# Robust Downlink Transmission for 6G LEO-MIMO Satellite Systems

Authors: Jiamin Li; Zifeng Zhang; Rui Liu; Chulong Liang; Pengcheng Zhu; Dongming Wang
Journal: Wireless Communications and Mobile Computing
 (2022)

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## Abstract

In this paper, we investigate downlink transmission schemes robust to the Doppler frequency offset (DFO) induced by high mobility for low earth orbit multiple-input and multiple-output satellite communication systems. Considering the impact of DFO, we derive the closed-form expressions for the downlink achievable rates with maximum ratio transmission (MRT) and maximum average signal-to-leakage-plus-noise ratio (MASLNR) precoding. Numerical results show that the derived closed-form expressions are accurate and the DFO severely affects system performance. In order to relieve the impact of DFO, we propose an adaptive DFO compensation algorithm based on the precompensation design and beam alignment. Numerical results verify the effectiveness of the proposed DFO compensation algorithm.


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## Body

## 1. Introduction

In recent years, the application of satellite communication has been increasing. In the 6G space-air-earth integrated network, a low earth orbit (LEO) satellite communication network will be combined with a ground communication network to provide wide coverage and high dynamic. However, there are many challenges in LEO multiple-input and multiple-output (LEO-MIMO) satellite communication systems. Firstly, due to the long distance of transmission links, it always results in large channel attenuation. Then Doppler frequency offset (DFO) inevitably causes intersymbol interference (ISI) and fast time variation of the channel [1]. Therefore, the receiver is difficult to obtain the instantaneous channel state information (iCSI) [2]. Many works have been conducted toward the development of DFO estimation. In [3], a geographic method to estimate DFO in a condition that satellites communicate with fixed earth stations was proposed. Yet, it assumed the angular velocity of the satellite relative to the user is constant, which was unrealistic. In [4], by adding a cyclic prefix (CP) and training data in orthogonal frequency-division multiplexing (OFDM) systems, a high-dynamic DFO estimation approach was presented. But the estimation range of the algorithm was less than half of a subcarrier interval. Some works improved the accuracy of estimation by adding CP [5], which can be applied to the scenario of large frequency offset. In [6], frequency offset and the Doppler shift joint estimation algorithm in high mobility environment based on orthogonal angle domain subspace projection was proposed. Paper [7] discussed a maximum frequency shift estimation algorithm based on the level pass rate, which needs loss information of transmission process to calculate the average power of the signal. However, the solutions discussed above cannot be substantially applied to high-speed systems. Thus, to achieve better performance in LEO-MIMO satellite systems, a lightweight and effective adaptive DFO compensation algorithm is needed.MIMO has been widely applied in the terrestrial wireless network. MIMO can substantially increase the number of antennas and spatial resolution and achieve a high transmission rate. However, incorporating MIMO into satellite communication systems is still a challenge.In this paper, we investigate robust downlink transmission for LEO-MIMO satellite systems. Taking DFO into account, we analyze the SE performance of the system and design an adaptive DFO compensation algorithm. The contributions of this work are summarized as follows.(i)
Considering the DFO effects, we derived the approximation expressions of achievable rate with maximum ratio transmission (MRT) and maximum average signal-to-leakage-plus-noise ratio (MASLNR) precoding. The SE of LEO-MIMO satellite communication systems is analyzed(ii)
We propose an adaptive DFO compensation algorithm. Based on information of orbit, user location, and velocity, precompensation is performed in the transmitter. In order to improve the estimation accuracy, we adopt the beam alignment (BA) algorithm for fine compensation in the receiver(iii)
Simulation results verified the correctness of approximation expressions, and the average system SE can be improved after using the proposed compensation algorithm in LEO-MIMO satellite systemsThe remainder of this paper is organized as follows. Section2 introduces the system model including system configuration and satellite channel model. Section 3 considers the DFO effects, deriving the approximation expressions of achievable rate with MRT and MASLNR precoding. Section 4 proposes an adaptive DFO compensation algorithm in LEO-MIMO satellite systems. Representative numerical results are given in Section 5 before we conclude the paper in Section 6.


