# On Channel Estimation in LTE-Based Downlink Narrowband Internet of Things Systems
## 1 Introduction
Three traditional channel estimator's (zero forcing, LMMSE, and linear phase) performance for NB-IoT uplink has been considered by [PERSON] et al. in [18]. In this work, the authors presented the dependency of the channel estimation quality on channel coherence time. However, there is no justification for the throughput achievements of these channel estimators. Channel equalization with interference problem in the coexistence of NB-IoT and LTE was considered for NB-IoT uplink by [PERSON] et al. in [19]. A one-tap channel equalization is presented to facilitate the coexistence of LTE and NB-IoT in this work. There is no discussion about the effect of phase noise under different types of channel conditions. [PERSON] et al. studied the DFT-based channel estimation in [20] for the uplink NB-IoT systems, depending on random sorting least square (RS-LS) and denoising least square (D-LS) along with LMMSE approximation (LMMSE-A). This research is lacking from the study of the increasing interference and communication latency because of applying additional operations in RS-LS and D-LS methods. Frequency domain channel equalization in [21] for NB-IoT uplink system under mismatched baseband processing and sampling rate is presented by [PERSON] et al. In this work, the authors presented a one-tap equalizer in the frequency domain to boost the performance of NB-IoT channel estimation. However, there is no study about the resultant complexity of this method. Channel estimation error suppression based on least square method hasbeen investigated by [PERSON] et al. in [22] for the downlink NB-IoT systems by applying additional operations. [PERSON] et al. also investigated narrowband demodulation reference signal (NDMRS)-based least square (LS) and minimum mean square error (MMSE) channel estimation for the NB-IoT uplink systems in [23]. In this work, the authors also investigated the peak to average power ratio (PAPR) reduction based on pulse shaping as an uplink transmit filter. However, these works are apart from the study of the justifications under different channel conditions in NB-IoT platform. Channel estimation and equalization with LMMSE and zero forcing are investigated by [PERSON] et al. in [24] for NB-IoT uplink. The authors studied the effectiveness of the LMMSE channel estimator with the MMSE equalizer as a good fit. A reduced complexity MMSE (sequential MMSE) channel estimation in the presence of random phase noise for the downlink NB-IoT systems has been well studied by [PERSON] et al. in [25]. However, the effect of coherence time variation is not discussed in both works above. Movable LS (MLS) technique with very insufficient investigations of different interpolation methods for NB-IoT downlink is studied in [26; 27]. Both of these methods possess poor performance because of exploiting LS estimator. Recently, a pilot-based hybrid channel estimation method based on the time domain wiener filter and frequency domain maximum likelihood estimator (MLE) was studied by the authors in [28] for downlink NB-IoT systems. The complexity is mentionably reduced by the hybrid application of MLE and wiener filter method while offering a good performance. The MMSE technique is more efficient than the LS method but possesses very high computational complexity, and hence it is very power-hungry as well. As we know that NB-IoT is a low power and low complexity system, a low complexity channel estimation is a prime demand for the receiver performance. In this regard, the linear minimum mean square error (LMMSE) technique, which has less complexity than MMSE, can be the right choice. However, LMMSE also possesses higher complexity than the LS method. Dividing the channel matrix depending on the coherence bandwidth method was studied for the complexity reduction of the LMMSE technique in [29] by [PERSON] et al. for the traditional OFDM systems. Furthermore, the well-known singular value decomposition (SVD) scheme was studied to ease the complexity of the LMMSE technique for a traditional OFDM system by [PERSON] et al.