# A Baseband Wireless Spectrum Hypervisor for Multiplexing Concurrent OFDM Signals
## 7 Simulation Results  
In this section, we present some simulation results carried out to validate and assess the functionality of the proposed architecture. For the first validation, we want to check the average Mean Squared Error (MSE) related to the multiplexing of several vPHY modulated signals by the hypervisor. In this simulation, we measure the error between OFDM symbols, \(y_{\text{PHY}}\), created with an 128-point IFFT, \(N_{\text{PHY}}\), plus 9-sample long CP, \(N_{\text{PHY}}^{CP}\), and the OFDM symbols received from the hypervisor after down-conversion and channelization, \(y_{\text{vPHY}}\). Each (v)PHY signal modulates 72 consecutive subcarriers, which translates into a useful transmission bandwidth of 1.08 [MHz] when a subcarrier spacing, \(\Delta_{f}\), of 15 [KHz] is used. We average the MSE error over \(10^{5}\) iterations, where at each iteration, we have the single PHY and all the vPHYs modulated with the same randomly generated data. For this simulation, we use 12 vPHYs, i.e., \(M=12\), where each vPHY has its signals mapped into 128 consecutive subcarriers, \(N_{\text{vPHY}}\), totaling 1536 subcarriers, which is the number of points used by the IFFT block in the hypervisor, \(\text{N}\). At the receiver side, we use a polyphase FFT analysis filter-bank [18] to split the wide-band signal into multiple uniformly spaced narrow sub-bands. It has a 180 [dB] stop-band attenuation and 512 filter coefficients per sub-band. The frequency response of the polyphase filter-bank used in all the simulations presented in this section is depicted in Figure 10. As can be seen in the figure, the physical spectrum band, \(R_{\text{hyper}}\), which is provided by the RF front-end, is split into \(M\) equal-bandwidth equally spaced sub-bands of 1.92 [MHz] (i.e., \(\frac{N\times\Delta_{f}}{M}\)). It is important to mention that the stop-band attenuation and filter order parameters play an important role in the fidelity of the multiplexed signals to the single PHY one [18]. The MSE for the \(k\)-th vPHY is calculated as defined by (13) below. \[ \begin{split}\text{MSE}_{k}=\mathbb{E}\left[\frac{1}{(N_{\text{PHY }}+N_{\text{PHY}}^{CP})}\right.\\ \times\left.\sum_{n=0}^{(N_{\text{PHY}}+N_{\text{PHY}}^{CP})-1} \left|y_{\text{PHY}}(n)-y_{\text{vPHY}}(n)\right|^{2}\right].\end{split}  \tag{13}\]  
Figure 10: Frequency response of a polyphase filter-bank with stop-band attenuation of 180 [dB] and 512 filter coefficients per sub-band. We additionally, we have also calculated the average Modulation Error Ratio (MER) to compare the error introduced by the multiplexing to the vPHY modulated signal. The MER compares the error between the modulated data symbols (i.e., the BPSK, QPSK, etc. symbols used to modulate the OFDM subcarriers) and the demodulated data symbols at the receiver side after all the vPHY, hypervisor and channelizer processing. Here we also take the MER average over \(10^{5}\) iterations. The MER for the \(k\)-th vPHY is calculated as follows  
\[\text{MER}_{k}=\mathbb{E}\left[10\log_{10}\left(\frac{\sum_{n=0}^{N_{\text{ vPHY}}^{U}-1}{(l_{k}^{2}+Q_{k}^{2})}}{\sum_{k=0}^{N_{\text{vPHY}}^{U}-1}{(l_{k}- I_{k})^{2}+(Q_{k}-\tilde{Q}_{k})^{2}}}\right)\right], \tag{14}\]  
where \(N_{\text{vPHY}}^{U}\) represents the number of useful subcarriers (i.e., the subcarriers that are modulated with the data symbols), \(I_{k}\) is the In-phase value corresponding to the \(k\)-th reference symbol, \(Q_{k}\) is the Quadrature phase value corresponding to the \(k\)-th reference symbol, \(\tilde{I}_{k}\) is the In-phase value corresponding to the \(k\)-th received symbol, and \(\tilde{Q}_{k}\) is the Quadrature phase value corresponding to the \(k\)-th received symbol. The MER can be seen as a signal-to-noise ratio (SNR) measurements, where it calculates the distortion/interference caused by the multiplexing operation performed by the hypervisor. In Table 3 we show the MSE and MER averages for several different modulation schemes.