CN109510650B - Combined pre-coding method of multi-user bidirectional AF MIMO relay system - Google Patents

Combined pre-coding method of multi-user bidirectional AF MIMO relay system Download PDF

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CN109510650B
CN109510650B CN201910041755.9A CN201910041755A CN109510650B CN 109510650 B CN109510650 B CN 109510650B CN 201910041755 A CN201910041755 A CN 201910041755A CN 109510650 B CN109510650 B CN 109510650B
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CN109510650A (en
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禹永植
郭立民
彭立群
刘鲁涛
张未坤
侯培迟
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Harbin Engineering University
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    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04B—TRANSMISSION
    • H04B7/00—Radio transmission systems, i.e. using radiation field
    • H04B7/02—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas
    • H04B7/04—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas using two or more spaced independent antennas
    • H04B7/0413—MIMO systems
    • H04B7/0452—Multi-user MIMO systems
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04B—TRANSMISSION
    • H04B7/00—Radio transmission systems, i.e. using radiation field
    • H04B7/02—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas
    • H04B7/04—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas using two or more spaced independent antennas
    • H04B7/0413—MIMO systems
    • H04B7/0456—Selection of precoding matrices or codebooks, e.g. using matrices antenna weighting

Abstract

The invention discloses a joint pre-coding method of a multi-user bidirectional AF MIMO relay system, belonging to the technical field of communication. The invention adopts minimum sum mean square error (MSMSMSSE) as a design criterion under the power limit of all nodes, designs a Four-Step iterative algorithm (Four-Step iterative algorithm) to respectively solve the non-convex optimization problems of a combined source, a relay and multiple users, converts the initial non-convex optimization problem into a sub-optimization problem to be solved separately, and then alternately and iteratively optimizes each sub-problem based on the convex optimization design of the standard. The invention verifies that the multi-user bidirectional AF MIMO relay communication system has better sum-MSE performance, and the proposed algorithm has a rapid convergence speed under the condition of low signal-to-noise ratio.

Description

Combined pre-coding method of multi-user bidirectional AF MIMO relay system
Technical Field
The invention belongs to the technical field of communication, and particularly relates to a joint precoding method of a multi-user bidirectional AF MIMO relay system.
Background
In recent years, the development of wireless communication technology follows the pace of the information age, new wireless communication technology is in endless, and the development of wireless communication technology today is more urgent and needs to have the characteristics of large capacity, high frequency spectrum utilization rate, high transmission rate, high reliability and the like. In order to solve the requirements, a relay-assisted multiple-input-multiple-output (MIMO) technology is adopted, relay nodes are considered in an original wireless communication system, and a typical three-node communication system structure of 'source-relay-sink' is formed.
Bidirectional relays are gaining more and more attention as they can make up for the lack of spectral efficiency loss caused by unidirectional relays. In addition, in an actual bidirectional communication system, an amplitude-and-forwarding (AF) relay protocol is widely considered due to its features of low complexity, small processing delay, and low implementation cost. For a bidirectional AF MIMO relay system in which two source nodes exchange information through one relay node, documents Wang C L, Chen J Y, Jheng J J.A decoder design for two-way amplification-and-forward MIMO relay systems with linear receivers [ C ]// Vehicular Technology reference (VTC Fall) based on MSMSMSMSMSMSMSSE criterion propose a relay precoding design scheme, 2014 IEEE 80th. IEEE,2014: 1-5P. A precoding design of a joint source and relay based on a convex optimization design concept is proposed in the documents Fang B, Qian Z, Zhong W, et al. Joint design for source and relay precoding in AF-based MIMO two-way relay networks [ C ]/Signal and Information Processing (China SIP),2015 IEEE Chinese sum and International Conference on. IEEE 2015:953-957P, and the algorithm makes the sum mean square error (sum-MSE) of the system minimum. However, there are only few documents to date that research into joint precoding design of a bidirectional AF MIMO relay system in which one source node and a plurality of users exchange information through one relay node, and this system model is more practical in actual communication.
