# Performance Analysis of NR-DCSK Based Copper Cable Model for G.fast Communication
## 3 Performance Analysis of NR-DCSK over Copper Channel  
In this section, the BER analysis of NR-DCSK system over copper channel is presented. In this work, the second-order Chebyshev polynomial function (CPF) \(x_{k+1}=1-2x_{k}^{2}\) is utilized. The chaotic signal is normalized to zero mean and unity variance. During this analysis \(T_{c}=1\) for simplicity. The total BER of the NR-DCSK over copper channel including impulsive noise and AWGN can be written as \(BER_{T}=\gamma P_{b}^{I}+(\gamma-1)P_{b}\), where \(P_{b}^{I}\) and \(P_{b}\) are the error probability in the existence of impulsive noise, and the error probability with only AWGN, respectively. The scale \(\gamma\) is the probability influenced by the impulsive noise with one bit transmitted. To derive the expressions \(P_{b}^{I}\) and \(P_{b}\), we need to find the mean and the variance of Equation (8). It can be noticed that the sequences are uncorrelated as the channel gains, the chaotic, and the noise signals are all statistically independent. Thus, the mean of the \(i\)th decision variable is written as  
\[\mathbb{E}(D_{i})=b_{i}\sum_{l=1}^{\mathcal{L}}|\lambda_{l}|^{2}\sum_{k=1}^{R }\mathbb{E}(x_{k,i}^{2}), \tag{11}\]  
where \(\mathbb{E}(\cdot)\) is the expectation operation. If we define the energy of the transmitted NR-DCSK as \(E_{b}=2P\sum_{k=1}^{\mathcal{B}/P}\mathbb{E}(x_{k,i}^{2})\), then Equation (9) can be written in terms of \(E_{b}\) as  
\[\mathbb{E}(D_{i})=\frac{1}{2P}b_{i}\sum_{l=1}^{\mathcal{L}}|\lambda_{l}|^{2}E _{b}, \tag{12}\]  
Similarly, the variance of the \(i\)th decision variables in terms of \(E_{b}\) can be calculated as  
\[\mathbb{V}(D_{i})=\frac{1}{2P^{2}}\sum_{l=1}^{\mathcal{L}}|\lambda|^{2}E_{b}( N_{g}+N_{l})+\frac{R}{2P^{2}}(N_{g}^{2}+N_{l}^{2}). \tag{13}\]  
where \(\mathbb{V}(\cdot)\) is the variance operation. If we considered the bit energy as constant for large reference length \(R\) and according to the central limit theorem, a Gaussian distribution can approximate the decision variable. The \(P_{b}^{I}\) formula can be expressed as:  
\[P_{b}^{I}=\frac{1}{2}erfc\bigg{(}\bigg{(}\frac{2\mathbb{V}(D_{i})}{\mathbb{E} (D_{i})^{2}}\bigg{)}^{-0.5}\bigg{)} \tag{14}\]  
where \(erfc(x)=2/\sqrt{\pi}\int_{x}^{\infty}\exp{(-\alpha^{2})}d\alpha\) is the complementary error function (CEF). By plugging Equations (10) and (11) into (12), the BER of NR-DCSK over copper channel in the presence of impulsive noise and AWGN can be written as  
\[P_{b}^{I}=\frac{1}{2}erfc\bigg{(}\bigg{(}\frac{4(N_{g}+N_{l})}{\sum_{l=1}^{ \mathcal{L}}|\lambda|^{2}E_{b}}+\frac{2R(N_{g}^{2}+N_{l}^{2})}{(\sum_{l=1}^{ \mathcal{L}}|\lambda|^{2}E_{b})^{2}}\bigg{)}^{-0.5}\bigg{)} \tag{15}\]  
By substituting \(N_{l}=0\), the BER in the presence of AWGN \(P_{b}\) is obtained as \[P_{b}^{I}=\frac{1}{2}erfc\bigg{(}\bigg{(}\frac{4(N_{g})}{\sum_{l=1}^{\mathcal{L}}| \lambda|^{2}E_{b}}+\frac{2R(N_{g}^{2})}{(\sum_{l=1}^{\mathcal{L}}|\lambda|^{2}E _{b})^{2}}\bigg{)}^{-0.5}\bigg{)} \tag{16}\]  
Finally, the total BER can be calculated as  
\[BER_{T} =\frac{\gamma}{2}erfc\bigg{(}\bigg{(}\frac{4(N_{g}+N_{I})}{\sum_{l =1}^{\mathcal{L}}|\lambda|^{2}E_{b}}+\frac{2R(N_{g}^{2}+N_{I}^{2})}{(\sum_{l= 1}^{\mathcal{L}}|\lambda|^{2}E_{b})^{2}}\bigg{)}^{-0.5}\bigg{)}\] \[+\frac{1-\gamma}{2}erfc\bigg{(}\bigg{(}\frac{4(N_{g})}{\sum_{l=1 }^{\mathcal{L}}|\lambda|^{2}E_{b}}+\frac{2R(N_{g}^{2})}{(\sum_{l=1}^{\mathcal{ L}}|\lambda|^{2}E_{b})^{2}}\bigg{)}^{-0.5}\bigg{)} \tag{17}\]  
Equation (17) shows that there are two separate components; the first component represents the combined effect of impulsive noise and AWGN, while the second component only takes into account the existence of AWGN.