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 PDFInfo
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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
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 userThe 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
WhereinIs a refined received sample steering vector matrix at the user side,is a refined transmission sampling guide vector matrix at the base station side.Refined steering vector matrix from vertical directionAnd fine guide vector matrix in horizontal directionThe 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 ofWhen 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
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
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):
When single time slot precoding is considered, the time slot number m is omitted, and the channel (5) can be further simplified into
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
Wherein P iskM being the kth UEt×dkDimension precoding matrix, zkIs a distribution ofThe complex gaussian random noise vector of (a),for each element of the variance of the noise vector,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:
Pkis the power of the user k and,for the channel covariance matrix of user k,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 pairCorresponding 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:
let RkRepresents z'kThe covariance matrix of (2) is:
wherein the expectation functionPresentation 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 functionThe 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:
whereinAlso 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:
defining functionsAnd 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
Wherein, wkIs the weighting factor for the kth user and P is the total power constraint.
Obviously, the precoding matrix of the userVector spaceCan 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
WhereinFor the flow pattern, eachIs a factor manifold thereof. Precoding sets that can prove to satisfy a total power constraintIs thatOne embedded sub-manifold. By usingIs shown in the embedding spaceThe objective function of (3) isRepresenting constraints in embedded sub-manifoldThe objective function of (1). The problem (14) is transformed into manifoldThe unconstrained problem on (1):
for theTwo tangent vectors of any point PAnddefinition ofThe Riemann measure ofThen f (P) can be deduced to beThe above Riemann gradient is gradf (P) ═ gradf (P)1),gradf(P2),...,gradf(PK) In which the component on the kth factor manifold is:
wherein
Maximization by utilizationThe 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)
whereinIs a matrix pair (A)k,Bk+ mu I) corresponds to the diagonal matrix formed by generalized eigenvalues, without loss of generality, andmatrix AkCan be viewed as a weighted channel covariance matrix, of user kIs 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 kCan 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)
whereinIs to satisfy the orthogonality conditionThe generalized eigenvector matrix of (a) is,a diagonal matrix is assigned to the power. Substituting (25) into (17) and reusing the first order requirement can derive SkIt should satisfy:
let vk,iTo representThe ith diagonal element of (1), taking into account the total power constraintS calculated by equation (26)kIt should also satisfy:
3. minimizing Rayleigh quotient in quotient flow
For signal to leakage and noise ratio precoding, signalling
Is a generalized Rayleigh quotient molecule matrix,
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).
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 vectorsIn thatAbove defined Riemann metric
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,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,collectionAn equivalence class called x. Business spaceIs a full spaceIs called Grassmann manifold. Natural projection (classical projection) is defined asIt will beElement x in (1) is mapped toElement [ x ] of (1)]. Generalized Rayleigh quotient over the entire spaceAlso transformed to commodity manifold via natural projectionThe method comprises the following steps:
suppose xi is the cut spaceX is the equivalence class pi-1([x]) For any given satisfaction ofTangent vector ofCan be regarded as a representation of xi, whereDenotes a pi (x) edgeThe directional derivative of (a). Because forComprises the following steps:
this makes there infinite legitimacy at point xMay be used to represent ξ. Any business form in abstractionThe designed algorithm is finally required to be in the corresponding full spaceIn the above, the cutting space of the full space isIn determining the tangent space of the quotient flowThe unique legal and effective representation of the arbitrary tangent vector is essential for the design implementation of the final algorithm.
Cutting space of full spaceCan be divided into vertical spacesFrom horizontal spaceThe direct sum of (a):
the vertical space is defined as the tangent space of the equivalence class:
in a way of view, the utility model,is when we try to cut the space in the full spaceWhere denotes the redundant part at ξ. Once the straight sum resolution shown in equation (36) is determined, there is one and only oneSatisfy the requirement ofCan be arranged inDenotes 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,for definition in full spaceArbitrary Riemann metric ofDo not necessarily satisfy
If equation (38) holds, then the Riemann gradient over the quotient flow can be derived from a trivial expansion in full space
WhereinFurther, if the horizontal space is defined as the vertical space with respect to the Riemann gradientQuadrature complement of
Is called asThe Riemannian commodity flow shape, the natural projection pi is called Riemannian subdivision.
