CN111786708A - Joint channel information acquisition method of large-scale MIMO system - Google Patents
Joint channel information acquisition method of large-scale MIMO system Download PDFInfo
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Abstract
The invention belongs to the technical field of wireless communication, and particularly relates to a joint channel information acquisition method of a large-scale MIMO system. The method takes the minimum mean square error as a target, and constructs the beam forming design and the channel estimation problem into a multivariable matrix solving problem. In order to solve the problem more effectively, a framework based on joint optimization is considered, and a beamforming matrix and a channel information matrix are acquired in sequence. Firstly, aiming at a beamforming design, a scheme based on a MADMM (multi-stream alternating multiplier method) is provided for carrying out alternating iterative update on variables, a Riemann manifold and a new Oblique manifold are introduced, the problems of discrete constant modulus constraint, unit norm constraint and the like are solved based on the idea of manifold search, and after an optimal beamforming matrix is converged, sparse recovery is carried out on channel information.
Description
Technical Field
The invention belongs to the technical field of wireless communication, and relates to millimeter-wave communication (millimeter-wave) communication, hybrid beam forming, Riemannian manifold, and Multiple Input Multiple Output (MIMO), in particular to a joint channel information acquisition method of a large-scale MIMO system.
Background
Because the millimeter wave is seriously influenced by the environment, the signal attenuation is large, the scattering and penetrating capabilities are poor, the millimeter wave is easy to block, and the overcoming of the free space path loss and the signal attenuation of signal transmission is a great key of millimeter wave communication. Based on the short wavelength and high frequency of millimeter wave, massive MIMO technology can be adopted to obtain higher gain and resist fading. The large-scale MIMO is the expansion and extension based on the MIMO, and the large-scale antenna array is configured at the base station end, so that higher spatial gain and multiplexing gain are obtained, the beam energy is more concentrated, the beam forming gain is generated to overcome certain path loss, and the energy efficiency is improved under the condition of not increasing the transmitting power. The application of the large-scale MIMO enables the communication between the base station and multiple users to use the same time-frequency resource, and the interference between the users can be not considered to a certain extent by considering that the channel vectors between the users have the characteristic of mutual orthogonality, and the frequency spectrum efficiency is improved by applying a simple beam forming technology so as to obtain good system performance.
Disclosure of Invention
In consideration of system capacity and estimation accuracy, the method takes minimum Mean Square Error (MSE) as a target, and constructs a beam forming design and a channel estimation problem into a multivariable matrix solving problem. In order to solve the problem more effectively, a framework based on joint optimization is considered, and a beamforming matrix and a channel information matrix are acquired in sequence. Firstly, aiming at a beamforming design, a scheme based on a MADMM (multi-stream alternating multiplier method) is provided for carrying out alternating iterative update on variables, a Riemann manifold and a new Oblique manifold are introduced, the problems of discrete constant modulus constraint, unit norm constraint and the like are solved based on the idea of manifold search, and after an optimal beamforming matrix is converged, sparse recovery is carried out on channel information.
The technical scheme of the invention is that the method for acquiring the joint channel information of the large-scale MIMO system is used for a millimeter wave large-scale MIMO single user HBF (Hybrid Beamforming, HBF) system using a phase shifter network, and N is configured at a base station end in the systemtA transmitting antenna anda radio frequency link to which the base station is configuredNrAn antenna anduser side transmission N of radio frequency linksA strip data stream, and satisfyNsThe first pass dimension of a stripe data stream isDigital processor FBBThen pass throughA radio frequency link connected to a dimension ofAnalog beamforming processor FRFAfter being transmitted by the antenna, the signals respectively pass through an analog combined processor at a receiving endAnd digital combined processorTreating and reducing to obtain NsA stripe data stream; the method comprises the following steps:
s1, establishing a model:
assuming that the millimeter wave channel used is a quasi-static channel, the channel information matrix H is regarded as constant for T times, and the vector expression of the received signal is as follows:
wherein, P represents the transmitting power of the transmitting end, n represents an additive white Gaussian noise vector, and obeysNormal distribution with mean 0, squareThe difference is Is a unit matrix, s represents a transmitted pilot symbol;
targeting at minimum mean square error while letting mean square error beThe following model was established:
Tr(PHP)≤P
wherein the diagonal matrix P is a power allocation matrix, (F)RF)i,jAnd (W)RF)i,jAre respectively a matrix FRFAnd matrix WRFRow i and column j;
