Sparse array phase-only excitation pattern synthesis method without template constraint

Abstract

The invention discloses a sparse array phase-only excitation pattern synthesis method without template constraint, and relates to the field of array signal processing. The method comprises the following steps: firstly, establishing a model of a sparse array phase-only excitation pattern comprehensive algorithm without template constraint, and on the premise of giving an array sparse rate, taking a main lobe level minimum value and a side lobe level maximum value as objective functions and simultaneously realizing optimization of array element selection and excitation phase; using 10 norm and 0 / 1 constraint to limit sparsity and constant modulus characteristic of excitation; the main lobe minimum value and the side lobe maximum value are used for limiting the response of the array directional diagram, and main lobe ripples are controlled; and then solving the model by using an alternating direction multiplier method to obtain a sparse and constant modulus excitation vector. According to the method, the problem that the sparse excitation position and the main lobe ripple are not optimized in an existing method is solved, template-free phase-only excitation pattern synthesis of the sparse array can be realized under the condition of given sparseness, and control on the main lobe ripple is realized.

Description

Technical Field

[0001] The invention relates to the field of array signal processing, in particular to a sparse array phase-only excitation pattern synthesis method without template constraints. Background Art

[0002] In the pattern synthesis technology of array antennas, the pattern synthesis method of sparse array phase-only excitation can select the array element of the transmitting signal according to the given array sparsity rate while ensuring the constant excitation amplitude, and obtain the desired pattern by only changing the phase of the excitation. This type of method can simplify the array structure, reduce the mutual coupling between array elements, and improve the energy utilization efficiency. In the study of pattern synthesis methods, a template is usually given to constrain the main lobe level, main lobe ripple and side lobe level. In the study of phase-only excitation pattern synthesis methods, since the degree of freedom of the excitation vector is limited, when the constraint conditions are set unreasonably (the main lobe / side lobe level is set too high / low, the main lobe ripple is set too small), the pattern synthesis for a given template may be difficult to achieve, so it is necessary to study a pattern synthesis method without template constraints. At present, there are few studies on the optimization of pattern synthesis methods without template constraints, and the typical methods are as follows: J. Liang et al. proposed a pattern synthesis method without template constraints. This method uses the ratio of the minimum main lobe level to the maximum side lobe level as the objective function for design, and uses the constant amplitude as a constraint to restrict the excitation vector. The Lagrange multiplier method is used to transform the non-convex optimization problem into a convex optimization problem for solution. However, this method does not optimize the main lobe ripples and sparse excitation positions of the pattern, and the main lobe of the generated pattern fluctuates greatly. The excitation position can only be randomly selected, and the optimal sparse excitation cannot be obtained. The main problem of the above method is that the main lobe ripples and sparse excitation positions of the pattern are not optimized, and the main lobe of the generated pattern fluctuates greatly and the optimal sparse excitation cannot be obtained. Summary of the invention

[0003] In view of this, the present invention proposes a method for synthesizing sparse array phase-only excitation patterns without template constraints. The method realizes the pattern synthesis of sparse array constant modulus excitation based on the alternating direction multiplier method, solves the problem that the existing method does not optimize the sparse excitation position, and can realize the template-free phase-only excitation pattern synthesis of sparse array under the condition of given sparsity, and realizes the control of main lobe ripple.

[0004] In order to achieve the above object, the technical solution adopted by the present invention is as follows:

[0005] A method for synthesizing phase-only excitation patterns of sparse arrays without template constraints comprises the following steps:

[0006] Step 1: Establish a model for the sparse array phase-only excitation pattern synthesis algorithm without template constraints:

[0007]

[0008] Among them, G and Q are constraints on the minimum value of the main lobe and the maximum value of the side lobe, respectively, w is the complex excitation vector of the array, ||·|| 0 for l 0 norm, a(θ m ) is the main lobe steering vector of the array, Θ ML represents the main lobe area, a(θ s ) is the sidelobe steering vector of the array, Θ SL represents the sidelobe area, δ is the constraint coefficient of the main lobe ripple; K represents the sparsity constraint of the array complex excitation vector, 0<K≤N, and K is a positive integer, and N is the number of array elements;

