CN113765553A - Multi-beam satellite communication system robust precoding method based on machine learning - Google Patents

Multi-beam satellite communication system robust precoding method based on machine learning Download PDF

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CN113765553A
CN113765553A CN202111053477.2A CN202111053477A CN113765553A CN 113765553 A CN113765553 A CN 113765553A CN 202111053477 A CN202111053477 A CN 202111053477A CN 113765553 A CN113765553 A CN 113765553A
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王闻今
刘彦浩
王一彪
伍诗语
任博文
丁睿
尤力
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Abstract

The invention discloses a machine learning-based multi-beam satellite communication system robust precoding method, which comprises the steps of constructing a multi-beam satellite downlink channel vector model containing user position positioning uncertainty; obtaining a statistical channel model related to a channel autocorrelation matrix; constructing a multi-beam satellite system and a robust precoding optimization design problem with maximized rate; equivalently converting a multi-beam satellite system and a rate-maximized robust precoding optimization design problem into a power minimization problem under the user signal-to-interference-and-noise ratio guarantee and single antenna power constraint; combining a Lagrange function of an equivalent optimization problem and a KKT condition of the Lagrange function to obtain an optimal precoding vector; a method combined with machine learning predicts Lagrangian multipliers required by an optimization problem based on a channel autocorrelation matrix. The invention can reduce the complexity of the realization of the channel autocorrelation matrix prediction problem algorithm and obviously improve the transmission performance of the multi-beam satellite communication system and the robustness of the positioning angle estimation error.

Description

Multi-beam satellite communication system robust precoding method based on machine learning
Technical Field
The invention relates to the technical field of satellite communication, in particular to a multi-beam satellite communication system robust precoding method based on machine learning.
Background
Currently, multi-beam satellite systems have shown great potential for ubiquitous global wireless access in 5G networks as well as future 6G networks. Where downlink precoder design plays a crucial role in satellite communications, existing precoder design methods are typically based on well-known channel state information and total power constraints. Due to high-speed mobility of a satellite and satellite attitude jitter, ideal channel state information is usually difficult to obtain on a low-orbit multi-beam satellite, and the performance of satellite downlink precoding is reduced by the existing precoder design method, so that the service quality of a user is reduced. In addition, with the general trend toward miniaturization of satellites and the dramatic increase in energy consumption during information transmission processing, power consumption in satellite communication systems is a factor that needs to be heavily considered in system design. Therefore, how to achieve low power consumption and robust information transmission is a trend in current satellite communication system design.
In recent years, artificial intelligence technology has been rapidly developed and widely used in the field of communications and the like. The technology can effectively reduce the complexity of algorithm realization through a mode of off-line training of the neural network and direct on-line prediction of results. In an actual satellite communication system, the signal transmission distance is long, which results in high transmission delay, and therefore, the real-time performance of signal transmission is very important in the system design process. How to combine the machine learning technology, reduce the implementation complexity and the system processing time of the precoding method, and then promote the signal transmission real-time of the satellite communication system, reduce the power consumption of the satellite-borne equipment is the problem that needs to be solved in the current satellite communication system design.
Disclosure of Invention
The purpose of the invention is as follows: in order to overcome the defects in the prior art, the invention provides a machine learning-based multi-beam satellite communication system robust precoding method, which can effectively reduce the adverse effect caused by the inaccuracy of the positioning angle of a multi-beam satellite user and improve the real-time performance and the transmission performance of the multi-beam satellite communication system.
The technical scheme is as follows: in order to achieve the above object, the present invention provides a robust precoding method for multi-beam satellite communication system based on machine learning, comprising the following steps,
step 1, constructing a multi-beam satellite downlink channel vector model containing user position positioning uncertainty based on position angle estimation errors of a multi-beam satellite for each user and public angle errors caused by satellite attitude orbit control;
step 2, solving mathematical expectations of the sum rate of the multi-beam satellite system with respect to positioning angle error variables to obtain the traversal transmission rate of the kth user, and obtaining a statistical channel model with respect to a channel autocorrelation matrix;
step 3, constructing a multi-beam satellite system and a robust precoding optimization design problem with maximized rate, wherein the constraint condition is that the power value of each antenna array element of the multi-beam satellite is smaller than a certain threshold value;
step 4, equivalently converting the robust precoding optimization design problem of the multi-beam satellite system and the rate maximization into a power minimization problem under the user signal-to-interference-and-noise ratio guarantee and the single antenna power constraint;
step 5, modeling the optimal structure of the robust precoding as the solution of the generalized characteristic value problem by combining the Lagrange function of the equivalent optimization problem and the KKT condition thereof, and obtaining the optimal precoding vector, namely the optimal solution of the original problem;
and 6, predicting a Lagrange multiplier required by the optimization problem based on the channel autocorrelation matrix by combining a machine learning method.
Further, in the present invention: the constructing of the multi-beam satellite downlink channel vector model in the step 1 further includes constructing an angle relationship as follows:
Figure RE-GDA0003324231820000021
Figure RE-GDA0003324231820000022
wherein, thetakThe exact angle of user k with respect to the y-axis of the low orbit satellite planar array antenna,
Figure RE-GDA0003324231820000023
planar array antenna for user k relative to low orbit satellitexThe exact angle of the shaft is such that,
Figure RE-GDA0003324231820000024
and
Figure RE-GDA0003324231820000025
indicating the satellite position angle estimation error for the k-th user,
Figure RE-GDA0003324231820000026
and
Figure RE-GDA0003324231820000027
representing common angle detection errors due to satellite attitude orbit control,
Figure RE-GDA0003324231820000028
expressed as the estimated angle of user k with respect to the y-axis of the low-orbit satellite planar array antenna,
Figure RE-GDA0003324231820000029
planar array antenna denoted as user k with respect to low orbit satellitexThe estimated angle of the axis.
Further, in the present invention: in the step 1, the corresponding vector of the planar array antenna
Figure RE-GDA00033242318200000210
The following relationship is satisfied:
Figure RE-GDA00033242318200000211
wherein the content of the first and second substances,
Figure RE-GDA00033242318200000212
representing a planar array antenna inxThe array element on the axis is a response vector,
Figure RE-GDA00033242318200000213
the array element response vector of the planar array antenna on the y axis is represented, and the two satisfy respectively:
Figure RE-GDA00033242318200000214
Figure RE-GDA00033242318200000215
further, in the present invention: the traversal and rate of the multi-beam satellite communication system of the kth user in the step 2
Figure RE-GDA0003324231820000031
Comprises the following steps:
Figure RE-GDA0003324231820000032
wherein the content of the first and second substances,
Figure RE-GDA0003324231820000033
and K is the total number of users,
Figure RE-GDA0003324231820000034
it is shown that the mathematical expectation symbol is calculated,
Figure RE-GDA0003324231820000035
representing the noise power value, wkPrecoding vector for the k-th user, wiPrecoding vector for ith user, hkDownlink channel parameter vector h for multi-beam satellite to kth useriDownlink channel parameter vector for multibeam satellite to ith user, superscript (·)HRepresenting a conjugate transpose operation on a vector or matrix,
Figure RE-GDA0003324231820000036
Figure RE-GDA0003324231820000037
representing a vector wiAnd hiThe conjugate transpose of (1);
SINRkSINR for the kth userkSatisfies the following conditions:
Figure RE-GDA0003324231820000038
wherein, wkFor the kth user's precoding vector, superscript (. cndot.)HDenotes the conjugate transpose, hkChannel correspondence vectors for user angle information for the k-th user from the multi-beam satellite.