### 1.1. Notations

The following notations are used. All boldface letters stand for vectors (lower case) or matrices (upper case). Italic letters (e.g.,X or x) denote scalars. The transpose and Hermitian transpose are denoted by ·T and ·H, respectively. ℂm×n denotes the set of m×n complex-valued matrices. x is the absolute value of a scalar x, and X is the spectral norm of a matrix X. x~CN0,σ2 means that x is a circularly symmetric complex Gaussian random variable with mean zero and variance σ2. E· denotes the expectation operator.


## 1.1. Notations

The following notations are used. All boldface letters stand for vectors (lower case) or matrices (upper case). Italic letters (e.g.,X or x) denote scalars. The transpose and Hermitian transpose are denoted by ·T and ·H, respectively. ℂm×n denotes the set of m×n complex-valued matrices. x is the absolute value of a scalar x, and X is the spectral norm of a matrix X. x~CN0,σ2 means that x is a circularly symmetric complex Gaussian random variable with mean zero and variance σ2. E· denotes the expectation operator.


## 2. System Model

We consider a LEO-MIMO satellite communication system whereNu single-antenna ground users (GUs) simultaneously are serviced by a satellite which is equipped with uniform linear array (ULA) composed of M antennas in Figure 1. All GUs are distributed within the coverage of the satellite. Due to different GUs are usually spatially separated by a few wavelengths, different GUs’ channels are assumed to be independent [8].Figure 1
System model.


### 2.1. Channel Model

The multipath channel model between satellite andk-th user can be given by
(1)Gkdl=∑p=0Pk−1gk,pdlej2πvksat+vkut,atOvkut,aakφk,where Pk represents the total number of channel propagation paths for k-th user, gk,pdl is the complex gain of the p-th path, akφk=1ej2πd/λcosφk⋯ej2πd/λM−1cosφkT is the array response vector, φk represents the ideal angle of department (AOD), λ is wavelength, and d is antenna spacing which is set as (λ/2). M represents the number of antennas.The overall DFO can be divided into two independent parts, DFO at satellitevksat and DFO at GUs vkut. vkut is further divided into absolute DFO vkut,a=fcvk/c caused by the long propagation distance and relative DFO caused by antenna spacing. The influence of GUs’ relative DFO can be expressed as a diagonal matrix:
(2)Ovkut,a=diag1ej2πvkut,aΔt⋯ej2πM−1vkut,aΔt,where fc is carrier frequency and vk is velocity of k-th GU. Then, the downlink channel model can be rewritten as
(3)Gkdl=∑p=0Pk−1gk,pdlej2πvksat+vkut,atakφk′,where φk′=arccos1+vk/ccosφkcosφk represents the deviated AOD, which is caused by relative DFO. In practice, the deviation of AOD will also take a negative impact on the AOA of receive signal in GUs.


## 2.1. Channel Model

The multipath channel model between satellite andk-th user can be given by
(1)Gkdl=∑p=0Pk−1gk,pdlej2πvksat+vkut,atOvkut,aakφk,where Pk represents the total number of channel propagation paths for k-th user, gk,pdl is the complex gain of the p-th path, akφk=1ej2πd/λcosφk⋯ej2πd/λM−1cosφkT is the array response vector, φk represents the ideal angle of department (AOD), λ is wavelength, and d is antenna spacing which is set as (λ/2). M represents the number of antennas.The overall DFO can be divided into two independent parts, DFO at satellitevksat and DFO at GUs vkut. vkut is further divided into absolute DFO vkut,a=fcvk/c caused by the long propagation distance and relative DFO caused by antenna spacing. The influence of GUs’ relative DFO can be expressed as a diagonal matrix:
(2)Ovkut,a=diag1ej2πvkut,aΔt⋯ej2πM−1vkut,aΔt,where fc is carrier frequency and vk is velocity of k-th GU. Then, the downlink channel model can be rewritten as
(3)Gkdl=∑p=0Pk−1gk,pdlej2πvksat+vkut,atakφk′,where φk′=arccos1+vk/ccosφkcosφk represents the deviated AOD, which is caused by relative DFO. In practice, the deviation of AOD will also take a negative impact on the AOA of receive signal in GUs.


