An Efficient Random Access Reception Algorithm for ToA Estimation in NB-IoT

Abstract

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.

1. Introduction

Narrowband Internet of Things (NB-IoT) aims to interconnect a massive number of device with improved power efficacy and reduced complexity [1]. In this technology, a waveform is designed to realize uplink synchronization through time-of-arrival (ToA) estimation, i.e., narrowband physical random access channel (NPRACH). In uplink synchronization, NPRACH is the first transmitted signal from user equipment (UE) to the base station (BS), so that the BS can perform timing advance according to the ToA estimated from the NPRACH, keeping the orthogonality among the UE [2]. However, the requirement of accurate ToA is crucial due to the larger cell size of IoT compared to traditional cellular networks, and the deviation of ToA estimation would cause a performance erosion of the system [3]. ToA estimation has emerged as a prominent research area in recent years. The authors of [4] presented an enhanced eigenvector technique to reduce the detrimental effects of cross-correlation components during ToA estimation. The author of [5] reported a fuzzy support vector machine-based adaptive subspace classification method for subspace-based ToA estimation in a multipath channel. The authors of [6] introduced an acceleration technique for compressed sensing, utilizing both the Kronecker product and zero element removal to estimate both the direction-of-arrival (DoA) and ToA. The authors of [7] presented a novel approach for fast joint delay and angle estimation of multipath signals using an improved propagator method, with the objective of reducing the computational complexity of the two-dimensional multiple signal classification (2D-MUSIC) algorithm. The authors of [8] proposed a change point detection algorithm, as an alternative to the traditional threshold crossing technique, for accurate ToA estimation based on energy detection. The aim of this approach was to achieve precise estimation of the first ToA path in complex indoor environments. The authors of [9] utilized the structure of the CP-OFDM radar signal model and employed properties such as circulant matrix, fast Fourier transform, and Hadamard multiplication to solve the optimization problem of joint ToA and angle-of-arrival (AoA) recovery. The authors of [10] utilized an iterative-adaptive approach to resolve multipath signals in the ToA domain, followed by a conventional beamformer to retrieve the desired line-of-sight DoA. The authors of [11] proposed a de-noising technique with constant false alarm rate detection, which was based on sinc filter banks. The aim of this technique was to optimize the estimation accuracy deterioration that occurs when wavelet de-noising was applied in the estimation of ToA. The authors of [12] proposed a novel joint estimation method for AoA and ToA, referred to as JADE. The authors of [13] proposed a scheme for the joint estimation of ToA and DoA with carrier frequency offset compensation, utilizing large-scale receive antenna arrays. The authors of [14] proposed a joint estimation scheme based on the two-dimensional multiple signal classification (MUSIC) algorithm to reduce array modelling errors in estimating DOA and ToA for fifth generation (5G) signals. The authors of [15] presented a method for estimating multiple users simultaneously using a leaky coaxial cable based on ToA. In [16], a technique utilizing phase-mode expansions and near frequency-invariant elliptical arrays was proposed for estimating joint DoA and ToA in multipath environments using ultra-wideband signals. A method was presented for estimating ToA in [17] by replicating the transmitted ISNAC signal and calculating the cross-correlation function (CCF) between the template and received signals. The authors of [18] proposed a high-precision ToA estimation method based on 5G downlink signals, which can be implemented through edge computing. However, NB-IoT has key feature of low power consumption and low-cost. Adding additional signals to improve performance will bring extra energy expenditure to this system and the requirement for high computing power is also impractical for low-cost NB-IoT devices.

For NPRACH, the residual carrier frequency offset (RCFO) impacts the phase domain on the received complex signal, resulting in a degradation of the ToA estimation [19]. Therefore, enhancing the performance of ToA estimation while restraining the effect of the RCFO has been widely studied. The schemes can be divided into two categories, i.e., decreasing the impact of the RCFO through the design of the receiver or to perform joint ToA and RCFO estimation to rectify frequency error. The former mainly relies on a pre-set frequency hopping pattern to perform phase difference of signals to obtain the ToA, possessing low computational complexity, and the latter also requires FFT for joint ToA and RCFO estimation, possessing higher accuracy and complexity than the former.

