CN115459821B - Low-complexity convex relaxation optimization hybrid precoding method based on matrix multiplication decomposition - Google Patents
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Abstract
A low-complexity convex relaxation optimization hybrid precoding method based on matrix multiplication decomposition belongs to the field of millimeter wave large-scale MIMO system mobile communication. The method aims at the problem of high algorithm complexity of alternating minimization based on convex relaxation optimization. Comprising the following steps: constructing an analog precoding matrix F RF Assigning an initial value to each element; according to the analog precoding matrix F RF Solving digital precoding matrix F by using least square method BB The method comprises the steps of carrying out a first treatment on the surface of the Alternate minimized iterative solution of analog precoding matrix F RF Digital precoding matrix F BB The method comprises the steps of carrying out a first treatment on the surface of the Until the change amounts of the current objective function and the adjacent previous objective function meet a preset change amount threshold value, and ending the iteration; the finally obtained analog precoding matrix F RF As final analog precoding matrix F RF The method comprises the steps of carrying out a first treatment on the surface of the According to the power constraint condition, the finally obtained digital precoding matrix F BB Normalization to obtain final digital precoding matrix F BB The method comprises the steps of carrying out a first treatment on the surface of the The method and the device are used for realizing the mixed precoding of the transmitting end signals in the MIMO system.
Description
Technical Field
The invention relates to a low-complexity convex relaxation optimization hybrid precoding method based on matrix multiplication decomposition, and belongs to the field of millimeter wave large-scale MIMO system mobile communication.
Background
The large-scale MIMO technology and the millimeter wave communication technology are used as key technologies of new-generation mobile communication, multiplexing gain of a large-scale antenna array and spectrum resources rich in millimeter waves are utilized, the system capacity can be effectively improved, and ultra-high-speed and ultra-low-time-delay data transmission is realized. The combination of the large-scale MIMO technology and the millimeter wave communication technology not only can overcome the defect of millimeter wave high path loss, but also can reduce the difficulty of large-scale antenna array integration. Therefore, the millimeter wave massive MIMO system is widely applied to civil, industrial and military fields such as mobile communication and unmanned aerial vehicle communication.
In a millimeter wave massive MIMO system, as the number of antenna elements increases, antenna coupling and channel correlation increase, resulting in a decrease in reliability of system transmission. In order to solve the above problems, researchers begin to process signals at the transmitting end by using a precoding technology, which not only can reduce the complexity of signal processing at the receiving end, but also can reduce the influence of channel correlation, thereby improving the spectral efficiency of the system and reducing the bit error rate. In addition, in order to overcome the limitations of the conventional digital precoding and analog beamforming techniques, researchers have proposed hybrid precoding techniques, i.e., the information preprocessing process is completed by combining low-dimensional digital precoding and high-dimensional analog precoding. The hybrid precoding technique will typically simulate the precoding matrix F RF Digital precoding matrix F BB And optimal all-digital precoding matrix F opt As an objective function, by minimizing the euclidean distance to achieve a hybrid precoding process. At the same time, due to the analog precoding matrix F RF Implemented in hardware by means of a phase shifter, the matrix thus has a constant modulus value aboutThe beam conditions are such that the optimization problem is a non-convex optimization.
The mixed pre-coding algorithm of the millimeter wave large-scale MIMO system mainly comprises two types, wherein the first type is a mixed pre-coding algorithm based on orthogonal matching pursuit and depending on channel estimation information, the sparse reconstruction is used as a theoretical basis of mixed pre-coding by utilizing the channel structural characteristics of the millimeter wave, and the channel information is utilized to simulate a pre-coding matrix F RF Reconstruction is carried out on the digital precoding matrix F by using a least square method BB Solving to approach the optimal all-digital precoding matrix F opt Thereby completing the hybrid precoding process. The second type is a hybrid precoding algorithm based on alternate minimization independent of channel estimation information, and the optimal all-digital precoding matrix F is directly realized by utilizing an optimization theory opt Is used for completing the optimization target of the analog precoding matrix F RF Digital precoding matrix F BB Is used for the alternate optimization solution of (3).