## 3. Performance Analysis

In this section, we analyze the downlink spectral efficiency in LEO-MIMO satellite communication systems with DFO.


### 3.1. Channel Estimation with DFO

To further investigate the statistical properties of line-of-sight (LoS) path and non-LOS (NLoS) propagation, the channel gainαk=∑p=0Pk−1gk,pdlej2πvksat+vk,put,at exhibits the Rician fading distribution with Rician factor Kk and power Ωk [9]. Thus, the multipath channel without GUs’ relative DFO effects gkdl=αkakφk can be regarded as a large-scale fading coefficient and Rice model. The channel model (1) can be rewritten as
(4)Gkdl,D=Ovkut,agkdl,where Ovkut,a represents GUs’ relative DFO effects matrix:
(5)gkdl=βkhk.βk represents the large-scale fading part, and the small-scale fading part hk can be given by
(6)hk=γkKkKk+1hk+1Kk+1hw,k,where Kk represents the Rice factor of the k-th user, γk=Ehk2 represents the average channel power, hk represents the line-of-sight (LoS) component which is equal to aφk, and hw,k represents non-LoS (NLoS) component, the real part and imaginary part which is subject to the cyclic complex Gaussian random process with zero mean and variance of 1/2 independently.We use pilot-assisted minimum mean square error (MMSE) channel estimation for the NLoS component ofgkdl and consider DFO effects on LoS component of gkdl. The estimated channel can be given by
(7)Gkdl,D=Ovkut,agkdl=βkγkKkKk+1Ovkut,ahk+1Kk+1hw,k.hw,k represents the estimated channel of NLoS component. ξkdl=Gkdl,D−Gkdl,D is channel estimation error and its variance σerror2=Eξk2 is as follows [10].
(8)σerror2=βk1+ppβkKk+1,where pp is the transmitting power of the pilot signal.


### 3.2. Downlink Spectral Efficiency Analysis with DFO

The received signal at userk is expressed as
(9)rkdl=pkdlGkdl,DHbkdlskdl+∑i≠k∩i∈KpidlGkdl,DHbidlsidl−∑i∈KpidlξkdlHbidlsidl+zkdl,where pkdl represents the transmitting power of the k-th downlink user, bkdl represents downlink precoding, skdl is transmitting signal, and zkdl~CN0,1 represents noise item. We assume that GUs only use statical channel state information (sCSI) for signal decoding. Thus, the downlink achievable SE with DFO is expressed as
(10)Rdl,D=∑k∈Klog21+SINRkdl,D,where
(11)SINRkdl,D=EGkdl,DHbkdl2∑i≠ki∈Kpidl/pkdlEGkdl,DHbidl2+∑i∈Kpidl/pkdlEbidl2σerror2+1/ρdl,and ρdl is the downlink signal-to-noise ratio.In this paper, we adopt the MRT precoding and MASLNR precoding. For MRT precoding(12)bkmrt,D=Gkdl,D/Gkdl,D,we have the following theorem.Theorem 1.
With MRT precoding, the approximation expression of downlink achievable rate with DFO is given by(13)Rmrt,D=∑k∈Klog21+SINRkmrt,D,where
(14)SINRkmrt,D=βk2ζk2I1∑i≠ki∈Kpidl/pkdlβkβiζkζiI2+∑i∈Kpidl/pkdlζiTkσerror2+Tk/ρdl,Tk=βkMKkKk+1+MηkKk+1,(15)I1=βk2Kk+12M2Kk2+2MKk+2M2Kkηk+M2+Mηk2,(16)I2=βkβiKk+1Ki+1KkKiΦk,i2+MKkηi+MKiηk+Mηkηi,
InTk=βkMKk/Kk+1+Mηk/Kk+1, ζk denotes the DFO effects item, and I1 and I2 are represented by (15) and (16).Proof.
Please see AppendixA.When DFO effects are eliminated, we can obtain the approximation expression of a downlink achievable rate without DFO:(17)Rmrt,ND=∑k∈Klog21+SINRkmrt,ND,where
(18)SINRkmrt,ND=βk2I1∑i≠ki∈Kpidl/pkdlβkβiI2+∑i∈Kpidl/pkdlTkσerror2+Tk/ρdl.As shown in (3), the GUs’ relative DFO causes the deviation of angle in array response vector. Considering the DFO effects, the MASLNR precoding can be given by Proposition 1 in [10].
(19)bkmaslnr,D=1ηkDΛD−1aφk′,where
(20)ηkD=ΛD−1aφk′,(21)ΛD=∑iβiΩiaφi′aφi′H+1ρdlIM,and we have the following theorem.Theorem 2.
With MASLNR precoding, the approximation expression of downlink achievable rate with DFO is given by(22)Rmaslnr,D=∑k∈Klog21+SINRkmaslnr,D,where
(23)SINRkmaslnr,D=βkΩkΓk,k∑i≠k,i∈KβiΩiΓk,k+ηkD2/ρdl,(24)Γk,k=1∑m=0M−1Re2+Im2,(25)Re=∑i∈KβiΩicos2πm−1cosφk′+1ρdl,(26)Im=∑i∈KβiΩisin2πm−1cosφk′.Proof.
Please see AppendixB.When DFO effects are compensated absolutely, we can obtain the approximation expression of the downlink achievable rate without DFO.(27)Rmaslnr,ND=∑k∈Klog21+SINRkmaslnr,ND,where
(28)SINRkmaslnr,ND=βkΩkΓk,k′∑i≠k,i∈KβiΩiΓk,k′+ηkD2/ρdl,(29)Γk,k′=1∑m=0M−1Re′2+Im′2,(30)Re′=∑i∈KβiΩicos2πm−1cosφk+1ρdl,(31)Im′=∑i∈KβiΩisin2πm−1cosφk.


