CN116155325A - Mixed precoding design method based on element-by-element iteration in large-scale MIMO system - Google Patents

Mixed precoding design method based on element-by-element iteration in large-scale MIMO system Download PDF

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CN116155325A
CN116155325A CN202310027929.2A CN202310027929A CN116155325A CN 116155325 A CN116155325 A CN 116155325A CN 202310027929 A CN202310027929 A CN 202310027929A CN 116155325 A CN116155325 A CN 116155325A
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姜园
李俊杰
赵磊
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Sun Yat Sen 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
    • 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
    • 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
    • H04B7/06—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas using two or more spaced independent antennas at the transmitting station
    • H04B7/0613—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas using two or more spaced independent antennas at the transmitting station using simultaneous transmission
    • H04B7/0615—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas using two or more spaced independent antennas at the transmitting station using simultaneous transmission of weighted versions of same signal
    • H04B7/0619—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas using two or more spaced independent antennas at the transmitting station using simultaneous transmission of weighted versions of same signal using feedback from receiving side
    • H04B7/0621—Feedback content
    • H04B7/0632—Channel quality parameters, e.g. channel quality indicator [CQI]
    • 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
    • H04B7/08—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas using two or more spaced independent antennas at the receiving station
    • H04B7/0837—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas using two or more spaced independent antennas at the receiving station using pre-detection combining
    • H04B7/0842—Weighted combining
    • H04B7/0848—Joint weighting
    • H04B7/0854—Joint weighting using error minimizing algorithms, e.g. minimum mean squared error [MMSE], "cross-correlation" or matrix inversion
    • 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
    • H04B7/08—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas using two or more spaced independent antennas at the receiving station
    • H04B7/0837—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas using two or more spaced independent antennas at the receiving station using pre-detection combining
    • H04B7/0842—Weighted combining
    • H04B7/0848—Joint weighting
    • H04B7/0857—Joint weighting using maximum ratio combining techniques, e.g. signal-to- interference ratio [SIR], received signal strenght indication [RSS]
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    • Y02—TECHNOLOGIES OR APPLICATIONS FOR MITIGATION OR ADAPTATION AGAINST CLIMATE CHANGE
    • Y02D—CLIMATE CHANGE MITIGATION TECHNOLOGIES IN INFORMATION AND COMMUNICATION TECHNOLOGIES [ICT], I.E. INFORMATION AND COMMUNICATION TECHNOLOGIES AIMING AT THE REDUCTION OF THEIR OWN ENERGY USE
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    • Y02D30/70—Reducing energy consumption in communication networks in wireless communication networks

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Abstract

The invention discloses a mixed precoding design method based on element-by-element iteration in a large-scale MIMO system, which adopts a full-connection type mixed precoding structure at both receiving and transmitting ends, namely a structure that each radio frequency link is connected with all antennas, and provides a mixed precoding method aiming at reducing error rate in a single-user large-scale MIMO system. The invention carries out mathematical modeling on the design problem of the mixed precoding with minimized error rate, and provides a mixed precoding algorithm with low complexity and excellent performance based on the thought of element-by-element iteration.

Description

Mixed precoding design method based on element-by-element iteration in large-scale MIMO system
Technical Field
The invention relates to the technical field of communication, in particular to a mixed precoding design method based on element-by-element iteration in a large-scale MIMO system.
Background
With the development of globalization of information, the number of access terminals of a wireless network is increased, and spectrum resources are increasingly tensioned. The large-scale Multiple-Input Multiple-Output (MIMO) technology is an important technical support of the fifth generation mobile communication system (5G), and the network capacity improving technology and method based on the large-scale MIMO can improve the throughput of the communication system and alleviate the problem of spectrum resource shortage.
The precoding technology can adjust array elements of an antenna array to generate directional beams, and enhance the strength of signals in an expected direction to improve the system capacity, which is a key technology for improving the system capacity in a 5G communication system. The traditional all-digital precoding technology needs to be equipped with a large number of expensive radio frequency links, and has high hardware circuit cost and large power consumption, so that the traditional all-digital precoding technology is difficult to apply to a practical large-scale MIMO system. Thus, hybrid precoding techniques are widely studied as a low cost alternative. The idea of the hybrid pre-coding technique is to replace the all-digital pre-coder with a low-dimensional digital pre-coder which can be amplitude-modulated and phase-modulated and a high-dimensional analog pre-coder which can only be amplitude-modulated, and the number of the required radio frequency links is determined by the dimension of the digital pre-coder, so that the hybrid pre-coder only needs a small number of radio frequency links, has low cost and low power consumption and is easier to realize practically. In general, the design of the hybrid precoding scheme needs to comprehensively consider four variables at the receiving and transmitting ends, namely, a digital precoder and an analog precoder at the transmitting end, a digital combiner and an analog combiner at the receiving end, and because the analog precoder and the analog combiner are implemented by a phase shifter, non-convex constant-mode constraint can be introduced during the design, and the difficulty of the design is increased.
