A phase-only synthesis method for sparse arrays based on maximization of directivity coefficient

Abstract

The present invention discloses a sparse array phase-only synthesis method based on maximizing directivity coefficient. First, a model of a sparse array phase-only excitation pattern synthesis algorithm based on maximizing directivity coefficient is established. Under the premise of a given array sparsity rate, the array directivity coefficient is used as the objective function to simultaneously optimize the array formation and the excitation phase; the 0 / 1 constraint is used to limit the sparsity and constant mode characteristics of the excitation; the main lobe gain is used to limit the array pattern response, and the main lobe ripple and sidelobe level are controlled; and then the model is solved using the alternating direction multiplier method to obtain a sparse and constant mode excitation vector. The present invention solves the problem of the existing method that the position and constant mode characteristics of the sparse excitation are not constrained. Under the condition of a given sparsity, an accurate approximation of the desired pattern can be achieved, while ensuring the constant mode of the array excitation and the maximization of the directivity coefficient.

Description

Technical Field

[0001] The invention belongs to the field of array signal processing, and in particular relates to a sparse array phase-only synthesis method based on maximization of directivity coefficients. Background Art

[0002] In the pattern synthesis technology of array antenna, the pattern synthesis method of sparse array phase-only excitation obtains the desired pattern by only changing the phase of the excitation under the premise of ensuring the constant excitation amplitude. This type of method can simplify the array structure, reduce the mutual coupling between array elements, and improve the energy utilization efficiency. In the research of pattern synthesis methods, there are more studies on sidelobe level optimization and pattern shape optimization, but because the optimization problem of maximizing the directivity coefficient is relatively complex, there are fewer studies on the optimization of array directivity coefficient. At present, there is only one typical method. Y. Zhang et al. proposed a pattern synthesis method based on maximizing the array directivity coefficient. This method uses the directivity coefficient as the objective function for design, uses the pattern shape as the constrained synthetic pattern, and converts the non-convex optimization problem into a convex optimization problem for solution. However, this method does not constrain the position and constant mode characteristics of sparse excitation, can only randomly select the excitation position, cannot obtain the optimal sparse excitation, and the excitation amplitude varies greatly. The main problem of the above method is that the position and constant mode characteristics of sparse excitation are not constrained, the excitation position can only be randomly selected, the optimal sparse excitation cannot be obtained, and the excitation amplitude varies greatly. Summary of the invention

[0003] The purpose of the present invention is to provide a sparse array phase-only synthesis method based on maximizing the directivity coefficient, which solves the problem of the existing methods that the position and constant mode characteristics of the sparse excitation are not constrained. Under the condition of given sparsity, the accurate approximation of the desired directivity pattern can be achieved while ensuring the maximum directivity coefficient of the array.

[0004] The technical solution adopted by the present invention is:

[0005] A sparse array phase-only synthesis method based on maximizing directivity coefficients includes the following steps:

[0006] Step 1: Establish a model for the sparse array phase-only excitation pattern synthesis algorithm based on maximizing the directivity coefficient:

[0007] Assume that the uniform linear array consists of N array elements, and the directional pattern array factor is:

[0008] S(θ)=w H a(θ) (1)

[0009] Where w is the complex excitation vector of the array, including the amplitude and phase of each element excitation, a(θ) is the steering vector of the array, and S(θ) is the synthesized directional pattern;

[0010] The directivity coefficient of the array at θ0 is expressed as:

[0011]

[0012] In the formula, is a positive Hermitian matrix; B0 is decomposed into B0 = B H B, where

[0013] The optimization of the directivity coefficient is achieved by constraining the numerator and denominator separately, and the following model is constructed:

[0014]

[0015] st G≤|w H a(θ m )|≤δG,θ m ∈Θ ML ,m=1,2,…,M (3b)

[0016]

[0017] |w H a(θ s )|≤ρ0G,θ s ∈Θ SL ,s=1,2,...,S (3d)

[0018] ||w||0=K (3e)

[0019] |w n |∈{0,1},n=1,...,N (3f)

