## 2 Fbmc-OQAM System Model
Thus, the analytical expression for the FBMC-OQAM group signal can be written as:  
\[s(k)=\sum_{m=0}^{M-1}\sum_{n\in Z}a_{m,n}g(k-nN)e^{j\frac{2\pi}{M}m(k-\frac{L_{q }-1}{2})}e^{j\phi_{m,n}} \tag{3}\]  
where \(a_{m,n}\) denotes transmitted PAM symbols, which are obtained by extracting the real and imaginary parts from QAM symbols, \(N=M/2\) is a discrete shift in the time domain by half the length of the baseband symbol, \(L_{g}\) is the filter prototype IR length, \(\phi_{m,n}=\frac{\pi}{M}(m+n)\) is the phase shift for the \(m\)-th subcarrier and \(n\)-th symbol, and \(M\) is the number of subcarriers. Thanks to the filtration of subcarrier signals, the FBMC-OQAM group signal has significantly better power spectral density (PSD) localization compared to CP-OFDM. To clearly demonstrate this difference, two signals were generated using each of the technologies, where half of the subcarriers located in the center of the main frequency range contained random data symbols (the same for both signals), and the symbols of the remaining subcarriers were equal to zero. The total number of subcarriers was set to 512. The power spectral densities of these signals are presented in Figure 2, which shows that for two subcarriers offset from useful signal frequencies, FBMC-OQAM provides a 28 dB lower out-of-band signal level compared to CP-OFDM. Figure 1: PHYDYAS prototype filter ambiguity function. The demodulated symbols are in turn calculated as:  
\[\hat{a}_{m,n}=\sum_{k=0}^{\infty}\sum_{m=0}^{M-1}y\bigg{[}k-\frac{nM}{2}\bigg{]}g \bigg{[}\frac{nM}{2}-k\bigg{]}e^{-\frac{l2\pi(k-\frac{l_{g}-1}{2})}{n}}e^{-j \phi_{m,n}}, \tag{4}\]  
where \(y\) is the received baseband signal. Thus, the block scheme of the described model, including the equalization and selection of the symbol's real part, can be represented as shown in Figure 3. Data symbols are multiplied by the phase shift factor \(\phi_{m,n}\), and then subcarrier signals become frequency-modulated in the inverse Fourier transform. Further narrowband channels are filtered by the set of synthesis filters with transfer functions \(G_{0}[z],G_{1}[z],\ldots,G_{M-1}[z]\), corresponding to impulse responses \(g(k-nN),g(k-nN)e^{j\frac{2\pi}{M}(k-\frac{Lg-1}{2})},\ldots,g(k-nN)e^{j\frac{2 \pi(M-1)}{M}(k-\frac{Lg-1}{2})}\), whose output signals are multiplexed in the modulated time-domain FBMC signal. The demodulation part works similarly but in reverse order: demultiplexing of the time-domain signal, filtering with a set of analysis filters (which, as was shown in [9], are equivalent to synthesis filters for the FMBC system), frequency demultiplexing by the FFT transform, the cancellation of the additional phase shift, the equalization stage and, finally, the separation of the received symbol's real part, which holds transmitted information. At the same time, despite the fact that information is contained only in the real part of the symbols, in the case of a change in the phase of the symbols when passing through the channel, the imaginary part must be taken into account when estimating the channel and synchronization [22], since in the presence of phase shifts, the imaginary part of the interference affects the real part. Figure 2: FBMC-OQAM and CP-OFDM power spectral density comparison.