s2, obtaining a channel matrix H by solving the model established in the step S1, wherein the specific method comprises the following steps:
firstly, randomly generating a channel matrix H which accords with the narrowband millimeter wave channel characteristics, converting a model into an HBF matrix optimization design problem containing discrete constant modulus constraint and power control constraint, and solving by using a plurality of manifold-assisted MADMM methods, wherein the method comprises the following steps:
introducing an auxiliary variable F ═ FRFFBBThe model is re-represented as:
F=FRFFBB
Tr(PHP)≤P
the augmented Lagrangian function expression of the above equation is:
where α denotes a penalty parameter scalar,representing a Lagrange operator matrix, defining an augmented Lagrange function as an objective function, and solving F through the objective functionRF、WRF、FBB、WBB(ii) a When the objective function is iterated for the nth time, the iterative solution steps are as follows:
s.t.Tr(PHP)≤P
riemannian manifold-based conjugate gradient line search method updating variable FRFAnd WRFUpdating variable F based on the steepest gradient descent method of Oblique manifold, and regarding F by an objective functionBBAnd WBBUpdating F of gradient expressionsBBAnd WBBUpdating the variable P after obtaining other variables; after the optimal beamforming matrix solution is obtained, the method substitutes the model established in the step S1 to obtain:
orientation quantization operation:
let ARAnd ATRespectively representing the sets of all receiving antenna array vectors and transmitting antenna array vectors, D represents the product of the normalization factor of the channel and each sub-path gain, i.e. the channel information matrix isObtaining:
thereby constructing a sparse channel matrix with the recovery problem:
s.t.||D||0=Np
and solving the problem through an OMP algorithm to obtain a channel matrix H.
Further, the Riemannian manifold-based conjugate gradient line search method updates the variable FRFAnd WRFThe specific method comprises the following steps:
due to the analog beam forming matrix FRFEach element in (a) satisfies the unit mode limit, and the limited area is regarded as an embedded sub-manifold space, so that x is vec (F)RF) I.e. form aGiven the inner product, the Riemannian sub-manifold of (1) is expressed as:
firstly, the objective function is simplified, and the objective function is rewritten to only contain FRFExpression (2)
Derivation of the above formula to obtain Euclidean gradient expression
Based on the nature of Riemann manifold, manifoldThe Riemann gradient at the upper point x is represented as the Euclidean gradientThe projection onto the tangent space is:
let αtDenotes the search step size, dtRepresenting the search direction, and moving points on the Riemannian manifold along the tangent vector by using retraction operation to obtain a mapping relation expression from the tangent space to the Riemannian manifold:
obtaining search step size α using Armijo backtracking methodtThe expression is
Wherein c is>0, a and b are respectively scalar quantities of values between 0 and 1, the minimum integer l meeting the formula is taken, and the search step length is αt=abl(ii) a Using a transmission operation for realizing the merging of two different tangent space tangent vectors for the tangent spaceThe tangent vector on is mapped to another tangent spaceThe above problem, the expression of the transfer operation is as follows:
in summary, F is obtained by using the cutoff space defined in the riemann manifold, the riemann gradient, the Armijo backtracking method, the retracting operation and the transmission algorithm, and by iterating the conjugate gradient algorithm based on the riemann manifold optimization, so that the cutoff vector converges to a critical pointRFA local optimal solution of;
in the same way, W can be obtainedRFThe local optimum solution of.
Further, the specific method for updating the variable F based on the steepest gradient descent method of the obique manifold is as follows:
first, a defined expression of the Obblique manifold is given as
Obblique manifoldViewed as a complex spaceIn the above-mentioned embedded sub-manifold space, taking the matrixObtaining a plurality of obique manifoldsThe tangent space at point F is expressed as
Derivation of the objective function yields an Euclidean gradient expression of
Order to represent an inclusion manifoldFor any point X ∈, the projection of the point onto the manifold is equivalent to the shortest distance of the point X to the manifold
Mapping the vector to the manifold cut space by projection, defining the gradient of point F on the manifold as the Euclidean gradient at that point to the manifold cut spaceTo obtain a gradient expression
Let d(k)The search direction of the kth iteration is shown, and the k-th iteration is obtained based on the idea of steepest gradient descentLet α(k)Representing the search step length of the kth iteration, adopting an Armijo line search method to satisfy the expression
Wherein, cdecIs a scalar quantity with a value ranging from 10-4 to 0.1,representing a functionAlong a search direction α(k)d(k)Is defined as point F at the manifoldInner product of gradient of (d) and search direction:
in summary, using the gradient descent algorithm based on the oblique manifold optimization, point F is then in the manifoldThe update expression of
And after iteration is carried out until convergence, the optimal solution of F can be obtained.
