CN111988069B - Large-scale MIMO generalized eigenvector structure precoding solving method and device - Google Patents

Large-scale MIMO generalized eigenvector structure precoding solving method and device Download PDF

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CN111988069B
CN111988069B CN202010684702.1A CN202010684702A CN111988069B CN 111988069 B CN111988069 B CN 111988069B CN 202010684702 A CN202010684702 A CN 202010684702A CN 111988069 B CN111988069 B CN 111988069B
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CN111988069A (en
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卢安安
王晨
高西奇
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Southeast 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

Abstract

The invention discloses a precoding solving method and a precoding solving device for a large-scale MIMO generalized eigenvector structure, wherein the method is used for equating a generalized eigenvalue problem to an optimization problem of generalized Rayleigh quotient by calculating the generalized Rayleigh quotient corresponding to each column in an initial precoding matrix of each user, so that matrix inversion caused by the conversion of the generalized eigenvalue problem to a standard eigenvalue problem is avoided; iterative optimization of the generalized Rayleigh quotient is carried out on the quotient manifold by adopting a Riemann conjugate gradient method, so that each row is updated in sequence, invalid search in the European space during iterative optimization is avoided, and the convergence speed of the algorithm is ensured; and finally, distributing power to different columns, and generating a pre-coding matrix according to the updated generalized eigenvector matrix. The invention can solve the problem of high complexity of a large-scale MIMO precoding solving algorithm.

Description

Large-scale MIMO generalized eigenvector structure precoding solving method and device
Technical Field
The invention relates to a precoding solving method and a precoding solving device, in particular to a precoding solving method for a large-scale/super-large-scale MIMO generalized eigenvector structure.
Background
In a large-scale Multiple-Input Multiple-output (M-MIMO) technology, a large number of antennas are arranged in a base station, so that not only can Multiple users be served simultaneously on the same time-frequency resource, but also higher frequency spectrum efficiency and energy efficiency can be achieved, and the technology becomes a key technology of a 5G physical layer. With the increasing shortage of low-frequency resources, the research on millimeter wave and Terahertz (THz) frequency band communication technology is imperative. It is expected that terahertz frequency band communication will play an important role in future 6G wireless communication, and Ultra-large-scale Multiple-Input Multiple-output (UM-MIMO) technology is a main means for overcoming path loss thereof.
As with multi-user MIMO, multi-user interference exists in a typical M-MIMO/UM-MIMO system, and thus its performance depends greatly on the precoding design of each user by the base station. Signal-to-Leakage-and-Noise Ratio (SLNR) precoding is widely used in practice because a precoding matrix is designed by maximizing useful Signal power and interference plus Noise power, and interference between users can be effectively reduced while maintaining simple implementation as compared with nonlinear precoding. Based on the SLNR precoding, the useful Signal covariance matrix, the interference Signal covariance matrix and the Noise covariance matrix can be Weighted, and Weighted-Signal-to-Leakage-Noise Ratio (WSLNR) precoding is designed by maximizing Weighted useful Signal power and Weighted interference plus Noise power. By designing the weighting factors, the WSLNR precoding can effectively enhance the effective power of the SLNR precoding lost due to interference avoidance between users, and improve the total transmission rate.
The analytical solution of the SLNR and WSLNR precoding problem can be summarized as the solution of the generalized eigenvalue problem. The traditional design method converts the generalized eigenvalue problem into a standard eigenvalue problem through matrix inversion and solves the problem. In a large-scale and ultra-large-scale MIMO system, along with the rapid increase of the number of days of a base station, the complexity of the third power of the number of base station antennas caused by matrix inversion cannot be borne.
Disclosure of Invention
The purpose of the invention is as follows: the invention provides a method for solving a precoding matrix with low complexity and high efficiency, which is suitable for a large-scale MIMO system. Another object of the invention is to provide a computer device based on the method.
The technical scheme is as follows: the invention relates to a precoding solving method of a large-scale MIMO generalized characteristic vector structure, which comprises the following steps:
(1) generating an initial precoding matrix of each user, wherein the number of rows is the number of base station antennas, and the number of columns is the number of user data streams;
(2) calculating generalized Rayleigh quotient corresponding to each column in each user initial pre-coding matrix, and performing iterative optimization on the generalized Rayleigh quotient on the quotient manifold by adopting a Riemannian conjugate gradient method to obtain an optimized generalized eigenvector matrix;
(3) and carrying out power distribution on different columns, and generating a precoding matrix according to the optimized generalized eigenvector matrix.
The method equates the generalized eigenvalue problem to the optimization problem of the generalized Rayleigh quotient, and matrix inversion caused by the conversion of the generalized eigenvalue problem to the standard eigenvalue problem is avoided.
Further, the generalized rayleigh quotient solving process corresponding to each column includes:
for each user, determining a numerator matrix and a denominator matrix of the generalized Rayleigh quotient, and if the current column is not the first column of the initial precoding matrix, performing deflmation operation on the numerator matrix of the generalized Rayleigh quotient;
multiplying the molecular matrix of the generalized Rayleigh quotient by the conjugate transpose of the current column on the left side, and then multiplying the current column on the right side to obtain a molecule;
the denominator matrix of the generalized Rayleigh quotient is multiplied by the conjugate transpose of the current column on the left side, and then multiplied by the current column on the right side to obtain a denominator;
the generalized Rayleigh quotient is obtained by dividing the numerator and the denominator.
The deflections operation on the generalized rayleigh quotient molecular matrix can be implicitly performed, that is, the deflected generalized rayleigh quotient molecular matrix is not directly calculated, and the deflected generalized rayleigh quotient molecular matrix is used for replacing the original molecular matrix only when the operation on the generalized rayleigh quotient molecular matrix is needed.
Further, if the signal-to-leakage-and-noise ratio is pre-coded, the molecular matrix of the generalized Rayleigh quotient is a channel covariance matrix of the current user; and if the weighted signal-to-leakage-and-noise ratio is pre-coded, the molecular matrix of the generalized Rayleigh quotient is a weighted channel covariance matrix of the current user.
