A Design Method for Reconfigurable Sparse Arrays of Beam Patterns under the Constraint of Dynamic Range Ratio

Abstract

The present invention discloses a method for designing a beam pattern reconfigurable sparse array under the constraint of dynamic range ratio, which relates to technical fields such as wireless communication, radar, and remote sensing. First, the present invention describes the beam pattern of an array with antenna activation and imposes a dynamic range ratio constraint on the complex excitation weight vector; then, an optimization problem is designed, and the alternating minimization method is used to decouple the variables in the optimization problem; finally, the sparse array composed of the activated array elements and each beam pattern are obtained according to the solved variables. Under the conditions of beam pattern nulls, the dynamic range of the array complex weighting vector, and antenna selection constraints, the present invention aims to minimize the minimum-maximum beam pattern weighted matching error, and realizes the design of a beam pattern reconfigurable sparse antenna array based on the dynamic range ratio constraint. The method of the present invention is simple and efficient, easy to implement, and can effectively solve the problem of infeasible initial values.

Description

Technical Field

[0001] The present invention relates to the technical fields of wireless communication, radar, remote sensing, etc., and particularly to a design method of a beam pattern reconfigurable sparse antenna array under the constraint of dynamic range ratio. Background Art

[0002] The reconfigurable antenna array realizes the diversity of the array beam pattern through an electronic control method (adjusting the complex weighting coefficients of the array). Usually, different beam patterns are controlled by corresponding excitation phases, but share the same excitation amplitude. The antenna array with the characteristic of beam pattern reconfigurability can achieve different beam patterns only through phase control according to different task requirements, without the need to dynamically change its excitation amplitude. This excellent characteristic enables the array system to use only one power divider, thus reducing the system cost and the complexity of the array design.

[0003] On the other hand, for a uniform linear antenna array, the antenna element spacing needs to be less than half a wavelength to prevent grating lobes from appearing in the array beam pattern. However, the limitation of the half-wavelength element spacing requires the antenna array to have a sufficient number of antenna elements to achieve the desired aperture and resolution. The sparse antenna array design technology can design the array distribution to increase the aperture size and resolution and prevent the generation of grating lobes in the beam pattern under the same number of elements. This technology has received extensive research and attention in array processing.

[0004] In addition, research shows that reducing the dynamic range ratio (DRR) of the excitation amplitude of the array elements can reduce the mutual coupling between adjacent elements and the cost of the array feeding network design. However, there is no existing design scheme for a beam pattern reconfigurable sparse array applying the dynamic range ratio constraint. Summary of the Invention

[0005] In view of this, the present invention proposes a design method of a beam pattern reconfigurable sparse array under the constraint of dynamic range ratio, which can realize the design of a beam pattern reconfigurable sparse array while controlling the dynamic range ratio of the element excitation amplitude.

[0006] To achieve the above object, the technical solution adopted by the present invention is as follows:

[0007] A design method of a beam pattern reconfigurable sparse array under the constraint of dynamic range ratio, which is applied to a uniform linear array composed of N antenna elements, and the positions of the elements are z1, …, z N , and there are antenna elements in the array for performing communication or detection tasks, and the antenna array radiates I different beam patterns; the method includes the following steps:

[0008] Step 1, describe the beam pattern of the array at direction θ as:

[0009] p(θ) = |aH (θ)w| 2 (1)

[0010] wherein, is the complex excitation weight vector corresponding to the array, represents the array steering vector, the superscript H represents conjugate transpose, and the superscript T represents transpose;

[0011] Define the activation vector of the antenna as If the element n is to be activated, the corresponding element s n is 1. If the element n is not activated, the corresponding element s n is 0, where n = 1, 2,..., N; The element at position z1 is used as the reference element and is always in the activated state, i.e., s1 = 1; Then the beam pattern with antenna activation is described as:

[0012] p(θ) = |a H (θ)diag(s)w| 2 (4)

[0013] diag() represents constructing a diagonal matrix;

[0014] Apply a dynamic range ratio constraint to the complex excitation weight vector w, and the dynamic range ratio is defined as follows:

[0015]

[0016] wherein, DRR represents the dynamic range ratio, and represent the maximum and minimum values of the modulus of the elements in the vector w respectively;

