Maximizing mainlobe gain method for analog-digital hybrid arrays for radar communication systems

Abstract

This invention discloses a method for maximizing the main lobe gain of a hybrid analog-digital array used in radar communication systems. The invention investigates a method for jointly designing analog and digital weights using an alternating direction method of multipliers (ADMM) framework, with the objective function being the maximization of the ratio of the minimum main lobe level to the peak sidelobe level. Under constraints of main lobe ripple, constant modulus constraint on analog weights, and power constraint on digital weights, the Riemann conjugate gradient method is used to solve the analog weight subproblem with constant modulus constraint; the Lagrange multiplier method is used to solve the digital weights. Numerical results show that the method of this invention can produce good main lobe ripple and low peak sidelobe levels, exhibiting excellent beam pattern control performance.

Description

Technical Field

[0001] This invention belongs to the field of array signal processing technology. Specifically, it is a method for maximizing the main lobe gain of a hybrid analog digital (HAD) array used in radar communication systems. Background Technology

[0002] Massive MIMO (Massively Multi-Area Arrays) are crucial in modern radar and millimeter-wave communication systems, enhancing target signal strength and mitigating the high path loss in the millimeter-wave band. However, fully digital (FD) arrays suffer from high hardware costs and significantly increased system complexity due to the need for independent RF links for each antenna. Therefore, hybrid analog-digital (HAD) architectures offer a compelling alternative by achieving near-full digital array performance while reducing hardware costs. Their operation involves digital beamforming using a small number of RF links and analog beamforming achieved by connecting antenna elements to the RF links using numerous low-cost analog phase shifters.

[0003] However, maximizing the main lobe gain of a hybrid analog-digital array presents numerous challenges. Due to the constant modulus constraint on analog weights and the severe coupling between analog and digital weights in the objective function, transmit pattern synthesis becomes an extremely challenging optimization problem. Current optimization algorithms have limitations in handling such problems, such as difficulty in effectively processing high-dimensional variables. Therefore, to achieve precise control of the transmit pattern of a hybrid analog-digital array, it is urgent to develop a novel algorithm capable of efficiently solving high-dimensional, non-convex optimization problems. This algorithm needs to possess excellent computational efficiency, quickly finding numerical solutions that satisfy the constraints within limited computational resources; simultaneously, it must exhibit good convergence and stability to ensure reliable and precise control of the transmit pattern in various application scenarios. Summary of the Invention

[0004] The purpose of this invention is to propose a method for maximizing the main lobe gain of an analog-digital hybrid array for radar communication systems.

[0005] The technical solution to achieve the objective of this invention is as follows: a method for maximizing the main lobe gain of an analog-digital hybrid array for a radar communication system, comprising the following steps:

[0006] Step 1: Construct a signal processing scenario for an analog-digital hybrid antenna array applied to radar or wireless communication systems. Using the criterion of maximizing the minimum main lobe level, establish a beamforming optimization problem model that maximizes the main lobe gain.

[0007]

[0008] In the formula, From the angle of the main lobe region, Main lobe region For angle The guide vector at that location, For a mixed weight vector, For numerical weights, The weights are simulated, M is the number of digital subarrays, and N is the number of elements in each simulated subarray. express A dimensional vector of all 1s;

[0009] Step 2: Apply peak sidelobe level constraints to the sidelobes and control the main lobe ripple within a set range. Introduce auxiliary variables to further express the non-convex optimization problem model of Step 1 as follows:

[0010]

[0011] In the formula, Indicates the minimum main lobe level. Indicates the relative peak sidelobe level. Indicates the main lobe ripple. Let be the angle of the sidelobe region located at the discrete point s. The angle of the main lobe region located at discrete point q is represented by S and Q, which represent the number of discrete angle points of the side lobe region and the main lobe region, respectively.

