Joint Doppler Shift and Channel Estimation for UAV mmWave System with Massive ULA

Yanan Li,Yue Zhu,Tiankui Zhang,Dian Fan,*

1 The China Academy of Information and Communications Technology,Beijing 100191,China

2 School of Information and Communication Engineering,Beijing University of Posts and Telecommunications,Beijing 100876,China

3 Beijing key engineering laboratory of testing and validation for smart terminal and hardware

Abstract:A joint Doppler shift and channel estimation method for the millimeter-wave communication system of an unmanned aerial vehicle(UAV)equipped with a large-scale uniform linear antenna(ULA)array has been proposed.Since Doppler shift induces intercarrier interference,the parameters of the channel paths have been decomposed into the Doppler shift and the channel information.In order to obtain the Doppler shift,a new estimation algorithm based on a combination of discrete Fourier transform and phase rotation has been proposed,which can determine the appropriate number of antennas.In addition to estimating the channel information,a low-complexity joint Doppler shift and channel estimation method has been designed that can quickly obtain accurate estimates.Furthermore,the achievable sum rate,the theoretical bounds of the mean squared errors,and the Cram´er-Rao lower bounds of the estimation method have been derived.The analysis and simulation results prove that the performance of the proposed approach is close to the theoretical inference.

Keywords:ULA;Doppler shift estimation;channel estimation;achievable sum rate

I.INTRODUCTION

In recent years,owing to the increasing attention received by space resources,unmanned aerial vehicle(UAV)communication technology has developed rapidly.Due to their compact size and flexible configuration,UAVs play a vital role in environmental monitoring,disaster rescue,and emergency communications[1,2].Therefore,the wireless communication system of a UAV has gradually become a topic of immense interest in both academia and industry[3,4].

Since UAVs can quickly build short line-of-sight(LOS)links,they can provide wireless communication connections for unsignalized users[5,6],and can adaptively regulate the system transmission capacity to meet different transmission needs[7,8].The most important point is that the UAVs are located at high altitudes.This gives them more opportunities to establish LOS links with users than ground base stations[9],thus making them more cost-effective.

A combination of UAV communication with millimeter-wave(mmWave)-based large-scale antenna array technology can achieve higher system throughput[10–12].MmWave can provide larger spectral bandwidth and faster transmission rate,relieving the pressure of insufficient frequency band and frequency conflict[13–15].By using many-to-one redundant antennas,large-scale antenna arrays can increase the mmWave channel gain and reduce path loss,thus compensating for the vulnerability of mmWave to environmental effects[16,17].Therefore,the UAV mmWave communication system with massive multiple-input multiple-output(MIMO)has received widespread attention[18–20].

One key factor that influences the transmission performance of the UAV wireless communication systems is channel estimation.The accuracy of channel estimation directly affects the transmission reliability and communication quality[21,22],especially in high-speed mobile scenarios.When a UAV travels at high speed,object movement will cause spatial and temporal changes in the channel[23–25],thus generating a substantial Doppler shift.The Doppler shift may result in significant performance degradation for channel estimation,and this needs to be estimated and compensated.The common algorithms used for Doppler shift estimation include the correlation receiver estimation method based on the decision feedback equalizer[26],an estimation method based on fuzzy function[27],estimation methods based on channel correlation characteristics[28–30],etc.Unfortunately,the first two types of methods are generally only used for small Doppler shift changes.The latter methods are susceptible to the influence of channel parameters,which lead to increased complexity.

Many studies have also been conducted on UAV channel estimation.[31]estimated the initial channel information by using iterative cascade filters and proposed an algorithm based on sphere decoding to determine the relative position between the UAVs.[32]designed an angle estimation algorithm based on the UAV motion track and spatiotemporal correlation function,which balanced the unstable factors in the channel.According to the codebook structure of the antenna array and the Doppler effect generated by the movement of the UAV,the mmWave cellular network propagation channel estimation of UAVs was studied by[33].[34]proposed a new three-dimensional model of UAV mmWave communication system and performed channel estimation using it.In these studies on the channel estimation algorithms for UAVs,the scattering paths and the beam positions have been treated as the unknown information that needs to be calculated.Therefore,the complexities of these algorithms are greater,and the UAV systems will incur additional hardware costs and path energy.However,a system with too high complexity has no practical application significance.

