CN115459821A - 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 invention aims at the problem of high complexity of an alternation minimization algorithm based on convex relaxation optimization. The method comprises the following steps: constructing an analog precoding matrix F RF And assigning an initial value to each element; according to an analog precoding matrix F RF Solving the digital precoding matrix F by using a least square method BB (ii) a Iterative solution of an analog precoding matrix F with alternating minimization RF And a digital precoding matrix F BB (ii) a Until the variation of the current objective function and the adjacent previous objective function meets a preset variation threshold, ending the iteration; the finally obtained analog precoding matrix F RF As a final analog precoding matrix F RF (ii) a According to the power constraint condition, the finally obtained digital pre-coding matrix F BB Normalizing to obtain a final digital precoding matrix F BB (ii) a The invention is used for realizing the mixed pre-coding of the signals of the transmitting terminal 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, and the multiplexing gain of the large-scale antenna array and rich frequency spectrum resources of millimeter waves are utilized, so that the system capacity can be effectively improved, and the data transmission with ultrahigh speed and ultralow time delay is realized. The combination of the large-scale MIMO technology and the millimeter wave communication technology can overcome the defect of millimeter wave high path loss and reduce the difficulty of large-scale antenna array integration. Therefore, the millimeter wave large-scale MIMO system is widely applied to civil, industrial and military fields such as mobile communication, unmanned aerial vehicle communication and the like.
In the 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 perform signal processing at the transmitting end by using a precoding technique, 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 spectrum 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 a hybrid precoding technique, in which a low-dimensional digital precoding and a high-dimensional analog precoding are combined to complete an information preprocessing process. Hybrid precoding techniques will typically simulate a precoding matrix F RF A digital precoding matrix F BB And optimal all-digital precoding matrix F opt As an objective function, by minimizing the euclidean distance, a hybrid precoding process is achieved. At the same time, since the precoding matrix F is simulated RF The hardware is realized by phase shifters, so the matrix has a constant modulus constraint condition, and the optimization problem is non-convex optimization.
Of millimeter-wave massive MIMO systemsThe hybrid precoding algorithm mainly comprises two types, the first type is a hybrid precoding algorithm based on orthogonal matching pursuit and relying on channel estimation information, sparse reconstruction is used as a theoretical basis of hybrid precoding by utilizing the structural characteristics of millimeter waves, and a channel information pair simulation precoding matrix F is utilized RF Reconstructing, using least square method to the digital pre-coding matrix F BB Solving to approximate the optimal all-digital precoding matrix F opt Thereby completing the hybrid precoding process. The second type is a hybrid precoding algorithm based on alternation minimization and independent of channel estimation information, and an optimal full-digital precoding matrix F is directly realized by utilizing an optimization theory opt To complete the simulation of the precoding matrix F RF And a digital precoding matrix F BB And (4) alternately optimizing and solving.
Because the hybrid precoding algorithm based on the alternation minimization does not depend on channel estimation and directly realizes an optimization target by utilizing an optimization theory, the system performance can approximately approach the optimal all-digital precoding algorithm. However, the optimization theories related to the algorithms are high in calculation complexity, wherein the alternating minimization algorithm based on convex relaxation optimization relaxes non-convex constraints to convex constraints, and then solves the hybrid precoding problem by using the convex optimization theory, but matrix inversion operation with high calculation complexity is existed, and hardware implementation is not facilitated.
Disclosure of Invention
Aiming at the problem of high complexity of an alternation minimization algorithm based on convex relaxation optimization, the invention provides a low-complexity convex relaxation optimization hybrid precoding method based on matrix multiplication decomposition.
The invention relates to a low-complexity convex relaxation optimization hybrid precoding method based on matrix multiplication decomposition, which comprises the following steps,
the method comprises the following steps: constructing an analog precoding matrix F RF And assigning an initial value to each element;
step two: according to an analog precoding matrix F RF Solving the digital precoding matrix F by using a least square method BB ；
Step three: alternating the digital precoding matrix obtained by the current calculationF BB Minimizing iterative solution of next simulation precoding matrix F RF And using the next time analog precoding matrix F RF Minimizing iterative solution of next digital precoding matrix F BB ；
Step four: according to the analog pre-coding matrix F obtained in the third step RF And a digital precoding matrix F BB Calculating a current target function until the variation of the current target function and the variation of the adjacent previous target function meet a preset variation threshold, and ending iteration;
step five: the analog pre-coding matrix F obtained in the third step RF As a final analog precoding matrix F RF (ii) a According to the power constraint condition, the digital pre-coding matrix F obtained in the step three BB Normalizing to obtain a final digital precoding matrix F BB (ii) a The hybrid precoding is completed.