[0009] Step 2: Introduce the following auxiliary variable x m ,y s With z n :

[0010] x m =w H a(θ m ),;

[0011] y s =w H a(θ s ),;

[0012] Z n =w n ;

[0013] And construct the augmented Lagrangian function:

[0014]

[0015] Where x is x m The vector formed by y is y s The vector formed by z is z n The vector composed of M , Α S are respectively the main lobe steering vector a(θ m ) and the sidelobe steering vector a(θ s ), λ, γ, μ are dual variables, ρ 1 , ρ 2 , ρ 3 is a custom iteration step size;

[0016] Step 3: Set the variable t indicating the number of iterations, let t = 0, G 0 =G,Q 0 =Q,w 0 =w,λ0 =λ,γ 0 =γ,μ 0 =μ;

[0017] Step 4: Solve the above problem using the alternating direction multiplier method. Specifically, perform the following steps:

[0018] Step 401, update G and x m :

[0019]

[0020] Among them, G t+1 represents the iteration result of G in the tth iteration, Indicates x in the tth iteration m The iterative result is represents the dual variable λ in the tth iteration t The mth element of 1 Indicates to use | b m |≤G t The set of m that holds true, p 1 P 1 The elements in Denotes the set P 1 The number of elements in P 2 Indicates that |b m |≥δG t The set of m that holds true, p 2 P 2 The elements in Denotes the set P 2 The number of elements in ;

[0021] Step 402, update Q and y s :

[0022]

[0023]

[0024] Among them, Q t+1 represents the iteration result of Q in the tth iteration, Indicates that y in the tth iteration s The iterative result is represents the dual variable μ in the tth iteration t The sth element of 3 Indicates that |c s |≥Q t The set of s that holds true, p 3 P 3 The elements in Denotes the set P 3 The number of elements in ;

[0025] Step 403, update z n :

[0026]

[0027] in, Indicates the z in the tth iteration n The iterative result is represents the dual variable γ in the tth iteration t The nth element of 4 Indicates d n The set of n corresponding to the largest K items in ;

[0028] Step 404, update w:

[0029]

[0030]

[0031] in, represents the pseudo-inverse of the matrix, I N is the N-order identity matrix, x t+1 ,y t+1 With z t+1 They are and The vector formed;

[0032] Step 405, update λ, μ, γ:

[0033]

[0034] γ t+1 =γ t +z t+1 -w t+1 ;

[0035] Step 406, let t = t + 1, return to execute steps 401 to 405, until the differences before and after the update of vectors x, y, z and w converge to the expected error, at which point w is the final array excitation vector, completing the sparse array phase-only excitation pattern synthesis method without template constraints.

[0036] Due to the adoption of the above technical solution, the beneficial effects of the present invention compared with the prior art are:

[0037] Under the premise of a given array sparsity rate, the present invention takes the minimum main lobe level and the maximum side lobe level as the objective function to optimize the array element selection and the excitation phase at the same time; using l 0The norm and 0 / 1 constraints limit the sparsity and constant mode characteristics of the excitation; the minimum mainlobe and maximum sidelobe are used to limit the array pattern response, and the mainlobe ripple is controlled; the problem that the existing methods do not optimize the sparse excitation position and mainlobe ripple is solved, and the template-free phase-only excitation pattern synthesis of sparse arrays can be achieved under the condition of given sparsity. BRIEF DESCRIPTION OF THE DRAWINGS

[0038] Figure 1 The figure is a comparison between the method proposed by J. Liang in the background technology and the flat-top pattern obtained by the present invention.

[0039] Figure 2 It is the normalized amplitude of the excitation corresponding to the method proposed by J. Liang in the background technology.

[0040] Figure 3 is the normalized amplitude of the excitation corresponding to the present invention. DETAILED DESCRIPTION

[0041] The present invention will be further described below in conjunction with the accompanying drawings and specific embodiments.