Further, the update variable FBBAnd WBBThe specific method comprises the following steps:
obtaining an augmented Lagrangian function from an objective function with respect to FBBIs expressed as
Let the above formula be 0, directly obtain the matrix FBBIs expressed as
Obtaining W by the same methodBBIs expressed as
Further, the specific method for updating the variable P is as follows:
when given the remaining variables and only the P-dependent computation terms are retained, the expression for the problem P is found to be
s.t.Tr(PHP)≤P
Since this expression is convex, and the constraint on power P is also convex, P is obtained by using CVX in matlab tool to solve this problem.
The invention has the advantages of improving the energy efficiency under the condition of not increasing the transmitting power, simultaneously obtaining higher space gain and multiplexing gain, further concentrating the beam energy, and generating the beam forming gain to overcome certain path loss.
Drawings
FIG. 1 shows a millimeter wave large MIMO single-user HBF system based on a phase shifter network
FIG. 2 shows a Riemannian manifold and a cutting space
Detailed Description
The technical solution of the present invention is described in detail below with reference to the accompanying drawings.
S1, System model and optimization objectives
The present invention contemplates a millimeter wave large MIMO single user HBF system using a phase shifter network as shown in figure 1. At the base station end, N is configuredtA transmitting antenna anda radio frequency link provided with NrAn antenna anduser side transmission N of radio frequency linksA strip data stream, and satisfyAs shown in FIG. 1, NsThe first pass dimension of a stripe data stream isDigital processor FBBThen pass throughA radio frequency link connected to a dimension ofAnalog beamforming processor FRFAfter the signal channel H is transmitted by antenna, the signal channel H is passed through analog combined processor at receiving endAnd digital combined processorTreating and reducing to obtain NsA striped data stream. Assuming that the millimeter wave channel used is a quasi-static channel, let ARAnd ATRespectively representing all receiving antenna array vectors and a set of transmitting antenna array vectors, D represents the product of the normalization factor of the channel and each sub-path gain, and the channel information matrix H is
The channel matrix is considered constant for T times, and a vector expression of the received signal can be obtained as follows
Wherein, P represents the transmitting power of the transmitting end, n represents an additive white Gaussian noise vector, and obeysHas a mean value of 0 and a variance ofIs normally distributed. s denotes a transmission pilot symbol.
In the present invention, it is not assumed that the Arrival angle (AOA) and the departure Angle (AOD) of the channel are known, and based on the millimeter wave channel expression, only the gain factor corresponding to the main path, i.e. the diagonal matrix D, needs to be estimated, so that the complete channel information matrix H can be recovered. Under the premise, the invention considers the joint recovery of the unknown CSI matrix H and the beam space representation thereof through the unknown sparse channel gain matrix D, and formulates a constrained multivariable optimization problem as follows
Tr(PHP)≤P
(1-3)
Wherein the diagonal matrix P is a power allocation matrix, and if so, the order is such that the influence of noise is minimizedRepresenting the mean square error, then the problem (1-3) can be further represented as
Wherein (F)RF)i,jAnd (W)RF)i,jAre respectively a matrix FRFAnd matrix WRFRow i and column j.
The problems (1-4) comprise non-convex constraint conditions and objective functions and are difficult to solve, and by utilizing the sparsity of millimeter waves, a channel information matrix can be solved based on sparse channel recovery, so that the whole joint optimization problem is considered to be divided into two parts for solving. Under the condition of assuming quasi-static channel, a scheme based on the MADMM is proposed to iteratively converge to the optimal solution of the HBF matrix, and then a measurement matrix is generated based on the optimized HBF matrix to recover channel information. The joint solution process for problems (1-4) will be explained below.
S2 beamforming design based on MADMM
If a channel information matrix is given, the problems (1-4) are converted into an HBF matrix optimization design problem comprising discrete constant modulus constraints, power control constraints and the like. For this sub-module optimization problem, the present invention proposes a MADMM scheme that uses multiple manifold-assisted solutions. Based on the constraint conditions, an embedded Riemann sub manifold space and an embedded Obblique sub manifold space are respectively defined, and searching is carried out on the manifolds so as to realize rapid convergence.