## 3.1. Channel Estimation with DFO

To further investigate the statistical properties of line-of-sight (LoS) path and non-LOS (NLoS) propagation, the channel gainαk=∑p=0Pk−1gk,pdlej2πvksat+vk,put,at exhibits the Rician fading distribution with Rician factor Kk and power Ωk [9]. Thus, the multipath channel without GUs’ relative DFO effects gkdl=αkakφk can be regarded as a large-scale fading coefficient and Rice model. The channel model (1) can be rewritten as
(4)Gkdl,D=Ovkut,agkdl,where Ovkut,a represents GUs’ relative DFO effects matrix:
(5)gkdl=βkhk.βk represents the large-scale fading part, and the small-scale fading part hk can be given by
(6)hk=γkKkKk+1hk+1Kk+1hw,k,where Kk represents the Rice factor of the k-th user, γk=Ehk2 represents the average channel power, hk represents the line-of-sight (LoS) component which is equal to aφk, and hw,k represents non-LoS (NLoS) component, the real part and imaginary part which is subject to the cyclic complex Gaussian random process with zero mean and variance of 1/2 independently.We use pilot-assisted minimum mean square error (MMSE) channel estimation for the NLoS component ofgkdl and consider DFO effects on LoS component of gkdl. The estimated channel can be given by
(7)Gkdl,D=Ovkut,agkdl=βkγkKkKk+1Ovkut,ahk+1Kk+1hw,k.hw,k represents the estimated channel of NLoS component. ξkdl=Gkdl,D−Gkdl,D is channel estimation error and its variance σerror2=Eξk2 is as follows [10].
(8)σerror2=βk1+ppβkKk+1,where pp is the transmitting power of the pilot signal.