Further, if the signal-to-leakage-and-noise ratio is precoding, the denominator matrix of the generalized rayleigh quotient is a sum matrix of a channel covariance matrix and a noise covariance matrix of other users except the current user; and if the precoding is weighted signal-to-leakage-and-noise ratio precoding, the denominator matrix of the generalized Rayleigh quotient is a sum matrix of weighted channel covariance matrixes and weighted noise covariance matrixes of other users except the current user.
Further, the commodity manifold is selected to be a riemann commodity manifold. By iteratively optimizing the generalized Rayleigh quotient on the Riemannian quotient manifold, invalid search in iterative optimization in a Euclidean space is avoided, and the convergence speed of the algorithm is ensured.
Further, the step (2) includes:
(21) initializing the current column of the generalized eigenvector matrix by using the ith column in the initial precoding matrix of the current user
Figure BDA0002587118180000021
The initial value of the generalized characteristic value serial number i is 1, and the initial value of the search time serial number j is 0; calculating a generalized Rayleigh quotient corresponding to the current column and a Riemann gradient direction thereof, and setting the current conjugate gradient direction as a negative direction of the Riemann gradient direction of the generalized Rayleigh quotient;
(22) calculating an optimal step length by using the current column and the current conjugate gradient direction, and updating the current column according to the optimal step length;
(23) judging whether the iteration number of the set conjugate gradient method is reached, if the iteration number of the set conjugate gradient method is reached and the current column is not the last column of the precoding matrix, stepping to the next column of the initial precoding matrix, returning to the step (21), and otherwise, entering the step (24); and (4) if the current column is the last column, ending the iteration and jumping to the step (3).
(24) Calculating the updated generalized Rayleigh quotient corresponding to the current column and the Riemann gradient direction of the generalized Rayleigh quotient;
(25) calculating vector transport in the conjugate gradient direction by using the current column, the optimal step length and the current conjugate gradient direction;
(26) updating the conjugate gradient coefficient;
(27) and (4) updating the current conjugate gradient direction by using the Riemann gradient direction of the generalized Rayleigh quotient, the conjugate gradient coefficient and the vector transportation of the conjugate gradient direction, and returning to the step (23).
Further, the optimal step size is a step size of the current column along the current conjugate gradient direction such that the generalized rayleigh quotient is minimum.
Further, the step (22) comprises:
if the optimal step length exists, the product of the optimal step length and the current conjugate gradient direction is added with the current column vector to serve as an updated current column; and if the optimal step length does not exist, updating the current column to be the current conjugate gradient direction.
Further, when the optimal step size does not exist, the conjugate gradient coefficient is set to zero. When the optimal step length exists, the conjugate gradient coefficient has a plurality of definition modes, and according to different definitions, the conjugate gradient coefficient can be calculated by the current feature vector, the current conjugate gradient direction, the Riemannian gradient of the generalized Rayleigh quotient and the Riemannian Hessian of the generalized Rayleigh quotient.
The invention relates to a precoding solving device of a large-scale MIMO generalized characteristic vector structure, which comprises the following components: the massive MIMO generalized eigenvector structure precoding solving method comprises the following steps of a memory, a processor and a computer program stored on the memory and executable, wherein the computer program realizes all or part of the steps of the massive MIMO generalized eigenvector structure precoding solving method when being executed by the processor.
Has the advantages that: the method is particularly suitable for the conditions of a large-scale/super-large-scale MIMO system, the calculation complexity of the traditional algorithm for solving the generalized eigenvalue problem is reduced from 3 th power to 2 th power, and the Riemann conjugate gradient method provided by the invention can be used for calculating the precoding matrix of the generalized eigenvector structure more efficiently.
Drawings
FIG. 1 is a flow chart of a method of an embodiment of the present invention;
FIG. 2 is a graph of weighted signal-to-leakage-and-noise ratio precoding and rate performance with an RZF precoding matrix as an initial precoding matrix;
fig. 3 is a graph of weighted signal to leakage plus noise ratio precoding and rate performance with a random precoding matrix as the initial precoding matrix.
Detailed Description
The technical solution of the present invention is further described below with reference to the accompanying drawings and examples.
Referring to fig. 1, it shows a flowchart of a low-complexity efficient solution method for precoding large-scale MIMO generalized eigenvector structure according to the present invention, which includes:
(1) generating an initial precoding matrix of each user, wherein the number of rows is the number of base station antennas, and the number of columns is the number of user data streams;
(2) the first column of the initial precoding matrix is used as the current column of the generalized eigenvector, the generalized Rayleigh quotient corresponding to each column in the initial precoding matrix of each user is calculated, iteration optimization is carried out on the generalized Rayleigh quotient on the quotient manifold by adopting a Riemannian conjugate gradient method, and the optimized generalized eigenvector matrix is obtained:
(21) for each user, determining a denominator matrix and a numerator matrix of the generalized Rayleigh quotient; if the current column is not the first column of the initial precoding matrix, deflmation operation is carried out on the molecular matrix of the generalized Rayleigh quotient;
calculating the generalized Rayleigh quotient and the Riemann gradient direction thereof corresponding to the current column in the initial precoding matrix of the current user according to the denominator matrix and the numerator matrix of the generalized Rayleigh quotient, and setting the current conjugate gradient direction as the negative direction of the Riemann gradient direction of the generalized Rayleigh quotient:
(22) calculating an optimal step length by using the current column and the current conjugate gradient direction, and updating the current column according to the optimal step length;
(23) judging whether the iteration number of the set conjugate gradient method is reached, if the iteration number of the set conjugate gradient method is reached and the current column is not the last column of the precoding matrix, stepping the current column to the next column of the initial precoding matrix, returning to the step (21), and otherwise, entering the step (24); and (4) if the current column is the last column, ending the iteration and jumping to the step (3).
