## 2 System Model
### Uplink Channel Estimation  
The channel can be estimated at the APs when \(K\) UEs simultaneously start to transmit their pilot sequences of \(\tau_{p}\). The received pilot sequence at \(m_{th}\) AP from \(K\) UEs, is expressed by  
\[Y_{m}=\sqrt{\tau_{p}\rho_{p}}\sum_{k=1}^{K}h_{k,m}\varphi_{k}^{H}+n_{m}, \tag{7}\]  
where \(\rho_{p}\) denotes the transmission power of each pilot symbol sent by \(k_{th}\) UE, \(\varphi_{k}\) gives \(\tau_{p}\times 1\) pilot signal assigned to \(k_{th}\) UE with \(||\varphi_{k}||^{2}=1\), and \(n_{m}\in\mathbb{C}^{N_{r}\times\tau_{p}}\) is known as a matrix of independent identically distributed (i.i.d.) received noise samples, with each entry distributed as \(\mathcal{CN}(0,\sigma^{2})\), in which \(\sigma^{2}\) is the noise power that can be computed as \(\sigma^{2}=-174\frac{\mathrm{dBm}}{\mathrm{Hz}}+10\log_{10}(B)+NF\), where \(B\) is the system bandwidth, and \(NF\) is the noise figure. It is worth noting that the vast majority of practical scenarios hold \(K>\tau_{p}\); hence, several UEs are allocated to a given pilot sequence. That leads to the pilot contamination phenomenon [28, 29]. Therefore, this work focuses on the case of \(K\leq\tau_{p}\). Based on \(Y_{m}\), the \(m_{th}\) AP can estimate the channel \(h_{k,m}\). Then, \(y_{k,m}\) is the projection of \(Y_{m}\) onto \(\varphi_{k}\), which is given as  
\[y_{k,m}\triangleq Y_{m}\varphi_{k}=\sqrt{\tau_{p}\rho_{p}}(h_{k,m}+\sum_{i\
eq k }^{K}h_{i,m}\varphi_{i}^{H}\varphi_{k})+n_{m}\varphi_{k}. \tag{8}\]Thus, the minimum mean square error (MMSE) is utilized to obtain the estimated channel \(\hat{h}_{k,m}\) under the assumption of the knowledge of \(\mathbb{E}\{\mathcal{A}_{k,m}\mathcal{A}_{k,m}^{H}\}\), which is the correlation matrix for all UEs is available at \(m_{th}\) AP [7]. For the MMSE estimation technique, we assume that the signals received at all of the AP's antennas are available. As a result, the low-complexity MMSE estimator can be used to estimate the full channel state information (CSI) associated with all \(N_{r}\) antennas. There is also a compressed sensing-based technique, as given in [24], that can be used to extract the entire CSI in the situation of sparse channels and very slow fading; however, this approach is highly complex, especially in cell-free massive MIMO systems. Thus, \(\hat{h}_{k,m}\) can be derived as [30]  
\[\begin{split}\hat{h}_{k,m}&=\mathbb{E}\{h_{k,m}y_{ k,m}^{H}\}(\mathbb{E}\{y_{k,m}y_{k,m}^{H}\})^{-1}y_{k,m}\\ &=\sqrt{\tau_{p}\rho_{p}}\big{(}\frac{G_{a}N_{r}}{\beta_{k,m}P_{ k,m}}\big{)}\mathbb{E}\{\mathcal{A}_{k,m}\mathcal{A}_{k,m}^{H}\}\\ &(\tau_{p}\rho_{p}\sum_{i=1}^{K}\frac{G_{a}N_{r}}{\beta_{i,m}P_{ i,m}}\mathbb{E}\{\mathcal{A}_{i,m}\mathcal{A}_{i,m}^{H}\}|\varphi_{i}^{H} \varphi_{k}|^{2}+\sigma^{2}I_{N_{r}})^{-1}y_{k,m}.\end{split} \tag{9}\]  
As a consequence, the total estimated channels between APs and \(K\) UEs is given as \(\hat{H}_{m}=\big{[}\hat{h}_{1,m},\hat{h}_{2,m},\ldots,\hat{h}_{K,m}\big{]} \in\mathbb{C}^{N_{r}\times K}\) and \(\hat{H}=\big{[}\hat{H}_{1},\hat{H}_{1},\ldots,\hat{H}_{M}\big{]}^{T}\in \mathbb{C}^{MN_{r}\times K}\). The next section explains how to employ the estimated channels \(\hat{H}\) and \(\hat{H}_{m}\) for the hybrid combining design in uplink data transmission.