The invention considers the multi-user bidirectional AF MIMO relay system, and takes minimum sum mean square error (MSMSMSMSMSE) as the pre-coding design problem of the joint information source, the relay and the multi-user of the design criterion under the limit of all node transmitting power. The combined precoding problem of the multi-user bidirectional AF MIMO relay system is researched based on an MSMSMSSE design criterion, an equivalent sub-optimization problem is solved by using an alternative iteration method, a semi-definite programming (SDP) design and a square constraint quadratic programming (QCQP) design, and the complexity of mathematical operation is simplified.
Disclosure of Invention
The invention aims to provide a combined pre-coding method of a multi-user bidirectional AF MIMO relay system, which is based on a four-step iterative algorithm of an MSMSMSSE design criterion and further improves the performance of the multi-user bidirectional AF MIMO relay communication system.
The purpose of the invention is realized by the following technical scheme:
the method aims at the joint pre-coding problem of the multi-user bidirectional MIMO relay system, solves the non-convex optimization problem of joint information sources, relays and multiple users by using a four-step iterative algorithm, and considers a multi-user bidirectional AF MIMO relay system model which is formed by allocating N information source nodes and K usersrRelay node exchange information composition of antenna, supposing source node equipped with NbAn antenna, each user being provided with Nk(K ═ 1,2, …, K) antennas, as shown in fig. 1. It is assumed herein that the system operates in half-duplex mode, the whole communication process occurs in two transmission slots, and it is assumed that Channel State Information (CSI) remains unchanged in each slot.
A joint precoding method of a multi-user bidirectional AF MIMO relay system comprises the following steps:
(1) respectively calculating received signals of three nodes of a transmitting end, a relay and a receiving end in two transmission time slots;
(2) calculating a mean square error expression of signal waveforms of an information source node and a kth user, and constructing a joint precoding optimization problem expression of the multi-user bidirectional AF MIMO relay system based on MSMSMSSE design criteria, wherein k is a positive integer;
(3) fixed source precoding matrix B1User k precoding matrix B2,kAnd a user k precoding matrix F for respectively calculating the information source receiving filter matrix W1And user k receives the filter matrix W2,k；
(4) Fixing B1，B2,k，W1And W2,kUpdating a user k precoding matrix F by solving an SDP problem;
(5) fixing B2,k，F，W1And W2,kUpdating the Source precoding matrix B by solving the SDP problem1；
(6) According to updated B1，F，W1And W2,kObtaining an updated user k precoding matrix B by solving the QCQP problem2,k；
(7) And (4) judging whether a termination standard is met, if so, ending the iteration, otherwise, skipping to the step (3) to continue the iteration until a convergence condition is met.
The step (2) specifically comprises the following steps:
(2.1) the mean square error matrix of the signal waveform estimate of the source node can be expressed as:
Figure BDA0001947793200000021
estimation of Mean Square Error (MSE) of signal waveform at kth user2,k) The matrix can be represented as:
Figure BDA0001947793200000022
wherein the content of the first and second substances,
Figure BDA0001947793200000023
as source node equivalent noise vector
Figure BDA0001947793200000024
The covariance matrix of (a) is determined,
Figure BDA0001947793200000031
for the equivalent noise vector at the k-th user
Figure BDA0001947793200000032
The covariance matrix of (a);
(2.2) under the condition of limiting the power of all nodes, the joint precoding optimization problem of the multi-user bidirectional AF MIMO relay system based on the MSMSMSSE design rule is represented as follows:
Figure BDA0001947793200000033
Figure BDA0001947793200000034
Figure BDA0001947793200000035
Figure BDA0001947793200000036
the step (3) specifically comprises the following steps:
according to the wiener filtering theory, the MSE of the expression (1) is enabled1Minimum source receive filter matrix W1And making MSE of expression (2)2,kThe smallest k-th user receives the filter matrix W2,kThe specific expression is as follows:
Figure BDA0001947793200000037
Figure BDA0001947793200000038
the step (4) specifically comprises the following steps:
further finishing the formulas (1) and (2), MSE1And MSE2,kThe following expressions are given, respectively:
Figure BDA0001947793200000039
Figure BDA00019477932000000310
wherein there are the following variable substitutions
Figure BDA00019477932000000311
Figure BDA00019477932000000312
Figure BDA00019477932000000313
Figure BDA00019477932000000314
The relay power limitation condition may be further modified to:
Figure BDA00019477932000000315
the step (5) specifically comprises the following steps:
using the schulk's theorem, the initial optimization problem (3) - (6) can be further translated into the SDP problem for the matrix variable F as follows:
Figure BDA0001947793200000041
Figure BDA0001947793200000042
Figure BDA0001947793200000043
Figure BDA0001947793200000044
wherein the auxiliary variable p1Satisfies p1≥MSE1，p2,kSatisfies p2,k≥MSE2,k。
The step (6) specifically comprises the following steps:
(6.1) and matrix variable B1Related MSE2,kThe expression can be converted into:
Figure BDA0001947793200000045
wherein:
Figure BDA0001947793200000046
Figure BDA0001947793200000047
(6.2) Using the Schuler's theorem, the initial optimization problems (3) - (6) can be further translated as follows with respect to the equivalent variable vec (B)1) SDP problem of (1):
Figure BDA0001947793200000048
Figure BDA0001947793200000049
Figure BDA00019477932000000410
wherein the auxiliary variable
Figure BDA0001947793200000051
Satisfy the requirement of
Figure BDA0001947793200000052
(6.3) and matrix variable B2,kRelated MSE1The expression can be converted into:
Figure BDA0001947793200000053
wherein the content of the first and second substances,
Figure BDA0001947793200000054
Dkkis formed by a matrix DkFrom the first to
Figure BDA0001947793200000055
Go to
Figure BDA0001947793200000056
A matrix of rows;
(6.4) define the following variable substitutions:
Figure BDA0001947793200000057
Figure BDA0001947793200000058
Figure BDA0001947793200000059
based on the above analysis, the initial optimization problems (3) - (6) can be further translated as follows with respect to the equivalent variable b2QCQP problem of (a):
Figure BDA00019477932000000510
Figure BDA00019477932000000511
Figure BDA00019477932000000512
wherein
Figure BDA00019477932000000513
And
Figure BDA00019477932000000514
Figure BDA00019477932000000515
at the same time
Figure BDA00019477932000000516
QCQP problems (14) - (16) can solve the equivalent variable b through a convex optimization tool box CVX2To obtain an optimized variable B2,k(K ═ 1,2, …, K) for optimum values.
The invention has the beneficial effects that:
the invention provides a novel joint precoding design method for a multi-user bidirectional AF MIMO relay system, a four-step iterative algorithm is designed to solve the non-convex optimization problem of a joint information source, a relay and multiple users, the initial optimization problem is converted into a sub-optimization problem to be solved independently, and then each sub-problem is optimized alternately and iteratively based on a standard convex optimization design. Under the condition of power limitation of all nodes, the MSMSMSSE is used as a design criterion, and the user bidirectional AF MIMO relay system has good performance. Meanwhile, the four-step iterative algorithm designed by the invention has good convergence under the condition of low signal-to-noise ratio.
Drawings
Fig. 1 is a model of a multi-user bidirectional AF MIMO relay system;
FIG. 2 is a plot of system and minimum mean square error (sum-MSE) performance versus iteration number.
Detailed Description
The following further describes embodiments of the present invention with reference to the accompanying drawings:
the method comprises the following steps: and respectively calculating the received signals of the three nodes of the transmitting end, the relay and the receiving end in two transmission time slots.