(24) Calculating the updated generalized Rayleigh quotient corresponding to the current column and the Riemann gradient direction of the generalized Rayleigh quotient;
(25) calculating vector transport in the conjugate gradient direction by using the current column, the optimal step length and the current conjugate gradient direction;
(26) updating the conjugate gradient coefficient;
(27) updating the current conjugate gradient direction by utilizing the Riemann gradient direction, the conjugate gradient coefficient and the vector transport of the conjugate gradient direction of the generalized Rayleigh quotient, and returning to the step (23);
(3) and carrying out power distribution on different columns, and generating a precoding matrix according to the optimized generalized eigenvector matrix.
The method is mainly suitable for a large-scale and ultra-large-scale MIMO system with a large-scale and ultra-large-scale antenna array arranged on a base station side to serve a plurality of users simultaneously. The following describes a specific implementation process of the method in detail with reference to a specific communication system example, and it should be noted that the method of the present invention is not only applicable to the specific system model exemplified in the following example, but also applicable to system models of other configurations.
First, system configuration
Consider a massive MIMO system equipped with a Uniform area Array (UPA) operating in Time Division Duplex (TDD) mode. The uniform area array has a total of Mt＝Mz×MxRoot antenna, wherein the vertical direction MzRoot, horizontal direction MxAnd (4) root. K subscribers are each provided with a device MkA Uniform Linear Array (ULA) of root antennas. For different users, MkThe values of (a) may be different. Assuming that the channel is flat block fading, the system time resource is divided into several time slots, each time slot includes NbThe channel remains unchanged over a time block. For simplicity, it is assumed that only uplink channel training and downlink transmission phases exist, and downlink transmission includes pre-coded field pilot and data signaling. In each time slot, the uplink pilot signal is transmitted only in the first time block. 2 nd to NbThe time block is used for transmitting the pilot frequency and the data signal of the downlink pre-coding domain. Each time slot obtains channel information for transmission of the time slot. For a Frequency Division Duplex (FDD) mode, the uplink channel training phase may be replaced with the downlink channel feedback phase, and the downlink transmission phase remains the same. Specifically, a downlink omni-directional pilot signal is transmitted in a first block, and mobile terminal feedback is received.
Second, refined wave beam domain posterior statistical channel model
The refined beam domain prior statistical channel model of the user k in the nth time block of the mth time slot can be written as
Figure BDA0002587118180000051
Wherein
Figure BDA0002587118180000052
Is a refined received sample steering vector matrix at the user side,
Figure BDA0002587118180000053
is a refined transmission sampling guide vector matrix at the base station side.
Figure BDA0002587118180000054
Refined steering vector matrix from vertical direction
Figure BDA0002587118180000055
And fine guide vector matrix in horizontal direction
Figure BDA0002587118180000056
The Kronecker product of (a). Gk，m，n＝(Mk⊙Wk，m，n) Is an element independent refined beam field channel matrix, where &denotesa hadamard product. Each row of the beam forming system corresponds to a refined beam domain on the user side, and each column of the beam forming system corresponds to a refined beam domain on the two-dimensional space of the base station side, MkTo refine the beam-domain channel amplitude matrix, Wk，m，nThe random matrix is composed of independent and identically distributed complex Gaussian random variables, and the elements of the random matrix are zero mean unit variance. Defining a refinement factor of
Figure BDA0002587118180000057
When the refinement factor is larger than 1, the number of cosine in the sampling direction is more than that of the antenna, and compared with the traditional wave beam domain prior statistical channel model based on the DFT matrix, the refined wave beam domain statistical model has more statistical characteristic directions, so that the actual physical channel model can be more accurately characterized. Defining a large-scale MIMO system channel refined beam domain energy matrix omegakIs omegak＝Mk⊙Mk。
To describe the time-dependent characteristics of massive MIMO, a first-order Gaussian Markov model is used to describe the time-dependent model. Under the model, the refined beam domain channel on the nth time block of the mth time slot can be expressed as
Figure BDA0002587118180000058
Wherein gamma isk，m(n-1) is channel Gk，m，nAnd Gk，m，1The function is a time dependent factor related to the speed of movement of the user. Correlation factor gammak，mThere are several methods of obtaining, here assuming that the correlation factor is known. In practice, empirical correlation factors of channel samples may be used, and correlation factor γ based on Jakes autocorrelation model, which is commonly used in the literature, may also be usedk，mIs calculated by a method of (i.e. gamma)k，m(n)＝J0(2πvkfcnT τ/c), wherein J0(. cndot.) denotes a first class of zero-order Bessel function, τ denotes the time corresponding to a time interval, vkRepresents the moving speed of the k-th user, fcRepresenting the carrier frequency and c the speed of light. In this embodiment, in order to consider the complexity of system implementation, precoding is performed on the entire slot m. For simplicity, it is assumed that the refined beam-domain channel matrix G can be obtained without considering channel estimation errorsk，m，1The posterior statistical information of the refined beam channel on the time slot m is obtained as
Figure BDA0002587118180000059
Wherein deltak，mAnd gamma over the whole time slot mk，mIn this regard, it is possible to take all the correlation factors γ over the time slotk，mRoot mean square (rms):
Figure BDA0002587118180000061
further, let
Figure BDA0002587118180000062
A refined posterior statistical model on the time slot m can be obtained as
Figure BDA0002587118180000063
When single time slot precoding is considered, the time slot number m is omitted, and the channel (5) can be further simplified into
Figure BDA0002587118180000064
For the precoding problem, we assume that δ is already obtained at the base station sidek，
Figure BDA0002587118180000065
And Ωk。
Triple, precoding design
1. Signal model
Considering the transmission on a single slot, the slot number m is omitted. Let xkD representing the k-th user terminal (UE)kThe x 1-dimensional transmit vector has a covariance matrix as a unit matrix. Reception signal y of kth UEkCan be expressed as
Figure BDA0002587118180000066
Wherein P iskM being the kth UEt×dkDimension precoding matrix, zkIs a distribution of
Figure BDA0002587118180000067
The complex gaussian random noise vector of (a),
Figure BDA0002587118180000068
for each element of the variance of the noise vector,
Figure BDA0002587118180000069
is Mk×MkAnd (4) an identity matrix. Because of the precoding matrix PkBased on a posterior statistical model of a refined beam domainThe method can adapt to various typical large-scale MIMO mobile scenes, namely has robustness, so the method is called as refined beam domain downlink robust precoding. The transmitted robust pre-coding domain pilot signals are on the same time frequency resource, and each user pilot frequency does not need to be orthogonal, namely pilot frequency multiplexing can be carried out. Specifically, the pre-coding domain pilot signal transmitted by the base station to each user is a frequency domain signal generated by modulating the ZC sequence or the ZC sequence group. After receiving the pilot signal, the mobile terminal performs channel estimation on the robust precoding domain equivalent channel, wherein the robust precoding domain equivalent channel is HkPk. For simplicity, it is assumed that the UE side can obtain perfect CSI with the respective robust precoding domain equivalent channel matrix. After each user receives the data signal, robust pre-coding domain signal detection can be carried out by using the received data signal.
