# Active Noise Control Using Modified FsLMS and Hybrid PSOFF Algorithm
## 4 Proposed Method
### MFsLMS Algorithm  
The secondary path is filtered by the non-linear expansion of each element in the FsLMS algorithm. The computational complexity of the FsLMS algorithm is lower due to the reduced non-linearity. Therefore, it is effectively utilized in the ANC system. The works developed recently are done with the help of the FsLMS algorithm, mostly. The FsLMS algorithm is viewed as a non-linear controller, and its input noise becomes non-linear that is produced from a disorganized source.  
This algorithm is derived with \(p1,p2,\ldots\ldots,pL\) reference microphones, \(q1,q2,\ldots\ldots,qM\) loudspeakers and \(e1,e2,\ldots\ldots\ldots,eN\) error microphones. In the present approach, the following terms are considered  
\(p_{l}(n)\) is the signal used as input at the \(l\)-th reference microphone at the instant \(n\); \(1<l<L\).  
\(q_{m}(n)\) is defined as the output signal at the loudspeaker \(m\) at the instant \(n\); \(1<m<M\).  
\(w_{l,n}(n)\) is the adjustable weight vector, which connects \(l\)-th reference microphone to \(n\)-th error microphone.  
\(s_{i}(n)\) is the signal vector, which is functionally expansion of the input signal \(p_{l}(n)\).  
\(e(n)\) is the noise or the error signal sensed by the reference microphone.  
\(f(n)\) is the primary noise signal at the \(n\)-th error microphone.  
\(a(n)\) is the impulse response.  
\[e(n)=f(n)+a(n)*q_{m}(n). \tag{1}\]  
Equation (1) shows the expression of error signal.  
This modified algorithm aims to reduce the noise at the output of the system. For that, the function of cost is defined as  
\[\chi=\sum_{n=1}^{N}{e_{n}}^{2}(n). \tag{2}\]  
The above function is needed to be reduced for the effective design of the ANC systems.  
Figure 1: Schematic of MFsLMS based on multichannel ANC.  
The output signal is estimated as expressed by  
\[q_{m}(n)=\sum_{l=1}^{L}w_{l,n}(n)*s_{i}(n) \tag{3}\]  
where \"\(\omega\)\" is the convolution operator. The adjustable weight vector \(w_{l,n}(n)\) can be computed through the successive relation  
\[w_{l,n}(n+1)=w_{l,n}(n)-\frac{\mu}{2}\overset{\wedge}{\
abla}(n). \tag{4}\]  
where \(\mu\) is the stability factor; which is also used for denoting the convergence of the algorithm. If the stability factor holds a small value, it denotes the considerable enhancement in stability but the reduction in convergence. If the stability factor has a larger value, it means faster convergence with reduced stability. \(\overset{\wedge}{\
abla}\) is the estimation (instantaneous) of the gradient of \(\chi\) with respect to the weight vector \(w_{l,n}(n)\).  
The Maclaurin series is employed for the functional expansion of \(s_{i}(n)\). The series expansion according to the modified algorithm is expressed by  
\[s_{i}(n)=\left\{\begin{array}{l}x_{i}(n),\frac{h}{2}\pi_{i}x^{\prime}(n), \frac{h^{2}}{2!}x_{i^{\prime\prime}}(n),\ldotsldots.x_{i}(n-1),\frac{h}{2!}x_{ i^{\prime}}(n-1),\\ \frac{h}{2!}x_{i^{\prime\prime}}(n-1),\ldots\ldots.x_{i}(n-m+1),\frac{h}{2!}x _{i}^{\prime}(n-m+1),\frac{h}{2!}x^{\prime\prime}(n-m+1)\end{array}\right\}. \tag{5}\]  
The series expansion term is utilized to evaluate the computational cost of the proposed system. The modification is given through the functional expansion term; which reduces the computational complexity of the algorithm. Figure 2 shows the ANC with the MFsLMS algorithm.