At present, research on hybrid precoding technology is mainly focused on improving the spectral efficiency or energy efficiency of a system, and these indexes are important, but for a communication system with a slightly high information transmission reliability requirement, a hybrid precoder designed with the aim of improving the spectral efficiency or energy efficiency may not meet the requirement.
Disclosure of Invention
The invention provides a mixed precoding design method based on element-by-element iteration in a large-scale MIMO system, aiming at reducing error rate, and provides a corresponding mixed precoding scheme based on the basic idea of element-by-element iteration.
In order to solve the technical problems, the technical scheme of the invention is as follows:
a mixed precoding design method based on element-by-element iteration in a massive MIMO system comprises the following steps:
s1: obtaining a channel matrix H;
s2: constructing an error rate minimization optimization problem and converting the error rate minimization optimization problem into a form which is easy to solve;
s3: initializing an analog combination matrix W and an auxiliary variable P, wherein the auxiliary variable P replaces a mixed pre-coding matrix T, and enabling the mixed pre-coder to approach the auxiliary variable P after the analog combination matrix W and the auxiliary variable P are alternately optimized;
s4: fixing the auxiliary variables P, and updating the simulation combination matrix W column by column and element by element;
s5: fixing a simulation combination matrix W, and updating an auxiliary variable P according to Karush-Kuhn-Tucker conditions;
s6: step S4 and step S5 are alternately carried out until a termination condition is triggered, and then an analog precoding matrix F is initialized;
s7: solving an analog precoding matrix F;
s8: solving a digital precoding matrix B;
s9: solving a digital combination matrix D;
s10: according to the results of steps S3 to S9, a final hybrid precoding matrix t=fb and a hybrid combining matrix r=wd are obtained.
Preferably, in step S2, an error rate minimization optimization problem is constructed and converted into a form that is easy to solve, specifically:
the ith data stream in the processed signal y at the receiving end has the following form:
Figure BDA0004045958800000021
wherein y is i 、d i 、b i 、s i Respectively representing the ith data stream of the signal y processed by the receiving end, the ith column of the digital combination matrix D, the ith column of the digital precoding matrix B and the ith data stream of the signal s to be transmitted;
average bit error rate P of each data stream e Expressed as:
Figure BDA0004045958800000022
in the method, in the process of the invention,
Figure BDA0004045958800000023
function representing a characteristic error rate related to signal-to-interference-plus-noise ratio of a signal, SINR i Representing the signal-to-interference-and-noise ratio, N, of the ith data stream s Representing the dimension of the transmitted data stream;
the receiving end adopts a digital minimum mean square error combination matrix, which is expressed as:
D＝(σ 2 W H W+W H HTT H H H W) -1 W H HT
the signal-to-interference-plus-noise ratio SINR of the ith data stream i Can be expressed as:
Figure BDA0004045958800000031
in the method, in the process of the invention,
Figure BDA0004045958800000032
represents an MSE matrix, [ E ]] [i,i] Elements representing the ith row and column of MSE matrix E, < >>
Figure BDA0004045958800000033
Represents N s Order unit array, sigma 2 Representing noise variance->
Figure BDA0004045958800000034
Figure BDA0004045958800000035
Representing the pseudo-inverse of the analog combining matrix W;
the average bit error rate of each data stream has a lower bound as follows:
Figure BDA0004045958800000036
wherein the inequality sign is given by the qin inequality, which takes the equality sign when the diagonal elements of the matrix E are all equal. When the number of antennas is large, the analog combining matrix tends to satisfy the orthogonal constraint
Figure BDA0004045958800000037
The MSE matrix E is approximately expressed as:
Figure BDA0004045958800000038
wherein E is 1 Representing the approximated MSE matrix, the bit error rate minimization hybrid precoder design problem to be solved is modeled in the following form, taking into account the constant modulus constraint introduced by the phase shifter and the power constraint at the transmit end:
Figure BDA0004045958800000039
/>
Figure BDA00040459588000000310
Figure BDA00040459588000000311
tr(FBB H F H )≤P max
wherein tr (E) 1 ) Representation E 1 Is [ F ]] [i,j] Representing an element on the ith row and jth column of the analog precoding matrix F, [ W ]] [i,j] Representing elements on the ith row and jth column of the simulated combining matrix W, P max Indicating the maximum transmit power at the transmitting end.
Preferably, in step S3, the analog combining matrix W and the auxiliary variable P are initialized, specifically:
the initialization of the analog combination matrix W adopts a mode of randomly selecting phases, and the initialization of P of auxiliary variables is given by Karush-Kuhn-Tucker conditions, and the form is as follows:
Figure BDA00040459588000000312
in the method, in the process of the invention,
Figure BDA0004045958800000041
representing equivalent channel->
Figure BDA0004045958800000042
Front N of right singular matrix after singular value decomposition s Matrix of columns, Λ H，1 Front N of singular value matrix after singular value decomposition of channel matrix H s Front N of line s Matrix of column elements>
Figure BDA0004045958800000043
Represents N s An order DFT matrix.