In the first transmission time slot, the source node and K users simultaneously transmit respective signals x1＝B1s1And x2,k＝B2,ks2,kTo the relay node, adding noise nrThe expression is:
Figure BDA0001947793200000061
wherein Hr1Is a channel matrix from a source node to a relay node, GrkFor the channel matrix between the k-th user and the relay node, nrDefined as complex Additive White Gaussian Noise (AWGN) at the relay node,
Figure BDA0001947793200000062
is the noise power at the relay node. Order to
Figure BDA0001947793200000063
The relay receives the signal vector yrIt can be further rewritten that:
Figure BDA0001947793200000064
in the second transmission time slot, the relay node processes the received signal y through the relay preprocessing matrix FrProcessing the signal vector x after linear processingrAnd simultaneously forwarding to the source node and all users:
Figure BDA0001947793200000065
Figure BDA0001947793200000066
wherein the content of the first and second substances,
Figure BDA0001947793200000067
Prthe maximum transmit power at the relay node. In addition, the power limits at the source node and the kth user respectively satisfy
Figure BDA0001947793200000068
And
Figure BDA0001947793200000069
Ps1and Ps2Defined as the maximum transmit power at the source node and the kth user, respectively. Received signal vector y at source node1And a received signal vector y at the k-th user2,kCan be expressed as follows:
Figure BDA00019477932000000610
Figure BDA0001947793200000071
wherein H1rFor MIMO channel matrix between relay node to source node, GkrIs the MIMO channel matrix between the relay node to the kth user. In addition, n1Defined as the complex AWGN at the source node,
Figure BDA0001947793200000072
defined as the complex AWGN at the kth user,
Figure BDA0001947793200000073
and
Figure BDA0001947793200000074
at the source node and the k-th user, respectivelyThe noise power.
The received signal vector at the source node and the kth user can be further represented as:
Figure BDA0001947793200000075
Figure BDA0001947793200000076
wherein the content of the first and second substances,
Figure BDA0001947793200000077
and
Figure BDA0001947793200000078
in addition, the first and second substrates are,
Figure BDA0001947793200000079
is an equivalent noise vector at the source node,
Figure BDA00019477932000000710
the equivalent noise vector at the kth user,
Figure BDA00019477932000000711
adjacent Interference (AI) for users other than the kth user.
Step two: and calculating a Mean Square Error (MSE) expression of signal waveforms at the information source node and the kth user, and constructing a joint precoding optimization problem expression of the multi-user bidirectional AF MIMO relay system based on an MSMSMSMSMSSE design criterion.
Signal waveform estimation Mean Square Error (MSE) of source node1) Estimation of Mean Square Error (MSE) of matrix and signal waveform at kth user2,k) The matrices can be directly represented as:
Figure BDA00019477932000000712
Figure BDA00019477932000000713
wherein the content of the first and second substances,
Figure BDA00019477932000000714
as source node equivalent noise vector
Figure BDA00019477932000000715
The covariance matrix of (a) is determined,
Figure BDA00019477932000000716
for the equivalent noise vector at the k-th user
Figure BDA00019477932000000717
The covariance matrix of (2).
Under the condition of limiting the power of all nodes, the joint precoding optimization problem of the multi-user bidirectional AF MIMO relay system based on the MSMSMSMSE design criterion is expressed as follows:
Figure BDA00019477932000000718
Figure BDA0001947793200000081
Figure BDA0001947793200000082
Figure BDA0001947793200000083
step three: fixing B1，B2,kAnd F, respectively calculating the information source receiving filter matrixes W by using the MSE expression1And user k receives the filter matrix W2,k。
According to wienerFiltering theory such that MSE of expression (9)1Minimum source receive filter matrix W1And making MSE of expression (10)2,kThe smallest k-th user receives the filter matrix W2,kThe specific expression is as follows:
Figure BDA0001947793200000084
Figure BDA0001947793200000085
step four: fixing B1，B2,k，W1And W2,kUpdating a relay forwarding matrix F by solving an SDP problem;
MSE1and MSE2,kThe following expressions are given, respectively:
MSE1:
Figure BDA0001947793200000086
Figure BDA0001947793200000087
Figure BDA0001947793200000088
MSE2,k(k＝1,2,…,K):
Figure BDA0001947793200000089
Figure BDA00019477932000000810
Figure BDA00019477932000000811
Figure BDA00019477932000000812
wherein the content of the first and second substances,
Figure BDA00019477932000000813
and
Figure BDA00019477932000000814
by working out the above equation, the following expression can be obtained:
Figure BDA00019477932000000815
Figure BDA0001947793200000091
wherein, for expressions (26) and (27), the following variables are substituted:
Figure BDA0001947793200000092
the relay power limitation condition may be further modified to:
Figure BDA0001947793200000093
according to the above analysis, order
Figure BDA0001947793200000094
Using the schulk's theorem, the initial optimization problem (11) - (14) can be further translated into the SDP problem for the matrix variable F as follows:
Figure BDA0001947793200000095
Figure BDA0001947793200000096
Figure BDA0001947793200000097
Figure BDA0001947793200000098
wherein the auxiliary variable p1Satisfies p1≥MSE1，p2,kSatisfies p2,k≥MSE2,k. Obviously, the problems (30) - (33) are standard convex optimization problems and the optimized values of the optimization variable F can be solved by the convex optimization toolset CVX.
