## 5 Simulation Results  
LTE-MTC devices can be implemented to have only one receive antenna and a limited BW of 1.4 MHz. Therefore, the assignment of resources for LTE-MTC devices is based upon non-overlapping narrowbands consisting of 6 RBs. In our simulation, consider a 1.4 MHz LTE-MTC system with extended CP. More specifically, the carrier frequency is set to \(f_{c}=2\) GHz, the DFT size to \(N=128\), and subcarrier spacing to \(\Delta_{f}=15\) kHz. Therefore, the baseband signal is sampled with frequency \(1/T_{s}=1.92\) MHz. The propagation scenarios are compliant with the Pedestrian A and Vehicular A channel models, which are represented by a maximum delay spread of 0.41 \(\upmu\)s and 2.15 \(\upmu\)s, respectively [23]. For initial synchronization, it is assumed that the MTC UE may have reference clock uncertainty of \(\pm 20\) ppm, which corresponds to \(M=3\)[16; 17]. If not otherwise noted, we set \(M=3\). Figure 3 shows \(P_{f}\) of existing and proposed detectors in the AWGN channel. To verify the accuracy the analysis presented in the previous section, the theoretical result is compared with the simulated result. From Figure 3a, it is shown that analytical results using numerical integral completely match simulated curves for both HCJD and RCJD methods when \(M=1\), which validates the accuracy of performance analysis. In Figure 3b, the presence of the IFO degrades the performance of all detectors, which is more prominent in the case of the RCSD scheme. Not surprisingly, the performance of the detection methods is deteriorated as the local oscillator's stability is getting worse, i.e., \(M\) increases. Figure 4 presents the probabilities \(\text{Prob}\{\hat{v}\
eq v\}\) and \(\text{Prob}\{\hat{n}\
eq i\}\), respectively, where the Pedestrian A channel is adopted with the MTC UE speed of \(3\) km/h. To meet the operating requirement at the coverage enhancement SNR of \(-14.2\) dB [4], the decision statistic of the metrics of conventional and proposed schemes is averaged every \(5\) ms because PSS is transmitted twice every \(10\) ms radio frame. As a performance indicator, the number of PSS repetitions required to meet a target error probability of \(10\%\) at \(-14.2\) dB SNR is defined as \(N_{r}\). It is evident that the performance of the HCJD scheme is almost the same as that of the RCJD scheme in detecting the IFO and SID, respectively. For both HCJD and RCJD schemes, it can be seen that the IFO detector requires \(N_{r}=140\) to get an error probability of \(10\%\), while \(N_{r}=60\) is needed to detect the SID. It is worth pointing out that the adoption of the proposed scheme is more beneficial in estimating the SID than the IFO. For the derivation of existing and proposed detectors, it has been assumed that the STO is completely removed. In practice, the existence of multipath channel may substantially affect the performance of the STO estimation scheme and small STO still remains uncompensated. The effect of residual STO on the probability of detection failure \(P_{f}=\text{Prob}\{(\hat{v},\hat{n})\
eq(v,i)\}\) is shown in Figure 5, Assume that residual STO is uniformly generated from \([0,\tau_{max}]\) with \(\tau_{max}\) being the maximum value of residual STO. When compared to the result in Figure 4, it is observed that the overall performance of the joint detection scheme is primarily dependent on the performance of the IFO detection scheme. Thus, to obtain the target rate at the required SNR in both HCJD and RCJD methods, \(N_{r}=140\) PSS repetition is required. Although not shown in this figure, it follows that the RCSD method requires \(N_{r}=15,\!000\) in the case of \(\tau_{max}=0\). This observation indicates that MTC UE have to store \(70\) radio  
Figure 3: Performance of the joint detection methods versus signal-to-noise ratio in the additive white Gaussian noise channel: (**a**) \(M=1\); (**b**) \(M=2\); and \(M=3\). frames for successful synchronization in both HCJD and RCJD methods, which is equivalent to about 1/100 of the frame number needed for the RCSD method.