[0017] Let and represent the main lobe, sidelobe, and null region of the i-th beam pattern, and let represent the beam pattern template corresponding to the i-th beam pattern; wherein, if then d i (θ k ) = 1, otherwise d i (θ k ) = 0, K i represents the number of discrete angle grid points of the i-th beam pattern, θ k represents the k-th discrete angle, k = 1, 2,..., K i ;

[0018] Step 2, design the following optimization problem:

[0019]

[0020]

[0021]

[0022]

[0023] wherein, represents the k-th angle θ corresponding to the i-th beam pattern k of the weighting coefficient, w i represents the array weighting vector corresponding to the i-th beam pattern; is the beam pattern null constraint, is the null depth specified by the user; δ is the dynamic range ratio threshold defined by the user;

[0024] Step 3, use the alternating minimization method to decouple the variables s and in the optimization problem, that is, alternately optimize the variables s and The specific method is as follows:

[0025] S301, define the objective function as follows:

[0026]

[0027] Set the threshold σ, assume the objective function of the q-th alternating minimization iteration is obj(q); given the initial value of the excitation weight and the initial value s(0) of the antenna selection vector; in subsequent iterations, the excitation weight is written as the antenna selection vector is written as s(t), where t is the iteration variable;

[0028] S302, in each iteration, calculate

[0029] Given s(t) and solve the following problem using the convex optimization toolbox CVX to obtain

[0030]

[0031]

[0032]

[0033]

[0034]

[0035]

[0036] In the above formula, α = [α1, α2,... α N ∈ RN×1 ξ and γ are auxiliary variables, and Re[] represents taking the real part. represents taking the conjugate of. represents the estimated value of the variable at the t-th iteration, η w is an auxiliary variable. is a slack variable λ introduced to handle potential infeasible point problems. w is a penalty parameter defined by the user;

[0037] Next, calculate

[0038] At and given s(t), use the convex optimization toolkit CVX to solve the following problem to obtain s(t + 1):

[0039]

[0040]

[0041]

[0042]

[0043]

[0044] In the above formula, η s is an auxiliary variable, μ is a slack variable λ introduced to handle potential infeasible initial point problems. s is a penalty parameter defined by the user;

[0045] Thus, obtain the objective function at the q-th iteration:

[0046]

[0047] Let

[0048] Update the iteration variables: q = q + 1, t = t + 1

[0049] Start a new round of iteration until the absolute value of the difference between the objective functions obtained in two adjacent iterations is less than the threshold σ, and output and s;

[0050] Step 4, obtain the sparse array composed of the activated array elements according to s; according to the beam pattern formula p(θ) = |a H (θ)diag(s)w| 2 , substitute w i and s to obtain the i-th beam pattern.

[0051] The beneficial effects of the present invention are as follows:

[0052] 1. Under the conditions of beam pattern null, array complex weight vector dynamic range, and antenna selection constraint, the present invention aims to minimize the minimum-maximum beam pattern weighted matching error, and realizes the design of a beam pattern reconfigurable sparse antenna array based on the dynamic range ratio constraint.

[0053] 2. The present invention uses the alternating minimization (AM) method to decouple the complex weight vector and antenna position selection vector of the array, decomposes the original problem into two sub-problems for iterative solution, and greatly simplifies the solution process.

[0054] 3. The present invention uses the successive convex approximation (SCA) method to transform the non-convex sub-optimization problem into a convex optimization problem, and uses the convex optimization toolbox (CVX) for solution, further simplifying the solution process.

[0055] 4. The present invention draws on the feasible point pursuit successive convex approximation (FPP-SCA) algorithm, and effectively solves the problem of infeasible initial values by introducing dynamically changing auxiliary variables and slack variables. Description of the Drawings

[0056] Figure 1 It is the beam pattern result of Example 1 in the embodiments of the present invention. Among them, (a) is the beam pattern, (b) is the array excitation amplitude, and (c) is the antenna activation position.

[0057] Figure 2 It is the beam pattern result of Example 2 in the embodiments of the present invention. Among them, (a) is the beam pattern, (b) is the array excitation amplitude, and (c) is the antenna activation position. Detailed Embodiments

[0058] The following further describes the present invention in detail with reference to the drawings.

