# A Simplistic Downlink Channel Estimation Method for NB-IoT
## 3 Evaluation Methodology  
In order to assess the performance of the proposed channel estimation scheme, appropriate quality indicators are required. Some authors use the bit error rate (BER) vs. SNR curves for this purpose. However, this approach is adequate mainly for simulation research, where bit sequences contained in transport blocks are known a priori and may be compared with sequences at the channel decoder output. In the case of signals originating from an eNB or a radio communication tester, this information is not available. For this reason, proper evaluation of BER in the physical layer is cumbersome as information bits are not recovered independently but as whole transport blocks, the validity of which is determined through cyclic redundancy check (CRC).  
In this paper, alternative indicators of channel estimation quality, which may be applied directly on the UE side without any prior knowledge about the transmitted downlink signal, are proposed. The quality assessment is based on the analysis of restored QPSK constellation. The error of symbol recovery is evaluated by comparing the symbols from channel equalization output with their original constellation points, which are restored by reverting the downlink signal processing chain, as shown in Figure 3.  
Full downlink physical layer processing is implemented, starting from time-frequency synchronization and OFDM FFT demodulation (not included in Figure 3) to channel decoding and transport block recovery. If the transport block is received correctly, which is verified by CRC check, its bits are again channel encoded and processed further until a set of reference QPSK symbols is obtained with the values from the original QPSK constellation, i.e., \(\pm 1\pm 1i\).  
Figure 2: Interpolated channel coefficients on NRS subcarriers (EPA channel profile).  
Two indicators are proposed to evaluate the relation between the received symbols and the restored QPSK symbols. The first one is the root-mean-square error (RMSE) of the symbol phase. It is calculated as follows:  
\[\Delta\theta_{RMS}=\sqrt{\frac{1}{N_{symb}}\sum_{i=1}^{N_{symb}}(\theta_{i}- \theta_{i,ref})^{2}}\ \ [rad], \tag{4}\]  
where \(\theta\) represents the symbol phase and \(N_{symb}\) is the number of the analyzed complex symbols.  
However, evaluating only the phase error may not be sufficient for implementations which make use of soft convolutional decoding, where symbol amplitudes affect path metric values in maximum likelihood sequence estimation (MLSE). Thus, a more comprehensive measure is proposed, namely the symbol correlation coefficient, which combines the impacts of phase and amplitude errors. It is calculated as follows:  
\[\rho_{IQ}=\frac{1}{\alpha}\cdot\sum_{i=1}^{N_{symb}}(I_{i}\cdot I_{i,ref}+Q_{ i}\cdot Q_{i,ref}), \tag{5}\]  
Figure 3: Restoring original QPSK symbols for channel estimation quality assessment.  
where \(\alpha\) is the normalization coefficient which limits the \(\rho_{IQ}\) values to the range between \(-1\) and \(1\):  
\[\alpha=\sum_{i=1}^{N_{symb}}{(|I_{i}|+|Q_{i}|)}. \tag{6}\]  
Normally the \(\rho_{IQ}\) ranges between \(0\) and \(1\). The higher value reflects more accurate symbol reception, whereas negative values would indicate the rotation between the received and reference symbol constellation. To provide a view on the relationship between \(\rho_{IQ}\) and the accuracy of the received symbols, a few example constellation graphs are presented in Figure 4. For lower cases, some symbol points are not visible on the graph due to axes limitations introduced to keep equal scales of all the subplots. These graphs were obtained for the proposed channel estimation method, in the additive white Gaussian noise (AWGN) channel at SNR values of \(-5\) dB, \(0\) dB, \(5\) dB, and \(10\) dB, respectively.  
Figure 4: Received symbols represented as points on IQ plane for different values of symbol accuracy coefficient: (**a**) \(\rho_{IQ}\) = 0.45, (**b**) \(\rho_{IQ}\) = 0.809, (**c**) \(\rho_{IQ}\) = 0.966, (**d**) \(\rho_{IQ}\) = 0.993.