\beta_{k_{i}}}x_{ai}(n)+\overset{\wedge}{a _{k_{i}}}x_{bi}(n) \tag{20}\]  
where  
\[\overset{\wedge}{a_{k_{i}}}=\sum\
olimits_{m=1}^{m=M}\overset{\wedge}{s}_ {m}\cos[(m+k)\omega_{r_{i}}]\overset{\wedge}{\beta_{k_{i}}}=\sum\
olimits_{m =1}^{m=M}\overset{\wedge}{s}_{m}\sin[(m+k)\omega_{r_{i}}] \tag{21}\]  
Based on the LMP cost function \(J(n)=1/3|e(n)|^{3}\), the frequency-related coefficients may be updated using the steepest descent gradient expression:  
\[c_{i}(n+1)=c_{i}(n)-\mu_{c_{i}}\
abla_{c_{i}(n)}J(n) \tag{22}\]  
This algorithm is derived as follows:  
\[ \begin{array}{rl}\
abla_{c_{i}(n)}I(n)&=-|e(n)|(e(n))^{2}\frac{\partial y_{ p}(n)}{\partial c_{i}^{2}(n)}\\ &=-|e(n)|(e(n))^{2}\frac{\partial\left(\sum\
olimits_{m=1}^{m}\sum\
olimits_ {i=M}^{n}s_{m}y_{i}(n-m)\right)}{\partial c_{i}(n)}\\ &=-|e(n)|(e(n))^{2}\frac{\partial\left[\sum\
olimits_{m=1}^{m-M}s_{m}\left( \psi_{0}(n)-m+\psi_{1}(^{-k})(n-m)\right)\right]}{\partial c_{i}(n)}\\ &=-|e(n)|(e(n))^{2}s_{0}\left[\frac{\partial y_{0}(n)}{\partial\overset{ \wedge}{x_{ai}}(n)}{\partial\overset{\wedge}{x_{ai}}(n)}\frac{\partial x_{ ai}(n)}{\partial c_{i}(n)}+\frac{\partial y_{i}(0)}{\partial\overset{ \wedge}{x_{bi}}(n)}\frac{\partial\overset{\wedge}{x_{bi}}(n)}{\partial c_{i}(n )}\right.\\ &\left.+\frac{\partial y_{i}(^{-k})(n-k)}{\partial\overset{\wedge}{x_{ai}}(n-k )}\frac{\partial x_{ai}(n-k)}{\partial c_{i}(n)}+\frac{\partial y_{i}(^{-k})(n -k)}{\partial\overset{\wedge}{x_{bi}}(n-k)}\frac{\partial\overset{\wedge}{x_ {bi}}(n-k)}{\partial c_{i}(n)}\right]\end{array}