[0059] A beam pattern reconfigurable sparse array design method under dynamic range ratio constraint is applied to a uniform linear array composed of N antenna elements, and the positions of the elements are z1,…,z N , and there are antenna elements in the array for performing communication or detection tasks, and the antenna array radiates I different beam patterns; it includes the following steps:

[0060] Step 1, describe the beam pattern of the array at direction θ as:

[0061] p(θ) = |a H (θ)w| 2 (1)

[0062] Among them, is the complex excitation weight vector corresponding to the array, represents the array steering vector, the superscript H represents the conjugate transpose, and the superscript T represents the transpose;

[0063] Define the activation vector of the antenna as If element n of the array is to be activated, the corresponding element s n is 1. If element n of the array is not activated, the corresponding element s n is 0, where n = 1, 2,..., N; The element at position z1 is used as the reference element and is always in the activated state, that is, s1 = 1; Then the beam pattern with antenna activation is described as:

[0064] p(θ) = |a H (θ)diag(s)w| 2 (4)

[0065] diag() represents constructing a diagonal matrix;

[0066] Apply a dynamic range ratio constraint to the complex excitation weight vector w, and the dynamic range ratio is defined as follows:

[0067]

[0068] Among them, DRR represents the dynamic range ratio, and represent the maximum and minimum values of the modulus of the elements in the vector w respectively;

[0069] Let and represent the main lobe, side lobe, and null region of the i-th beam pattern, and let represent the beam pattern template corresponding to the i-th beam pattern; Among them, if then d i (θ k ) = 1, otherwise d i (θ k ) = 0, K i represents the number of discrete angular grid points of the i-th beam pattern, θ k represents the k-th discrete angle, where k = 1, 2,..., K i ;

[0070] Step 2, design the following optimization problem:

[0071]

[0072]

[0073]

[0074]

[0075] Among them, represents the weighted coefficient \(w_{ik}\) corresponding to the \(k\)-th angle \(\theta\) of the \(i\)-th beam pattern k ; and \(\mathbf{w}_i\) i represents the array weighting vector corresponding to the \(i\)-th beam pattern; is the beam pattern null constraint, where \(\xi\) is the specified null depth; \(\delta\) is the user-defined dynamic range ratio threshold;

[0076] Step 3, use the alternating minimization method to decouple the variables \(s\) and in the optimization problem, that is, alternately optimize the variables \(s\) and The specific method is as follows:

[0077] S301, define the objective function as follows:

[0078]

[0079] Set the threshold \(\sigma\) (usually set to be less than \(10\) -2 ), assume that the objective function of the \(q\)-th alternating minimization iteration is \(obj(q)\); given the initial value of the excitation weight and the initial value \(s(0)\) of the antenna selection vector; in subsequent iterations, the excitation weight is written as and the antenna selection vector is written as \(s(t)\), where \(t\) is the iteration variable;

[0080] S302, in each iteration, calculate

[0081] Given \(s(t)\) and , use the convex optimization toolbox CVX to solve the following problem to obtain

[0082]

[0083]

[0084]

[0085]

[0086]

[0087]

[0088] In the above formula, \(\alpha = [\alpha_1, \alpha_2, \ldots, \alpha_{ N}] \in \mathbb{R}^{ N×1} and \(\gamma\) are auxiliary variables, and \(Re[\cdot]\) represents taking the real part. denotes taking the conjugate of denotes the estimated value of the t-th iteration of the variable , η w is the auxiliary variable, , λ is the slack variable introduced to handle potential infeasible point problems w is the penalty parameter defined by the user;

[0089] Next, calculate

[0090] Given and s(t), solve the following problem using the convex optimization toolkit CVX to obtain s(t + 1):

[0091]

[0092]

[0093]

[0094]

[0095]

[0096] In the above formula, η s is the auxiliary variable, μ is the slack variable introduced to handle potential infeasible initial point problems, λ s is the penalty parameter defined by the user;

[0097] Thus, the objective function of the q-th iteration is obtained:

[0098]

[0099] Let

[0100] Update the iteration variables: q = q + 1; t = t + 1

[0101] Start a new round of iteration until the absolute value of the difference between the objective functions obtained in two adjacent iterations is less than the threshold σ, and output and s;

[0102] Step 4, obtain the sparse array composed of the activated array elements according to s; according to the beam pattern formula p(θ) = |a H (θ)diag(s)w| 2 , substitute w i and s into it to obtain the i-th beam pattern.