2. Signal-to-leakage-and-noise ratio precoding design
Order to
Hk＝[H1，H2，...，Hk-1，Hk+1，...，HK]， (8)
Defining the statistical signal-to-leakage-noise ratio as:
Figure BDA0002587118180000071
Pkis the power of the user k and,
Figure BDA0002587118180000072
for the channel covariance matrix of user k,
Figure BDA0002587118180000073
is the sum of the channel covariance matrices of users other than user k plus the noise covariance matrix of user k. The signal to leakage and noise ratio precoding can be obtained by maximizing the expression (9). Specifically, the precoding matrix of the maximization formula (9) is a matrix pair
Figure BDA0002587118180000074
Corresponding to the maximum generalized eigenvalue ofThe generalized eigenvectors of (3).
2. Weighted signal-to-leakage-and-noise ratio precoding design method
Total interference noise z 'of each UE'kConsidered as gaussian noise:
Figure BDA0002587118180000075
let RkRepresents z'kThe covariance matrix of (2) is:
Figure BDA0002587118180000076
wherein the expectation function
Figure BDA0002587118180000077
Presentation based on user-side long-term statistics pair HkIs desired. According to the channel reciprocity, the long-term statistical channel information of the user side is consistent with the long-term statistical channel information of the base station end given in the formula (6). Therefore, the expectation function
Figure BDA0002587118180000078
The calculation can be performed according to equation (6). Suppose user k knows RkAt this time, the traversal rate of user k can be expressed as:
Figure BDA0002587118180000079
wherein
Figure BDA00025871181800000710
Also represents the result obtained from the posterior model in equation (6) for HkIs a function of the conditional expectation. Since log det (·) is a concave function, from the Jensen inequality, one upper bound on the velocity of user k can be found as:
Figure BDA00025871181800000711
defining functions
Figure BDA0002587118180000081
And expressing the weighted sum of each user and the rate upper bound, namely the weighted sum of each user and the rate upper bound calculated according to the established refined beam domain posterior statistical channel model, wherein K is the number of the users. By designing a precoding matrix P1，P2，...，PKMaximizing the weighted sum of individual users and the upper bound on the rate can be written as an optimization problem
Figure BDA0002587118180000082
Wherein, wkIs the weighting factor for the kth user and P is the total power constraint.
Obviously, the precoding matrix of the user
Figure BDA0002587118180000083
Vector space
Figure BDA0002587118180000084
Can be viewed as a linear manifold. Considering the precoding of all users as a whole, we define P ═ (P)1，P2，...，PK) Then there is
Figure BDA0002587118180000085
Wherein
Figure BDA0002587118180000086
For the flow pattern, each
Figure BDA0002587118180000087
Is a factor manifold thereof. Precoding sets that can prove to satisfy a total power constraint
Figure BDA0002587118180000088
Is that
Figure BDA0002587118180000089
One embedded sub-manifold. By using
Figure BDA00025871181800000810
Is shown in the embedding space
Figure BDA00025871181800000811
The objective function of (3) is
Figure BDA00025871181800000812
Representing constraints in embedded sub-manifold
Figure BDA00025871181800000813
The objective function of (1). The problem (14) is transformed into manifold
Figure BDA00025871181800000814
The unconstrained problem on (1):
Figure BDA00025871181800000815
for the
Figure BDA00025871181800000816
Two tangent vectors of any point P
Figure BDA00025871181800000817
And
Figure BDA00025871181800000818
definition of
Figure BDA00025871181800000819
The Riemann measure of
Figure BDA00025871181800000820
Then f (P) can be deduced to be
Figure BDA00025871181800000821
The above Riemann gradient is gradf (P) ═ gradf (P)1)，gradf(P2)，...，gradf(PK) In which the component on the kth factor manifold is:
Figure BDA00025871181800000822
wherein
Figure BDA00025871181800000823
Figure BDA0002587118180000091
Figure BDA0002587118180000092
Figure BDA0002587118180000093
Figure BDA0002587118180000094
Further, it can be deduced
Figure BDA0002587118180000095
Has a Riemann gradient of
Figure BDA0002587118180000096
Wherein
Figure BDA0002587118180000097
Maximization by utilization
Figure BDA0002587118180000098
The first order requirement of the optimal point can be used to obtain the problem (16) that the optimal precoding satisfies the generalized eigenvector structure:
AkPk＝(Bk+μI)PkΛk k＝1，2，...，K (24)
wherein
Figure BDA0002587118180000099
Is a matrix pair (A)k，Bk+ mu I) corresponds to the diagonal matrix formed by generalized eigenvalues, without loss of generality, and
Figure BDA00025871181800000910
matrix AkCan be viewed as a weighted channel covariance matrix, of user k
Figure BDA00025871181800000911
Is the weighted signal covariance matrix for user k; matrix BkThe weighted sum matrix of the weighted channel covariance matrices of users other than user k, μ I is the weighted noise covariance matrix of user k
Figure BDA00025871181800000912
Can be viewed as a weighted leakage plus noise covariance matrix for user k. The precoding matrix satisfying equation (24) can be regarded as weighted signal-to-leakage-and-noise ratio precoding. It is known from equations (18), (19) and (23) to design an optimal precoding matrix P ═ P using a generalized eigenvector structure (24)1，P2，...，PK) And the users are coupled together and need to be iteratively calculated. Using the generalized eigenvector structure (24), let the precoding matrix be:
Pk＝QkSk k＝1，2，...K (25)
wherein
Figure BDA00025871181800000913
Is to satisfy the orthogonality condition
Figure BDA00025871181800000914
The generalized eigenvector matrix of (a) is,
Figure BDA00025871181800000915
a diagonal matrix is assigned to the power. Substituting (25) into (17) and reusing the first order requirement can derive SkIt should satisfy:
Figure BDA0002587118180000101
let vk，iTo represent
Figure BDA0002587118180000102
The ith diagonal element of (1), taking into account the total power constraint
Figure BDA0002587118180000103
S calculated by equation (26)kIt should also satisfy:
Figure BDA0002587118180000104
3. minimizing Rayleigh quotient in quotient flow
For signal to leakage and noise ratio precoding, signalling
Figure BDA0002587118180000105
Is a generalized Rayleigh quotient molecule matrix,
Figure BDA0002587118180000106
is a generalized Rayleigh quotient denominator matrix. For weighted signal-to-leakage-and-noise ratio precoding, signalling
N＝-Ak (30)
Is a generalized Rayleigh quotient molecule matrix,
D＝Bk+μI (31)
a denominator matrix of generalized rayleigh quotient. When d iskThe precoding matrix for user k can be obtained by minimizing the generalized rayleigh quotient (rayleigh quotient). In other words, PkIs a solution to the problem of equation (32).
