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.
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 systems
The remainder of this paper is organized as follows. Section 2 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., or ) denote scalars. The transpose and Hermitian transpose are denoted by and , respectively. denotes the set of complex-valued matrices. is the absolute value of a scalar , and is the spectral norm of a matrix . means that is a circularly symmetric complex Gaussian random variable with mean zero and variance . denotes the expectation operator.
2. System Model
We consider a LEO-MIMO satellite communication system where single-antenna ground users (GUs) simultaneously are serviced by a satellite which is equipped with uniform linear array (ULA) composed of 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].
2.1. Channel Model
The multipath channel model between satellite and -th user can be given by where represents the total number of channel propagation paths for -th user, is the complex gain of the -th path, is the array response vector, represents the ideal angle of department (AOD), is wavelength, and is antenna spacing which is set as (). represents the number of antennas.
The overall DFO can be divided into two independent parts, DFO at satellite and DFO at GUs . is further divided into absolute DFO 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: where is carrier frequency and is velocity of -th GU. Then, the downlink channel model can be rewritten as where 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 gain exhibits the Rician fading distribution with Rician factor and power [9]. Thus, the multipath channel without GUs’ relative DFO effects can be regarded as a large-scale fading coefficient and Rice model. The channel model (1) can be rewritten as where represents GUs’ relative DFO effects matrix:
represents the large-scale fading part, and the small-scale fading part can be given by where represents the Rice factor of the -th user, represents the average channel power, represents the line-of-sight (LoS) component which is equal to , and 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 independently.
We use pilot-assisted minimum mean square error (MMSE) channel estimation for the NLoS component of and consider DFO effects on LoS component of . The estimated channel can be given by
represents the estimated channel of NLoS component. is channel estimation error and its variance is as follows [10]. where is the transmitting power of the pilot signal.
3.2. Downlink Spectral Efficiency Analysis with DFO
The received signal at user is expressed as where represents the transmitting power of the -th downlink user, represents downlink precoding, is transmitting signal, and 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 where and is the downlink signal-to-noise ratio.
In this paper, we adopt the MRT precoding and MASLNR precoding. For MRT precoding we have the following theorem.
Theorem 1. With MRT precoding, the approximation expression of downlink achievable rate with DFO is given by where In , denotes the DFO effects item, and and are represented by (15) and (16).
Proof. Please see Appendix A.
When DFO effects are eliminated, we can obtain the approximation expression of a downlink achievable rate without DFO: where
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]. where and we have the following theorem.
Theorem 2. With MASLNR precoding, the approximation expression of downlink achievable rate with DFO is given by where
Proof. Please see Appendix B.
When DFO effects are compensated absolutely, we can obtain the approximation expression of the downlink achievable rate without DFO. where
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, can be divided into satellite’s DFO and GU’s absolute DFO . 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]. where
represents the working frequency, represents the speed of light, represents the radius of the earth, represents the distance between the satellite and the user, and 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 -th GU :
The compensated signal is given by
Based on the definition of GU’s absolute DFO, it can be estimated in satellite. Therefore, the second compensation matric of -th GU can be expressed as
The compensated signal is given by
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:
represents the estimation of the channel, represents DFT matrix with points, indicates angle compensation matrix, and is the angle compensation of -th GU in -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:
The direction of the largest channel gain is the aligned direction, and we can get the optimal rotation angle . Meanwhile, the ideal AOD and the true AOD can be obtained by solving Equations (38) and (39). Thus, the deviation of AOD caused by residual DFO is compensated.
As shown in Algorithm 1, the algorithm consisted of three stages which can be seen in Figure 2. Firstly, when signal 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 by (35).
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The first-stage estimation data within frames can be denoted as , , , and . Then, the variance of DFO can be calculated as and compare it with preset control factors and . is defined as the average estimation value in frames. When , we need to implement the second-stage DFO compensation, which is processed by (37). When , the BA algorithm is required to eliminate the rotation of AOD caused by the GU’s relative DFO. Meanwhile, since accurate true AOD 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 by .
SE against the number of antennas with MRT and MASLNR precoding is shown in Figure 3. 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.
(a)
(b)
Figure 4 shows the simulation results of SE with MRT and MASLNR precoding with . 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.
As seen from Figure 5, 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 . 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.
It also can be found that, from Figures 4 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.
Figure 6 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.
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.
Appendix
A. Proof of Theorem 1
The derivation of approximation expression of the downlink achievable rate with the MRT precoding is shown as follows. By bringing the MRT precoding (12) into the simulation expression of the downlink achievable rate (11), we can obtain
First, we derive : where results from the independence of and , , . By applying Equation in [10], we can get where
can be obtained by summing up the imaginary part and real part that effects to where
For , where . By applying Lemma 5 in [10], we obtain
Finally, for , we have where . By applying Equation (70) in [10], we can get where
We can obtain the approximation expression of downlink achievable rate with DFO by substituting (A.2), (A.8), and (A.10) into (A.1).
Similarly, under the condition of no DFO, we can obtain
The approximation expression of the downlink achievable rate without DFO can be obtained by substituting (A.13), (A.15), and (A.14) into the simulation expression of the downlink achievable rate.
B. Proof of Theorem 2
To derive the approximation expression of the downlink achievable rate with the MASLNR precoding, by bringing MASLNR precoding into the simulation expression of SE under the condition of perfect CSI, we can obtain where , represents the channel power, as we know in Section 2.1, which is defined as .
The derivation of is shown as follows. Firstly, is a diagonal matrix and which yields that is a vector and
Thus, we can obtain where
Similarly, we can conduct . So the approximation expression of the downlink achievable rate with the MASLNR precoding under the condition of DFO is acquired.
Considering the condition of no DFO, it only takes the impact on . So we can obtain where
Then the approximation expression of the downlink achievable rate with the MASLNR precoding under the condition of no DFO is acquired.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported in part by the National Key Research and Development Program of China under grant 2021YFB2900300 and by the National Natural Science Foundation of China (NSFC) under grants 61971127, 61871465, and 61871122.