## 7 Simulation Results
### Performance Assessment  
Figure 8 illustrates the performance of normalization factors under the condition of the AWGN channel and \(-10\) dB SNR. All the curves are normalized to the maximum of each. \(\beta_{MMOD}^{{}^{\prime}}\) and \(\beta_{MMOD}^{{}^{\prime\prime}}\) are compared in Figure 8a, and the latter gives a deeper notch. As shown in Equations (23) and (28), \(\beta_{MMOD}^{{}^{\prime\prime}}\) is the FFO elimination version of \(\beta_{MMOD}^{{}^{\prime}}\). For this reason, when the correct timing is absent, the attached FFO compensation makes no contribution. When the correct timing is present, the pre-estimated FFO helps eliminate primary phase error between the two differential components. Figure 8b indicates the promotion of noise elimination. The relative notch after noise elimination factor \(\beta_{NE}\) is more obvious than \(\beta_{MMOD}^{{}^{\prime\prime}}\). It is easy to deduce that this promotion will be more significant with lower SNR. The normalized mean square error (NMSE) of pre-estimated FFO is shown in Figure 9. This simulation is conducted with different channels and SNRs. The initial FFO is set as 7.5\(\mathrm{KHz}\), so \(\varepsilon_{F}\) is 0.5. According to Equation (22) and considering the worst condition of \(V=10\), the residual FFO should be less than \(5\times 10^{-2}\). The result shows that even under extended coverage SNR, this condition can be fulfilled within 18 radio frames. Even the theory of this FFO pre-estimator is similar to [7], and increases computation to some extent. The pre-estimated FFO can make the differential normalization effective, which is proved invalid under large-frequency offset in [12]. Additionally, a typical residual FFO is 50\(\mathrm{Hz}\), which means that \(6.7\times 10^{-3}\) should be accomplished. We can see that this target is achieved within 40 radio frames, which outperforms [7]. Figure 10 presents the NPSS cumulative probability distribution (CDF) vs. the number of processed frames. The simulation SNR is \(-5\) dB and \(-15\) dB. The simulation channel of Figure 10a-c is AWGN, ETU-1 Hz and EPA-1 Hz, respectively. The operation frequency of the proposed method and [8] is 1.92 MHz, and the solution in [7] uses 240 KHz frequency to reduce power consumption. From the simulation results, the proposed method outperforms the others. In particular, when the NB-IoT terminals work with extended coverage \(\text{SNR}=-15\) dB, our method can achieve a target CDF of 0.9 with ten more radio frames less than [7,8]. When \(\text{SNR}=-5\) dB, the proposed method can also provide considerable performance gain. That is because the proposed NF uses the pre-estimated FFO and noise power to accelerate the detection process. However, as nothing can be complete in both respects, we have declared that in order to achieve rapid NPSS detection, acceptable hardware resources and computation complexity are sacrificed in Section 1. In particular, when compared with the solution in [7], the working frequency of our method is eight times its size, and the noise power estimator, FFO pre-estimator, and normalization factor almost quadruple the computation complexity. Nevertheless, there are two aspects that support that such a compromise can save more power than  
Figure 8: Performance of the proposed normalization factors: (**a**) Comparison of modified MOD vs. frequency offset-compensated MOD normalization factors; (**b**) Comparison of frequency offset-compensated MOD vs. frequency offset-compensated and noise-eliminated MOD normalization factors. Figure 9: Performance of the FFO pre-estimator with different channel and SNR conditions. is consumed. One is that the initial NPSS detection only needs to be conducted at the start of each transmission. Then, we can see from Figure 10 that in order to achieve 0.9 CDF, our method can save more than 15 NB-IoT frames (150 ms), and during such a long time, the RF transceiver will consume much more power than baseband digital circuits.