Under the first scheme, a frequency hopping pattern of the NPRACH preamble was proposed in [3], and the authors utilized all feasible hopping distances under the resource configuration of 12 sub-carriers. The method improved the accuracy of ToA estimation. Nevertheless, no discussion was provided on how to estimate the ToA through the designed frequency hopping pattern. In contrast, under the second scheme, the authors of [20] used a 2D FFT to jointly estimate RCFO and ToA. However, high complexity is a non-negligible issue. In [21], the authors discussed the detailed mathematical model of NPRACH, and a 1D FFT method was presented to calculate the ToA after the frequency offset was corrected. Unfortunately, this method only supported small ToA values, limited to NPRACH format 0.

To address above problems, a method compatible with the standard system, supports NB-IoT cell size, and has acceptable complexity is desirable. In this paper, considering the low complexity of IoT devices, we inherited the first category and propose an efficient NPRACH reception algorithm, which effectively improves the accuracy of ToA estimation and suppress the effect of RCFO. The proposed scheme makes full use of the fixed frequency hopping pattern by taking two traditional basic units as a whole unit to construct a new hopping distance. Note, the novel scheme still follows the traditional frequency hopping pattern, meaning the newly designed hopping distance is logical. Therefore, the proposed scheme is compatible to the frame structure of the NB-IoT standard on the basis of not changing the frequency hopping pattern. The comparison shows that the reduced complexity of the proposed scheme.

The rest of the paper is organized as follow. Section 2 describes the system model of the NPRACH preamble. In Section 3, an efficient reception algorithm is proposed along with its complexity analysis. Section 4 analyses the numerical results. Section 5 gives the conclusions.

2. System Model

In this section, we first recap the standard of NPRACH designed in Release 13 [22]. Then the progression of the NPRACH preamble is illustrated.

2.1. Basics in Random Access of the NB-IoT

As shown in Figure 1, the NPRACH preamble consists of four symbol groups (SGs), where each one has five identical symbols, and five symbols share a single cyclic prefix (CP). In addition, based on different cell sizes, the CP lengths can either be 66.7 μs or 266.7 μs [22]. Four SGs are regarded as one basic unit, which can be transmitted repeatedly $2^k,k=0,1,\dots ,7$, times to improve the coverage [22].

The whole frequency band of the NPRACH is 45 kHz, which can be divided into 12 sub-carriers, and each SG occupies one tone of 3.75 kHz in the frequency domain. To avoid collision from UE, the NPRACH adopts frequency hopping, the pattern of which can be partitioned into two schemes. One adopts the fixed hopping distance between the SGs in the basic unit, and the other uses the pseudo-random hopping distance between the basic units. Specifically, in the fixed hopping pattern, single sub-carrier hopping is adopted between the 1st and 2nd SGs and the 3th and 4th SGs, while six sub-carriers hopping is applied between the 2nd and 3th SGs. In another schemes, the hopping distance between the basic units is pseudo-random, defined in [22].

2.2. Random Access Procedure

Based on [19], the equivalent baseband signal for the NPRACH preamble from the transmitter can be written as

$$\begin{array}{cc}\hfill S_{p,\mu }\left[n\right]& =\sum _{m=0}^{M-1}{a}_{p,\mu }\left[m\right]e^{j2\pi \frac{m}{M}n},\hfill \\ \hfill n& =N_{p,\mu }-N_{cp},\cdots ,N_{p,\mu }+M-1,\hfill \end{array}$$

(1)

where $S_{p,\mu }\left[n\right]$ is the n-th time domain sample of the p-th symbol in the $\mu$-th SG, $a_{p,\mu }\left[m\right]$ denotes the p-th symbol in the $\mu$-th SG on the m-th sub-carrier, M stands for FFT size, and $N_{cp}$ represents the size of the CP. Moreover, $N_{p,\mu }=\mu N_g+pM$, where $N_g$ is the size of an SG. As stated above, five symbols occupy one CP, i.e., $N_g=N_{cp}+5M$.

