# A Simplistic Downlink Channel Estimation Method for NB-IoT
## 2 NB-IoT Channel Estimation Procedure  
This section presents the proposed algorithm for downlink channel estimation whose objective is to compute channel coefficients. A channel coefficient, denoted by \(h\), is a complex number whose modulus and argument correspond to the attenuation and phase shift introduced during signal transmission. Each \(h\) value refers to an OFDM resource element on a specific subcarrier \(k\) of a given symbol \(n\)  
\[y_{n,k}=h_{n,k}\cdot x_{n,k}+\eta_{n,k}, \tag{1}\]  
where \(x\), \(y\) and \(\eta\) represent a transmitted resource element, a received resource element and additive interference, respectively. A set of channel coefficients is calculated per frame, i.e., every 10 ms. The first step is to calculate \(h\) values for NRS resource elements only. The \(LS\) method is used, according to which the estimate of channel coefficient \(\hat{h}\) is calculated as follows [14; 20]:  
\[\hat{h}_{n,k}^{LS}=x_{n,k}^{-1}\cdot y_{n,k}, \tag{2}\]where \(\{\hat{n},\bar{k}\}\) pairs address NRS resource elements. The component \(x(\hat{n},\bar{k})\) is known here as it is the element of an NRS sequence. In short, NRS channel coefficients are estimated by complex division of the received NRS resource elements by respective reference NRS resource elements. It is a simple calculation, not requiring matrix inversion, as opposed to the MMSE method. In the next steps of the algorithm, channel coefficients are estimated for all the OFDM resource elements other than NRS ones. Firstly, time-domain interpolation is performed on NRS subcarriers. In order to keep the calculation complexity low, linear interpolation was chosen. Let \(\bar{n}^{\prime}\) and \(\bar{n}^{\prime\prime}\) be OFDM symbol indices corresponding to subsequent NRS resource elements on subcarrier \(\bar{k}\). Channel coefficients for the resources lying between these elements are calculated as follows:  
\[\hat{h}_{n,\bar{k}}=\hat{h}_{n^{\prime},\bar{k}}+\frac{n-\bar{n}^{\prime}}{\bar {n}^{\prime\prime}-\bar{n}^{\prime}}\cdot(\hat{h}_{n^{\prime\prime},\bar{k}}- \hat{h}_{n^{\prime},\bar{k}})\qquad n=\bar{n}^{\prime}+1,\ldots,\bar{n}^{ \prime\prime}-1. \tag{3}\]  
Channel coefficients are complex numbers, so the interpolation is performed separately for their moduli and arguments. An additional phase unwrapping step is necessary prior to the actual interpolation. However, with this method, the resulting phase transitions are more smooth than in the case of the interpolation of real/imaginary components. It should be noted that symbol spacing between consecutive NRS elements varies. In most cases \(\bar{n}^{\prime\prime}\) is offset from \(\bar{n}^{\prime}\) by seven symbols (see Figure 1). However, some subframes do not contain NRS component. This refers to non NB-IoT subframes (e.g., LTE subframes in the in-band operation mode) and synchronization subframes. In such cases, the linear interpolation is performed at adequately longer intervals. As mentioned before, channel estimation is performed on a per-frame basis. This might cause discontinuities in channel coefficients between consecutive frames. To solve this problem, channel estimation is performed on 147 OFDM symbols, including one full frame (140 symbols) and the first slot of the following frame (7 symbols). Channel coefficients partially determined for this slot are passed to the next iteration of the channel estimation procedure. Figure 2 presents an example of linear interpolation of NB-IoT channel coefficients on NRS subcarriers for two consecutive frames. The circle markers represent coefficients calculated for NRS symbols using LS. The dot markers represent interpolated coefficients. The larger spacing between the circles corresponds to NPSS and NSSS subframes. As can be seen, the estimation follows the amplitude and phase variations smoothly and continuously. These variations reflect propagation in the channel modelled according to the Extended Pedestrian A profile with the maximum carrier Doppler shift of 5 Hz (EPA 5 Hz), without additive interference.