Preferably, in step S4, the auxiliary variables P are fixed, and the analog combining matrix W is updated column by column and element by element, specifically:
rewriting an objective function of the error rate minimization optimization problem:
Figure BDA0004045958800000044
wherein N is r Indicating the number of receive antennas and,
Figure BDA0004045958800000045
Figure BDA0004045958800000046
represents N r An array of order units,
Figure BDA0004045958800000047
representing the submatrices obtained by removing the j-th column from the analog combined matrix W, W j Representation ofSimulating the j-th column of the combined matrix W, the above optimization problem is equivalent to:
Figure BDA0004045958800000048
Figure BDA0004045958800000049
in the method, in the process of the invention,
Figure BDA00040459588000000410
according to the split programming theory, by introducing an auxiliary variable tau j The above problems can be translated into:
Figure BDA00040459588000000411
Figure BDA00040459588000000412
variable τ j 、w j The following optimal solutions are respectively provided:
Figure BDA00040459588000000413
Figure BDA00040459588000000414
/>
in [ w ] j ] [i] Representing a column vector w j Is (x) =x/|x| represents a phase extraction operation;
updating τ alternately according to the optimal solution j And w j Until convergence, the j-th column of the simulation combination matrix W is updated, and other columns of the simulation combination matrix W are updated in the same step to complete the simulation combination matrixUpdating W.
Preferably, in step S5, the analog combining matrix W is fixed, and the auxiliary variable P is updated according to Karush-Kuhn-turner conditions, specifically:
Figure BDA0004045958800000051
preferably, the termination condition in step S6 is set such that the absolute value of the difference between the objective functions of the two iterations is smaller than a preset constant.
Preferably, in step S6, the analog precoding matrix F is initialized, specifically:
for equivalent channels
Figure BDA0004045958800000052
Front N of right singular matrix after singular value decomposition s Matrix of columns->
Figure BDA0004045958800000053
The phase extraction operation is performed for each element in the matrix, and the result is used as an initial value of the analog precoding matrix F.
Preferably, in step S7, the analog precoding matrix F is solved, specifically:
introduction of auxiliary variable B 1 Make FB 1 Approximation is made
Figure BDA0004045958800000054
I.e. solving the following problem:
Figure BDA0004045958800000055
Figure BDA0004045958800000056
in this problem, the auxiliary variable B 1 There are the following optimal solutions:
Figure BDA0004045958800000057
solving the analog precoding matrix F by adopting the ideas of updating column by column and element by element, so as to enable
Figure BDA0004045958800000058
And let symbol f j 、/>
Figure BDA0004045958800000059
c j 、
Figure BDA00040459588000000510
The method comprises the steps of respectively representing a j-th column of the analog precoding matrix F, a submatrix obtained by removing the j-th column of the analog precoding matrix F, a j-th column of the matrix C, and a submatrix obtained by removing the j-th column of the matrix C, and obtaining the update formulas of all elements in the j-th column of the analog precoding matrix F as follows:
Figure BDA00040459588000000511
alternately updating the auxiliary variable B 1 And simulating the precoding matrix F until convergence to obtain a final precoding matrix F.
Preferably, in step S8, the digital precoding matrix B is solved, specifically:
in the case of a fixed analog precoding matrix F, an analog combining matrix W, the digital precoding matrix B is given by:
Figure BDA00040459588000000512
in the method, in the process of the invention,
Figure BDA00040459588000000513
representing an equivalent channel matrix->
Figure BDA00040459588000000514
Right singular matrix after singular value decomposition, power distribution matrix/>
Figure BDA00040459588000000515
Given by the formula:
Figure BDA00040459588000000516
after the optimal solution of the digital precoding matrix is substituted into the matrix E, diagonal elements of the matrix E are equal, so that an equal sign is taken on the basis of the organ generation inequality of the error rate and the lower bound thereof, namely, the lower bound of the optimized error rate in the method is equivalent to the optimized error rate.
Preferably, in step S9, the digital combination matrix D is solved, specifically:
in the case of a fixed analog precoding matrix F, an analog combining matrix W, a digital precoding matrix B, the digital combining matrix D is given by:
D＝(σ 2 W H W+W H HTT H H H W) -1 W H HT。
compared with the prior art, the technical scheme of the invention has the beneficial effects that:
the invention discloses a hybrid precoding method aiming at reducing error rate in a single-user large-scale MIMO system, which adopts a full-connection type hybrid precoding structure at both receiving and transmitting ends, namely a structure that each radio frequency link is connected with all antennas. The invention carries out mathematical modeling on the design problem of the mixed precoding with minimized error rate, and provides a mixed precoding algorithm with low complexity and excellent performance based on the thought of element-by-element iteration. Compared with other mixed pre-coding algorithms taking maximized spectrum efficiency or MMSE as optimization targets, the scheme of the invention can greatly reduce the error rate of the system without losing too much spectrum efficiency and ensure the reliability of information transmission.