Since the hybrid precoding algorithm based on the alternate minimization does not depend on channel estimation and directly realizes the optimization target by utilizing the optimization theory, the system performance can approximate to the optimal all-digital precoding algorithm. However, the optimization theory related to the algorithm has higher calculation complexity, wherein the alternating minimization algorithm based on convex relaxation optimization relaxes the non-convex constraint to the convex constraint, so that the mixed precoding problem is solved by utilizing the convex optimization theory, but matrix inversion operation with higher calculation complexity exists, and the hardware implementation is not facilitated.
Disclosure of Invention
Aiming at the problem of high complexity of an alternating minimization algorithm based on convex relaxation optimization, the invention provides a low-complexity convex relaxation optimization mixed precoding method based on matrix multiplication decomposition.
The invention relates to a low-complexity convex relaxation optimization mixed pre-coding method based on matrix multiplication decomposition, which comprises the following steps of,
step one: constructing an analog precoding matrix F RF Assigning an initial value to each element;
step two: according to the analog precoding matrix F RF Solving digital precoding matrix by least square methodF BB ；
Step three: digital precoding matrix F obtained by alternately using current calculation BB Minimizing the analog precoding matrix F for the next iteration solution RF And using the next analog precoding matrix F RF Minimizing the next digital precoding matrix F for iterative solution BB ；
Step four: the analog precoding matrix F obtained according to the step three RF Digital precoding matrix F BB Calculating a current objective function until the variation of the current objective function and the adjacent previous objective function meets a preset variation threshold value, and ending the iteration;
step five: the analog precoding matrix F obtained in the step three is processed RF As final analog precoding matrix F RF The method comprises the steps of carrying out a first treatment on the surface of the According to the power constraint condition, the digital precoding matrix F obtained in the step three is obtained BB Normalization to obtain final digital precoding matrix F BB The method comprises the steps of carrying out a first treatment on the surface of the And (5) completing mixed precoding.
According to the low-complexity convex relaxation optimization hybrid precoding method based on matrix multiplication decomposition, a precoding matrix F is simulated in the first step RF Amplitude of initial value of medium elementThe following relationship is satisfied:
representing an analog precoding matrix F RF Elements of the ith row and the jth column; n (N) t The number of the antenna array elements at the transmitting end of the MIMO system;
analog precoding matrix F RF Is of the dimension ofThe number of the radio frequency chains;
the phase of each element is randomly generated.
According to the low-complexity convex relaxation optimization hybrid precoding method based on matrix multiplication decomposition, in the second step, the digital precoding matrix F obtained by solving by using a least square method is adopted BB The method comprises the following steps:
in the middle ofIs F RF Transposed conjugate matrix of F opt Is the optimal all-digital precoding matrix.
According to the low-complexity convex relaxation optimization mixed pre-coding method based on matrix multiplication decomposition, in the third step, the matrix multiplication decomposition theory is utilized to decompose the objective function, the power constraint condition is ignored, and the objective function is expressed as:
in the middle ofAn optimal solution for the analog precoding matrix;
according to the non-convexity of the constraint condition of the objective function, the constraint condition is relaxed, and the objective function is converted into:
wherein beta is a preset constant in the convex relaxation process;
search solution is carried out on the feasible domain of the constraint condition through relaxation, and an analog precoding matrix F is obtained RF Normalization is carried out:
wherein arg represents a plurality of argument;
then F is carried out by utilizing matrix multiplication decomposition theory RF F BB The rewrites are the sum of the series of submatrices:
wherein m is an integer and the value range isA row index or a column index representing a matrix;
further, a residual matrix F is obtained res ：
F in the formula m Is an auxiliary matrix;
according to the formula (6), converting the optimal solution optimization target intoA sub-problem, expressed as:
in the middle ofAnd 1 is a unit vector of a corresponding dimension.
