## 4 System Model  
We consider a wireless communication link between \(N_{t}\) data sensing nodes (DSNs) serving as one Virtual MIMO transmitting side node, and one Virtual MIMO receiving side node which consists of one single-antenna data gathering node (DGN) and \(N_{r}-1\) data assisting nodes (DANs), each with one antenna as shown in Figure 4.  
Figure 3: Real and imaginary components of BPSK-16 WOFDM, 16 WOFDM, BPSK-16 FOFDM, and 16 FOFDM.  
In our system model, we consider V-BLAST signal processing by DGN at the receiving side with the assumption that it can cope with more computational complexity than its DANs. Moreover, no local communication and processing are essential among the DSNs. It is assumed that \(N_{t}\) DSNs are transmitting their data simultaneously over a flat fading MIMO channel to DGN (referred to as long-haul communications [8,14]). In addition, there are \(N_{r}-1\) DANs in close proximity of DGN to form one virtual receiving node of size \(N_{r}\), including the DGN itself. All DANs transmit their data using time-division-multiple access (TDMA) to DGN (referred as local communication on receiving side) to form received signal vector \(Rec\) as shown in the following equation:  
\[Rec=HS+\zeta \tag{4}\]  
where \(Rec\) is an \(N_{r}\times 1\) vector, \(S\) is \(N_{t}\times 1\) vector, \(\zeta\) is an \(N_{r}\times 1\) noise vector whose elements are complex Gaussian random variables with zero mean and variance \(N_{o}\), \(H\) is an \(N_{r}\times N_{t}\) channel matrix. It is assumed in this article that \(N_{t}\leq N_{r}\)[14].  
At each DSN, a serial-to-parallel converter is used to form the input for WOFDM modulator. Every \(k_{th}\) input is first up-sampled by \(2^{P}\) and then filtered by sub-channel impulse response \(f_{k}(n)\). Received signal vectors at DGN are detected using QR decomposition detection algorithm [26]. Denoting channel response matrix \(H=QR\), where \(Q\) is \(N_{r}\times N_{t}\) unitary matrix composed of orthonormal columns with unit norm and \(R\) is \(N_{t}\times N_{t}\), upper triangular matrix, the received signal expression in Equation (4) can be modified to detect the transmitted signals by multiplying it with \(Q^{t}\) (transpose of \(Q\)) as follows:  
\[\widetilde{Rec}=Q^{t}Rec=Q^{t}(HS+\zeta)=Q^{t}QRS+\eta \tag{5}\]  
\[\widetilde{Rec}=RS+\eta \tag{6}\]  
where \(Q^{t}Q=I\) (I is identity matrix), and \(\eta=Q^{t}\zeta\) is statistically identical to \(\zeta\). Due to upper triangular structure of R, the \(i_{\text{th}}\) element of \(\widetilde{Rec}\) is given by:  
\[\widetilde{rec}_{t}=r_{ii}+d_{i}+\eta_{i} \tag{7}\]  
where \(d_{i}=\sum_{j=i+1}^{N_{t}}r_{ij}\,s_{i}\) is the interference term. The interference free signal element is given by:  
Figure 4: Communication between Transmitting and Receiving side Virtual MIMO nodes.  
\[z_{i}=\tilde{r}\overline{e}c_{i}-d_{i} \tag{8}\]  
and the detected signal \(\widehat{s_{i}}=\frac{z_{i}}{r_{ii}}\), corresponding to each receiving antenna, is demodulated using WOFDM demodulator. The detected signal stream is first filtered by sub-channel impulse response \(h_{k}(n)\) and then down-sampled by \(2^{P}\). For BPSK-WOFDM system, at each DSN, the bit stream is first modulated using BPSK modulator and then fed to a serial-to-parallel converter to form the input for WOFDM modulator.