With the aim of addressing the above mentioned issues,enhanced Doppler shift and channel estimation algorithms have been developed in this study for wireless communications in UAVs.According to the physical characteristics of the massive ULA,we decompose the channel parameters into Doppler shift related to DOA and channel information.A new Doppler shift estimation method based on the angle domain has been proposed to obtain the robust Doppler shift estimate and to determine the appropriate antenna allocation scheme.Through the proposed algorithm,the optimal number of antennas corresponding to different channel angles and the number of users can be calculated,so the respective antenna allocation schemes can be determined through a small amount of experiments.This algorithm uses angle rotation,which often relies on angle perception.When the angle perception is not sensitive,this algorithm may have estimation errors.Further,a low-complexity joint Doppler shift and channel estimation method has been designed,which utilizes the spatiotemporal correlation property of the channel matrix to reduce the effective channel dimension.

This paper focuses on the joint Doppler shift and channel estimation for a UAV mmWave system equipped with a massive uniform linear antenna(ULA).The main contributions of this study are as follows:

1.Because high-speed movement of UAV changes the angle parameter of the channel between users and UAV,a discrete Fourier transform(DFT)and phase rotation algorithm(i.e.,DFTPR algorithm)has been proposed to obtain the angle information and the Doppler shift.

2.Based on the UAV mmWave channel characteristics and the DFTPR algorithm,a high-throughput,low-cost antenna allocation scheme has been proposed.

3.In order to obtain an accurate joint Doppler shift and channel estimation,a low-complexity joint estimation algorithm,namely,the channel and Doppler orthogonal matching pursuit(CDOMP)algorithm,has been developed.

The remainder of the paper has been organized as follows: The system model and the channel model of UAV communication have been described in Section II.The joint Doppler shift and channel estimation algorithm has been described in Section III.The achievable sum rate(SR),the mean squared error(MSE)and the Cram´er-Rao lower bound(CRLB)have been analyzed in Section IV.Simulation results have been presented in Section V and conclusions drawn from this study have been given in Section VI.

II.SYSTEM MODEL

We consider a UAV mmWave wireless communication system where the UAV is equipped with a ULA ofNantennas.The adjacent antenna elements are spaced at half-wavelength,as shown in Figure.1.The UAV servesIsingle-antenna users simultaneously.We assume the channels between the UAV and the different users are flat fading channels.Here,consider the uplink channel between the UAV and the user in the frequency division duplex(FDD)system[35].

Figure 1.The system model of UAV mmWave wireless communication.

For the mmWave wireless communication system,the path loss factor is variable in different channel environments[36].In the LOS channel environment,its value fluctuates around 1.45 dB.Whereas in the nonline-of-sight(NLOS)channel environment,its range is 5 dB～10 dB[37].In addition,owing to the highaltitude operation of UAVs,the probability of scatterers around it is very small.Thus,there are only few mmWave scattering paths and the following discussion only considers the LOS paths[38].According to the Saleh-Valenzuela channel model[39],the channel between a UAV and the users can be expressed as

III.JOINT DOPPLER SHIFT AND CHANNEL ESTIMATION

In this section,for estimating the channel parameter between the UAV and the users,two algorithms,DFTPR and CDOMP,have been proposed in this study.The algorithm based on the angle domain can effectively reduce the complexity while maintaining high-precision DOA estimation[40].Therefore,the DFTPR algorithm can quickly obtain an initial Doppler shift estimate via the DFT transform and angular rotation of the channel matrix.Meanwhile,the DFTPR algorithm can determine the optimal number of antennas to maximize the link throughput.

However,DFTPR algorithm is based on the angular domain and its accuracy depends on the angular accuracy.Therefore,the CDOMP algorithm has been proposed for estimating the joint Doppler shift and channel parameters.The estimated channel can be obtained on the basis of the initial Doppler shift estimation according to Eq.(8).

