A NB-IoT Random Access Scheme Based on Change Point Detection in NTNs

In order to satisfy the higher demand raised by 6G, the major part of which refers to further extension of the coverage and achievements of ubiquitously massive connections, the integration of Non-Terrestrial Networks (NTNs) and the evolved version of Narrow Bandwidth Internet of Things (NB-IoT) are introduced as a potential solution. However, one of the main challenges still open is the Random Access (RA) procedure which refers to preamble detection and uplink synchronization. With the large Time of arrival (ToA) and Carrier frequency offset (CFO), traditional methods designed for Terrestrial Networks (TNs) can not provide accurate estimation performance. Some studies adopt Global Navigation Satellite System (GNSS) to pre-compensate the large ToA and CFO but it may not work well for NB-IoT devices in practice. Consequently, a complete system without GNSS is designed for RA in this paper. A system-level method is firstly applied to pre-compensate the CFO. Then we focus on tackling the ToA and residual CFO. A pre-process method for received signal is proposed and its phase series are analyzed. Then we propose a novel random access scheme based on change point detection (CPD) of phase series with machine learning to estimate ToA and CFO. Simulation results demonstrate that the designed overall system is practical in NTNs. Meanwhile, the proposed random access scheme achieves higher estimation accuracy of ToA and CFO compared with existing methods with the same range of ToA and CFO under low signal-noise ratio (SNR) conditions.


I. INTRODUCTION
I N RECENT years, fifth-generation New Radio (5G NR) has made great progress in commercial development, providing users with high quality services and better experiences.However, many challenges still exist, including global coverage, ultra-high data rate transmission, ultra-low latency, low power consumption, etc [1].In this context, several researches summarize the higher requirements expected to be explored and resolved in sixth generation (6G) [2], [3], [4], [5].Among the raised factors, global coverage has been widely concerned.The future target in 6G vision is to build an integrated space-air-groundsea communication system, achieving completely ubiquitous connections.
Then Non-Terrestrial networks (NTNs) are introduced to help make global coverage come true.The primary function of NTNs here is to help provide reliable communication services for devices in remote areas where terrestrial networks can not cover [6].In NTNs, Low and very Low Earth Orbit (LEO and vLEO) satellite are widely recognized as the most suitable way to achieve global coverage with its lower power consumption and propagation delay compared with Geostationary Earth Orbit (GEO) satellite.In the Third Generation Partnership Project (3GPP) Release 17 [7], normative work on Internet of Things (IoT) was started to pave the way to introduce Narrow Bandwidth Internet of Things (NB-IoT) support for satellites.And several use cases have been presented through satellite access in [8].
NB-IoT is a cellular technology standardized by 3GPP in release-16 [9] which enables devices to communicate with each other, connect to existing network receivers and finally contribute to application of massive Machine Type of Communication (mMTC).Typical use cases include mobile health, smart cities, smart homes, vehicle tracking and smart phones [10].NB-IoT has been deployed over 100 countries and regions with the main advantages of low power consumption, low throughput and wide coverage.However, with the rapid growth of number of NB-IoT devices, traditional Terrestrial networks (TNs) can not cover the devices deployed in remote areas such as desert, forest, or ocean.Consequently, it is necessary to further extend the coverage of NB-IoT through NTNs.
