# On Channel Estimation in LTE-Based Downlink Narrowband Internet of Things Systems
## 4 NB-IoT Downlink Channel Estimation
### LMMSE Estimation with SVD and Banded Technique  
As we know that the LMMSE technique uses channel correlation and noise weighting property, the complexity reduction by SVD application is not significant enough to apply it in the low complexity NB-IoT system. Therefore, depending on the available bandwidth, the channel vector is further subdivided to reduce the complexity as in [29; 31]. Hence, we can represent the channel vector by  
\[\mathbf{\
abla}=\left[\mathbf{\
abla}_{1}^{T},\mathbf{\
abla}_{2}^{T},\dots\mathbf{\
abla}_{ R}^{T}\right]^{T} \tag{20}\]  
If every OFDM symbol contains \(L\) number of NRSs, and the desired subvector size is \(S\), then for Equation (20) we get \(R=L/S\), and \(\mathbf{\
abla}_{r}=\left[\mathbf{\
abla}_{S(r-1)+1},..,\mathbf{\
abla}_{S(r-1)+S}\right] ^{T}\) which is the \(r^{th}\) subvector of \(\mathbf{\
abla}\) for \(r=1,\dots,R\). Therefore, the correlation matrix \(\mathbf{\Phi}_{\mathbf{\
abla}\mathbf{\
abla}}\) from Equation (16) becomes  
\[\mathbf{\Phi}_{\mathbf{\
abla}\mathbf{\
abla}}=\left( \begin{array}{cccc}\mathbf{\Phi}_\mathbf{{\
abla}_{1},\mathbf{\
abla}_{1}}&\mathbf{\Phi}_{\mathbf{\
abla}_{1},\mathbf{\
abla}_{2}}& \dots&\mathbf{\Phi}_{\mathbf{\
abla}_{1},\mathbf{\
abla}_{R}}\\ \mathbf{\Phi}_{\mathbf{\
abla}_{2},\mathbf{\
abla}_{1}}&\mathbf{\Phi}_{\mathbf{\
abla}_{2},\mathbf{ \
abla}_{2}}&\dots&\mathbf{\Phi}_{\mathbf{\
abla}_{2},\mathbf{\
abla}_{R}}\\ \vdots&\vdots&&\vdots\\ \mathbf{\Phi}_{\mathbf{\
abla}_{R},\mathbf{\
abla}_{1}}&\mathbf{\Phi}_{\mathbf{\
abla}_{R},\mathbf{ \
abla}_{2}}&\dots&\mathbf{\Phi}_{\mathbf{\
abla}_{R},\mathbf{\
abla}_{R}}\end{array} \right) \tag{21}\]  
The largely correlated components of this matrix will always take a better influence on the channel estimation operation. In this case, less correlated components will have a minor impact. As the less correlated components are comparatively at a distance from the coherence bandwidth, they will take a minor influence on the channel estimation operation. Hence, we can set the value of these components to zero, as shown below:  
\[\mathbf{\Phi}_{\mathbf{\
abla}\mathbf{\
abla}}=\left( \begin{array}{cccc}\mathbf{\Phi}_\mathbf{{\
abla}_{1},\mathbf{\
abla}_{1}}&0\dots&0\\ 0&\mathbf{\Phi}_{\mathbf{\
abla}_{2},\mathbf{\
abla}_{2}}&\dots&0\\ \vdots&\vdots&&\vdots\\ 0&0\dots&\mathbf{\Phi}_{\mathbf{\
abla}_{R},\mathbf{\
abla}_{R}}\end{array} \right) \tag{22}\]Therefore, by taking only diagonal values, we get the approximate banded matrix as given below:  
\[\mathbf{\Phi}_{\texttt{TV}}\approx diag(\mathbf{\Phi}_{\texttt{\Psi}_{1},\texttt{\Psi}_{ 1}},\mathbf{\Phi}_{\texttt{\Psi}_{2},\texttt{\Psi}_{2}},\ldots,\mathbf{\Phi}_{\texttt{ \Psi}_{E},\texttt{\Psi}_{E}}) \tag{23}\]  
Now, from Equations (16), (18), and (23), we obtain the approximation of the LMMSE estimation as calculated below:  
\[\widetilde{\textbf{\Psi}}_{LMMSE}=\left( \begin{array}{c}\textbf{\Psi}_{1, LMMSE}\\ \textbf{\Psi}_{2,LMMSE}\\ \vdots\\ \textbf{\Psi}_{R,LMMSE}\end{array} \right)\approx\left( \begin{array}{c} \textbf{U}_{1}diag(\Delta_{1},\ldots,0)\textbf{U}_{1}^{H}\textbf{\Psi}_{1,LS} \\ \textbf{U}_{2}diag(\Delta_{2},\ldots,0)\textbf{U}_{2}^{H}\textbf{\Psi}_{2,LS} \\ \vdots\\ \textbf{U}_{R}diag(\Delta_{R},\ldots,0)\textbf{U}_{R}^{H}\textbf{\Psi}_{E, LS}\end{array}