CN111786708A - Joint channel information acquisition method of large-scale MIMO system - Google Patents

Joint channel information acquisition method of large-scale MIMO system Download PDF

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CN111786708A
CN111786708A CN202010625345.1A CN202010625345A CN111786708A CN 111786708 A CN111786708 A CN 111786708A CN 202010625345 A CN202010625345 A CN 202010625345A CN 111786708 A CN111786708 A CN 111786708A
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刘红
赵柏睿
张忠培
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University of Electronic Science and Technology of China
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    • H04B7/00—Radio transmission systems, i.e. using radiation field
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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

Joint channel information acquisition method of large-scale MIMO system
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 and
Figure BDA0002566390930000021
a radio frequency link to which the base station is configuredNrAn antenna and
Figure BDA0002566390930000022
user side transmission N of radio frequency linksA strip data stream, and satisfy
Figure BDA0002566390930000023
NsThe first pass dimension of a stripe data stream is
Figure BDA0002566390930000024
Digital processor FBBThen pass through
Figure BDA0002566390930000025
A radio frequency link connected to a dimension of
Figure BDA0002566390930000026
Analog beamforming processor FRFAfter being transmitted by the antenna, the signals respectively pass through an analog combined processor at a receiving end
Figure BDA0002566390930000027
And digital combined processor
Figure BDA0002566390930000028
Treating 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:
Figure BDA0002566390930000029
wherein, P represents the transmitting power of the transmitting end, n represents an additive white Gaussian noise vector, and obeys
Figure BDA00025663909300000210
Normal distribution with mean 0, squareThe difference is
Figure BDA00025663909300000211
Figure BDA00025663909300000212
Is a unit matrix, s represents a transmitted pilot symbol;
targeting at minimum mean square error while letting mean square error be
Figure BDA00025663909300000213
The following model was established:
Figure BDA00025663909300000214
Figure BDA00025663909300000215
Figure BDA00025663909300000216
Figure BDA00025663909300000217
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:
Figure BDA0002566390930000031
Figure BDA0002566390930000032
Figure BDA0002566390930000033
Figure BDA0002566390930000034
F＝FRFFBB
Tr(PHP)≤P
the augmented Lagrangian function expression of the above equation is:
Figure BDA0002566390930000035
where α denotes a penalty parameter scalar,
Figure BDA0002566390930000036
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:
Figure BDA0002566390930000037
Figure BDA0002566390930000038
Figure BDA0002566390930000041
Figure BDA0002566390930000042
Figure BDA0002566390930000043
Figure BDA0002566390930000044
Figure BDA0002566390930000045
Figure BDA0002566390930000046
Figure BDA0002566390930000047
s.t.Tr(PHP)≤P
Figure BDA0002566390930000048
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:
Figure BDA0002566390930000049
orientation quantization operation:
Figure BDA00025663909300000410
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 is
Figure BDA00025663909300000411
Obtaining:
Figure BDA0002566390930000051
thereby constructing a sparse channel matrix with the recovery problem:
Figure BDA0002566390930000052
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 a
Figure BDA0002566390930000053
Given the inner product, the Riemannian sub-manifold of (1) is expressed as:
Figure BDA0002566390930000054
giving a manifold
Figure BDA0002566390930000055
The tangent space expression of the upper arbitrary point x is as follows
Figure BDA0002566390930000056
Wherein the vector
Figure BDA0002566390930000057
And is orthogonal to x;
firstly, the objective function is simplified, and the objective function is rewritten to only contain FRFExpression (2)
Figure BDA0002566390930000058
Figure BDA0002566390930000059
Derivation of the above formula to obtain Euclidean gradient expression
Figure BDA0002566390930000061
Based on the nature of Riemann manifold, manifold
Figure BDA0002566390930000062
The Riemann gradient at the upper point x is represented as the Euclidean gradient
Figure BDA0002566390930000063
The projection onto the tangent space is:
Figure BDA0002566390930000064
wherein the content of the first and second substances,
Figure BDA0002566390930000065
is a vectorized representation of the euclidean gradient;
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:
Figure BDA0002566390930000066
Figure BDA0002566390930000067
obtaining search step size α using Armijo backtracking methodtThe expression is
Figure BDA0002566390930000068
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 space
Figure BDA0002566390930000069
The tangent vector on is mapped to another tangent space
Figure BDA00025663909300000610