Aiming to integrate NB-IoT and NTNs into a unified communication system, one of the major challenges is the NB-IoT uplink synchronization for RA which refers to preamble detection and accurate time of arrival (ToA) plus carrier frequency offset (CFO) estimation in particular.A number of traditional researches concentrate on handling this challenge in terrestrial networks (TNs) [11], [12], [13], [14].In [11], Brute Force (BF) algorithm based on differential correlation is proposed to detect ToA while its accuracy degrades with the presence of residual CFO.Reference [12] exploits frequency hopping rules in RA to eliminate the impact of CFO when estimating ToA but its estimation range is limited by preamble format.Hence, if directly applies the original estimation method [11], [12], [13], [14] in NTNs, they will suffer from large propagation delay and large doppler shift.In recent years, a few studies [15], [16], [17], [18] adopt Global Navigation Satellite System (GNSS) in user devices to pre-compensate the large propagation delay and Doppler shift and then tackle the issue with small residual ToA and CFO after pre-compensation.For example, [18] uses cumulative sum algorithm (CUSUM) to detect the change point of wavelet to estimate the ToA and CFO but its performance heavily depends on how well the actual data follows the assumed distribution.
However, the assumption of GNSS-assisted devices with high accuracy positioning for NB-IoT in NTNs is questionable.Major challenges include [19], [20], [21]: 1) Equipping devices with GNSS shortens the battery lifetime which is not suitable for NB-IoT devices with limited power consumption.2) Natural radio propagation impairments and building blockage can undermine the positioning performance of GNSS.
3) The time synchronization algorithm in presence of GNSS is vulnerable to different types of intentional attacks.Such impacts heavily interfere the pre-compensation performance of GNSS.Hence the residual ToA and CFO may exceed the estimation range, leading to overall performance degradation.Consequently, it is apparent that existing solutions depending on GNSS in NTNs may be not practical for power limited devices in NB-IoT.Thus we focus here on UEs without GNSS and design a complete system for Random Access (RA).A system-level method based on the information from downlink synchronization is firstly applied to reduce the CFO to relatively small range.Then a pre-process method for received signal is designed.After anlayzing the relavance of its phase series and ToA or CFO, a Random Access scheme consists of a two-stage estimation method based on change point detection (CPD) is designed.
Here we use the autoencoders with a time-invariant representation (TIRE) CPD method.Finally, based on the estimation of ToA and CFO, 2-D-FFT preamble detection method is presented.In details, our main contributions in this paper are summarized as follows: 1) A pre-process method is designed at the receiver side.Through summing symbols and conjugate multiplication, the phase series of preamble signal after pre-process can be extracted.The periodicity of phase series reflects CFO and the location of each period reflects ToA. 2) A two stage estimation method based on change point detection is designed.The coarse estimation stage is based on the first two detected change points to precompensate the large propagation delay to a small scale.Then the fine estimation stage based on the change points in whole preamble signal after preprocess achieves accurate estimation of relatively small residual ToA and CFO. 3) Change point detection based on autoencoders with a time-invariant representation (TIRE) is applied to detect the change points of phase series for above two stage estimation method.It can be found that the slope of phase series changes abruptly at the junction between each period caused by phase ambiguity.TIRE method offers accurate detection ability of the slope change despite the data is not generated from parametric probability distribution which outperforms traditional CPD method.Besides, the sensitivity of detection performance to common noise can be mitigated during the process of feature learning by autoencoders.The remainder of this paper is summarized as follows.Section II models the complete system and describes the Random Access preamble structure and transmission of preamble in NTNs.Section III proposes the random access scheme based on change point detection of phase series.The simulation results are presented in Section IV and the conclusion of this paper is in Section V.

