## 4 Simulation
### Periodic Sinusoids: \(\omega_{2,3}=n\omega_{1}\)
It is thought that this phenomenon occurs as the frequency component increases with the stage, and is attributed to the characteristics of the multi-stage structure of the algorithm. At a point \(k\), when the conventional LMS algorithm was applied to four incommensurate sinusoids, there was an unexpected frequency other than the frequency component of the input signal, due to the unintentional modulation process between filter input \(x[k]\) and error \(e[k]\). Some were observed. This frequency component can be confirmed by analyzing the \(x[k]\cdot e[k]\) as follows: In all cases, the frequency of the reference signal is assumed to be \(\omega_{2}\), the phase difference. The frequency components at the first sampling point \(k=1\) are as follows:  
\[x[k]\cdot e[k]=\cos(\omega_{2}t)\cdot\{\cos(\omega_{1}t)+\cos(\omega_{2}t)+ \cos(\omega_{3}t)\} \tag{24}\]  
\[y[x[k]\cdot e[k]]=\tfrac{1}{2}\pi[\delta(\omega+\omega_{1}-\omega_{2})+\delta (\omega-\omega_{1}-\omega_{2})]\]  
\[ \begin{array}{l}+\tfrac{1}{2}\pi[\delta(\omega-\omega_{3}+\omega_{2})+ \delta(\omega-\omega_{3}-\omega_{2})]+\tfrac{1}{2}\pi[\delta(\omega-2\omega_{ 2})+\delta(\omega)]\end{array}  \tag{25}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Control Algorithm**} & \multirow{2}{*}{**Error Analysis**} & \multicolumn{3}{c}{**Time**} \\ \cline{3-5}  & & **2.0–3.0 s** & **3.0–3.5 s** & **3.5–4.0 s** \\ \hline \multirow{2}{*}{**LMS Algorithm**} & Maximum & 0.1963 & 0.1941 & 0.1930 \\  & Minimum & \(-0.1915\) & \(-0.1917\) & \(-0.1946\) \\  & Peak-to-Peak & 0.3878 & 0.3858 & 0.3876 \\ \hline \multirow{2}{*}{**MSA-LMS Algorithm**} & Maximum & 0.1176 & 0.0928 & 0.0558 \\  & Minimum & \(-0.1456\) & \(-0.0622\) & \(-0.0393\) \\ \cline{1-1}  & Peak-to-Peak & **0.2632 (32\%\(\downarrow\))** & **0.1550 (60\%\(\downarrow\))** & **0.0951 (75\%\(\downarrow\))** \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Peak-to-Peak value analysis for the estimation errors (“Periodic Sinusoids” case). Figure 7: Comparison of predicted estimation errors. Key: –  
where \(y[]\) is the Fourier transform and Dirac delta function. At the second sampling point \(k\geq 2\), the effect of the reference input was added. The frequency components are summarized as follows:  
\[x[k]\cdot([x[k]\cdot e[k]]) \tag{26}\] \[=\cos(\omega_{2}t)\cdot(\cos(\omega_{2}t)\cdot\{\cos(\omega_{1}t)+ \cos(\omega_{2}t)+\cos(\omega_{3}t)\})\]  
\[y[x[k]\cdot([x[k]\cdot e[k]])] =\tfrac{1}{4}\pi[\delta(\omega+\omega_{1}-2\omega_{2})+\delta( \omega-\omega_{1}-2\omega_{2})+2\delta(\omega-\omega_{1})] \tag{27}\] \[+\tfrac{1}{4}\pi[\delta(\omega+\omega_{3}-2\omega_{2})+\delta( \omega-\omega_{3}-2\omega_{2})+2\delta(\omega-\omega_{3})]\] \[+\tfrac{1}{2}\pi\delta(\omega)+\tfrac{1}{4}\pi[\delta(\omega- \omega_{2})+\delta(\omega-2\omega_{3})]\]  
As can be seen in Equations (25) and (27), the controlled signal includes additional frequency components, such as \(\omega_{j}=\omega_{2}\pm\omega_{1}\), \(\omega_{3}\pm\omega_{2}\), \(2\omega_{2}\), \(3\omega_{2}\), \(2\omega_{2}\pm\omega_{1}\), \(2\omega_{2}\pm\omega_{3}\), and so on, \(\omega_{i}=\omega_{1}\), \(2\omega_{2}\), \(\omega_{3}\), which were previously included in the control signal. It is also apparent that additional side bands are located at \(\Delta\omega_{a}=\omega_{2}-\omega_{1}\), \(\Delta\omega_{b}=\omega_{3}-\omega_{2}\), and \(|\Delta\omega_{a}-\Delta\omega_{b}|\). It can be observed from Figure 8 that these additional frequency components occurred at \(|2\omega_{2}-\omega_{1}|/2\pi=\)48 Hz and \(|\omega_{2}-\omega_{1}|/2\pi=\)18 Hz. Here, it is determined that the error 2 Hz occurred because the object to be controlled was not the input signal but the response of the secondary system, and the main frequency component of the signal was an estimated value rather than an actual value (presumed to be a nearby value).