Due to low accuracy of low-cost commercial IoT devices and the limited spectrum of random access [3], the wireless channels can be partitioned into two impulse function. The first is the RCFO caused by imperfect oscillators or synchronization errors, whereas the second is multipath channel which can be modelled as one-tap [23]. Therefore, the received signal can be written as

$$\begin{array}{cc}\hfill y_{p,\mu }\left[n\right]& =h_{\mu }{e}^{j2\pi f(n-D)}\sum _{m=0}^{M-1}{a}_{p,\mu }\left[m\right]e^{j2\pi \frac{m}{M}(n-D)}+{\eta }_{p,\mu },\hfill \\ \hfill n& =N_{p,\mu }-N_{cp},\cdots ,N_{p,\mu }+M-1,\hfill \end{array}$$

(2)

where $h_{\mu }$ stands for the one-tap channel impulse response, f is the RCFO normalized by the sampling rate, D denotes the round-trip delay (RTD) normalized by the symbol duration within $[0,N_{cp}-1]$, and ${\eta }_{p,\mu }$ represents the zero-mean complex additive white Gaussian noise (AWGN) with variance ${\sigma }^2$.

At the receiver, the BS discards the CP and then performs M-point FFT on each symbol to restore the data. Then, we obtain

$$\begin{array}{cc}\hfill Y_{p,\mu }\left[k\right]=& h_{\mu }\sum _{n=N_{p,\mu }}^{N_{p,\mu }+M-1}{e}^{j2\pi f(n-D)}\sum _{m=0}^{M-1}{a}_{p,\mu }\left[m\right]\hfill \\ \hfill & · e^{j2\pi \frac{m}{M}(n-D)}{e}^{-j2\pi \frac{k}{M}n}+w_{p,\mu },\hfill \end{array}$$

(3)

where $Y_{p,\mu }\left[k\right]$ is the p-th symbol in the $\mu$-th SG on the k-th sub-carrier, and $w_{p,\mu }$ is the noise term. Note that the inter-carrier interference (ICI) is introduced in the signal owing to the RCFO. Assuming that ICI is negligible, when $k=m$, the p-th symbol of receiver becomes

$$\begin{array}{c}\hfill Y_{p,\mu }\left[k\right]=h_{\mu }{e}^{j2\pi f(N_{p,\mu }-D)}{a}_{p,\mu }\left[k\right]e^{-j2\pi D\frac{k}{M}}\frac{1-e^{j2\pi fM}}{1-e^{j2\pi f}}+w_{p,\mu }.\end{array}$$

(4)

To enhance the performance in a low signal-to-noise ratio (SNR) area, a coherent combination can be performed to improve the received SNR. Based on [22], the symbols $a_{p,\mu }\left[k\right]$ in the NPRACH are constant sequences, i.e., all the symbols are equal to 1. Therefore, combining the signal within the $\mu$-th SG under the invariant channel, it can be written as

$$\begin{array}{cc}\hfill Y_{\mu }\left[k\right]& =h_{\mu }{e}^{-j2\pi D\frac{k}{M}}\frac{1-e^{j2\pi fM}}{1-e^{j2\pi f}}\sum _{p=0}^4{e}^{j2\pi f(N_{p,\mu }-D)}+W_{\mu }\hfill \\ \hfill & =h_{\mu }{e}^{j2\pi f(N_{p,\mu }-D)}{e}^{-j2\pi D\frac{k}{M}}\frac{1-e^{j2\pi f5M}}{1-e^{j2\pi f}}+W_{\mu },\hfill \end{array}$$

(5)

where $Y_{\mu }\left[k\right]$ is the sum of the $\mu$-th SG on the k-th sub-carrier and $W_{\mu }$ denotes the sum of the noise. Since the Doppler frequency defined for NB-IoT is minor (1 Hz), it is noteworthy that the channel remains constant for at least 111 SGs in the time domain with sufficient flatness [19].

3. Proposed Method for ToA Estimation

In this section, we first study the effect of the hopping distance on ToA estimation. With which, we articulate our proposed reception algorithm for ToA estimation together with its complexity analysis.