II. SYSTEM MODEL
The complete system design for NB-IoT Random Access in NTNs at the receiver side is illustrated in Fig. 1.The flowchart aims to provide readers with a rough framework of different steps which are described in detail in the following.To be specific, in step 2, a system level solution based on the information of initial downlink synchronization is used to reduce the large CFO to a relatively small scale [22].In most cases, the residual frequency offset is about 600Hz [23] which is still much larger than estimation range of existed studies [11], [12], [13], [14], [15], [16], [17], [18].Therefore, we consider the frequency uncertainty as 600Hz in the following analysis.And the detailed random access scheme is introduced in step 3-9.Then to clarify the context, in this section, random access preamble structure in NB-IoT is described in Section II-A and II-B models the transmission signal when integrating NB-IoT with NTNs.

A. RANDOM ACCESS PREAMBLE STRUCTURE
NB-IoT adopts a compatible sampling rate of 1.92MHz and the narrow operating bandwidth of 180KHz [24].Orthogonal Frequency Division Multiple Access (OFDMA) is used in NB-IoT downlink channels and for uplink access and communication.Random Access procedure is widely used to establish a wireless link and enable the data transmission between user equipments (UEs) and satellite base station.For RA in NB-IoT, a waveform with single tone based on frequency hopping pattern is designed as NB-IoT Random Access (RA) preamble transmitted on Narrowband Physical Random Access Channel (NPRACH).NPRACH supports two subcarrier spacings of 3.75kHz and 15kHz with 48 and 12 sub-carriers respectively.The preamble format design is introduced to reduce the peak-to-average power ratio (PAPR).When PAPR exceeds the threshold, a large amount of signal passing through the power amplifier will produce large nonlinear distortion, which affects the efficiency of the power amplifier.
In NB-IoT, the RA preamble is transmitted on Narrowband Physical Random Access Channel (NPRACH).TABLE 1 lists main parameters defined by protocols of NB-IoT.Format denotes three commonly used formats tagged by format 0 to format 2 for RA preamble in FDD mode.The starting time of NPRACH transmission is set by N start .Several symbol groups constitute a basic unit preamble and  the number of preamble units in whole RA procedure is determined by N rep .One symbol group (SG) consists of one cyclic prefix (CP) plus identical symbols.In frequency domain, the total number of sub-carriers is denoted as N sc .The initial location of sub-carrier for NPRACH is defined as N off .
The size of target cell determines the choice of three predefined formats with different sub-carrier spacing, CP length, SG length and symbol length.To illustrate, TABLE 2 shows the parameters of different preamble formats.f denotes the sub-carrier spacing and T sym denotes the length of one symbol.For instance, in format 1, a SG consists of one CP plus five identical symbols with length of 266.67 μs.Four SGs are combined as a basic unit.In frequency domain, each SG occupies one subcarrier with the bandwidth of 3.75KHz.
The frequency spacing between SGs follows a predefined hopping pattern and the frequency spacing from one unit preamble to another follows a pseudo-random selection principle summarized in [25].Symbol groups in preamble format 0 and 1 hop by one or six subcarriers in frequency, whereas symbol groups in format 2 (with 1.25-kHz SCS) hop by one, three, or eighteen subcarriers in frequency.In fact, the allocation of frequency resources in one unit preamble only depends on the subcarrier location of first SG.Fig. 2 shows an example of preamble structure in format 1.

B. PREAMBLE TRANSMISSION IN NTNs
In this work, we consider the NTNs reference scenario defined in TR 38.821 (Release 16) [26].Specifically, our system is operated in S-band at a carrier-frequency of 2 GHz, deployed at the altitude of 600km.
Then the baseband signal transmitted by NPRACH can be expressed as (2) After transmitting to the receiver on NPRACH, the n-th sample of p-th symbol of the received signal in NTNs can be written as where f off denotes the CFO normalized by sampling frequency and D denotes the ToA normalized by the symbol duration.w m,p (n) denotes the noise term and h m denotes the channel coefficient for the m-th SG.Here we consider the impact of Doppler rate.The changing rate of the frequency offset over time changes from f off to f off + α(n − D) with the presence of Doppler rate α normalized by squared sampling frequency.In this context, the received signal can be changed as The following assumptions are made based on the considered scenario: i) the terrestrial UEs are directly connected to LEO satellite.ii) CFO remains constant during the RA procedure.iii) The Doppler rate of each geographical location can be estimated by LEO satellite.Most researches model the transmission channel as Additive White Gaussian Noise (AWGN) channel [11], [12], [13], [14], [15], [16] and a few [17], [18] also consider Tapped Delay Line-C (TDL-C) channel.For a more comprehensive analysis, both two kinds of transmission channels are considered in the following analysis.

III. RANDOM ACCESS SCHEME BASED ON CHANGE POINT DETECTION
In this section, detailed random access theme (from step 3 to step 9 in Fig. 1) is introduced.

A. PRE-PROCESS OF RECEIVED SIGNAL
In this subsection, we deduce the expression of received signal after pre-process.Based on the above analysis, the received signal can be rewritten into another form (step 1): where S m,p (k) is the Discrete Fourier Transform (DFT) form of s m,p (n).Through removing CP and performing Fast Fourier Form (FFT), the received signal y m,p (n) can be transmitted to Y m,p (s): Only when s = N sc (m) (N sc (m) represents the location of sub-carrier occupied by m-th SG), S m,p (N sc (m)) = 1, while for other values of s, the expression is equal to 0. Then we get the expression ( 7) from ( 6): × e jπα (7) where Because the parameter α normalized by squared sampling frequency is far less than 1.So the expression containing α (i.e., e jπα ) can be neglected when received signal is analyzed.Then we combine the signal within the same m-th SG (Step 4): 1 − e j2π (foff +αmN g) LN   1 − e j2π (foff +αmN g) (8) As specified above, L denotes the number of symbol in one SG.Identically, here we still neglect the impact of Doppler rate α (i.e., e jπα[D 2 −2mN g D+(mN g ) 2 ] ).Then similar to [12], we multiply the m-th SG with the complex conjugated (m+1)th SG to get R m (Step 5): The phase series extracted from R m (Step 6) is expressed as In our scenario, the Doppler rate varies from 0 to −620 Hz [26], the parameter α normalized by squared sampling frequency is far less than f off normalized by sampling frequency.So α can be neglected when phase series is analyzed.The impact of w to phase series can be weakened by smoothing the mean of received signal.Then the phase series can be simplified as To better highlight the characteristic of this sequence, we replace N g with n.It can be seen that the phase series is a linear function of time sample n with the slope as 2π f off and intercept as 2π N sc (m) N D, manifested as periodic change of phase series in the range of [−π, π].

