## 2 Foundations on RISs
### RIS-Assisted Communication System Model  
Following the scheme in Figure 2, the transmitter at the BS has \(m\) inputs, the encoded bits \(\mathbf{a}_{k}=\{d_{n}\}_{n=0}^{m}\), where \(d_{n}\in\{0,1\}\). The outputs are the signals \(\mathbf{x}_{k}\), which come from modulated bits. The system can use \(M\)-ary quadrature amplitude modulation (QAM) or any other modulation scheme. The signal received by the \(k\)th user via the \(n\)th RIS and the direct path is defined as [39]  
Figure 2: RIS-aided wireless communication system scheme. \[ \begin{split}&\mathbf{y}_{k}=\sqrt{\gamma_{k}}\mathbf{H}_{1,k}\mathbf{ \Theta}_{1}\mathbf{G}_{1}\mathbf{H}_{1}^{SR}\mathbf{x}_{k}+\sqrt{\gamma_{k}} \mathbf{H}_{2,k}\mathbf{\Theta}_{2}\mathbf{H}_{2}^{SR}\mathbf{G}_{2}\mathbf{x}_ {k}+\cdots\\ +\sqrt{\gamma_{k}}\mathbf{H}_{l,k}\mathbf{\Theta}_{l}\mathbf{H}_{l}^{ SR}\mathbf{G}_{l}\mathbf{x}_{k}+\cdots+\sqrt{\gamma_{k}}\mathbf{H}_{l,k}\mathbf{ \Theta}_{l}\mathbf{G}_{l}\mathbf{x}_{k}+\sqrt{\gamma_{k}}\mathbf{H}_{k}^{DP} \mathbf{x}+\mathbf{n}_{k},\end{split}  \tag{1}\]  
where \(\mathbf{\Theta}_{l}\) is the \(l\)th matrix of phase shifts, defined as \(\mathbf{\Theta}_{l}=\text{diag}(e^{\theta_{l,1}},e^{\theta_{l,2}},\cdots,e^{ \theta_{l,N_{k}}})\). In the given expression, \(e^{\theta_{l,i}}\) denotes the phase shift of the \(i\)th reflecting element of the \(l\)th RIS, and \(\mathbf{n}_{k}\in\mathbb{C}^{N_{k}\times 1}\) stands for the noise whose samples are assumed to be independent and identically distributed (i.i.d.) with \(\mathcal{CN}(0,\sigma^{2})\). Moreover, the term \(\sqrt{\gamma_{k}}\) corresponds to the destination's signal-to-noise ratio (SNR). The expression (1) can also be written as  
\[\mathbf{y}_{k}=\sqrt{\gamma_{k}}\Bigg{(}\sum_{i=1}^{N}\mathbf{H}_{l,k}\mathbf{ \Theta}_{l}\mathbf{G}_{l}+\mathbf{H}_{k}^{DP}\Bigg{)}\mathbf{x}_{k}+\mathbf{ n}_{k}. \tag{2}\]  
Let us consider Equation (1); the received signal \(\mathbf{y}_{k}\) is directly affected by the matrix channels \(\mathbf{H}_{l,k}\). These links are difficult to estimate if a passive RIS is considered without signal processing facilities [40]. Two common reference models for the channel are the Rayleigh [41] and the Rician fading channels. However, one of the principal open problems for RIS-assisted communications' enhancement is channel estimation. In [42], the authors propose an ON/OFF protocol to estimate the cascaded channel \(\mathbf{H}_{l,k}\) dividing the process into stages, where each stage only estimates one column vector associated with one RIS element. More specifically, the cascaded channel is represented by \(N_{s}\) columns as  
\[\mathbf{H}_{l,k}=[\mathbf{h}_{l,k}^{(1)},\mathbf{h}_{l,k}^{(2)}\cdots\mathbf{ h}_{l,k}^{(N_{s})}], \tag{3}\]  
where \(\mathbf{h}_{l,k}^{(i)}\) represents a column vector associated with the individual channel corresponding to the \(i\)th RIS element. At the \(i\)th stage, only the \(i\)th RIS element is turned on, while the remaining elements are turned off. The procedure of channel estimation is defined as follows: the direct channel \(\mathbf{H}_{k}^{DP}\) is estimated by adopting a conventional wireless transmission strategy among users, such as sending pilot signals. Then, the RIS-element's associated channels \(\mathbf{h}_{l,k}^{(i)}\) are estimated independently using various methods. However, the channel estimation accuracy can be degraded because only one RIS element reflects the pilot signal to the BS. Another estimation strategy considers the power in all RIS elements but still divides the channel matrix \(\mathbf{H}_{l,k}\) into columns to improve the estimation accuracy [43]. The matrices \(\mathbf{G}_{l}\) and \(\mathbf{\Theta}_{l}\) in Equation (1) can be adjusted to control the impinging signal amplitude and phase through the RIS elements, respectively. One of the most important RIS communications' advantages is called _beamforming_. Beamforming enhances RIS communications by allowing the control of the signal reflection towards a desired direction to improve the reception quality.