# A Low Complexity Channel Estimation and Detection for Massive MIMO Using SC-FDE
## 3 Channel Estimation
Since different data blocks are uncorrelated and the data symbols have usually zero mean, this approach tends to be very efficient. Therefore, the channel is initially estimated from the pilots, and then we remove the interference (pilots) from the received signal to detect the data symbols. The procedure for obtaining the channel estimates follow iteratively the following three steps (see indexes in the chains of Figure 3):  
1. Obtain the channel estimates from the pilots (remove the data from the received signal, estimated from the previous iteration, if not the first iteration);
2. Obtain the channel estimates from the data, after removing the pilots from the received signal (which represents interference);
3. Combine the channel estimates obtained from 1. and 2., to improve the estimate of the channel, and repeat the process. For the first iteration (step 1), an initial channel estimation is obtained by correlating the received signal \(Y_{k}^{(r)Rx}\) with the pilots, by using Equations (13) and then (16). As described above, if this is not the first iteration, then the data symbols are removed from the received blocks, as:  
\[Y_{k}^{(r)P}=Y_{k}^{(r)Rx}-Y_{k}^{(r)D}=Y_{k}^{(r)Rx}-\widehat{H}_{k}^{(t,r)} \widetilde{X}_{k}^{(t)}, \tag{20}\]  
and the channel estimation is calculated by using Equations (13) and (16). For the second iteration (step 2), the pilots are removed from the received blocks, as  
\[Y_{k}^{(r)D}=Y_{k}^{(r)Rx}-Y_{k}^{(r)P}=Y_{k}^{(r)Rx}-\widehat{H}_{k}^{(t,r)} X_{k}^{(t)P}, \tag{21}\]  
and the average values of the data symbols will be used as pilots for obtaining the channel frequency response estimate, as in [11]  
\[\widetilde{H}_{k}^{(t,r)}=\frac{Y_{k}^{(r)}X_{k}^{(t)P*}}{2\sigma_{P}^{2}}, \tag{13}\]0  
\[\widetilde{H}_{k}^{(t,r)}=\frac{Y_{k}^{(r)}X_{k}^{(t)P*}}{2\sigma_{P}^{2}}, \tag{13}\]1 Block diagram of m-MIMO receiver. Since using \(\alpha=0\) might lead to noise enhancement effects in the channel estimates when \(\left|X_{k}^{(i)}\right|^{2}\) is small, we will consider \(\alpha\) as defined in (4). If we have moderate to high SNR then  
\[\widetilde{H}_{k}^{(t,r)}=\frac{Y_{k}^{(r)}X_{k}^{(t)P*}}{2\sigma_{P}^{2}}, \tag{13}\]2  
and we could use \(\alpha=0\). Note that we employ pilots with relatively low power, and we average the pilots over a large number of blocks so as to obtain accurate channel estimates. The computation of Equation (22) corresponds to an improved channel estimate obtained with the data estimated from the previous iteration is very effective because the data symbols have usually zero mean and different data blocks are uncorrelated. Naturally, there are limitations on the length of this averaging window, since the channel should be constant within it. Finally, following step 3, the channel estimates obtained from the pilots (\(\widetilde{H}_{k}^{p}\)) and from the data aided (\(\widetilde{H}_{k}^{pD}\)), can be combined to provide the normalized channel estimates with minimum error variance [14], as defined by  
\[\widetilde{H}_{k}^{(t,r)}=\frac{Y_{k}^{(r)}X_{k}^{(t)P*}}{2\sigma_{P}^{2}}, \tag{13}\]3  
where  
\[\widetilde{H}_{k}^{(t,r)}=\frac{Y_{k}^{(r)}X_{k}^{(t)P*}}{2\sigma_{P}^{2}}, \tag{13}\]4