Further, in the present invention: the system traversal and rate in the step 3
Figure RE-GDA0003324231820000039
Without closed form expressions, enabling direct processing
Figure RE-GDA00033242318200000310
Are relatively difficult and are therefore determined here according to the Jackson inequality
Figure RE-GDA00033242318200000311
Upper limit of (2)
Figure RE-GDA00033242318200000312
Is composed of
Figure RE-GDA00033242318200000313
The robust precoding optimization design problem P1 modeling of the system and the rate upper limit maximization is represented as follows:
Figure RE-GDA00033242318200000314
Figure RE-GDA00033242318200000315
wherein P isnIs as followsnMaximum allowed transmit power of individual antenna elements, aggregate
Figure RE-GDA00033242318200000316
Satisfy the requirement of
Figure RE-GDA00033242318200000317
Collection
Figure RE-GDA00033242318200000318
Satisfy the requirement of
Figure RE-GDA00033242318200000319
Further, in the present invention: in the step 4, the optimization problem P1 with the maximum system and speed under the constraint of single antenna power is equivalently converted into a power minimization problem P2 under the constraint of user signal-to-interference-and-noise ratio and single antenna power,
Figure RE-GDA0003324231820000041
Figure RE-GDA0003324231820000042
Figure RE-GDA0003324231820000043
user signal-to-interference-and-noise ratio guarantee constraint E { SINRkThe expression of can be approximated as:
Figure RE-GDA0003324231820000044
wherein the constraint threshold value of the target SINR is
Figure RE-GDA0003324231820000045
Figure RE-GDA0003324231820000046
To achieve the maximum sum rate for the problem P1,
Figure RE-GDA0003324231820000047
the derivation of the expression requires mathematical expectation of the angle error variables.
Further, in the present invention: in the step 4, based on the multi-beam satellite downlink channel parameter model, the channel autocorrelation matrix is obtained
Figure RE-GDA0003324231820000048
The simplification is as follows:
Figure RE-GDA0003324231820000049
wherein the corresponding vector of the planar array antenna
Figure RE-GDA00033242318200000410
To (1) anThe items may be represented as:
Figure RE-GDA00033242318200000411
n＝0,…,MxMy-1,
Figure RE-GDA00033242318200000412
wherein the channel autocorrelation matrix
Figure RE-GDA00033242318200000413
Row m and column n elements of (1)
Figure RE-GDA00033242318200000414
Can be expressed as:
Figure RE-GDA00033242318200000415
wherein the content of the first and second substances,
Figure RE-GDA00033242318200000416
representing channel power values [. ]]m,nTo express taking the matrixmGo to the firstnElements of a column, (.)nThe representation takes the nth element of the vector,
Figure RE-GDA00033242318200000417
representing a pair vector
Figure RE-GDA00033242318200000418
The m-th element of (1)
Figure RE-GDA00033242318200000419
And conjugates of the nth element
Figure RE-GDA00033242318200000420
The product of (a) and (b) to obtain a mathematical expectation;
and:
Figure RE-GDA00033242318200000421
m＝0,…,MxMy-1,
Figure RE-GDA00033242318200000422
n＝0,…,MxMy-1,
Figure RE-GDA0003324231820000051
wherein a and b are a first intermediate variable and a second intermediate variable respectively;
if the satellite estimates the error of the position angle of the k user
Figure RE-GDA0003324231820000052
And
Figure RE-GDA0003324231820000053
obey a uniform distribution, namely:
Figure RE-GDA0003324231820000054
Figure RE-GDA0003324231820000055
wherein, thetaLAnd thetaURespectively representing angle errors
Figure RE-GDA0003324231820000056
Upper limit of the value andthe lower limit of the amount of the organic solvent,
Figure RE-GDA0003324231820000057
and
Figure RE-GDA0003324231820000058
respectively representing angle errors
Figure RE-GDA0003324231820000059
Upper and lower value limits;
the above formula represents
Figure RE-GDA00033242318200000510
In the interval U [ theta ]L,θU]Subject to a uniform distribution of the flux in the flux,
Figure RE-GDA00033242318200000511
in the interval
Figure RE-GDA00033242318200000512
Subject to a uniform distribution, further define:
Δθ＝θU-θL
Figure RE-GDA00033242318200000513
wherein, DeltaθAnd
Figure RE-GDA00033242318200000514
respectively representing angle errors
Figure RE-GDA00033242318200000515
And
Figure RE-GDA00033242318200000516
the difference between the upper and lower limits of (d);
then it is possible to obtain:
Figure RE-GDA00033242318200000517
will be in the above expression
Figure RE-GDA00033242318200000518
And
Figure RE-GDA00033242318200000519
integration is performed to obtain:
Figure RE-GDA00033242318200000520
wherein each intermediate variable is defined as
Figure RE-GDA00033242318200000521
Figure RE-GDA00033242318200000522
Figure RE-GDA00033242318200000523
Wherein A, B and Z are the third intermediate variable, the fourth intermediate variable and the fifth intermediate variable, respectively;
if it is
Figure RE-GDA00033242318200000524
Obey mean value of muθ,kVariance of
Figure RE-GDA00033242318200000525
The distribution of the gaussian component of (a) is,
Figure RE-GDA00033242318200000526
obey mean value of
Figure RE-GDA00033242318200000527
Variance of
Figure RE-GDA00033242318200000528
A Gaussian distribution of
Figure RE-GDA00033242318200000529
And
Figure RE-GDA00033242318200000530
is expressed as:
Figure RE-GDA0003324231820000061
Figure RE-GDA0003324231820000062
wherein the content of the first and second substances,
Figure RE-GDA0003324231820000063
to represent
Figure RE-GDA0003324231820000064
Is determined by the probability density function of (a),
Figure RE-GDA0003324231820000065
to represent
Figure RE-GDA0003324231820000066
A probability density function of;
at this time
Figure RE-GDA0003324231820000067
Can be expressed as:
Figure RE-GDA0003324231820000068
for the above
Figure RE-GDA0003324231820000069
Is calculated byLine-invariant integration yields:
Figure RE-GDA00033242318200000610
wherein each intermediate variable is defined as
Figure RE-GDA00033242318200000611
Figure RE-GDA00033242318200000612
Figure RE-GDA00033242318200000613
Figure RE-GDA00033242318200000614
Figure RE-GDA00033242318200000615
Figure RE-GDA00033242318200000616
Wherein P, Q, D, F, C, E are the sixth intermediate variable, the seventh intermediate variable, the eighth intermediate variable, the ninth intermediate variable, the tenth intermediate variable and the eleventh intermediate variable, respectively
Further, in the present invention: lagrangian function of the equivalence optimization problem P2 in the step 5
Figure RE-GDA00033242318200000617
Comprises the following steps:
Figure RE-GDA0003324231820000071
wherein the content of the first and second substances,
Figure RE-GDA0003324231820000072
for lagrange multipliers corresponding to SINR constraints,
Figure RE-GDA0003324231820000073
corresponding to NTLagrange multiplier of single antenna power constraint, enThe dimension of the vector which represents the nth element is 1 and other elements are 0 is determined by the multiplication matrix,
Figure RE-GDA0003324231820000074
represents a pair vector enIs transposed, PnRepresenting the upper limit value of the power of the nth antenna array element;
to simplify the calculation, let
Figure RE-GDA0003324231820000075
Then:
Figure RE-GDA0003324231820000076
wherein, the diagonal matrix
Figure RE-GDA0003324231820000077
For lagrange function
Figure RE-GDA0003324231820000078
The derivation yields:
Figure RE-GDA0003324231820000079
optimal precoding vector based on KKT condition
Figure RE-GDA00033242318200000710
The requirements are satisfied:
Figure RE-GDA00033242318200000711
it is thus possible to obtain:
Figure RE-GDA00033242318200000712
order matrix
Figure RE-GDA00033242318200000713
And assumes an optimal precoding vector
Figure RE-GDA00033242318200000714
Figure RE-GDA00033242318200000715
