Phased array transmitting directional diagram synthesis method for maximizing wide main lobe minimum power gain

Abstract

The invention provides a phased array transmitting directional diagram synthesis method for maximizing the minimum power gain of a wide main lobe, and a phase-only transmitting directional diagram is synthesized and described as a non-convex constraint fraction programming problem with a maximum and minimum fraction objective function and constant modulus constraint by taking the design criterion of obtaining higher power gain in an expected wide main lobe. According to the problem, the minimum power gain in the main lobe region can be maximized, and the power radiation efficiency can be effectively improved on the premise of not losing the synthetic performance of the directional diagram. According to the method, an effective solving algorithm is designed by using an alternating direction multiplier method and a successive convex approximation technology, and a non-convex constraint fractional programming problem is converted into a series of convex problems to be solved. Simulation results show that the method provided by the invention can generate a power gain directional diagram with better performance than a typical SBPS beamforming method.

Description

Technical Field

[0001] The invention belongs to the technical field of array signal processing, and in particular relates to a phased array transmission pattern synthesis method for maximizing wide main lobe minimum power gain. Background Art

[0002] The Power Gain Pattern Synthesis (PGPS) problem is a major concern in array antenna applications. When the array antenna is installed on the top of a mobile platform such as a vehicle, ship or aircraft to receive satellite signals, the random movement of the platform requires the array antenna to have a wide beam main lobe above a certain threshold. Since the complexity of the phase-only array is greatly reduced, the corresponding wide beam main lobe synthesis problem has also become a hot topic in many studies. Previous studies mainly focused on the Shaped Beam Pattern Synthesis (SBPS) problem, directly using the amplitude gain expression as the cost function, and trying to synthesize a transmission pattern similar to the desired beam by optimizing the phase-only weights.

[0003] The power gain of an array antenna in a certain direction is determined by the total radiated power and the array radiated power density, both of which are affected by the weights of the transmit beamformer. However, the total radiated power is not considered based on the SBPS problem, so the optimal power gain diagram cannot be directly obtained. In order to overcome these shortcomings, many literatures have focused on the synthesis of power gain patterns in recent years. "S.Lei, Y.Yang, H.Hu, Z.Zhao, B.Chen, and X.Qiu, "Power gain optimization method for wide-beam array antenna via convex optimization," IEEE Transactions on Antennas and Propagation, vol.67, no.3, pp.1620-1629, 2018." The basic PGPS problem of maximizing the minimum power gain in a wide main lobe was first proposed, but because the desired sidelobe level could not be achieved, "S.Lei, H.Hu, B.Chen, J.Tian, ​​P.Tang, and X.Qiu, "Power gain pattern synthesisfor wide-beam array antenna via linear programming," IEEE Antennas and Wireless Propagation Letters,vol.19,no.12,pp.2300-2304,2020.” and “S.Lei,B.Chen,Z.Lin,W.Yang,J.Tian,and H.Hu,“Sidelobe-level minimization with power gain constraint via a wide-beam antenna array,”IEEE Antennas and WirelessPropagation Letters,vol.22,no.2,pp.422-426,2022.” The constraints of sidelobe power gain are further added to the basic PGPS problem, but they are all based on traditional digital arrays. At present, the PGPS problem of phase-only arrays is still blank.

[0004] Phased array antennas have attracted wide attention for their flexible beam steering capabilities and have been widely used in many fields such as radar, sonar, navigation, and wireless communications. The power gain pattern is the focus of many applications, specifically, the required power gain pattern is formed in a given angle area. In order to improve the defect that the traditional Shaped Beam Pattern Synthesis (SBPS) method cannot obtain the optimal power pattern in the phased array transmission system. Summary of the invention

[0005] The present invention proposes a phased array transmission pattern synthesis method for maximizing wide main lobe minimum power gain.

[0006] The technical solution to achieve the purpose of the present invention is: a phased array transmission pattern synthesis method that maximizes the minimum power gain of a wide main lobe, comprising:

[0007] Establish a model for maximizing the minimum power gain of a wide main lobe in a phase-only transmitting array;

[0008] The Dinkelbach transformation is used to process the non-convex constrained fractional quadratic function maximization and minimization problem, and the fractional quadratic objective function is converted into an integer quadratic objective function.