[0012] Step 3: Introduce auxiliary variables and ,definition , Constructing the augmented Lagrange function as follows:

[0013]

[0014] in, , and These are introduced auxiliary variables. and It is a dual variable. and Let the penalty factor be ; then the optimization problem model after step 2 is rewritten as:

[0015]

[0016] Step 4: Based on the manifold ADMM framework, the optimization problem model in Step 3 is decomposed into several easily solvable subproblems, specifically as follows:

[0017]

[0018]

[0019]

[0020]

[0021] Among them, all variables with superscript k+1 are the results of k+1 iterations, and all variables with superscript k are the results of k-th iterations;

[0022] Step 5: Solve the subproblems in Step 4. The simulation weight subproblem is solved iteratively using the Riemann conjugate gradient method, while the digital weights and minimum main lobe level auxiliary variables are obtained using closed-form solutions. Finally, solve the problem of maximizing the main lobe gain of the HAD array.

[0023] Compared with existing technologies, the significant advantages of this invention are: It achieves a high-gain, stable main lobe by maximizing the main lobe gain; and it reduces computational complexity and improves the feasibility of practical applications through optimized algorithm design. Due to the superior performance of the ADMM framework, the proposed algorithm also exhibits superior performance in maximizing main lobe gain compared to other representative methods. Attached Figure Description

[0024] Figure 1 This is a flowchart of the method for maximizing the main lobe gain of an analog-digital hybrid array for radar communication systems proposed in this invention.

[0025] Figure 2 This is a single main lobe flat-top beam pattern.

[0026] Figure 3 This is a diagram of the HAD array structure. Detailed Implementation

[0027] Hybrid analog-digital (HAD) arrays have been widely used in radar and communication systems due to their low hardware complexity. However, the constant-mode constraint of the analog weights makes the transmit beamforming problem of HAD arrays a non-convex optimization problem. Furthermore, the coupling between analog and digital weights further complicates the direct solution of the transmit beamforming problem.

[0028] The present invention, namely a method for maximizing the main lobe gain of HAD arrays, is further described below with reference to the accompanying drawings and examples.

[0029] This invention relates to a method for maximizing the main lobe gain of HAD arrays. The method proposes an ADMM-based transmit beamforming algorithm. Under constant-mode analog weights, main lobe ripple constraints, and digital weight power constraints, the algorithm synthesizes the desired beam pattern by maximizing the minimum main lobe level. The algorithm first introduces auxiliary variables to simplify fractional constraints, and then, within the ADMM framework, introduces auxiliary variables to alternately optimize the analog and digital weights. In each optimization, the Riemann conjugate gradient method is used to solve the subproblem with constant-mode constraints. The specific implementation steps are as follows:

[0030] Step 1: Consider the far-field narrowband signal of a HAD uniform linear array with a partially connected structure, where each analog subarray is composed of... It consists of isotropic element antennas, with the spacing between each element being [missing information]. Each simulated subarray at an angle The guidance vector at that location is represented as

[0031]

[0032] in Indicates the carrier wavelength; each analog subarray is connected to one radio frequency link. The steering vector of the HAD array composed of the above-mentioned simulated subarrays is represented as follows:

[0033]

[0034] Therefore, the total steering vector of the HAD array is expressed as:

[0035]

[0036] Define the weight of the nth element on the mth simulated subarray as: , , The simulated weights of the HAD array are then expressed as:

[0037]

[0038] Define the RF link weight corresponding to the m-th analog subarray as: The digital weights of the HAD array are then expressed as:

[0039]

[0040] Therefore, the hybrid weight of the HAD array is equivalently represented as

[0041]

[0042] The HAD array transmission pattern is

[0043]

[0044] To concentrate the energy of the transmitted signal more in the main lobe, this invention establishes a model for maximizing the main lobe gain of the HAD array based on the criterion of maximizing the ratio of the minimum main lobe level to the peak sidelobe level. The specific optimization model is as follows:

[0045]

[0046] In the formula, From the angle of the main lobe region, Main lobe region For angle The guide vector at that location, For a mixed weight vector, For numerical weights, M represents the number of digital subarrays, and N represents the number of elements in each simulated subarray.

[0047] Step 2: To enhance the stability of target detection performance in the main lobe region of the transmitted beam, the main lobe ripple needs to be controlled within a set range. Represent the main lobe ripple and define... Minimum main lobe level, This represents the peak sidelobe level. Therefore, the optimization problem model in step 1 can be further expressed as:

[0048]

[0049] In the formula, Indicates the minimum main lobe level. This represents the peak sidelobe level.