[0042] The technical solution adopted by the sparse array phase-only excitation pattern synthesis algorithm without template constraints in this embodiment is to first establish a model of the sparse array phase-only excitation pattern synthesis algorithm without template constraints. Under the premise of a given array sparsity rate, the model uses the minimum main lobe level and the maximum side lobe level as objective functions to simultaneously optimize the array element selection and the excitation phase; using l 0 The norm and 0 / 1 constraints limit the sparsity and constant modulus characteristics of the excitation; the minimum main lobe and the maximum side lobe are used to limit the array pattern response and control the main lobe ripple; since the model is a highly non-convex model and difficult to solve directly, the alternating direction of multipliers method (ADMM) is used to solve the model and obtain a sparse and constant modulus excitation vector. The details are as follows:

[0043] Step 1: Establish a model for the sparse array phase-only excitation pattern synthesis algorithm without template constraints:

[0044] Assume there is a uniform linear array consisting of N array elements, and its directional pattern factor is

[0045] S(θ)=w H a(θ) (1)

[0046] 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.

[0047] Under the premise of a given sparsity rate, a sparse array can be realized by selecting the elements of a uniform linear array. 0 The array pattern synthesis can be realized by constraining the main lobe and side lobe, so the following model can be constructed:

[0048]

[0049] stG≤|w H a(θ m )|≤δG,θ m ∈Θ ML ,m=1,2,...,M (2b)

[0050] |w H a(θ s )|≤Q,θ s ∈Θ SL ,s=1,2,...,S (2c)

[0051] ||w|| 0 =K (2d)

[0052] |w n |∈{0,1},n=1,2,...,N (2e)

[0053] Among them, G and Q are constraints on the minimum value of the main lobe and the maximum value of the side lobe, respectively, w is the complex excitation vector of the array, ||·|| 0 for l 0 norm, a(θ) is the steering vector of the array, Θ ML represents the main lobe area, Θ SL Represents the sidelobe area, δ constrains the mainlobe ripple, and its typical value is 0.6dB; K represents the sparsity constraint of the array complex excitation vector, 0<K≤N, and K is a positive integer, and N is the number of array elements. The meaning of formula (2a) is to minimize the difference between the maximum level of the sidelobe and the minimum level of the mainlobe, that is, to minimize the sidelobe level; the meaning of formula (2b) is to control the mainlobe gain to be higher than G and the mainlobe ripple to be lower than δG; the meaning of formula (2c) is to limit the sidelobe level to be lower than Q; formula (2e) is a 0 / 1 constraint, and formulas (2d) and (2e) jointly limit the sparsity and constant modulus of the excitation. Therefore, formula (2) can optimize the array element selection and excitation phase while satisfying the constraint of minimizing the array sidelobe level, and realize the synthesis of sparse array phase-only excitation pattern without template constraints.

[0054] Step 2, equivalent transformation of the problem:

[0055] Formula (2) 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:

[0056]

[0057] stx m =w H a(θ m ),θ m ∈Θ ML (3b)

[0058] G≤|x m |≤δG,m=1,2,...,M (3c)

[0059] y s =w H a(θ s ),θ s ∈Θ SL (3d)

[0060] |y s |≤Q,s=1,2,...,S (3e)

[0061] z=w(3f)

[0062] ||z|| 0 =K (3g)

[0063] |z n |∈{0,1},N=1,...,N (3h)

[0064] Among them, ||·|| 0 for l 0 Norm, x m ,y s , z n , z is the introduced auxiliary variable. The following augmented Lagrangian function can be further constructed:

[0065]

[0066] Among them, Α M , Α S are respectively the main lobe steering vector a(θ m ) and the sidelobe steering vector a(θ s ), λ, γ, μ are dual variables, ρ 1 , ρ 2 , ρ 3 A custom iteration step size.