B. TWO STAGE ESTIMATION METHOD OF PHASE SERIES
In this subsection, we analyze the phase series and propose our two stage estimation method.After system level compensation in Step 2, the range of residual CFO needed to tackle is [−600 Hz,600 Hz].In order to prevent the linearity of phase series from being undermined in extreme cases, a fixed frequency offset can be added to received signal in Step 3.Here we set the predetermined frequency offset as 900Hz (600Hz for positive CFO and −600Hz for negative CFO).Then the residual CFO uncertainty turns to [(−1500,−900),(900,1500)]Hz.During the NPRACH process, based on the assumptions ii), the slope of phase series remains constant and the intercept changes with the SG.Fig. 3 shows the phase series with the ToA of 200 sample points and the CFO of 1500 Hz.
Next, we reveal that the information of ToA and CFO can be reflected in phase series.Determined by configuration parameters and formulation (11), the initial phase at the start of NPRACH transmission can be expressed as Then the ToA and CFO can be calculated as T ph denotes the period of phase series and n l denotes the timing location of the first period and the sign of π depends on the sign of CFO.Now the estimation of ToA and CFO refers to the estimation of T ph and n l .We use the change point detection method which would be introduced in following part to determine T ph and n l .Due to the 2π phase ambiguity, the solution to equation ( 13) is not unique, resulting several ToA candidates to be selected.A discrimination method based on Doppler rate [22] is utilized to select the true ToA among candidates.Because both the Doppler rate and ToA are related to the position of UE, the precise positioning information is not required.To illustrate, assuming the true ToA is 104.7μs and the solutions to formulation (13)  Then in ToA coarse estimation process (Step 7), the distance of first two detected change points is recorded as T ph .Coarse ToA values which are used to compensate large ToA can be calculated based on the T ph .Noting that the coarse T ph estimation would result in wrong ToA candidates selection, leading major ToA estimation error in coarse estimation process.After coarse estimation of ToA, the accurate CFO is estimated based on the whole preamble signal (Step 8).In order to eliminate the impact of different intercept of phase series in each SGs, we divide the phase series into segments with the length of one symbol group and then calculate the distances in each segments, which is shown in Fig. 4. The estimation of CFO relies on the average distances of change points in each segments and we use the Tree Sigma Guidelines as the pre-process scheme to exclude anomalous values of distances to increase estimation accuracy.Then fine estimation of ToA can be conducted.

C. CHANGE POINT DETECTION METHOD
In this subsection, we explain our specific change point detection method.In order to obtain better robustness to noise when detecting the change points accurately, we design a customized autoencoders network with a time-invariant representation (TIRE) [27] to detect slope change points of phase series rather than traditional CUSUM method.Here autoencoders are utilized to extract features of hidden layer from the consecutive windows.Based on the change points, the T ph and n l can be calculated accurately.
The structure of TIRE is shown in Fig. 5.We first segment phase series into consecutive windows with the length of N.And the features of hidden layer can be divided into time-invariant features s n and instantaneous features u n (16).h n denotes the encoded output from encoder layer of the n-th window.Invariant features refer to the statistical characteristics that change only when change point exists in consecutive windows.That means the differences of time-invariant features between time windows can manifest the abrupt change.The differences are summarized by the defined dissimilarity measure D N denotes the time-domain window size.The change points are located at the peaks of dissimilarity measure D n .In order to reduce the alarming rate, we exploit the prominence of peaks to determine the location of change points by comparing with a pre-defined threshold: In our context, we set the number of time invariant features as one in time domain which refers to the slope of phase series.Fig. 6 demonstrates the phase series and its corresponding dissimilarity measure.The blue lines represent the prominence of peaks.Change points can be detected when the prominence exceeds predefined threshold.