According to the low-complexity convex relaxation optimization hybrid precoding method based on matrix multiplication decomposition, a precoding matrix F is simulated in the step one RF Amplitude of initial value of medium element
The following relation is satisfied:
analog precoding matrix F RF Has the dimension of
Is the number of radio frequency chains;
the phase of each element is randomly generated.
Matrix-based multiplication according to the inventionIn the second step, a least square method is utilized to solve the obtained digital precoding matrix F BB Comprises the following steps:
in the formula
Is F RF Transposed conjugate matrix of (1), F opt Is the optimal all-digital precoding matrix.
According to the low-complexity convex relaxation optimization hybrid precoding method based on matrix multiplication decomposition, in the third step, a matrix multiplication decomposition theory is utilized to decompose an objective function, power constraint conditions are ignored, and the objective function is expressed as follows:
in the formula
Simulating an optimal solution of the precoding matrix;
according to the non-convexity of the constraint condition of the objective function, relaxing the constraint condition, and converting the objective function into:
wherein beta is a preset constant in the convex relaxation process;
searching and solving on the feasible domain of the relaxed constraint condition, and simulating a pre-coding matrix F RF And (3) carrying out normalization:
wherein arg represents a complex argument;
then using matrix multiplication decomposition theory to convert F RF F BB Rewrite as the sum of the series of submatrices:
wherein m is an integer having a value range of
A row index or a column index representing a matrix;
further, a residual matrix F is obtained res ：
In the formula F m Is an auxiliary matrix;
according to the formula (6), the optimal solution optimization target is converted into
Sub-questions, expressed as:
in the formula
For the optimal solution of the digital precoding matrix, 1 is a unit vector of the 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 alternative minimization iteration solving method comprises the following steps:
step three, firstly: solving the kth auxiliary matrix
In the formula
For the k-th time residual matrix,
for the simulation of the precoding matrix for the k-th time,
precoding a matrix for the k-th order digit; k =1,2,3, … …;
step three: fixing the k-th digital precoding matrix during the k-th iteration
Simulating precoding matrix for k-th time by utilizing convex optimization theory
Solving is carried out;
step three: fixing the k-th analog precoding matrix in the k-th iteration process
Precoding matrix for k-th order digit by using least square method
And (3) solving:
step three and four: in the k iteration process, solving the k residual error matrix
Step three and five: returning to the step three to the step one, and repeating
Then, completing the digit pre-coding matrix F in the k iteration process BB And an analog precoding matrix F RF And (4) solving.
According to the low-complexity convex relaxation optimization hybrid precoding method based on matrix multiplication decomposition, a digital precoding matrix F BB Normalizing to obtain a final digital precoding matrix F BB The method comprises the following steps:
N s the number of data streams is transmitted for the communication system.
The invention has the beneficial effects that: the method relaxes the non-convex constraint of the precoding problem into convex constraint, and alternately optimizes the analog and digital precoding matrixes by using a convex optimization theory and a least square method; the hybrid precoding optimization problem is converted into a plurality of optimization sub-problems by using a matrix multiplication decomposition theory, so that the digital and analog precoding matrixes are optimized and solved line by line and column by column respectively, the matrix multiplication dimensionality is reduced, the matrix inversion operation with higher complexity is avoided, and the purpose of reducing the algorithm complexity is achieved.
The method is applied to a millimeter wave large-scale MIMO system, can reduce the matrix multiplication dimensionality and avoid the matrix inversion process on the premise of ensuring no performance loss such as the system spectral efficiency, the bit error rate and the like, reduces the complexity of hybrid precoding, improves the realizability of the hybrid precoding, and completes the signal preprocessing process.