Figure BDA0002587118180000107
Figure BDA0002587118180000108
Representing a complex space with the origin removed. When d isk> 1, each column of the precoding matrix for user k can be solved for d by DeflationkThe above problem is solved. Further, for arbitrary tangent vectors
Figure BDA0002587118180000109
In that
Figure BDA00025871181800001010
Above defined Riemann metric
Figure BDA00025871181800001011
Wherein
Figure BDA0002587118180000111
The representation takes the real part.
One solution x to the problem (32)minObviously cxminIs also a solution where c is any complex number other than zero. Further, for any point x,
Figure BDA0002587118180000112
any method that does not distinguish between x and cx when solving the problem (32) is inefficient.
Defining the equivalence relation "-" as: x-y if and only if y ═ cx,
Figure BDA0002587118180000113
collection
Figure BDA0002587118180000114
An equivalence class called x. Business space
Figure BDA0002587118180000115
Is a full space
Figure BDA0002587118180000116
Is called Grassmann manifold. Natural projection (classical projection) is defined as
Figure BDA0002587118180000117
It will be
Figure BDA0002587118180000118
Element x in (1) is mapped to
Figure BDA0002587118180000119
Element [ x ] of (1)]. Generalized Rayleigh quotient over the entire space
Figure BDA00025871181800001110
Also transformed to commodity manifold via natural projection
Figure BDA00025871181800001111
The method comprises the following steps:
Figure BDA00025871181800001112
suppose xi is the cut space
Figure BDA00025871181800001113
X is the equivalence class pi-1([x]) For any given satisfaction of
Figure BDA00025871181800001114
Tangent vector of
Figure BDA00025871181800001115
Can be regarded as a representation of xi, where
Figure BDA00025871181800001116
Denotes a pi (x) edge
Figure BDA00025871181800001117
The directional derivative of (a). Because for
Figure BDA00025871181800001118
Comprises the following steps:
Figure BDA00025871181800001119
this makes there infinite legitimacy at point x
Figure BDA00025871181800001120
May be used to represent ξ. Any business form in abstraction
Figure BDA00025871181800001121
The designed algorithm is finally required to be in the corresponding full space
Figure BDA00025871181800001122
In the above, the cutting space of the full space is
Figure BDA00025871181800001123
In determining the tangent space of the quotient flow
Figure BDA00025871181800001124
The unique legal and effective representation of the arbitrary tangent vector is essential for the design implementation of the final algorithm.
Cutting space of full space
Figure BDA00025871181800001125
Can be divided into vertical spaces
Figure BDA00025871181800001126
From horizontal space
Figure BDA00025871181800001127
The direct sum of (a):
Figure BDA00025871181800001128
the vertical space is defined as the tangent space of the equivalence class:
Figure BDA00025871181800001129
in a way of view, the utility model,
Figure BDA00025871181800001130
is when we try to cut the space in the full space
Figure BDA00025871181800001131
Where denotes the redundant part at ξ. Once the straight sum resolution shown in equation (36) is determined, there is one and only one
Figure BDA0002587118180000121
Satisfy the requirement of
Figure BDA0002587118180000122
Can be arranged in
Figure BDA0002587118180000123
Denotes ξ, which is called horizontal lift of ξ at the x point.
It is most intuitive to expand the whole-space Riemannian metric to commodity manifold trivia. However, in general, any two points p, q ∈ π in the same equivalence class-1([x]) Their horizontal lift of any two tangent vectors in the respective horizontal space,
Figure BDA0002587118180000124
for definition in full space
Figure BDA0002587118180000125
Arbitrary Riemann metric of
Figure BDA0002587118180000126
Do not necessarily satisfy
Figure BDA0002587118180000127
If equation (38) holds, then the Riemann gradient over the quotient flow can be derived from a trivial expansion in full space
Figure BDA0002587118180000128
Wherein
Figure BDA0002587118180000129
Further, if the horizontal space is defined as the vertical space with respect to the Riemann gradient
Figure BDA00025871181800001210
Quadrature complement of
Figure BDA00025871181800001211
Figure BDA00025871181800001212
Is called as
Figure BDA00025871181800001213
The Riemannian commodity flow shape, the natural projection pi is called Riemannian subdivision.