[0020] Where G and Q are constraints on the array directivity coefficient, || ||0 is the l0 norm, and a(θ m ) is the main lobe steering vector, a(θ s ) is the sidelobe steering vector, Θ ML Represents the main lobe area of ​​the desired pattern, divided into M angles; Θ SL represents the sidelobe area, which is divided into S angles; δ and ρ0 constrain the mainlobe ripple and sidelobe level according to the desired pattern; N is the number of array elements, w is the N×1 array complex excitation vector, which has K non-zero elements, and w nis the nth element of w; Formula (3a) means to use the array directivity coefficient as the objective function to simultaneously optimize the array formation and the excitation phase; Formula (3b) means to control the main lobe gain to be higher than G and the main lobe ripple to be lower than δG; Formulas (3b) and (3c) constrain the numerator and denominator of the array directivity coefficient; Formula (3d) means to limit the sidelobe level to be lower than ρ0G; Formula (3f) is a 0 / 1 constraint, and Formulas (3e) and (3f) jointly limit the sparsity and constant modulus of the excitation;

[0021] Step 2, equivalent transformation of the problem:

[0022] Introduce the following auxiliary variables:

[0023]

[0024] stx m =w H a(θ m ),θ m ∈Θ ML (4b)

[0025] G≤|x m |≤δG,m=1,2,...,M (4c)

[0026] y s =w H a(θ s ),θ s ∈Θ SL (4d)

[0027] |y s |≤ρ0G,s=1,2,...,S (4e)

[0028] z=Bw (4f)

[0029]

[0030] u=w (4h)

[0031] ||u||0=K (4i)

[0032] |w n |∈{0,1},n=1,...,N (4j)

[0033] In the formula, || ||0 is the l1 norm, x m ,y s and z are the introduced auxiliary variables;

[0034] Construct the following augmented Lagrangian function:

[0035]

[0036] In the formula, Α M and Alpha S are respectively the main lobe steering vector a(θ m ) and the sidelobe steering vector a(θ s ), λ, γ, μ and is the dual variable, ρ1, ρ2, ρ3 and ρ4 are the custom iteration steps, x and y are x in equation (4b) and equation (4d) respectively. m and s The vector formed;

[0037] Step 3, use the alternating direction multiplier method to solve the problem in step 2:

[0038] (1) Update x, y and G:

[0039] Ignore the variables that are not related to x and y in equation (5) and define We can get:

[0040]

[0041] The first quadratic term in the objective function of formula (6) can be decomposed into M sub-problems, and the closed-form solution is:

[0042]

[0043] Where b m is the mth element of b;

[0044] The second quadratic term in the objective function of formula (6) can be decomposed into S sub-problems, and the closed-form solution is:

[0045]

[0046] In the formula, c s is the sth element of c;

[0047] Substituting equations (7) and (8) into equation (6) yields a quadratic function only about G, with the coefficients of the quadratic term, linear term, and constant term being:

[0048]

[0049] In the formula, P1 represents |b m |≤ the number of items in G, P2 represents |b m |≥δG, P3 represents |c s |≥ρ0The number of items in G; According to the characteristics of quadratic functions, the optimal solution of G is:

[0050]

[0051] Substituting equation (12) into equation (7) and equation (8) yields the elements of x and y;

[0052] (2) Update z and Q:

[0053] Ignoring the variables in equation (5) that are independent of z and defining d = Bw-γ, we can obtain

[0054]

[0055]

[0056] (3) Update u:

[0057] Ignore the variables in equation (5) that are not related to u, and define The i-th element u of u i It is expressed as:

[0058]

[0059] In the formula, e i is the i-th element of e;

[0060] In order to satisfy the constraint of ||u||0=K, i Sort: If e is greater than 1 / 2 i If the number of items is greater than or equal to K, then take K items and let the corresponding |u i |=1; if it is greater than 1 / 2 of e i If the number of terms is P and P < K, then select the largest KP term from the remaining terms to obtain a complete closed-form solution for u;

[0061] (4) Update w:

[0062] Ignoring the variables in equation (5) that are not related to u, we get the optimization problem:

[0063]

[0064] in,

[0065]

[0066] In the formula, I N is an N×1 unit matrix; the closed-form solution of equation (15) is:

[0067]

[0068] (5) Update λ, μ, γ and

[0069]

[0070]

[0071] In the formula, λ, μ, γ and Indicates the value before updating. and Indicates the updated value;

[0072] Step 4, repeat steps (1)-(5) in step 3 until the difference between variables x, y, z, u and w before and after the update converges to the expected error, and the optimal array excitation vector w is obtained, which is completed based on the sparse array phase-only synthesis method based on maximizing the directivity coefficient.