S21, MADMM iteration process
The MADMM may be regarded as an extended application of ADMM (Alternating orientation Method of polymorphisms, ADMM). As a variation of the Augmented Lagrange multiplier Method (ALM), ADMM achieves convergence without the need for specific assumptions such as strict convexity on the objective function, as compared to some other optimal solution methods. ADMM was not widely known in the early days and, after a new discussion of it by Bord et al, 2011, it began to be gradually applied to large-scale distributed optimization problems. ADMM can be considered as a special solution computation framework in practice, which is suitable for convex optimization problems, especially those with separable structures. Since the ADMM method can converge well to an optimal solution and the processing speed is fast, it has been widely used in fields such as machine learning and image restoration. Using ADMM can decompose a large, difficult-to-solve global problem into a number of smaller, easy-to-solve local sub-problems by decomposing the coordination operations, and then obtain a solution to the global large problem by coordinating the solutions of the sub-problems.
In order to construct an iteration problem expression based on the MADMM, the derivation of the ADMM needs to be used for reference, and the ADMM is applied, the concept of a dual-rise method is introduced firstly, and a convex optimization problem is given
Wherein,the variables of the optimization are represented by a table,andrepresenting a constant, the objective function f (x) is a convex function with respect to x, and Ax ═ c can be viewed as a co-written form of p constraints. Can be constructed to have a Lagrangian function of
Where λ represents a dual variable, also known as the lagrangian operator. The original constraint-containing problem (1-6) is an unconstrained dual problemInstead, the optimal solution of both problems is equivalent. By utilizing a dual-ascending method, the solving process is divided into two steps, and an iterative solving expression can be obtained based on the ideas of dual decomposition and gradient descending
Where ρ iskRepresenting the search step size. The dual-rise method requires the objective function f (x) to have strict convex characteristics, and in order to relax the requirement and increase the convergence rate, the ALM is proposed to be applied by increasing the penalty parameter tau>0, constructive augmented Lagrangian function
The problem is solved. The addition of penalty terms makes the dual function more general, however, the addition of square terms does not facilitate solving for variable separation, and hence, the ADMM method is proposed.
ADMM combines the advantages of the dual-rise method such as the decomposability and the weak condition convergence of ALM, and is generally used for solving a multivariable optimization problem containing equality constraint
Wherein,the variables of the optimization are represented by a table,is a constant and g (z) is a convex function with respect to z. Giving an augmented Lagrangian function expression of formulas (1-10)
If the multiplier method is adopted for solving, the iterative expression is
In an iterative process, uniformly solving for variable xk+1And zk+1. If the ADMM method is adopted, the alternative optimization is similar, other variables are fixed, only one variable is updated each time, and the iterative expression is (step mark in the original document is wrong and is modified)
The ADMM is more convenient to realize in terms of the updating process of the ADMM and therefore the method is more widely applied. In addition, while ADMM has previously been applied primarily to the solution of some convex optimization problems, recent studies have demonstrated good performance in some non-convex matrix factorization problems as well.
The basic framework of MADMM is similar to ADMM, but uses manifold subspaces to assist in sub-problem optimization. For the problem (1-4), if the solution is based on the MADMM, the auxiliary variable F ═ F is introduced firstRFFBBRe-expression of the problems (1-4) as
The introduction of the auxiliary variable F enables the HBF matrix constraint in the problems (1-17) to be converted into a univariate constraint, and the iterative solution of the MADMM is convenient to use. Similar to ADMM, first, the augmented Lagrangian function expressions of the given formulas (1-17) are derived as follows
Where α denotes a penalty parameter scalar,representing a lagrange operator matrix. Based on the idea of alternating optimization, for the objective functions (1-18), the iterative solution at the nth iteration can be given as follows
Under the framework of the MADMM, the problems (1-19) to (1-25) need to be solved in order to solve the problem (1-17).
S22 updating variable F based on Riemannian manifoldRFAnd WRF
With respect to the matrix FRFAnd WRFThe problems (1-20) and (1-21) need to be solved. First, the objective functions (1-18) are simplified and rewritten to include only FRFAnd WRFCan respectively obtain the problem expressions
And
however, the discrete constant amplitude constraints contained in equations (1-26) and (1-27) make the minimization problem still difficult to solve, and there is no currently available method for perfectly solving the problem. On the basis, the invention uses an effective algorithm based on manifold optimization to find the local optimal solution of the formulas (1-26) and (1-27) by searching on the manifold.