3.1 Doppler Shift Estimation

In this section,a new Doppler shift estimation algorithm,designed for the ULA,has been described.From Eq.(5),it can be seen that the Doppler shift is related to the DOA.Therefore,the estimation of the Doppler shift can be transformed into the estimation of the DOA.Traditional DOA estimations,e.g.,multiple signal classification(MUSIC)and estimation of signal parameters via rotational invariance technique(ESPRIT)require subspace identification to obtain valid DOA[41,42].However,the principle of these subspace-based methods is eigen-decomposition,so for massive MIMO systems,the complexity is too high to be realized[43].In this paper,DFTPR uses DFT to process massive ULA structures and the coarse DOA can be obtained.After DFT,the precise DOA can be acquired by phase rotation.First,the DFT matrix

Algorithm 1.DFTPR based doppler shift estimation.Require: Aihi,FN,Ri Ensure: ˆβi,ˆfi 1: ni =argmax ni‖[FN]nAihi‖2 2: for ψ =-π N : 2π N2 : πN do 3: for l=1:L do 4:ψ =argmax ψ‖[FN]n[Ri]n[Ai]nhi‖2images/BZ_74_1743_814_1744_817.png5: end for 6: end for 7: Obtain ˆβi,ˆfi from Eq.(17)and Eq.(18)

3.2 CDOMP-based Channel Estimation

The algorithms used for solving the optimization model(Eq.(24))are generally divided into two categories: an algorithm that is converted to a convex optimization problem,such as the BP algorithm,and an algorithm based on a greedy iterative algorithm,such as an orthogonal matching pursuit(OMP)algorithm.

In order to reduce the complexity,the CDOMP algorithm based on OMP has been proposed in this work.It can use a shorter pilot sequence to perform a lowcomplexity channel estimation and is more accurate than the matching pursuit(MP)algorithm.

As shown in Algorithm 2,the CDOMP algorithm iteratively selects the atom(base signal)that is closest to the current residual value(i.e.,obtains the nearest value).When the final residual satisfies the constraint,the resultingis the last coefficient vector estimation.The complexity can be calculated asO(pMN),anddenotes the complexity of the BP algorithm.

The UAV wireless communication system adopts the FDD working mode,and thus it is necessary to estimate the uplink and downlink channels separately.In Section III,the Doppler shift and DOA of the uplink channel has been accurately estimated.According to the angular reciprocity of the antenna array theory,the DOA of the uplink and downlink channels are identical.Therefore,when estimating the downlink channel,only the channel gain parameter needs to be adjusted to obtain the estimated parameters.Such a downlink channel estimation method eliminates the redesigning step and greatly reduces the cost of channel estimation.

Algorithm 2.Channel and Doppler orthogonal matching pursuit algorithm.Require: Measurement vector γ with noise,dictionary matrix D,the maximum number of iterations p,error threshold ε to stop iteration Ensure: Doppler shift estimation ˆf,channel estimation ˆH 1: Initialization:Support set T =∅,coefficient vector w=0,residual r =γ,current iteration number p=0 2: for k =1:p do 3: for‖γ-Dw‖2l2＞ε do 4:j =argmax j〈r,Dj〉5:T =T ∪{j}6:w=D†T γ 7:r =γ-DT w 8: end for 9: end for 10: obtain ˆf from Eq.(4)and Eq.(8),and obtain ˆH from Eq.(8)

IV.PERFORMANCE ANALYSIS

4.1 Performance Analysis of the Achievable Sum Rate

The achievable SR index has been proposed in this work for measuring the accuracy of DFTPR algorithm.The reason is that the more accurate the Doppler shift estimation is,the higher the SR calculated based on it.The following is the detailed analysis.The SR can be expressed as

where SINRidenotes the signal to interference plus noise ratio.SINRican be expressed as

As is well known,under the premise of not losing the system SR,the fewer the number of antennas,the lower is the manufacturing cost and energy consumption of the system.Based on the above analysis,the optimal number of antennas corresponding to different angles and number of users situations is different.We can determine the corresponding antenna allocation schemes under different conditions based on a small amount of experiments,thereby improving the transmission efficiency of the system.Therefore,it is necessary to select the appropriate number of antennas.