[0073] The beneficial effects of the present invention are as follows: the model proposed by the present invention optimizes the array formation and the excitation phase at the same time under the premise of a given array sparsity rate with the array directivity coefficient as the objective function; uses 0 / 1 constraints to limit the sparsity and constant mode characteristics of the excitation; uses the main lobe gain to limit the array pattern response, and controls the main lobe ripple and side lobe level; solves the problem of the existing method that the position and constant mode characteristics of the sparse excitation are not constrained, and can achieve accurate approximation of the desired pattern under the condition of a given sparsity, while ensuring that the array directivity coefficient is maximized. BRIEF DESCRIPTION OF THE DRAWINGS

[0074] Figure 1 It is a comparison diagram of the flat-top radiation pattern obtained by the method proposed by Y. Zhang and the method of the present invention.

[0075] Figure 2 It is the normalized amplitude diagram of the excitation corresponding to the present invention.

[0076] Figure 3 Normalized amplitude diagram of the excitation corresponding to the method proposed by Y.Zhang. DETAILED DESCRIPTION

[0077] The present invention is further described below in conjunction with the accompanying drawings and specific embodiments.

[0078] The technical solution adopted by the sparse array phase-only synthesis method based on maximizing the directivity coefficient of the present invention is to first establish a model of a sparse array phase-only excitation pattern synthesis algorithm based on maximizing the directivity coefficient. Under the premise of a given array sparsity rate, the model uses the array directivity coefficient as the objective function to simultaneously optimize the array formation and the excitation phase; uses 0 / 1 constraints to limit the sparsity and constant mode characteristics of the excitation; uses the main lobe gain to limit the array pattern response, and controls the main lobe ripple and sidelobe level; because the model is a highly non-convex model and is difficult to solve directly, the alternating direction of multipliers method (ADMM) is used to solve the model to obtain a sparse and constant mode excitation vector. The details are as follows:

[0079] Step 1: Establish a model for the sparse array phase-only excitation pattern synthesis algorithm based on maximizing the directivity coefficient:

[0080] Assume that the uniform linear array consists of N array elements, and the directional pattern array factor is:

[0081] S(θ)=w H a(θ) (1)

[0082] Where w is the complex excitation vector of the array, including the amplitude and phase of each element excitation, a(θ) is the steering vector of the array, and S(θ) is the synthesized directional pattern;

[0083] The directivity coefficient of the array at θ0 is expressed as:

[0084]

[0085] In the formula, is a positive Hermitian matrix; B0 is decomposed into B0 = B H B, where

[0086] Since the expression of the array directivity coefficient is relatively complex, its numerator and denominator both contain the complex excitation vector w to be optimized, and the optimization problem of maximizing the directivity coefficient cannot be expressed by a convex function. Therefore, the optimization of the directivity coefficient can be achieved by constraining its numerator and denominator respectively. Under the premise of a given sparsity rate, a sparse array can be achieved by selecting the array elements of a uniform linear array. A given sparsity rate can be achieved by constraining the excitation l0 norm, and the array's directional pattern synthesis can be achieved by constraining the main lobe and side lobes, so the following model can be constructed:

[0087]

[0088] st G≤|w Ha(θ m )|≤δG,θ m ∈Θ ML ,m=1,2,...,M (3b)

[0089]

[0090] |w H a(θ s )|≤ρ0G,θ s ∈Θ SL ,s=1,2,...,S (3d)

[0091] ||w||0=K (3e)

[0092] |w n |∈{0,1},n=1,...,N (3f)