3.1. Effect of the Hopping Distance on ToA Estimation

The frequency hopping pattern is designed to have a structure of paired upward and downward hoppings, in which the timing estimation can be accomplished based on the phase difference between the signals. There are two ToA estimation methods: (1) the coarse ToA estimation, which adopts a smaller hopping distance of 3.75 kHz; (2) the fine ToA estimation, in which a large hopping distance with 22.5 kHz is applied. In the following, we take the fine estimation as an example to illustrate the effect of the hopping distance on ToA estimation.

Assuming the channel is flat within 12 sub-carriers, and for convenience, we temporarily ignore noise $W_{\mu }$. Then, the differential result between the $(\mu +2)$-th and $(\mu +1)$-th SG within a unit can be obtained as

$$\begin{array}{cc}\hfill Z_{1,2}\left(r\right)& =Y_{\mu +1}\left[k_{\mu +1}\right]Y_{\mu +2}^{*}\left[k_{\mu +2}\right]\hfill \\ \hfill & ={\left|h_{\mu }\right|}^2{\left|\frac{1-e^{j2\pi f5M}}{1-e^{j2\pi f}}\right|}^2{e}^{-j2\pi f{N}_g}{e}^{\frac{j2\pi D(k_{\mu +2}-k_{\mu +1})}{M}},\hfill \end{array}$$

(6)

where $Z_{1,2}\left(r\right)$ denotes the difference between the 1st and 2nd SG in the r-th basic unit and $k_{\mu +2}-k_{\mu +1}$ is the hopping distance between the $(\mu +2)$-th and $(\mu +1)$-th SG. The differential result between the $\mu$-th and $(\mu +3)$-th in the same unit can be expressed as

$$\begin{array}{cc}\hfill Z_{0,3}\left(r\right)& =Y_{\mu }^{*}\left[k_{\mu }\right]Y_{\mu +3}\left[k_{\mu +3}\right]\hfill \\ \hfill & ={\left|h_{\mu }\right|}^2{\left|\frac{1-e^{j2\pi f5M}}{1-e^{j2\pi f}}\right|}^2{e}^{j2\pi 3f{N}_g}{e}^{\frac{j2\pi D(k_{\mu }-k_{\mu +3})}{M}},\hfill \end{array}$$

(7)

where $k_{\mu }-k_{\mu +3}$ represents the hopping distance between the $\mu$-th and $(\mu +3)$-th SG, the value of which is the inverse of $k_{\mu +2}-k_{\mu +1}$. Note that the distance in the time domain between $Z_{1,2}\left(r\right)$ and $Z_{0,3}\left(r\right)$ is not equal. Therefore, it requires phase compensation. Then, we can obtain

$$\begin{array}{c}\hfill Z_{0,3}^{{}^{\prime }}\left(r\right)=Z_{0,3}^{*}\left(r\right)·\left(Z_{1,2}\left(r\right)Z_{0,3}\left(r\right)\right),\end{array}$$

(8)

$$\begin{array}{c}\hfill Z_{1,2}^{{}^{\prime }}\left(r\right)=Z_{1,2}\left(r\right)·\left(Z_{1,2}\left(r\right)Z_{0,3}\left(r\right)\right),\end{array}$$

(9)

where $Z_{0,3}^{{}^{\prime }}\left(r\right)$ and $Z_{1,2}^{{}^{\prime }}\left(r\right)$ are obtained by phase compensation of $Z_{1,2}\left(r\right)$ and $Z_{0,3}\left(r\right)$, respectively. We denote the hopping distance as $\phi$$(\phi \in \mathbb{Z})$. Summing (8) and (9) together, we have

$$\begin{array}{c}\hfill U\left(r\right)=2G·\cos \left(2\pi f{N}_g\right)e^{\frac{j2\pi D\phi }{M}},\end{array}$$

(10)

$$\begin{array}{c}\hfill G={\left[{\left|h_{\mu }\right|}^2{\left|\frac{1-e^{j2\pi f5M}}{1-e^{j2\pi f}}\right|}^2\right]}^2,\end{array}$$