Drawings
FIG. 1 is a schematic flow chart of the method of the present invention.
Fig. 2 is a schematic diagram of a downlink single-user MIMO system.
Fig. 3 is a graph showing the spectral efficiency of different algorithms according to the signal-to-noise ratio.
Fig. 4 is a schematic diagram showing the variation of bit error rate with signal to noise ratio according to different algorithms provided in the embodiments.
Detailed Description
The drawings are for illustrative purposes only and are not to be construed as limiting the present patent;
for the purpose of better illustrating the embodiments, certain elements of the drawings may be omitted, enlarged or reduced and do not represent the actual product dimensions;
it will be appreciated by those skilled in the art that certain well-known structures in the drawings and descriptions thereof may be omitted.
The technical scheme of the invention is further described below with reference to the accompanying drawings and examples.
Example 1
The embodiment provides a hybrid precoding design method based on element-by-element iteration in a massive MIMO system, as shown in fig. 1, comprising the following steps:
s1: obtaining a channel matrix H;
s2: constructing an error rate minimization optimization problem and converting the error rate minimization optimization problem into a form which is easy to solve;
s3: initializing an analog combination matrix W and an auxiliary variable P, wherein the auxiliary variable P replaces a mixed pre-coding matrix T, and enabling the mixed pre-coder to approach the auxiliary variable P after the analog combination matrix W and the auxiliary variable P are alternately optimized;
s4: fixing the auxiliary variables P, and updating the simulation combination matrix W column by column and element by element;
s5: fixing a simulation combination matrix W, and updating an auxiliary variable P according to Karush-Kuhn-Tucker conditions;
s6: step S4 and step S5 are alternately carried out until a termination condition is triggered, and then an analog precoding matrix F is initialized;
s7: solving an analog precoding matrix F;
s8: solving a digital precoding matrix B;
s9: solving a digital combination matrix D;
s10: according to the results of steps S3 to S9, a final hybrid precoding matrix t=fb and a hybrid combining matrix r=wd are obtained.
Example 2
The present embodiment continues to disclose the following on the basis of embodiment 1:
in this embodiment, consider a downlink single-user MIMO system as shown in fig. 2, where the dimension of the data stream to be transmitted is N s Transmitting side configuration L t The radio frequency links are connected to N t Root transmitting antenna, receiving end configuration L r The radio frequency links are connected to N r The transmitting end outputs N finally by the root receiving antenna s And (5) maintaining the data stream. The signal to be transmitted, the signal processed by the receiving end, the zero-mean Gaussian white noise signal, the analog precoding matrix, the digital precoding matrix, the analog combining matrix, the digital combining matrix, the channel matrix, the noise variance and the maximum transmitting power of the transmitting end are respectively represented by the symbols s, y, n, F, B, W, D, H and sigma 2 、P max And (3) representing. In order to reduce the number of radio frequency links required to reduce hardware cost and power consumption, consider the case where the number of radio frequency links at the transceiver end is equal to the dimension of the data stream, denoted as L t ＝L r ＝N s . For the sake of simplicity of mathematical formulation, let the symbol [ A ]] [i，j] 、[A] [i，：] 、[A] [：，j] 、[a] [i] Representing the elements on the ith row and jth column of any matrix A, the ith row and jth column of matrix A and the elements on the ith column of any column vector a respectively; let t=fb, r=wd denote a hybrid precoder, a hybrid combiner, respectively.
