In this work, we study the problem of Doppler shift and channel estimation for wireless communication systems on high-speed railways (HSRs). We focus on tunnel scenario, one of the classical scenarios of HSRs. We first build up the mathematical system model, design a joint Doppler shift and channel estimator, and compare its performance with the typical Moose algorithm. We show that our estimator outperforms the Moose algorithm in Doppler estimation. Besides, since wireless channels in tunnel scenarios often contain several or multiple taps, we suggest an adaptive frame structure to improve transmission efficiency. Simulations are then provided to corroborate our proposed studies.

1. Introduction

During the past two decades, Chinese government has obtained a world famous achievement: the fast development and deployment of high-speed rails (HSRs) [1]. By the end of 2020, almost 40,000 kilometers of HSRs have been built in the vast mainland of China. HSRs greatly boost the economy and significantly change the lives of people.

Accordingly, research and applications of wireless communications on HSRs arouse extensive interests [2, 3]. Different from other scenarios of wireless communication systems, the scenario of HSRs has three special features:(1)High mobility of train(2)Large penetration loss of the signals passing through train carriages [4](3)Access requirement of large users within a short period

The three special characteristics give rise to a series of research challenges for the wireless communications on HSRs, including channel modeling, Doppler shift compensation, time-varying channel estimation, fast handover, large and quick access methods, adaptive beamforming, and signal detection [1, 2, 5]. Besides, the applications and future visions of the 5th and 6th generation technologies on HSRs, such as massive multiple input multiple output (massive MIMO) [4], millimeter-wave (mmWave) [6], visible light communications, reconfigurable intelligent surfaces, newly emergent backscatter communications [7, 8], and smartly connected world, also bring about new open challenges for both academic research and engineering realization [9, 10].

High mobility will result in Doppler shifts that can cause channels to be fast time-varying [11, 12]. Doppler shifts can destroy the orthogonality of subcarriers of orthogonal frequency division multiplexing (OFDM) systems [13]. Therefore, estimation and compensation of Doppler shifts are of vital importance for wireless OFDM systems.

The most famous Doppler shift estimator is the Moose algorithm proposed by Moose in 1994 [14]. It requires the transmitter to send two identical OFDM symbols, and the receiver uses the difference of the received signals to estimate the CFO, that is, the Moose algorithm exploits the fact that each subcarrier in the frequency domain has the same phase frequency shift.

Based on the Moose algorithm, Schmidl and Cox presented another CFO estimator in 1997 [15]. They divided the frequency offset into fractional and integer multiples, used the relation of two identical parts of an OFDM symbol, and defined two symbols with a specific pilot format to estimate the fractional and integer multiple frequency offset separately.

In 1999, Morelli and Mengali proposed the Morelli-Mengali (MM) algorithm [16]. It supposes the transmitter transmits several same signals, and the receiver divides the received information into identical parts and then calculates the autocorrelation between the parts, so as to construct the linear relationship between the autocorrelation and the frequency offset. After that, the frequency offset is estimated by a specially designed best linear unbiased estimation algorithm.

If the Doppler shift cannot be compensated or there exists a difference between the transmitter oscillator and receiver oscillator, the channels will become time-varying [1721]. In such cases, time-varying channel estimators can be applied which transfer many channel parameters into a few parameters through basis expansion models (BEMs) [2224], autoregressive models [25, 26], array signal processing models [27, 28], or exploiting channel sparse feature [21, 29].

Tunnels are one of the classic scenarios of HSRs. It is known that the wireless channels in tunnel scenarios are rich of taps, that is, multiple wireless paths often exist in tunnels on HSRs [30]. Recently, it is revealed that if the base station (BS) is located in the middle of a tunnel, there are more wireless paths in the both ends of the tunnel than those in the middle of the tunnel [31]. Such feature arouses our interest and motivates this study.

In this study, we focus on the tunnel scenarios and investigate the Doppler shift and channel estimation problems for wireless OFDM communication systems. Specifically, we build up the mathematical system model, exploit the feature of wireless channels in tunnel scenarios, design a joint Doppler shift and channel estimator, and compare its performance with the typical Moose algorithm.

The rest of the study is organized as follows. Section 2 introduces the mathematical system model. Section 3 designs a joint Doppler shift and channel estimator. Section 4 provides simulation results to corroborate the proposed studies, and Section 5 summarizes the whole study.

Notations: vectors and matrices are boldface small and capital letters; the transpose, Hermitian, inverse, and pseudoinverse of the matrix are denoted by , , , and , respectively; denotes a diagonal matrix with the diagonal elements constructed from , denotes the statistical expectation, is the integer ceiling, and the entry indices of vectors and matrices start from 1.

2. System Model

Consider the wireless communication system in the tunnel, as shown in Figure 1. The system consists of a transmitter with one base station (BS) antenna and a receiver with one train antenna.