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FIG. 1 is a flow chart of a low-complexity convex relaxation optimization hybrid precoding method based on matrix multiplication decomposition according to 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, an Optimal Full-Digital represents an Optimal Full-Digital pre-coding algorithm, which is a classical method; OMP represents an algorithm based on orthogonal matching pursuit, and is a classic algorithm of mixed precoding, CVXR represents the existing convex relaxation method, and MMD _ CVXR represents the method;
fig. 4 is a diagram of bit error rate analysis in an embodiment of the present invention.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be obtained by a person skilled in the art without making any creative effort based on the embodiments in the present invention, belong to the protection scope of the present invention.
It should be noted that the embodiments and features of the embodiments may be combined with each other without conflict.
The invention is further described with reference to the following drawings and specific examples, which are not intended to be limiting.
Detailed description of the inventionas shown in fig. 1 and fig. 2, the present invention provides a low-complexity convex relaxation optimization hybrid precoding method based on matrix multiplication decomposition, which includes,
the method comprises the following steps: constructing an analog precoding matrix F RF And assigning an initial value to each element; the element amplitude satisfies the constraint condition of constant modulus value;
step two: according to an analog precoding matrix F RF Solving the digital precoding matrix F by using a least square method BB ；
Step three: alternating with the digital precoding matrix F obtained by the current calculation BB Minimizing iterative solution of next simulation precoding matrix F RF And using the next time analog precoding matrix F RF Minimizing iterative solution of next digital precoding matrix F BB ；
Step four: according to the analog pre-coding matrix F obtained in the third step RF And a digital precoding matrix F BB Calculating a current target function until the variation of the current target function and the variation of the adjacent previous target function meet a preset variation threshold, and ending 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 pre-coding matrix F obtained in the third step RF As a final analog precoding matrix F RF (ii) a According to the power constraint condition, the digital pre-coding matrix F obtained in the third step BB Normalizing to obtain a final digital precoding matrix F BB (ii) a The hybrid pre-coding is completed.
Further, since the precoding matrix F is simulated RF Is realized by phase shifters, so that the matrix is constant modulus limited, i.e. simulating the precoding matrix F in step one RF Amplitude of initial value of medium element
The following relation is satisfied:
analog precoding matrix F RF Has a dimension of
Is the number of radio frequency chains;
the phase of each element is randomly generated.
Solving the obtained digital precoding matrix F by using a least square method in the step two BB Comprises the following steps:
in the formula
Is F RF Transposed conjugate matrix of (1), F opt Is the optimal full digital precoding matrix.
Since the design problem of hybrid precoding takes into account the power constraints of the system, the analog precoding matrix F RF The constant modulus constraint amplitude of the method does not influence the system performance and the solving process. Under the condition of temporarily ignoring system power constraints, the objective function can be expressed as:
in the formula
Simulating an optimal solution of the precoding matrix;
is an objective function;
the constraint condition of the above formula objective function has non-convexity, so that the objective function is converted into:
wherein beta is a preset constant in the convex relaxation process;
after the relaxed constraint condition can be searched and solved in the domain, in order to meet the original non-convex constraint condition, the analog pre-coding moment is requiredArray F RF And (4) normalization is carried out:
wherein arg represents a complex argument;
on the basis, in order to simplify the optimization target, the matrix multiplication decomposition theory is reused to convert F into RF F BB Rewrite as the sum of the series of submatrices:
wherein m is an integer having a value range of
A row index or a column index representing a matrix;
further, a residual matrix F is obtained res ：
In the formula F m Is an auxiliary matrix;
according to the formula (6), the optimal solution optimization target is converted into
Sub-questions, denoted as:
in the formula
For the optimal solution of the digital precoding matrix, 1 is a unit vector of the corresponding dimension.
And (4) carrying out optimization solution on the optimization target of the formula (7) by adopting an alternative minimization method. Firstly, the methodCan be regarded as F BB (m,: remain unchanged, then with respect to F RF The optimization problem of (m) can be expressed as:
using convex optimization theory to solve to obtain F RF After (m), keeping it unchanged, with respect to F BB The optimization goal of (m,: may be expressed as:
for the optimization problem, the least square method is adopted to F BB (m,: solving:
for is to
The optimization sub-problems are solved in sequence according to the optimization method, and the digital pre-coding matrix F can be realized BB And an analog precoding matrix F RF The optimization line by line and column by column is as follows:
in step three, the kth iteration process of the alternating minimization iterative solution method comprises the following steps:
step three, firstly: solving the kth auxiliary matrix
In the formula
For the k-th time residual matrix,
for the simulation of the precoding matrix for the k-th time,
precoding a matrix for the k-th order digit; k =1,2,3, … …;
step two: fixing the kth digital pre-coding matrix in the kth iteration process
Simulating precoding matrix for k-th time by utilizing convex optimization theory
Solving is carried out;
step three: fixing the k-th analog precoding matrix in the k-th iteration process
Precoding matrix for k-th order digit by using least square method
And (3) solving:
step three and four: in the k iteration process, solving the k residual error matrix
Step three and five: returning to the step three to the step one, and repeating
Then, completing the digit pre-coding matrix F in the k iteration process BB And an analog precoding matrix F RF And (4) solving.
