# A Baseband Wireless Spectrum Hypervisor for Multiplexing Concurrent OFDM Signals
## 7 Simulation Results
As shown in the figure, it is possible to give independent gains to each vPHY, which is accomplished by multiplying the \(N_{\rm vPHY}^{\rm UI}\) useful modulation symbols by a multiplication factor varying from 0 (no transmission power at all) to 1 (maximum transmission power used by the physical RF front-end). Figure 13 gives a closer look at the boost factor, showing that it is possible to vary the transmitted power of each individual vPHY just by changing the factor used to multiple the useful subcarriers, \(N_{\rm vPHY}^{\rm UI}\). The legend on the figure shows the attenuation given to the default transmission power of a vPHY, which is around 0 [dBW] as shown by the vPHY centered around 0 [Hz]. Figure 11: Comparison between OFDM symbols generated by a plain/single PHY and 2 vPHYs. Figure 12: Frequency-domain representation of the wide baseband signal generated by the hypervisor. Figure 13: Closer look at the frequency-domain boost factor. Figure 14 presents uncoded BER results for a setup where 12 vPHYs concurrently have their signals multiplexed by the hypervisor and employ QPSK, 16 QAM, and 64 QAM modulation schemes. In each one of the sub-figures, we compare the BER for the 12 vPHYs against the BER achieved by a single PHY, i.e., there is no other signal being transmitted along with that of the PHY under test. In this simulation, we consider an Additive White Gaussian Noise (AWGN) channel. The single PHY and each one of the 12 vPHYs modulate 72 (i.e., \(N_{\mathrm{vPHY}}^{U}\)) out of 128 (i.e., \(N_{\mathrm{vPHY}}\)) consecutive subcarriers by using QPSK, 16 QAM, and 64 QAM modulation schemes. The hypervisor's IFFT length, \(N\), is set to 1536 and the cyclic prefix length, \(N_{\mathrm{CP}}\), is set to 108 samples. For each SNR point, \(1\times 10^{6}\) iterations were run, where the total number of wrongly decoded bits and transmitted bits were calculated for the BER calculation. As can be seen, the vPHY BER curves exactly match the single PHY BER curve (dashed black curve with squares along it), meaning that there is no interference between the current transmissions. Additionally, it is also worth mentioning that all curves match the theoretical BER curves (red-dashed curve with dots along it), which can be approximated by (15) [58]. \[P_{b}\approx\frac{2(\sqrt{\gamma}-1)}{\sqrt{\gamma}\log_{2}(\gamma)}\mathrm{ erfc}\left(\sqrt{\frac{3\log_{2}(\gamma)(E_{b}/N_{0})}{2(\gamma-1)}}\right)\,, \tag{15}\]  
where \(\gamma\) is the modulation order and \(E_{b}/N_{0}\) is the bit energy over the power spectrum density. The results presented in Figure 14, are very important as they prove that the proposed architecture provides perfect isolation among all the signals being multiplexed. The perfect isolation is due to the orthogonality provided by the IFFT processing in the hypervisor, which guarantees that every single subcarrier, spaced of \(\Delta_{f}\) [Hz], is mutually orthogonal to all other ones. Figure 15 shows the frequency-domain representation of the wide baseband signal generated by the hypervisor for 3 vPHYs with different but correlated numerologies. These 3 vPHYs employ frequency spacings, \(\Delta_{f}\) of 15, 7.5, and 3.75 [KHz], and CP lengths of 9, 18, and 36 samples, FFT/IFFT lengths of 128, 512, and 2048 points, and sampling rates of 1.92, 3.84, and 7.68 [MSps], respectively. In this simulation, we use a hypervisor with an IFFT of 6144 points with a sampling rate of 23.04 [MSps], which results in a subcarrier frequency spacing of 3.75 [KHz]. For all 3 vPHYs, we modulate 72 subcarriers and leave the others unused. As can be seen, even though the 3 vPHYs have only 72 useful subcarriers, we notice that due to the different frequency spacings, \(\Delta_{f}\), each one of them has a different occupied bandwidth. In the case of vPHY#0, the useful subcarriers appear every 4 points so that the final subcarrier frequency spacing equals 15 [KHz].