The above problem, the expression of the transfer operation is as follows:
Figure BDA00025663909300000611
Figure BDA00025663909300000612
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
Figure BDA0002566390930000071
Obblique manifold
Figure BDA0002566390930000072
Viewed as a complex space
Figure BDA0002566390930000073
In the above-mentioned embedded sub-manifold space, taking the matrix
Figure BDA0002566390930000074
Obtaining a plurality of obique manifolds
Figure BDA0002566390930000075
The tangent space at point F is expressed as
Figure BDA0002566390930000076
Derivation of the objective function yields an Euclidean gradient expression of
Figure BDA0002566390930000077
Order to represent an inclusion manifold
Figure BDA0002566390930000078
For any point X ∈, the projection of the point onto the manifold is equivalent to the shortest distance of the point X to the manifold
Figure BDA0002566390930000079
Figure BDA00025663909300000710
Point F goes to manifold
Figure BDA00025663909300000711
Is projected as
Figure BDA00025663909300000712
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 space
Figure BDA00025663909300000713
To obtain a gradient expression
Figure BDA0002566390930000081
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 descent
Figure BDA0002566390930000082
Let α(k)Representing the search step length of the kth iteration, adopting an Armijo line search method to satisfy the expression
Figure BDA0002566390930000083
Wherein, cdecIs a scalar quantity with a value ranging from 10-4 to 0.1,
Figure BDA0002566390930000084
representing a function
Figure BDA0002566390930000085
Along a search direction α(k)d(k)Is defined as point F at the manifold
Figure BDA0002566390930000086
Inner product of gradient of (d) and search direction:
Figure BDA0002566390930000087
in summary, using the gradient descent algorithm based on the oblique manifold optimization, point F is then in the manifold
Figure BDA00025663909300000811
The update expression of
Figure BDA0002566390930000088
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
Figure BDA0002566390930000089
Let the above formula be 0, directly obtain the matrix FBBIs expressed as
Figure BDA00025663909300000810
Obtaining W by the same methodBBIs expressed as
Figure BDA0002566390930000091
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
Figure BDA0002566390930000092
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 and
Figure BDA0002566390930000093
a radio frequency link provided with NrAn antenna and
Figure BDA0002566390930000094
user side transmission N of radio frequency linksA strip data stream, and satisfy
Figure BDA0002566390930000095
As shown in FIG. 1, NsThe first pass dimension of a stripe data stream is
Figure BDA0002566390930000096
Digital processor FBBThen pass through
Figure BDA0002566390930000101
A radio frequency link connected to a dimension of
Figure BDA0002566390930000102
Analog beamforming processor FRFAfter the signal channel H is transmitted by antenna, the signal channel H is passed through analog combined processor at receiving end
Figure BDA0002566390930000103
And digital combined processor
Figure BDA0002566390930000104
Treating 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
Figure BDA0002566390930000105
The channel matrix is considered constant for T times, and a vector expression of the received signal can be obtained as follows
Figure BDA0002566390930000106
Wherein, P represents the transmitting power of the transmitting end, n represents an additive white Gaussian noise vector, and obeys
Figure BDA0002566390930000107
Has a mean value of 0 and a variance of
Figure BDA0002566390930000108
Is 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
Figure BDA0002566390930000109
Figure BDA00025663909300001010
Figure BDA00025663909300001011
Figure BDA00025663909300001012
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 minimized
Figure BDA0002566390930000111
Representing the mean square error, then the problem (1-3) can be further represented as
Figure BDA0002566390930000112
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
Figure BDA0002566390930000121
Wherein the content of the first and second substances,
Figure BDA0002566390930000122
the variables of the optimization are represented by a table,
Figure BDA0002566390930000123
and
Figure BDA0002566390930000124
representing 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
Figure BDA0002566390930000125
Where λ represents a dual variable, also known as the lagrangian operator. The original constraint-containing problem (1-6) is an unconstrained dual problem
Figure BDA0002566390930000126
Instead, 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
Figure BDA0002566390930000127
Figure BDA0002566390930000128
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
Figure BDA0002566390930000129
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
Figure BDA0002566390930000131
Wherein the content of the first and second substances,
Figure BDA0002566390930000132
the variables of the optimization are represented by a table,