According to the low-complexity convex relaxation optimization hybrid precoding method based on matrix multiplication decomposition, in the third step, the kth iteration process of the alternate minimized iteration solving method comprises the following steps:
step three: solving the kth secondary matrix
In the middle ofFor the kth residual matrix,/th order>For the kth analog precoding matrix, +.>A digital precoding matrix for a kth order; k=1, 2,3, … …;
step three, two: fixing the kth order digital precoding matrix during the kth iterationThe kth analog precoding matrix is +.>Solving;
and step three: during the kth iteration, the kth analog precoding matrix is fixedUse of least square method for k-th order digital precoding matrix +.>And (3) carrying out solving:
and step three, four: in the kth iteration, calculateSolving the kth residual matrix
Step three, five: returning to the third step, repeatingThe next time, the digital precoding matrix F in the kth iteration process is completed BB And an analog precoding matrix F RF Is a solution to (c).
According to the low-complexity convex relaxation optimization hybrid precoding method based on matrix multiplication decomposition, a digital precoding matrix F BB Normalization to obtain final digital precoding matrix F BB The method of (1) comprises:
N s the data stream is transmitted for the communication system.
The invention has the beneficial effects that: the method of the invention relaxes the non-convex constraint of the precoding problem into a convex constraint, and alternately optimizes the analog and digital precoding matrix by utilizing a convex optimization theory and a least square method; the matrix multiplication decomposition theory is utilized to convert the mixed precoding optimization problem into a plurality of optimization sub-problems, so that the line-by-line and column-by-column optimization solution of the digital and analog precoding matrixes is respectively realized, the multiplication dimension of the matrixes is reduced, matrix inversion operation with higher complexity is avoided, and the purpose of reducing the algorithm complexity is realized.
The method is applied to a millimeter wave large-scale MIMO system, can reduce the multiplication dimension of a matrix and avoid the matrix inversion process on the premise of ensuring the spectral efficiency, the bit error rate and other performance losses of the system, reduces the complexity of mixed precoding, improves the realizability of the mixed precoding and completes the signal preprocessing process.
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FIG. 1 is a flow chart of a low complexity convex relaxation optimized hybrid precoding method based on matrix multiplication decomposition in accordance with the present invention;
FIG. 2 is a schematic diagram of a system in an embodiment of the invention;
FIG. 3 is a graph of spectral efficiency analysis in an embodiment of the present invention; in the figure, optimal Full-Digital represents an Optimal all-Digital precoding algorithm, and is a classical method; OMP represents an algorithm based on orthogonal matching pursuit, is a classical algorithm of mixed precoding, CVXR represents the existing convex relaxation method, and MMD_CVXR represents the method of the invention;
fig. 4 is a chart of bit error rate analysis in an embodiment of the present invention.
Detailed Description
The following description of the embodiments of the present invention will be made clearly and completely with reference to the accompanying drawings, in which it is apparent that the embodiments described are only some embodiments of the present invention, but not all embodiments. All other embodiments, which can be made by those skilled in the art based on the embodiments of the invention without making any inventive effort, are intended to be within the scope of the invention.
It should be noted that, without conflict, the embodiments of the present invention and features of the embodiments may be combined with each other.
The invention is further described below with reference to the drawings and specific examples, which are not intended to be limiting.
The present invention provides a low complexity convex relaxation optimized hybrid precoding method based on matrix multiplication decomposition, as shown in fig. 1 and 2, comprising,
step one: constructing an analog precoding matrix F RF Assigning an initial value to each element; the element amplitude meets the constraint condition of a constant module value;
step two: according to the analog precoding matrix F RF Solving digital precoding matrix F by using least square method BB ；
Step (a)Thirdly,: digital precoding matrix F obtained by alternately using current calculation BB Minimizing the analog precoding matrix F for the next iteration solution RF And using the next analog precoding matrix F RF Minimizing the next digital precoding matrix F for iterative solution BB ；
Step four: the analog precoding matrix F obtained according to the step three RF Digital precoding matrix F BB Calculating a current objective function until the variation of the current objective function and the adjacent previous objective function meets a preset variation threshold value, and ending the iteration; the iteration end condition can be selected according to the actual system performance requirement, and the embodiment takes the variation of the objective function as the iteration end condition
Step five: the analog precoding matrix F obtained in the step three is processed RF As final analog precoding matrix F RF The method comprises the steps of carrying out a first treatment on the surface of the According to the power constraint condition, the digital precoding matrix F obtained in the step three is obtained BB Normalization to obtain final digital precoding matrix F BB The method comprises the steps of carrying out a first treatment on the surface of the And (5) completing mixed precoding.
