# Communication Fault Maintenance Decision of Information System Based on Inverse Symmetry Algorithm
## 2 Definition of Algorithm  
### Principle of Inverse Symmetry Algorithm  
The basic idea of the inverse symmetric algorithm (PISM) is to divide the channel in frequency domain or time domain, so that the noise between adjacent frequency bands or adjacent time slots has strong correlation [9; 10; 11; 12], and then use the correlation of band limited noise and its digital characteristics to eliminate the noise. The basic principle block diagram of the inverse symmetry algorithm is shown in Figure 1.  
In Figure 1, C\({}_{1}\) and C\({}_{2}\) are two adjacent subchannels, which can be adjacent frequency bands, adjacent time slots, or adjacent space. The so-called \"inverse phase\" refers to sending two mutually opposite signals, \(s_{1}(t)\) and \(s_{2}(t)\), into two adjacent channels, C\({}_{1}\) and C\({}_{2}\) respectively, at the transmitting end, where \(s_{1}(t)=-s_{2}(t)=s(t)\). The so-called \"symmetry\" means that the transmission characteristics of the two channels are basically the same. In order to simplify the calculation, let the gain of these two channels be one, and assume that they have zero mean additive noise [13], which is recorded as \(n_{1}(t)\) and \(n_{2}(t)\). If the channel attenuation and delay are ignored, the output of C\({}_{1}\) and C\({}_{2}\) are respectively. \[r_{1}(t)=\sum_{i=1,3,5}^{\infty}\frac{4}{\pi}\frac{\delta}{i}\sin(i\alpha)\cos (i\theta)+n_{1}(t) \tag{1}\]  
\[r_{2}(t)=\frac{4}{\pi}\frac{\delta}{i}\sin(i\alpha)=\frac{4}{\pi}\frac{\delta} {i}\sin(i\alpha_{p}\frac{\pi}{2})+n_{2}(t) \tag{2}\]  
where \(\sin(i\alpha)\) and \(\cos(i\theta)\) are the included angles of corresponding adjacent time slots and adjacent frequency bands respectively, and \(\delta\) is the harmonic amplitude. In the process of denoising in frequency domain, the elimination effect of the channel depends on the attenuation power of the subchannel. The signal output result can be calculated according to the following formula.Send them to the subtracter and the total output is:  
\[r(t)=r_{1}(t)\left[\sum_{j=1,2,3}^{\infty}\frac{\sin(j\beta)}{j}\cos(j\theta) \right]-r_{2}(t)\beta S_{o}(t)+n_{0}(t) \tag{3}\]  
Figure 1: Basic principle block diagram of inverse symmetric algorithm. \(s_{o}(t)\) and \(n_{o}(t)\) are the total output signal and noise, respectively. According to the output results, the whole cluster center is obtained, and the average value of training samples in cluster set \(\beta\) is calculated. The final basis function center of the RBF (Radial Basis Function) neural network is obtained, and the total output signal power is:  
\[s_{o}=E\Big{[}(2s(t))^{2}\Big{]}=4E\Big{[}s^{2}(t)\Big{]}\approx 4S \tag{4}\]  
In Equation (4), \(s\approx E\Big{[}s^{2}(t)\Big{]}\) is the power of single output signal. The total output noise power is:  
\[N_{o}=E\Big{[}(n_{1}(t)-n_{2}(t))^{2}\Big{]} \tag{5}\] \[=N_{1}+N_{2}-2\rho(N_{1}N_{2})^{1/2}\]  
In Equation (5), \(N_{1}\) and \(N_{2}\) are the output noise power of C1 and C2, respectively, and \(\rho\) is the correlation coefficient of the two channels of noise. Then, ADMM (Alternating Direction Methodof Multipliers) alternating multiplier method is used to update the center frequency, modal components, and \(\theta(\mathrm{t})\) alternately to find the saddle point of the above formula. According to the gradient of demodulation signal, the bandwidth is:  
\[\mathrm{N_{1}}=\frac{N_{0}-\sum\limits_{k=1}^{K}\sin(i\alpha)+\frac{\theta(t )}{2}}{1+2\alpha(i\alpha-j\theta)} \tag{6}\]  
Suppose that the noise power of two channels is equal, that is to say, substituting in Equation (5) can get:  
\[N_{o}=2(1-\rho)N \tag{7}\]  
According to the above formula, the output SNR formula is as follows:  
\[r_{1}(t)=\sum_{i=1,3,5}^{\infty}\frac{4}{\pi}\frac{\delta}{i}\sin(i\alpha)\cos (i\theta)+n_{1}(t) \tag{1}\]0  
As the channel gain is one, the total input signal power of the subtractor is \(S_{i}\approx 2S\), and the total input noise power of the subtractor is \(N_{i}\approx 2N\).