MIMO Array Sparse Antenna Position Design Method Based on Greedy Algorithm

Abstract

The present invention discloses a method for designing the positions of sparse antennas in a MIMO array based on a greedy algorithm, belonging to the field of radar technology. In the present invention, based on the optimized transmitted signal and the ideal radiation pattern, the problem of antenna position design is modeled as minimizing the squared difference between the ideal transmitted radiation pattern and the designed transmitted radiation pattern, subject to the integer constraint of the antenna position vector. Based on the characteristic that the greedy algorithm can effectively solve the integer constraint problem in the design of the positions of sparse antennas in a MIMO array, the antenna position vector under the integer constraint is obtained, and the deployment result of the antenna positions is obtained. The present invention first updates the value of the shrinkage factor, then generates a set of possible solutions, and finally updates the feasible solutions, so as to obtain the antenna position vector based on the solution result. The method of the present invention solves the integer constraint problem based on the characteristics of the greedy algorithm, and selects appropriate antenna positions from the given number of grid points to improve the matching performance of the system transmitted radiation pattern.

Description

Technical Field

[0001] The present invention belongs to the technical field of radar, and relates to a method for designing the sparse antenna positions of a MIMO array based on a greedy algorithm Background Art

[0002] MIMO arrays have better target recognition and interference suppression capabilities due to their ability to transmit mutually independent waveforms, which has attracted extensive attention from researchers. The selection of sparse antenna positions is a key technology widely used in MIMO array systems. Since the available antennas are placed in a wider emission field, additional degrees of freedom are introduced into the system. As the degrees of freedom increase, the transmit radiation pattern of the MIMO array can achieve better accuracy using the same number of antennas. In this case, the power consumption of the system can be greatly reduced, and a similar transmit radiation pattern can be achieved using a smaller number of antennas. Therefore, the design of sparse antenna positions for MIMO arrays has received extensive attention

[0003] Currently, the existing methods can be roughly divided into two categories: the first category solves this problem through a relaxation method, but the performance of the transmit radiation pattern will decline; the second category solves this problem through an exhaustive method, but its computational cost is relatively high

[0004] In the first category of methods, typically, the literature "M. Deng, Z. Cheng, and Z. He, 'Co-Design of Waveform Correlation Matrix and Antenna Positions for MIMO Radar Transmit Beampattern Formation,' IEEE SENSORS JOURNAL, VOL. 20, NO. 13, JULY 1, 2020" proposed a convex relaxation (CR) method to transform the integer constraint relaxation into a convex constraint optimization problem. However, accuracy loss occurs during the convex relaxation process, resulting in a decline in the performance of the transmit radiation pattern after array selection. To improve the performance of the transmit radiation pattern, the literature "K.D., Z. Xu, and A.P., 'Sparse Antenna Array Design for MIMO Radar Using Softmax Selection,' IEEE SIGNAL PROCESSING LETTERS, arXiv preprint arXiv:2102.05092 (2021)." proposed a soft selection network based on an orthogonality criterion constraint to solve the integer constraint problem. However, this constraint criterion requires successive approximation of an integer value, resulting in a large amount of computation

[0005] In the second type of methods, typically, the literature "W.R., L.Xu, J.Li and P.S., 'Sparse Antenna Array Design for MIMO Active Sensing Applications,' IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL.59, NO.3, MARCH 2011." proposed the Cyclic Algorithm (CA). However, due to the need to traverse the entire matrix vector, it has a relatively high complexity. To reduce the computational amount, the literature "Wanxin Shi, Qian He, and Li Sun., 'Transmit antenna placement based on graph theory for MIMO radar detection,' IEEE Transactions on Signal Processing, vol.70, pp.3993 - 4005, Jun 2022." proposed a radar antenna selection method based on graph theory by assigning different weights to different antennas. However, since this method is based on a linear search method, it is difficult to obtain a globally optimal solution. Summary of the Invention

[0006] The object of the present invention is to provide a method for designing the sparse antenna positions of a MIMO array based on a greedy algorithm, so as to achieve better performance for the transmit pattern of the MIMO array.

