# A Baseband Wireless Spectrum Hypervisor for Multiplexing Concurrent OFDM Signals
## 3 Multiplexing Concurrent OFDM Signals
At the receiver side, the wide pass-band signal is translated into a wide baseband signal by the receiver RF front-end. Here we do not consider any channel influence, i.e., the transmitted signal does not suffer from any impairment, and consequently, it is perfectly received at the receiver side. Next, after the frequency translation, the received wide baseband signal \(y_{\rm wBB}^{\prime}(n)=y_{\rm wBB}(n)\)which contains the \(M\) multiplexed sub-band (narrow-band) signals, is digitally translated/split into \(M\) narrow baseband signals (each one centered around 0 [Hz]), bandwidth restricted by low-pass digital Finite Impulse Response (FIR) filters, and finally, submitted to a sample rate reduction. These steps are depicted in Figure 5, which shows the architecture of a conventional channelizer as described in [18]. As shown in the figure, the output of the \(k\)-th channel, which is denoted by \(y_{\text{wBB}}^{\text{\tiny${{}_{\text{R}}}$}}\), prior to the down-sampling, is a convolution as defined by  
\[ \begin{split} y_{\text{wBB}}^{\text{\tiny${{}_{\text{L}}}$}}(n)& =\left[y_{\text{wBB}}^{\text{\tiny${{}_{\text{R}}}$}}(n)e^{i\theta_{ \text{k}}n}\right]\otimes h(n)\\ &=\sum_{l=0}^{N_{\text{FIR}}-1}y_{\text{wBB}}^{\text{\tiny${{}_{ \text{R}}}$}}(n-l)e^{i\theta_{\text{k}}(n-l)}h(l),\\ & 0\leq n<N+N_{\text{FIR}}-1\end{split}  \tag{4}\]  
where \(N_{\text{FIR}}\) is the length of the FIR filter, \(\theta_{\text{k}}=\frac{2\pi\delta_{\text{k}}n}{N}\) is the angle corresponding to the digital frequency offset, and \(\delta_{\text{k}}\) is the center frequency of the \(k\)-th narrow sub-band, \(B_{\text{k}}\), in number of subcarrier spacings, \(\Delta_{f}\). After substituting (3) into (4) we have  
\[ \begin{split} y_{\text{wBB}}^{\text{\tiny${{}_{\text{R}}}$}}(n) =\sum_{l=0}^{N_{\text{FIR}}-1}\frac{1}{N}\sum_{k=0}^{M-1}\sum_{i^{\prime}=0}^{ N_{\text{wBB}}-1}\rho_{\text{k}}B_{\text{k}}(i^{\prime})e^{\frac{-2\pi\Theta_{ \text{f}}\text{f}_{\text{k}}(n-l)}{N}}h(l),\\ -N_{\text{CP}}\leq n\leq N-1,\end{split}  \tag{5}\]  
where \(\Theta_{i^{\prime},k}=(i^{\prime}+kN_{\text{vPHY}}-\delta_{\text{k}})\mod N\). The multiplication of \(y_{\text{wBB}}^{\text{\tiny${{}_{\text{R}}}$}}(n)\) by \(e^{i\theta_{\text{k}}n}\) centers the \(k\)-th narrow sub-band, \(B_{\text{k}}\), around 0 [Hz] and the convolution with the low-pass filter makes sure that only the signal components belonging to the \(k\)-th narrow sub-band go through it while the other sub-bands are filtered out. This way, (5) can be re-written as  
\[ \begin{split} y_{\text{wBB}}^{\text{\tiny${{}_{\text{L}}}$}}(n) =\frac{1}{N}\sum_{i=0}^{N-1}\begin{cases}\rho_{\text{k}}B_{\text{k}}(i)e^{ \frac{-2\pi i}{N}},&0\leq i\leq\frac{N_{\text{wBB}}}{2},\\ \rho_{\text{k}}B_{\text{k}}(i)e^{\frac{-2\pi i}{N}},&N-\frac{N_{\text{vBB}}}{ 2}+1\leq i\leq N-1,\\ 0,&\text{otherwise},\\ &-N_{\text{CP}}\leq n\leq N-1.\end{cases}\end{split}  \tag{6}\]  
As can be seen by inspecting (6), out of \(N\) subcarriers/bins, only the lower and upper \(\frac{N_{\text{SPIV}}}{2}\) bins contain useful values, i.e., values different from zero. We consider in (6) that the filters have a perfect window-shaped frequency response, which is not true in practice but can be approximated by well-designed high order low-pass filters [19]. Figure 5: Architecture of a conventional channelizer: frequency offset to baseband, low-pass filters, and down-samplers. Next, the wide baseband representation of the \(k\)-th narrow sub-band signal, \(y_{\text{wBB}}^{\text{\tiny{$\mathcal{I}_{k}$}}}(n)\), is subject to a down-sampling operation, as depicted in Figure 5, in order to have a narrow baseband representation of the sub-band signal, which can then be fed into a PHY for data decoding. After down-sampling, the signal expressed by (6), can be written as  
\[y_{\text{wBB}}^{\text{\tiny{$\mathcal{I}_{k}$}}}(nM)=\frac{1}{N}\sum_{i=0}^{N -1}F(i)e^{\frac{-2\pi ni}{N/M}},\ \ -N_{\text{CP}}\leq nM\leq N-1. \tag{7}\]  
where the function \(F(i)\) is defined as  
\[F(i)=