Step five: fixing B2,k，F，W1And W2,kUpdating the Source precoding matrix B by solving the SDP problem1。
Order to
Figure BDA0001947793200000099
According to expressions (17) and (18), and a matrix variable B1Related MSE2,kThe expression can be converted into:
Figure BDA00019477932000000910
Figure BDA00019477932000000911
Figure BDA00019477932000000912
expressions (35) to (37) are substituted in (10), MSE2,kCan be expressed byConversion to:
Figure BDA0001947793200000101
wherein the content of the first and second substances,
Figure BDA0001947793200000102
Figure BDA0001947793200000103
and
Figure BDA0001947793200000104
according to the above analysis, order
Figure BDA0001947793200000105
Using the Schuler's theorem, the initial optimization problems (11) - (14) can be further translated as follows with respect to the equivalent variable b1SDP problem of (1):
Figure BDA0001947793200000106
Figure BDA0001947793200000107
Figure BDA0001947793200000108
wherein the auxiliary variable
Figure BDA0001947793200000109
Satisfy the requirement of
Figure BDA00019477932000001010
Obviously, the SDP problems (39) - (41) are a standard convex optimization problem and the equivalent variable b can be solved by the convex optimization toolkit CVX1To obtain an optimized variable B1The optimum value of (c).
Step six: according to updated B1，F，W1And W2,kObtaining an updated user k precoding matrix B by solving the QCQP problem2,k；
Order to
Figure BDA00019477932000001011
By definition
Figure BDA00019477932000001012
Then the and matrix variable B2,kRelated MSE1The expression can be converted into:
Figure BDA00019477932000001013
wherein the content of the first and second substances,
Figure BDA0001947793200000111
Dkkis formed by a matrix DkFrom the first to
Figure BDA0001947793200000112
Go to
Figure BDA0001947793200000113
A matrix of rows. In addition, the following variable substitutions are defined:
Figure BDA0001947793200000114
based on the above analysis, the initial optimization problems (11) - (14) can be further translated as follows with respect to the equivalent variable b2QCQP problem of (a):
Figure BDA0001947793200000115
Figure BDA0001947793200000116
Figure BDA0001947793200000117
wherein
Figure BDA0001947793200000118
And
Figure BDA0001947793200000119
Figure BDA00019477932000001110
at the same time
Figure BDA00019477932000001111
QCQP problems (44) - (46) can solve the equivalent variable b through a convex optimization tool box CVX2To obtain an optimized variable B2,k(K ═ 1,2, …, K) for optimum values.
Step seven: and judging whether the termination standard is met, if so, ending the iteration, otherwise, skipping to the third step and continuing the iteration until the convergence condition is met.
The performance of the algorithm was further verified by simulation in conjunction with fig. 1:
experimental scenario
The noise power at all nodes is the same, i.e.