Further, due to the analog precoding matrix F RF Is implemented by means of a phase shifter, so that the matrix is constrained by a constant modulus value, i.e. the analog precoding matrix F in step one RF Amplitude of initial value of medium elementThe following relationship is satisfied:
representing an analog precoding matrix F RF Elements of the ith row and the jth column; n (N) t The number of antenna array elements at a transmitting end of a large-scale MIMO system is calculated;
analog precoding matrix F RF Is of the dimension ofThe number of the radio frequency chains;
the phase of each element is randomly generated.
In the second step, the least square method is used for solving the obtained digital precoding matrix F BB The method comprises the following steps:
in the middle ofIs F RF Transposed conjugate matrix of F opt Is the optimal all-digital precoding matrix.
Since the design of hybrid precoding takes into account the power constraints of the system, the precoding matrix F is modeled RF The constant modulus constraint amplitude of (2) does not affect the system performance and the solving process. The objective function may be expressed as, under the condition of temporarily ignoring the system power constraint:
in the middle ofAn optimal solution for the analog precoding matrix; />Is an objective function;
the constraint condition of the objective function has non-convexity, so that the constraint condition is considered to be relaxed to convert the objective function into:
wherein beta is a preset constant in the convex relaxation process;
after searching and solving on the feasible domain of the constraint condition which is relaxed, in order to meet the original non-convex constraint condition, an analog precoding matrix F is needed RF Normalization is carried out:
wherein arg represents a plurality of argument;
based on the optimization target, the matrix multiplication decomposition theory is utilized to make F RF F BB The rewrites are the sum of the series of submatrices:
wherein m is an integer and the value range isA row index or a column index representing a matrix;
further, a residual matrix F is obtained res ：
F in the formula m Is an auxiliary matrix;
according to the formula (6), converting the optimal solution optimization target intoA sub-problem, expressed as:
in the middle ofAn optimal solution of the digital precoding matrix, 1 is the corresponding oneA unit vector of dimensions.
And (3) carrying out optimization solution on the optimization target of the formula (7) by adopting an alternating minimization method. First, it can be considered that F BB (m: remain unchanged), then about F RF The optimization problem of (: m) can be expressed as:
solving by utilizing convex optimization theory to obtain F RF After m), it is kept unchanged, then about F BB The optimization objective of (m,:) can be expressed as:
for the optimization problem, the least square method is adopted for F BB (m, step) is solved:
for a pair ofThe optimization sub-problems are solved sequentially according to the optimization method, and the digital precoding matrix F can be achieved BB And an analog precoding matrix F RF Is optimized row by row and column by column, in particular as follows:
in the third step, the kth iteration process of the alternate minimized iteration solution method includes:
step three: solving the kth secondary matrix
In the middle ofFor the kth residual matrix,/th order>For the kth analog precoding matrix, +.>A digital precoding matrix for a kth order; k=1, 2,3, … …;
step three, two: fixing the kth order digital precoding matrix during the kth iterationThe kth analog precoding matrix is +.>Solving;
and step three: during the kth iteration, the kth analog precoding matrix is fixedUse of least square method for k-th order digital precoding matrix +.>And (3) carrying out solving:
and step three, four: in the kth iteration process, solving the kth residual matrix
Step three, five: returning to the third step, repeatingThe next time, the digital precoding matrix F in the kth iteration process is completed BB And an analog precoding matrix F RF Is a solution to (c).
Digital precoding matrix F BB Normalization to obtain final digital precoding matrix F BB The method of (1) comprises:
N s the data stream is transmitted for the communication system.
Specific examples:
the method is applied to a millimeter wave large-scale MIMO system, and the actual system schematic diagram is shown in FIG. 2: the transmitting end adopts a uniform planar antenna array structure, wherein the transmitting end is provided with 12×12 antenna elements, 4 data streams are transmitted to the receiving end provided with 6×6 antenna elements, and the number of radio frequency chains of the transmitting end is 4.