For the optimal precoding vector, the optimal precoding vector can be obtained
Figure RE-GDA00033242318200000716
Is regarded as a corresponding matrix pair (S)k,Nk) The maximum generalized eigenvalue of the eigenvector of the problem, the maximum generalized eigenvalue being γk，
Figure RE-GDA00033242318200000717
And gammakRespectively as follows:
γk＝max.generalized eigenvalue(Sk,Nk)
Figure RE-GDA00033242318200000718
while Lagrange multiplier
Figure RE-GDA00033242318200000719
And
Figure RE-GDA00033242318200000720
after determinationDirection of the optimal precoder
Figure RE-GDA00033242318200000721
And gammakCan be uniquely determined and its SINR constraint satisfies E { SINR when the equivalent optimization problem P3 gets the optimal solutionk}＝γkAccording to the condition, the optimal power distribution can be obtained, namely:
Figure RE-GDA0003324231820000081
where ρ is a power allocation vector and ρ ═ ρ1,…,ρK]TThe dimension of the matrix F is K × K, and the elements in the kth row and the ith column are:
Figure RE-GDA0003324231820000082
at this time, the optimal precoding vector can be calculated
Figure RE-GDA0003324231820000083
Further, in the present invention: in said step 5, to determine the Lagrangian multiplier
Figure RE-GDA0003324231820000084
And
Figure RE-GDA0003324231820000085
the optimal value of (a) is obtained by adopting a sub-gradient projection technology and combining a KKT condition qn≥0,
Figure RE-GDA0003324231820000086
And q isnPn＝0,
Figure RE-GDA0003324231820000087
Get about
Figure RE-GDA0003324231820000088
The iterative expression Q of (a) is:
Figure RE-GDA0003324231820000089
wherein Q isiExpressed as lagrange multiplier matrix corresponding to single antenna power constraint in the ith iteration process
Figure RE-GDA00033242318200000810
tiRepresents the iteration step size, and ti＝1/i；
According to the lagrangian function and the dual function of the equivalent optimization problem P2 and the strong dual relationship between the two, the following can be obtained:
Figure RE-GDA00033242318200000811
wherein, PTActual transmit power for the system; rearranging the above calculation formulas
Figure RE-GDA00033242318200000812
Equation for the dead point of (1):
Figure RE-GDA00033242318200000813
wherein the optimal Lagrange multiplier matrix
Figure RE-GDA00033242318200000814
In practical application, the constraint threshold value gamma of the target SINRkIt is desirable to maximize the value as much as possible, and determine v in conjunction with the convex optimization problem described belowkThe value of (c):
Figure RE-GDA00033242318200000815
Figure RE-GDA00033242318200000816
Figure RE-GDA00033242318200000817
wherein, PtotalRepresents the system maximum transmission power; to reduce the complexity of solving the above optimization problem, we will refer to vkThe equation for the dead point of (a) is approximated as:
Figure RE-GDA0003324231820000091
wherein the cofactor beta is calculatedk＝||hk||2The optimization problem is solved by a water injection algorithm.
Further, in the present invention: said step 6 further comprises the step of,
step 6-1, constructing a convolutional neural network;
step 6-2, collecting training samples and training a convolutional neural network;
6-3, training the neural network to obtain a channel autocorrelation parameter matrix
Figure RE-GDA0003324231820000092
With the real and imaginary parts of the lagrange multiplier as inputs
Figure RE-GDA0003324231820000093
And
Figure RE-GDA0003324231820000094
as an output;
step 6-4, inputting channel autocorrelation parameter matrix
Figure RE-GDA0003324231820000095
Direct acquisition of lagrange via neural networksA multiplier.
Has the advantages that: compared with the prior art, the invention has the beneficial effects that: by solving mathematical expectations for the system and the rate and combining power constraints on single antenna array elements, the adverse effects caused by inaccuracy of the positioning angle of a multi-beam satellite user are effectively reduced, and the real-time performance and the transmission performance of the multi-beam satellite communication system are improved under the assistance of an artificial intelligence technology compared with the traditional method without considering the angular error of satellite positioning;
(1) the method integrates the angle uncertainty existing when a satellite positions a user and the detection angle error caused by the satellite attitude jitter into channel modeling, and establishes a statistical channel model which can realize the robustness of precoding performance to the positioning angle error according to the characteristic that the angle error variable obeys a certain probability distribution;
(2) according to the method, traversal and rate maximization of a satellite communication system are realized under the power constraint of a single antenna array element, and an optimal robust precoding structure is modeled into a solution of a generalized characteristic value problem according to a Lagrangian function of an optimization problem and a KKT condition of the optimal robust precoding structure. Meanwhile, the invention develops an efficient iterative algorithm to solve the optimal precoding direction and power distribution, thereby obtaining the optimal precoding vector
(3) The method combines the machine learning technology, realizes direct prediction of the Lagrange multiplier in the solving process by an off-line training mode, replaces an algorithm iteration process in the solving process, meets the requirements on the calculation complexity and the system real-time performance in the satellite communication system, and reduces the system realization complexity and the system power consumption.
Drawings
Fig. 1 is a schematic overall flow chart of a machine learning-based multi-beam satellite communication system robust precoding method proposed by the present invention;
fig. 2 is a schematic diagram of a multi-beam satellite system equipped with a planar array antenna according to the present invention;
FIG. 3 is a schematic diagram of a positioning angle error of a satellite-side antenna array for a user according to the present invention;
fig. 4 is a schematic diagram of a convolutional neural network structure for predicting lagrangian multipliers in the present invention.
Detailed Description
The technical scheme of the invention is further explained in detail by combining the attached drawings:
the present invention may be embodied in many different forms and should not be construed as limited to the embodiments set forth herein. Rather, these embodiments are provided so that this disclosure will be thorough and complete, and will fully convey the scope of the invention to those skilled in the art.
As shown in fig. 1, an overall flow chart of a robust precoding method for a multi-beam satellite communication system based on machine learning is shown, where the method includes the following steps:
step 1, constructing a multi-beam satellite downlink channel vector model containing user position positioning uncertainty based on position angle estimation errors of a multi-beam satellite for each user and public angle errors caused by satellite attitude orbit control;
referring to the schematic diagram of fig. 2, fig. 2 is a schematic diagram of a multi-beam satellite system, assuming that K single-antenna users are served simultaneously in a multi-beam satellite downlink transmission scenario, and the number N of antenna elements of a planar array antenna provided for a multi-beam satelliteTComprises the following steps:
NT＝Mx×My
wherein M isxAnd MyThe antenna array element numbers on the x axis and the y axis of the area array antenna are respectively, the x axis represents the horizontal direction, and the y axis represents the vertical direction.