[0050] Step 3: Introduce auxiliary variables and ,definition , Constructing the augmented Lagrange function :

[0051]

[0052] in, , and It is a dual variable. and Let be the penalty factor. Then the optimization problem model in step 2 can be equivalently rewritten as:

[0053]

[0054] Step 4: Based on the manifold ADMM framework, the optimization problem model in Step 3 is decomposed into several easily solvable subproblems, specifically as follows:

[0055]

[0056]

[0057]

[0058]

[0059] In this context, all variables with a superscript of k+1 represent the results of the k+1th iteration, and all variables with a superscript of k represent the results of the kth iteration.

[0060] Step 5: Solve each sub-problem in Step 4 to obtain the optimal simulated weights and digital weights. The specific process is as follows:

[0061] Step 5.1: Ignore the constant term and... Equivalent representation is In the formula, all variables with superscript k represent the k-th iteration, and ,

[0062] ,

[0063] ;

[0064] The Riemann conjugate gradient method is used to solve the transformed subproblem to obtain the simulated weights for the (k+1)th iteration. The specific solution process is as follows:

[0065] Step 5.1.1: Obtain the objective function for the j-th iteration. The gradient is Its negative gradient direction is The Riemann gradient is obtained using the following formula. :

[0066]

[0067] in, This indicates element-wise operation. This indicates the operation of taking the real part;

[0068] Step 5.1.2: Calculate the conjugate parameters of the conjugate gradient descent method using the following formula. :

[0069]

[0070] in, Represents the Riemannian inner product. This represents a vector transfer operation, specifically expressed as:

[0071]

[0072] Then, the descent direction of the j-th round is calculated. :

[0073]

[0074] Step 5.1.3: Adaptively update the step size using backtracking search. The specific method is as follows:

[0075] Initialize step size Using the formula Iteration step size, This is the step size coefficient. When the backtracking condition is met, the iteration stops, and the current iteration position is set to... The final step size for the j-th round is defined by the backtracking condition as follows:

[0076]

[0077] Indicates the descent parameter. Indicates the vector Each element is projected onto the unit circle, i.e. ;

[0078] Step 5.1.4: Solve for the simulated weights in the (j+1)th iteration using the following formula:

[0079]

[0080] Determine whether the iteration termination condition is met, i.e., whether it is satisfied. , This is the algorithm's termination tolerance; if it is met, the loop stops, and the current value is set to zero. Use the optimal simulation weight for the (k+1)th round; otherwise, return to step 5.1.1.

[0081] Step 5.2: Solve To obtain the numerical weights for the (k+1)th iteration, the specific steps are as follows:

[0082] Ignoring the constant term, the subproblem can be equivalently represented as

[0083]

[0084] in, , , Represents an N x 1 column vector. Indicates the vector Diagonalize to a matrix. Define the Lagrange function as:

[0085]

[0086] in, These are Lagrange multipliers. The first-order optimality condition of this Lagrange function is:

[0087]

[0088] According to the above formula, we can obtain

[0089]

[0090] definition The matrix decomposition is as follows: ,in, A matrix composed of eigenvectors. Let represent a diagonal matrix composed of eigenvalues, and Redefining The Lagrange optimality condition equation can be transformed into

[0091]

[0092] make Then the equation can be simplified to

[0093]

[0094] Because the left side of the equation is Since time is a monotonic function, the optimal Lagrange multipliers can be found using the line search method. Substitute the Lagrange multipliers into the following formula:

[0095]

[0096] Obtain the numerical weights for the (k+1)th iteration. .

[0097] Step 5.3: Obtain the auxiliary variables for the (k+1)th iteration. Auxiliary variables and minimum main lobe level The specific method is as follows:

[0098] Ignore irrelevant terms and let , Transform the subproblem into:

[0099]

[0100] First fix The following formula can be used to solve for it. and :

[0101]

[0102]

[0103] Substituting the result of the above formula into the objective function of the subproblem, we obtain an optimization problem concerning a single variable:

[0104]

[0105] If ,but ;otherwise, ;like ,but ;otherwise, ;like ,but ;otherwise .

