# Wireless Multi-Node uRLLc B5G/6G Networks for Critical Services in Electrical Power Systems
## 3 Analysis of OFDM/FHMA System Based on Mini-Slot Frame Structure
### Signal Analysis for Reliable OFDM/FHMA System  
The transceiver block of the OFDM/FH system proposed in this paper is shown in Figure 2. As mentioned in Section 2, the subcarrier in each subchannel of OFDM is hopped over the corresponding \(N_{h}\) frequency slots. To guarantee the orthogonality among these hopped subcarriers, the first frequencies of these \(N_{f}\) sub-channels are assumed to be  
\[f_{l}=IN_{h}f_{d},\quad l=0,1,2,\ldots,N_{f}-1 \tag{2}\]  
where \(f_{d}=1/T_{s}\) represents the minimum frequency spacing between adjacent FH frequency slots. \(T_{s}\) denotes the symbol interval on the sub-channel. On the \(l\)-th subchannel of the \(k\)-th user, the instantaneous FH frequency slot over the \(i\)-th interval is denoted as \(c_{l,i}^{(k)}\), which is assumed to follow an independently and identically uniform distribution over the integers set \([0,N_{h}-1]\) for any \(k\), \(l\), and \(i\). The transmitted OFDM/FH signal of the \(k\)-th user is expressed as  
\[s_{k}(t)=\sum_{i}\sum_{l=0}^{N_{f}-1}\sqrt{2P}X_{l,i}^{(k)}e^{j2\pi\left(f_{l} +c_{l,i}^{(k)}f_{d}\right)\left(t-iT_{s}\right)}P_{T_{s}}(t-iT_{s}) \tag{3}\]  
where \(X_{l,i}^{(k)}\) is the complex symbol after operating a binary phase-shift-keying (BPSK) mapping. \(\sqrt{2P}\) represents the average power of the OFDM symbol. The average transmitted energy per bit on the sub-channel is equal to \(PT_{s}\). The function \(P_{\lambda}(t)\) is a unit rectangular pulse signal, for which the interval width is spanned over \([0,\lambda]\). In practice, the locations of service nodes in the power system are normally fixed. Particularly in urban distribution power networks, there is a large number of scatters distributed around the desired nodes, and there is non-light-of-sight between the nodes (transmitter) and the transformer substation (receiver). It is reasonable to assume that the wireless channel suffers from flat Rayleigh fading and additive white Gaussian noise (AWGN) in this paper. The complex channel gain of the \(l\)-th sub-channel is assumed to be \(g_{l,i}^{(k)}=\alpha_{l,i}^{(k)}e^{j\theta_{l,i}^{(k)}}\), where the amplitude \(\alpha_{l,i}^{(k)}\) follows a Rayleigh distribution with the mean-squared deviation \(\overline{\alpha_{l,l}^{(k)^{2}}}=\Omega\), and the phase \(\theta_{l,i}^{(k)}\) follows a uniform distribution over \([0,2\pi)\)[18]. The variables \(\alpha_{l,i}^{(k)}\) and \(\theta_{l,i}^{(k)}\) are i.i.d. for any \(k\), \(l\), and \(i\). In such a wireless channel, the received signal of the \(k\)-th user is expressed as  
\[r_{rec}^{(k)}(t)= \sum_{l}\sum_{l=0}^{N_{f}-1}g_{l,i}^{(k)}\sqrt{2P}X_{l,i}^{(k)}e^{j2 \pi\left(f_{l}+c_{l,i}^{(k)}f_{d}\right)\left(t-iT_{s}\right)}P_{T_{s}}(t-iT_{ s}) \tag{4}\]  
In the following analysis, we consider \(N_{u}\) nodes connecting to the power system at the same time, then the received signal of multi-node OFDM/FHMA is denoted as  
\[\begin{array}{l}r(t)=\sum_{k=1}^{N_{u}}r_{rec}^{(k)}(t)+v(t)\\ \hskip 14.226378 pt=\hskip-14.226378 pt\sum_{k=1}^{N_{u}}\sum_{i}^{N_{f}-1}\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!  
The model of the receiver is shown in Figure 2. Without loss of generality, the data of the first node in the \(i=0\) interval is considered as the data of interest due to the orthogonality and symmetry of the branch in the receiver, and the paired transceiver of the desired nodes is perfectly synchronized. The signal after the dehopping operation at the \(m\)-th branch in the receiver can be written as  
\[q_{m}(t)=r(t)e^{-\beta\pi\left(f_{m}+c_{m,0}^{(1)}f_{s}\right)t}=\sum_{k=1}^{ N_{u}}\sum_{l=0}^{N_{l}-1}g_{l}^{(k)}X_{l}^{(k)}\sqrt{2p}e^{\beta\pi\left(f_{l}-f_{m }+\left(c_{m,0}^{(k)}-c_{m,0}^{(1)}\right)f_{s}\right)t}+v(t)e^{-\beta\pi\left( f_{m}+c_{m,0}^{(1)}f_{s}\right)t} \tag{6}\]  
where the variables \(g_{l,0}^{(k)}\) and \(X_{l,0}^{(k)}\) are abbreviated as \(g_{l}^{(k)}\) and \(X_{l}^{(k)}\) for simplicity.