The embodiment is described with reference to fig. 1, which includes the following specific implementation steps:
step 1: constructing an analog precoding matrix F with dimensions of 128×4 RF Assigning an initial value to each element to obtain an initial analog precoding matrix
Due to the analog precoding matrix F RF Is implemented by a phase shifter, so that the matrix is limited by a constant modulus value, i.e. the amplitude of the element needs to satisfyTo sum up, the initial analog precoding matrix +.>The amplitude of the element meets the constraint condition, and the phase is randomly generatedAnd (3) obtaining the product.
Step 2: digital precoding matrix F using least square method BB And (3) carrying out solving:
step 3: alternate minimized iterative solution of analog precoding matrix F RF Digital precoding matrix F BB ；
In the present embodiment, the iteration end condition is that the variation of the objective function is less than 10 -3 。
Solving the 4 optimization sub-problems sequentially according to the optimization method, and realizing the digital precoding matrix F BB And an analog precoding matrix F RF The specific steps are as follows:
step 3.1: in the kth iteration, solving the auxiliary variable
Step 3.2: in the kth iteration, the digital precoding matrix is fixedAnalog precoding matrix using convex optimization>Solving;
step 3.3: in the kth iteration, the pre-coding matrix is fixedDigital precoding matrix using least square method>And (3) carrying out solving:
step 3.4: in the kth iteration process, solving the residual matrix
Step 3.5: repeating steps 3.1, 3.2, 3.3, 3.4 for 4 times to complete the digital precoding matrix F during the kth iteration BB And an analog precoding matrix F RF Is a solution to (c).
Step 4: repeating the step 3 until reaching the iteration ending condition;
step 5: according to the power constraint condition, the digital precoding matrix F BB Normalization is carried out:
in combination with the method of the invention shown in fig. 3, the traditional convex relaxation optimization method and the classical mixed precoding algorithm spectrum efficiency performance comparison analysis, the method of the invention utilizes matrix multiplication decomposition theory to decompose the objective function intoThe sub-problems are that the digital and analog pre-coding matrixes are optimized column by column, the dimensionality of matrix multiplication is reduced, meanwhile, the matrix inversion process is avoided, and the purpose of reducing algorithm complexity is achieved. FIG. 4 is a graph of bit error rate performance versus analysis of the method of the present invention versus a conventional convex relaxation optimization method and a classical hybrid precoding algorithm; table 1 shows the complexity analysis of the CVXR-AltMin algorithm, and Table 2 shows the complexity analysis of the MMD-CVXR-AltMin algorithm:
TABLE 1 CVXR-AltMin Algorithm complexity analysis
TABLE 2 MMD-CVXR-AltMin Algorithm complexity analysis
Table 1 shows the complexity analysis of the hybrid precoding method by adopting the conventional CVXR-AltMin algorithm, and Table 2 shows the complexity analysis of the hybrid precoding method by adopting the MMD-CVXR-AltMin algorithm; r in Table 1 m Is an auxiliary variable in the CVXR-AltMin algorithm; the comparison of tables 1 and 2 shows that the method of the present invention achieves a reduction in algorithm complexity.
In connection with fig. 3 and 4, it is shown that the method of the present invention performs consistently with the conventional convex relaxation optimization method and better approximates the optimal all-digital precoding algorithm. As can be seen from simulation results, the spectrum efficiency and the bit error rate performance of the method are approximately the same as those of the traditional convex relaxation optimization method, but the dimension of matrix multiplication can be reduced, the matrix inversion process is avoided, and the complexity of an algorithm is reduced.
In summary, the method of the invention can reduce the complexity of the mixed precoding under the premise of ensuring the performances of the system such as frequency spectrum efficiency, bit error rate and the like, and has the characteristics of good system performance, low complexity and the like
Although the invention herein has been described with reference to particular embodiments, it is to be understood that these embodiments are merely illustrative of the principles and applications of the present invention. It is therefore to be understood that numerous modifications may be made to the illustrative embodiments and that other arrangements may be devised without departing from the spirit and scope of the present invention as defined by the appended claims. It should be understood that the different dependent claims and the features described herein may be combined in ways other than as described in the original claims. It is also to be understood that features described in connection with separate embodiments may be used in other described embodiments.