Can prove that the space is full
Figure BDA00025871181800001214
When the above Riemann gradient is defined by formula (33), Grassmann manifold
Figure BDA00025871181800001215
Is a riemann commodity. Then the problem (32) may be
Figure BDA00025871181800001216
The solution is more efficiently performed:
Figure BDA00025871181800001217
under the definition of Riemann gradient according to equation (33), the horizontal space is
Figure BDA00025871181800001218
Cutting space of full space
Figure BDA00025871181800001219
Any tangent vector xi in the vector can be divided into two parts which are orthogonal
Figure BDA00025871181800001220
Wherein
Figure BDA00025871181800001221
Respectively, projected parts of xi to horizontal space and vertical space. Derived for arbitrary
Figure BDA00025871181800001222
Its projection into horizontal space is
Figure BDA00025871181800001223
Wherein
Figure BDA0002587118180000131
By utilizing the property of Riemann quotient manifold, rho ([ x)]) Has a Riemann gradient of
Figure BDA0002587118180000132
To simplify the calculation, an approximate Riemannian sea may be defined as
Figure BDA0002587118180000133
Wherein
Figure BDA0002587118180000134
4. Riemann conjugate gradient method
The riemann conjugate gradient method is an extension of the riemann manifold from the conjugate gradient method in the european space, and is different from the conjugate gradient method in the european space in some details. First, the j-th iteration of the Riemann conjugate gradient method is actually in the tangent space
Figure BDA0002587118180000135
The above is carried out, the result (x)(j)+ηj) Is also in the cutting space
Figure BDA0002587118180000136
In (2), a Recraction function is required to be used to convert (x)(j)+ηj) Mapping a manifold
Figure BDA0002587118180000137
The extraction on the Grassmann manifold can be chosen as
Figure BDA0002587118180000138
In addition, because no addition is defined between different tangent spaces, the more recent change of the direction of the conjugate gradient is also changed into
Figure BDA0002587118180000139
Wherein
Figure BDA00025871181800001310
Is a tangent vector etajFrom the cutting space of itself to alphajηjCan be calculated as
Figure BDA00025871181800001311
For the direction coefficient of conjugate gradient betaj+1According to Fletcher-Reeves (FR) type, Polak-Ribier (PR) type
Figure BDA00025871181800001312
Figure BDA0002587118180000141
Calculating, or, assuming
Figure BDA0002587118180000142
And
Figure BDA0002587118180000143
about
Figure BDA0002587118180000144
Conjugation, i.e.
Figure BDA0002587118180000145
Then there is
Figure BDA0002587118180000146
5. Implementation of signal-to-leakage-to-noise ratio precoding algorithm
Step a): randomly generating or using RZF precoding as an initial precoding matrix
Figure BDA0002587118180000147
And for a user k, setting the sequence number of the generalized characteristic value as i to 1, and giving the maximum iterative search time M.
Step b): when i is less than or equal to dkLet the number j of the conjugate gradient search times be 0, so as to
Figure BDA0002587118180000148
Initializing the current column
Figure BDA0002587118180000149
And calculating an initial maximum eigenvalue
Figure BDA00025871181800001410
The initial conjugate gradient direction is a negative gradient direction
Figure BDA00025871181800001411
Wherein
Figure BDA00025871181800001412
Figure BDA00025871181800001413
If i > 1, deflmation is performed, if
Figure BDA00025871181800001414
Wherein
Figure BDA0002587118180000151
When i > dkTo obtain the maximum front dkAngular matrix of generalized eigenvalues
Figure BDA0002587118180000152
And corresponding orthogonalized generalized eigenvector matrix
Figure BDA0002587118180000153
Step c): using the current column
Figure BDA0002587118180000154
Current conjugate gradient direction ηjCalculating an optimum step size αj. Computing
Figure BDA0002587118180000155
Figure BDA0002587118180000156
Figure BDA0002587118180000157
Figure BDA0002587118180000158
Figure BDA0002587118180000159
Figure BDA00025871181800001510
Figure BDA00025871181800001511
Figure BDA00025871181800001512
Figure BDA00025871181800001513
If i > 1, then
Figure BDA00025871181800001514
The defllation similar to equation (58) is required for all calculations. If it is not
Figure BDA00025871181800001515
Figure BDA00025871181800001516
If it is not
Figure BDA00025871181800001517
And is
Figure BDA00025871181800001518
Figure BDA00025871181800001519
If it is not
Figure BDA00025871181800001520
And is
Figure BDA00025871181800001521
Then alpha isjIs absent.
Step d): if α isjIf so, the current column is updated
Figure BDA00025871181800001522
If α isjIf not, the current column is updated
Figure BDA00025871181800001523
If j is M-1, then
Figure BDA00025871181800001524
To pair
Figure BDA00025871181800001525
Is subjected to orthogonalization to obtain
Figure BDA0002587118180000161
Then unitized, have
Figure BDA0002587118180000162
Let q bei＝q″iI +1, return to step b).
Step e): calculating the updated maximum eigenvalue, namely the updated generalized Rayleigh quotient
Figure BDA0002587118180000163
Riemann gradient corresponding to updated generalized eigenvector
Figure BDA0002587118180000164
If i > 1, then
Figure BDA0002587118180000165
The calculation of (c) requires deflmation similar to equation (58).
Step f): if α isjPresence, first calculate vector move
Figure BDA0002587118180000166
Next, the riemann hessian is calculated:
Figure BDA0002587118180000167
wherein
Figure BDA0002587118180000168
Figure BDA0002587118180000169
The coefficient of conjugate gradient is
Figure BDA0002587118180000171
When alpha isjIs absent, then
Figure BDA0002587118180000172
Step g): update Riemann conjugate gradient direction to
Figure BDA0002587118180000173
And let j equal j +1, return to step c).