Digital precoding matrix F BB Normalizing to obtain a final digital precoding matrix F BB The method comprises the following steps:
N s the number of data streams is transmitted for the communication system.
The specific embodiment is as follows:
the method of the invention is applied to a millimeter wave large-scale MIMO system, and the actual system schematic diagram is shown in FIG. 2: the transmitting and receiving end adopts a uniform plane antenna array structure, wherein the transmitting end is provided with 12 multiplied by 12 antenna elements, 4 data streams are transmitted to the receiving end provided with 6 multiplied by 6 antenna elements, and the number of radio frequency chains of the transmitting and receiving end is 4.
The embodiment is described with reference to fig. 1, and the specific implementation steps are as follows:
step 1: constructing an analog precoding matrix F with dimensions 128 x 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 realized by phase shifters, the matrix is thus limited by constant modulus values, i.e. the amplitude of the elements needs to be sufficient
In summary, the initial analog precoding matrix
The amplitude of the element meets the constraint condition, and the phase is randomly generated.
And 2, step: digital precoding matrix F using least squares BB And (3) solving:
and step 3: iterative solution of an analog precoding matrix F with alternating minimization RF And a digital precoding matrix F BB ；
In this embodiment, the iteration end condition is that the variation of the objective function is less than 10 -3 。
The 4 optimization sub-problems are solved in sequence according to the optimization method, and the digital pre-coding matrix F can be realized BB And an analog precoding matrix F RF The method comprises the following steps of optimizing row by row and column by column:
step 3.1: solving for auxiliary variables during the kth iteration
Step 3.2: fixing the digital precoding matrix during the kth iteration
Modeling precoding matrices using convex optimization
Solving is carried out;
step 3.3: fixing the analog precoding matrix during the kth iteration
Digital precoding matrix using least squares
And (3) solving:
step 3.4: during the k-th iteration, the residual error matrix is solved
Step 3.5: repeating the steps 3.1, 3.2, 3.3 and 3.4 for 4 times to complete the digital precoding matrix F in the k iteration process BB And an analog precoding matrix F RF And (4) solving.
And 4, step 4: repeating the step 3 until an iteration end condition is reached;
and 5: according to the power constraint condition, a digital pre-coding matrix F BB And (3) carrying out normalization:
combining the method of the present invention shown in FIG. 3 with the conventional convex relaxation optimization method and the spectral efficiency performance comparison analysis of the classical hybrid pre-coding algorithm, the method of the present invention decomposes the objective function into the target function by using the matrix multiplication decomposition theory
And the sub-problems are that the digital and analog pre-coding matrixes are optimized column by column, the dimensionality of matrix multiplication is reduced, the matrix inversion process is avoided, and the purpose of reducing algorithm complexity is achieved. FIG. 4 is a bit error rate performance comparison analysis of the method of the present invention with 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 implemented by using the existing CVXR-AltMin algorithm, and table 2 shows the complexity analysis of the hybrid precoding method implemented by using the MMD-CVXR-AltMin algorithm of the present invention; r in Table 1 m Auxiliary variables in the CVXR-AltMin algorithm; in comparison with tables 1 and 2, it is shown that the method of the present invention achieves a reduction in algorithm complexity.
The combination of fig. 3 and fig. 4 shows that the performance of the method of the present invention is consistent with that of the conventional convex relaxation optimization method and better approaches to the optimal all-digital precoding algorithm. The simulation result shows that 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 dimensionality of matrix multiplication can be reduced, the matrix inversion process is avoided, and the algorithm complexity is reduced.
