## 2 System Model
### M-MIMO Channel Model  
Consider an uplink multi-user massive multiple input-multiple output (M-MIMO) scenario such as the one shown in Figure 3, assuming \(Q\) active users and \(N_{t}\) antennas in each user's device. Now, the number of antennas at the transmitting end and receiving end are \(N_{T}\) and \(N_{R}\), respectively, where \(N_{T}\) is equal to \(QN_{t}\) and \(N_{R}\) is far outweighs \(N_{T}\). The transmitted signal and the base station-received signal can be expressed as \(\mathbf{x}=[x_{1},x_{2},\cdots,x_{N_{T}}]^{T}\) and \(\mathbf{y}=[y_{1},y_{2},\cdots,y_{N_{R}}]^{T}\), respectively. Then, the channel model is described as  
\[\mathbf{y}=\mathbf{H}\mathbf{x}+\mathbf{n}, \tag{5}\]  
where \(\mathbf{H}\in\mathbb{C}^{N_{R}\times N_{T}}\) is the channel matrix and \(\mathbf{n}\) is the noise vector with size \(N_{R}\times 1\). For ease of comprehension, the component form of the model can be re-expressed as follows:  
Figure 2: Block diagram of UFMC. \[ \begin{bmatrix}y_{1}\\ y_{2}\\ \vdots\\ y_{i}\\ \vdots\\ y_{N_{R}}\end{bmatrix} = \begin{bmatrix}h_{11}&h_{12}&\cdots&\cdots&h_{1N_{T}}\\ h_{21}&\ddots&&&\vdots\\ \vdots&&h_{ij}&&\vdots\\ \vdots&&\ddots&\vdots\\ h_{N_{R}1}&\cdots&\cdots&\cdots&h_{N_{R}N_{T}}\\ \end{bmatrix}\begin{bmatrix}x_{1}\\ x_{2}\\ \vdots\\ x_{j}\\ \vdots\\ x_{N_{T}}\end{bmatrix} + \begin{bmatrix}n_{1}\\ n_{2}\\ \vdots\\ n_{i}\\ n_{N_{R}}\end{bmatrix} , \tag{6}\]  
where \(h_{ij}\) is the impulse response channel path between the \(jth\) transmitting antenna and \(ith\) receiving antenna and where \(n_{i}\) is the noise element received by the \(ith\) receiving antenna. To set up the outdoor transmission environment in this study, we establish a channel with two independent and identically distributed (i.i.d.) paths and conform to a Gaussian distribution with unit variance and zero mean. We also assume that no line-of-sight (LoS) path exists between transmitting and receiving antennas and utilize the Rayleigh fading channel to simulate the channel matrix \(\mathbf{H}\)[47]. In addition, the noise vector employs additive white Gaussian noise (AWGN) which is i.i.d. and follows a complex Gaussian distribution. Among the popular pilot insertion methods [41] there are block-type, comb-type, and lattice-type versions. We adopt the comb-type scheme [41,48] due to its simple structure and ease of implementation at the base station for Rayleigh fading channel estimation, with pilots regularly inserted into the subcarriers. Furthermore, this pilot tone information is applied to the least-square (LS) method [49] for channel estimation, which is shown as follows:  
\[\mathbf{\hat{H}}_{LS}=\left(\mathbf{X}_{p}^{H}\mathbf{X}_{p}\right)^{-1}\mathbf{ X}_{p}^{H}\mathbf{Y}_{p}=\mathbf{X}_{p}^{-1}\mathbf{Y}_{p} \tag{7}\]  
where \(\mathbf{\hat{H}}_{LS}\) represents the channel matrix estimated by considering the data containing the pilot tone through the LS method and \(\mathbf{X}_{p}\) and \(\mathbf{Y}_{p}\) respectively represent the transmit and receive signals, including the response to the pilot tone. To ensure that \(\left(\mathbf{X}_{p}^{H}\mathbf{X}_{p}\right)^{-1}\) exists, \(\mathbf{X}_{p}\) must be of full column rank. Here, we consider a detector with the goal of restoring the transmitted signal vector \(\mathbf{x}\) as much as possible according to the received signal vector \(\mathbf{y}\). In line with [50], in an uplink MIMO environment, the performance of the traditional MMSE detector has been proven to be the benchmark among linear detectors, and its estimated transmitted signal can be expressed as follows:  
\[\mathbf{\hat{x}}_{MMSE}=\left(\mathbf{\hat{H}}_{LS}^{H}\mathbf{\hat{H}}_{LS}+ \sigma^{2}\mathbf{I}_{N_{T}}\right)^{-1}\mathbf{\hat{H}}_{LS}^{H}\mathbf{y}= \mathbf{W}^{-1}\mathbf{y}^{MF} \tag{8}\]  
Figure 3: Uplink M-MIMO architecture diagram.