Narrowband Internet of Things (NB-IoT) aims to provide wide coverage for a massive number of low-cost devices. Therefore, an NB-IoT physical random access channel (NPRACH) preamble based on a single tone signal with frequency hopping was designed, enabling the base station to estimate the time-of-arrival (ToA) values for realizing uplink synchronization among multiple users. However, due to residual carrier frequency offset (RCFO) in the NPRACH preamble, it is crucial to keep the accuracy of the ToA estimation. Recognizing this urgency, in this paper we first judiciously investigate the effect of the hooping distance on ToA estimation. With which, we propose an efficient receiving algorithm to improve the accuracy of ToA estimation. The main aim of the algorithm is to treat two consecutive symbol groups as a whole and then carry out difference calculations on the two newly constituted symbol groups to construct a more logical frequency hopping distance. Extensive numerical results validate the superiorly of our proposed scheme compared against conventional strategies, showing that the probability of the ToA estimation obtained by this method is 99% within the acceptable error range.

The traditional multi-level grid multiple-relaxation-time lattice Boltzmann method (MRT-LBM) requires interpolation calculations in time and space. It is a complex and computationally intensive process. By using the buffer technique, this paper proposes a new multi-level grid MRT-LBM which requires only spatial interpolation calculations. The proposed method uses a center point format to store multi-level grid information. The grid type determination in the flow field calculation domain is done using the axis aligned bounding box (AABB) triangle overlap test. According to the calculation characteristics of MRT-LBM, the buffer grid is proposed for the first time at the interface of different levels of grids, which is used to remove the temporal interpolation calculation and simplify the spatial interpolation calculation. The corresponding multi-level grid MRT-LBM algorithm is also presented for two-dimensional and three-dimensional flow field calculation problems. For the two-dimensional problem of flow around a circular cylinder, the simulation results show that a four-level grid MRT-LBM proposed in this paper can accurately obtain the aerodynamic coefficients and Strouhal number at different Reynolds numbers, and it has about 1/9 of the total number of grids as a single-level grid MRT-LBM and is 6.76 times faster. For the three-dimensional flow calculation problem, the numerical experiments of flow past a sphere are simulated to verify the numerical precision of the presented method at Reynolds numbers = 100, 200, 250, 300, and 1000. With the streamlines and velocity contours, it is demonstrated that the multi-level grid MRT-LBM can be calculated accurately even at the interface of different size grids.