## 6 Analytical Assessment of SES  
SES uses an exceptional method to represent a bit in three possible signal states. The amplitude \(\mathbf{(a_{k},a_{k+n})}\) may be used for 1 s, having \"A\" amplitude and 0 levels, while a logic 0 can illustrate the signal level \"-A\" and 0. The low state of 0 is used in both cases of logic (0, 1). The probability of 0 is \(\frac{1}{2}\) and the probability of +A and -A is \(\frac{1}{4}\). Therefore, SES uses a different adjacent bit pattern with various past line codes. It is due to the symmetric level of SES, which affects the behavior of autocorrelation. The possibilities of digital bit transmission are equal to \(\frac{1}{2}\). PSD is based on two main factors: one is the pulse shape (line coder) and the second is autocorrelation [31]. Equation (6) shows the simplified form of autocorrelation, and Equation (7) represents the physical waveform of line encoding. The simplified form of the PSD digital signal is shown in Equation (1). There are three possible levels of signal representation in SES. The signal demonstration has the specific levels +A for 1 and -A for 0, and 0 symbols are used for both 1 s and 0 s for a second consecutive bit with a probability of \(\frac{1}{2}\). Table 2 shows the probabilities of inputs of digital bits and a description of the bits with their likelihood. For n = 0  
Figure 7: Line Coder System. \[ \begin{array}{l}R0=A.A\frac{1}{4}+\left(-A.-A\frac{1}{4}\right)+0.0\frac{1}{2}\\ =\frac{A^{2}}{4}+\frac{A^{2}}{4}+0\\ =\frac{2A^{2}}{4}=\frac{A^{2}}{2}\end{array}  \tag{8}\]  
Table 2 shows the case of n = 1 with consecutive adjacent bits. The autocorrelation necessitates the difference for \(|\)n\(|\) = 1. Next is the last case of \(|\)n\(|\) > 1, which has a possible combination for n > 1. The combinations of bits in three-bit logic are (000, 001, 010... 111). Now, all of these bits have equal probability. (\(\mathbf{a_{k}}\), \(\mathbf{a_{k+n}}\)) = 0. Equation (9) shows all the results of the autocorrelation \(R_{A}(\mathbf{n})\). The autocorrelation results are calculated for n = 0, n = 1, and n > 1. \[R_{A}(n)=\left[ \begin{array}{l}\frac{a^{2}}{2},n=0\\ \frac{-\sigma^{2}}{4},n=+A\\ \frac{\sigma^{2}}{4},n=-A\\ 0,n>1\end{array} \right] \tag{9}\]  
Equation (10) shows the general format of PSD with autocorrelation and line code. The PSD is formulated for SES after applying autocorrelation and probability factors. \[ \begin{array}{l}S_{x}(f)=\frac{1}{T_{b}}\bigg{[}T_{b}\bigg{(}\frac{sin\pi fT _{b}}{\pi fT_{b}}\bigg{)}^{2}\bigg{]}\bigg{[}R_{A}(-1)e^{j2\pi fTb}\bigg{]}+R_ {A}(0)+\Big{[}R_{A}(1)e^{-j2\pi fTb}\bigg{]}+R_{A}(0)\\ S_{x}(f)=\bigg{[}T_{b}\bigg{(}\frac{sin\pi fT_{b}}{\pi fT_{b}}\bigg{)}^{2} \bigg{]}\bigg{[}\frac{\sigma^{2}}{2}-\frac{\sigma^{2}}{4}\bigg{(}e^{j2\pi fTb} +e^{-j2\pi fTb}\bigg{)}\bigg{]}\\ S_{x}(f)=\bigg{[}\frac{\pi^{2}T_{b}}{2}\bigg{(}\frac{sin\pi fT_{b}}{\pi fT_{b}} \bigg{)}^{2}\bigg{]}\bigg{[}1-\cos(2\pi fTb)\bigg{]}\\ S_{x}(f)=\bigg{[}\frac{\pi^{2}T_{b}}{2}\bigg{(}\frac{sin\pi fT_{b}}{\pi fT_{b}} \bigg{)}^{2}\bigg{]}\bigg{[}2 sin^{2}(\pi fT_{b})\bigg{]}\\ S_{x}(f)=\bigg{[}a^{2}T_{b}\bigg{(}\frac{sin\pi fT_{b}}{\pi fT_{b}}\bigg{)}^{2} \bigg{]}\bigg{[}sin^{2}(\pi fT_{b})\bigg{]}\end{array}  \tag{10}\]  
Equation (4) shows that each \(\mathbf{T_{b}}\) requires double bandwidth in bipolar RZ and AMI. SES has similarities of signal modulation with RZ and AMI, except for dual bandwidth usage. SES does not include \(\mathbf{T_{b/2}}\), so the equation of SES for PSD is shown in Equation (11). \[P_{SES}(f)=\frac{A^{2}T_{b}}{4}\bigg{(}\frac{sin\pi fT_{b}}{\pi fT_{b}}\bigg{)} ^{2}sin^{2}(\pi fT) \tag{11}\] \begin{table}
\begin{tabular}{c c c} \hline
**Input** & **Probability** & **Description** \\ \hline \(P(Ak=+a)\) & \(\frac{1}{4}\) & It is for bit 1 \\ \hline \(P(Ak=-a)\) & \(\frac{1}{4}\) & It is for bit 0 \\ \hline \(P(Ak=0)\) & \(\frac{1}{2}\) & It is for second consecutive bit 0 or 1 \\ \hline \end{tabular}
\end{table} 
Table 2: Digital bit probabilities in SES.