## 3 Bit Error Rate Analysis of NR-ODBR-DCSK System  
In this section, Gaussian approximation (GA) is used to derive the bit error rate of the NR-ODBR-DCSK system under AWGN channel conditions, using the chaotic mapping as a normalized improved logistic mapping, which is a smooth chaotic sequence used as noise interference in the AWGN channel, with \(n_{i}\) mean of 0, variance of \(N_{0}/2\), and statistical independence.  
Table 2 is a compilation of several important mathematical symbols and the meanings they denote that are applied in this chapter.  
Here, we only consider the BER performance of message \(a_{k}\) and use it as a substitute for the BER performance of the whole system, so expanding \(Z_{i,1}\) in Equation (11) yields  
\[Z_{i,1}=\sum_{k=1}^{\beta/P}(\underbrace{a_{k}x_{i}^{2}}_{required signal}+\underbrace{b_{k}x_{i}x^{\prime}+\frac{1}{P}\sum_{p=1}^{p}n_{i}x_{i}+a_{k}x_{i} \frac{1}{P}\sum_{p=1}^{p}n_{i,P+k}+b_{k}x_{i}^{\prime}\frac{1}{P}\sum_{p=1}^{ p}n_{i,P+k}}_{internal interference}+\underbrace{\frac{1}{P}\sum_{p=1}^{p}n_{i,P+k}}_{ noise}\frac{1}{P}\sum_{p=1}^{p}n_{i}) \tag{13}\]  
From the signal frame structure format of the NR-ODBR-DCSK system, the signal energy within one frame can be expressed as  
\[E_{b}=P\sum_{k=1}^{\beta/P}E[x_{i}^{2}] \tag{14}\]  
If \"+1\" is transmitted for the lth user, (4) becomes  
\[Z_{i,1}=\sum_{k=1}^{\beta/P}(\underbrace{x_{i}^{2}}_{required signal}+ \underbrace{b_{k}x_{i}x^{\prime}+\frac{1}{P}\sum_{p=1}^{p}n_{i}x_{i}+x_{i} \frac{1}{P}\sum_{p=1}^{p}n_{i,P+k}+b_{k}x_{i}^{\prime}\frac{1}{P}\sum_{p=1}^ {p}n_{i,P+k}}_{internal interference}+\underbrace{\frac{1}{P}\sum_{p=1}^{p}n_{i, P+k}\frac{1}{P}\sum_{p=1}^{p}n_{i})}_{noise} \tag{15}\]  
Similarly, when sending \"\(-1\)\", the required signal term becomes negative.  
The means and variances of \(Z_{i,1}\), given a \"+1\" or \"\(-1\)\" is sent, can be evaluated by numerical simulations. Denote the respective means and variances by \(E[Z_{i,1}]\) and \(Var[Z_{i,1}]\). Since \(Z_{i,1}\) is the sum of a large number of random variables, we may assume that it is normally distributed. Hence, the approximate error probability of the \(i\)th transmitted bit is  
\[\begin{array}{ll}BER_{NR-ODBR-DCSK}&=2\text{Prob}(a_{k}=+1)\times\text{Prob} (Z_{i,1}\leq 0|a_{k}=+1)\\ &=\frac{1}{2}erfc[\frac{E[Z_{i,1}(a_{k}=+1)]}{\sqrt{2 Var[Z_{i,1}(a_{k}=+1)]} }]\end{array} \tag{16}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Mathematical Symbol** & **Connotation** \\ \hline \(Z_{i,1}\) & Judgment signal \\ \(E_{b}\) & Signal energy \\ \(E[\cdot]\) & Mean value operator \\ \(Var[\cdot]\) & Variance operator \\ \(erfc[\cdot]\) & Complementary error function \\ \(\text{Prob}[\cdot]\) & Probability function \\ \hline \hline \end{tabular}
\end{table}
Table 2: Math operators used in this chapter and their meanings.  
In order to obtain the BER, we first obtain the values of the terms (expected values, variances and covariances) on the right side of Equation (15) by numerical simulations. Then, the values of mean and variance can be computed and substituted into (16) to obtain the BER. Since both numerical simulation and analytical method are involved in obtaining the BER, we refer to this method as a mixed analysis-simulation (MAS) technique.  
Based on the statistical properties of chaotic sequences and signal energies, it is not difficult to conclude that  
\[E[Z_{i,1}|(a_{k}=+1)\,]=2\sum_{k=1}^{\beta/P}E[x_{i}^{2}]=2\frac{E_{b}}{P} \tag{17}\]  
\[Var[Z_{i,1}|(a_{k}=+1)\,]=\frac{4E_{b}N_{0}}{P^{2}}+\frac{\beta N_{0}^{2}}{4P^ {3}} \tag{18}\]  
Substituting Equations (17) and (18) into Equation (16) yields the theoretical BER equation for NO-ODBR-DCSK under AWGN:  
\[BER_{NR-ODBR-DCSK}=\frac{1}{2}erfc[(\frac{2N_{0}}{E_{b}}+\frac{\beta N_{0}^{2}} {8 PE_{b}^{2}})^{-\frac{1}{2}}] \tag{19}\]