[0007] The technical solution adopted by the present invention is as follows:

[0008] A method for designing the sparse antenna positions of a MIMO array based on a greedy algorithm, the method comprising the following steps:

[0009] Step 1, set the optimization model for the sparse antenna positions of the MIMO array as:

[0010]

[0011]

[0012] where J() represents the cost function, K represents the number of points used to cover the entire airspace range, r represents the r-th point in the airspace range, δ represents the contraction factor, ω r represents the weight coefficient corresponding to the airspace angle of the r-th point, and the K weight coefficients ω r constitute the weight vector ω = [ω1,..., ω r ,..., ω K, d = [d1,..., d r ..., d K T represents the ideal radiation pattern, d r represents the ideal radiation pattern corresponding to the spatial angle of the r-th point. The sparse antenna position vector p = [p1,..., p m ..., p M T , where is a binary vector set containing N non-zero elements and a total of M elements. M is the number of half-wavelength grid points, and N is the number of antennas. p m represents the antenna position symbol at the m-th half-wavelength grid point, and p m takes values of 0 or 1. When p m is 1, it means an antenna element is arranged at the m-th half-wavelength grid point. When p m is 0, it means no antenna element is arranged at the m-th half-wavelength grid point;

[0013] Define p(θ r , p) to represent the energy of the transmit radiation pattern of the MIMO array at the spatial angle θ r :

[0014]

[0015] where, I L represents the identity matrix, is the steering vector at the spatial angle θ r , and s is the transmit signal;

[0016] Use the greedy algorithm to solve the optimization model of the sparse antenna position of the MIMO array, obtain the optimal solution of the antenna position vector p based on the solution result, and obtain the deployment result of the sparse antenna position of the MIMO array.

[0017] Furthermore, step 2 is specifically as follows:

[0018] Step 201, initialize the transmit signal s, the ideal radiation pattern d, the weight coefficient vector ω, and initialize the antenna position vector p, and ||p||1 = M;

[0019] Step 202, in the i-th iteration process, perform the following steps, where the initial value of i is 1;

[0020] (1) Regard p as a fixed value and update the shrinkage factor of the current iteration process

[0021] where, the intermediate parameter

[0022] (2) Generate the possible solution set of the antenna position: ​​

[0023] wherein, p i-1 respectively represent the possible solution sets of the antenna positions generated by the i-th and (i - 1)-th iterative processes, and H(·) represents the Hamming distance between two vectors;

[0024] (3) Update the feasible solution:

[0025] (4) Determine whether the iterative convergence condition is satisfied. If so, terminate the iterative process to obtain the solution result; otherwise, continue to execute step 202;

[0026] wherein, the iterative convergence condition is: ||p i ||1 = N.

[0027] The technical solution provided by the present invention at least brings the following beneficial effects:

[0028] The present invention assumes that there already exist optimized transmission signals and ideal radiation patterns (these prior information can be obtained through existing waveform optimization methods). On this basis, the antenna position design problem is modeled as minimizing the squared difference between the ideal transmission radiation pattern and the designed transmission radiation pattern, while satisfying the integer constraint of the antenna position vector. Based on the characteristic that the greedy algorithm can effectively solve the integer constraint problem in the sparse antenna position design of MIMO arrays, the antenna position vector p under the integer constraint is obtained to obtain the deployment result of the antenna position. The present invention proposes a method for sparse antenna position design of MIMO arrays based on the greedy algorithm. In this method, the value of the contraction factor is first updated, then the possible solution set is generated, and finally, the feasible solution is updated, so as to obtain the antenna position vector based on the solution result. The method of the present invention solves the integer constraint problem based on the characteristics of the greedy algorithm, and selects appropriate antenna positions from the given number of grid points to improve the matching performance of the system transmission radiation pattern. BRIEF DESCRIPTION OF THE DRAWINGS

[0029] In order to more clearly illustrate the technical solutions in the embodiments of the present invention, the following will briefly introduce the drawings required for the description of the embodiments. Obviously, the following drawings are only some embodiments of the present invention. For those of ordinary skill in the art, without creative efforts, other drawings can be obtained based on these drawings.

[0030] Figure 1 is the comparison curve of the transmission radiation patterns after sparse array selection in the embodiment;

[0031] Figure 2 is the antenna position diagram after sparse array selection in the embodiment;

[0032] Figure 3It is the mean square error (MSE) change curve provided in the embodiments when different numbers of grid points are used. Detailed implementation manners

[0033] To make the objectives, technical solutions and advantages of the present invention clearer, the following will further describe the embodiments of the present invention in detail with reference to the accompanying drawings.

[0034] For convenience of description, the following definitions are made first:

[0035] Consider a centralized MIMO array with N transmit antennas and M half-wavelength grid points. Let s = [s(1),..., s(ML)] represent the transmitted signal, and s(1) to s(ML) represent the values of each sampling point of the transmitted signal, where L represents the number of snapshots. The antenna position vector is expressed as:

[0036] p = [p1, p2,..., p M T , p m ∈{0, 1}, m∈{1,..., M}

[0037] p m indicates whether an array element is deployed at the m-th grid point. If it is 1, it means an array element is placed; if it is 0, it means no array element is placed.

