# Performance Analysis of NR-DCSK Based Copper Cable Model for G.fast Communication
## 2 System Model  
Figure 1 shows the block diagram of NR-DCSK transceiver system over copper cable channel. In the \(i\)th frame, the chaotic generator produces a signal \(x_{k/P,i}\) of length \(\beta/P\), and then repeats \(P\) times, which composes the reference chaotic sequence \(x_{x,i}\) of length \(\beta\). The reference chaotic sequence is placed in the first time slot, the second time slot contains the information chaotic sequence. The \(i\)th information chaotic signal is generated by multiplying the \(i\)th information bit \(b_{i}\in\{-1,+1\}\) by \(x_{k,i}\). In NR-DCSK system, the frame interval length for each bit \(T_{NR-DCSK}=2\beta T_{c}\), where \(T_{c}\) is the chip-time and \(\beta\) is replicated reference chaotic sequence length as shown in Figure 2 The \(i\)th output of the transmitter \(s_{k,i}\) is written as  
\[s_{k,i}= \begin{cases}x_{[k/P],i},&0<k<\beta\\ b_{i}x_{[k/P]-\beta,i},&\beta<k<2\beta\end{cases}  \tag{1}\]  
where \(x_{[k/P],i}\) is the \(i\)th replicated chaotic sequence of length \(\beta/P=R\). As illustrated in Figure 1, the channel model represented in this work is the copper cable channel that is developed in [14; 16; 30] with impulse response,  
\[h(t)=\sum_{l=1}^{\mathcal{L}}\lambda_{l}\delta(t-\tau_{l}) \tag{2}\]  
where \(\delta(t)\) is the Dirac delta function, \(\lambda_{l},\tau_{l}\) represent \(l\)th gain and the delay of the channel, respectively, and \(\mathcal{L}\) is the path number. The frequency response of this channel based on Chen model can be written as  
\[H=e^{-L\psi(f)} \tag{3}\]  
where, \(\psi(f)=\varepsilon(f)+j\omega(f)\) is the propagation constant and \(L\) is the length in meter. while the variables \(\varepsilon(f)\) and \(\omega(f)\) are the attenuation and phase coefficients and can be expressed as  
\[\varepsilon(f)=K_{1}\sqrt{f}+K_{2}\] \[\omega(f)=K_{3}f \tag{4}\]where, \(K_{1}\), \(K_{2}\), and \(K_{3}\) are the constants of the model. The copper cable channel is similar to the power line channel in containing two additive noise types, impulsive and Additive White Gaussian Noise (AWGN). Hence, the total noise signal, \(n_{k}\), can be expressed as,  
\[n_{k}=g_{n}+I_{n} \tag{5}\]  
where \(g_{n}\) is a complex AWGN with variance \(N_{g}\), and \(I_{n}\) is the impulse complex noise with variance \(N_{l}\). All the signals in Equation (3) are mutually independent. In this work, Middleton noise model is considered to perform the impulsive noise for copper cable channel. The probability density function (PDF) of \(n_{k}\) is performed as a summation of zero-mean Gaussian components with weighting depends on a Poisson distribution [30; 31]:  
\[P_{r}(n_{k})=\sum_{m=0}^{\infty}\frac{\gamma^{m}e^{-m}}{m!}\frac{1}{\sqrt{2\pi \sigma_{m}^{2}}}e^{-\frac{\pi_{k}^{2}}{2\sigma_{k}^{2}}} \tag{6}\]  
The variance \(\sigma^{2}\) is calculated as  
\[\sigma_{m}^{2}=\sigma^{2}\frac{(\frac{mA}{\gamma}+1)}{1+A}, \tag{7}\]  
where \(\sigma^{2}=\sigma_{g}^{2}+\sigma_{I}^{2}\), in which \(\sigma_{g}^{2}\) and \(\sigma_{I}^{2}\) represent the variance of AWGN and impulsive noise, respectively. The value \(\gamma\) is an impulsive noise density and \(A=\frac{\sigma_{I}^{2}}{\sigma_{g}^{2}}\). The \(i\)th received signal \(r_{k,i}\) is expressed as  
\[r_{k,i}=\sum_{l=1}^{\mathcal{L}}\lambda_{l}s_{k-\gamma_{l}}+n_{k}. \tag{8}\]  
The received sequence is first averaged according to  
\[r_{k,i}=\frac{1}{P}\sum_{\rho=1}^{P}r_{k+\rho R,i},\qquad k=1,\ldots,2R \tag{9}\]  
The noise signal after averaging process \(\bar{n}_{k,i}=\frac{1}{P}\sum_{\rho=1}^{P}n_{k+\rho R,i}\) with the variance \(\sigma^{2}/P\). Then, the correlation between the averaged reference and information sequence over the duration \(RT_{c}\) is occurred. The purpose of the averaging process is to reduce the variance of the noise by \(P\) value, and hence improvement of the BER is obtained. In this work, we assumed that the maximum channel delay, \(\tau\) is less than the chaotic sequence duration, \(T_{c}R\), to neglect the inter-symbol interference (ISI) [29; 31].