Referring to the illustration of fig. 3, assuming that the satellite has an angle positioning error for the user, after introducing a positioning angle error variable, a parameter vector h of a downlink channel from the multi-beam satellite to the kth userkComprises the following steps:
Figure RE-GDA0003324231820000101
where t represents time, f represents system frequency,
Figure RE-GDA0003324231820000102
representing the corresponding vector of the planar array antenna,
Figure RE-GDA0003324231820000103
representing the Doppler frequency offset, tau, due to multi-beam satellite movementk,lRepresenting the propagation delay of the multi-beam satellite to the ith propagation path of the kth user,
Figure RE-GDA0003324231820000104
representing propagation delay tauk,lMinimum value of (i), i.e.
Figure RE-GDA0003324231820000105
Figure RE-GDA0003324231820000106
Represents the downlink channel gain of the k-th user, and
Figure RE-GDA0003324231820000107
satisfies the following conditions:
Figure RE-GDA0003324231820000108
wherein L iskRepresents the total number of propagation paths from the multi-beam satellite to the k-th user,
Figure RE-GDA0003324231820000111
represents the complex gain of the ith propagation path to the kth user,
Figure RE-GDA0003324231820000112
indicating the doppler shift due to the movement of the kth user on the ith propagation path,
Figure RE-GDA0003324231820000113
represents a user-side transmission delay and
Figure RE-GDA0003324231820000114
satisfies the following conditions:
Figure RE-GDA0003324231820000115
in a multibeam satellite communication scenario, the signal propagates along a direct path, thus the downlink channel gain of the kth user of the satellite
Figure RE-GDA0003324231820000116
Can be regarded as obeying a Rice factor of KkPower value of
Figure RE-GDA0003324231820000117
The rice distribution of (a).
Specifically, constructing the multi-beam satellite downlink channel vector model further comprises constructing the following angle relationship:
Figure RE-GDA0003324231820000118
Figure RE-GDA0003324231820000119
wherein, thetakThe exact angle of user k with respect to the y-axis of the low orbit satellite planar array antenna,
Figure RE-GDA00033242318200001110
the exact angle of user k with respect to the x-axis of the low orbit satellite planar array antenna,
Figure RE-GDA00033242318200001111
and
Figure RE-GDA00033242318200001112
indicating the satellite position angle estimation error for the k-th user,
Figure RE-GDA00033242318200001113
and
Figure RE-GDA00033242318200001114
representing common angle detection errors due to satellite attitude orbit control,
Figure RE-GDA00033242318200001115
expressed as the estimated angle of user k with respect to the y-axis of the low-orbit satellite planar array antenna,
Figure RE-GDA00033242318200001116
expressed as the estimated angle of user k with respect to the x-axis of the low orbit satellite planar array antenna. Further corresponding vectors of the planar array antenna
Figure RE-GDA00033242318200001117
The following relationship is satisfied:
Figure RE-GDA00033242318200001118
wherein the content of the first and second substances,
Figure RE-GDA00033242318200001119
the array element response vector of the plane array antenna on the x axis is shown,
Figure RE-GDA00033242318200001120
the array element response vector of the planar array antenna on the y axis is represented, and the two satisfy respectively:
Figure RE-GDA00033242318200001121
Figure RE-GDA00033242318200001122
step 2, solving mathematical expectations of the sum rate of the multi-beam satellite system with respect to positioning angle error variables to obtain the traversal transmission rate of the kth user, and obtaining a statistical channel model with respect to a channel autocorrelation matrix;
in particular, the traversal and rate of the multi-beam satellite communication system for the kth user
Figure RE-GDA00033242318200001123
Comprises the following steps:
Figure RE-GDA00033242318200001124
wherein the content of the first and second substances,
Figure RE-GDA00033242318200001125
k is the total number of users, an
Figure RE-GDA00033242318200001126
Figure RE-GDA00033242318200001127
Representing the mathematical expectation symbol,
Figure RE-GDA00033242318200001128
representing the noise power value, wkPrecoding vector for the k-th user, wiPrecoding vector for ith user, hkDownlink channel parameter vector h for multi-beam satellite to kth useriDownlink channel parameter vector for multibeam satellite to ith user, superscript (·)HRepresenting a conjugate transpose operation on a vector or matrix,
Figure RE-GDA0003324231820000121
representing a vector wiAnd hiThe conjugate transpose of (1);
SINRksignal to interference plus noise ratio (SINR) for the k-th userkSatisfies the following conditions:
Figure RE-GDA0003324231820000122
step 3, constructing a multi-beam satellite system and a robust precoding optimization design problem with maximized rate, wherein the constraint condition is that the power value of each antenna array element of the multi-beam satellite is smaller than a certain threshold value;
wherein, the optimization target of the multi-beam satellite system and the robust precoding optimization design problem P1 with the rate maximization is
Figure RE-GDA0003324231820000123
In particular, due to system traversal and rate
Figure RE-GDA0003324231820000124
Without closed form expressions, enabling direct processing
Figure RE-GDA0003324231820000125
Are relatively difficult and are therefore determined here according to the Jackson inequality
Figure RE-GDA0003324231820000126
Upper limit of (2)
Figure RE-GDA0003324231820000127
And is represented as
Figure RE-GDA0003324231820000128
Then the multi-beam satellite system and rate-maximizing robust precoding optimization design problem P1 can be modeled as:
Figure RE-GDA0003324231820000129
Figure RE-GDA00033242318200001210
wherein, PnFor maximum allowed transmit power of the nth antenna element, assembling
Figure RE-GDA00033242318200001211
Collection
Figure RE-GDA00033242318200001212
Step 4, equivalently converting the robust precoding optimization design problem of the multi-beam satellite system and the rate maximization into a power minimization problem under the user signal-to-interference-and-noise ratio guarantee and the single antenna power constraint;
specifically, in order to obtain a solution structure of the optimal robust precoding, the multi-beam satellite system and the robust precoding optimization design problem P1 with the maximized rate in step 3 are equivalently converted into a power minimization problem P2 under the user signal-to-interference-and-noise ratio guarantee and the single antenna power constraint, that is:
Figure RE-GDA0003324231820000131
Figure RE-GDA0003324231820000132
Figure RE-GDA0003324231820000133
wherein the content of the first and second substances,
Figure RE-GDA0003324231820000134
representing the maximum sum rate value that the system can achieve when the problem P1 achieves an optimal solution. Therefore, when problem P2 achieves the same maximum sum rate as problem P1
Figure RE-GDA0003324231820000135
When the transmission power of the system can be guaranteed to be equal to or less than the transmission power value required by the problem P1, the problem P2 can realize the same or better optimal precoding vector as the problem P1.
To solve the problem P2, the maximum sum rate is calculated
Figure RE-GDA0003324231820000136
Approximation by the jensen inequality is:
Figure RE-GDA0003324231820000137
the following equivalence relations are obtained:
Figure RE-GDA0003324231820000138
wherein, γkIs the target signal-to-interference-and-noise ratio value of the kth user.