(11)

where $U\left(r\right)$ is independent from the RCFO in the phase domain. Inspired by the above analysis, we found that the impact of the RCFO can be mitigated by differential operation. Finally, we can obtain the RTD estimation as

$$\begin{array}{c}\hfill D=\frac{\mathrm{angle}\left(U\right(r\left)\right)·M}{2\pi \phi },\end{array}$$

(12)

where $\mathrm{angle}(·)$ denotes the phase angle return. Based on [24] and (12), it can be observed that the higher the value of $\phi$, the finer the resolution of D, and the smaller variance in the estimation.

Next, combined with [3], we discuss the effect of the hopping distance on phase ambiguity. Based on (10), after the effect of the RCFO is eliminated, the phase of the differential signal measured by the BS can be obtained as

$$\begin{array}{c}\hfill \theta =\frac{D\phi }{M}-2\pi \beta ,\end{array}$$

(13)

where $\theta$ (within $[0,2\pi ]$) is phase difference, $\beta$ depicts the quotient obtained by the true phase divided by $2\pi$, i.e., $\beta =⌊\frac{D\phi }{M}⌋$, and $⌊x⌋$ denotes the integer smaller than or equal to x. For notation simplicity, we consider a line-of-sight (LOS) channel between the UE and BS. The distance estimation is derived by $d=D·c$, where c is the radio propagation speed. Then we have

$$\begin{array}{c}\hfill d=\frac{(\theta +2\pi \beta )Mc}{2\pi \phi },\end{array}$$

(14)

where d is the distance between the UE and BS. It is worth noting that D is the taps of the RTD normalized by symbol duration. Then, it can be obtained that

$$\begin{array}{c}\hfill \beta =\frac{d\phi f_s}{Mc}-\frac{\theta }{2\pi },\end{array}$$

(15)

where $f_s$ is the system sampling frequency. Obviously, the range of $\beta$ is $\left[0,⌊\frac{d\phi f_s}{Mc}-\frac{\theta }{2\pi }⌋\right]$, which means we need to determine the optimal $\beta$ in this range.

We use an example to show how $\beta$ is chosen. Consider a typical NB-IoT cell, where $d=30$ km, $\phi =6$ tones, $f_s/M=3.75$ kHz, $c=3\times {10}^8$ m/s, and $\theta =0.5\pi$. In fine estimation, the range of $\beta$ is [0, 2], which means that there are three feasible values for $\beta$, i.e., $\beta =0$, $d=3.33$ km, $\beta =1$, $d=16.67$ km, and $\beta =2$, $d=30.0$ km, but it is difficult for the BS to choose the right d. However, in coarse estimation which adopts single sub-carrier hopping, since $⌊\frac{d\phi f_s}{Mc}-\frac{\theta }{2\pi }⌋<1$, the value of $\beta$ is unique. In the above situation, due to the certainty of coarse estimation, the BS can take the coarse estimation results as the reference to choose the correct d to avoid phase ambiguity in fine estimation. Therefore, in ToA estimation, the coarse estimation is always carried out first and then fine estimation can be obtained. Finally, since the fine estimation value is more accurate and its range is smaller than the coarse estimation [3], we regard the fine estimation as the estimated value of ToA.

3.2. Proposed Reception Algorithm

As stated above, the larger the hopping tone value, the finer the resolution in the estimation. Therefore, we aim to propose a novel reception algorithm to construct a lager hopping distance to enhance the accuracy of the ToA estimation. We acknowledged that, in the NPRACH, the total number of basic unit repetitions is to the power of 2. Therefore, taking every eight SGs as a basic unit is feasible in practice and does not emerge as a separate traditional unit. Based on eight SG units, more available novel hopping steps can be introduced. For example, as shown in Figure 2, the hopping distance of PRACH#0 between the SG numbered “0” and the 2nd SG is a seven sub-carrier spacing, and the hopping distance between the 1st and 3th SG is a five sub-carrier spacing.