Figure BDA0002566390930000133
is a constant and g (z) is a convex function with respect to z. Giving an augmented Lagrangian function expression of formulas (1-10)
Figure BDA0002566390930000134
If the multiplier method is adopted for solving, the iterative expression is
Figure BDA0002566390930000135
Figure BDA0002566390930000136
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)
Figure BDA0002566390930000137
Figure BDA0002566390930000138
Figure BDA0002566390930000139
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
Figure BDA0002566390930000141
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
Figure BDA0002566390930000142
Where α denotes a penalty parameter scalar,
Figure BDA0002566390930000143
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
Figure BDA0002566390930000144
Figure BDA0002566390930000145
Figure BDA0002566390930000146
Figure BDA0002566390930000151
Figure BDA0002566390930000152
Figure BDA0002566390930000153
Figure BDA0002566390930000154
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
Figure BDA0002566390930000155
And
Figure BDA0002566390930000156
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, manifold
Figure BDA0002566390930000161
The tangent space of the last given point x
Figure BDA0002566390930000162
Tangent 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 space
Figure BDA0002566390930000163
One 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 given
Figure BDA0002566390930000164
The above Euclidean measure is defined as follows
Figure BDA0002566390930000165
The formula (1-28) introduces inner product operation in the complex plane
Figure BDA0002566390930000166
Proceed like real space
Figure BDA0002566390930000167
The above operation. Then correspondingly, the complex plane can be formed
Figure BDA0002566390930000168
One circle on is shown as
Figure BDA0002566390930000169
For the
Figure BDA00025663909300001610
For 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
Figure BDA00025663909300001611
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
Figure BDA0002566390930000171
Equations (1-31) transform the bounding regions in equations (1-26) into a multiplication of m circles on the complex planeIs accumulated to form one
Figure BDA0002566390930000172
Given the Riemannian sub-manifold of the inner product. Generally speaking, in a manifold
Figure BDA0002566390930000173
The 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 taking
Figure BDA0002566390930000174
If each element in z and the corresponding element in x satisfy
Figure BDA0002566390930000175
Then the vector z is considered orthogonal to x and is considered to be manifold
Figure BDA0002566390930000176
The 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 given
Figure BDA0002566390930000177
The tangent space expression of the upper arbitrary point x is as follows
Figure BDA0002566390930000178
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
Figure BDA0002566390930000179
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, manifold
Figure BDA0002566390930000181
The Riemann gradient at the upper point x may be represented as a Euclidean gradient
Figure BDA0002566390930000182
Projection in the tangential space, the expression being
Figure BDA0002566390930000183
Wherein the content of the first and second substances,
Figure BDA0002566390930000184
is a vectorized representation of equation (1-34).
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
Figure BDA0002566390930000185
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
Figure BDA0002566390930000186
Figure BDA0002566390930000191
Armijo backtracking is used in algorithm 1-1 to obtain the search step αtFor the expression
Figure BDA0002566390930000192
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 merged
Figure BDA0002566390930000193
The tangent vector on is mapped to another tangent space
Figure BDA0002566390930000194
The expression of the transfer operation is as follows
Figure BDA0002566390930000195
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
Figure BDA0002566390930000196
Vector taking
Figure BDA0002566390930000201
Can give manifold
Figure BDA0002566390930000202
The tangent space expression of the upper arbitrary point x is as follows
Figure BDA0002566390930000203
By deriving the equations (1-27) the euclidean gradient expression can be found as
Figure BDA0002566390930000204
Based on projection operation, Riemann manifold can be obtained
Figure BDA0002566390930000205
The Riemann gradient at the upper point x is expressed as
Figure BDA0002566390930000206
Wherein the content of the first and second substances,
Figure BDA0002566390930000207
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
Figure BDA0002566390930000208
Therefore, the concept of complex Obblique manifold can be introduced to represent the constraint region. A defined expression giving an Obblique manifold is
Figure BDA0002566390930000209
Obblique manifold
Figure BDA00025663909300002010
Can be regarded as a complex space
Figure BDA00025663909300002011