Figure BDA00019477932000001112
The maximum transmission power of all transmitting nodes is the same, i.e. Ps1＝Ps2＝PsAnd all channel matrices are quasi-statically distributed. In the following discussion, the signal-to-noise ratio (SNR) of all channel links is expressed as SNR, i.e., SNR is SNRsr＝SNRrd. In addition, considering that the system model consists of two users, i.e., K is 2, the number of antennas provided for all nodes is: n is a radical ofb＝Nr＝4，
Figure BDA00019477932000001113
Wherein all areNumber of user antennas satisfies
Figure BDA00019477932000001114
Analysis of Experimental content
Under the condition that the SNR is 10dB, the curve of the sum-MSE performance of the system changes along with the iteration number, and as can be clearly seen from the simulation result, the sum-MSE performance curve of the proposed Four-Step iteration algorithm (Four-Step algorithm) has a rapid descending trend and is close to the lower limit of the system. In particular, the proposed Four-Step iterative algorithm (Four-Step algorithm) has good convergence performance in low SNR situations.
The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention, and various modifications and changes may be made by those skilled in the art. Any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the protection scope of the present invention.

Claims (6)

1. A joint precoding method of a multi-user bidirectional AF MIMO relay system is characterized by comprising the following steps:
(1) respectively calculating received signals of three nodes of a transmitting end, a relay and a receiving end in two transmission time slots;
(2) calculating a signal waveform mean square error expression of an information source node and a kth user, and constructing a joint precoding optimization problem expression of the multi-user bidirectional AF MIMO relay system under a minimum sum mean square error (MSMSMSMSSE) design criterion, wherein k is a positive integer;
(3) fixed source precoding matrix B1User k precoding matrix B2,kAnd a relay forwarding matrix F for respectively calculating the source receiving filter matrix W1And user k receives the filter matrix W2,k；
(4) Fixing B1，B2,k，W1And W2,kUpdating a relay forwarding matrix F by solving a semi-definite programming (SDP) problem;
(5) fixing B2,k，F，W1And W2,kUpdating the source precoding matrix B by solving a semi-definite programming (SDP) problem1；
(6) According to updated B1，F，W1And W2,kObtaining an updated user k precoding matrix B by solving a Quadratic Constraint Quadratic Programming (QCQP) problem2,k；
(7) Judging whether a termination standard is met, if so, ending iteration, otherwise, skipping to the step (3) to continue iteration until a convergence condition is met;
the step (2) specifically comprises the following steps:
(2.1) the mean square error matrix of the signal waveform estimate of the source node can be expressed as:
Figure FDA0003010217590000011
estimation of Mean Square Error (MSE) of signal waveform at kth user2,k) The matrix can be represented as:
Figure FDA0003010217590000012
wherein the content of the first and second substances,
Figure FDA0003010217590000013
as source node equivalent noise vector
Figure FDA0003010217590000014
The covariance matrix of (a) is determined,
Figure FDA0003010217590000015
for the equivalent noise vector at the k-th user
Figure FDA0003010217590000016
Covariance matrix of H1rFor MIMO channel matrix between relay node to source node, GkrFor relaying between node and kth userA matrix of a MIMO channel is formed,
Figure FDA0003010217590000017
and
Figure FDA0003010217590000018
noise power at the source node and the kth user respectively,
Figure FDA0003010217590000019
is the noise power at the relay node;
(2.2) under the condition of limiting the power of all nodes, the joint precoding optimization problem of the multi-user bidirectional AF MIMO relay system based on the MSMSMSSE design rule is represented as follows:
Figure FDA00030102175900000110
Figure FDA0003010217590000021
Figure FDA0003010217590000022
Figure FDA0003010217590000023
wherein, PrMaximum transmit power, P, at the relay nodes1And Ps2Respectively defining the maximum transmitting power at the source node and the k user;
the step (4) specifically comprises the following steps:
further converting the initial optimization problems (3) - (6) into SDP problems about the matrix variable F by using the schuler's complement theorem;
the step (5) specifically comprises the following steps:
to and moment variable B1Related MSE2,kConverting the expression;
the initial optimization problems (3) - (6) are further converted into the equivalent variable vec (B) by using the Schuler's complement theorem1) The SDP problem of (1);
the step (6) specifically comprises the following steps:
to and matrix variable B2,kRelated MSE1Converting the expression;
translating the initial optimization problem into a relation to an equivalent variable b2The QCQP problem of (1);
solving the equivalent variable b of the QCQP problem through a convex optimization tool box CVX2To obtain an optimized variable B2,kThe optimum value of (c).