4.2 Analysis of the Joint Doppler Shift and Channel Estimation

From(62)and(63),it can be seen that the MSE decreases as the SNR increases for the channel h andf.The error of the first part is related to the Doppler shift,whereas the error of the second part is related only to the channel noise.When the Doppler shift estimation is accurate,it reduces the estimation error of the channel.

4.3 CRLB Analysis

From the previous subsection,we know that the expectation of the Doppler shiftfand the channel h are zero and unbiased estimates.Further constraint estimation with the help of the CRLB is required[47].One way to determine the quality of an estimator is by comparing its MSE with the theoretical performance limit.If the MSE approaches the theoretical limit,it indicates that the estimator has good performance.

Because the CRLB is ML in the case of high SNR,the received signal after vectorization can be obtained from(35)

It can be seen that the CRLBs offand h are consistent with their corresponding variance and covariance matrices,indicating that the channel estimation MSEs based on CDOMP are nearly identical to those corresponding to the CRLBs in the case of a high SNR.

V.SIMULATION RESULTS

Numerical examples have been adopted to verify the validity of the joint Doppler shift and channel estimation algorithm.The UAV was equipped with a ULA array of half-wavelength spacing,and the number of antennas was preset toN.The sampling interval wastd= 5×10-6s,and the UAV traveled at the speed of 10km/h.Users were randomly distributed within the hotspot area.The ray-tracing method was used for modeling the 30 GHz mmWave channels.The joint Doppler shift and channel matrix of different users were formulated according to(1)and(8).The rotation angle was adjusted with accuracy as small as.The main simulation parameters are given in Table 1.

Table 1.Main simulation parameters.

Figure 2 shows the SRs performance of the DFTPR method for a different number of antennas with different users,for the SNR set to 10 dB and 20 dB.The angles between the UAV and users were[5.7◦,11.5◦]and[5.7◦,11.5◦,17.2◦,22.9◦]for two and four users respectively.It can be seen that the SR trends of the UAV system are cyclical and the difference ofNcorresponding to the adjacent peaks is roughly 20.This is because the presence of a specific node in the period can minimize the interference of other users.Furthermore,it can be seen that the SR corresponding to high SNR is larger than that corresponding to low SNR.In addition,asNgradually increases,the difference between the peaks and valleys of the SRs gradually decreases.This is because the higher the number of antennas,the finer is the angular accuracy of the signal reception and the higher is the sensitivity of the initial Doppler shift estimation method.Nevertheless,the peaks of SR do not differ much and increase infinitely.

Figure 3 plots the SRs of the Doppler shift estimation obtained using the DFTPR method and the real CSI as a function of the SNR,respectively.The channel gains were set toαi=1 for two users and the angles between UAV and users were[5.7◦,11.5◦].It can be seen that the performance of our proposed DFTPR method is very close to that of the real CSI,which verifies the accuracy of the estimation method.We chooseN= 128,N= 140,N= 192 for comparison,because these are special points calculated according to the set angle.That is,whenN ∈[128,192),the SR corresponding toN=128 is the largest.Furthermore,when there is only one user interference,SRN=128and SRN=192are straight lines,and both increase with a larger SNR.AsNis at a specific node,the system can completely eliminate the interference of another user by angle rotation.

Figure 4 indicates the performance of initial Doppler estimation in terms of the SRs obtained using the DFTPR method for four users,a case consisting of more users compared to the case shown in Figure.3.The angles between the UAV and the users were[5.7◦,11.5◦,17.2◦,22.9◦].It can be seen that as the SNR increases gradually,the SRs rapidly increase first and then progressively attain an almost constant value.This is because the method corresponding to a specific node cannot completely balance the interference of the remaining users.Therefore,when the SNR becomes higher,the signal strengths of other users will increase and the user interference will also increase,resulting in a larger percentage of useless signals.

