# Joint Doppler Shift and Channel Estimation for High-Speed Railway Wireless Communications in Tunnel Scenarios

Authors: Zhigang Wang; Zihao Wang; Fusheng Zhu; Zezhou Luo; Fang Li; Haopeng Liu
Journal: Wireless Communications and Mobile Computing
 (2022)

---

## Abstract

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.


---

## Body

## 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 periodThe 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 [17–21]. In such cases, time-varying channel estimators can be applied which transfer many channel parameters into a few parameters through basis expansion models (BEMs) [22–24], 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. Section2 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 matrixA are denoted by AT, AH, A−1, and A†, respectively; diaga denotes a diagonal matrix with the diagonal elements constructed from a, E⋅ 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 Figure1. The system consists of a transmitter with one base station (BS) antenna and a receiver with one train antenna.Figure 1
System model.Suppose BS transmits OFDM symbols to the train antenna. Each OFDM symbol containsN discrete signals sn, 1≤n≤N, and duration of each signal is Ts. Define s=s0,s1,…,sN−1T. After inverse Fourier transform, we can have x=FHs, where F represents the discrete Fourier transform matrix.The BS modulates the signalsx to the carrier frequency fc0 and transmits them via the antenna, that is, the BS will transmit the analog signals:(1)bt=xtej2πfc0t+θ0,where θ0 denotes the initial phase of the transmitter oscillator [32].Assume that the fading channels between the BS antenna and the train antenna isht. The received signals at the train antenna can be obtained as(2)rt=htej2πfdt∗bt+wt=ht∗xtej2πfc0+fdt+θ0+wt,where wt denotes the white Gaussian noise.The first step for the receiver is to down-convert the signalsrt to the baseband signals, which is given by(3)yt=rte−j2πfc1t+θ1,where fc1 is the carrier frequency of the receiver oscillator, and θ1 denotes its initial phase.Substituting (2) into (3), we can further obtain(4)yt=ht∗xtej2πfc0−fc1+fdt+θ0−θ1+wt.Define(5)gt=htejθ0−θ1.In the case offc0=fc1, i.e., the transmitter and the receiver are of the same carrier frequency, and we can rewrite (4) as(6)yt=ej2πfdtgt∗xt+wt.Clearly, our goal is to estimate the Doppler shiftfd and the channels gt, which is the focus of our next section.Remark 1.
The CFO is defined as(7)v=fc0−fc1+fd,which indicates that two factors can result in CFO: (1) the Doppler shift fd caused by movement of the transceiver; (2) the difference fc0−fc1 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 estimatefd and gt.


### 3.1. Channel Coherence Time

When the train speedvt is 360 km/hour, i.e., vt=100m/s, we can find the maximum Doppler shift:(8)maxfd=vtλ=vtc/fc0,where λ denotes the wavelength and c=3∗108m/s. In the case of fc0=3∗109, we can find maxfd=1000Hz. Accordingly, the channel coherence time Tc is(9)Tc=12maxfd=5×10−4,which 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 channelsgt contains L taps, that is, the channels gt=0 when t≥LTs.After sampling, we can obtain (6) as(10)ynTs=yt|t=nTs=ej2πfdnTs∑l=0L−1glxnTs−lTs+wnTs,where 0≤n≤N−1 and(11)glTs=gl,0≤l≤L−1.For conciseness, we assumeTs=1 and can further have(12)yn=ej2πfdn∑l=0L−1glxn−l+wn.Define(13)y=y0,y1,⋅,yN−1T,x=x0,x1,⋅,xN−1T,w=w0,w1,⋅,wN−1T,Γfd=diag1,ej2πfd,ej2π2fd,…,ej2πN−1fd.We can rewrite (12) as(14)y=ΓfdGx+w,where G is the circulant matrix.(15)G=g00⋯0gL−1⋯g1g1g0⋯00⋯g2⋮⋮⋯⋮⋮⋯⋮gL−1gL−2⋯g00⋯00gL−1gL−2⋯g0⋯0⋮⋮⋯⋮⋮⋯⋮00gL−1gL−2⋯⋯g0.Suppose(16)g=g0,g1,…,gL−1T,g¯=gT,0,0,…,0︸N−LT.It can be readily checked that(17)G=FHDF,where(18)D=diagd=diagFg.SinceΓfdΓ−fd=I, we can rewrite (14) as(19)Γ−fdy=Gx+Γ−fdw.Substituting (17) into (19) and using FHF=I and x=FHs, we can obtain(20)Γ−fdy=FHDFx+Γ−fdw=FHDFFHs+Γ−fdw=FHDs+Γ−fdw.Multiplying both sides of equation (20) with F will produce(21)FΓ−fdy=Ds+w˜,where(22)w˜=FΓ−fdw.Define(23)z=FΓ−fdy.It is worth noting that the vectorz contains only one unknown parameter fd. We can have(24)z=Ds+w˜=Sd+w˜,(25)=SFg+w˜,where d=diagD=Fg.Suppose the number of pilots in one OFDM symbol isM and the pilot order in the OFDM symbol is p0,p1,…,pM−1. We choose the following symbols from (25) and construct the following new vectors and matrixes:(26)zp=zp0,zp1,…,zpM−1T,sp=sp0,sp1,…,spM−1T,dp=dp0,dp1,…,dpM−1T,w˜p=w˜p0,w˜p1,…,w˜pM−1T.Therefore, we can have(27)zp=Spdp+w˜p,where Sp=diagsp.It is worth noting that the matrixF is square and orthogonal. Let us choose the first L columns and M rows of F as Fp(28)Fp=F1:M,1:L.Then, we can obtain(29)dp=Fpg.Substituting (29) into (27), we can find(30)zp=SpFpg+w˜p.Accordingly, our estimator aims to obtain(31)f^d,g^=argminzp−SpFpg2.It can be readily checked that(32)g^=FpHSpSpFp−1FpHSpzp.Subsequently, the Doppler shiftfd can be estimated through one-dimensional search:(33)f^d=argminzp−SpFpFpHSpSpFp−1FpHSpzp2.Next, we can estimateg using (32).With estimatedg^, we can recover the matrix using (18).Remark 2.
The receiver will discard the cyclic prefix (CP) part as shown in Figure2. 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.Figure 2
Structure of one OFDM symbol.