Suppose BS transmits OFDM symbols to the train antenna. Each OFDM symbol contains discrete signals , , and duration of each signal is . Define . After inverse Fourier transform, we can have , where represents the discrete Fourier transform matrix.

The BS modulates the signals to the carrier frequency and transmits them via the antenna, that is, the BS will transmit the analog signals:where denotes the initial phase of the transmitter oscillator [32].

Assume that the fading channels between the BS antenna and the train antenna is . The received signals at the train antenna can be obtained aswhere denotes the white Gaussian noise.

The first step for the receiver is to down-convert the signals to the baseband signals, which is given bywhere is the carrier frequency of the receiver oscillator, and denotes its initial phase.

Substituting (2) into (3), we can further obtain


In the case of , i.e., the transmitter and the receiver are of the same carrier frequency, and we can rewrite (4) as

Clearly, our goal is to estimate the Doppler shift and the channels , which is the focus of our next section.

Remark 1. The CFO is defined aswhich indicates that two factors can result in CFO: (1) the Doppler shift caused by movement of the transceiver; (2) the difference between the transmitter oscillator and the receiver oscillator.

3. Joint Doppler Shift and Channel Estimator

This section aims to design a joint Doppler shift and channel estimator to estimate and .

3.1. Channel Coherence Time

When the train speed is 360 km/hour, i.e., , we can find the maximum Doppler shift:where denotes the wavelength and . In the case of , we can find . Accordingly, the channel coherence time iswhich indicates that during the period of 0.5 ms, and the channels between the BS antenna and the train antenna can be considered as static or quasistatic.

3.2. Design of Joint Estimator

Suppose the channels contains taps, that is, the channels when .

After sampling, we can obtain (6) aswhere and

For conciseness, we assume and can further have


We can rewrite (12) aswhere is the circulant matrix.


It can be readily checked thatwhere

Since , we can rewrite (14) as

Substituting (17) into (19) and using and , we can obtain

Multiplying both sides of equation (20) with will producewhere


It is worth noting that the vector contains only one unknown parameter . We can havewhere .

Suppose the number of pilots in one OFDM symbol is and the pilot order in the OFDM symbol is . We choose the following symbols from (25) and construct the following new vectors and matrixes:

Therefore, we can havewhere .

It is worth noting that the matrix is square and orthogonal. Let us choose the first columns and rows of as

Then, we can obtain

Substituting (29) into (27), we can find

Accordingly, our estimator aims to obtain

It can be readily checked that

Subsequently, the Doppler shift can be estimated through one-dimensional search:

Next, we can estimate using (32).

With estimated , we can recover the matrix using (18).

Remark 2. The receiver will discard the cyclic prefix (CP) part as shown in Figure 2. It is worth noting that the length of the CP part can be changed so as to achieve adaptive transmission. In the places where less wireless paths exist, for example, the middle of the tunnel, the CP contains less symbols, while the CP will include more symbols in the case of multiple paths.

4. Simulation Results

This section provides numerical examples to evaluate the proposed studies. Here, we set , unless otherwise specified. We assume that the channel follows Rayleigh distribution.

We use the simulation platform CloudRT, a ray-tracing channel simulation platform, to generate the wireless channels in tunnels and investigate the feature of channels (CloudRT is a high performance computing (HPC) cloud-based platform. More details can be found in the website https://www.raytracer.cloud).

Figure 3 shows the channel taps of two scenarios: tunnels and open areas. Our BS is located in the middle of the tunnel, i.e., 500 m of the tunnel length. It can be seen from Figure 3 that the number of channels taps is more than 10 at both ends of the tunnel while only about 2 taps at the middle of the tunnel, which is a sharp contrast to the open area scenario that only has 2 or 3 paths.

Figure 4 shows the Doppler estimation MSEs of our estimator. For comparison, the MSE of the Moose algorithm is also provided. It can be seen from Figure 4 that our joint estimator outperforms the Moose algorithm and that more pilots can enhance the estimation accuracy.

Figure 5 shows the channel estimation MSEs of our estimator. We set number of pilots as 4, 8, 16, and 32 separately. It can be found from Figure 5 that more pilots can result in better estimation performance.

5. Conclusion

We focused on the tunnels, a classic scenario of HSRs, and investigated the Doppler shift and channel estimation problems. Specifically, we built up the mathematical model for the wireless OFDM system in tunnels, designed a joint Doppler shift and channel estimation algorithm, and compared its performance with the typical Moose algorithm. It is found that our estimator could outperform the Moose algorithm in Doppler estimation and that more pilots would achieve better estimation performance. Finally, simulations were provided to corroborate our proposed studies.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that there are no conflicts of interest.


This work was funded by Key-Area Research and Development Program of Guangdong Province (2018B010124001).