At this time, the optimization target is converted into
Figure RE-GDA0003324231820000139
The constraint conditions are respectively user signal-to-interference-and-noise ratio guarantee constraints
Figure RE-GDA00033242318200001310
And single antenna power constraints
Figure RE-GDA00033242318200001311
Wherein, PnIs the maximum allowable transmission power of the nth antenna element, and
Figure RE-GDA00033242318200001312
at this time, the question P2 can be expressed as:
Figure RE-GDA00033242318200001313
Figure RE-GDA00033242318200001314
Figure RE-GDA00033242318200001315
wherein, the user signal-to-interference-and-noise ratio guarantee constraint E { SINRkIt can be approximated as:
Figure RE-GDA00033242318200001316
wherein the constraint threshold value gamma of the target SINRkIs composed of
Figure RE-GDA00033242318200001317
Figure RE-GDA00033242318200001318
To achieve the maximum sum rate for the problem P1,
Figure RE-GDA00033242318200001319
is a channel autocorrelation matrix, and
Figure RE-GDA00033242318200001320
the derivation of the expression requires mathematical expectation of the angle error variables.
Further, assume that the multibeam satellite can only obtain imperfect channel information about the departure angle AOD of each service user without loss of generality, and the position angle estimation error of the satellite for the kth user
Figure RE-GDA0003324231820000141
And
Figure RE-GDA0003324231820000142
while obeying a certain probability distribution, such as uniform distribution or Gaussian distribution, and common angle detection error caused by satellite attitude orbit control
Figure RE-GDA0003324231820000143
And
Figure RE-GDA0003324231820000144
is usually 10-3rad scale of, due toThe channel autocorrelation matrix
Figure RE-GDA0003324231820000145
Can be ignored in the calculation process
Figure RE-GDA0003324231820000146
And
Figure RE-GDA0003324231820000147
channel autocorrelation matrix based on multi-beam satellite downlink channel parameter model
Figure RE-GDA0003324231820000148
The simplification is as follows:
Figure RE-GDA0003324231820000149
wherein the corresponding vector of the planar array antenna
Figure RE-GDA00033242318200001410
Item n of (1)
Figure RE-GDA00033242318200001411
Can be expressed as:
Figure RE-GDA00033242318200001412
n＝0,…,MxMy-1,
Figure RE-GDA00033242318200001413
wherein the channel autocorrelation matrix
Figure RE-GDA00033242318200001414
Row m and column n elements of (1)
Figure RE-GDA00033242318200001415
Can be expressed as:
Figure RE-GDA00033242318200001416
wherein the content of the first and second substances,
Figure RE-GDA00033242318200001417
representing channel power values [. ]]m,nThe expression takes the element of the m-th row and n-th column of the matrix, (-)nThe representation takes the nth element of the vector.
Figure RE-GDA00033242318200001418
Representing a pair vector
Figure RE-GDA00033242318200001419
The m-th element of (1)
Figure RE-GDA00033242318200001420
And conjugates of the nth element
Figure RE-GDA00033242318200001421
The product of (a) and (b) to obtain a mathematical expectation;
and:
Figure RE-GDA00033242318200001422
m＝0,…,MxMy-1,
Figure RE-GDA00033242318200001423
n＝0,…,MxMy-1,
Figure RE-GDA00033242318200001424
wherein a and b are a first intermediate variable and a second intermediate variable respectively;
if the satellite estimates the error of the position angle of the k user
Figure RE-GDA00033242318200001425
And
Figure RE-GDA00033242318200001426
obey a uniform distribution, namely:
Figure RE-GDA00033242318200001427
Figure RE-GDA0003324231820000151
wherein, thetaLAnd thetaURespectively representing angle errors
Figure RE-GDA0003324231820000152
The upper and lower limits of the values are,
Figure RE-GDA0003324231820000153
and
Figure RE-GDA0003324231820000154
respectively representing angle errors
Figure RE-GDA0003324231820000155
Upper and lower value limits;
the above formula represents
Figure RE-GDA0003324231820000156
In the interval U [ theta ]L,θU]Subject to a uniform distribution of the flux in the flux,
Figure RE-GDA0003324231820000157
in the interval
Figure RE-GDA0003324231820000158
Subject to a uniform distribution, further define:
Δθ＝θU-θL
Figure RE-GDA0003324231820000159
wherein, DeltaθAnd
Figure RE-GDA00033242318200001510
respectively representing angle errors
Figure RE-GDA00033242318200001511
And
Figure RE-GDA00033242318200001512
the difference between the upper and lower limits of (d);
then it is possible to obtain:
Figure RE-GDA00033242318200001513
will be in the above expression
Figure RE-GDA00033242318200001514
And
Figure RE-GDA00033242318200001515
integration is performed to obtain:
Figure RE-GDA00033242318200001516
wherein the content of the first and second substances,
Figure RE-GDA00033242318200001517
Figure RE-GDA00033242318200001518
Figure RE-GDA00033242318200001519
where A, B and Z are the third intermediate variable, the fourth intermediate variable, and the fifth intermediate variable, respectively.
If it is
Figure RE-GDA00033242318200001520
Obey mean value of muθ,kVariance of
Figure RE-GDA00033242318200001521
The distribution of the gaussian component of (a) is,
Figure RE-GDA00033242318200001522
obey mean value of
Figure RE-GDA00033242318200001523
Variance of
Figure RE-GDA00033242318200001524
A Gaussian distribution of
Figure RE-GDA00033242318200001525
And
Figure RE-GDA00033242318200001526
is expressed as:
Figure RE-GDA00033242318200001527
Figure RE-GDA00033242318200001528
wherein the content of the first and second substances,
Figure RE-GDA00033242318200001529
to represent
Figure RE-GDA00033242318200001530
Is determined by the probability density function of (a),
Figure RE-GDA00033242318200001531
to represent
Figure RE-GDA00033242318200001532
Is determined.
At this time
Figure RE-GDA0003324231820000161
Can be expressed as:
Figure RE-GDA0003324231820000162
for the above
Figure RE-GDA0003324231820000163
The calculation of (a) is integrated indefinitely to obtain:
Figure RE-GDA0003324231820000164
wherein:
Figure RE-GDA0003324231820000165
Figure RE-GDA0003324231820000166
Figure RE-GDA0003324231820000167
Figure RE-GDA0003324231820000168
Figure RE-GDA0003324231820000169
Figure RE-GDA00033242318200001610
wherein P, Q, D, F, C, E are the sixth intermediate variable, the seventh intermediate variable, the eighth intermediate variable, the ninth intermediate variable, the tenth intermediate variable, and the eleventh intermediate variable, respectively.
It is to be understood that no intermediate variables used in the present invention have any specific meaning.
Step 5, combining a Lagrangian function of an equivalent optimization problem P2 and a KKT condition thereof, modeling an optimal structure of robust precoding as a solution of a generalized eigenvalue problem thereof, and obtaining an optimal precoding vector, namely an optimal solution of an original problem P1;
in particular, the Lagrangian function of the equivalence optimization problem P2
Figure RE-GDA00033242318200001611
Comprises the following steps:
Figure RE-GDA00033242318200001612
wherein v iskLagrange multiplier, q, representing the SINR constraint for the kth usernLagrange multiplier, e, corresponding to the nth single antenna power constraintnA vector representing that the nth element is 1 and other elements are 0, and the dimensionality of the vector is determined by a multiplication matrix of the nth element and the nth element; (.)TIndicating a transpose operation on a vector, then
Figure RE-GDA0003324231820000171
Represents a pair vector enIs transposed, PnAnd the upper limit value of the power of the nth antenna element is shown.
