## 3 PA Nonlinear Model  
Most recent works focusing on M-MIMO networks neglect the impact of hardware impairments, such as the nonlinear characteristics of PAs. However, it has been shown that in the worst-case scenario, the distortion generated by the PAs installed at the M-MIMO's RU can be steered toward the UE with the same precoding gain as the desired signal. To model this phenomenon in this paper, we consider the so-called Rapp model of the PA [18].  
As described in Section 2, every RU is equipped with a fully digital antenna array, meaning that there are \(M_{l}\) transceiver chains and the same number of PAs installed at each RU. While the PA operates on a passband analog signal, our digital signal model creates equivalent distortions. Let us denote the \(n\)-th complex sample input to the PA from the \(m\)-th transceiver chain at the \(l\)-th RU as \(\tilde{x}_{m,l}(n)\). This sample has already passed all stages of physical layer processing like precoding, modulation, and IFFT. The output complex sample \(\tilde{y}_{m,l}(n)\) of the Rapp-modeled PA is a result of the following nonlinear transformation of the input signal sample \(\tilde{x}_{m,l}(n)\)[18]:  
\[\tilde{y}_{m,l}(n)=\frac{G\tilde{x}_{m,l}(n)}{1+\left(\frac{\left|\tilde{x}_{ m,l}(n)\right|^{2p}}{p_{\max,l}^{p}}\right)^{\frac{1}{2p}}}, \tag{6}\]  
where \(G\) is the amplifier gain (without loss of generality, \(G=1\) is assumed from now on); \(P_{\max,l}\) is the saturation power at the output of each PA installed at RU \(l\); and \(p\) is the smoothing factor. The higher the \(p\), the closer the PA's characteristic to the so-called soft limiter. Under the assumption that a complex Gaussian signal is transmitted (valid for OFDM signals [19]), based on the [PERSON] theorem, the output sample of PA \(\tilde{y}_{m,l}(n)\) can be decomposed into the desired signal sample \(\tilde{x}_{m,l}(n)\) scaled by \(\alpha_{m,l}\) and the nonlinear distortion sample \(\tilde{n}_{\text{dis},m,l}(n)\), which is uncorrelated with \(\tilde{x}_{m,l}(n)\):  
\[\tilde{y}_{m,l}(n)=\alpha_{m,l}\tilde{x}_{m,l}(n)+\tilde{n}_{\text{dis},m,l}( n). \tag{7}\]The scaling factor is given by:  
\[\alpha_{m,l}=\frac{\mathbb{E}[\tilde{y}_{m,l}(n)\tilde{x}_{m,l}(n)^{*}]}{\mathbb{E }[\tilde{x}_{m,l}(n)\tilde{x}_{m,l}(n)^{*}]}, \tag{8}\]  
where \(\mathbb{E}[\,]\) denotes expectation and \({}^{*}\) is the complex conjugate. For the considered Rapp model of the PA, there is no closed-form expression to calculate \(\alpha_{m,l}\). However, both the scaling factor \(\alpha_{m,l}\) and the power of nonlinear distortion can be calculated numerically by simulating the processing of input samples by the Rapp model of the PA. Based on the output nonlinear distortion associated with each RU and taking into account the effect of the radio channel coefficients, one can estimate the power of nonlinear distortion \(\sigma^{2}_{\text{dis},l,r}\) for the \(i\)-th UE within the band of the \(r\)-th RB:  
\[\sigma^{2}_{\text{dis},l,r}=\mathbb{E}\left[\left|\sum_{l=1}^{\text{N}_{\text{ RU}}}\sum_{m=1}^{M_{l}}h_{i,l,r,m}\mathcal{F}\{\tilde{n}_{\text{dis},m,l}(n)\}_{r} \right|^{2}\right], \tag{9}\]  
where \(\mathcal{F}\{\}_{r}\) denotes the Fourier transform at RB \(r\). Similarly, one can estimate the values of \(\alpha_{m,l}\) that constitute the diagonal matrix \(\mathbf{A}_{l}\). Both \(\mathbf{A}_{l}\), and \(\sigma^{2}_{\text{dis},i,r}\), are necessary for the SINR\({}_{i,r}\) calculation in Equation (4).