[0103] This method fully considers the outstanding advantages of the reconfigurability, sparsity, and controllability of the dynamic range ratio (DRR) of the array beam pattern, and realizes the design of a sparse array with reconfigurable beam pattern while controlling the dynamic range ratio of the element excitation amplitude.

[0104] The following is a more specific example:

[0105] Suppose a uniform linear array consists of N antenna elements, and the positions of the elements are z1, …, z N . Then, the beam pattern of this array at direction θ is given by:

[0106] p(θ) = |a H (θ)w| 2 (1)

[0107] where is the complex excitation weight vector corresponding to the array, represents the array steering vector.

[0108] Suppose there are only antenna elements in the array for performing communication or detection tasks.

[0109] Define the antenna activation vector

[0110]

[0111] That is, for the array element n to be activated, the corresponding s n element is 1, otherwise it is 0 (not activated). In addition, the element at position z1 is taken as the reference element, that is, it is always in the activated state (i.e., s1 = 1). According to the above definition, equation (2) can be written as:

[0112]

[0113] Furthermore, the beam pattern of equation (1) with antenna activation can be described as:

[0114] p(θ) = |a H (θ)diag(s)w| 2 .(4)

[0115] In practical applications, imposing DRR constraints on the excitation weight w can reduce the system cost and the mutual coupling between adjacent elements. The DRR is defined as follows:

[0116]

[0117] Usually, a single array will radiate multiple different beam patterns to meet the requirements of different communication or detection tasks. Suppose an antenna array radiates I different beam patterns, and let and Denote the main lobe, sidelobe, and null region of the \(i\)-th beam pattern. Let denote the beam pattern template corresponding to the \(i\)-th beam pattern, where otherwise it is 0, K i denotes the number of discrete angular grid points of the \(i\)-th beam pattern.

[0118] Based on the above definitions and analyses, the present invention proposes the following optimization problem:

[0119]

[0120]

[0121]

[0122]

[0123] where is the weighting coefficient corresponding to the \(k\)-th angle of the \(i\)-th beam pattern, i.e., \(\theta\) k ; is the beam pattern null constraint. To reduce the signal-related interference from a specified angular direction, where is the user-specified null depth; \(\delta\) is the user-defined DRR threshold.

[0124] The main difficulty of the optimization problem (6) is the non-convex objective function with coupled variables and \(s\), the Boolean constraint imposed on \(s\), the amplitude constraint imposed on the complex weighting coefficient \(w\) i and the DRR constraint. To handle the above difficulties, the present invention will handle these difficult problems one by one according to the alternating minimization method (AM) and the successive convex approximation (SCA) method. First, use the alternating minimization method to decouple the variables \(s\) and

[0125] i.e., alternately optimize the variables \(s\) and i.e., alternately optimize the variables \(s\) and The detailed steps are given in Algorithm 1:

[0126]

[0127]

[0128] The objective function in Algorithm 1 is defined as follows:

[0129]

[0130] Next, solve the solutions of the sub-optimization problems (8) and (7) through the successive convex approximation algorithm.

[0131] 1. Continuous Convex Approximation of Optimization Problem (7)

[0132] The main difficulties in optimization problem (7) are the modulus operation and amplitude modulus constraint in the objective function, as well as the DRR constraint. Expanding the objective function of optimization problem (7), we get:

[0133]

[0134]

[0135] It can be seen that the non-convexity of (10) is due to -2d i (θ k )|a H (θ k )diag(s(t))w i |. Let w i (t) be the t-th estimate of w i , that is, w i (t) lies within the feasible region of optimization problem (7). By the Cauchy-Schwarz inequality, we have

[0136]

[0137] The equality in the above equation holds when w i = w i (t).

[0138] Replacing the modulus term in (10) with the right side of (11), we get

[0139]

[0140] The equality in the above equation holds when .

