## 3 General Formulation
### Generic Receiver  
The block diagram for the generic receiver is shown in Figure 3b. First, the received block at the receiver input, given by (1) with \(\mathbf{x}_{l-i}\), and equal to (6), is submitted to the removal of the CP, which is accomplished with the matrix \(\mathbf{\Psi}_{\text{r}}=[\mathbf{0}_{2N\times L_{\text{CP}}}\ \ \mathbf{I}_{2N}]\). After that, the normalized DFT matrix, \(\mathbf{F}\), and the frequency domain equalizer \(\mathbf{E}\) apply. In the sequel, the estimated \(2N\)-length symbol block is obtained after using the inverse of the precoding matrix \(\mathbf{Q}^{\dagger}\). Consequently, the generic receiver matrix is defined as  
\[\mathbf{R}\triangleq\mathbf{Q}^{\dagger}\mathbf{E}\mathbf{F}\mathbf{\Psi}_{ \text{r}}. \tag{7}\]  
Figure 3: Block diagrams of the generic transmitter and receiver. (**a**) Generic transmitter; (**b**) generic receiver.  
Depending on the type of frequency domain equalizer, distinct levels of interference mitigation can be accomplished. According to [27], frequency domain equalizers offering three distinct levels of interference mitigation are given by  
\[\mathbf{E}=\begin{cases}\mathbf{B}_{0}^{\dagger}\Big{(}\sum_{i=-1}^{I}\mathbf{B} _{i}^{\dagger}\mathbf{B}_{i}\Big{)}^{-1},&\text{for C-ZF}\\ (\mathbf{B}_{0}^{\dagger}\mathbf{B}_{0})^{-1}\mathbf{B}_{0}^{\dagger},&\text{ for M-ZF}\\ \mathbf{\Lambda}_{\mathbf{B}_{0}}^{-1},&\text{for ST-ZF}\end{cases}, \tag{8}\]  
where \(\mathbf{\Lambda}_{\mathbf{B}_{0}}\) is obtained assuming that a full matrix \(\mathbf{Z}\) is decomposed into its diagonal (\(\mathbf{\Lambda}_{\mathbf{Z}}=\mathbf{diag}\{\mathbf{Z}\}\)) and off-diagonal components (\(\mathbf{\overline{\Lambda}}_{\mathbf{Z}}=\mathbf{Z}-\mathbf{\Lambda}_{\mathbf{ Z}}\)), and \(\mathbf{B}_{i}=\mathbf{F}\mathbf{r}_{\mathbf{r}}\mathbf{H}_{-i}\mathbf{r}_{ \mathbf{r}}\mathbf{F}^{\dagger}\). Note that the C-ZF scheme attains the best interference mitigation and demands the highest processing power. In contrast, ST-ZF offers the worst interference mitigation and demands the lowest processing power. Moreover, M-ZF achieves a performance (and demands a processing power) between C-ZF and ST-ZF.  
Applying (7) in (1) while considering (6) results in the estimated mapped vector, which corresponds to an estimation of \(\mathbf{X}_{\zeta,l}\), given by  
\[\mathbf{\widehat{X}}_{\zeta,l} =\sum_{i=-1}^{I}\mathbf{RH}_{-i}\mathbf{TX}_{\zeta,l-i}+\mathbf{R }\mathbf{v}_{l} \tag{9}\] \[=\mathbf{RH}_{0}\mathbf{TX}_{\zeta,l}+\sum_{\begin{subarray}{c}i=- 1\\ i\
eq 0\end{subarray}}^{I}\mathbf{RH}_{-i}\mathbf{TX}_{\zeta,l-1}+\mathbf{Q}^{ \dagger}\mathbf{EV}_{l}\] \[=\underbrace{\mathbf{\Lambda}_{\mathbf{A}_{0}}\mathbf{X}_{\zeta,l }}_{\text{attenuated vector}}+\underbrace{\mathbf{\overline{\Lambda}}_{\mathbf{A}_{0}} \mathbf{X}_{\zeta,l}+\sum_{\begin{subarray}{c}i=-1\\ i\
eq 0\end{subarray}}^{I}\mathbf{A}_{i}\mathbf{X}_{\zeta,l-1}}_{\text{ interference vector}}+\underbrace{\mathbf{GV}_{l}}_{\text{enhanced-noise}}.\]  
where \(\mathbf{A}_{i}=\mathbf{RH}_{-i}\mathbf{T}\), \(\mathbf{G}=\mathbf{Q}^{\dagger}\mathbf{E}\) is the noise-enhancement matrix, \(\mathbf{V}_{l}=\mathbf{F}\mathbf{\overline{\Psi}}_{\mathbf{r}}\mathbf{v}_{l}\) is the noise in the frequency domain. The final step of the receiver is the demapping process.  
Based on mathematical descriptions of the generic transmitter and receiver, Figure 4 shows how to derive the transmitter and receiver of HS-OFDM, OCDM, SCCP, and OTFDM from the generic transmitter and receiver. In other words, the detailed mathematical formulation shows that the similar compositions of these four data communication schemes opens up the opportunity for a future compact and unified implementation (i.e., low hardware resources) in a PLC transceiver, which can be dynamically switched to deal with specific characteristics of a data communication media, sensed by the PLC system.