Based on (5), the derivation of the differential result between two SGs can be written as

$$\begin{array}{cc}\hfill Z_{\mu ,\mu +2}\left(r\right)& =Y_{\mu }\left[k_{\mu }\right]Y_{\mu +2}^{*}\left[k_{\mu +2}\right]\hfill \\ \hfill & ={\left|h_{\mu }\right|}^2{\left|\frac{1-e^{j2\pi f5M}}{1-e^{j2\pi f}}\right|}^2{e}^{-j2\pi f2N_g}{e}^{\frac{j2\pi D(k_{\mu +2}-k_{\mu })}{M}},\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill Z_{\mu +4,\mu +6}\left(r\right)& =Y_{\mu +4}^{*}\left[k_{\mu +4}\right]Y_{\mu +6}\left[k_{\mu +6}\right]\hfill \\ \hfill & ={\left|h_{\mu }\right|}^2{\left|\frac{1-e^{j2\pi f5M}}{1-e^{j2\pi f}}\right|}^2{e}^{j2\pi f2N_g}{e}^{\frac{j2\pi D(k_{\mu +4}-k_{\mu +6})}{M}}.\hfill \end{array}$$

(17)

Since pseudo-random hopping affects the results of the differential procedure, the hopping pattern for a novel basic unit can be divided into four categories, i.e., whether $k_{\mu +2}-k_{\mu +1}=-(k_{\mu +6}-k_{\mu +5})$ or $\left|k_{\mu +2}-k_{\mu }\right|=\left|k_{\mu +6}-k_{\mu +4}\right|$.

For the first case $k_{\mu +2}-k_{\mu +1}=-(k_{\mu +6}-k_{\mu +5})$ and $k_{\mu +2}-k_{\mu }=-(k_{\mu +6}-k_{\mu +4})$ as shown in PRACH#0 in Figure 2, (16) and (17) can be rewritten as

$$\begin{array}{c}\hfill Z_{\mu ,\mu +2}\left(r\right)={\left|h_{\mu }\right|}^2{\left|\frac{1-e^{j2\pi f5M}}{1-e^{j2\pi f}}\right|}^2{e}^{-j2\pi f2N_g}{e}^{\frac{j2\pi D{\phi }_{57}}{M}},\end{array}$$

(18)

$$\begin{array}{c}\hfill Z_{\mu +4,\mu +6}\left(r\right)={\left|h_{\mu }\right|}^2{\left|\frac{1-e^{j2\pi f5M}}{1-e^{j2\pi f}}\right|}^2{e}^{j2\pi f2N_g}{e}^{\frac{j2\pi D{\phi }_{57}}{M}}.\end{array}$$

(19)

where ${\phi }_{57}$ is the hopping step and could be $\pm 5$ or $\pm 7$. Then in adding them together, we can obtain

$$\begin{array}{c}\hfill U_1\left(r\right)={\left|h_{\mu }\right|}^2{\left|\frac{1-e^{j2\pi f5M}}{1-e^{j2\pi f}}\right|}^2\cos \left(2\pi f2N_g\right)e^{\frac{j2\pi D{\phi }_{57}}{M}},\end{array}$$

(20)

where $U_1\left(r\right)$ is independent from the RCFO in the phase domain. The ToA estimation can be obtained as

$$\begin{array}{cc}\hfill D_{57}\left(r\right)& =\frac{\mathrm{angle}\left(U_1\left(r\right)\right)M}{2\pi {\phi }_{57}},\hfill \end{array}$$

(21)

where $D_{57}\left(r\right)$ is the estimation of the RTD for the first case.