To simplify the calculation, let
Figure RE-GDA0003324231820000172
Then:
Figure RE-GDA0003324231820000173
wherein the content of the first and second substances,
Figure RE-GDA0003324231820000174
for lagrange function
Figure RE-GDA0003324231820000175
The derivation yields:
Figure RE-GDA0003324231820000176
optimal precoding vector based on KKT condition
Figure RE-GDA0003324231820000177
The requirements are satisfied:
Figure RE-GDA0003324231820000178
it is thus possible to obtain:
Figure RE-GDA0003324231820000179
order matrix
Figure RE-GDA00033242318200001710
And assumes an optimal precoding vector
Figure RE-GDA00033242318200001711
Figure RE-GDA00033242318200001712
For the optimal precoding vector, the optimal precoding vector can be obtained
Figure RE-GDA00033242318200001713
Is regarded asCorresponding matrix pair (S)k,Nk) The maximum generalized eigenvalue of the eigenvector of the problem, the maximum generalized eigenvalue being γk，
Figure RE-GDA00033242318200001714
And gammakRespectively as follows:
γk＝max.generalized eigenvalue(Sk,Nk)
Figure RE-GDA00033242318200001715
for convenience of presentation, the present embodiment utilizes
Figure RE-GDA00033242318200001716
Indicating that the value of the subscript K ranges from 1 to K,
Figure RE-GDA00033242318200001717
the value of the middle subscript N ranges from 1 to NTWhen lagrange multiplier
Figure RE-GDA00033242318200001718
And
Figure RE-GDA00033242318200001719
after determination, the direction of the optimal precoder
Figure RE-GDA00033242318200001720
And gammakCan be uniquely determined and its SINR constraint when the equivalent optimization problem P3 gets the optimal solution
Figure RE-GDA00033242318200001721
Will take equal sign, i.e. E { SINR at this timek}＝γkAccording to the condition, the optimal power distribution can be obtained, namely:
Figure RE-GDA0003324231820000181
where ρ is a power allocation vector and ρ ═ ρ1,…,ρK]TThe dimension of the matrix F is K, the K-th row and i-th column of the element [ F]k,iComprises the following steps:
Figure RE-GDA0003324231820000182
at this time, the optimal precoding vector can be calculated
Figure RE-GDA0003324231820000183
Further, to determine the Lagrangian multiplier
Figure RE-GDA0003324231820000184
And
Figure RE-GDA0003324231820000185
the present embodiment adopts a sub-gradient projection technique and combines the KKT condition qn≥0,
Figure RE-GDA0003324231820000186
And q isnPn＝0,
Figure RE-GDA0003324231820000187
Get about
Figure RE-GDA0003324231820000188
Satisfies the iterative expression Q:
Figure RE-GDA0003324231820000189
wherein Q isiExpressed as lagrange multiplier matrix corresponding to single antenna power constraint in the ith iteration process
Figure RE-GDA00033242318200001810
tiRepresents the iteration step size, and ti＝1/i。
According to the lagrangian function and the dual function of the equivalent optimization problem P3 and the strong dual relationship between the two, the following can be obtained:
Figure RE-GDA00033242318200001811
wherein, PTThe actual transmit power of the system.
Rearranging the above calculation formulas
Figure RE-GDA00033242318200001812
Equation for the dead point of (1):
Figure RE-GDA00033242318200001813
wherein the optimal Lagrange multiplier matrix
Figure RE-GDA00033242318200001814
In practical application, the constraint threshold value gamma of the target SINRkIt is desirable to maximize the value as much as possible, and determine v in conjunction with the convex optimization problem described belowkThe value of (c):
Figure RE-GDA00033242318200001815
Figure RE-GDA00033242318200001816
Figure RE-GDA00033242318200001817
wherein, PtotalRepresenting the maximum transmission power of the system.
In order to reduce the above optimization problemWill be with respect to vkThe equation for the dead point of (a) is approximated as:
Figure RE-GDA0003324231820000191
wherein the cofactor beta is calculatedk＝||hk||2And the optimization problem is solved through a water injection algorithm.
The optimization problem is solved by a water injection algorithm.
And 6, predicting a Lagrange multiplier required by the optimization problem based on the channel autocorrelation matrix by combining a machine learning method.
Specifically, the step 6 further comprises the following steps,
step 6-1, constructing a convolutional neural network; referring to the schematic diagram of fig. 4, the schematic diagram of the structure of the convolutional neural network includes an input layer, a convolutional layer, a batch normalization layer, an activation layer, a flat layer, a full connection layer, and an output layer.
Step 6-2, collecting training samples and training a convolutional neural network; the training samples comprise a channel autocorrelation parameter matrix
Figure RE-GDA0003324231820000192
And lagrange multiplier
Figure RE-GDA0003324231820000193
And
Figure RE-GDA0003324231820000194
and training the convolutional neural network in an off-line training mode.
6-3, training the neural network to obtain a channel autocorrelation parameter matrix
Figure RE-GDA0003324231820000195
With the real and imaginary parts of the lagrange multiplier as inputs
Figure RE-GDA0003324231820000196
And
Figure RE-GDA0003324231820000197
as an output;
step 6-4, inputting channel autocorrelation parameter matrix
Figure RE-GDA0003324231820000198
The Lagrange multiplier can be directly obtained through the neural network, an iterative process in an algorithm implementation process is avoided, algorithm implementation complexity is reduced, and the optimal precoding vector meeting the original problem is directly calculated.
It should be noted that the above-mentioned examples only represent some embodiments of the present invention, and the description thereof should not be construed as limiting the scope of the present invention. It should be noted that, for those skilled in the art, various modifications can be made without departing from the spirit of the present invention, and these modifications should fall within the scope of the present invention.

Claims (10)

1. A multi-beam satellite communication system robust precoding method based on machine learning is characterized in that: comprises the following steps of (a) carrying out,
step 1, constructing a multi-beam satellite downlink channel vector model containing user position positioning uncertainty based on position angle estimation errors of a multi-beam satellite for each user and public angle errors caused by satellite attitude orbit control;
step 2, solving mathematical expectations of the sum rate of the multi-beam satellite system with respect to positioning angle error variables to obtain the traversal transmission rate of the kth user, and obtaining a statistical channel model with respect to a channel autocorrelation matrix;
step 3, constructing a multi-beam satellite system and a robust precoding optimization design problem with maximized rate, wherein the constraint condition is that the power value of each antenna array element of the multi-beam satellite is smaller than a certain threshold value;
step 4, equivalently converting the robust precoding optimization design problem of the multi-beam satellite system and the rate maximization into a power minimization problem under the user signal-to-interference-and-noise ratio guarantee and the single antenna power constraint;
step 5, modeling the optimal structure of the robust precoding as the solution of the generalized characteristic value problem by combining the Lagrange function of the equivalent optimization problem and the KKT condition thereof, and obtaining the optimal precoding vector, namely the optimal solution of the original problem;
and 6, predicting a Lagrange multiplier required by the optimization problem based on the channel autocorrelation matrix by combining a machine learning method.