Step h): assume that each user's power allocation matrix is
Figure BDA0002587118180000174
And is provided with
Figure BDA0002587118180000175
Each user's final signal to leakage and noise ratio is precoded into
Pk＝QkSk，k＝1，2，...，K (81)
6. Implementation of weighted signal-to-leakage-to-noise ratio precoding algorithm
Step a): randomly generating or using RZF precoding as an initialization precoding matrix
Figure BDA0002587118180000176
The sequence number of the external iteration times is set as d to be 0, and the maximum external iteration time is set as Mo。
Step b): when d is less than or equal to MoCalculating the weighted channel covariance matrix A of each userkWeighted sum matrix B of weighted channel covariance matrices for other userskI.e. calculating RkAnd phil(Cl). Firstly, calculating the beam domain precoding matrix of each user
Figure BDA0002587118180000177
Then, calculating the energy coupling matrix of the beam field precoding matrix of each user and the sum matrix thereof
Figure BDA0002587118180000178
Figure BDA0002587118180000181
Where |, indicates the Hadamard product of the matrix. Then the noise plus interference covariance matrix R for each userkIs calculated as
Figure BDA0002587118180000182
Wherein
Figure BDA0002587118180000183
Is a column vector of all 1's. Further, the method can be used for preparing a novel materialCalculating Al P l1, 2, K is
Figure BDA0002587118180000184
Can be calculated to obtain
Figure BDA0002587118180000185
Further, there are
Figure BDA0002587118180000186
Calculating weighted noise covariance matrix by using weighted channel covariance matrix of each user and weighted sum matrix of weighted channel covariance matrices of other users, i.e. using Ak、BkCalculate μ I. First calculate BkPkK is 1, 2, K is
Figure BDA0002587118180000187
It is noted that
Figure BDA0002587118180000188
Has been calculated in step b), then there is
Figure BDA0002587118180000189
Mu.s of(d)< 0, by a small positive number e, e.g. 10-5。
For a user K, 1, 2, 1, K, the generalized eigenvalue index is 1, and the maximum number M of intra-iteration searches is giveni。
When d > MoAnd finishing the updating to obtain the external iteration number of MoThe number of internal iterations is MiEach user precoding matrix
Figure BDA0002587118180000191
Step c): when i is less than or equal to dkLet the number j of the conjugate gradient search times be 0, so as to
Figure BDA0002587118180000192
Initializing the current column
Figure BDA0002587118180000193
And calculating an initial maximum eigenvalue
Figure BDA0002587118180000194
The initial conjugate gradient direction is a negative gradient direction
Figure BDA0002587118180000195
Wherein
Figure BDA0002587118180000196
And
Figure BDA0002587118180000197
is calculated as equation (86) and equation (88). If i > 1, deflmation is performed, if
Figure BDA0002587118180000198
When i > dkTo obtain the maximum front dkAngular matrix of generalized eigenvalues
Figure BDA0002587118180000199
And corresponding orthogonalized generalized eigenvector matrix
Figure BDA00025871181800001910
Step d): using the current column
Figure BDA00025871181800001911
Current conjugate gradient direction ηjCalculating an optimum step size αj. Computing
Figure BDA00025871181800001912
Figure BDA00025871181800001913
Figure BDA00025871181800001914
Figure BDA00025871181800001915
Figure BDA00025871181800001916
Figure BDA00025871181800001917
Figure BDA00025871181800001918
Figure BDA00025871181800001919
Figure BDA0002587118180000201
If i > 1, then
Figure BDA0002587118180000202
The calculation of (A) requires deflmation similar to the formula (91). If it is not
Figure BDA0002587118180000203
Figure BDA0002587118180000204
If it is not
Figure BDA0002587118180000205
And is
Figure BDA0002587118180000206
If it is not
Figure BDA0002587118180000207
And is
Figure BDA0002587118180000208
Then alpha isjIs absent.
Step e): if α isjIf so, the current column is updated
Figure BDA0002587118180000209
If α isjIf not, the current column is updated
Figure BDA00025871181800002010
If j is equal to Mi-1, then
Figure BDA00025871181800002011
Orthogonalization
Figure BDA00025871181800002012
Is provided with
Figure BDA00025871181800002013
Then unitized, have
Figure BDA00025871181800002014
Let q bei＝q″iI is i +1, return to step d).
Step f): calculating the updated maximum eigenvalue, namely the updated generalized Rayleigh quotient
Figure BDA00025871181800002015
Riemann gradient corresponding to updated generalized eigenvector
Figure BDA00025871181800002016
If i > 1, then
Figure BDA00025871181800002017
The calculation of (2) requires deflmation similar to the formula (91).
Step g): if α isjPresence, first calculate vector move
Figure BDA00025871181800002018
Next, the riemann hessian is calculated:
Figure BDA0002587118180000211
wherein
Figure BDA0002587118180000212
The coefficient of conjugate gradient is
Figure BDA0002587118180000213
When alpha isjIs absent, then
Figure BDA0002587118180000214
Step h): update Riemann conjugate gradient direction to
Figure BDA0002587118180000215
And let j equal j +1, return to step d).
Step i): respectively calculating power distribution matrix of each user
Figure BDA0002587118180000216
Setting the unnormalized power distribution matrix as
Figure BDA0002587118180000217
Then there is
Figure BDA0002587118180000218
Further, power normalization is performed
Figure BDA0002587118180000219
Step j): updating the coding matrix for each user
Figure BDA0002587118180000221
And d is equal to d +1, and the step b) is returned.
Fourth, effect of implementation
In order to make those skilled in the art better understand the scheme of the present invention, the following provides a specific system configuration for performing traversal of precoding transmission by using a low-complexity efficient solution method for precoding with a large-scale MIMO generalized eigenvector structure in this embodimentAnd rate performance display. The system is configured as M t128, K40 and MkLarge scale MIMO system of 1, wherein base station antenna configuration is Mx＝8，Mz16. For simplicity, the moving speed of all users is set to be the same. The refinement factors at the base station are set to F respectivelyx＝2，Fz＝2。
Fig. 2 and fig. 3 show the comparison of the rate performance of the weighted signal-to-leakage-and-noise ratio precoding transmission based on the conjugate gradient method and the riemann conjugate gradient method with the random value as the initial precoding matrix and the RZF precoding matrix as the initial precoding matrix, respectively, under the condition that the 20dB user moving speed is 250 km per hour. In the figure, the ordinate represents the sum rate of all users in the system, and the abscissa represents the number of iterations of the conjugate gradient method/riemann conjugate gradient method; "CG" represents the sum-rate performance curve of the inner iteration using the weighted leakage-to-noise ratio precoding of the conjugate gradient method, and "RCG" represents the sum-rate performance curve of the inner iteration using the weighted leakage-to-noise ratio precoding of the riemann conjugate gradient method. The solid line represents the precoding and rate performance curves for 3 outer iterations and the dashed line represents the precoding and rate performance curves for 1 outer iteration. Regardless of the choice of initial precoding matrix, there is a performance difference between the conjugate gradient method and the riemann gradient method, and the performance difference is more significant in the case of insufficient outer iteration. Under the condition of insufficient external iteration, no matter how the initial precoding matrix is selected, the Riemann conjugate gradient method can still obtain better performance, and the high efficiency of the algorithm is shown; the use of the conjugate gradient rule may degrade performance in the case of a poor initial value.