Embedded sub-manifold space above. Taking a particular matrix
Figure BDA0002566390930000211
Can give manifold
Figure BDA0002566390930000212
The tangent space at point F is expressed as
Figure BDA0002566390930000213
Similar to Riemann manifold optimization, derivation of equations (1-18) can result in a Euclidean gradient expression of
Figure BDA0002566390930000214
If the order represents an inclusive manifold
Figure BDA0002566390930000215
For any point X ∈, the projection of the point onto the manifold is equivalent to the shortest distance of the point X to the manifold
Figure BDA0002566390930000216
Then the point F is substituted into the formula (1-47), and the solution can obtain the point F to manifold
Figure BDA0002566390930000217
Is projected as
Figure BDA0002566390930000218
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 space
Figure BDA0002566390930000219
Can obtain a gradient expression
Figure BDA00025663909300002110
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 a
Figure BDA00025663909300002111
The direction is decreased fastest. Thus, for a point b, if the expression is
Figure BDA0002566390930000221
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
Figure BDA0002566390930000222
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 obtain
Figure BDA0002566390930000223
Let α(k)Representing the search step length of the kth iteration, adopting an Armijo line search method to satisfy the expression
Figure BDA0002566390930000224
Wherein, cdecIs a scalar quantity with a value ranging from 10-4 to 0.1,
Figure BDA0002566390930000225
representing a function
Figure BDA0002566390930000226
Along a search direction α(k)d(k)Can be defined as point F in the manifold
Figure BDA0002566390930000227
Inner product of gradient of (d) and search direction
Figure BDA0002566390930000228
It can be found that point F is in manifold when searched by gradient descent method
Figure BDA0002566390930000229
The update expression above is as follows
Figure BDA00025663909300002210
TABLE 1-2 gradient descent algorithm based on oblique manifold optimization for F
Figure BDA00025663909300002211
Figure BDA0002566390930000231
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 directions
Figure BDA0002566390930000232
The 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
Figure BDA0002566390930000233
By directly making equation (1-55) equal to 0, a matrix F can be obtainedBBIs expressed as
Figure BDA0002566390930000234
Similarly, for problems (1-23), an augmented Lagrangian function can be derived with respect to WBBThe gradient expression is as follows
Figure BDA0002566390930000241
When the value of the equation (1-57) is 0, the matrix W can be obtainedBBIs expressed as
Figure BDA0002566390930000242
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
Figure BDA0002566390930000243
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
Figure BDA0002566390930000244
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 on
Figure BDA0002566390930000245
The operation rule of (1) to (60) can be obtained by performing an orientation quantization operation on the calculation term
Figure BDA0002566390930000251
By substituting the formula (1-1), the formula (1-61) can be further transformed
Figure BDA0002566390930000252
A new channel sparse gain vector estimation problem can be constructed based on (1-60) and (1-62)
Figure BDA0002566390930000253
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
Figure BDA0002566390930000254
Figure BDA0002566390930000261

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 and
Figure FDA0002566390920000011
a radio frequency link, a base station is configured with NrAn antenna and
Figure FDA0002566390920000012
user side transmission N of radio frequency linksA strip data stream, and satisfy
Figure FDA0002566390920000013
NsThe first pass dimension of a stripe data stream is
Figure FDA0002566390920000014
Digital processor FBBThen pass through
Figure FDA0002566390920000015
A radio frequency link connected to a dimension of
Figure FDA0002566390920000016
Analog beamforming processor FRFAfter being transmitted by the antenna, the signals respectively pass through an analog combined processor at a receiving end
Figure FDA0002566390920000017
And digital combined processor
Figure FDA0002566390920000018
Treating 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:
Figure FDA0002566390920000019
wherein, P represents the transmitting power of the transmitting end, n represents an additive white Gaussian noise vector, and obeys
Figure FDA00025663909200000110
Has a mean of 0 and a variance of
Figure FDA00025663909200000111
Figure FDA00025663909200000112
Is a unit matrix, s represents a transmitted pilot symbol;
targeting at minimum mean square error while letting mean square error be
Figure FDA00025663909200000113
The following model was established:
Figure FDA00025663909200000114
Figure FDA00025663909200000115
Figure FDA00025663909200000116
Figure FDA00025663909200000117
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:
Figure FDA0002566390920000021
