## 3 Massive-MIMO Sparse Uplink Channel Estimation Using a First-Order Statistics Based Approach  
For the uplink scenario of a massive-MIMO communication system, if each mobile user is assigned with a specific pilot sequence that is superimposed on the information sequence, a first order statistics of the received signal can be used to estimate the CIR as outlined in [50]. In this section, an SiT based channel estimation technique of [26] is extended for the estimation of massive-MIMO sparse uplink channels. For \(n_{t}\)-th transmitter, the training sequence \(c_{n_{t}}(k)\) is periodic with period \(P=\tilde{P}N_{t}\), where \(\tilde{P}\) is a positive integer. The training sequence \(c_{n_{t}}(k)\) is given as,  
\[c_{n_{t}}(k)=\sum_{i=0}^{P-1}c_{i,n_{t}}\,\mathrm{e}^{j(2\pi i/P)k},\qquad \forall k, \tag{6}\]  
where \(j=\sqrt{-1}\) and  
\[c_{i,n_{t}}=\frac{1}{P}\sum_{k=0}^{P-1}c_{n_{t}}\,\mathrm{e}^{-j(2\pi i/P)k}. \tag{7}\]  
The pilot sequence \(c_{n_{t}}(k)\) is selected such that only \(\tilde{P}\) coefficients among \(P\) are non-zero. Hence, the training sequence \(c_{n_{t}}(k)\) for \(n_{t}\)-th UE is given by,  
\[c_{n_{t}}(k)=\sum_{i=0}^{\tilde{P}-1}c_{i,n_{t}}^{\prime}\,\mathrm{e}^{j \mathrm{i}\alpha_{i,n_{t}}k},\qquad\forall k, \tag{8}\]  
where \(\alpha_{i,n_{t}}=2\pi(iN_{t}+n_{t}-1)/P\), and \(c_{i,n_{t}}^{\prime}\) are suitably chosen coefficients for \(1\leq n_{t}\leq N_{t}\) and \(0\leq i\leq\tilde{P}-1\). The training sequence \(c_{n_{t}}(k)\) can be designed by first choosing a periodic base sequence \(c_{o}(k)\) with a period of \(\tilde{P}\)[26] such that,  
\[\bar{c}_{i,o}=\frac{1}{\tilde{P}}\sum_{k=0}^{\tilde{P}-1}\bar{c}_{o}(k)\, \mathrm{e}^{-j(2\pi i/\tilde{P})k}. \tag{9}\]  
The training sequence \(\bar{c_{1}}(n_{t})\) with period \(P\) is then generated by replicating \(\bar{c}_{o}(k)\) for \(N_{t}\) times. Therefore, for \(n_{t}\)-th UE, the training sequence is obtained as [26]  
\[c_{n_{t}}(k)=\sigma_{cn_{t}}\,\bar{c}_{1}(k)\,\mathrm{e}^{j(2\pi/P)(n_{t}-1)k} \text{ for }n_{t}=1,2,\cdots,N_{t}. \tag{10}\]  
Taking the expected value of the received signal \(y_{n_{t}}(k)\) at \(n_{t}\)-th receive antenna gives  
\[E\{y_{n_{t}}(k)\}=\sum_{n_{t}=1}^{N_{t}}\sum_{i=0}^{\tilde{P}-1}\left[\sum_{ \ell=0}^{L}c_{i,n_{t}}^{\prime}h_{n_{t}n_{t}}^{\ell}\mathrm{e}^{-j\kappa_{i,n_{ t}}\ell}\right]\,\mathrm{e}^{j\kappa_{i,n_{t}}k}. \tag{11}\]  
For \(n_{1}\
eq n_{2}\), we have \(\alpha_{i,n_{1}}\
eq\alpha_{i,2,n_{2}}\) for any \(\{i_{1},i_{2}\}\in 0,1,...,\tilde{P}-1\). Let \(\mathbf{d}_{n_{t}n_{t}}=[d_{n_{t}n_{t},0},d_{n_{t}n_{t},1},\cdots,d_{n_{t}n_{t},(\tilde{P}-1)}]^{T}\), where \(d_{n_{t}n_{t},i}\) is given by  
\[d_{n_{t}n_{t},i}=\sum_{\ell=0}^{L}c_{i,n_{t}}^{\prime}h_{n_{t}n_{t}}^{\ell} \mathrm{e}^{-j\kappa_{i,n_{t}}\ell}. \tag{12}\]  
The mean square consistent estimate \(\mathbf{\hat{d}}_{n_{t}n_{t}}=[d_{n_{t}n_{t},0},\hat{d}_{n_{t}n_{t},1},\cdots, \hat{d}_{n_{t}n_{t},(\tilde{P}-1)}]^{T}\) of \(\mathbf{d}_{n_{t}n_{t}}\) is computed as in [26] and is given by  
\[\hat{d}_{n_{t}n_{t},i}=\frac{1}{M}\sum_{k=0}^{M-1}y_{n_{t}}(k)\,\mathrm{e}^{-j \kappa_{i,n_{t}}k}, \tag{13}\]where \(M\) is the total number of received symbols, as \(M\rightarrow\infty\), \(\hat{d}_{n_{t}n_{t},i}\to d_{n_{t}n_{t},i}\). The vector form of (13) is given below,  
\[\hat{\mathbf{d}}_{n_{t}n_{t}}=\mathbf{C}_{n_{t}}\mathbf{h}_{n_{t},n_{t}}, \tag{14}\]  
where the matrix \(\mathbf{C}_{n_{t}}\) is computed as  
\[\mathbf{C}_{n_{t}}=diag\ \{c^{\prime}_{0,n_{t}},c^{\prime}_{1,n_{t}},\cdots,c^{ \prime}_{(\tilde{p}-1),n_{t}}\}\mathbf{V}_{n_{t}}, \tag{15}\]  
where \(\mathbf{V}_{n_{t}}\) is the Vandermonde matrix given by  
\[\mathbf{V}_{n_{t}}=\left[