[0106] make The objective function of this subproblem can be represented as a piecewise function:

[0107]

[0108] in

[0109]

[0110] It is defined in The k-th sub-function within. Define them respectively. , and for , and The set after sorting in ascending order and removing duplicate sequences. It is , and The result is obtained after merging, sorting in ascending order, and removing duplicate sequences.

[0111] By Represented as

[0112]

[0113] in

[0114] , , Optimal Determined by the following formula:

[0115]

[0116] Will Substitute and solve and In the formula, we obtain the auxiliary variable for the (k+1)th iteration. and .

[0117] Step 5.4: Obtain the dual variable for the (k+1)th iteration using the following formula. , :

[0118]

[0119] Step 5.5: Determine whether the iteration termination condition is met, i.e., whether it is satisfied. and ,in , , This is the termination tolerance of the algorithm. If it is satisfied, the loop stops, and the simulated weights obtained in the current iteration are used. and numerical weights Use the best simulated weights and digital weights; otherwise, return to step 5.1.

[0120] Example

[0121] The method for maximizing the main lobe gain of the analog-digital hybrid array used in radar communication systems is further illustrated through Matlab simulation.

[0122] 1) Simulation system parameter settings

[0123] Unless otherwise specified, the HAD array used in each simulation is... , All array elements are evenly distributed and the spacing between array elements is [missing information]. . Spatial angular domain Divided into 181 intervals Discrete grid, and set , , , , and The array's mixed weight variables are randomly initialized. In addition, the FD array and FA array are selected as references for the HAD array.

[0124] 2) Beam plotting

[0125] To visually demonstrate the transmit beamforming effect of the HAD array, this embodiment uses the method proposed in this invention to plot the transmit beam pattern, and compares it with the typical HAD transmit pattern synthesis method: the two-stage method. The horizontal axis of the beam pattern represents the angular range of [-90°, 90°], and the unit of the vertical axis of the beam pattern is dB.

[0126] 3) Measurement indicators

[0127] In this invention, it is necessary to measure the effect of the final beamforming. The minimum mainlobe level (MML) is used as an indicator to quantitatively measure the mainlobe level of the transmitted beamforming. The definition of MML is as follows:

[0128]

[0129] in: Main lobe region This is the level formula. Under the same initial conditions, the larger the MML, the better the main lobe detection performance of the radar and communication system's transmitted beam.

[0130] 4) Results Analysis

[0131] This invention presents an example simulation, which is a single main lobe flat-top beam. Figure 2 It is a single main lobe flat-top beam.

[0132] pass Figure 2 As can be seen, both the two-stage method and the proposed method can effectively form the main lobe and side lobes. The MML of the proposed method reaches 23.05 dB with a ripple of 0.2 dB. On the other hand, the MMLs of FA and FD arrays are 22.17 dB and 23.22 dB, respectively. The algorithm proposed in this invention can effectively form a flat-top beam in HAD arrays, and with a hardware complexity 66.67% lower than that of FD arrays, the MML is only 0.17 dB lower than that of all-digital arrays. Therefore, HAD arrays achieve a good trade-off between transmit beam performance and hardware complexity.

[0133] In summary, the method described in this invention exhibits excellent overall performance. The proposed method can effectively form a beam and maximize the main lobe gain. When applied to radar and communication systems, this invention can significantly reduce hardware costs and computational complexity with minimal performance loss, demonstrating high practical value.