[0141]

[0142]

[0143] It is easy to verify that is a convex function. Then, based on the relation (12), the convex approximation function corresponding to formula (7) is:

[0144]

[0145]

[0146]

[0147]

[0148] The main difficulties of (15) are the non-convex modulus constraint and the DRR constraint. In addition, w 1 is restricted by these two types of constraints. By introducing auxiliary variables these two constraints can be equivalently written as:

[0149]

[0150]

[0151] In addition, by introducing a new variable γ≥0, the DRR constraint can be written as:

[0152]

[0153] According to (16)–(18), equation (15) can be rewritten as the following equivalent optimization problem:

[0154]

[0155] The remaining difficulty is the non-convex modulus constraint, that is First, it is written as the following equivalent inequality constraint:

[0156]

[0157]

[0158] It can be seen that the non-convexity is due to terms exist.

[0159] According to the Cauchy-Schwarz inequality, we have:

[0160]

[0161] where the equality holds when w n i = w n i (t). Therefore, constraint (21) can be replaced by the following linear constraints:

[0162]

[0163] Therefore, the final convex approximation problem of the optimization problem (7) is:

[0164]

[0165] 2. Successive convex approximation of the optimization problem (8):

[0166] Using the same method as in (10)–(13), the first convex approximation problem of (8) can be obtained as:

[0167]

[0168]

[0169]

[0170] Among them

[0171]

[0172] Note that s(t) in (26) must be a feasible solution to the optimization problem (8).

[0173] Next, the non-convex Boolean constraint is processed by the convex approximation method. First, this constraint is equivalently written as:

[0174]

[0175]

[0176] It can be seen that the non-convexity of the Boolean constraint is mainly due to Similar to (20) and (21), replace the equality constraint with the following two inequality constraints

[0177]

[0178]

[0179] Assume that s(t) is the t-th estimate of the variable s. By Expanding this formula gives:

[0180]

[0181] Based on the discussion in (27)-(31), similar to (22)–(24), the final convex approximation of the optimization problem (8) is as follows:

[0182]

[0183]

[0184]

[0185]

[0186] (24) and (32) can both be converted into second-order cone programming, and their global optimal solutions can be solved by the interior point method. Here, the CVX toolbox can be used to solve (24) and (32). By summarizing the above steps, Algorithm 2 can be obtained:

[0187]

[0188]

[0189] Algorithm 2 requires a feasible point as the initialization value, i.e., the excitation weights and the antenna selection vector s(0) must be feasible, which is usually difficult to obtain. To this end, inspired by the feasible point pursuit successive convex approximation (FPP-SCA) algorithm, the present invention introduces the auxiliary variable η w , η s and the slack variable μ (where ) to handle potential infeasible point problems and modifies Algorithm 2 to Algorithm 3:

[0190]

[0191]

[0192] In Algorithm 3, and λ s are penalty parameters defined by the user to balance the objective function and the slack variable. The optimization problems (33) and (34) are always feasible, and when μ = 0, the optimal solution of Algorithm 3 is also the optimal solution of Algorithm 2. For detailed information on relaxation, reference can be made to (O. Mehanna, K. Huang, B. Gopalakrishnan, A. Konar, and N. D. Sidiropoulos, “Feasible point pursuit and successive approximation of non-convex QCQPs,” IEEE Signal Process. Lett., vol. 22, no. 7, pp. 804–808, 2015.). In addition, different from O. Mehanna who uses a constant penalty parameter throughout the algorithm, this method gradually increases the penalty parameter value, i.e., The experimental results show that the method of gradually increasing the penalty parameter has a faster convergence rate than using a constant penalty parameter.

[0193] The following uses a simulation example to prove the effectiveness of the method of the present invention:

[0194] In the simulation, it is assumed that there are L candidate antenna positions with a position spacing of and the azimuth sampling interval is 1°. Other parameters are shown in Table I:

[0195] Table I Beam pattern parameter settings

[0196]

[0197] The experimental parameters and beam pattern results are shown in Table II below:

[0198] Table II Experimental Parameters and Beam Pattern Results

[0199]

[0200] Figure 1 and Figure 2 are schematic diagrams of the beam pattern synthesis results, the amplitudes of the array excitations, and the positions of the activated antennas for Examples 1 and 2 in Table II. As can be seen from Table II and Figure 1 、 Figure 2 , the method has good results in the peak sidelobe level (PSL) and the dynamic range ratio (DRR) of the excitation amplitude of the obtained beam pattern.