Based on the constant modulus constraint, the idea of manifold can be introduced to define a multi-dimensional Riemannian manifold. A manifold may be generally considered to be a space that has euclidean space properties in part, where references to manifolds herein refer to topological manifolds, similar to euclidean space or some other relatively simple space in part. At each point on the manifold there is a field, which can be mapped to euclidean space by certain rules.
Manifold classifications are numerous, and Riemann manifold is a relatively common one. FIG. 2 shows a typical Riemann manifold space, manifoldThe tangent space of the last given point xTangent ξ to curve y passing through point xxAnd (4) forming. Riemann manifold refers to a differential manifold having a Riemann metric in space, and Riemann metric refers to a differential manifold defined in tangential spaceOne inner product of (a). That is, it is considered that a manifold has a symmetric and positive second-order covariant tensor field, that is, an orthometric quadratic form is provided in a tangent space of each point, and geometric quantities such as a length and a volume can be measured by using a metric and integrated. Since the riemann metric is defined on the riemann manifold, with rich geometric properties, such that a series of operations such as gradients can be defined, the optimization performed on the riemann manifold will locally resemble the optimization on the euclidean space with smooth constraints. Thus, some optimization algorithms for Euclidean space may be transformed for use on a particular Riemannian manifold.
In order to facilitate the application of manifold optimization, some manifold related concepts need to be introduced. First, a complex plane is givenThe above Euclidean measure is defined as follows
The formula (1-28) introduces inner product operation in the complex planeProceed like real spaceThe above operation. Then correspondingly, the complex plane can be formedOne circle on is shown as
For theFor a given point x, the direction in which it can move can be characterized by a tangent vector.
And the tangent space at the x point may be defined as
For problems (1-26), due to the analog beamforming matrix FRFEach element in (a) satisfies the unit mode limit, and the limited region can be regarded as an embedded sub-manifold space, if x is equal to vec (F)RF) Equivalent to a complex circular manifold
Equations (1-31) transform the bounding regions in equations (1-26) into a multiplication of m circles on the complex planeIs accumulated to form oneGiven the Riemannian sub-manifold of the inner product. Generally speaking, in a manifoldThe upper process optimization problem is less convenient than in euclidean vector space, while the cut space of the riemann manifold has a smoothly varying inner product, which can help solve the problem. Vector takingIf each element in z and the corresponding element in x satisfy
Then the vector z is considered orthogonal to x and is considered to be manifoldThe tangent vector at the upper point x. The set of all tangent vectors at point x constitutes the tangent space at point x, and thus a manifold can be givenThe tangent space expression of the upper arbitrary point x is as follows
Since the field of each point on the manifold resembles euclidean space, some optimization algorithms applied to euclidean space may also be applied locally in the riemann manifold. The riemann manifold-based tangent space provides convenience for optimization problems, and certain line search methods can be adopted for solving, so that a conjugate gradient-based line search method is provided in this chapter for solving equations (1-26).
First, by deriving the formula (1-26), the Euclidean gradient expression can be obtained as
Similar to euclidean space, among all the tangent vectors, the tangent vector associated with the negative riemann gradient is considered to characterize the direction of descent of the function which is the fastest. Based on the nature of Riemann manifold, manifoldThe Riemann gradient at the upper point x may be represented as a Euclidean gradientProjection in the tangential space, the expression being
The retract operation is an important means in manifold optimization by which a vector can be mapped from the tangent space to the manifold itself, when a point is moved along a tangent vector, the position on the manifold after the point is moved can be found by retracting, let αtDenotes the search step size, dtRepresenting the search direction, if the point on the Riemannian manifold is moved along the tangent vector, the expression of the mapping relation from the tangent space to the Riemannian manifold can be obtained as follows
In summary, using the cutoff space, the Riemann gradient and the retraction operation defined in the Riemann manifold, the Riemann manifold optimization-based conjugate gradient algorithm as shown in Table 1-1 can be obtained to find FRFThe local optimum solution of.