In the step S2, an error rate minimization optimization problem is constructed and converted into a form easy to solve, specifically:
the ith data stream in the processed signal y at the receiving end has the following form:
Figure BDA0004045958800000081
wherein y is i 、d i 、b i 、s i Respectively representing the ith data stream of the signal y processed by the receiving end, the ith column of the digital combination matrix D, the ith column of the digital precoding matrix B and the ith data stream of the signal s to be transmitted;
average bit error rate P of each data stream e Expressed as:
Figure BDA0004045958800000082
in the method, in the process of the invention,
Figure BDA0004045958800000083
function representing a characteristic error rate related to signal-to-interference-plus-noise ratio of a signal, SINR i Representing the signal-to-interference-and-noise ratio, N, of the ith data stream s Representing the dimension of the transmitted data stream;
the receiving end adopts a digital minimum mean square error combination matrix, which is expressed as:
D＝(σ 2 W H W+W H HTT H H H W) -1 W H HT
the signal-to-interference-plus-noise ratio SINR of the ith data stream i Can be expressed as:
Figure BDA0004045958800000084
in the method, in the process of the invention,
Figure BDA0004045958800000085
represents an MSE matrix, [ E ]] [i，i] Elements representing the ith row and column of MSE matrix E, < >>
Figure BDA0004045958800000086
Represents N s Order unit array, sigma 2 Representing noise variance->
Figure BDA0004045958800000087
Figure BDA0004045958800000088
Representing the pseudo-inverse of the analog combining matrix W;
the average bit error rate of each data stream has a lower bound as follows:
Figure BDA0004045958800000089
wherein the inequality sign is given by the qin inequality, which takes the equality sign when the diagonal elements of the matrix E are all equal. It can be seen that it is difficult to directly optimize the bit error rate itself, and indirectly reducing the bit error rate by minimizing the lower bound of the bit error rate is a good solution. Due to the function
Figure BDA00040459588000000810
As a monotonically decreasing function, minimizing the lower error rate bound is equivalent to minimizing the trace of the MSE matrix E. When the number of antennas is large, the analog combining matrix tends to satisfy the orthogonal constraint +.>
Figure BDA00040459588000000811
The MSE matrix E is approximately expressed as:
Figure BDA00040459588000000812
wherein E is 1 Representing the approximated MSE matrix, the bit error rate minimization hybrid precoder design problem to be solved is modeled in the following form, taking into account the constant modulus constraint introduced by the phase shifter and the power constraint at the transmit end:
Figure BDA0004045958800000091
Figure BDA0004045958800000092
Figure BDA0004045958800000093
tr(FBB H F H )≤P max
wherein tr (E) 1 ) Representation E 1 Is [ F ]] [i，j] Representing an element on the ith row and jth column of the analog precoding matrix F, [ W ]] [i，j] Representing elements on the ith row and jth column of the simulated combining matrix W, P max Indicating the maximum transmit power at the transmitting end.
In step S3, the analog combining matrix W and the auxiliary variable P are initialized, specifically:
in order to reduce the complexity of the algorithm, the invention firstly introduces an auxiliary variable P to replace the hybrid precoder t=fb, and after alternately optimizing the analog combining matrix W and the auxiliary variable P, the hybrid precoder t=fb is made to approach the auxiliary variable P.
The initialization of the analog combination matrix W adopts a mode of randomly selecting phases, and the initialization of P of auxiliary variables is given by Karush-Kuhn-Tucker conditions, and the form is as follows:
Figure BDA0004045958800000094
in the method, in the process of the invention,
Figure BDA0004045958800000095
representing equivalent channel->
Figure BDA0004045958800000096
Front N of right singular matrix after singular value decomposition s Matrix of columns, Λ H，1 Front N of singular value matrix after singular value decomposition of channel matrix H s Front N of line s Matrix of column elements>
Figure BDA0004045958800000097
Represents N s An order DFT matrix.
In step S4, the auxiliary variables P are fixed, and the analog combination matrix W is updated column by column and element by element, specifically:
in order to optimize the analog combining matrix W column by column, the objective function of the bit error rate minimization optimization problem is rewritten:
Figure BDA0004045958800000098
wherein N is r Indicating the number of receive antennas and,
Figure BDA0004045958800000099
Figure BDA00040459588000000910
represents N r An array of order units,
Figure BDA00040459588000000911
representing the submatrices obtained by removing the j-th column from the analog combined matrix W, W j The j-th column of the analog combining matrix W is represented. It is easy to find w j Independent of A j Therefore, assuming that only the j-th column of the analog combining matrix W is optimized, the above optimization problem is equivalent to:
Figure BDA0004045958800000101
/>
Figure BDA0004045958800000102
in the method, in the process of the invention,
Figure BDA0004045958800000103
according to the split programming theory, by introducing an auxiliary variable tau j The above problems can be translated into:
Figure BDA0004045958800000104
Figure BDA0004045958800000105
variable τ j 、w j The following optimal solutions are respectively provided:
Figure BDA0004045958800000106
Figure BDA0004045958800000107
in [ w ] j ] [i] Representing a column vector w j Is (x) =x/|x| represents a phase extraction operation;
updating τ alternately according to the optimal solution j And w j And (3) until convergence, finishing updating the j-th column of the analog combination matrix W, and performing the same step updating on other columns of the analog combination matrix W to finish updating the analog combination matrix W.
In step S5, the simulation combining matrix W is fixed, and the auxiliary variable P is updated according to Karush-Kuhn-turner conditions, specifically:
Figure BDA0004045958800000108
the termination condition in step S6 is set such that the absolute value of the difference between the objective functions of the two iterations is smaller than a preset constant.
In step S6, an analog precoding matrix F is initialized, specifically:
for equivalent channels
Figure BDA0004045958800000109
Front N of right singular matrix after singular value decomposition s Matrix of columns->
Figure BDA00040459588000001010
The phase extraction operation is performed for each element in the matrix, and the result is used as an initial value of the analog precoding matrix F.