Claims

1. A method for designing a beam pattern reconfigurable sparse array under the constraint of dynamic range ratio, characterized in that, Applied to a uniform linear array composed of N antenna elements, the positions of the elements are z1, …, z N , and there are antenna elements in the array for performing communication or detection tasks, and the antenna array radiates I different beam patterns; the method includes the following steps: Step 1, describe the beam pattern of the array at direction θ as: p(θ) = |a H (θ)w| 2 (1) Among them, is the complex excitation weight vector corresponding to the array, represents the array steering vector, the superscript H represents the conjugate transpose, and the superscript T represents the transpose; Define the activation vector of the antenna as If the element n is to be activated, the corresponding element s n is 1. If the element n is not activated, the corresponding element s n is 0, where n = 1, 2, ..., N; Take the element at position z1 as the reference element, which is always in the activated state, i.e., s1 = 1; Then the beam pattern with antenna activation is described as: p(θ) = |a H (θ) diag(s) w| 2 (4) diag() represents constructing a diagonal matrix; Apply a dynamic range ratio constraint to the complex excitation weight vector w, and the dynamic range ratio is defined as follows: where DRR represents the dynamic range ratio, and respectively represent the maximum value and the minimum value of the modulus of the elements in the vector w; Let and represent the main lobe, sidelobe, and null region of the i-th beam pattern, and let denotes the beam pattern template corresponding to the i-th beam pattern; where, if then d i (θ k ) = 1, otherwise d i (θ k ) = 0, K i denotes the number of discrete angular grid points of the i-th beam pattern, θ k denotes the k-th discrete angle, k = 1, 2,..., K i ; Step 2, design the following optimization problem: Among them, represents the k-th angle θ corresponding to the i-th beam pattern k of the weighting coefficient, w i represents the array complex excitation weight vector corresponding to the i-th beam pattern; is the beam pattern null constraint, is the null depth specified by the user; δ is the dynamic range ratio threshold defined by the user; Step 3, use the alternating minimization method to decouple the variables s and in the optimization problem and Step 4, obtain the sparse array composed of the activated array elements according to s; according to the beam pattern formula p(θ) = |a H (θ)diag(s)w| 2 , substitute w i and s into it to obtain the i-th beam pattern.

2. The method for designing a beam pattern reconfigurable sparse array under the constraint of dynamic range ratio according to claim 1, characterized in that, Sparse array design that can reconfigure the beam pattern while controlling the dynamic range ratio of the array element excitation amplitude; The specific method of Step 3 is as follows: S301, Define the objective function as follows: Set the threshold σ, and assume that the objective function for the q-th alternating minimization iteration is obj(q); given the initial value of the excitation weight and the initial value s(0) of the antenna selection vector; in subsequent iterations, the excitation weight is written as the antenna selection vector is written as s(t), where t is the iteration variable; S302, in each iteration, calculate Given \(s(t)\) and the following problem is solved using the convex optimization toolbox CVX to obtain In the above formula, and γ are auxiliary variables, Re[] represents taking the real part, represents taking the conjugate of, represents the estimated value of the variable at the t-th iteration, η w is an auxiliary variable, , is a slack variable introduced to handle potential infeasible point problems, λ w is a penalty parameter defined by the user; Next, calculate Given and s(t), solve the following problem using the convex optimization toolbox CVX to obtain s(t + 1): In the above formula, η s is an auxiliary variable, μ is a slack variable introduced to handle potential infeasible initial point problems, and λ s is a penalty parameter defined by the user; Thus, obtain the objective function for the q-th iteration: Let λ v (q + 1) = 2λ v (q), λ s (q + 1) = 2λ s (q), Update the iteration variables: q = q + 1;, t = t + 1 Start a new round of iteration until the absolute value of the difference between the objective functions obtained in two adjacent times is less than the threshold σ, and output and s.

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