## 4 MAMZI with Phase Noise Stabilization
### Analysis of the Results in Phase Noise Stabilization
Values for \(\delta\) lower than three were discarded since they caused the system to lose controllability. Furthermore, an increase in \(\delta\) accelerates the control system's response, although it increases the error, indicating a less precise operation. This same trend is reflected in the visibility and crosstalk graphs, where it can be seen that, for minimum values for \(\delta\), the system shows less contamination by crosstalk and greater visibility of interference. However, by increasing \(\delta\), the control system works faster when seeking system stabilization, at the cost of more significant crosstalk contamination and a loss in visibility quality. Based on the above, improving the system's performance is possible by adapting the step (\(\delta\)) to increase when more speed is required and reduce when greater precision is needed. To implement this adaptive control, it is proposed that the step be proportional to the error, i.e., \(\delta[n]=e[n]\cdot A+\delta_{min}\), where \(A\) is a system parameter that regulates the impact of the signal \(e[n]\) on the value \(\delta[n]\), according to previous results, \(\delta_{min}=3\). Figure 7 compares the mean and standard deviation values of RMSE and \(cv\) as a function of \(N\) and \(A\), using the proposed adaptive control. It is observed that both the  
Figure 6: (**a**) RMSE for orders \(N=2,3,4,5,6,7\), and 8. (**b**) Standard deviation of RMSE for the range \(1<\delta<20\) using the P&O algorithm. (**c**) Control velocity \(cv\) for orders \(N=2\) to \(N=8\). (**d**) Deviation of \(cv\) for \(1<\delta<20\). (**e**) Visibility for orders \(N=2\) to \(N=8\). (**f**) Variation of visibility within \(1<\delta<20\). (**g**) \(Cr\) parameter for orders \(N=2\) to \(N=8\). (**h**) Standard deviation of \(Cr\) over the interval \(1<\delta<20\). RMSE and its standard deviation decrease as \(A\) increases for systems with \(N>4\). On the other hand, for systems with \(N\leq 4\), the improvements are minimal. Furthermore, the speed and its standard deviation increase with \(A\), and, as expected, less complex systems (fewer arms) perform better in terms of RMSE and speed. Regarding visibility, it is observed that adaptive control maintains practically constant values, regardless of \(A\), for systems with \(N\leq 5\). For systems with a more significant number of arms (\(N>5\)), an increase in visibility is observed, accompanied by a decrease in its standard deviation as \(A\) increases (Figure 7). It is relevant to note that almost all visibility values exceed 0.99. The application of adaptive control maintains practically constant values of visibility, regardless of the increase in speed for all the systems studied, and a decrease in its standard deviation is observed as \(A\) increases (Figure 7), which implies that the system is more stable despite being accelerated. It is relevant to highlight that almost all visibility values exceed 0.99. Similarly to visibility, crosstalk shows a mean value that tends to remain constant for all values of \(N\) as \(A\) increases, following a similar trend in its standard deviation. In addition, it presents lower values concerning \(P\&O\) with fixed steps, reflecting a clear improvement in the control process. In summary, the results obtained by the adaptive steps show that the steady-state error remains below 1% for systems with \(N<5\) and below 2% for higher-order systems, even with higher \(A\). This behavior directly influences the optical parameters, with visibility remaining above 0.99 and crosstalk below \(-27\) dB in all cases analyzed, regardless of the system order. We theoretically demonstrate the controllability of a complex \(N\times N\) optical system using a simple Perturbation and Observation (P&O) strategy on systems up to \(8\times 8\) to validate the functionality of the proposed method. However, future work  
Figure 7: (**a**) RMSE for orders \(N=2,3,4,5,6,7\), and 8. (**b**) Standard deviation of RMSE. (**c**) Control parameter \(cv\). (**d**) Standard deviation of \(cv\).