# On Channel Estimation in LTE-Based Downlink Narrowband Internet of Things Systems
## 7 Simulation Results
### Mean Square Error (MSE)  
For the justification of the effectiveness and performance of the channel estimator, we have utilized the averaged channel MSE, which is defined at the \(n^{th}\) repetition copy as  
\[MSE=\frac{1}{N}E\|\overline{\mathbf{\
abla}}-\mathbf{\
abla}\|^{2} \tag{33}\]  
where \(\overline{\mathbf{\
abla}}\) and \(\overline{\mathbf{\
abla}}\) are the actual and estimated channel values for all the estimators used in this paper. Furthermore, we have averaged \(N=10^{4}\) independent channel realizations to take all the MSE curves. In this case, the simulation SNR is set as \(-15\) to \(5\) dB to ensure a noisy environment and deep penetration of coverage according to the features of NB-IoT systems. Figure 9a,b depict the MSE comparison of the proposed channel estimation technique for both conventional and proposed interpolation method with LS, theoretical and practical LMMSE, and its other counterpart versions for TU 1 Hz and EPA 1 Hz channel models. It is clear from Figure 9a,b that the LS estimator possesses the worst MSE among all techniques. It is very obvious that the LMMSE will have the best performance, which is also clear in our simulation. When SVD is applied on the channel matrix then we can see that the MSE performance is degraded by \(1.2\) dB SNR at \(10^{-3}\) from LMMSE in the TU 1 Hz channel. The reason is that the low-rank SVD estimator always possesses an irreducible error floor due to the part of the channel that does not include some specific subspace of the whole channel [30]. In this case, the space of the matrix dimension for time and frequency components leads the channel matrix to a low-rank approximation while still possessing an optimal performance by sacrificing a negligible performance degradation [30]. In the banded SVD estimator, we observe that the MSE is degraded by \(2.2\) dB at \(10^{-3}\) compared to the SVD LMMSE. The reason for the performance degradation is the utilization of less correlated components of the channel matrix for edge subcarriers. The MSE for the proposed overlap banded technique with the conventional interpolation overcomes this performance degradation by about \(1.3\) dB. With the application of the proposed interpolation method in the proposed overlap banded LMMSE channel estimation, this performance loss is recovered by almost \(2.1\) dB, which means it offers almost the same MSE performance of the SVD LMMSE. The overall MSE performance degradation caused by overlap banded SVD LMMSE from traditional LMMSE is about \(2.1\) dB, while traditional interpolation is used but offers a very low computationally complex algorithm. With the application of the proposed interpolation, the MSE performance loss is only \(1.3\) dB from LMMSE, and the computational complexity is further reduced. Especially, this attribute makes the proposed algorithm a good fit for NB-IoT systems compared to other concerned methods in this work. Similar performance is observed in EPA \(1\) Hz channel also. The received SNR of the MSE at \(10^{-3}\) is recorded in Table 4 for both TU and EPA channel models for better understanding and comparison. It is clear from Table 4 that the TU 1 Hz channel shows better performance than the EPA \(1\) Hz channel in the context of deep coverage (where \(SNR<<0\)), and the similar performance of this two channel models is also observed in [43]. With the application of the polyphase decomposed FIR filter interpolation method in the proposed overlap banded SVD LMMSE, almost \(1\) dB additional power is saved compared to the conventional interpolation method in both TU and EPA channel scenarios. In summary, the slight performance penalty of the proposed overlap banded LMMSE with the proposed interpolation method from the optimum LMMSE is only \(1.3\) dB and \(1.1\) dB for TU 1 Hz and EPA 1 Hz channels, respectively, which is nominal in the case of NB-IoT scenarios. Twenty subvectors (S = 20) were considered for this simulation for the case of overlap banded SVD LMMSE.