## 3.2. Downlink Spectral Efficiency Analysis with DFO

The received signal at userk is expressed as
(9)rkdl=pkdlGkdl,DHbkdlskdl+∑i≠k∩i∈KpidlGkdl,DHbidlsidl−∑i∈KpidlξkdlHbidlsidl+zkdl,where pkdl represents the transmitting power of the k-th downlink user, bkdl represents downlink precoding, skdl is transmitting signal, and zkdl~CN0,1 represents noise item. We assume that GUs only use statical channel state information (sCSI) for signal decoding. Thus, the downlink achievable SE with DFO is expressed as
(10)Rdl,D=∑k∈Klog21+SINRkdl,D,where
(11)SINRkdl,D=EGkdl,DHbkdl2∑i≠ki∈Kpidl/pkdlEGkdl,DHbidl2+∑i∈Kpidl/pkdlEbidl2σerror2+1/ρdl,and ρdl is the downlink signal-to-noise ratio.In this paper, we adopt the MRT precoding and MASLNR precoding. For MRT precoding(12)bkmrt,D=Gkdl,D/Gkdl,D,we have the following theorem.Theorem 1.
With MRT precoding, the approximation expression of downlink achievable rate with DFO is given by(13)Rmrt,D=∑k∈Klog21+SINRkmrt,D,where
(14)SINRkmrt,D=βk2ζk2I1∑i≠ki∈Kpidl/pkdlβkβiζkζiI2+∑i∈Kpidl/pkdlζiTkσerror2+Tk/ρdl,Tk=βkMKkKk+1+MηkKk+1,(15)I1=βk2Kk+12M2Kk2+2MKk+2M2Kkηk+M2+Mηk2,(16)I2=βkβiKk+1Ki+1KkKiΦk,i2+MKkηi+MKiηk+Mηkηi,
InTk=βkMKk/Kk+1+Mηk/Kk+1, ζk denotes the DFO effects item, and I1 and I2 are represented by (15) and (16).Proof.
Please see AppendixA.When DFO effects are eliminated, we can obtain the approximation expression of a downlink achievable rate without DFO:(17)Rmrt,ND=∑k∈Klog21+SINRkmrt,ND,where
(18)SINRkmrt,ND=βk2I1∑i≠ki∈Kpidl/pkdlβkβiI2+∑i∈Kpidl/pkdlTkσerror2+Tk/ρdl.As shown in (3), the GUs’ relative DFO causes the deviation of angle in array response vector. Considering the DFO effects, the MASLNR precoding can be given by Proposition 1 in [10].
(19)bkmaslnr,D=1ηkDΛD−1aφk′,where
(20)ηkD=ΛD−1aφk′,(21)ΛD=∑iβiΩiaφi′aφi′H+1ρdlIM,and we have the following theorem.Theorem 2.
With MASLNR precoding, the approximation expression of downlink achievable rate with DFO is given by(22)Rmaslnr,D=∑k∈Klog21+SINRkmaslnr,D,where
(23)SINRkmaslnr,D=βkΩkΓk,k∑i≠k,i∈KβiΩiΓk,k+ηkD2/ρdl,(24)Γk,k=1∑m=0M−1Re2+Im2,(25)Re=∑i∈KβiΩicos2πm−1cosφk′+1ρdl,(26)Im=∑i∈KβiΩisin2πm−1cosφk′.Proof.
Please see AppendixB.When DFO effects are compensated absolutely, we can obtain the approximation expression of the downlink achievable rate without DFO.(27)Rmaslnr,ND=∑k∈Klog21+SINRkmaslnr,ND,where
(28)SINRkmaslnr,ND=βkΩkΓk,k′∑i≠k,i∈KβiΩiΓk,k′+ηkD2/ρdl,(29)Γk,k′=1∑m=0M−1Re′2+Im′2,(30)Re′=∑i∈KβiΩicos2πm−1cosφk+1ρdl,(31)Im′=∑i∈KβiΩisin2πm−1cosφk.