Can prove that the space is fullWhen the above Riemann gradient is defined by formula (33), Grassmann manifoldIs a riemann commodity. Then the problem (32) may beThe solution is more efficiently performed:
under the definition of Riemann gradient according to equation (33), the horizontal space is
Cutting space of full spaceAny tangent vector xi in the vector can be divided into two parts which are orthogonal
WhereinRespectively, projected parts of xi to horizontal space and vertical space. Derived for arbitraryIts projection into horizontal space is
WhereinBy utilizing the property of Riemann quotient manifold, rho ([ x)]) Has a Riemann gradient of
To simplify the calculation, an approximate Riemannian sea may be defined as
Wherein
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 spaceThe above is carried out, the result (x)(j)+ηj) Is also in the cutting spaceIn (2), a Recraction function is required to be used to convert (x)(j)+ηj) Mapping a manifoldThe extraction on the Grassmann manifold can be chosen as
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
For the direction coefficient of conjugate gradient betaj+1According to Fletcher-Reeves (FR) type, Polak-Ribier (PR) type
Then there is
5. Implementation of signal-to-leakage-to-noise ratio precoding algorithm
Step a): randomly generating or using RZF precoding as an initial precoding matrixAnd 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 toInitializing the current columnAnd calculating an initial maximum eigenvalueThe initial conjugate gradient direction is a negative gradient direction
Wherein
If i > 1, deflmation is performed, if
Wherein
When i > dkTo obtain the maximum front dkAngular matrix of generalized eigenvaluesAnd corresponding orthogonalized generalized eigenvector matrix
Step c): using the current columnCurrent conjugate gradient direction ηjCalculating an optimum step size αj. Computing
If i > 1, thenThe defllation similar to equation (58) is required for all calculations. If it is not If it is notAnd is If it is notAnd isThen alpha isjIs absent.
Step d): if α isjIf so, the current column is updated
If α isjIf not, the current column is updated
Then unitized, have
Let q bei=q″iI +1, return to step b).
Step e): calculating the updated maximum eigenvalue, namely the updated generalized Rayleigh quotientRiemann gradient corresponding to updated generalized eigenvector
Step f): if α isjPresence, first calculate vector move
Next, the riemann hessian is calculated:
wherein
The coefficient of conjugate gradient is
When alpha isjIs absent, then
Step g): update Riemann conjugate gradient direction to
And let j equal j +1, return to step c).
Step h): assume that each user's power allocation matrix isAnd is provided withEach 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 matrixThe 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
Then, calculating the energy coupling matrix of the beam field precoding matrix of each user and the sum matrix thereof
Where |, indicates the Hadamard product of the matrix. Then the noise plus interference covariance matrix R for each userkIs calculated as
WhereinIs a column vector of all 1's. Further, the method can be used for preparing a novel materialCalculating Al P l1, 2, K is
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
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
Step c): when i is less than or equal to dkLet the number j of the conjugate gradient search times be 0, so as toInitializing the current columnAnd calculating an initial maximum eigenvalueThe initial conjugate gradient direction is a negative gradient direction
When i > dkTo obtain the maximum front dkAngular matrix of generalized eigenvaluesAnd corresponding orthogonalized generalized eigenvector matrix
Step d): using the current columnCurrent conjugate gradient direction ηjCalculating an optimum step size αj. Computing
If i > 1, thenThe calculation of (A) requires deflmation similar to the formula (91). If it is not If it is notAnd isIf it is notAnd isThen alpha isjIs absent.
Step e): if α isjIf so, the current column is updated
If α isjIf not, the current column is updated
Then unitized, have
Let q bei=q″iI is i +1, return to step d).
Step f): calculating the updated maximum eigenvalue, namely the updated generalized Rayleigh quotientRiemann gradient corresponding to updated generalized eigenvector
Step g): if α isjPresence, first calculate vector move
Next, the riemann hessian is calculated:
wherein
The coefficient of conjugate gradient is
When alpha isjIs absent, then
Step h): update Riemann conjugate gradient direction to
And let j equal j +1, return to step d).
Step i): respectively calculating power distribution matrix of each userSetting the unnormalized power distribution matrix asThen there is
Further, power normalization is performed
Step j): updating the coding matrix for each user
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 userThe 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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