2. The joint precoding method of the multi-user bidirectional AF MIMO relay system as claimed in claim 1, wherein said step (3) specifically comprises:
according to the wiener filtering theory, the MSE of the expression (1) is enabled1Minimum source receive filter matrix W1And making MSE of expression (2)2,kThe smallest k-th user receives the filter matrix W2,kThe specific expression is as follows:
Figure FDA0003010217590000024
Figure FDA0003010217590000025
3. the joint precoding method of the multi-user bidirectional AF MIMO relay system as claimed in claim 1, wherein said step (4) specifically comprises:
further finishing the formulas (1) and (2), MSE1And MSE2,kThe following expressions are given, respectively:
Figure FDA0003010217590000026
Figure FDA0003010217590000027
wherein there are the following variable substitutions
Figure FDA0003010217590000031
Figure FDA0003010217590000032
Figure FDA0003010217590000033
Figure FDA0003010217590000034
The relay power limitation condition may be further modified to:
Figure FDA0003010217590000035
4. the joint precoding method of the multi-user bidirectional AF MIMO relay system as claimed in claim 1, wherein said step (4) specifically comprises:
using the schulk's theorem, the initial optimization problem (3) - (6) can be further translated into the SDP problem for the matrix variable F as follows:
Figure FDA0003010217590000036
Figure FDA0003010217590000037
Figure FDA0003010217590000038
Figure FDA0003010217590000039
wherein the auxiliary variable p1Satisfies p1≥MSE1，p2,kSatisfies p2,k≥MSE2,k。
5. The joint precoding method of the multi-user bidirectional AF MIMO relay system as claimed in claim 1, wherein the step (5) specifically comprises:
(5.1) and matrix variable B1Related MSE2,kThe expression can be converted into:
Figure FDA00030102175900000310
wherein:
Figure FDA00030102175900000311
Figure FDA00030102175900000312
Hr1a channel matrix from the information source node to the relay node;
(5.2) Using the Schuler's theorem, the initial optimization problems (3) - (6) can be further translated as follows with respect to the equivalent variable vec (B)1) Is/are as followsSDP problem:
Figure FDA0003010217590000041
Figure FDA0003010217590000042
Figure FDA0003010217590000043
wherein the auxiliary variable
Figure FDA0003010217590000044
Satisfy the requirement of
Figure FDA0003010217590000045
6. The joint precoding method of the multi-user bidirectional AF MIMO relay system as claimed in claim 1, wherein the step (6) specifically comprises:
(6.1) and matrix variable B2,kRelated MSE1The expression can be converted into:
Figure FDA0003010217590000046
wherein the content of the first and second substances,
Figure FDA0003010217590000047
Dkkis formed by a matrix DkFrom the first to
Figure FDA0003010217590000048
Go to
Figure FDA0003010217590000049
Matrix of rows, Gr1A channel matrix from the 1 st user of the user side to the relay node;
(6.2) define the following variable substitutions:
Figure FDA00030102175900000410
Figure FDA00030102175900000411
Figure FDA00030102175900000412
based on the above analysis, the initial optimization problems (3) - (6) can be further translated as follows with respect to the equivalent variable b2QCQP problem of (a):
Figure FDA00030102175900000413
Figure FDA00030102175900000414
Figure FDA00030102175900000415
wherein
Figure FDA00030102175900000416
And
Figure FDA00030102175900000417
Figure FDA00030102175900000418
at the same time
Figure FDA00030102175900000419
GrkFor the channel matrix between the kth user and the relay node, the QCQP problems (14) - (16) can solve an equivalent variable b through a convex optimization tool box CVX2To obtain an optimized variable B2,kK is 1, 2.
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