Claims

1. A method for maximizing the main lobe gain of a hybrid analog-digital array for a radar communication system, characterized in that: Includes the following steps: Step 1: Construct a signal processing scenario for an analog-digital hybrid antenna array applied to radar or wireless communication systems. Using the criterion of maximizing the minimum main lobe level, establish a beamforming optimization problem model that maximizes the main lobe gain as follows: ， In the formula, From the angle of the main lobe region, Main lobe region For angle The guide vector at that location, For a mixed weight vector, For numerical weights, The weights are simulated, M is the number of digital subarrays, and N is the number of elements in each simulated subarray. express A dimensional vector of all 1s; Step 2: Apply peak sidelobe level constraints to the sidelobes and control the main lobe ripple within a set range. Introduce auxiliary variables to further express the non-convex optimization problem model of Step 1 as follows: ， In the formula, Indicates the minimum main lobe level. Indicates the relative peak sidelobe level. Indicates the main lobe ripple. Let be the angle of the sidelobe region located at the discrete point s. The angle of the main lobe region located at discrete point q is represented by S and Q, which represent the number of discrete angle points of the side lobe region and the main lobe region, respectively. Step 3: Introduce auxiliary variables and ,definition , Constructing the augmented Lagrange function as follows: ， in, , and These are introduced auxiliary variables. and It is a dual variable. and Let the penalty factor be ; then the optimization problem model after step 2 is rewritten as: ， Step 4: Based on the manifold ADMM framework, the optimization problem model in Step 3 is decomposed into several easily solvable subproblems, specifically as follows: ， ， ， ， Among them, all variables with superscript k+1 are the results of k+1 iterations, and all variables with superscript k are the results of k-th iterations; Step 5: Solve the subproblems in Step 4. The simulation weight subproblem is solved iteratively using the Riemann conjugate gradient method, while the digital weights and minimum main lobe level auxiliary variables are obtained using closed-form solutions. Finally, solve the problem of maximizing the main lobe gain of the HAD array.

2. The method for maximizing the main lobe gain of an analog-digital hybrid array for a radar communication system according to claim 1, characterized in that: In step 1, the far-field narrowband signal of a HAD uniform linear array with a partially connected structure is considered, wherein each analog subarray is composed of... It consists of isotropic element antennas, with the spacing between each element being [missing information]. ; Each simulation subarray is at an angle The guidance vector at that location is represented as ， in Indicates the carrier wavelength; each analog subarray is connected to one radio frequency link. The steering vector of the HAD array composed of the above-mentioned simulated subarrays is represented as follows: The total steering vector of the HAD array is represented as follows: ， Define the weight of the nth element on the mth simulated subarray as: , , The simulated weights of the HAD array are then expressed as: ， Define the RF link weight corresponding to the m-th analog subarray as: The digital weights of the HAD array are then expressed as: ， Therefore, the hybrid weight of the HAD array is equivalently represented as: ， The HAD array transmission pattern is 。 3. The method for maximizing the main lobe gain of an analog-digital hybrid array for a radar communication system according to claim 2, characterized in that: In step 2, the specific process of problem transformation is as follows: Consider applying a peak sidelobe level constraint, specifically as follows: ， in, This represents the minimum gain in the main lobe region, and S represents the total number of angular discrete points in the side lobe region. Consider applying a main lobe ripple constraint, specifically as follows: ， in, This represents the ripple in the main lobe region, and Q represents the total number of angular discrete points in the main lobe region. After considering the main lobe ripple constraint and the peak sidelobe level constraint, the optimization problem in step 1 is further expressed as: ， Introducing auxiliary variables Let the minimum gain in the main lobe region be represented. Then the above problem is transformed into: 。 4. The method for maximizing the main lobe gain of an analog-digital hybrid array for a radar communication system according to claim 3, characterized in that: The specific transformation process of the optimization problem after the equivalence in step 3 is as follows: For the non-convex optimization problem in step 2, the alternating direction multiplier method is used for solution. The specific process is as follows: First, we introduce auxiliary variables. and ,make , The non-convex optimization problem in step 2 can then be rewritten as: ， Then, the sidelobe level constraints and main lobe ripple constraints are added as penalty terms to the objective function, resulting in the following Lagrangian function for the above problem: ， in , and It is a dual variable. and The penalty factor is used; finally, the optimization problem model after step 2 is rewritten as: 。 5. The method for maximizing the main lobe gain of an analog-digital hybrid array for a radar communication system according to claim 1, characterized in that: The specific process of solving each sub-problem in step 4 to obtain the optimal simulated weights and digital weights is as follows: Step 5.1: Ignore the constant term and... Equivalent representation is In the formula, all variables with superscript k represent the k-th iteration, and , ， ； The Riemann conjugate gradient method is used to solve the transformed subproblem to obtain the simulated weights for the (k+1)th iteration. Step 5.2: Solve Obtain the numerical weight for the (k+1)th iteration; Step 5.3: Solve Obtain the auxiliary variables for the (k+1)th iteration. Auxiliary variables and minimum main lobe level ; Step 5.4: Obtain the dual variable for the (k+1)th iteration using the following formula. , : ， Step 5.5: Determine whether the iteration termination condition is met, i.e., whether it is satisfied. and ,in , , This is the termination tolerance of the algorithm. If it is satisfied, the loop stops, and the simulated weights obtained in the current iteration are used. and numerical weights Use the best simulated weights and digital weights; otherwise, return to step 5.