Claims (2)
1. A low-complexity convex relaxation optimization mixed pre-coding method based on matrix multiplication decomposition is characterized by comprising the following steps of,
step one: constructing an analog precoding matrix F RF Assigning an initial value to each element;
step two: according to the analog precoding matrix F RF Solving digital precoding matrix F by using least square method BB ；
Step three: digital precoding matrix F obtained by alternately using current calculation BB Minimizing the analog precoding matrix F for the next iteration solution RF And using the next analog precoding matrix F RF Minimizing the next digital precoding matrix F for iterative solution BB ；
Step four: the analog precoding matrix F obtained according to the step three RF Digital precoding matrix F BB Calculating a current objective function until the variation of the current objective function and the adjacent previous objective function meets a preset variation threshold value, and ending the iteration;
step five: the analog precoding matrix F obtained in the step three is processed RF As final analog precoding matrix F RF The method comprises the steps of carrying out a first treatment on the surface of the According to the power constraint condition, the digital precoding matrix F obtained in the step three is obtained BB Normalization to obtain final digital precoding matrix F BB The method comprises the steps of carrying out a first treatment on the surface of the Completing mixed pre-coding;
step one simulates a precoding matrix F RF Amplitude of initial value of medium elementThe following relationship is satisfied:
representing an analog precoding matrix F RF Elements of the ith row and the jth column; n (N) t The number of the antenna array elements at the transmitting end of the MIMO system;
analog precoding matrix F RF Is of the dimension ofThe number of the radio frequency chains;
the phase of each element is randomly generated;
in the second step, the least square method is used for solving the obtained digital precoding matrix F BB The method comprises the following steps:
in the middle ofIs F RF Transposed conjugate matrix of F opt The optimal all-digital precoding matrix;
in the third step, the matrix multiplication decomposition theory is utilized to decompose the objective function, the power constraint condition is ignored, and the objective function is expressed as:
in the middle ofAn optimal solution for the analog precoding matrix;
according to the non-convexity of the constraint condition of the objective function, the constraint condition is relaxed, and the objective function is converted into:
wherein beta is a preset constant in the convex relaxation process;
search solution is carried out on the feasible domain of the constraint condition through relaxation, and an analog precoding matrix F is obtained RF Normalization is carried out:
wherein arg represents a plurality of argument;
then F is carried out by utilizing matrix multiplication decomposition theory RF F BB The rewrites are the sum of the series of submatrices:
wherein m is an integer and the value range isA row index or a column index representing a matrix;
further, a residual matrix F is obtained res ：
F in the formula m Is an auxiliary matrix;
according to the formula (6), converting the optimal solution optimization target intoA sub-problem, expressed as:
in the middle ofThe method comprises the steps that 1 is a unit vector of a corresponding dimension, namely an optimal solution of a digital precoding matrix;
in the third step, the kth iteration process of the alternate minimized iteration solution method includes:
step three: solving the kth secondary matrix
In the middle ofFor the kth residual matrix,/th order>For the kth analog precoding matrix, +.>A digital precoding matrix for a kth order; k=1, 2,3, … …;
step three, two: fixing the kth order digital precoding matrix during the kth iterationThe kth analog precoding matrix is +.>Solving;
and step three: during the kth iteration, the kth analog precoding matrix is fixedUsing least square method pairDigital precoding matrix of the kth degree->And (3) carrying out solving:
and step three, four: in the kth iteration process, solving the kth residual matrix
Step three, five: returning to the third step, repeatingThe next time, the digital precoding matrix F in the kth iteration process is completed BB And an analog precoding matrix F RF Is a solution to (c).
2. The low complexity convex relaxation optimized hybrid precoding method based on matrix multiplication decomposition of claim 1, wherein,
digital precoding matrix F BB Normalization to obtain final digital precoding matrix F BB The method of (1) comprises:
N s the data stream is transmitted for the communication system.
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