TABLE 1-1 on FRFBased on RiemaConjugate gradient algorithm for nnian manifold optimization
Armijo backtracking is used in algorithm 1-1 to obtain the search step αtFor the expression
Is provided with c>0, and a and b are respectively determined scalars with values between 0 and 1, and the minimum integer l satisfying the formula (1-37) is taken to obtain the search step length of αt=abl. Meanwhile, in the algorithm 1-1, a Polak-Ribiere parameter is used to assist the updating of the search direction, so that the objective function remains non-constructive in each iteration. In addition, algorithm 1-1 uses an operation called transmission for merging two different tangent space tangent vectors for the tangent space to be mergedThe tangent vector on is mapped to another tangent spaceThe expression of the transfer operation is as follows
By iteration of algorithm 1-1, a critical point can be converged on.
Similarly, for problems (1-27), the combining matrix W can be based on simulationsRFThe unit mode in (1) is restricted, an embedded sub-manifold space is introduced, if x is equal to vec (W)RF) Equivalent to a complex circular manifold
Vector takingCan give manifoldThe tangent space expression of the upper arbitrary point x is as follows
By deriving the equations (1-27) the euclidean gradient expression can be found as
Based on projection operation, Riemann manifold can be obtainedThe Riemann gradient at the upper point x is expressed as
Wherein,is a vectorized representation of equation (1-41). In summary, in combination with the operations of retraction, transmission, etc., the matrix W can be found by using a conjugate gradient algorithm based on manifold optimization similar to that shown in Table 1-1RFThe local optimum solution of.
S23, updating variable F based on Oblique manifold
For problems (1-19), the unit norm constraint of the auxiliary variable is actually derived from power control, and in a large MIMO system, considering that each transmit antenna has equal maximum transmit power in order to improve the use efficiency of the power amplifier, an equivalent expression with respect to the constraint condition can be obtained as
Therefore, the concept of complex Obblique manifold can be introduced to represent the constraint region. A defined expression giving an Obblique manifold is
Obblique manifoldCan be regarded as a complex spaceEmbedded sub-manifold space above. Taking a particular matrixCan give manifoldThe tangent space at point F is expressed as
Similar to Riemann manifold optimization, derivation of equations (1-18) can result in a Euclidean gradient expression of
If the order represents an inclusive manifoldFor any point X ∈, the projection of the point onto the manifold is equivalent to the shortest distance of the point X to the manifold
Then the point F is substituted into the formula (1-47), and the solution can obtain the point F to manifoldIs projected as
The vectors can also be mapped to the manifold cut space by projection, defining the gradient of a point F on the manifold as the euclidean gradient at that point to the manifold cut spaceCan obtain a gradient expression
Some optimization algorithm expansion on euclidean space can be applied to the sub-manifold space solution (1-19), this section using a gradient descent method.
The basic idea of the gradient descent method is to consider the direction of the negative gradient to be the direction of the fastest descent, and use the direction of the negative gradient at the starting point of each iteration as the search direction of the iterative optimization, so the method is also called the steepest descent method. For a point a, if the function F (x) is defined and differentiable at point a, the function F (x) follows at point aThe direction is decreased fastest. Thus, for a point b, if the expression is
For a sufficiently small gamma greater than zero, there is F (b). ltoreq.F (a). Consider a office from function f (x)Initial point x of minimum0Starting from, a set of sequences is taken to satisfy
May be represented by the sequence xnThe iteration of the method converges gradually to a desired minimum.
Generalizing to manifold subspace, if d(k)The search direction of the kth iteration is represented, and based on the idea of steepest gradient descent, the method can obtainLet α(k)Representing the search step length of the kth iteration, adopting an Armijo line search method to satisfy the expression
Wherein, cdecIs a scalar quantity with a value ranging from 10-4 to 0.1,representing a functionAlong a search direction α(k)d(k)Can be defined as point F in the manifoldInner product of gradient of (d) and search direction
It can be found that point F is in manifold when searched by gradient descent methodThe update expression above is as follows
TABLE 1-2 gradient descent algorithm based on oblique manifold optimization for F
In summary, the Oblique shape optimization algorithm based on gradient descent as shown in Table 1-2 can be obtained for algorithm 1-2, the parameter β can also be introduced based on modified Hesteees-Stiefel law, etckAnd using the transmission operation Tran (-) to obtain a new search direction by the combination of two search directionsThe algorithm is transformed into a conjugate gradient descent method.
S24, updating variable FBBAnd WBB
For the solution of the problem (1-22), firstly, the augmented Lagrangian function is obtained from the objective function about FBBIs expressed as
By directly making equation (1-55) equal to 0, a matrix F can be obtainedBBIs expressed as
Similarly, for problems (1-23), an augmented Lagrangian function can be derived with respect to WBBThe gradient expression is as follows
When the value of the equation (1-57) is 0, the matrix W can be obtainedBBIs expressed as
S25, updating variable P
Given the remaining variables and retaining only P-dependent computations, the expression of the problem (1-24) can be found as
Since the objective function in the problem (1-59) is convex, and the constraint on the power P is also convex, the sub-problem can be solved by using CVX.