[0067] Step 3: Solve the above problem using the alternating direction multiplier method

[0068] (1) Update x, G:

[0069] x is x in formula (3b)m Since G is coupled with x in both the objective function of equation (3a) and the constraint condition of equation (3c), joint optimization must be performed. Ignore the variables that are not related to x and y in equation (4) and define Available

[0070]

[0071] The first quadratic term in the objective function of formula (5) can be decomposed into M sub-problems:

[0072]

[0073] The closed-form solution is

[0074]

[0075] Substituting equation (7) into equation (5) yields a quadratic function only about G:

[0076] αG 2 +βG+χ(8)where,

[0077]

[0078] Among them, P 1 Indicates |b m |≤The number of terms in G, P 2 Indicates |b m |≥δG. According to the characteristics of quadratic functions, the optimal solution of G is

[0079]

[0080] Substituting equation (12) into equation (7) yields x.

[0081] (2) Update y, Q:

[0082] Since Q is coupled with variable z in both the objective function of (3a) and the constraint of (3e), joint optimization is necessary. Ignore the variables unrelated to y in (4) and define Available

[0083]

[0084] The quadratic term in formula (13) can be decomposed into S sub-problems

[0085]

[0086] Its closed-form solution is

[0087]

[0088] Substituting equation (15) into equation (13) yields the quadratic function only about Q:

[0089] αQ 2 +βQ+χ(16)where,

[0090]

[0091] Where P represents |c s |≥Q. Further, the optimal solution of Q is

[0092]

[0093] Substituting equation (18) into equation (15) yields y.

[0094] (3) Update z:

[0095] Ignoring the variables in equation (4) that are independent of z and defining d = w - γ, we can obtain

[0096]

[0097] Formula (19) can be decomposed into N sub-problems

[0098]

[0099] Obviously, the closed-form solution of equation (20) is

[0100]

[0101] In order to simultaneously satisfy ||z|| 0 =K constraint, we also need to i Sort. If d is greater than or equal to 1 / 2 i If the number of items is greater than or equal to K, the larger K items should be taken as z i phase angle, and let the corresponding |z i |=1; if d is greater than 1 / 2 i The number of items is P 0 And P 0 <K, then it should be from |d i |<1 / 2 Select the larger KP from the remaining items 0 The closed-form solution of formula (19) can be expressed as:

[0102]

[0103] (4) Update w:

[0104] Ignoring the variables in equation (5) that are not related to w, we can obtain the optimization problem

[0105]

[0106] in,

[0107]

[0108] The closed-form solution of equation (23) is

[0109]

[0110] (5) Update λ, μ, γ:

[0111]

[0112] γ t+1 =γ t +z t+1 -w t+1 (27)

[0113] Repeat step 3 until the differences before and after the update of vectors x, y, z, and w converge to the expected error. At this time, w is the final array excitation vector, and the sparse array phase-only excitation pattern synthesis method without template constraints is completed.

[0114] The following is a more specific embodiment:

[0115] The experiment in this embodiment is a uniform linear array of 80 elements, the element spacing is 1 / 4 wavelength corresponding to the operating frequency, and 50 elements are enabled in the array; the flat-top radiation pattern with a beam center of 0° and a beam width of 30° is designed using this embodiment, wherein the main lobe area is [-15°, 15°], the main lobe level fluctuation range is set to 0.6dB, and the side lobe area is [-90°, -20°] ∪ [20°, 90°]. The optimal excitation vector that meets the sparsity requirements can be obtained through equations (5) to (27). The radiation pattern calculated in this way can obtain both sparse and constant-mode excitation while achieving the desired radiation pattern. Figure 1 As shown, both the method of this embodiment and the reference method can synthesize the desired directional pattern according to the template, but the reference method does not optimize the main lobe ripple, and the main lobe amplitude fluctuates greatly. The main lobe level fluctuation range in the directional pattern synthesis result of the method of this embodiment does not exceed 0.6dB, and the main lobe level fluctuation range in the reference method synthesis result is approximately 13.5dB at most. At the same time, the reference method does not optimize the sparse excitation position and cannot obtain the optimal sparse excitation. The main-to-side ratio in the directional pattern synthesis result is approximately 4dB higher than that of the method of this embodiment. The method of this embodiment can simultaneously obtain sparse and constant-mode excitation, and the obtained excitation selects 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, such as Figure 3 shown.

