## 2 Massive-MIMO System Model for Uplink Communications  
The uplink communications system model for massive-MIMO is assumed to consist of \(N_{t}\) single antenna UEs communicating with \(N_{r}\) antennas at the BS, as depicted in Figure 1. The information sequence to be transmitted by \(n_{t}\)-th UE is represented by \(\mathbf{b}_{n_{t}}=[b_{n_{t}}(0),b_{n_{t}}(1),\cdots,b_{n_{t}}(M-1)]^{T}\), such that \(\mathbf{b}_{n_{t}}\) is zero-mean, statistically independent of other UEs, and with \(E\{\left|\ b_{n_{t}}(k)\right.\left|^{2}\right\}=1\). Let \(\mathbf{c}_{n_{t}}=[c_{n_{t}}(0),c_{n_{t}}(1),\cdots,c_{n_{t}}(M-1)]^{T}\) represent a low power and periodic training sequence for \(n_{t}\)-th UE. The training sequence \(\mathbf{c}_{n_{t}}\) is periodic with period \(P\) such that \(c_{n_{t}}(k)=c_{n_{t}}(k+aP)\), for \(k\) and \(a\) being any integers, and is assumed to also be known at the BS side. After superimposing the training sequence \(\mathbf{c}_{n_{t}}\) over the information sequence, \(\mathbf{b}_{n_{t}}\), the resultant transmit signal for \(n_{t}\)-th UE is given as  
\[\mathbf{x}_{n_{t}}=\mathbf{b}_{n_{t}}+\mathbf{c}_{n_{t}}. \tag{1}\]  
The sequence \(\mathbf{x}_{n_{t}}=[x_{n_{t}}(0),x_{n_{t}}(1),\cdots,x_{n_{t}}(M-1)]^{T}\) is then transmitted by the \(n_{t}\)-th UE over the massive-MIMO sparse uplink channel. Between \(n_{t}\)-th UE and \(n_{t}\)-th receive antenna at the BS, a frequency selective and time invariant channel is assumed with CIR denoted by a sparse vector \(\mathbf{h}_{n_{t},n_{t}}=[h_{n_{t},n_{t}}^{0},h_{n_{t},n_{t}}^{1},\cdots,h_{n_ {t},n_{t}}^{L-1}]^{T}\). The number of resolvable mutipaths is represented by \(L\). Moreover, the CIR vector \(\mathbf{h}_{n_{t},n_{t}}\) is assumed to be \(Q\) sparse i.e., \(\{Q=\left\|\mathbf{h}_{mm}\right\|_{\ell_{0}}\}\ll L\), and has support \(\tilde{\mathbf{p}}=[\tilde{p}_{0},\tilde{p}_{1},\cdots,\tilde{p}_{Q-1}]\) such that  
\[h_{n_{t},n_{t}}^{\ell}=\left\{\begin{array}{ll}\
eq 0&;\;\ell\in\tilde{ \mathbf{p}};\\ =0&;\;\text{otherwise}.\end{array}\right. \tag{2}\]  
The signal received at time instant \(k\) by the \(n_{r}\)-th antenna element of receiver array at the BS is given below,  
\[y_{n_{r}}(k)=\sum_{n_{t}=1}^{N_{t}}\sum_{\ell=0}^{L-1}h_{n_{t},n_{t}}^{\ell}x_ {n_{t}}(k-\ell)+z_{n_{r}}(k), \tag{3}\]where \(z_{n_{r}}(k)\) denotes \(k\)th sample of zero-mean, additive white Gaussian noise (AWGN) with variance \(\sigma_{z}^{2}\). At time instant \(k\), the overall combined received signal by all antennas at the BS is denoted by \(\mathbf{y}(k)=[y_{1}(k),y_{2}(k),\cdots,y_{N_{t}}(k)]^{T}\) and given by,  
\[\mathbf{y}(k)=\sum_{\ell=0}^{L-1}\mathbf{H}^{\ell}\mathbf{x}(k-\ell)+\mathbf{ z}(k), \tag{4}\]  
where \(\mathbf{x}(k-\ell)=\left[x_{1}(k-\ell),x_{2}(k-\ell),\cdots,x_{N_{t}}(k-\ell) \right]^{T}\), \(\mathbf{z}(k)=\left[z_{1}(k),z_{2}(k),\cdots,z_{N_{r}}(k)\right]^{T}\) and the \(N_{r}\times N_{t}\) channel matrix \(\mathbf{H}^{\ell}\) for \(\ell\)th tap delay is given by,  
\[\mathbf{H}^{\ell}=\left[\begin{array}{cccc}h_{11}^{\ell}&h_{12}^{\ell}& \cdots&h_{1N_{t}}^{\ell}\\ h_{21}^{\ell}&h_{22}^{\ell}&\cdots&h_{2N_{t}}^{\ell}\\ \vdots&\ddots&\ddots&\vdots\\ h_{N_{t}1}^{\ell}&h_{N_{t}2}^{\ell}&\cdots&h_{N_{t}N_{t}}^{\ell}\end{array} \right]. \tag{5}\]  
The channel estimator (CE) block, shown in Figure 1, is implemented by using the SiT and first-order statistics based least squares (SiT-LS) technique presented in [50] and the proposed SiT-StOMP and SiT-GP techniques. Once the CIR is estimated, the effect of superimposed training sequence is removed from the information sequence at the receiver side by the training effect remover (TER) block, as shown in Figure 1. After eliminating the superimposed training sequence effect, the output of TER is fed as an input to the LMMSE equalizer.  
Figure 1: Proposed system model for massive multiple-input multiple-output (massive-MIMO) uplink communications.