# A Baseband Wireless Spectrum Hypervisor for Multiplexing Concurrent OFDM Signals
## 3 Multiplexing Concurrent OFDM Signals  
In this section, we present a mathematical analysis of the working of the hypervisor proposed in this work and depicted in Figure 4.  
Figure 4: Architecture of the proposed baseband wireless hypervisor. Figure 3: Flexible and dynamic spectrum access example. Communications spectrum bandwidths can be flexibly and dynamically changed according to the availability of spectrum as well as application requirements. For this analysis, we consider that complex modulated samples, \(s_{i}\), carrying information are grouped into several narrow sub-bands, \(B_{k}(i)\), which are composed of \(N_{\rm vPHY}\) subcarriers. The complex samples, \(s_{i}\), are symbols drawn from a digital modulation scheme such as BPSK, QPSK, 16 QAM, etc. A maximum number of \(M=\lfloor N/N_{\rm vPHY}\rfloor\) narrow sub-bands, containing \(N_{\rm vPHY}\) subcarriers can be used, where \(N\) is the length of the IFFT used in the hypervisor and \(\lfloor c\rfloor\) is known as the floor operator and gives the largest integer value that is less than or equal to \(c\). Out of the \(N_{\rm vPHY}\) subcarriers, a narrow sub-band might have, \(N_{\rm vPHY}^{ll}\) carrying useful information, i.e., the modulation symbols \(s_{i}\), one Direct Current (DC) subcarrier, \(N_{\rm vPHY}^{DC}\), which is set to zero, and \(N_{\rm vPHY}^{Null}\) subcarriers that are also set to zero. The nulled subcarriers can be used as frequency guard bands at both edges of the narrow sub-band, allowing for the straightforward realization of anti-aliasing filters. On the other hand, the DC subcarrier allows the use of the simpler and cheaper direct-conversion, also known as zero intermediate-frequency (IF), RF front ends. The narrow sub-band signal can be defined as  
\[B_{k}(i)= \begin{cases}0,&i=0,\\ s_{i},&1\leq i\leq\frac{N_{\rm vPHY}^{ll}}{2},\\ 0,&(\frac{N_{\rm vPHY}^{ll}}{2})+1\leq i\leq(\frac{N_{\rm vPHY}^{ll}}{2})+N_{ \rm vPHY}^{Null},\\ s_{i},&(\frac{N_{\rm vPHY}^{ll}}{2})+N_{\rm vPHY}^{Null}+1\leq i\leq N_{\rm vPHY }-1.\end{cases}  \tag{1}\]  
For the sake of clarity, we consider that \(N\) is split into \(M\) sub-bands with \(N_{\rm vPHY}\) subcarriers and that the subcarrier mapping block in the hypervisor maps complex samples into contiguous sub-bands. Please note that the mapping of the \(M\) sub-bands, \(B_{k}(i)\), into the IFFT subcarriers/bins corresponds to a frequency translation (mixing) of the sub-band signal, which is centered around 0 [Hz], to a frequency offset defined by the bin number times the subcarrier spacing, \(\Delta_{f}=R_{\rm hyper}/N\), where \(R_{\rm hyper}\) is the sampling rate used by the hypervisor. The hypervisor's sampling rate, \(R_{\rm hyper}\), defines the amount of physical spectrum bandwidth that it can virtualize. The set, \(S(i)\), containing the concatenation of all the \(M\) narrow sub-bands, \(B_{k}(i)\), and of length \(N\) is transformed into time domain using an IFFT of size \(N\) as  
\[ \begin{split} s(n)&=\frac{1}{N}\sum_{i=0}^{N-1}S(i )e^{-j2\pi in/N}\\ &=\frac{1}{N}\sum_{k=0}^{M-1}\sum_{i^{\prime}=0}^{N_{\rm vPHY} -1}B_{k}(i^{\prime})e^{-j2\pi in/N},\end{split}  \tag{2}\]  
where \(i=i^{\prime}+kN_{\rm vPHY}\). Then, based on (2), the transmitted wide-band signal, \(y_{\rm wBB}(n)\), including a gain factor and the appending of a cyclic prefix can be represented by  
\[ \begin{split} y_{\rm wBB}(n)=\frac{1}{N}\sum_{k=0}^{M-1}\sum_{i ^{\prime}=0}^{N_{\rm vPHY}-1}\rho_{k}B_{k}(i^{\prime})e^{-j2\pi(i^{\prime}+kN _{\rm vPHY})n/N},\\ -N_{CP}\leq n\leq N-1,\end{split}  \tag{3}\]  
where \(\rho_{k}\) is the frequency amplification factor (or frequency gain), varying from 0 to 1, applied individually to each sub-band and \(N_{CP}\) is the CP length, which has to be an integer multiple of \(M\). As can be noticed, \(y_{\rm wBB}(n)\) is a wide-band signal containing all the \(M\) multiplexed vPHY signals. As the final step in the transmission chain, the wide-band signal \(y_{\rm wBB}(n)\) is sent to the RF front-end, which will translate the signal into the desired pass-band frequency.