Figure FDA0002566390920000022
Figure FDA0002566390920000023
Figure FDA0002566390920000024
F＝FRFFBB
Tr(PHP)≤P
the augmented Lagrangian function expression of the above equation is:
Figure FDA0002566390920000025
where α denotes a penalty parameter scalar,
Figure FDA0002566390920000026
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:
Figure FDA0002566390920000027
Figure FDA0002566390920000028
Figure FDA0002566390920000029
Figure FDA00025663909200000210
Figure FDA0002566390920000031
Figure FDA0002566390920000032
Figure FDA0002566390920000033
Figure FDA0002566390920000034
Figure FDA0002566390920000035
s.t.Tr(PHP)≤P
Figure FDA0002566390920000036
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:
Figure FDA0002566390920000037
orientation quantization operation:
Figure FDA0002566390920000038
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 is
Figure FDA0002566390920000039
Obtaining:
Figure FDA00025663909200000310
thereby constructing a sparse channel matrix with the recovery problem:
Figure FDA0002566390920000041
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 a
Figure FDA0002566390920000042
Given the inner product, the Riemannian sub-manifold of (1) is expressed as:
Figure FDA0002566390920000043
giving a manifold
Figure FDA0002566390920000044
The tangent space expression of the upper arbitrary point x is as follows
Figure FDA0002566390920000045
Wherein the vector
Figure FDA0002566390920000046
And is orthogonal to x;
firstly, the objective function is simplified, and the objective function is rewritten to only contain FRFExpression (2)
Figure FDA0002566390920000047
Figure FDA0002566390920000048
Derivation of the above formula to obtain Euclidean gradient expression
Figure FDA0002566390920000049
Based on the nature of Riemann manifold, manifold
Figure FDA00025663909200000410
The Riemann gradient at the upper point x is represented as the Euclidean gradient
Figure FDA00025663909200000411
The projection onto the tangent space is:
Figure FDA0002566390920000051
wherein the content of the first and second substances,
Figure FDA0002566390920000052
is a vectorized representation of the euclidean gradient;
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:
Figure FDA0002566390920000053
Figure FDA0002566390920000054
obtaining search step size α using Armijo backtracking methodtThe expression is
Figure FDA0002566390920000055
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 space
Figure FDA0002566390920000056
The tangent vector on is mapped to another tangent space
Figure FDA0002566390920000057
The above problem, the expression of the transfer operation is as follows:
Figure FDA0002566390920000058
Figure FDA0002566390920000059
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
Figure FDA0002566390920000061
Obblique manifold
Figure FDA0002566390920000062
Viewed as a complex space
Figure FDA0002566390920000063
In the above-mentioned embedded sub-manifold space, taking the matrix
Figure FDA0002566390920000064
Obtaining a plurality of obique manifolds
Figure FDA0002566390920000065
The tangent space at point F is expressed as
Figure FDA0002566390920000066
Derivation of the objective function yields an Euclidean gradient expression of
Figure FDA0002566390920000067
Order to represent an inclusion manifold
Figure FDA0002566390920000068
For an arbitrary point, is
Figure FDA0002566390920000069
The projection of a point onto the manifold is equivalent to the shortest distance of the point X to the manifold
Figure FDA00025663909200000610
Figure FDA00025663909200000611
Point F goes to manifold
Figure FDA00025663909200000612
Is projected as
Figure FDA00025663909200000613
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 space
Figure FDA00025663909200000614
To obtain a gradient expression
Figure FDA00025663909200000615
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 descent
Figure FDA00025663909200000616
Let α(k)Representing the search step length of the kth iteration, adopting an Armijo line search method to satisfy the expression
Figure FDA00025663909200000617
Wherein, cdecIs a scalar quantity with a value ranging from 10-4 to 0.1,
Figure FDA0002566390920000071
representing a function
Figure FDA0002566390920000072
Along a search direction α(k)d(k)Is defined as point F at the manifold
Figure FDA0002566390920000073
Inner product of gradient of (d) and search direction:
Figure FDA0002566390920000074
in summary, using the gradient descent algorithm based on the oblique manifold optimization, point F is then in the manifold
Figure FDA0002566390920000075
The update expression of
Figure FDA0002566390920000076
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
Figure FDA0002566390920000077
Let the above formula be 0, directly obtain the matrix FBBIs expressed as
Figure FDA0002566390920000078
Obtaining W by the same methodBBIs expressed as
Figure FDA0002566390920000079
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
Figure FDA0002566390920000081
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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