## 4 Analysis of Computational Complexity
### Simulation of a Single-Channel ANC System  
\(\bullet\) First experiment: Multitonal noise input  
We carried out two evaluations to validate the model operation using a fixed scaling factor, which was chosen by trail and error, and a dynamic adjustment of the scaling factor, as shown in Table 7. The parameters were chosen to provide the fastest convergence speed of the FXAPL-I algorithm and the lowest steady-state MSE in the FXLMS algorithm. As shown in Figure 3, the FXA-AP algorithm exhibits the best performance in terms of convergence speed and MSE level by paying a penalty in terms of computational cost. In contrast, the proposed filtering scheme achieves a significant reduction in the steady-state MSE, i.e., of approximately 5 dB and 3 dB when compared to the existing FXAPL-I using a fixed and dynamic adjustment of the scaling factor \(sf\), respectively. The proposed algorithm also reduces the computational cost when compared to the existing FXAPL-I algorithm. In the proposed filtering scheme, the FXAPL-I algorithm updates its coefficients 59,170 times (2.95%). Enabling the dynamic adjustment of the scaling factor, the FXAPL-I algorithm updates its coefficients 26,075 times (1.3%). When the steady-state MSE is reached, the FXLMS algorithm carries out the filtering process by updating its coefficients for the remaining part of the process. As shown in Figure 3, the steady-state MSE is reached more rapidly when the dynamic adjustment of the scaling factor is enabled. However, the MSE is higher when compared to the performance of the proposed filtering algorithm with the fixed scaling factor. As can be observed in Table 8, our proposal with fixed and dynamic scaling factor expend approximately 2.57, 2.52, 8.64, and 8.48 times fewer multiplications and 3.42, 3.27, 8.44, and 8.06 times fewer additions than do the FXAPL-I and FXA-AP algorithms, respectively. On the other hand, our proposal with fixed and dynamic scaling factor require approximately 2.42 and 2.47 more multiplications and 2.20 and 2.30 more additions than are required by the FXLMS algorithm. However, our proposal exhibits better convergence speed when compared with FXLMS algorithms. This factor is crucial in ANC applications since this factor determines the speed at which the noise is cancelled. From the engineering perspective, the use of FXLMS algorithms allows the development of easy and fast implementations. However, their application in ANC systems could be ineffective due to their low performance. Therefore, our proposal offers a good balance between convergence speed and computational cost. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Parameters**} & \multicolumn{2}{c}{**Single-Channel ANC**} & \multicolumn{2}{c}{**Multi-Channel ANC**} \\ \cline{2-5}  & **Multi-Tonal Input** & **Gaussian Noise Input** & **Multi-Tonal Input** & **Gaussian Noise Input** \\ \hline \(sf\) & 0.2 & 0.1 & 0.001 & 0.01 \\ \(sf_{max}\) & 0.1 & 0.1 & 0.5 & 0.5 \\ \(\mu\) & 0.00001 & 0.0006 & 0.00001 & 0.0007 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Simulation parameters for computer experiments. * Second experiment: Gaussian noise input We consider a Gaussian noise input to validate the robustness of the proposed filtering scheme. As shown in Figure 4, the FXA-AP algorithm presents the best MSE level compared with other algorithms. However, the convergence speed of the FXA-AP algorithm is as good as the convergence speed of our proposal. On the other hand, the convergence speed of the proposed filtering scheme is similar to the convergence speed of the existing FXAPL-I algorithms. However. the proposed filtering scheme requires fewer updates when compared with the existing FXAPL-I algorithm, i.e., the FXAPL-I algorithm with fixed and dynamic scaling factor updates its coefficients 98,765 times ( 4.93%) and 98,400 times (4.92%), respectively, whereas the existing FXAPL-I algorithm updates its coefficients during the entire filtering process. As previous case, the performance of our proposal was verified in terms of operations.