In step S7, the analog precoding matrix F is solved, specifically:
introduction of auxiliary variable B 1 Make FB 1 Approximation is made
Figure BDA00040459588000001011
I.e. solving the following problem:
Figure BDA00040459588000001012
Figure BDA00040459588000001013
in this problem, the auxiliary variable B 1 There are the following optimal solutions:
Figure BDA0004045958800000111
solving the analog precoding matrix F by adopting the ideas of updating column by column and element by element, so as to enable
Figure BDA0004045958800000112
And let symbol f j 、/>
Figure BDA0004045958800000113
c j 、
Figure BDA0004045958800000114
The method comprises the steps of respectively representing a j-th column of the analog precoding matrix F, a submatrix obtained by removing the j-th column of the analog precoding matrix F, a j-th column of the matrix C, and a submatrix obtained by removing the j-th column of the matrix C, and obtaining the update formulas of all elements in the j-th column of the analog precoding matrix F as follows:
Figure BDA0004045958800000115
alternately updating the auxiliary variable B 1 And simulating the precoding matrix F until convergence to obtain a final precoding matrix F.
In step S8, the digital precoding matrix B is solved, specifically:
in the case of a fixed analog precoding matrix F, an analog combining matrix W, the digital precoding matrix B is given by:
Figure BDA0004045958800000116
in the method, in the process of the invention,
Figure BDA0004045958800000117
representing an equivalent channel matrix->
Figure BDA0004045958800000118
Right singular matrix after singular value decomposition, power allocation matrix +.>
Figure BDA0004045958800000119
Given by the formula:
Figure BDA00040459588000001110
after the optimal solution of the digital precoding matrix is substituted into the matrix E, diagonal elements of the matrix E are equal, so that an equal sign is taken on the basis of the organ generation inequality of the error rate and the lower bound thereof, namely, the lower bound of the optimized error rate in the method is equivalent to the optimized error rate.
In step S9, the digital combination matrix D is solved, specifically:
in the case of a fixed analog precoding matrix F, an analog combining matrix W, a digital precoding matrix B, the digital combining matrix D is given by:
D＝(σ 2 W H W+W H HTT H H H W) -1 W H HT。
example 3
This embodiment provides the following specific examples on the basis of embodiment 1 and embodiment 2:
the simulation parameters are set as follows: the channel model adopts a Saleh-Valenzuela model, wherein the number of scatterers is set to be 20; transmitting side configuration L t =8 radio frequency links connected to N t =144 transmit antennas; receiving end is equipped with L r =8 radio frequency links connected to N r =36 receive antennas; the dimension of the transmission data stream is N s =8; the signal is modulated using quadrature phase shift keying (Quadrature Phase Shift Keying, QPSK). Definition of Signal-to-noise ratio (SNR) to 10log 10 (P max /σ 2 ) The method comprises the steps of carrying out a first treatment on the surface of the Spectral efficiency (Spectral Efficiency) is represented by
Figure BDA0004045958800000121
The Bit Error Rate (Bit Error Rate) is calculated as a function of the ratio of the number of transmission Error bits to the total number of transmission bits.
The new method (Proposed method) provided by the invention is compared with the full digital bit error rate minimization algorithm (full-digital), the ARAB algorithm taking the spectrum efficiency as an optimization target and the GEVD algorithm taking the MMSE as an optimization target, and the performance of the method provided by the invention is slightly lower than that of full digital precoding regardless of the change of the signal to noise ratio and is slightly better than that of the GEVD algorithm and the ARAB algorithm, as shown in figure 3. It can be seen that the all-digital precoding method performs best, but is difficult to be practically applied because of its excessive hardware cost, which is only referred to herein as a performance reference. In addition, although the ARAB algorithm takes the spectrum efficiency as an optimization target, the performance is lost to a certain extent because the method adopts a mode of decoupling in an analog domain and a digital domain in the design process, and the method provided by the invention adopts a combined optimization thought, so that the performance is ensured.
The above embodiments show that although the method of the present invention aims at optimizing the bit error rate, the method still has good performance in terms of spectrum efficiency.
The bit error rate performance of the method of the present invention is discussed on the basis of the simulation parameters of the above embodiment. The method comprises the following steps: according to the method shown in fig. 4, the difference between the error rate and the full digital precoding performance is about 1dB, and compared with the GEVD algorithm, the performance improvement of about 0.5dB is achieved, and particularly compared with the ARAB algorithm taking the spectrum efficiency as an optimization target, the method has a great advantage in the error rate.
The above embodiments show that, compared with other hybrid precoding algorithms, the method of the present invention has a certain advantage in terms of bit error rate, and can further improve the reliability of information transmission and ensure the service quality of the communication system.
The same or similar reference numerals correspond to the same or similar components;
the terms describing the positional relationship in the drawings are merely illustrative, and are not to be construed as limiting the present patent;
it is to be understood that the above examples of the present invention are provided by way of illustration only and not by way of limitation of the embodiments of the present invention. Other variations or modifications of the above teachings will be apparent to those of ordinary skill in the art. It is not necessary here nor is it exhaustive of all embodiments. Any modification, equivalent replacement, improvement, etc. which come within the spirit and principles of the invention are desired to be protected by the following claims.

