## 4 Ring-Shaped Signaling System Algorithm
The modified signals produced by the phase-modifying factor and magnitude-modifying factor are shown in Figure 5a,b, respectively. The signal modified by the phase-modifying factor contains data points that are evenly distributed at certain degrees of angles according to the resolution of the intensity level and in the shape of a circle, as shown in Figure 5a. The signal modified by the magnitude-modifying factor has data points that are distributed with different magnitude values. They are displaced toward the inside of the circle, as shown in Figure 5b. The ring-shaped signaling system, simultaneously applied with the phase-modifying factor and magnitude-modifying factor, has a lower average signal energy than the original signal due to the reduced average magnitude, as shown in Equation (3). Since the modified signal has flexibly changing power as shown in Equation (3), and the signal bandwidth or  
Figure 4: Distribution of magnitude-modifying factor. Figure 5: Ring-shaped signaling processed I/Q constellation (**a**) only affected by phase-modifying factor and (**b**) only affected by magnitude-modifying factor. Figure 3: Phase-modifying factor generation process. data rate is not affected by the ring-shaped signaling system, the Shannon capacity limit is only affected by \(Im\), as shown in Equation (4). The reduction in signal energy makes it possible to increase the covertness performance, but the performance of the recovered communication system is degraded because it is more sensitive to the channel noise. The reduced magnitude is directly related to reduced decision-making distance, as shown in Equation (5). Therefore, the noise signaling system must be driven by adjusting the parameters for the best performance of the recovered communication system and the best covertness at the same time:  
\[P_{ring}=\frac{1}{T}\int_{0}^{T}\left|\frac{I_{m}+1}{2}Ae^{i\omega t}\right|^{2 }dt=\left(A\frac{I_{m}+1}{2}\right)^{2} \tag{3}\]  
\[C_{ring}=Blog_{2}\left(1+\frac{\left(\frac{I_{m}+1}{2}\right)^{2}S}{N}\right) \tag{4}\]  
\[BER_{ring}=\frac{\sum_{1}^{symbol\ length}2Q\left(\sqrt{\frac{\left(\frac{I_{ m}+1}{2}\right)^{2}E_{0}}{N_{0}}}\right)}{symbol\ length} \tag{5}\]  
To measure the constellation confusion degree, we use the information entropy of the constellation data (\(\gamma=a+j^{*}b\)):  
\[H^{\Delta}(\gamma):=\sum_{i=-\infty}^{\infty}\sum_{j=-\infty}^{\infty}\Gamma( i,j)\log_{2}\Gamma(i,j) \tag{6}\]  
where \(\Gamma(i,j)=\int_{j\Lambda}^{(i+1)\Delta}\int_{j\Lambda}^{(j+1)\Delta}p(a,b) dab\), \(p(a,b)\) is the joint probability density function for \(a\) and \(b\) and \(\Delta\) is the quantification accuracy. Larger information entropy means less leakage of information. The ring-shaped signaling affects \(p(a,b)\) and \(\Delta\), \(lp\) affects \(p(a,b)\), and \(Im\) affects \(\Delta\). As \(lp\) increases, \(p(a,b)\) decreases. For example, if the original system has QPSK, \(p(a,b)=1/4\) is modified to an 8 PSK and 16 PSK shape, and \(p(a,b)\) changes to \(1/8\) and \(1/16\), respectively. In addition, as \(Im\) decreases, the average power decreases as shown in Equation (3). The lowered power is directly related to a lower \(\Delta\), which also affects \(\Gamma(i,j)\). With the combinations of \(Im\) and \(lp\) values, the information entropy will increase with the minimum value, as with the original system's information entropy.