For the second case $k_{\mu +2}-k_{\mu +1}=-(k_{\mu +6}-k_{\mu +5})$ and $\left|k_{\mu +2}-k_{\mu }\right|\ne \left|k_{\mu +6}-k_{\mu +4}\right|$ as shown in PRACH#1 in Figure 2, (16) and (17) can be rewritten as

$$\begin{array}{c}\hfill Z_{\mu ,\mu +2}\left(r\right)={\left|h_{\mu }\right|}^2{\left|\frac{1-e^{j2\pi f5M}}{1-e^{j2\pi f}}\right|}^2{e}^{-j2\pi f2N_g}{e}^{\frac{j2\pi D{\phi }_0}{M}},\end{array}$$

(22)

$$\begin{array}{c}\hfill Z_{\mu +4,\mu +6}\left(r\right)={\left|h_{\mu }\right|}^2{\left|\frac{1-e^{j2\pi f5M}}{1-e^{j2\pi f}}\right|}^2{e}^{j2\pi f2N_g}{e}^{\frac{j2\pi D{\phi }_1}{M}}.\end{array}$$

(23)

where ${\phi }_0$ and ${\phi }_1$ are the hopping step and could be ($\pm 5,\pm 7$). Note that, ${\phi }_0$ and ${\phi }_1$ have the same sign. Then in multiplying them together, we can obtain

$$\begin{array}{c}\hfill U_2\left(r\right)=G{e}^{\frac{j2\pi D({\phi }_0+{\phi }_1)}{M}},\end{array}$$

(24)

where $U_2\left(r\right)$ is independent from the RCFO in the phase domain. The ToA estimation can be obtained as

$$\begin{array}{cc}\hfill D_{12}\left(r\right)& =\frac{\mathrm{angle}\left(U_2\left(r\right)\right)M}{2\pi ({\phi }_0+{\phi }_1)},\hfill \end{array}$$

(25)

where $D_{12}\left(r\right)$ is the estimation of the RTD in the second case. It is obvious that ${\phi }_0+{\phi }_1=\pm 12$, which means we construct a novel hopping distance of twelve.

For the third case $k_{\mu +2}-k_{\mu +1}=k_{\mu +6}-k_{\mu +5}$ and $\left|k_{\mu +2}-k_{\mu }\right|\ne \left|k_{\mu +6}-k_{\mu +4}\right|$ as shown in PRACH#2 in Figure 2, the ToA estimation can be obtained as

$$\begin{array}{cc}\hfill D_2\left(r\right)& =\frac{\mathrm{angle}\left(U_3\left(r\right)\right)M}{2\pi ({\phi }_0-{\phi }_1)},\hfill \end{array}$$

(26)

$$\begin{array}{c}\hfill U_3\left(r\right)=G{e}^{\frac{j2\pi D({\phi }_0-{\phi }_1)}{M}},\end{array}$$

(27)

where $D_2\left(r\right)$ is the estimation of the RTD in the third case and ${\phi }_0-{\phi }_1=\pm 2$, which means we construct a hopping distance of two that is not in the traditional pattern.

For the last case $k_{\mu +2}-k_{\mu +1}=k_{\mu +6}-k_{\mu +5}$ and $k_{\mu +2}-k_{\mu }=k_{\mu +6}-k_{\mu +4}$ as shown in PRACH#3 in Figure 2, (16) and (17) can be rewritten as

$$\begin{array}{c}\hfill Z_{\mu ,\mu +2}\left(r\right)={\left|h_{\mu }\right|}^2{\left|\frac{1-e^{j2\pi f5M}}{1-e^{j2\pi f}}\right|}^2{e}^{-j2\pi f2N_g}{e}^{\frac{j2\pi D{\phi }_{57}}{M}},\end{array}$$

(28)

$$\begin{array}{c}\hfill Z_{\mu +4,\mu +6}\left(r\right)={\left|h_{\mu }\right|}^2{\left|\frac{1-e^{j2\pi f5M}}{1-e^{j2\pi f}}\right|}^2{e}^{j2\pi f2N_g}{e}^{\frac{j2\pi D(-{\phi }_{57})}{M}}.\end{array}$$

(29)

Then by multiplying them together, we can obtain

$$\begin{array}{c}\hfill U_4\left(r\right)=G,\end{array}$$

(30)

where $U_4\left(r\right)$ is a constant without the RTD components in the phase domain. In this case, we adopt the traditional fine estimation to fill this gap.