[0038] Consider selecting N out of M half-wavelength grid points as transmit antennas to design the transmit pattern. Let represent the antenna selection vector, where is a set of binary vectors containing N non-zero elements and a total of M elements. Then the corresponding waveform at the target position with respect to the direction θ can be expressed as:

[0039]

[0040] where I L is the identity matrix, that is, the identity matrix of dimension L, a(θ) represents the steering vector matrix, ⊙ represents the Hadamard element-wise multiplication, represents the Kronecker product.

[0041] Therefore, the energy of the transmitted wave in the spatial angle θ r can be expressed as:

[0042]

[0043] where is the steering vector in the direction θ r .

[0044] ​The objective of the present invention is to minimize the error between the ideal radiation pattern and the designed radiation pattern by selecting appropriate antenna positions. Based on this objective, a signal model is established and expressed as:

[0045]

[0046]

[0047] where J(p,δ) represents the cost function, ω = [ω1,...,ω r ,...,ω K represents the weight coefficient, δ represents the shrinkage factor, d = [d1,...,d K T represents the ideal pattern, K represents the number of points used to cover the entire airspace (i.e., the discrete points of the airspace), and r represents the numbering of these points.

[0048] The working principle of the present invention is as follows:

[0049] Since the problem model is a quartic polynomial and there are integer constraints, the problem is non-convex, which increases the difficulty of solving the problem. Currently, the existing methods are roughly divided into two categories: The first category is to solve the problem by relaxation methods, but the performance of the radiation pattern will decline; the second category is to solve the problem by traversal methods, but their computational cost is relatively high. We note that the greedy algorithm is well-suited for solving non-convex problems, and the sparse antenna position vector is an integer constraint with values of 0 or 1. Based on this feature, the embodiments of the present invention propose a MIMO array sparse antenna position design based on the greedy algorithm to directly and effectively solve the problem. The algorithm steps are roughly as follows: First, update the value of the shrinkage factor; second, generate a possible solution set; finally, update the feasible solution.

[0050] As a possible implementation, in the embodiments of the present invention, the specific solution process of the signal model includes:

[0051] Step 1: Treat p as a fixed value and update the shrinkage factor δ i value at the i-th iteration:

[0052]

[0053] where the intermediate parameter ⊙ represents the Hadamard element-wise product, the superscript T represents the matrix transpose, and the initial value of the iteration number i is 1.

[0054] Step 2: Generate a possible solution set:

[0055] The sparse antenna position vector p is a binary vector with values of 0 or 1, and there are 2 MPossible results. Given the optimal solution \(p\) of the position vector at the \((i - 1)\)-th iteration, i-1 an optimal solution set containing all possible optimal solutions can be obtained, denoted as:

[0056]

[0057] where \(\|\cdot\|_1\) represents the 1-norm of a vector, and \(H(x, y)\) represents the Hamming distance between two vectors \(x\) and \(y\), that is, the number of positions \(i\) where \(x i \neq y i .

[0058] Generally speaking, given the optimal solution \(p\) at the \((i - 1)\)-th iteration, i-1 the optimal solutions in the newly generated optimal solution set differ from \(p i-1 by only one bit (only one non-zero element is missing). Therefore, the upper limit of the optimal solution set generated in each iteration is

[0059] Step 3: Update the feasible solution:

[0060] Updating the feasible solution is a selection process, that is, selecting the optimal feasible solution from the updated feasible solution set to enter the next iteration process. Here, the optimal feasible solution is the feasible solution with the minimum cost function value.

[0061] In the case of obtaining the optimal feasible solution set , the process of obtaining the optimal feasible solution is expressed as follows:

[0062]

[0063] where \(J(p)\) represents the cost function in the optimization model, and \(x=\arg\min(y)\) represents the \(x\) required to obtain the minimum \(y\).

[0064] To further verify the performance of the MIMO array sparse antenna position design method based on the greedy algorithm disclosed in the embodiments of the present invention, the waveforms are optimized using the existing scheme and the proposed scheme.

