## 3 Performance Parameters
### Selected Vectoring Pre-Coding and Cancellation Algorithms  
For DS transmissions, the vector of observables \(\mathbf{r}_{k}\) at the output of the \(N\leq M\) active receivers at sub-carrier frequency \(f_{k}\) is:  
\[\mathbf{r}_{k}=\mathbf{H}_{k}\mathbf{b}_{k}+\mathbf{n}_{k} \tag{6}\]  
where \(\mathbf{b}_{k}\) is the \(N\times 1\) vector of symbols transmitted by the \(N\) users on the sub-carrier frequency \(f_{k}\) and \(\mathbf{n}_{k}\) is the \(N\times 1\) vector accounting for background noise at each receiver. Symbols \(b_{i,k}\), \(i=1,\ldots,N\) are assumed to be zero mean and identically distributed with the same power \(P\). The \(N\times N\) matrix \(\mathbf{H}_{k}\) is the channel matrix at sub-carrier frequency \(f_{k}\). The main diagonal terms of \(\mathbf{H}_{k}\) account for direct propagation, i.e., \(h_{ii}^{(k)}=H_{D,k}(d_{i})\); the off-diagonal terms account for FEXT, i.e., \(h_{ij}^{(k)}=H_{F,k}(d_{i},l_{ij})\) for \(i\
eq j\) and  
\[H_{F,k}(d_{i},l_{ij})=\sqrt{\chi_{F}l_{ij}}f_{k}10^{X_{ij}/20}10^{\phi_{ij}}g_ {ij}(f_{k})H_{D,k}(d_{i}) \tag{7}\]  
In (7), \(g_{ij}(f_{k})\) is the frequency selectivity term of the FEXT transfer function accounting for random fluctuations with respect to the average FEXT level at frequency \(f_{k}\). We assume \(g_{ij}(f_{k})\) are complex Normal random variables and are identically distributed and statistically independent (i.i.d.) over \(i\) and \(j\) and for each \(k\); they have zero mean and unitary variance. The \(\phi_{ij}\) is a random phase term independent of \(k\) uniformly distributed in \([0,2\pi)\); \(l_{ij}\) is the coupling length between the \(i\)-th and \(j\)-th active users; finally \(\chi_{F}\) is the FEXT coupling coefficient. For a given reference distance \(d_{i}\), the coupling lengths \(l_{ij}\) can be easily obtained from the access network geometry such as that depicted in Figure 1. The \(X_{ij}\) (in dB) are random variables accounting for FEXT fluctuation with respect to the 1% FEXT condition [22]. The \(X_{ij}\) is assumed to be Normal (in dB) and its mean \(\mu_{dB}\) and standard deviation \(\sigma_{dB}\) do not depend on \(d\) and on the sub-carrier frequency. Furthermore, we consider the \(\{X_{ij}\}\) do not vary with frequency and are i.i.d. A discussion on the validity of Log-normal assumption even when \(X_{ij}\) are considered to be Beta distributed [23] has been presented in [24]. In the following we consider the random variables \(e^{y_{ij}}\) in place of \(10^{X_{ij}/20}\) i.e., \(y_{ij}=X_{ij}\ln(10)/20\) and \(y_{ij}=\mu+\sigma v_{ij}\) with \(\mu=-\mu_{dB}\ln 10/20\) and \(\sigma=\sigma_{dB}\ln 10/20\). Furthermore, for notational simplicity the explicit dependence of \(H_{F,k}(d_{i},l_{ij})\) on \(l_{ij}\) is omitted and we indicate \(H_{F,k}(d_{i},l_{ij})\) with \(H_{F,k}^{(ij)}(d_{i})\). Finally, from now on, the index \(i\) refer to the generic reference user. The channel matrix \(\mathbf{H}_{k}\) in (6) can be conveniently re-written as:  
\[\mathbf{H}_{k}=\mathbf{D}_{k}\left(\mathbf{I}+\mathbf{C}_{k}\right) \tag{8}\]  
**I** is the identity matrix; \(\mathbf{D}_{k}\) is a diagonal matrix containing the direct propagation terms along its diagonal; the matrix \(\mathbf{C}_{k}\) has zeros along the main diagonal and contains the FEXT terms normalized by row for the corresponding direct propagation term, i.e., \(c_{ij}^{(k)}=H_{F,k}^{(i,j)}(d_{i})/H_{D,k}(d_{i})\) for \(i\
eq j\). The pre-coding matrix to be applied to the DS symbols approximates the inverse of \(\mathbf{I}+\mathbf{C}_{k}\). Since \(|c_{ij}^{(k)}|\ll 1\) in \(\mathbf{C}_{k}\) a good approximation of the inverse of \(\mathbf{I}+\mathbf{C}_{k}\) is:\[\mathbf{L}_{k}^{(q)}=\mathbf{I}+\sum_{m=1}^{q-1}(-1)^{m}\mathbf{C}_{k}^{m} \tag{9}\]  
and \(q>1\) is referred as the order of the pre-coding matrix.