## 4. Adaptive DFO Compensation Algorithm

In this section, we propose an adaptive compensation algorithm based on the historical data. Noting that due to the high-speed feature of LEO satellites, only one-stage estimation and compensation cannot guarantee the quality of service (QoS) requirement of different scenarios [11]. Meanwhile, the LEO-MIMO satellite communication system is delay-sensitive, which needs a low-latency and low-complexity DFO estimation and compensation algorithm. According to the analysis of DFO above, vkc can be divided into satellite’s DFO vksat and GU’s absolute DFO vkut,a. vksat is caused by relative movement between satellites and GUs such that it can be computed by orbit information and GUs’ location information. By bringing the specific position information into the basic definition of the Doppler frequency shift, the estimation formula of the Doppler frequency shift of LEO satellite is obtained as follows [12].
(32)vksat=−fcrErwnsinwntcosλ~crE2+r2−2rEcosλ~coswnt,where
(33)λ~=arccosrErcosϕmax−ϕmax,fc represents the working frequency, c represents the speed of light, rE represents the radius of the earth, r represents the distance between the satellite and the user, and ϕmax represents the maximum elevation angle of communication. From the above formula, DFO is associated with time, relative altitude, and maximum elevation. Therefore, we can obtain the first-stage compensation matric of k-th GU Ψk1:
(34)Ψk1=1ejvksat⋯ejM−1vksatT.The compensated signal is given by(35)R1k=RkΨk1.Based on the definition of GU’s absolute DFO, it can be estimated in satellite. Therefore, the second compensation matric ofk-th GU Ψk2 can be expressed as
(36)Ψk2=1ejvkut,a⋯ejM−1vkut,aT.The compensated signal is given by(37)R2k=R1kΨk2.To improve the accuracy of the algorithm, the BA algorithm is performed to compensate for the angle rotation caused by the residual DFO [13] at the user equipment. Since relative DFO is related to the AOA of the signal, through applying DFT transformation into downlink estimated channel, we obtain the modified channel with DFO:
(38)Gkdl,D=FOϕk,prGkdl=∑p=0Pk−1gk,pdlFOϕk,prOvkut,aakφk.Gkdl represents the estimation of the channel, F represents DFT matrix with M points, Oϕk,pr indicates angle compensation matrix, and ϕk,pr is the angle compensation of k-th GU in p-th path. To obtain the optimal rotation angle and the optimal angle compensation matrix, the beam-scanning method is used by maximizing the absolute value of channel power in the specific direction as follows:
(39)ϕk,pr,o=argϕk,prmaxGkdl,D.The direction of the largest channel gain is the aligned direction, and we can get the optimal rotation angleϕk,pr,o. Meanwhile, the ideal AOD φk and the true AOD φk′ can be obtained by solving Equations (38) and (39). Thus, the deviation of AOD caused by residual DFO is compensated.As shown in Algorithm1, the algorithm consisted of three stages which can be seen in Figure 2. Firstly, when signal Rk is received at GUs, the coarse DFO compensation can be performed. Based on the satellite orbit and the user’s position information, we can obtain the data of first-stage estimation and the signal after compensation R1k by (35).Algorithm 1: Adaptive DFO estimation and compensation algorithm.
1: Initialize data counterflag=0, η1=0.005, η2=0.05, q=102:whileflag<10do.3:Input: receive the signal Rk.4: First-stage estimation and compensation.5:flag=flag+16:ifflag==q−1then7:flag=0, calculate v¯ksat=1/q∑i=1qvk,isat and σerr2=1/q∑i=1qvk,isat−v¯ksat.8:ifσerr2>η2then9: Connect interface E1 to interface B1 and connect interface E2 to interface B2.10:else ifη2≥σerr2≥η1then11: Connect interface E1 to interface B1 and connect interface E2 to interface A2.12:else13: Connect interface E1 to interface A1.14:end if15: store the historical data of first-stage and maintain the interface E connection.16:end if17: Second-stage estimation and compensation.18: Third-stage estimation and compensation.19: Return to step2.20:end whileFigure 2
Adaptive Doppler frequency offset compensation algorithm.The first-stage estimation data withinq frames can be denoted as vk,1sat, vk,2sat, ⋯, and vk,qsat. Then, the variance of DFO can be calculated as σerr2=1/q∑i=1qvk,isat−v¯ksat and compare it with preset control factors η1 and η2. v¯ksat is defined as the average estimation value in q frames. When η1≤σerr2≤η2, we need to implement the second-stage DFO compensation, which is processed by (37). When σerr2>η2, the BA algorithm is required to eliminate the rotation of AOD caused by the GU’s relative DFO. Meanwhile, since accurate true AOD φk′ can be obtained in the third-stage estimation, the accuracy of the second stage can be further optimized to achieve a closed-loop effect. The estimation control module is the most important part of this algorithm, that is, according to the variance of the estimation value in the first stage to judge whether to carry out the second- or third-stage estimation compensation. If the variation of DFO is small, it indicates that the environment is stable and we can precisely estimate the DFO by preestimation. So there is no need to carry out the following fine estimation operation, which greatly reduces the algorithm complexity. If the DFO changes greatly, the later stages are necessary to improve the accuracy. So the adaptive algorithm can efficiently adjust the compensation process under different DFO value scenarios.