[0093] Where G and Q are constraints on the array directivity coefficient, || ||0 is the l0 norm, and a(θ m ) is the main lobe steering vector, a(θ s ) is the sidelobe steering vector, Θ ML Represents the main lobe area of ​​the desired pattern, divided into M angles; Θ SL represents the sidelobe area, which is divided into S angles; δ and ρ0 constrain the mainlobe ripple and sidelobe level according to the desired pattern; N is the number of array elements, w is the N×1 array complex excitation vector, which has K non-zero elements, and w n is the nth element of w; Formula (3a) means to use the array directivity coefficient as the objective function to simultaneously optimize the array formation and the excitation phase; Formula (3b) means to control the main lobe gain to be higher than G and the main lobe ripple to be lower than δG; Formulas (3b) and (3c) constrain the numerator and denominator of the array directivity coefficient; Formula (3d) means to limit the sidelobe level to be lower than ρ0G; Formula (3f) is a 0 / 1 constraint, and Formulas (3e) and (3f) jointly limit the sparsity and constant modulus of the excitation. Therefore, Formula (3) can simultaneously optimize the array formation and the excitation phase while satisfying the constraints of the array directivity coefficient, and realize the synthesis of the sparse array phase-only excitation pattern with the maximization of the directivity coefficient.

[0094] Step 2, equivalent transformation of the problem:

[0095] Formula (3) is a highly non-convex problem and is difficult to solve directly by optimization. It can be solved by using the alternating direction multiplier method. The following auxiliary variables are introduced:

[0096]

[0097] x m =w Ha(θ m ),θ m ∈Θ ML (4b)

[0098] G≤|x m |≤δG,m=1,2,...,M (4c)

[0099] y s =w H a(θ s ),θ s ∈Θ SL (4d)

[0100] |y s |≤ρ0G,s=1,2,...,S (4e)

[0101] z=Bw (4f)

[0102]

[0103] u=w (4h)

[0104] ||u||0=K (4i)

[0105] |w n |∈{0,1},N=1,...,N (3j)

[0106] Among them, || ||0 is the l1 norm, x m ,y s , z n , z is the introduced auxiliary variable.

[0107] The following augmented Lagrangian function can be further constructed:

[0108]

[0109] In the formula, Α M and Alpha S are respectively the main lobe steering vector a(θ m ) and the sidelobe steering vector a(θ s ), λ, γ, μ and is the dual variable, ρ1, ρ2, ρ3 and ρ4 are the custom iteration steps, x and y are x in equation (4b) and equation (4d) respectively. m and s Vector composed.

[0110] Step 3, use the alternating direction multiplier method to solve the above problem:

[0111] (1) Update x, y, G:

[0112] Since G is coupled with x and y in both the objective function of equation (4a) and the constraints of equations (4c) and (4e), joint optimization is necessary. Ignore the variables that are not related to x and y in equation (5) and define Available

[0113]

[0114] The first quadratic term in the objective function of formula (6) can be decomposed into M sub-problems:

[0115]

[0116] The closed-form solution is:

[0117]

[0118] Where b m is the mth element of b;

[0119] Similarly, the second quadratic term in the objective function of formula (6) can be decomposed into S sub-problems:

[0120]

[0121] The closed-form solution is:

[0122]

[0123] Substituting equations (8) and (10) into equation (6) yields a quadratic function only about G:

[0124] αG 2 +βG+χ(11) where the coefficients of the quadratic term, linear term and constant term are:

[0125]

[0126] In the formula, P1 represents |b m |≤ the number of items in G, P2 represents |b m |≥δG, P3 represents |c s |≥ρ0The number of items in G; According to the characteristics of quadratic functions, the optimal solution of G is:

[0127]

[0128] Substituting equation (15) into equation (8) and equation (10) yields x and y.