Claims (10)

1. The mixed precoding design method based on element-by-element iteration in the massive MIMO system is characterized by comprising the following steps:
s1: obtaining a channel matrix H;
s2: constructing an error rate minimization optimization problem and converting the error rate minimization optimization problem into a form which is easy to solve;
s3: initializing an analog combination matrix W and an auxiliary variable P, wherein the auxiliary variable P replaces a mixed pre-coding matrix T, and enabling the mixed pre-coder to approach the auxiliary variable P after the analog combination matrix W and the auxiliary variable P are alternately optimized;
s4: fixing the auxiliary variables P, and updating the simulation combination matrix W column by column and element by element;
s5: fixing a simulation combination matrix W, and updating an auxiliary variable P according to Karush-Kuhn-Tucker conditions;
s6: step S4 and step S5 are alternately carried out until a termination condition is triggered, and then an analog precoding matrix F is initialized;
s7: solving an analog precoding matrix F;
s8: solving a digital precoding matrix B;
s9: solving a digital combination matrix D;
s10: according to the results of steps S3 to S9, a final hybrid precoding matrix t=fb and a hybrid combining matrix r=wd are obtained.
2. The method for designing hybrid precoding based on element-by-element iteration in massive MIMO system according to claim 1, wherein the error rate minimization optimization problem is constructed in step S2 and converted into a form easy to solve, specifically:
the ith data stream in the processed signal y at the receiving end has the following form:
Figure FDA0004045958790000011
wherein y is i 、d i 、b i 、s i Respectively representing the ith data stream of the signal y processed by the receiving end, the ith column of the digital combination matrix D, the ith column of the digital precoding matrix B and the ith data stream of the signal s to be transmitted;
average bit error rate P of each data stream e Expressed as:
Figure FDA0004045958790000012
in the method, in the process of the invention,
Figure FDA0004045958790000013
representing a characteristic bit error rate related to signal-to-interference-and-noise ratio of a signalFunction of (1), SINR i Representing the signal-to-interference-and-noise ratio, N, of the ith data stream s Representing the dimension of the transmitted data stream;
the receiving end adopts a digital minimum mean square error combination matrix, which is expressed as:
D＝(σ 2 W H W+W H HTT H H H W) -1 W H HT
the signal-to-interference-plus-noise ratio SINR of the ith data stream i Can be expressed as:
Figure FDA0004045958790000021
in the method, in the process of the invention,
Figure FDA0004045958790000022
represents an MSE matrix, [ E ]] [i,i] Elements representing the ith row and column of MSE matrix E, I Ns Represents N s Order unit array, sigma 2 Representing noise variance->
Figure FDA0004045958790000023
Figure FDA0004045958790000024
Representing the pseudo-inverse of the analog combining matrix W;
the average bit error rate of each data stream has a lower bound as follows:
Figure FDA0004045958790000025
wherein the inequality sign is given by the qin inequality, which takes the equality sign when the diagonal elements of the matrix E are all equal. When the number of antennas is large, the analog combining matrix tends to satisfy the orthogonal constraint
Figure FDA0004045958790000026
The MSE matrix E is approximately expressed as:
Figure FDA0004045958790000027
wherein E is 1 Representing the approximated MSE matrix, the bit error rate minimization hybrid precoder design problem to be solved is modeled in the following form, taking into account the constant modulus constraint introduced by the phase shifter and the power constraint at the transmit end:
Figure FDA0004045958790000028
Figure FDA0004045958790000029
Figure FDA00040459587900000210
tr(FBB H F H )≤P max
wherein tr (E) 1 ) Representation E 1 Is [ F ]] [i,j] Representing an element on the ith row and jth column of the analog precoding matrix F, [ W ]] [i,j] Representing elements on the ith row and jth column of the simulated combining matrix W, P max Indicating the maximum transmit power at the transmitting end.
3. The method for designing hybrid precoding based on element-wise iteration in massive MIMO system according to claim 2, wherein the initializing the analog combining matrix W and the auxiliary variable P in step S3 is specifically:
the initialization of the analog combination matrix W adopts a mode of randomly selecting phases, and the initialization of P of auxiliary variables is given by Karush-Kuhn-Tucker conditions, and the form is as follows:
Figure FDA0004045958790000031
in the method, in the process of the invention,
Figure FDA0004045958790000032
representing equivalent channel->
Figure FDA0004045958790000033
Front N of right singular matrix after singular value decomposition s Matrix of columns, Λ H,1 Front N of singular value matrix after singular value decomposition of channel matrix H s Front N of line s A matrix of column elements is formed,
Figure FDA0004045958790000034
represents N s An order DFT matrix.