## 5. Simulation Result

In this section, in order to analyze the performance of LEO-MIMO satellite systems, we verified the theoretical analysis through the simulation of the Monte Carlo method. The major simulation setup parameters are listed as follows. There are 50 GUs with a speed of 360 km/h served by the satellite. The Rice factor is uniformly distributed in the 20 dB and 40 dB intervals, and the distance between the satellite and users is uniformly distributed in 1000 km and 1200 km intervals. The gain of the beam is 3 dB. The working frequency is about 20 GHz. The relative azimuth between users and the satellite is distributed byU−π/3,π/3.SE against the number of antennas with MRT and MASLNR precoding is shown in Figure3. Considering the condition of the system with DFO and the system without DFO, the results verified the correctness of approximation expression of MRT precoding and MASLNR precoding which can be obtained by (14), (18), (23), and (28). It can be seen that the SE of the system without DFO is higher than that with DFO. This also reflects that it is important to relieve the influence of DFO.Figure 3
SE against numbers of antenna with MRT precoding and MASLNR precoding.
(a)(b)Figure4 shows the simulation results of SE with MRT and MASLNR precoding with SNR=10dB. We can see that the sum-rate of the users increases as the number of antennas in the transmitter increases regardless of whether we adopt the MRT precoding or MASLNR precoding. However, it is noteworthy that compared with the system without DFO, there is an obvious drop in the system with DFO. Moreover, as the number of antennas increases, the performance gap caused by DFO is larger. This is because the increase of the number of antennas causes the expansion of DFO.Figure 4
The sum-rate of users against the number of antennas.As seen from Figure5, the sum-rate of users increases as the SNR increases regardless of whether it is used by the MRT precoding or the MASLNR precoding. The number of antennas is 60. We can find a notable gap between the system without DFO and the system with DFO. But the reduction of SE caused by the frequency offset changes a little as the SNR increases. This means that DFO effects on the sum-rate of users remain stable under the different SNR conditions.Figure 5
The sum-rate of users against SNR.It also can be found that, from Figures4 and 5, the SE performance in the system using the MRT precoding is better than using the MASLNR precoding. That is because the CSI in the MRT precoding is instantaneous, but CSI in the MASLNR precoding is statistical. However, iCSI is more difficult to acquire than sCSI in a high-speed system.Figure6 illustrates the SE performance of the proposed DFO compensation algorithm in LEO-MIMO satellite systems with the MRT precoding under the condition of 3 dB SNR. Compared with the curve of the sum-rate with initial DFO, there is a significant improvement after the coarse stage compensations in (35) and (37). By using the BA algorithm, the gap between SE without DFO and SE after fine compensation is small. Thus, the coarse stage compensations require fewer computation overhead, and the fine stage compensation can achieve better performance. It is effective to make a balance between complexity and precision through the algorithm. Figure 7 makes a comparison of the proposed algorithm and estimation method based on CP. After three stages in the receiver, the adaptive DFO compensation algorithm achieves a better SE performance than the compensation algorithm based on CP.Figure 6
The performance of the adaptive DFO compensation algorithm.Figure 7
Comparison of the proposed algorithm and estimation method based on CP.


## 6. Conclusion

In this paper, we investigate the SE performance of LEO-MIMO satellite systems and design a robust transmission scheme. The MMSE channel estimation under the DFO effects was obtained. Considering the DFO effects and imperfect CSI, the approximation expressions of the achievable rate with maximum ratio transmission (MRT) and maximum average signal-to-leakage-plus-noise ratio (MASLNR) precoding is derived. To improve the quality of communication, an adaptive DFO compensation algorithm based on the historical data is designed. Numerical results have verified the correctness of approximation expressions and the effectiveness of the proposed algorithm.



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*Source: 6235241-2022-03-09.xml*