[0116] In summary, the present invention realizes the directional pattern synthesis of sparse array constant mode excitation based on the alternating direction multiplier method, which solves the problem that the existing method does not optimize the sparse excitation position. Under the condition of given sparsity, the template-free phase-only excitation directional pattern synthesis of sparse array can be realized, and the control of the main lobe ripple is realized.

[0117] Those skilled in the art will appreciate that the embodiments described are intended to help readers understand the principles of the present invention, and should be understood that the scope of protection of the present invention is not limited to the embodiments described. For those skilled in the art, the present invention may have various changes and variations. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention should be included in the scope of the claims of the present invention.

Claims

1. A method for synthesizing phase-only excitation patterns of sparse arrays without template constraints, characterized in that: The following steps are involved: Step 1: Establish a model for the sparse array phase-only excitation pattern synthesis algorithm without template constraints: stG≤|w H a(θ m )|≤δG,θ m ∈Θ ML ; |w H a(θ s )|≤Q,θ s ∈Θ SL ,; ||w||0=K; |w n |∈{0,1},n=1,2,......,N; Among them, G and Q are the constraints on the minimum value of the main lobe and the maximum value of the side lobe, respectively, w is the complex excitation vector of the array, ||·||0 is the l0 norm, and a(θ m ) is the main lobe steering vector of the array, Θ ML represents the main lobe area, a(θ s ) is the sidelobe steering vector of the array, Θ SL represents the sidelobe area, δ is the constraint coefficient of the main lobe ripple; K represents the sparsity constraint of the array complex excitation vector, 0<K≤N, and K is a positive integer, and N is the number of array elements; Step 2: Introduce the following auxiliary variable x m ,y s With z n : x m =w H a(θ m ),; y s =w H a(θ s ),; With n =in n ; And construct the augmented Lagrangian function: Where x is x m The vector formed by y is y s The vector formed by z is z n The vector composed of M , Α S are respectively the main lobe steering vector a(θ m ) and the sidelobe steering vector a(θ s ), λ, γ, μ are dual variables, ρ1, ρ2, ρ3 are user-defined iteration steps; Step 3: Set the variable t indicating the number of iterations, let t = 0, G 0 =G,Q 0 =Q,w 0 =w,λ 0 =λ,γ 0 =γ,μ 0 =μ; Step 4: Solve the above problem using the alternating direction multiplier method. Specifically, perform the following steps: Step 401, update G and x m : Among them, G t+1 represents the iteration result of G in the tth iteration, Indicates x in the tth iteration m The iterative result is represents the dual variable λ in the tth iteration t The mth element of P1 means that |b m |≤G t The set of m that holds true, p1 is the element in P1, represents the number of elements in set P1; P2 represents the number of elements in set P1; m |≥δG t The set of m that holds true, p2 is the element in P2, Represents the number of elements in set P2; Step 402, update Q and y s : Among them, Q t+1 represents the iteration result of Q in the tth iteration, Indicates that y in the tth iteration s The iterative result is represents the dual variable μ in the tth iteration t The sth element of P3 means that |c s |≥Q t The set of s that holds true, p3 is the element in P3, Represents the number of elements in set P3; Step 403, update z n : make in, Indicates the z in the tth iteration n The iterative result is represents the dual variable γ in the tth iteration t The nth element of n The set of n corresponding to the largest K items in ; Step 404, update w: in, represents the pseudo-inverse of the matrix, I N is the N-order identity matrix, x t+1 ,y t+1 With z t+1 They are and The vector formed; Step 405, update λ, μ, γ: c t+1 =c t +z t+1 -w t+1 ; Step 406, let t = t + 1, return to execute steps 401 to 405, until the differences before and after the update of vectors x, y, z and w converge to the expected error, at which point w is the final array excitation vector, completing the sparse array phase-only excitation pattern synthesis method without template constraints.

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