S26 and channel information H
After converging to the optimal beamforming matrix solution, the formula (1-4) is substituted, and the problem expression can be obtained as follows
The expression (1-60) only contains 1 variable H, and based on the sparse characteristic of millimeter waves, only gains related to the AoA and AoD of the main paths can be estimated to recover the channel matrix based onThe operation rule of (1) to (60) can be obtained by performing an orientation quantization operation on the calculation term
By substituting the formula (1-1), the formula (1-61) can be further transformed
A new channel sparse gain vector estimation problem can be constructed based on (1-60) and (1-62)
Thus, the channel path gain matrix D can be recovered using an OMP based algorithm, and since the channels AoA and AoD are assumed to be known, a can be usedTAnd ARA channel matrix H is constructed.
S27 joint optimization framework
In summary, the frames for implementing the MADMM-based joint channel information acquisition and beamforming design algorithm shown in tables 1 to 3 can be obtained. Firstly, a channel matrix H which accords with the narrowband millimeter wave channel characteristics is randomly generated and is used as a fixed quantity, algorithms 1-1, 1-2 and the like are called through the step 2 to sequentially and circularly iterate and update a beam forming matrix and the like, and when the maximum iteration times are reached or the difference between two iteration values is judged to be small, circulation is ended to obtain the optimal solution. And finally, constructing a sparse channel matrix recovery problem based on the convergence optimal solution, and acquiring a channel matrix H through an OMP algorithm.
Tables 1-3 MADMM-based channel information acquisition and beamforming design algorithm
Claims (5)
1. A joint channel information acquisition method of a large-scale MIMO system is used for a millimeter wave large-scale MIMO single-user HBF system using a phase shifter network, and N is configured at a base station end in the systemtA transmitting antenna anda radio frequency link, a base station is configured with NrAn antenna anduser side transmission N of radio frequency linksA strip data stream, and satisfyNsThe first pass dimension of a stripe data stream isDigital processor FBBThen pass throughA radio frequency link connected to a dimension ofAnalog beamforming processor FRFAfter being transmitted by the antenna, the signals respectively pass through an analog combined processor at a receiving endAnd digital combined processorTreating and reducing to obtain NsA stripe data stream; characterized in that the method comprises the following steps:
s1, establishing a model:
assuming that the millimeter wave channel used is a quasi-static channel, the channel information matrix H is regarded as constant for T times, and the vector expression of the received signal is as follows:
wherein, P represents the transmitting power of the transmitting end, n represents an additive white Gaussian noise vector, and obeysHas a mean of 0 and a variance of Is a unit matrix, s represents a transmitted pilot symbol;
targeting at minimum mean square error while letting mean square error beThe following model was established:
Tr(PHP)≤P
wherein the diagonal matrix P is a power allocation matrix, (F)RF)i,jAnd (W)RF)i,jAre respectively a matrix FRFAnd matrix WRFRow i and column j;
s2, obtaining a channel matrix H by solving the model established in the step S1, wherein the specific method comprises the following steps:
firstly, randomly generating a channel matrix H which accords with the narrowband millimeter wave channel characteristics, converting a model into an HBF matrix optimization design problem containing discrete constant modulus constraint and power control constraint, and solving by using a plurality of manifold-assisted MADMM methods, wherein the method comprises the following steps:
introducing an auxiliary variable F ═ FRFFBBThe model is re-represented as:
F=FRFFBB
Tr(PHP)≤P
the augmented Lagrangian function expression of the above equation is:
where α denotes a penalty parameter scalar,representing a Lagrange operator matrix, defining an augmented Lagrange function as an objective function, and solving F through the objective functionRF、WRF、FBB、WBB(ii) a When the objective function is iterated for the nth time, the iterative solution steps are as follows:
s.t.Tr(PHP)≤P
riemannian manifold-based conjugate gradient line search method updating variable FRFAnd WRFUpdating variable F based on the steepest gradient descent method of Oblique manifold, and regarding F by an objective functionBBAnd WBBUpdating F of gradient expressionsBBAnd WBBUpdating the variable P after obtaining other variables; after the optimal beamforming matrix solution is obtained, the method substitutes the model established in the step S1 to obtain:
orientation quantization operation:
let ARAnd ATRespectively representing the sets of all receiving antenna array vectors and transmitting antenna array vectors, D represents the product of the normalization factor of the channel and each sub-path gain, i.e. the channel information matrix isObtaining:
thereby constructing a sparse channel matrix with the recovery problem:
s.t.||D||0=Np
and solving the problem through an OMP algorithm to obtain a channel matrix H.