[0129] (2) Update z, Q:

[0130] Since Q is coupled with variable z in both the objective function of equation (4a) and the constraint condition of equation (4g), joint optimization must be performed. Ignoring variables unrelated to z in equation (5) and defining d = Bw-γ, we can obtain:

[0131]

[0132] Equation (16) can be solved by the Lagrangian function

[0133]

[0134] Separate order

[0135]

[0136] Combining equations (18a)-(18c) yields:

[0137]

[0138] (3) Update u:

[0139] Ignore the variables in equation (5) that are not related to u, and define We can get:

[0140]

[0141] Formula (20) can be decomposed into N sub-problems:

[0142]

[0143] Obviously, the closed-form solution of formula (21) is:

[0144]

[0145] In the formula, e i is the i-th element of e;

[0146] In order to satisfy the constraint of ‖u‖0=K at the same time, it is also necessary to i Sort: If e is greater than 1 / 2 i If the number of items is greater than or equal to K, then K items should be taken and the corresponding |u i |=1; if it is greater than 1 / 2 of e i If the number of terms is P and P < K, then the largest KP term should be selected from the remaining terms. The closed-form solution of the above equation is obtained.

[0147] (4) Update w:

[0148] Ignoring the variables in equation (5) that are not related to u, we can obtain the optimization problem:

[0149]

[0150] in,

[0151]

[0152] In the formula, I N is an N×1 unit matrix; the closed-form solution of equation (23) is:

[0153]

[0154] (5) Update λ, μ, γ,

[0155]

[0156] Step 4, repeat steps (1)-(5) in step 3 until the difference between variables x, y, z, u and w before and after the update converges to the expected error, and the optimal array excitation vector w is obtained, which is completed based on the sparse array phase-only synthesis method based on maximizing the directivity coefficient.

[0157] The following is a more specific embodiment:

[0158] The experiment of the present invention is a uniform linear array with 80 array elements, the array element spacing is 1 / 4 wavelength corresponding to the working frequency, and the array sparsity is 50, that is, 50 array elements are enabled; the present invention is used to design a flat-top radiation pattern with a beam center at 0° and a beam width of 30°, wherein the main lobe area is [-15°, 15°], the main lobe ripple is set to 0.3dB, the side lobe area is [-90°, -20°] ∪ [20°, 90°], and the side lobe level is suppressed to 10dB. The optimal excitation vector can be obtained by equations (6) to (29). The radiation pattern calculated in this way can obtain sparse and constant-mode excitation while achieving the desired radiation pattern, and ensure that the array directivity coefficient is maximized. Figure 1 As shown in FIG. 1 , both the method of the present invention and the reference method can synthesize the desired directional pattern according to the template. The method of the present invention can simultaneously obtain sparse and constant-mode excitations. The obtained excitations select 50 array elements according to the given conditions, such as Figure 2 As shown; the reference method can only randomly select sparse excitation positions during the optimization process, and cannot obtain the optimal sparse excitation, and cannot achieve constant modulus excitation, and the excitation amplitude fluctuates greatly, such as Figure 3 shown.

[0159] The present invention realizes the synthesis of the directional pattern of sparse array constant modulus excitation under the constraint of maximizing the directivity coefficient based on the alternating direction multiplier method, which solves the problem of the existing method that the position and constant modulus characteristics of sparse excitation are not constrained. Under the condition of given sparsity, the accurate approximation of the desired directional pattern can be achieved, while ensuring the maximization of the array excitation constant modulus and directivity coefficient.

