## 3 General Formulation
### Interference in the Data Communication System  
If the data communication system is operating with \(L_{\text{CP}}<L_{th}-1\) and \(\Delta\
eq 0\), two types of interference may be present in the data communication system. The first is the ISI caused by the \(2N\)-length blocks symbols \(\mathbf{X}_{\zeta,l-i},i=-1,1,\ldots,I\), to the desired \(2N\)-length symbol block, and \(\mathbf{\Lambda}_{\mathbf{A}_{i^{\prime}}}i=-1,1,\ldots,I\) contributes to this interference. The second interference varies with the scheme in use; it is called inter-carrier interference (ICI) for the HS-OFDM scheme, inter-chirp interference (ICpl) for the OCDM scheme, inter-slot interference (IStl) for the SCCP scheme, and inter-tile interference (ITI) for the OTFDM scheme. This second type of interference is caused by other subcarriers (HS-OFDM), sub-chirps (OCDM), subslots (SCCP), and subtiles (OTFDM) from the \((l-i)\)th \(2N\)-length blocks, \(i=-1,0,\ldots,I\). The matrices \(\mathbf{\overline{\Lambda}}_{\mathbf{A}_{i^{\prime}}}\), \(i=-1,0,\ldots,I\) comprise this second type of interference. Moreover, it is easy to see that from (2), a value of \(\Delta=0\) results in \(\mathbf{H}_{1}=\mathbf{0}_{\widetilde{N}\times\widetilde{N}}\), and, as a consequence, in \(\mathbf{A}_{-1}=\mathbf{0}_{2N\times 2N}\) regardless of the \(L_{\text{CP}}\) values. Moreover, regarding the channel spreading influence of other \(2N\)-length symbol blocks into the \(l\)th \(2N\)-length symbol block, the use of \(L_{\text{CP}}\geq L_{h}-1\) deals with this issue, i.e., \(\mathbf{A}_{i}=\mathbf{0}_{2N\times 2N}\), \(i=1,\ldots,I\). However, values of \(\Delta>0\) also decrease the channel spreading in the target \(2N\)-length symbol block, as shown in (2), which leads to \(\mathbf{A}_{i}=\mathbf{0}_{2N\times 2N}\), \(i=1,\ldots,I\) if \(L_{\text{CP}}+\Delta\geq L_{h}-1\) as Theorem 1 dictates. **Theorem 1**. : _Let \(\overline{\mathbf{H}}_{-i}=\mathbf{\Psi}_{r}\mathbf{H}_{-i}\mathbf{\Psi}_{t} \in\mathbb{R}^{2N\times 2N}\) be the so-called cropped channel CIR convolution matrix, with entries given by_  
\[\overline{\underline{H}}_{-i}[n,m]= \begin{cases}0,&\mu+n-m<0\\ h[\mu+n-m],&0\leq\mu+n-m\leq L_{h}-1\\ 0,&L_{h}-1<\mu+n-m,\end{cases}  \tag{10}\]  
_with \(n,m\in[0,2N-1]\). If \(\Delta\geq L_{h}-1-L_{\text{CP}}\), then \(\overline{\underline{\mathbf{H}}}_{-i}=\mathbf{0}_{2N\times 2N}\) for \(i=1,\ldots,I\)._  
Proof. : Suppose that \(\Delta\geq L_{h}-1-L_{\text{CP}}\) and \((n-m)\in[-2N+1,2N-1]\) with \(N\in\mathbb{N}^{*}\). To show that \(\overline{\underline{\mathbf{H}}}_{-i}=\mathbf{0}_{2N\times 2N}\), \(i=1,\ldots,I\), we need to prove that all of its entries are equal to 0. Meaning that the intersection of the interval \(\mathbb{A}=\{(n-m)\in\mathbb{Z}\mid\mu\leq n-m\leq L_{h}-1-\mu\}\) and interval \(\mathbb{B}=\{(n-m)\in\mathbb{Z}\mid-2N+1\leq n-m\leq 2N-1\}\) must be empty for \(i=1,\ldots,I\) and \(\Delta\geq L_{h}-1-L_{\text{CP}}\). We can prove that \(\mathbb{A}\cap\mathbb{B}=\emptyset\) by proving that \(\mathbb{A}\cap\mathbb{B}\
eq\emptyset\) is false. Therefore, we expand \(\mathbb{A}\), so that  
\[\mathbb{A}=\{n-m\in\mathbb{Z}\mid-i(2N+L_{\text{CP}})-\Delta\leq n-m\leq L_{h }-1-i(2N+L_{\text{CP}})-\Delta\}. \tag{11}\]  
Hence, for \(\mathbb{A}\cap\mathbb{B}\
eq\emptyset\) to be true, at least one extremity of \(\mathbb{B}\) needs to be inside the interval \(\mathbb{A}\). Applying the inferior extremity of \(\mathbb{B}\) first, we have that  
\[-i(2N+L_{\text{CP}})-\Delta\leq-2N+1 \tag{12}\]  
and  
\[-2N+1\leq L_{h}-1-i(2N+L_{\text{CP}})-\Delta. \tag{13}\]Therefore, (12) is a correct statement if \(\Delta\geq-(i-1)2N-iL_{\text{CP}}-1\) and since we already established that \(\Delta\geq L_{h}-1-L_{\text{CP}}\), then (12) is true.