# A Construction Method for the Random Factor-Based G Function
## 5 Simulation Results
### Two-Dimensional Continuity  
In the context of frequency-hopping communication, two-dimensional continuity is defined as the characteristic of adjacent frequency point changes, which is related to the regularity of hopping. The two-dimensional continuity of a frequency-hopping sequence can be evaluated by analyzing the transition probability between different frequency points in the sequence [33]. If the frequency points' transition in the frequency-hopping sequence exhibits good continuity, then the number of occurrences between different frequency points will tend to be balanced, thereby increasing the difficulty for an adversary to predict the frequency hopping pattern.  
It is expected that a two-dimensional continuity will be observed in an ideal differential frequency-hopping sequence, as follows:  
\[P(f_{i},f_{j})=1/N^{2},f_{i},f_{j}\in\Omega\in\{f_{0},f_{1},\cdots,f_{N-1}\} \tag{13}\]  
where P(f\({}_{i}\),f\({}_{j}\)) represents the joint probability density function, with f\({}_{i}\) denoting the current hopping frequency, f\({}_{j}\) the previous hopping frequency, and \(\Omega\) the frequency set. It is assumed that each frequency point is sent with equal probability.  
\[P(\text{f}_{i})=1/\text{N} \tag{14}\]  
In the case of the RFGF,  
\[P(f_{j}|f_{i})=\frac{1}{J\cdot K} \tag{15}\]  
In accordance with the Bayesian formula, the joint probability density of the G function can be expressed as follows:  
\[P(f_{i},f_{j})=P(f_{i})\cdot P(f_{j}|f_{i})=\frac{1}{N\cdot J\cdot K} \tag{16}\]  
In the case where N = J\(\cdot\)K, the probability of occurrence of a particular outcome, P(f\({}_{i}\),f\({}_{j}\)), may be expressed as 1/N\({}^{2}\). This value meets the ideal two-dimensional continuity requirement. Figure 9 illustrates the two-dimensional continuity test frequency statistical histograms for the four aforementioned methods of frequency-hopping sequences.  
In order to gain a more intuitive understanding of the uniformity of the two-dimensional continuity of the frequency-hopping sequences generated by the four methods, the standard deviation of the number of frequency point pairs is calculated. A larger standard deviation indicates a greater degree of discrete occurrence of the number of frequency point pairs.  
As evidenced in Table 6, the frequency-hopping sequence generated by the G function of the RFGF method exhibits a lower standard deviation in two-dimensional continuity performance when compared to other methods. This indicates that the frequency point pair distribution is more uniform and demonstrates diminished correlation between the adjacent hops. The standard deviation of the RFGF method is reduced by 2.9141 in comparison with  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **RFGF** & **TVID-GF** & **SA-GF** & **MRCS-GF** \\ \hline Maximum & 0.01623 & 0.01646 & 0.01672 & 0.01664 \\ Minimum & 0.01466 & 0.01474 & 0.01475 & 0.01465 \\ Range & 0.00157 & 0.00175 & 0.00197 & 0.00199 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Statistics of extreme values of frequency distribution.  
TVID-GF, which indicates that the two-dimensional continuity performance is enhanced by 36.87%.