## 2 Related Work
### System Model  
In the CF M-MIMO system, M APs and K UE are randomly dispersed over a sizable region, with L as the number of antennas per AP and N as UE. In the system the coherence time-frequency block size has no impact on the channel responses [24]. The system operates in time division duplex (TDD) mode; each block consists of a total of \(\tau_{\mathrm{c}}\) samples, with \(\tau_{p}\) samples set for UL and the remaining \(\tau_{\mathrm{u}}=\tau_{\mathrm{c}}-\tau_{p}\) samples used for data transmission. The complex-valued channel between the mth AP and the kth UE is given as \(H_{mk}\ \in\ \mathbb{C}^{L\times N}\), with the assumption that the channel coefficients \(\mathrm{H}_{mk}\) are not dependent for different AP-UE pairs and independent and identically distributed (IID) across different blocks. The system model for the CF M-MIMO system is shown in Figure 2. In this model, M APs and K UE are randomly dispersed over a sizable region, with L and N denoting the number of antennas per AP and the number of antennas per UE, respectively. Figure 2: System model for the M access point and K user equipment. The Weichselberger model is modeled as [15]  
\[\mathrm{H}_{mk}=\overline{\mathrm{H}}_{mk}e^{i\varphi_{mk}}+\underbrace{\mathrm{ U}_{mk,r}\left(\widetilde{\mathrm{W}}_{mk}\odot\mathrm{H}_{mk,\mathrm{idid}}\right) \mathrm{U}_{mk,t}^{H}}_{\widetilde{\mathrm{H}}_{mk}} \tag{1}\]  
First, the line of the LoS component of the channel \(\overline{\mathrm{H}}_{mk}=\left[\overline{\mathrm{h}}_{mk1},\ldots,\overline{ \mathrm{h}}_{mkN}\right]\) is modeled as a deterministic complex vector with L elements, where \(\overline{\mathrm{h}}_{mkn}\in\mathbb{C}^{L}\) is the channel between the n-th antenna of UE k and AP m, and \(\varphi_{mk}\sim\mathcal{U}[-\pi,\pi]\) is a uniformly distributed phase shift [17; 25]. Second, the NLoS component of the channel \(\widetilde{\mathrm{H}}_{mk}\) is modeled as a complex random vector with L elements, where \(\mathrm{H}_{mk,\ \mathrm{id}}\in\mathbb{C}^{L\times N}\) is a matrix of i.i.d. complex Gaussian entries with zero mean and unit variance. Third, the unitary matrices \(\mathrm{U}_{mk,r}\in\mathbb{C}^{L\times L}\) and \(\mathrm{U}_{mk,t}\in\mathbb{C}^{N\times N}\) are the eigen basis of the one-sided correlation matrices \(\mathrm{R}_{mk,r}\triangleq\mathbb{E}\left\{\widetilde{\mathrm{H}}_{mk} \widetilde{\mathrm{H}}_{mk}^{H}\right\}\) and \(\mathrm{R}_{mk,t}\triangleq\mathbb{E}\left\{\widetilde{\mathrm{H}}_{mk}^{T} \widetilde{\mathrm{H}}_{mk}^{H}\right\}\), respectively, which are defined as the expected values of the outer product of the NLoS component \(\widetilde{\mathrm{H}}_{mk}\)[26]. Fourth, the eigenmode coupling matrix \(\mathrm{W}_{mk}\triangleq\widetilde{\mathrm{W}}_{mk}\odot\widetilde{\mathrm{ W}}_{mk}\) is defined as the element-wise product of the matrix \(\mathrm{U}_{mk,r}\) and the conjugate transpose of the matrix \(\mathrm{U}_{mk,t}\), where the \((i,j)\)th element of \(\left[\mathrm{W}_{mk}\right]_{ij}\) specifies the average amount of power coupling from the ith column of \(\mathrm{U}_{mk,r}\) to the jth column of \(\mathrm{U}_{mk,t}\). Finally, the full correlation matrix \(\mathrm{R}_{mk}\) is calculated by taking the Kronecker product of the conjugate transpose of \(\mathrm{U}_{mk,t}\) and \(\mathrm{U}_{mk,r}\) and then multiplying it by a diagonal matrix formed by the vectorization of the matrix \(\mathrm{W}_{mk}\). The full correlation matrix is calculated as shown below by taking into consideration the channel between the UE and AP as \(\mathrm{h}_{mk}=\mathrm{vec}(\mathrm{H}_{mk})\in\mathbb{C}^{LN}\).