D. PREAMBLE DETECTION METHOD
Based on the estimation method of ToA (D * ) and CFO (f * off ) using phase series in Section III-C, we then apply 2-D-FFT detection method [28] to detect the preamble in this subsection.The 2-D-FFT formulation is as followed: Here, And Q denotes the splitting groups of preamble sequence.
Then we sum the squares of the amplitude of Consequently, with the estimation of ToA (D * ) and CFO (f * off ), the decision rule to determine whether RA preamble is correctly detected is the following: ξ denotes the predetermined threshold which can be obtained experimentally in practice.If the value of J[D * , f * off ] exceeds the preset threshold, the successful preamble detection is announced by base station.Otherwise, the base station announces that the preamble is not present.It can be concluded that the performance of preamble detection highly relies on the estimation accuracy of D * and f off .

IV. SIMULATION RESULTS
In this section, we firstly discuss the coarse estimation performance of ToA to determine the range of residual ToA through Monte Carlo simulations.Noting that after coarse estimation stage, the scale of residual ToA and CFO are same as which in TNs or NTNs with GNSS.Then residual ToA and CFO estimation results are presented in the fine estimation stage.Finally we present the preamble detection performance compared with the target detection rate.Here we compare our estimation method with Brute Force (BF) algorithm (designed for TNs) [11] and Stationary Discrete Wavelet Transform (S-DWT) algorithm (designed for NTNs with GNSS) [18].In order to ensure the fairness and rationality of comparison, the simulation results are compared in ToA fine estimation stage.The simulation parameters are listed in TABLE 3.

A. TOA COARSE ESTIMATION PERFORMANCE
The normalization of ToA and CFO is done on the symbol duration and sampling frequency.Table 4 reports the ToA coarse estimation performance in terms of absolute average estimation error and max estimation error.Based on the estimation performance, the range of residual ToA is set as [−100μs, 100μs] in the following simulation.It can be found that the absolute CFO estimation error in 99% cases is less than 1.92 Hz with SNR=3dB.However, the estimation error grows when decreasing the SNR.This can be solved by increasing the repetition of basic preamble units (N rep ).When N rep = 32, the max absolute CFO error is less than 4.5 Hz.The estimation accuracy of CFO grows with the N rep resulting from the amount of detected change points in phase series.The amount also increases with the growth of CFO.Hence, the estimation performance with large CFO can be guaranteed.

C. TOA FINE ESTIMATION PERFORMANCE
We simulate the ToA estimation process on two kinds of channels: AWGN channel and TDL-C channel.Here the Stationary Discrete Wavelet Transform (S-DWT) (designed for NTNs with GNSS) technique proposed in [18] and the Brute Force (BF) (designed for TNs) algorithm based on differential correlation [11] are compared with our method.In [18], the author exploits S-DWT to decompose the received signal into 8 levels.It can be found that the decomposed sequence y i follows two different distributions before and after time delay τ .Then the hypothesis of no change point in whole sequence is described as H 0 : θ = θ 0 for 1 ≤ i ≤ N and if one change point appears at i = τ , the hypothesis is presented as Based on cumulative sum (CUSUM) algorithm of change point detection, the log-likelihood ratio (LLR) of hypothesis reaches the maximum after the change.This algorithm is indicated as S-DWT in the following.In [11], the BF algorithm based on differential correlation detects the peak of cross correlation values.
In the following, Fig. 8-9 report the normalized ToA error with different SNR (SNR=3dB, SNR=0dB, SNR=−3dB) when the N rep is set as 8 on AWGN channel and TDL-C channel.It can be seen that the proposed method (CPD) outperforms the S-DWT and BF method both on AWGN channel and on TDL-C channel due to the steepest slope of CDF.The performance of differential correlation method is heavily ruined because of the presence of noise.The performance of S-DWT heavily depends on how well the actual data follows the assumed Gaussian distribution which can be proved by the simulation results that the estimation error of S-DWT increases apparently on TDL-C channel.Since we use the autoencoders with TIRE to detect the change points of phase series without advanced assumption of distribution, our method is applicable for various channels besides AWGN channel.
When the SNR decreases from 3dB to −3dB on AWGN channel and TDL-C channel, the curves follow the same trend.And the max absolute normalized ToA error changes from 2.1μs to 4.2μs on AWGN channel and from 2.7μs to 5.3μs on TDL-C channel which demonstrates the robustness against the noise.The whole estimation process is conducted in the time domain and both ToA and CFO are estimated in one process of change point detection in this method, which conserves the computational complexity of Discrete Fourier Transform (DFT).The structure of autoencoders with TIRE in this context can be pre-trained using the training data from actual preamble signals to speed up the uplink synchronization and reduce computational complexity.In addition, our method enlarges the estimation range of CFO compared   with existed methods.Thus, the results demonstrate that our method can achieve higher estimation accuracy and better overall performance compared with both BF and S-DWT method.