In summary, we add four different sub-carrier spacings to enhance the performance and discard the differential operation of a six tone hopping distance in most cases. In order to better explain the scheme, we take a 10 km target distance and 128 repetitions of classical units as an example, in which all kinds of ToA results can be obtained, and $D_2$ is non-existent phase ambiguity. Then, we can directly use A to replace the traditional method to eliminate the phase ambiguity of $D_2$ and obtain the actual of $D_{57}$. As mentioned above, more accurate results can be obtained with larger frequency hopping intervals. Therefore, we adopt $D_{57}$ to obtain $D_6$, and so on until we obtain $D_{12}$. Note that the coarse estimation based on a single tone is discarded in this situation, and only used when the number of repetitions is small. For a scenario where a traditional unit is transmitted repeatedly only once, $D_2$ may not exist in fine estimation. Hence, the coarse estimation results are used as a reference to distinguish $D_f$, where $D_f$ could be $D_5$, $D_6$, $D_7$ or $D_{12}$.

3.3. Complexity Analysis

In order to verify the practicability of using the proposed method, we carry out a complexity analysis of the fine estimation, which is the difference between two methods. The traditional scheme requires $N_{rep}$ times complex multiplications in (6) and $2N_{rep}$ in (8), where $N_{rep}$ is the number of repeated SG transmissions. For a frequency hopping distance of 5 or 7, the complex multiplications of (20) is $N_{rep}$ times. For other cases, the complex multiplication is $N_{rep}$ times in (24). We assume that the probability of occurrence of the four cases is the same. In general, for 12 UEs, considering the additional calculation of each hopping distance caused by phase ambiguity, the overall complexity in terms of complex multiplication of the novel scheme is $\mathcal{O}(12·5N_{rep})$, while the traditional scheme is $\mathcal{O}(12·8N_{rep})$. Clearly, the proposed method has less complexity compared to the classical scheme.

4. Simulation Results

In this section, numerical results are carried out to evaluate the performance of our proposed reception scheme. The simulations are presented under a one-tap channel and other parameters are set as follows: FFT size is $M=1024$, CP length is 266.7 μs, NPRACH band is 12 sub-carriers, number of active UE is 1, antenna configuration is 1 Tx with 1 space Rx, and the distance between the UE and BS is 30 km.

The resource configuration of the NPRACH will change according to the target maximum coupling loss (MCL). Therefore, in Figure 3, we configure the number of repetitions to be 8, 32 and 128 SGs, corresponding to three different MCL [20], to evaluate the new scheme in the same simulation environment. From the results, the centre of the cumulative distribution function (CDF) of the novel scheme is improved. The reason for this is that after phase ambiguity is eliminated, $D_{12}$ can provide finer resolution. Moreover, as the number of repetitions increases, the number of estimated values obtained through the 12 hopping distance increases, and the higher the accuracy of ToA estimation.

In the following, we take two units that is the minimum number of repetitions as an example to comprehensively show the novel scheme. Figure 4 shows that the proposed scheme increases the CDF tail. One explanation is that, $D_{12}$ cannot always be obtained in each fine estimation with one repetition, and $D_2$, the value range of which is larger, also becomes the result of the fine estimation, making the curve intersect with the traditional one. In practice, the NPRACH resource is only configured for retransmission once when the MCL is optimal. Therefore, even if the tail is elevated, the probability of ToA error can reach 99% in [−3.646∼3.646] μs which is the minimum requirement of the timing estimation in [25]. From these results, it is observed that the performance of the novel method is affected by the number of SG repetitions.

5. Conclusions

This paper first discussed the fine estimation process in detail according to the proposed protocol. In order to improve the performance of the algorithm in fine estimation, the factors affecting the accuracy of ToA estimation were analysed. Based on previous analysis, a receiving algorithm for ToA estimation was proposed in this paper. The main idea of the proposed algorithm was to explore more logical frequency hopping distances which could be used to improve the accuracy of ToA estimation. Simulation results and complexity analysis show that the proposed ToA algorithm reduces the computational complexity and error range of ToA estimation without changing the frequency-modulated pattern under the existing standard, meaning the proposed scheme has the potential to be directly applied to the standard.