## 2 NB-PLC and RF Characterization
### NB-PLC Characterization
#### 2.1.1 NB-PLC Attenuation  
The NB-PLC signals transmitted on power lines are affected by high attenuation due to the propagation of line losses. These depend on the topology of a network, environmental conditions, and behavior of connected appliances, exhibiting variable spectro-temporal characteristics [28]. As a function of loads and of a network's layout, PLC may induce high attenuation and deep fading effects. The generation of multipath signals is a result of impedance mismatches, which are caused by the presence of many loads connected to a network's termination points and multiple cable branches. Therefore, the reflected waves yield to multiple delayed echoes in the channel impulse response. In [29], authors develop a delay detection technique for the PLC multipath channel and examine the impact of the electric line length, number of branches, and impedances on the channel path delays. In [30], the authors investigate NB-PLC channel modeling and present a bottom-up approach, namely the voltage ratio approach (VRA), which is based on the topology of a network. A comparison of different NB-PLC models is presented in [31], in which it is demonstrated that the PLC transfer function depends on the network topology and the type of connected loads. Thus, in this paper, the authors opt to apply the VRA model as it accurately computes the transfer function. A single branch is defined in Figure 2a as the combination of three points: A is the transmitting point (Tx), B is the receiving  
Figure 1: General system overview. point (Rx), and C is the connected load which comprises another branch. The equivalent representation of a branch in terms of transmission line (TL) parameters is depicted in Figure 2b. In the representation in Figure 2b, \(l_{1}\) denotes the cable length from the Rx end of the path line to the tap point C, \(l_{2}\) denotes the length of the cable connected from the tap point C to the backbone, and \(l_{3}\) represents the cable length from the tap point C to the Tx end of the path line A. The input impedances, \(Z_{inB}\), \(Z_{inC}\), and \(Z_{in}\), are determined using Equations (1) and (2). Without the loss of generality, the authors assume that the direct backbone path (from A to B) consists of the same transmission line with a defined propagation \(\gamma\) and characteristic impedance \(Z_{0}\). The stub \(Z_{C}\) is represented by a different line with a defined propagation constant \(\gamma_{1}\) and characteristic impedance \(Z_{01}\). \[Z_{inB}=Z_{0}\frac{Z_{B}+Z_{0}\text{tanh}(\gamma l_{3})}{Z_{0}+Z_{0}\text{tanh }(\gamma l_{3})}\text{ and }Z_{inC}=Z_{01}\frac{Z_{C}+Z_{01}\text{tanh}(\gamma_{1}l_{2})}{Z_{01}+Z_{C} \text{tanh}(\gamma_{1}l_{2})} \tag{1}\]  
\[Z_{in}=Z_{0}\frac{(Z_{inB}//Z_{inC})+Z_{0}\text{tanh}(\gamma l_{1})}{Z_{0}+(Z_ {inB}//Z_{inC})\text{tanh}(\gamma l_{1})} \tag{2}\]  
In order to calculate the transfer function of a basic branch, the reflection coefficients of the end path (\(\Gamma_{1}\)) and from the tap point (\(\Gamma_{2}\)) are introduced. By applying the VRA method between access point A and end point B, the transfer function \(H_{e}(f)\) is given in Equation (3) as follows:  
\[H_{e}(f)=\frac{Z_{in}}{Z_{in}+Z_{g}}\times\frac{(1+\Gamma_{1})e^{-\gamma l_{3} }}{1+\Gamma_{1}e^{-2\gamma l_{3}}}\times\frac{(1+\Gamma_{2})e^{-\gamma l_{1}} }{1+\Gamma_{2}e^{-2\gamma l_{1}}} \tag{3}\]  
For multibranch network and using (1), the transfer function is written in Equation (4) as follows:  
\[H_{e,network}(f)=\prod_{k=1}^{N}H_{e}^{(k)}(f) \tag{4}\]  
where \(N\) represents the number of branches in the network and \(H_{e}^{(k)}(f)\) is the transfer function of the elementary branch. In this paper, the authors suppose that the network between the transmitter and the receiver is a cascade of three single branches where a cable's length depends on the area type: rural, urban, or industrial. Loads are frequency-selective and modeled using a resonant RLC circuit.