D. PREAMBLE DETECTION PERFORMANCE
Fig. 10-11 presents the correct preamble detection rate with the estimation ToA and CFO using 2-D-FFT detection method [28].We use Monte Carlo simulations to obtain detection results with N rep varying from 4 to 32.The red line in [28] denotes the target correct estimation rate (99%).When the correct preamble detection rate exceeds the target, a successful process of preamble detection is announced.It can be concluded that on AWGN channel (Fig. 10), when N rep = 4, the SNR that meets the requirements of correct estimation rate (99%) is -3.7dB.And the increase of N rep from 4 to 32 can bring a gain of 4dB.On TDL-C channel Fig. 11, the SNR which reaches the 99% estimation rate is 3.1dB when N rep = 4. Also, when N rep varies from 4 to 32, a gain of 5dB is obtained.The performance of preamble detection highly relies on the accuracy of ToA and CFO estimation.

V. CONCLUSION
In this paper, a complete system for NB-IoT random access is designed instead of exploiting GNSS system.After system-level pre-compensation, our random access scheme addresses residual ToA and CFO by analyzing the phase series of pre-processed received signal.The linearity of phase series allows us to identify the expression of ToA and CFO.Then a novel two stage estimation method (coarse estimation and fine estimation) based on TIRE is adopted in this context to detect the change points.Simulation results of the coarse estimation stage demonstrate that it can reduce the large ToA to a relatively small scale.Then through comparing with both BF and S-DWT method in ToA fine estimation stage, we can conclude that our estimation scheme obtains higher estimation accuracy.In addition, the performance of preamble detection reaches the target rate in most cases.These simulation results demonstrate the effectiveness and superiority of our theme when adopting NB-IoT in NTNs.
s m,p (n) denotes the n-th sample of the time domain waveform in p-th symbol of the m-th SG. S m,p [k] denotes the m-th symbol on the p-th subcarrier during the m-th SG.The range of sample point n belongs to [N m,p −N CP , N m,p +N−1], in which p ∈ [0, L − 1].L denotes the number of symbol in one SG and N m,p = mN g + pN, with N g = N CP + pN being the size of one SG, N CP denoting the length of CP and N denoting the length of one symbol.Based on [25], S m,p [k] = 1 k = N sc (m) 0 others , N sc (m) is the subcarrier index occupied by m-th SG, so the transmitted signal can be rewritten as

FIGURE 6 .
FIGURE 6. Correspondence of phase series and dissimilarity measure.

Fig. 7
Fig. 7 shows the cumulative distribution function (CDF) of normalized CFO error with different SNR (SNR=3dB, SNR=0dB, SNR=−3dB) when the N rep is set as 8 and the CDF of normalized CFO error with different N rep (N rep = 8, N rep = 16, N rep = 32) when SNR is set as −3dB.It can be found that the absolute CFO estimation error in 99% cases is less than 1.92 Hz with SNR=3dB.However, the estimation error grows when decreasing the SNR.This can be solved by increasing the repetition of basic preamble units (N rep ).When N rep = 32, the max absolute CFO error is less than 4.5 Hz.The estimation accuracy of CFO

FIGURE 8 .
FIGURE 8. CDF of normalized ToA estimation error on AWGN channel.

FIGURE 9 .
FIGURE 9. CDF of normalized ToA estimation error on TDL-C channel.

TABLE 2 . NPRACH preamble format.
are 104.7μs,371.3μs, 638.0μs with corresponding Doppler rate of −297 Hz/s, −252 Hz/s, −215 Hz/s respectively.If the Doppler estimation is −240 Hz/s, then correct ToA selection is 371.3μs because its Doppler rate is the nearest one.