4. The method for designing hybrid precoding based on element-by-element iteration in massive MIMO system according to claim 3, wherein the step S4 is to fix the auxiliary variable P, update the analog combining matrix W column by column and element by element, specifically:
rewriting an objective function of the error rate minimization optimization problem:
Figure FDA0004045958790000035
wherein N is r Indicating the number of receive antennas and,
Figure FDA0004045958790000036
Figure FDA0004045958790000037
represents N r Order unit array,/->
Figure FDA0004045958790000038
Representing the analog combining matrix W minus the firstSub-matrix obtained after j columns, w j Representing the j-th column of the simulated combining matrix W, the optimization problem described above is equivalent to: />
Figure FDA0004045958790000039
Figure FDA00040459587900000310
In the method, in the process of the invention,
Figure FDA00040459587900000311
according to the split programming theory, by introducing an auxiliary variable tau j The above problems can be translated into:
Figure FDA00040459587900000312
Figure FDA00040459587900000313
variable τ j 、w j The following optimal solutions are respectively provided:
Figure FDA00040459587900000314
Figure FDA00040459587900000315
in [ w ] j ] [i] Representing a column vector w j Is (x) =x/|x| represents a phase extraction operation;
updating τ alternately according to the optimal solution j And w j And (3) until convergence, finishing updating the j-th column of the analog combination matrix W, and performing the same step updating on other columns of the analog combination matrix W to finish updating the analog combination matrix W.
5. The method for designing hybrid precoding based on element-by-element iteration in massive MIMO system according to claim 4, wherein in step S5, the simulation combining matrix W is fixed, and the auxiliary variable P is updated according to Karush-Kuhn-Tucker conditions, specifically:
Figure FDA0004045958790000041
6. the method for designing hybrid precoding based on element-wise iteration in massive MIMO system according to claim 5, wherein the termination condition in step S6 is set such that the absolute value of the difference of the objective functions of the two iterations is smaller than a preset constant.
7. The method for designing hybrid precoding based on element-wise iteration in massive MIMO system according to claim 6, wherein the initializing of the analog precoding matrix F in step S6 is specifically:
for equivalent channels
Figure FDA0004045958790000042
Front N of right singular matrix after singular value decomposition s Matrix of columns->
Figure FDA0004045958790000043
The phase extraction operation is performed for each element in the matrix, and the result is used as an initial value of the analog precoding matrix F.
8. The method for designing hybrid precoding based on element-wise iteration in massive MIMO system according to claim 7, wherein the solving of the analog precoding matrix F in step S7 is specifically:
introduction of auxiliary variable B 1 Make FB 1 Approximation is made
Figure FDA0004045958790000044
I.e. solving the following problem:
Figure FDA0004045958790000045
Figure FDA0004045958790000046
in this problem, the auxiliary variable B 1 There are the following optimal solutions:
Figure FDA0004045958790000047
/>
solving the analog precoding matrix F by adopting the ideas of updating column by column and element by element, so as to enable
Figure FDA0004045958790000048
And let symbol f j 、/>
Figure FDA0004045958790000049
c j 、/>
Figure FDA00040459587900000410
The method comprises the steps of respectively representing a j-th column of the analog precoding matrix F, a submatrix obtained by removing the j-th column of the analog precoding matrix F, a j-th column of the matrix C, and a submatrix obtained by removing the j-th column of the matrix C, and obtaining the update formulas of all elements in the j-th column of the analog precoding matrix F as follows:
Figure FDA00040459587900000411
alternately updating the auxiliary variable B 1 And simulating the precoding matrix F until convergence to obtain a final precoding matrix F.
9. The method for designing hybrid precoding based on element-by-element iteration in massive MIMO system according to claim 8, wherein the solving of the digital precoding matrix B in step S8 is specifically:
in the case of a fixed analog precoding matrix F, an analog combining matrix W, the digital precoding matrix B is given by:
Figure FDA0004045958790000051
in the method, in the process of the invention,
Figure FDA0004045958790000052
representing an equivalent channel matrix->
Figure FDA0004045958790000053
Right singular matrix after singular value decomposition, power allocation matrix +.>
Figure FDA0004045958790000054
Given by the formula:
Figure FDA0004045958790000055
after the optimal solution of the digital precoding matrix is substituted into the matrix E, diagonal elements of the matrix E are equal, so that the equal sign is taken on the basis of the organ generation inequality of the error rate and the lower bound thereof, namely, the lower bound of the optimized error rate is equivalent to the optimized error rate.
10. The method for designing hybrid precoding based on element-by-element iteration in massive MIMO system according to claim 9, wherein the solving of the digital combining matrix D in step S9 is specifically:
in the case of a fixed analog precoding matrix F, an analog combining matrix W, a digital precoding matrix B, the digital combining matrix D is given by:
D＝(σ 2 W H W+W H HTT H H H W) -1 W H HT。
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