# MIMO Channel Estimation in an SDR Platform for Evaluation of D&F Relay Nodes
## 4 Channel Estimation for the Core Network: Link eNB-to-Relay Node  
Downlink channel estimation in LTE is carried out through Cell-specific Reference Signals (CRSs). Pilot symbols are inserted during subcarrier mapping in both time and frequency. CRSs are used both for de-modulation and feedback calculation [26].  
In this section, the channel between the eNB and Relay Node presented in Figure 1 is studied. Channel estimation in 3 GPP LTE is performed in two steps. Firstly, the frequency response is extracted for the subcarriers of the pilot symbols; which, in our case, is based on the LS estimator. Then, taking to account the first step, the frequency response of the subcarriers of the data symbols can be obtained using the three studied interpolation methods: Bi-Linear (BLI), Bi-Cubic (BCI) and Bi-Nearest Neighbors (BNNI). In Figure 3, the Channel Estimation and Equalization Block Diagram are presented, with respect to the received RG Y.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Symbols** & **Description** \\ \hline \(\text{N}_{Tx}\) & Number of transmitter antennas \\ \(\text{N}_{Rx}\) & Number of receiver antennas \\ \(\mathbf{y}_{c,s}^{i}\) & Received vector of symbol \\ \(\mathbf{y}_{c,s}^{i}\) & (for \(j\)-th antenna of the receiver) \\  & Channel Matrix \\ \(\mathbf{H}_{c,s}^{i,j}\) & (between \(i\)-th transmitter and \(j\)-th receiver antennas) \\ \(\mathbf{x}_{c,s}^{i}\) & Vector of the transmitted symbols \\ \(\mathbf{x}_{c,s}^{i}\) & (for \(i\)-th antenna of the transmitter) \\  & AWGN vector \\ \(\mathbf{n}_{c,s}^{i}\) & (for \(j\)-th antenna of the receiver) \\ \(\mathbf{Y}\) & Received Resource Grid Matrix \\ \(\mathbf{H}_{c,s}^{i}\) & Channel Matrix \\ \(\mathbf{H}\) & Channel Matrix \\ \(\mathbf{X}\) & Matrix of transmitted symbols \\ \(\mathbf{N}\) & AWGN Matrix \\ \(\mathbf{y}_{c,s}^{i}\) & Received vector of pilot symbols \\ \(\mathbf{y}_{c,s}^{i}\) & (for \(j\)-th antenna of the receiver) \\ \(\mathbf{Y}_{p}\) & Received Resource Grid Matrix for pilot symbols \\ \(\mathbf{H}_{p}\) & Channel Matrix of transmitted pilot symbols \\ \(\mathbf{P}_{p}^{i}\) & Matrix of transmitted pilot symbols \\ \(\mathbf{N}_{p}\) & AWGN Matrix for pilot symbols \\ \(\mathbf{p}_{p}^{H}\) & Matrix of the Least-Square Estimator \\ \(\mathbf{H}_{p}\) & Channel Matrix Estimate of transmitted pilot symbols \\ \(\hat{\mathbf{H}}_{c_{s}^{i},s_{s}^{i}}^{i}\) & Channel Matrix Estimate of the transmitted data symbols \\ \(\mathbf{M}_{MMSE}\) & (between \(i\)-th transmitter and \(j\)-th receiver antennas) \\ \(\mathbf{M}_{MMSE}\) & matrix of the MMSE equalization scheme \\ \(\mathbf{M}_{ZF}\) & matrix of the ZF equalization scheme \\ \(\hat{\mathbf{Y}}\) & Equalized Received RG \\ \hline \hline \end{tabular}
\end{table}
Table 1: Description of the Symbols.  
The received signal in (1) is comprised of both data and pilot symbols. In consideration of this assumption, the pilot signal can be expressed as  
\[\mathbf{y}_{cp,s_{p}^{i}}^{j}=\sum_{i=1}^{N_{Tx}}\mathbf{H}_{cp,s_{p}^{i}}^{i,j} \mathbf{p}_{cp,s_{p}^{i}}^{i}+\mathbf{n}_{cp,s_{p}^{i}}^{j}, \tag{3}\]  
where (\(c_{p}^{i},s_{p}^{i}\)) is the location of the pilot symbol \(\mathbf{p}_{cp,s_{p}^{i}}^{i}\) at the RG received by the \(j\)-th receiver antenna. Considering (2), (3) can be rewritten as  
\[\mathbf{Y}_{p}=\mathbf{H}_{p}\mathbf{P}_{p}+\mathbf{N}_{p}. \tag{4}\]  
The LS channel estimator is used to equalize the channel frequency responses at pilot locations sent from different transmitters at all receiver antennas, which can be formulated as (5) and is integrated into the Pilot Estimation block in Figure 3.  
\[\mathbf{\hat{H}}_{p}=\mathbf{P}_{p}^{H}\mathbf{Y}_{p}. \tag{5}\]