## 2 System Model
### Propagation Channel Abstraction  
Different from traditional one-way channels, backscattering channels exhibit composite fading effects, given that the received signal at a generic receiver is affected by the product of two cascade fading components that describe the forward link (i.e., illuminator-node link) and the backscattering link (i.e., node-receiver link), respectively.  
Under the settings described in assumptions \(A_{2}\) and \(A_{3}\), the power received by the \(k\)-th node through the forward link is given as  
\[\mathsf{P}_{f,k} = P_{\text{Rx}}\alpha_{k}e^{\mathsf{S}_{f,k}}, \tag{1}\]  
with \(k=1,\ldots,K\), where \(\alpha_{k}=c_{0}/d_{f,k}^{k_{0}}\), \(d_{f,k}\) is the distance between the illuminator and the \(k\)-th tag, \(c_{0}\) is the frequency-dependent path-loss at reference distance of 1 m, \(k_{0}\) is  
Figure 1: System architecture.  
the path-loss exponent, and \(\mathsf{S}_{f,k}{\sim}\mathcal{N}(0,\sigma^{2})\) is zero-mean Gaussian random variable that describes fading on the forward link.  
The level of the received power at the node determines whether or not the node is able to modulate the reflected signal. Specifically, we consider a simplified node abstraction, where the node is able to reflect all the incident power (in practice, due to loss phenomena of the harvesting circuitry, only a fraction of the incident power can be backscattered by the node. Moreover, the portion of the reflected power is, in general, a non-linear function of the incident power itself [45]. We remark that a more refined node abstraction can be safely included in our framework, without substantial modifications.) if \(\mathsf{P}_{f,k}\) is larger than the node's sensitivity, which we denote as \(\gamma_{h}\). On the contrary, when the received power is below the sensitivity threshold, the scattered power is equal to zero. Therefore, the received power at the \(m\)-th receiver can be expressed as  
\[\mathsf{P}_{b,k}^{(m)}=\begin{cases}\beta_{m,k}\mathsf{P}_{f,k}\mathsf{e}^{ \mathsf{S}_{b,k}^{(m)}}&\mathsf{P}_{f,k}>\gamma_{h}\\ 0&\text{otherwise},\end{cases} \tag{2}\]  
with \(m=1,\ldots,M\). Here, \(\beta_{m,k}=c_{0}/d_{b,k}^{k_{0}}\) where \(d_{b,k}\) is the distance between the \(k\)-th node and the \(m\)-th receiver and \(\mathsf{S}_{b,k}^{(m)}{\sim}\mathcal{N}(0,\sigma^{2})\) describes the fading in the backscattering link. By substituting Equation (1) in Equation (2), we get  
\[\mathsf{P}_{b,k}^{(m)}=\begin{cases}P_{\text{tx}}D_{r,k}\mathsf{e}^{\mathsf{Y }_{k}^{(m)}}&\mathsf{P}_{f,k}>\gamma_{h}\\ 0&\text{otherwise},\end{cases} \tag{3}\]  
where \(D_{m,k}=\alpha_{k}\beta_{m,k}\) and \(\mathsf{Y}_{k}^{(m)}=\mathsf{S}_{f,k}+\mathsf{S}_{b,k}^{(m)}\). Given the physical separation between the illuminator's antenna and the generic receiver's antenna, which in our settings is much higher than the operative wavelength, the fading components \(\mathsf{S}_{f,k}\) and \(\mathsf{S}_{b,k}^{(m)}\) can be considered uncorrelated [46] and, thus independent. Therefore, \(\mathsf{Y}_{k}^{(m)}\) is still a zero-mean Gaussian random variable, with variance \(2\sigma^{2}\). In addition, according to assumption \(A_{4}\), the random variables \(\mathsf{S}_{f,k}\) and \(\mathsf{S}_{b,k}^{(r)}\), for \(k=1,\ldots,N\) and \(m=1,\ldots,M\), are independent.