## 3.1. Channel Coherence Time

When the train speedvt is 360 km/hour, i.e., vt=100m/s, we can find the maximum Doppler shift:(8)maxfd=vtλ=vtc/fc0,where λ denotes the wavelength and c=3∗108m/s. In the case of fc0=3∗109, we can find maxfd=1000Hz. Accordingly, the channel coherence time Tc is(9)Tc=12maxfd=5×10−4,which 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 channelsgt contains L taps, that is, the channels gt=0 when t≥LTs.After sampling, we can obtain (6) as(10)ynTs=yt|t=nTs=ej2πfdnTs∑l=0L−1glxnTs−lTs+wnTs,where 0≤n≤N−1 and(11)glTs=gl,0≤l≤L−1.For conciseness, we assumeTs=1 and can further have(12)yn=ej2πfdn∑l=0L−1glxn−l+wn.Define(13)y=y0,y1,⋅,yN−1T,x=x0,x1,⋅,xN−1T,w=w0,w1,⋅,wN−1T,Γfd=diag1,ej2πfd,ej2π2fd,…,ej2πN−1fd.We can rewrite (12) as(14)y=ΓfdGx+w,where G is the circulant matrix.(15)G=g00⋯0gL−1⋯g1g1g0⋯00⋯g2⋮⋮⋯⋮⋮⋯⋮gL−1gL−2⋯g00⋯00gL−1gL−2⋯g0⋯0⋮⋮⋯⋮⋮⋯⋮00gL−1gL−2⋯⋯g0.Suppose(16)g=g0,g1,…,gL−1T,g¯=gT,0,0,…,0︸N−LT.It can be readily checked that(17)G=FHDF,where(18)D=diagd=diagFg.SinceΓfdΓ−fd=I, we can rewrite (14) as(19)Γ−fdy=Gx+Γ−fdw.Substituting (17) into (19) and using FHF=I and x=FHs, we can obtain(20)Γ−fdy=FHDFx+Γ−fdw=FHDFFHs+Γ−fdw=FHDs+Γ−fdw.Multiplying both sides of equation (20) with F will produce(21)FΓ−fdy=Ds+w˜,where(22)w˜=FΓ−fdw.Define(23)z=FΓ−fdy.It is worth noting that the vectorz contains only one unknown parameter fd. We can have(24)z=Ds+w˜=Sd+w˜,(25)=SFg+w˜,where d=diagD=Fg.Suppose the number of pilots in one OFDM symbol isM and the pilot order in the OFDM symbol is p0,p1,…,pM−1. We choose the following symbols from (25) and construct the following new vectors and matrixes:(26)zp=zp0,zp1,…,zpM−1T,sp=sp0,sp1,…,spM−1T,dp=dp0,dp1,…,dpM−1T,w˜p=w˜p0,w˜p1,…,w˜pM−1T.Therefore, we can have(27)zp=Spdp+w˜p,where Sp=diagsp.It is worth noting that the matrixF is square and orthogonal. Let us choose the first L columns and M rows of F as Fp(28)Fp=F1:M,1:L.Then, we can obtain(29)dp=Fpg.Substituting (29) into (27), we can find(30)zp=SpFpg+w˜p.Accordingly, our estimator aims to obtain(31)f^d,g^=argminzp−SpFpg2.It can be readily checked that(32)g^=FpHSpSpFp−1FpHSpzp.Subsequently, the Doppler shiftfd can be estimated through one-dimensional search:(33)f^d=argminzp−SpFpFpHSpSpFp−1FpHSpzp2.Next, we can estimateg using (32).With estimatedg^, we can recover the matrix using (18).Remark 2.
The receiver will discard the cyclic prefix (CP) part as shown in Figure2. 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.Figure 2
Structure of one OFDM symbol.


## 4. Simulation Results

This section provides numerical examples to evaluate the proposed studies. Here, we setN=512, unless otherwise specified. We assume that the channel gl 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 websitehttps://www.raytracer.cloud).Figure3 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 3
Channel taps of tunnels and open area.Figure4 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 4
Estimation MSE of the Doppler shift.Figure5 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.Figure 5
Estimation MSE of the channels.


## 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.



---

*Source: 6804412-2022-01-31.xml*