In conclusion, the method of the present invention can reduce the complexity of hybrid precoding on the premise of ensuring the performance of the system, such as spectrum efficiency, bit error rate, etc., and has the characteristics of good system performance, low complexity, etc
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 various dependent claims and the features described herein may be combined in ways different from those described in the original claims. It is also to be understood that features described in connection with individual embodiments may be used in other described embodiments.
Claims (6)
1. A low-complexity convex relaxation optimization hybrid precoding method based on matrix multiplication decomposition is characterized by comprising the following steps of,
the method comprises the following steps: constructing an analog precoding matrix F RF And assigning an initial value to each element;
step two: according to an analog precoding matrix F RF Solving the digital precoding matrix F by using a least square method BB ；
Step three: alternating with the digital precoding matrix F obtained by the current calculation BB Minimizing iterative solution of next simulation precoding matrix F RF And using the next time analog precoding matrix F RF Minimizing iterative solution of next digital precoding matrix F BB ；
Step four: according to the analog pre-coding matrix F obtained in the third step RF And a digital precoding matrix F BB Calculating a current target function until the variation of the current target function and the variation of the adjacent previous target function meet a preset variation threshold, and ending iteration;
step five: the analog precoding matrix F obtained in the step three RF As a final analog precoding matrix F RF (ii) a According to the power constraint condition, the digital pre-coding matrix F obtained in the step three BB Normalizing to obtain a final digital precoding matrix F BB (ii) a The hybrid pre-coding is completed.
2. The matrix factorization based low-complexity convex relaxation optimized hybrid precoding method of claim 1,
simulating a precoding matrix F in step one RF Amplitude of initial value of medium element
The following relation is satisfied:
analog precoding matrix F RF Has a dimension of
Is the number of radio frequency chains;
the phase of each element is randomly generated.
3. The matrix factorization based low-complexity convex relaxation optimized hybrid precoding method of claim 2,
solving the obtained digital precoding matrix F by using a least square method in the step two BB Comprises the following steps:
in the formula
Is F RF Transposed conjugate matrix of (1), F opt Is the optimal full digital precoding matrix.
4. The matrix factorization based low-complexity convex relaxation optimized hybrid precoding method of claim 3,
in the third step, the objective function is decomposed by using a matrix multiplication decomposition theory, and power constraint conditions are ignored, and the objective function is expressed as follows:
in the formula
Simulating an optimal solution of the precoding matrix;
according to the non-convexity of the constraint condition of the objective function, relaxing the constraint condition, and converting the objective function into:
wherein beta is a preset constant in the convex relaxation process;
searching and solving on the feasible domain of the relaxed constraint condition, and simulating a pre-coding matrix F RF And (3) carrying out normalization:
wherein arg represents a complex argument;
then using matrix multiplication decomposition theory to convert F RF F BB Rewrite as the sum of the series of submatrices:
wherein m is an integer having a value range of
A row index or a column index representing a matrix;
further, a residual matrix F is obtained res ：
In the formula F m Is an auxiliary matrix;
converting the optimal solution optimization target into the optimal solution optimization target according to the formula (6)
Sub-questions, expressed as:
in the formula
For the optimal solution of the digital precoding matrix, 1 is a unit vector of the corresponding dimension.
5. The matrix factorization based low-complexity convex relaxation optimized hybrid precoding method of claim 4, wherein,
in step three, the kth iteration process of the alternating minimization iterative solution method comprises the following steps:
step three, firstly: solving the kth auxiliary matrix
In the formula
For the k-th time residual matrix,
for the k-th simulation of the precoding matrix,
precoding a matrix for the k-th order digit; k =1,2,3, … …;
step three: fixing the kth digital pre-coding matrix in the kth iteration process
Simulating precoding matrix for k-th time by utilizing convex optimization theory
Solving is carried out;
step three: fixing the k-th analog precoding matrix in the k-th iteration process
Precoding matrix for k-th order digit by using least square method
And (3) solving:
step three and four: in the k iteration process, solving the k residual error matrix
Step three and five: returning to the step three to the step one, and repeating
Then, completing the digit pre-coding matrix F in the k iteration process BB And an analog precoding matrix F RF And (4) solving.
6. The matrix factorization based low-complexity convex relaxation optimized hybrid precoding method of claim 5, wherein,
digital precoding matrix F BB Normalizing to obtain a final digital precoding matrix F BB The method comprises the following steps:
N s the number of data streams is transmitted for the communication system.
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