Figure 5 compares the SRs of the proposed initial Doppler estimation method and the real CSI.It can be seen from the figure that for the sameN,in the case of a high SNR,our proposed Doppler shift estimation method performs slightly worse than that of the real bound.This is because when the radio frequency(RF)chain is in charge of a beam space,it can precisely adjust the signal phase and amplitude.However,when a single RF chain serves multiple beam spaces,the precision of its amplitude adjustment will be reduced.Interestingly,the SRN=103of the proposed method approximates to SRN=100of the real CSI,which indicates that our proposed method can strive to achieve a performance close to the real CSI by choosing the right value ofN.In summary,although the SR of the proposed estimation method is slightly reduced,the benefit achieved as a result of reducing the RF chains is more pronounced.

Figure 2.Performance of the initial Doppler shift estimation method in terms of the SRs for different number of antennas with SNR=10,I =2;SNR=20,I =2;SNR=10,I =4;SNR=20,I =4 respectively.

Figure 3.Comparison of the SRs of the real CSI and the DFTPR method with two users.

Figure 6 indicates the SRs of Doppler shift estimation as a function of SNR for the proposed method and beam-selection(BSE)method with different users.The special pointsN= 128 andN= 103 atI= 2 andI=4 are selected for comparison,respectively.It can be seen that the SR performance of the proposed method is better than that of the BSE method.Although BSE is also a Doppler shift estimation method based on the angle domain,it only selects the largest beam for channel transmission,which results in partial beam energy loss[48].In the case ofI=4,the difference between the two increases as the SNR increases.The proposed method can improve SR performance by aggregating energy through phase rotation,when the SNR becomes larger.In addition,compared to the case of I=2,the inter-user interference in the case of I=4 is more complicated,so the difference becomes larger.

Figure 4.The SRs obtained using the initial Doppler estimation method with four users.

Figure 7 illustrates the MSEs of channel estimation as a function of SNR for the CDOMP method,LS estimation,and CRLB.TheI= 20 users were evenly gathered into five spatially distributed clusters.The angles between the UAV and users of one group were set to[5.7◦,11.5◦,17.2◦,22.9◦],and the guard interval for user grouping was set to 5◦.From the figure,it can be seen that our proposed CDOMP algorithm performs slightly worse than CRLB but much better than the LS estimation method.Furthermore,whenNbecomes larger,the error of the estimation algorithm becomes smaller.This is because the smaller the proportion of the nonzero elements,i.e.,the greater the channel sparsity,the higher is the estimation accuracy of the multipath sparse channel.However,in order to obtain a higher benefit of the UAV system,we choose to slightly reduce the accuracy.That is,we obtain higher benefits by choosing the appropriate number of antennas,not the more antennas.

Figure 5.Comparison of the SRs corresponding to the real CSI and the proposed DFTPR method with four users.

Figure 6.Comparison of the SRs corresponding to the beam-selection method and the proposed DFTPR method with I =2;I =4,respectively.

Figure 7.Comparison of the MSEs of the CDOMP estimation and the LS estimation.

VI.CONCLUSION

In this paper,we proposed a joint Doppler shift and channel estimation method for the mmWave system of a UAV equipped with massive ULA.The joint estimation characteristics have been decomposed into the initial Doppler shift and joint information.The DFTPR algorithm was designed based on DFT and angle rotation,using which the initial Doppler shift estimation was achieved.A CDOMP algorithm based on CS was proposed,which could obtain the joint Doppler shift and channel estimation.Furthermore,the joint estimation algorithms have the advantage of low complexity and small hardware costs.To evaluate the validity of our proposed methods,the theoretical bounds of MSEs and CRLB of the joint Doppler shift and channel estimation in high SNR zone were estimated and found to be very close.

ACKNOWLEDGEMENT

This work was supported by National Natural Science Foundation of China(No.62101601,No.61971445).

NOTES

Small and upper bold-face letters denote vectors and matrices respectively;the superscripts(·)H,(·)T,(·)-1,(·)†stand for the conjugate-transpose,transpose,inverse,and pseudoinverse of a matrix respectively;[A]ijis the(i,j)th entry of A;Im{A}indicates the imaginary part of A;tr(A)indicates the trace of A;Diag(a)denotes a diagonal matrix with the diagonal element constructed from a;vec(·)stands for vectorization;E{A}denotes the statistical expectation.