## 4 Analysis of Computational Complexity
### Simulation of a Multi-Channel ANC System  
* First experiment: Multitonal noise input  
Table 7 lists the parameters used in this experiment. As shown in Figure 5, the FXA-AP algorithm presents the fastest convergence speed. However, its MSE level varies significantly when abrupt changes occurs. On the contrary, the proposed filtering scheme employs fixed and dynamic scaling factor \(sf\) to achieve a significant improvement in the steady-state MSE of approximately 3 dB when compared to the existing FXAPL-I and FXA-AP algorithms. The proposed filtering scheme exhibits a convergence speed similar to that of the existing FXAPL-I algorithm, by employing a dynamic scaling factor. In this case, the convergence speed is affected when the proposed filtering scheme with fixed scaling factor is used. As can be observed in Figure 5, the proposed algorithm with fixed scaling factor presents slow convergence speed at certain level of noise since the \(sf\) is not optimal, i.e, we adjusted it by trial and error. In contrast, the proposed algorithm with dynamic scaling factor presents high convergence speed during the entire filtering process since the use of this strategy intends to achieve the optimal value automatically at each iteration. In this experiment, the FXAPL-I algorithm with fixed and dynamic scaling factor updates its coefficients 31,835 times (1.59%) and 32,747 times (1.63%), respectively. As can be observed in Table 10, our proposal with fixed and dynamic scaling factor exprend approximately 2.64, 2.10, 8.37, and 6.66 times fewer multiplications and 2.23, 1.71, 5.89, and 4.51 times fewer additions than do the FXAPL-I and FXA-AP algorithms, respectively. In addition, our proposal guarantees similar convergence speed and lower MSE level when compared with the FXAPL-I algorithm, as shown in Figure 5. On the other hand, our proposal with fixed and dynamic scaling factor require approximately 2.86 and 3.59 more multiplications and 3.62 and 4.72 more additions than  
Figure 4: Learning curves for the FXAPL-I algorithm, the FXMS algorithm, the FXA-AP algorithm and the proposed algorithm for a Gaussian noise input in a single-channel system. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Algorithm** & **Multiplications** & **Additions** \\ \hline FXLMS & 660,000,000 & 652,000,000 \\ FXAPL-I & 4,122,000,000 & 4,922,000,000 \\ FXA-AP & 13,842,000,000 & 12,124,000,000 \\ Proposed algorithm with fixed scaling factor & 1,678,715,530 & 1,581,390,400 \\ Proposed algorithm with dynamic scaling factor & 1,717,833,600 & 1,582,122,590 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Total number of multiplications and additions of the FXAPL-I, FXLMS, FXA-AP and the proposed algorithms for a single-channel ANC system using a Gaussian noise input. are required by the FXLMS algorithm. However, our proposal exhibits better convergence speed and similar MSE level when compared with FXLMS algorithm. \(\bullet\) Second experiment: Gaussian noise input  
In Figure 6, the learning curves created by simulating the proposed filtering scheme with fixed and dynamic scaling factors are shown. The corresponding data are listed in Table 7. The data show that the convergence speed of the proposed filtering scheme is similar to that of the existing FXAPL-I and FXA-AP algorithms. However, it requires fewer updates than the existing FXAPL-I algorithm requires. Specifically, it requires 315,200 (15.76%) and 440,434 (22.02%) updates for the fixed and dynamic scaling factors, respectively. Table 11 shows the number of additions and multiplications required to perform each algorithm for a multi-channel ANC system processing a Gaussian noise signal. In this case, our proposal with fixed and dynamic scaling factor expend approximately 2.13, 1.59, 6.74, and 5.06 times fewer multiplications and 1.83, 1.49, 4.83, and 3.93 times fewer additions than do the FXAPL-I and FXA-AP algorithms, respectively.