[0065] Consider a uniform linear array with \(N = 10\) transmitting antennas, the number of snapshots \(L = 32\), and the number of half-wavelength grid points \(M=\{10, 13, 15, 20\}\). The angular range is \((-90^{\circ}, 90^{\circ})\), with an interval of \(1^{\circ}\) in between. The weight coefficients \(\omega_1=\cdots=\omega K = 1. Consider an ideal pattern with three main lobes, denoted as:

[0066]

[0067] Define the mean square error (MSE) expression as:

[0068]

[0069] Figure 1 The comparison curves of the radiation patterns after sparse array selection are given. From Figure 1 it can be seen that as the number of grid points M increases, the radiation pattern after sparse array selection has lower side lobes. In addition, compared with the radiation pattern without array selection (M = 10), the side lobes of the radiation pattern after array selection are significantly reduced, and the target detection performance is significantly improved.

[0070] Figure 2 The antenna position distribution diagrams after sparse array selection are given in . It can be seen from the figures that for different given numbers of grid points, i.e., M = 13, 15, 20, the MIMO array sparse antenna position design method based on the greedy algorithm provided by the embodiments of the present invention can select an appropriate number of antennas according to the constraint conditions and allocate them at reasonable positions.

[0071] Figure 3 The curves of the change of the mean square error (MSE) for different numbers of grid points are given in . It can be seen that as the number of grid points increases, the value of the mean square error continuously decreases, that is, the radiation pattern after sparse array selection gets closer and closer to the ideal radiation pattern. Therefore, providing more grid points is significantly helpful for improving radar performance.

[0072] Through the above simulations, the performance advantages of the MIMO array sparse antenna position design method based on the greedy algorithm provided by the present invention are verified: it can obtain a lower mean square error value, and also has the advantages of main lobe gain and null depth on the radiation pattern, which is more conducive to improving the target detection performance of the MIMO array.

[0073] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those of ordinary skill in the art should understand that they can still modify the technical solutions recorded in the foregoing embodiments, or perform equivalent replacements for some of the technical features; and these modifications or replacements do not make the essence of the corresponding technical solutions deviate from the spirit and scope of the technical solutions of the embodiments of the present invention.

[0074] The above are only some embodiments of the present invention. For those of ordinary skill in the art, without departing from the creative concept of the present invention, several deformations and improvements can still be made, and these all belong to the protection scope of the present invention.

Claims

1. A method for designing the sparse antenna positions of a MIMO array based on a greedy algorithm, characterized in that It includes the following steps: Step 1, set the optimization model for the sparse antenna positions of the MIMO array as: Among them, J() represents the cost function, K represents the number of points used to cover the entire airspace range, r represents the r-th point in the airspace range, δ represents the contraction factor, and ω r represents the weight coefficient corresponding to the airspace angle corresponding to the r-th point. The K weight coefficients ω r constitute the weight vector ω = [ω1,..., ω r ,..., ω K , d = [d1,..., d r ..., d K T represents the ideal radiation pattern, and d r represents the ideal radiation pattern of the airspace angle corresponding to the r-th point. The sparse antenna position vector p = [p1,..., p m ..., p M T , where is a binary vector set containing N non-zero elements and a total of M elements. M is the number of half-wavelength grid points, N is the number of antennas, and p m represents the antenna position symbol of the m-th half-wavelength grid point, and the value of p m is 0 or 1. When p m is 1, it means that an antenna element is arranged at the m-th half-wavelength grid point. When p m is 0, it means that no antenna element is arranged at the m-th half-wavelength grid point;​​ Define \(p(\theta\) r , p)\) to represent the energy of the transmit pattern of the MIMO array at the spatial angle \(\theta\) r : where, I L represents the identity matrix, is the steering vector at the spatial angle θ r and s is the transmitted signal; Step 2, use the greedy algorithm to solve the optimization model for the sparse antenna positions of the MIMO array, obtain the optimal solution of the antenna position vector p based on the solution result, and obtain the design result of the sparse antenna positions of the MIMO array; Among them, Step 2 is specifically: Step 201, initialize the transmitted signal s, the ideal pattern d, the weight coefficient vector ω, and initialize the antenna position vector p, and ||p||1 = M; Step 202, in the i-th iteration process, execute the following steps, where the initial value of i is 1; (1) Treat p as a fixed value and update the contraction factor being processed in the current iteration Among them, the intermediate parameter (2) Generate a possible solution set for the antenna position: wherein, p i-1 respectively represent the possible solution sets of the antenna positions generated by the i-th and (i-1)-th iterative processes, and H(·) represents the Hamming distance between two vectors; (3) Update the feasible solution: (4) Determine whether the iterative convergence condition is satisfied. If so, terminate the iterative process and obtain the solution result; otherwise, continue to execute Step 202; wherein, the iterative convergence condition is: ||p i ||1 = N.

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