Based on the same inventive concept, the embodiment of the invention also discloses a large-scale MIMO generalized eigenvector structure precoding solving device, which comprises a memory, a processor and a computer program stored on the memory and capable of running on the processor, wherein when the computer program is loaded to the processor, part or all of the steps of the large-scale MIMO generalized eigenvector structure precoding solving method are realized.

Claims (9)

1. A precoding solving method for a large-scale MIMO generalized eigenvector structure is characterized by comprising the following steps:
(1) generating an initial precoding matrix of each user, wherein the number of rows is the number of base station antennas, and the number of columns is the number of user data streams;
(2) calculating generalized Rayleigh quotient corresponding to each column in each user initial pre-coding matrix, and performing iterative optimization on the generalized Rayleigh quotient on the quotient manifold by adopting a Riemannian conjugate gradient method to obtain an optimized generalized eigenvector matrix;
(3) carrying out power distribution on different columns, and generating a precoding matrix according to the optimized generalized eigenvector matrix;
the step (2) comprises the following steps:
(21) initializing the current column of the generalized eigenvector matrix by using the ith column in the initial precoding matrix of the current user
Figure FDA0003221189170000011
The initial value of the generalized characteristic value serial number i is 1, and the initial value of the search time serial number j is 0; calculating a generalized Rayleigh quotient corresponding to the current column and a Riemann gradient direction thereof, and setting the current conjugate gradient direction as a negative direction of the Riemann gradient direction of the generalized Rayleigh quotient;
(22) calculating an optimal step length by using the current column and the current conjugate gradient direction, and updating the current column according to the optimal step length;
(23) judging whether the iteration number of the set conjugate gradient method is reached, if the iteration number of the set conjugate gradient method is reached and the current column is not the last column of the precoding matrix, stepping to the next column of the initial precoding matrix, returning to the step (21), and otherwise, entering the step (24); if the current column is the last column, finishing iteration, outputting the optimized generalized eigenvector matrix, and jumping to the step (3);
(24) calculating the updated generalized Rayleigh quotient corresponding to the current column and the Riemann gradient direction of the generalized Rayleigh quotient;
(25) calculating vector transport in the conjugate gradient direction by using the current column, the optimal step length and the current conjugate gradient direction;
(26) updating the conjugate gradient coefficient;
(27) and (4) updating the current conjugate gradient direction by using the Riemann gradient direction of the generalized Rayleigh quotient, the conjugate gradient coefficient and the vector transportation of the conjugate gradient direction, and returning to the step (23).
2. The method for solving precoding of a massive MIMO generalized eigenvector structure according to claim 1, wherein the solving process of the generalized Rayleigh quotient corresponding to each column comprises:
for each user, determining a numerator matrix and a denominator matrix of the generalized Rayleigh quotient, and if the current column is not the first column of the initial precoding matrix, performing deflmation operation on the numerator matrix of the generalized Rayleigh quotient;
multiplying the molecular matrix of the generalized Rayleigh quotient by the conjugate transpose of the current column on the left side, and then multiplying the current column on the right side to obtain a molecule;
the denominator matrix of the generalized Rayleigh quotient is multiplied by the conjugate transpose of the current column on the left side, and then multiplied by the current column on the right side to obtain a denominator;
the generalized Rayleigh quotient is obtained by dividing the numerator and the denominator.
3. The massive MIMO generalized eigenvector structure precoding solution method of claim 2, characterized in that: if the signal-to-leakage-and-noise ratio is pre-coded, the molecular matrix of the generalized Rayleigh quotient is a channel covariance matrix of the current user; and if the weighted signal-to-leakage-and-noise ratio is pre-coded, the molecular matrix of the generalized Rayleigh quotient is a weighted channel covariance matrix of the current user.
4. The massive MIMO generalized eigenvector structure precoding solution method of claim 2, characterized in that: if the signal-to-leakage-and-noise ratio is pre-coded, the denominator matrix of the generalized Rayleigh quotient is a sum matrix of a channel covariance matrix and a noise covariance matrix of other users except the current user; and if the precoding is weighted signal-to-leakage-and-noise ratio precoding, the denominator matrix of the generalized Rayleigh quotient is a sum matrix of weighted channel covariance matrixes and weighted noise covariance matrixes of other users except the current user.
5. The massive MIMO generalized eigenvector structure precoding solution of claim 1, characterized in that the quotient manifold is selected as a riemann quotient manifold.
6. The method according to claim 1, wherein the optimal step size is a step size at which the generalized Rayleigh quotient is minimized along the current conjugate gradient direction for the current column.
7. The massive MIMO generalized eigenvector structure precoding solution method according to claim 1, characterized in that the step (22) comprises:
if the optimal step length exists, the product of the optimal step length and the current conjugate gradient direction is added with the current column vector to serve as an updated current column; and if the optimal step length does not exist, updating the current column to be the current conjugate gradient direction.
8. The method of claim 1, wherein the conjugate gradient coefficients are set to zero when an optimal step size does not exist.
9. A precoding solving device of a large-scale MIMO generalized eigenvector structure is characterized by comprising the following components: memory, processor and computer program stored on the memory and executable, the computer program when executed by the processor implementing the steps of the massive MIMO generalized eigenvector structure precoding solving method according to any one of claims 1 to 8.
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