2. The machine learning based multi-beam satellite communication system robust precoding method of claim 1, wherein: the constructing of the multi-beam satellite downlink channel vector model in the step 1 further includes constructing an angle relationship as follows:
Figure RE-FDA0003324231810000011
Figure RE-FDA0003324231810000012
wherein, thetakThe exact angle of user k with respect to the y-axis of the low orbit satellite planar array antenna,
Figure RE-FDA0003324231810000013
planar array antenna for user k relative to low orbit satellitexThe exact angle of the shaft is such that,
Figure RE-FDA0003324231810000014
and
Figure RE-FDA0003324231810000015
indicating the satellite position angle estimation error for the k-th user,
Figure RE-FDA0003324231810000016
and
Figure RE-FDA0003324231810000017
representing common angle detection errors due to satellite attitude orbit control,
Figure RE-FDA0003324231810000018
expressed as the estimated angle of user k with respect to the y-axis of the low-orbit satellite planar array antenna,
Figure RE-FDA0003324231810000019
planar array antenna denoted as user k with respect to low orbit satellitexThe estimated angle of the axis.
3. The machine learning based multi-beam satellite communication system robust precoding method of claim 2, wherein: in the step 1, the corresponding vector of the planar array antenna
Figure RE-FDA00033242318100000110
The following relationship is satisfied:
Figure RE-FDA00033242318100000111
wherein the content of the first and second substances,
Figure RE-FDA00033242318100000112
representing a planar array antenna inxThe array element on the axis is a response vector,
Figure RE-FDA0003324231810000021
the array element response vector of the planar array antenna on the y axis is represented, and the two satisfy respectively:
Figure RE-FDA0003324231810000022
Figure RE-FDA0003324231810000023
4. the machine-learning based multi-beam satellite communication system robust precoding method of claim 3, wherein: the traversal and rate of the multi-beam satellite communication system of the kth user in the step 2
Figure RE-FDA0003324231810000024
Comprises the following steps:
Figure RE-FDA0003324231810000025
wherein the content of the first and second substances,
Figure RE-FDA0003324231810000026
and K is the total number of users,
Figure RE-FDA0003324231810000027
it is shown that the mathematical expectation symbol is calculated,
Figure RE-FDA0003324231810000028
representing the noise power value, wkPrecoding vector for the k-th user, wiPrecoding vector for ith user, hkDownlink channel parameter vector h for multi-beam satellite to kth useriDownlink channel parameter vector for multibeam satellite to ith user, superscript (·)HRepresenting a conjugate transpose operation on a vector or matrix,
Figure RE-FDA0003324231810000029
Figure RE-FDA00033242318100000210
representing a vector wiAnd hiThe conjugate transpose of (1);
SINRkSINR for the kth userkSatisfies the following conditions:
Figure RE-FDA00033242318100000211
wherein, wkFor the kth user's precoding vector, superscript (. cndot.)HDenotes the conjugate transpose, hkChannel correspondence vectors for user angle information for the k-th user from the multi-beam satellite.
5. The machine-learning based multi-beam satellite communication system robust precoding method of claim 4, wherein: the system traversal and rate in the step 3
Figure RE-FDA00033242318100000212
Without closed form expressions, enabling direct processing
Figure RE-FDA00033242318100000213
Are relatively difficult and are therefore determined here according to the Jackson inequality
Figure RE-FDA00033242318100000214
Upper limit of (2)
Figure RE-FDA00033242318100000215
Is composed of
Figure RE-FDA00033242318100000216
The robust precoding optimization design problem P1 modeling of the system and the rate upper limit maximization is represented as follows:
P1:
Figure RE-FDA00033242318100000217
Figure RE-FDA00033242318100000218
wherein P isnIs as followsnMaximum allowed transmit power of individual antenna elements, aggregate
Figure RE-FDA0003324231810000031
Satisfy the requirement of
Figure RE-FDA0003324231810000032
Collection
Figure RE-FDA0003324231810000033
Satisfy the requirement of
Figure RE-FDA0003324231810000034
6. The machine-learning based multi-beam satellite communication system robust precoding method of claim 5, wherein: in the step 4, the optimization problem P1 with the maximum system and speed under the constraint of single antenna power is equivalently converted into a power minimization problem P2 under the constraint of user signal-to-interference-and-noise ratio and single antenna power,
P2:
Figure RE-FDA0003324231810000035
Figure RE-FDA0003324231810000036
Figure RE-FDA0003324231810000037
user signal-to-interference-and-noise ratio guarantee constraint E { SINRkThe expression of can be approximated as:
Figure RE-FDA0003324231810000038
wherein the constraint threshold value of the target SINR is
Figure RE-FDA0003324231810000039
Figure RE-FDA00033242318100000310
To achieve the maximum sum rate for the problem P1,
Figure RE-FDA00033242318100000311
the derivation of the expression requires mathematical expectation of the angle error variables.
7. The machine-learning based multi-beam satellite communication system robust precoding method of claim 6, wherein: in the step 4, based on the multi-beam satellite downlink channel parameter model, the channel autocorrelation matrix is obtained
Figure RE-FDA00033242318100000312
The simplification is as follows:
Figure RE-FDA00033242318100000313
wherein the corresponding vector of the planar array antenna
Figure RE-FDA00033242318100000314
To (1) anThe items may be represented as:
Figure RE-FDA00033242318100000315
wherein the channel autocorrelation matrix
Figure RE-FDA00033242318100000316
Row m and column n elements of (1)
Figure RE-FDA00033242318100000317
Can be expressed as:
Figure RE-FDA00033242318100000318
wherein the content of the first and second substances,
Figure RE-FDA00033242318100000319
representing channel power values [. ]]m,nTo express taking the matrixmGo to the firstnElements of a column, (.)nThe representation takes the nth element of the vector,
Figure RE-FDA00033242318100000320
representing a pair vector
Figure RE-FDA00033242318100000321
The m-th element of (1)
Figure RE-FDA00033242318100000322
And conjugates of the nth element
Figure RE-FDA00033242318100000323
The product of (a) and (b) to obtain a mathematical expectation;
and:
Figure RE-FDA0003324231810000041
Figure RE-FDA0003324231810000042
Figure RE-FDA0003324231810000043
wherein a and b are a first intermediate variable and a second intermediate variable respectively;
if the satellite estimates the error of the position angle of the k user
Figure RE-FDA0003324231810000044
And
Figure RE-FDA0003324231810000045
obey a uniform distribution, namely:
Figure RE-FDA0003324231810000046
Figure RE-FDA0003324231810000047
wherein, thetaLAnd thetaURespectively representing angle errors
Figure RE-FDA0003324231810000048
The upper and lower limits of the values are,
Figure RE-FDA0003324231810000049
and
Figure RE-FDA00033242318100000410
respectively representing angle errors
Figure RE-FDA00033242318100000411
Upper and lower value limits;
the above formula represents
Figure RE-FDA00033242318100000412
In the interval U [ theta ]L,θU]Subject to a uniform distribution of the flux in the flux,
Figure RE-FDA00033242318100000413
in the interval
Figure RE-FDA00033242318100000414
Subject to a uniform distribution, further define:
Δθ＝θU-θL
Figure RE-FDA00033242318100000415
wherein, DeltaθAnd
