## 2 System Model
To combat the effect of multipath channels with \(L\) significant paths, a sequence \(\mathbf{x}_{d}=[x_{d_{1}},x_{d_{2}},\ldots,x_{d_{m}}]\) is appended at the beginning of the block. It is emphasized that in general \(m\geq L\). The block \(\mathbf{b}=[\mathbf{x}_{d}\ \mathbf{x}]^{T}\) is thus formed for transmission over the channel, which can be seen in Figure 2. The extended block is also shown in Figure 2. It is important to bear in mind that if the channel is modeled as linear convolution, then \(\mathbf{b}\) resembles the TDS-OFDM block structure; nevertheless, for circular convolution, a necessary condition \(\mathbf{x}_{i}=\mathbf{x}_{j}\) must be fulfilled, where \(\mathbf{x}_{i},\mathbf{x}_{j}\in\{\mathbf{x}_{d}\}\). The multipath channel with time-varying path gains is assumed for signal propagation, which can be expressed as  
\[h(t,\tau)=\sum_{l=1}^{L}\alpha_{l}(t)\delta(\tau-\tau_{l}(t)) \tag{2}\]  
where \(\alpha_{l}\) and \(\tau_{l}(t)\) are the time-varying amplitude and delay component of path \(l\), respectively, while \(\delta(t)\) is the Dirac delta function. The \(\tau_{l}(t)=\tau_{l}+\Delta_{l}t\) encompasses the effect of the time-varying Doppler scaling factor, \(\Delta_{l}\). The multipath CIR with the fixed Doppler scaling factor, \(\Delta\), for all paths can be expressed as  
\[h(t,\tau)=\sum_{l=1}^{L}\alpha_{l}(t)\delta(\tau-\tau_{l}+\Delta t)=\sum_{l=1 }^{L}h_{l} \tag{3}\]  
The multipath channel can be expressed in vector form as \(\mathbf{h}=[h_{1},h_{2},\ldots,h_{L}]\). The convolution of the OFDM block \(\mathbf{b}^{T}\) and \(\mathbf{h}\), i.e., \(\mathbf{r}=\mathbf{b}^{T}*\mathbf{x}_{d}\), yields a vector of the time domain received sequence \(\mathbf{r}=[r_{1},r_{2},\ldots,r_{N+m+L-1}]\). The data part from \(\mathbf{r}\) can be obtained as \(\mathbf{r}^{\prime}=[r_{m+1},r_{m+2},\ldots,r_{N+m}]\) after the subtraction of \(\mathbf{e}=\mathbf{h}^{T}*\mathbf{x}_{d}\) from \(\mathbf{r}\). The detection of \(\mathbf{x}_{d}\), due to zero cross-correlation properties, results in a reduced error probability of \(G\) symbols, which becomes more significant in highly frequency-selective channels, as can be observed in the subsequent section. The time-frequency view of the eOFDM block is also shown in Figure 3, where the subcarriers with indices from set \(B_{o}^{\prime}\) are labeled as data+pilot subcarriers. Each subcarrier with an encircled plus sign is actually a data subcarrier and can also be used as pilot, as the data symbols on these subcarriers are used to select the guard interval sequence of the block. The significance of this time-frequency view of the eOFDM block lies in the fact that although it is similar to the time-frequency view of scattered pilots in the OFDM block structure, there is no pilot insertion in the proposed eOFDM block. All the \(N\) subcarriers are data subcarriers; a sub-group of these with an encircled plus sign are linked to the selection of a PN sequence at the transmitter. Therefore, the data symbols on these subcarriers are detected before the FFT operation at the receiver as soon as PNSDR detects the PN sequence. It should be noted that these subcarriers inherit additional channel estimation capabilities within the eOFDM block, which is useful for fast and frequency-selective fading channels. Therefore, these are labeled as data+pilot subcarriers. Figure 1: Proposed model for OFDM system. Figure 2: Transmitted blocks of CP-OFDM, TDS-OFDM and eOFDM schemes.