Claims

1. A sparse array phase-only synthesis method based on maximizing directivity coefficient, characterized in that: The specific process includes the following: Step 1: Establish a model for the sparse array phase-only excitation pattern synthesis algorithm based on maximizing the directivity coefficient: Assume that the uniform linear array consists of N array elements, and the directional pattern array factor is: S(θ)=w H a(i) (1) Where w is the complex excitation vector of the array, including the amplitude and phase of each array element excitation, a(θ) is the steering vector of the array, and S(θ) is the synthesized directional pattern; The directivity coefficient of the array at θ0 is expressed as: In the formula, is a positive Hermitian matrix; B0 is decomposed into B0 = B H B, where The optimization of the directivity coefficient is achieved by constraining the numerator and denominator separately, and the following model is constructed: s.t.G≤|w H a(θ m )|≤δG,θ m ∈Θ ML ,m=1,2,...,M (3b) |w H a(θ s )|≤ρ0G,θ s ∈Θ SL ,s=1,2,...,S (3d) ||w||0=K (3e) |w n |∈{0,1},n=1,...,N (3f) Where G and Q are constraints on the array directivity coefficient, || ||0 is the l0 norm, and a(θ m ) is the main lobe steering vector, a(θ s ) is the sidelobe steering vector, Θ ML Represents the main lobe area of ​​the desired pattern, divided into M angles; Θ SL represents the sidelobe area, which is divided into S angles; δ and ρ0 constrain the mainlobe ripple and sidelobe level according to the desired pattern; N is the number of array elements, w is the N×1 array complex excitation vector, which has K non-zero elements, and w n is the nth element of w; Formula (3a) means to use the array directivity coefficient as the objective function to simultaneously optimize the array formation and the excitation phase; Formula (3b) means to control the main lobe gain to be higher than G and the main lobe ripple to be lower than δG; Formulas (3b) and (3c) constrain the numerator and denominator of the array directivity coefficient; Formula (3d) means to limit the sidelobe level to be lower than ρ0G; Formula (3f) is a 0 / 1 constraint, and Formulas (3e) and (3f) jointly limit the sparsity and constant modulus of the excitation; Step 2, equivalent transformation of the problem: Introduce the following auxiliary variables: stx m =w H a(θ m ),θ m ∈Θ ML (4b) G≤|x m |≤δG,m=1,2,…,M (4c) y s =w H a(θ s ),θ s ∈Θ SL (4d) |y s |≤ρ0G,s=1,2,…,S (4e) z=Bw (4f) u=w (4h) ||u||0=K (4i) |w n |∈{0,1},n=1,...,N (4j) where || ||0 is the l1 norm, x m ,y s and z are the introduced auxiliary variables; Construct the following augmented Lagrangian function: In the formula, Α M and Alpha S are respectively the main lobe steering vector a(θ m ) and the sidelobe steering vector a(θ s ), λ, γ, μ and is the dual variable, ρ1, ρ2, ρ3 and ρ4 are the custom iteration steps, x and y are x in equation (4b) and equation (4d) respectively. m and s The vector formed; Step 3, use the alternating direction multiplier method to solve the problem in step 2: (1) Update x, y and G: Ignore the variables that are not related to x and y in equation (5) and define We can get: The first quadratic term in the objective function of formula (6) can be decomposed into M sub-problems, and the closed-form solution is: Where b m is the mth element of b; The second quadratic term in the objective function of formula (6) can be decomposed into S sub-problems, and the closed-form solution is: In the formula, c s is the sth element of c; Substituting equations (7) and (8) into equation (6) yields a quadratic function only about G, with the coefficients of the quadratic term, linear term, and constant term being: In the formula, P1 represents |b m |≤ the number of items in G, P2 represents |b m |≥δG, P3 represents |c s |≥ρ0The number of items in G; According to the characteristics of quadratic functions, the optimal solution of G is: Substituting equation (12) into equation (7) and equation (8) yields the elements of x and y; (2) Update z and Q: Ignoring the variables in equation (5) that are independent of z and defining d = Bw-γ, we can obtain (3) Update u: Ignore the variables in equation (5) that are not related to u, and define The i-th element u of u i It is expressed as: In the formula, e i is the i-th element of e; In order to satisfy the constraint of ||u||0=K, i Sort: If e is greater than 1 / 2 i If the number of items is greater than or equal to K, then take K items and let the corresponding |u i |=1; if it is greater than 1 / 2 of e i If the number of terms is P and P < K, then select the largest KP term from the remaining terms to obtain a complete closed-form solution for u; (4) Update w: Ignoring the variables in equation (5) that are not related to u, we get the optimization problem: in, In the formula, I N is an N×1 unit matrix; the closed-form solution of equation (15) is: (5) Update λ, μ, γ and In the formula, λ, μ, γ and Indicates the value before updating. and Indicates the updated value; Step 4, repeat steps (1)-(5) in step 3 until the difference between variables x, y, z, u and w before and after the update converges to the expected error, and the optimal array excitation vector w is obtained, which is completed based on the sparse array phase-only synthesis method based on maximizing the directivity coefficient.

Images