Figure RE-FDA00033242318100000416
respectively representing angle errors
Figure RE-FDA00033242318100000417
And
Figure RE-FDA00033242318100000418
the difference between the upper and lower limits of (d);
then it is possible to obtain:
Figure RE-FDA00033242318100000419
will be in the above expression
Figure RE-FDA00033242318100000420
And
Figure RE-FDA00033242318100000421
integration is performed to obtain:
Figure RE-FDA00033242318100000422
wherein each intermediate variable is defined as
Figure RE-FDA00033242318100000423
Figure RE-FDA0003324231810000051
Figure RE-FDA0003324231810000052
Wherein A, B and Z are the third intermediate variable, the fourth intermediate variable and the fifth intermediate variable, respectively;
if it is
Figure RE-FDA0003324231810000053
Obey mean value of muθ,kVariance of
Figure RE-FDA0003324231810000054
The distribution of the gaussian component of (a) is,
Figure RE-FDA0003324231810000055
obey mean value of
Figure RE-FDA0003324231810000056
Variance of
Figure RE-FDA0003324231810000057
A Gaussian distribution of
Figure RE-FDA0003324231810000058
And
Figure RE-FDA0003324231810000059
is expressed as:
Figure RE-FDA00033242318100000510
Figure RE-FDA00033242318100000511
wherein the content of the first and second substances,
Figure RE-FDA00033242318100000512
to represent
Figure RE-FDA00033242318100000513
Is determined by the probability density function of (a),
Figure RE-FDA00033242318100000514
to represent
Figure RE-FDA00033242318100000515
A probability density function of;
at this time
Figure RE-FDA00033242318100000516
Can be expressed as:
Figure RE-FDA00033242318100000517
for the above
Figure RE-FDA00033242318100000518
The calculation of (a) is integrated indefinitely to obtain:
Figure RE-FDA00033242318100000519
wherein each intermediate variable is defined as
Figure RE-FDA00033242318100000520
Figure RE-FDA00033242318100000521
Figure RE-FDA00033242318100000522
Figure RE-FDA00033242318100000523
Figure RE-FDA00033242318100000524
Figure RE-FDA00033242318100000525
Wherein P, Q, D, F, C, E are the sixth intermediate variable, the seventh intermediate variable, the eighth intermediate variable, the ninth intermediate variable, the tenth intermediate variable, and the eleventh intermediate variable, respectively.
8. The machine-learning based multi-beam satellite communication system robust precoding method of claim 7, wherein: lagrangian function of the equivalence optimization problem P2 in the step 5
Figure RE-FDA0003324231810000061
Comprises the following steps:
Figure RE-FDA0003324231810000062
wherein the content of the first and second substances,
Figure RE-FDA0003324231810000063
for lagrange multipliers corresponding to SINR constraints,
Figure RE-FDA0003324231810000064
corresponding to NTLagrange multiplier of single antenna power constraint, enThe dimension of the vector which represents the nth element is 1 and other elements are 0 is determined by the multiplication matrix,
Figure RE-FDA0003324231810000065
represents a pair vector enIs transposed, PnRepresenting the upper limit value of the power of the nth antenna array element;
to simplify the calculation, let
Figure RE-FDA0003324231810000066
Then:
Figure RE-FDA0003324231810000067
wherein, the diagonal matrix
Figure RE-FDA0003324231810000068
For lagrange function
Figure RE-FDA0003324231810000069
The derivation yields:
Figure RE-FDA00033242318100000610
optimal precoding vector based on KKT condition
Figure RE-FDA00033242318100000611
The requirements are satisfied:
Figure RE-FDA00033242318100000612
it is thus possible to obtain:
Figure RE-FDA00033242318100000613
order matrix
Figure RE-FDA00033242318100000614
And assumes an optimal precoding vector
Figure RE-FDA00033242318100000615
Figure RE-FDA00033242318100000616
For the optimal precoding vector, the optimal precoding vector can be obtained
Figure RE-FDA00033242318100000617
Is regarded as a corresponding matrix pair (S)k,Nk) The maximum generalized eigenvalue of the eigenvector of the problem, the maximum generalized eigenvalue being γk，
Figure RE-FDA00033242318100000618
And gammakRespectively as follows:
γk＝max.generalized eigenvalue(Sk,Nk)
Figure RE-FDA0003324231810000071
while Lagrange multiplier
Figure RE-FDA0003324231810000072
And
Figure RE-FDA0003324231810000073
after determination, the direction of the optimal precoder
Figure RE-FDA0003324231810000074
And gammakCan be uniquely determined and its SINR constraint satisfies E { SINR when the equivalent optimization problem P3 gets the optimal solutionk}＝γkAccording to the condition, the optimal power distribution can be obtained, namely:
Figure RE-FDA0003324231810000075
where ρ is a power allocation vector and ρ ═ ρ1,…,ρK]TThe dimension of the matrix F is K × K, and the elements in the kth row and the ith column are:
Figure RE-FDA0003324231810000076
at this time, the optimal precoding vector can be calculated
Figure RE-FDA0003324231810000077
9. The machine-learning based multi-beam satellite communication system robust precoding method of claim 8, wherein: in said step 5, to determine the Lagrangian multiplier
Figure RE-FDA0003324231810000078
And
Figure RE-FDA0003324231810000079
optimum value of (2)By using sub-gradient projection technique in combination with KKT condition
Figure RE-FDA00033242318100000710
And
Figure RE-FDA00033242318100000711
get about
Figure RE-FDA00033242318100000712
The iterative expression Q of (a) is:
Figure RE-FDA00033242318100000713
wherein Q isiExpressed as lagrange multiplier matrix corresponding to single antenna power constraint in the ith iteration process
Figure RE-FDA00033242318100000714
tiRepresents the iteration step size, and ti＝1/i；
According to the lagrangian function and the dual function of the equivalent optimization problem P2 and the strong dual relationship between the two, the following can be obtained:
Figure RE-FDA00033242318100000715
wherein, PTActual transmit power for the system; rearranging the above calculation formulas
Figure RE-FDA00033242318100000716
Equation for the dead point of (1):
Figure RE-FDA00033242318100000717
among them, the optimum LagrangianMultiplier matrix
Figure RE-FDA00033242318100000718
In practical application, the constraint threshold value gamma of the target SINRkIt is desirable to maximize the value as much as possible, and determine v in conjunction with the convex optimization problem described belowkThe value of (c):
Figure RE-FDA0003324231810000081
Figure RE-FDA0003324231810000082
wherein, PtotalRepresents the system maximum transmission power; to reduce the complexity of solving the above optimization problem, we will refer to vkThe equation for the dead point of (a) is approximated as:
Figure RE-FDA0003324231810000083
wherein the cofactor beta is calculatedk＝||hk||2The optimization problem is solved by a water injection algorithm.
10. The machine-learning based multi-beam satellite communication system robust precoding method of claim 9, wherein: said step 6 further comprises the step of,
step 6-1, constructing a convolutional neural network;
step 6-2, collecting training samples and training a convolutional neural network;
6-3, training the neural network to obtain a channel autocorrelation parameter matrix
Figure RE-FDA0003324231810000084
With the real and imaginary parts of the lagrange multiplier as inputs
Figure RE-FDA0003324231810000085
And
Figure RE-FDA0003324231810000086
as an output;
step 6-4, inputting channel autocorrelation parameter matrix
Figure RE-FDA0003324231810000087
The lagrange multiplier is obtained directly through a neural network.
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