## 3 Simulation and Discussion
### Effect of Nyquist Frequency Shift on the Coupled-MTF  
The Nyquist frequency \(f_{N}\) is the highest spatial frequency that can be theoretically resolved. Let the spatial frequency of the input optical signal be \(f=f_{N}/l\), \(l\geq 1\), let \(L=1/l\) be the frequency offset, and \(0<L\leq 1\). The spatial frequency of the final input optical signal can be obtained as  
\[f=L\times f_{N} \tag{15}\]  
assuming that the fiber radius \(R=3.2\)\(\upmu\)m, the fiber core layer radius \(r=2.67\)\(\upmu\)m, the number of fibers is \(1000\times 1000\), the initial position deviation is set to \(0.1R\), the image coupling error along the x-direction is \(\Delta i=0.5R\), and the image coupling error along the y-direction is \(\Delta j=0.7R\). The simulation results of the oscillatory convergence of the coupled-MTF with the input frequency offset are obtained using Equation (14). The simulation results of the oscillation convergence of the coupled-MTF as the number of fibers in the fiber bundle increases; when the parameters of the hexagonally aligned fiber imaging bundle and the area array CCD, as well as its coupling errors, are the same as in the second case; and when the spatial frequency of the input cosine target deviates from the Nyquist frequency by \(2\%\), i.e., \(f=0.98f_{N}\), are shown in Figure 7.  
Figure 6: Simulation of oscillation convergence of the coupled-MTF with the number of fibers: (**a**) the number of fibers is \(10\times 10\); (**b**) the number of fibers is \(100\times 100\); (**c**) the number of fibers is \(1000\times 1000\); (**d**) the number of fibers is \(2000\times 2000\). The simulation results of the oscillation convergence of the coupled-MTF as the number of fibers in the fiber bundle increases; when the parameters of the hexagonally aligned fiber imaging bundle and the area array CCD, as well as its coupling errors, are the same as in the second case; and when the spatial frequency of the input cosine target deviates from the Nyquist frequency by 1%, i.e., \(f=0.99f_{N}\), are shown in Figure 8.  
The simulation results of the oscillation convergence of the coupled-MTF when the parameters of the hexagonally aligned fiber imaging bundle and the area array CCD, as well as its coupling errors, are the same as in the second case, and when the spatial frequency of the input cosine target deviates from the Nyquist frequency by 0.1% (i.e., \(f=0.999f_{N}\) as the number of fibers in the fiber bundle increases), are as shown in Figure 10.  
Figure 8: Simulation of oscillation convergence of the coupled-MTF with the number of fibers when the deviation between the cosine target spatial frequency and Nyquist frequency is 1%. Figure 7: Simulation of oscillation convergence of the coupled-MTF with the number of fibers when the deviation between the cosine target spatial frequency and Nyquist frequency is 2%. Figure 9: Simulation of oscillation convergence of the coupled-MTF with the number of fibers when the deviation between the cosine target spatial frequency and Nyquist frequency is 0.5%. The simulation results shown in Figures 8-10 demonstrate that the coupled-MTF exhibits a clear trend of gradually increasing the convergence rate as the deviation of the spatial frequency of the input cosine target from the Nyquist frequency gradually increases from 0.1% to 1%. The simulation results shown in Figures 7 and 8 demonstrate that, as the deviation of the spatial frequency of the input cosine target from the Nyquist frequency increases from 1% to 2%, the coupled-MTF oscillation amplitude converges faster, but the total oscillation amplitude is consistent, and all converge to a fixed value of approximately 0.5. According to Figures 7-10, as the deviation of the spatial frequency of the input cosine target from the Nyquist frequency gradually increases, the coupled-MTF exhibits a clear tendency to converge faster.