## 2 System Model  
A block diagram for the baseband system model in the discrete-time domain is illustrated in Figure 1. It comprises a generic transmitter, a filtered additive channel model, and a generic receiver. It considers a vector representation for modeling the data communication through an LTI PLC channel, in which the output is corrupted by the presence of an additive noise modeled by a zero-mean and wide-sense stationary (WSS) random process. The time-invariant assumption is made because the time interval for transmitting a \(\bar{N}\)-length symbol block is much shorter than the coherence time of PLC channels. The vector \(\mathbf{h}=[h[0],\ldots,h[L_{h}-1]]^{T}\), where \((\cdot)^{T}\) is the transpose operator and \(h[n]\) is the \(n\)th sample of the CIR, denotes a vector representation of an \(L_{h}\)-length PLC channel CIR. Note that the CFR of such a PLC channel is also supposed to be frequency-selective.  
To build the \(l^{\text{th}}\)\(\bar{N}\)-length transmitted block, we assume that an \(L_{\text{CP}}\)-length CP is pre-appended to the \(2N\)-length sequence of symbols, generating \(\mathbf{x}_{l}\in\mathbb{R}^{\bar{N}\times 1}\), where \(\bar{N}=2N+L_{\text{CP}}\) and \(L_{\text{CP}}\leq L_{h}-1\). Further, the vector representation of the CFR in the discrete frequency domain is given by \(\boldsymbol{\mathcal{H}}=\sqrt{2N}\mathbf{F}[\mathbf{h}^{\mathsf{T}}\mathbf{ \mathcal{H}}_{(2N-L_{h})\times 1}^{T}]^{T}\in\mathbb{R}^{2N\times 1}\)where \(\mathbf{F}\in\mathbb{C}^{2N\times 2N}\) is the normalized DFT matrix with elements \(\mathrm{F}[n,m]=\frac{1}{\sqrt{2N}}\mathrm{e}^{-\mathrm{j}\frac{2n}{2N}nm}\), \((m,n)=0,1,\ldots,2N-1\), and \(\mathbf{0}_{\mathrm{C}\times\mathrm{D}}\) is a (C\(\times\)D)-size matrix with entries equal to zero.  
Now, we assume a lack of perfect synchronization at the receiver and, consequently, the presence of STO, which is quantified by the variable \(\Delta\in\mathbb{Z}\). Considering the presence of inter-symbol interference (ISI), STO, and the additive noise, the vector representation of the \(l\)th \(\widetilde{N}\)-length received block can be expressed as [28]  
\[\mathbf{y}_{l}=\sum_{i=-1}^{l}\mathbf{H}_{-i}\mathbf{x}_{l-i}+\mathbf{v}_{l}, \tag{1}\]  
where \(\mathbf{x}_{l-i}\in\mathbb{R}^{\widetilde{N}\times 1}\) represents the \((l-i)\)th transmitted block, \(-1\leq i\leq I\), with \(I=\lceil L_{h}/\widetilde{N}\rceil\) being the number of transmitted blocks that may cause interference because of the channel spreading, and \(\lceil z\rceil=\min\{a\in\mathbb{Z}|a\geq z\}\) being the ceiling function; \(\mathbf{v}_{l}\in\mathbb{R}^{\widetilde{N}\times 1}\) is the vector representation of the additive noise. Moreover, \(\mathbf{x}_{l-i}\) and \(\mathbf{v}_{l}\) are independent and zero-mean WSS random vectors, and \(\mathbf{H}_{-i}\in\mathbb{R}^{\widetilde{N}\times\widetilde{N}}\) is the channel CIR convolution matrix with entries  
\[H_{-i}[n,m]=\begin{cases}0,&\mu+n-m<0\\ h[\mu+n-m],&0\leq\mu+n-m\leq L_{h}-1\\ 0,&L_{h}-1<\mu+n-m,\end{cases} \tag{2}\]  
where \(\mu=i\widetilde{N}+\Delta\) is the first sample of the \((l-i)\)th received block with reference to the \(l\)th received block. Figure 2 shows a visual representation of (1) with \(\widetilde{N}\gg L_{\mathrm{CP}}\) and \(\mathbf{v}_{l}=\mathbf{0}_{\widetilde{N}\times 1}\).  
In the following section, we provide a detailed general formulation from which four multiplexing techniques are derived. These four schemes have distinct characteristics for dealing with typical operating conditions, such as complete or incomplete channel state information (CSI), cyclic prefix violations, frequency selectivity of the magnitude of CFR, power spectral density (PSD) of the additive noise, and nSNR.