1.

6. The method for maximizing the main lobe gain of an analog-digital hybrid array for a radar communication system according to claim 5, characterized in that: The specific process of solving the transformed subproblem using the Riemann conjugate gradient method to obtain the simulated weights for the (k+1)th iteration is as follows: Step 5.1.1: Obtain the objective function for the j-th iteration. The gradient is Its negative gradient direction is The Riemann gradient is obtained using the following formula. : ， in, This indicates element-wise operation. This indicates the operation of taking the real part; Step 5.1.2: Calculate the conjugate parameters of the conjugate gradient descent method using the following formula. : ， in, Represents the Riemannian inner product, This represents a vector transfer operation, specifically expressed as: ， Then, the descent direction of the j-th round is calculated. : ， Step 5.1.3: Adaptively update the step size using backtracking search. The specific method is as follows: Initialize step size Using the formula Iteration step size, This is the step size coefficient. When the backtracking condition is met, the iteration stops, and the current iteration position is set to... The final step size for the j-th round is defined by the backtracking condition as follows: ， Indicates the descent parameter. Indicates the vector Each element is projected onto the unit circle, i.e. ; Step 5.1.4: Solve for the simulated weights in the (j+1)th iteration using the following formula: ， Determine whether the iteration termination condition is met, i.e., whether it is satisfied. , This is the algorithm's termination tolerance; if it is met, the loop stops, and the current value is set to zero. Use the optimal simulation weight for the (k+1)th round; otherwise, return to step 5.1.

1.

7. The method for maximizing the main lobe gain of an analog-digital hybrid array for a radar communication system according to claim 5, characterized in that: The specific process of solving the transformed digital weight subproblem using a closed-form solution to obtain the digital weight in the (k+1)th iteration is as follows: Ignoring the constant term, the subproblem can be equivalently represented as ， in, , , Represents an N x 1 column vector. Indicates the vector Diagonalize to a matrix; define the Lagrange function as: in, It is a Lagrange multiplier; the first-order optimality condition of the Lagrange function is: ， According to the above formula, we can obtain ， definition The matrix decomposition is as follows ,in, A matrix composed of eigenvectors. Let represent a diagonal matrix composed of eigenvalues, and Redefining The Lagrange optimality condition equation is transformed into ， make Then the equation simplifies to ， Because the left side of the equation is Since time is a monotonic function, this equation uses the line search method to find the optimal Lagrange multipliers. Substitute the Lagrange multipliers into the following formula: ， Obtain the numerical weights for the (k+1)th iteration. .

8. The method for maximizing the main lobe gain of an analog-digital hybrid array for a radar communication system according to claim 5, characterized in that: The following subproblems are solved using closed-form solutions. Obtain the auxiliary variables for the (k+1)th iteration. Auxiliary variables and minimum main lobe level The specific method is as follows: Ignore irrelevant terms and let , Transform the subproblem into: ， First fix The following formula can be used to solve for the problem. and : ， ， Substituting the result of the above formula into the objective function of the subproblem, we obtain an optimization problem concerning a single variable: ， If ,but ; otherwise, ;like ,but ; otherwise, ;like ,but ;otherwise ; make The objective function of this subproblem can be represented as a piecewise function: ， in ， It is defined in The k-th sub-function within; define respectively , and for , and The set sorted in ascending order and after removing duplicate sequences; It is , and The result obtained after merging, sorting in ascending order, and removing duplicate sequences; By Represented as ， in , , ; optimal Determined by the following formula: ， Will Substitute and solve and In the formula, we obtain the auxiliary variable for the (k+1)th iteration. and .

Images