# On Multi-Hop Decode-and-Forward Cooperative Relaying for Industrial Wireless Sensor Networks
## 2 System and Channel Model
### System Model  
In our work, we consider a multi-hop wireless relay system with L hops using DF relaying strategy. Figure 1 illustrates the investigated scenario, where the nodes \(R_{1}\) and \(R_{L+1}\) correspond to the source node and destination node, respectively; and the link between the nodes \(R_{\ell}\) and \(R_{\ell+1}\), \(\ell=1,\ldots,L\), is denoted as \(\ell\)-th hop with separation distance \(d_{\ell}\). Full-duplex mode of communication is assumed, in which the nodes can transmit and receive at the same time employing frequency division duplexing (FDD). In FDD mode, each node \(R_{\ell}\) use frequency bands \(B_{\ell}^{u}\) and \(B_{\ell}^{d}\) to transmit and receive data, respectively. Altogether, there are \(L\) such non-overlapping frequency pairs; and only a subsequent node can listen to the signal transmitted by a previous node, thus avoiding the interference in any hop owing to the transmissions occurring in the surrounding hops. The channels between adjacent nodes \(R_{\ell}\) and \(R_{\ell+1}\), \(\ell=1,\ldots,L\), are mutually independent and undergo Nakagami-\(m\) fading.  
Data transmission is done using the M-ary phase shift keying (M-PSK) symbols with equal a priori probabilities. The node \(R_{\ell}\), \(\ell=1,\ldots,L\), transmits the unit-energy M-PSK symbol \(s_{\ell}\) (i.e., \(E_{b}~{}=~{}\mathrm{E}[|s_{\ell}|^{2}]=1\)) with power \(P_{Tx,\ell}\). The corresponding received signal \(r_{\ell+1}\) at node \(R_{\ell+1}\) and the total transmit power \(P_{Tx}\) of the relay system can be written as  
\[r_{\ell+1} =\sqrt{P_{Rx,\ell+1}}\cdot h_{\ell}\cdot s_{\ell}+z_{\ell},~{}~{ }~{}~{}\ell=1,\ldots,L \tag{1}\] \[P_{Tx} =\sum_{\ell=1}^{L}P_{Tx,\ell}, \tag{2}\]  
where \(P_{Rx,\ell+1}\) is the received signal power by node \(R_{\ell+1}\); \(h_{\ell}\) and \(z_{\ell}\) are the channel fading amplitude and the additive noise of the \(\ell\)-th hop, respectively. The received signal power \(P_{Rx,\ell+1}\) by node \(R_{\ell+1}\) depends on the transmitted signal power \(P_{Tx,\ell}\) by the previous node \(R_{\ell}\) and the distance \(d_{\ell}\) between  
Figure 1: Illustration of a multi-hop relay system with \(L\) hops.  
the two nodes. A widely used model for the signal attenuation is the one-slope path-loss model [33]. According to the one-slope model, the relationship between \(P_{Rx,\ell+1}\) and \(P_{Tx,\ell}\) can be expressed as  
\[10\log_{10}(P_{Rx,\ell+1})=10\log_{10}(P_{Tx,\ell})-10\cdot\xi\cdot\log_{10}(d_ {\ell}),\hskip 14.226378 pt\ell=1,\ldots,L \tag{3}\]  
where \(\xi\) is the path-loss exponent, which reflects the rate at which the received power decreases with distance. The value of the parameter \(\xi\) in a specific environment could be simply obtained by field measurement [36; 37; 38] or estimated using numerical algorithms [39; 40]. The overall performance of the DF relaying network is limited by the worst link, it is thus straightforward to show that in order to obtain the best performance, the total power should be allocated among different hops in a way such that the received signal power is equal for each receiving node, i.e., \(P_{Rx,\ell}=P_{Rx,\ell+1}=P_{Rx,}\)\(\ell=2,\ldots,L\). From (3), this constraint of the received powers translates into the following relationship on the transmitted power of each node  
\[\frac{P_{Tx,1}}{(d_{1})^{\delta}}=\frac{P_{Tx,2}}{(d_{2})^{\delta}}=\cdots= \frac{P_{Tx,L}}{(d_{L})^{\delta}}. \tag{4}\]