2. The joint channel information acquisition method of massive MIMO system as claimed in claim 1 wherein the Riemannian manifold based conjugate gradient line search method updates variable FRFAnd WRFThe specific method comprises the following steps:
due to the analog beam forming matrix FRFEach element in (a) satisfies the unit mode limit, and the limited area is regarded as an embedded sub-manifold space, so that x is vec (F)RF) I.e. form aGiven the inner product, the Riemannian sub-manifold of (1) is expressed as:
firstly, the objective function is simplified, and the objective function is rewritten to only contain FRFExpression (2)
Derivation of the above formula to obtain Euclidean gradient expression
Based on the nature of Riemann manifold, manifoldThe Riemann gradient at the upper point x is represented as the Euclidean gradientThe projection onto the tangent space is:
let αtDenotes the search step size, dtRepresenting the search direction, and moving points on the Riemannian manifold along the tangent vector by using retraction operation to obtain a mapping relation expression from the tangent space to the Riemannian manifold:
obtaining search step size α using Armijo backtracking methodtThe expression is
Wherein c is>0, a and b are respectively scalar quantities of values between 0 and 1, the minimum integer l meeting the formula is taken, and the search step length is αt=abl(ii) a Using a transmission operation for realizing the merging of two different tangent space tangent vectors for the tangent spaceThe tangent vector on is mapped to another tangent spaceThe above problem, the expression of the transfer operation is as follows:
in summary, F is obtained by using the cutoff space defined in the riemann manifold, the riemann gradient, the Armijo backtracking method, the retracting operation and the transmission algorithm, and by iterating the conjugate gradient algorithm based on the riemann manifold optimization, so that the cutoff vector converges to a critical pointRFA local optimal solution of;
in the same way, W can be obtainedRFThe local optimum solution of.
3. The joint channel information acquisition method of the massive MIMO system according to claim 1 or 2, wherein the specific method for updating the variable F based on the steepest gradient descent method of the obique manifold is as follows:
first, a defined expression of the Obblique manifold is given as
Obblique manifoldViewed as a complex spaceIn the above-mentioned embedded sub-manifold space, taking the matrixObtaining a plurality of obique manifoldsThe tangent space at point F is expressed as
Derivation of the objective function yields an Euclidean gradient expression of
Order to represent an inclusion manifoldFor an arbitrary point, isThe projection of a point onto the manifold is equivalent to the shortest distance of the point X to the manifold
Mapping the vector to the manifold cut space by projection, defining the gradient of point F on the manifold as the Euclidean gradient at that point to the manifold cut spaceTo obtain a gradient expression
Let d(k)The search direction of the kth iteration is shown, and the k-th iteration is obtained based on the idea of steepest gradient descentLet α(k)Representing the search step length of the kth iteration, adopting an Armijo line search method to satisfy the expression
Wherein, cdecIs a scalar quantity with a value ranging from 10-4 to 0.1,representing a functionAlong a search direction α(k)d(k)Is defined as point F at the manifoldInner product of gradient of (d) and search direction:
in summary, using the gradient descent algorithm based on the oblique manifold optimization, point F is then in the manifoldThe update expression of
And after iteration is carried out until convergence, the optimal solution of F can be obtained.
4. The joint channel information acquisition method for massive MIMO system as in claim 3, wherein the updating variable FBBAnd WBBThe specific method comprises the following steps:
obtaining an augmented Lagrangian function from an objective function with respect to FBBIs expressed as
Let the above formula be 0, directly obtain the matrix FBBIs expressed as
Obtaining W by the same methodBBIs expressed as
5. The joint channel information acquisition method of massive MIMO system as claimed in claim 4, wherein the specific method for updating variable P is:
when given the remaining variables and only the P-dependent computation terms are retained, the expression for the problem P is found to be
s.t.Tr(PHP)≤P
Since this expression is convex, and the constraint on power P is also convex, P is obtained by using CVX in matlab tool to solve this problem.
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