[0009] Auxiliary variables are introduced to express the mini-max problem equivalently as a constant modulus plus non-convex quadratic constraint minimization problem;

[0010] Based on the framework of alternating direction multiplier method, the constant modulus and non-convex quadratic constraints are decoupled, and the non-convex quadratic constraint subproblem is solved using successive convex approximation technology and standard convex optimization algorithm. Finally, the closed-form optimal solution of the constant modulus constraint subproblem with respect to the phase-only emission weight vector is given.

[0011] Preferably, the wide main lobe minimum power gain maximization problem model under the phase-only transmitting array is established as follows:

[0012]

[0013] st|w n |=1,n=1,...,N

[0014] Where w is the phase-only emission weight vector, w n is the phase-only transmission weight vector of the nth unit, N is the total number of antenna array elements; D(θ m ) is θ m Directivity of the array antenna, Θ ml The main lobe points to the angle area.

[0015] Preferably, θ m The directivity of the array antenna in the direction is specifically expressed as:

[0016] 3.

[0017] In the formula, a(θ m ) is the steering vector, θ m is θ ml The discrete angle in .

[0018] Preferably, the integral quadratic objective function is specifically:

[0019]

[0020] In the formula, in the formula, a(θ m ) is the steering vector, Θ ml is the main lobe pointing angle area, θ m is θ ml The discrete angles in the image are: w is the phase-only transmit weight vector, N is the total number of antenna elements, and w is the phase-only transmit weight vector. n is the phase-only emission weight vector of the nth unit.

[0021] Preferably, an auxiliary variable is introduced to equivalently express the minimax problem as a constant modulus plus non-convex quadratic constraint minimization problem. The specific method is:

[0022] Introducing auxiliary variables The integral quadratic objective function is equivalently expressed as a constant modulus plus a non-convex quadratic constraint minimization objective function, specifically expressed as:

[0023]

[0024] Where M is the main lobe pointing angle area Θ ml The number of sampling points

[0025]

[0026] Introducing auxiliary variables Constructing the augmented Lagrangian function

[0027]

[0028] Among them, β>0 is the penalty factor, r=[r1,r2,...,r N ] T is the scaled dual variable;

[0029] The constant modulus plus non-convex quadratic constraint minimization objective function is transformed into the following form:

[0030]

[0031] Preferably, the specific method for decoupling the constant modulus and non-convex quadratic constraints based on the framework of the alternating direction multiplier method is:

[0032] The transformed constant modulus plus non-convex quadratic constraint minimization objective function is transformed into a sub-problem with non-convex power constraints, specifically:

[0033]

[0034] By neglecting the constant term, The equivalent expression is:

[0035]

[0036] definition Using SCA technology to constrain Perform a second-order Taylor expansion and transform it into the following form:

[0037]

[0038] Among them, w t is the phase-only emission weight vector obtained in the previous iteration;

[0039] Transformed into a series of sub-problems as follows:

[0040]

[0041] for By neglecting the constant term, it can be expressed equivalently as:

[0042]

[0043] Its closed-form solution is:

[0044]

[0045] Preferably, the specific method of solving the non-convex quadratic constraint subproblem by using the successive convex approximation technique and the standard convex optimization algorithm and giving the closed-form optimal solution of the constant modulus constraint subproblem is:

[0046] S1: Solve the following equation using a convex optimization algorithm:

[0047]

[0048] Get w t+1 ,q t+1

[0049] S2: By formula Solution

[0050] S3:

[0051] S4: Determine whether the iteration stop condition is met If not, set t = t + 1 and return to S1. If satisfied, set the last obtained w t+1 As the optimized phase-only emission weight w * .

[0052] Compared with the prior art, the present invention has the following significant advantages: the PGPS-based algorithm of the present invention provides a higher minimum power gain in the main lobe, and the performance loss introduced by the phase constraint is also within an acceptable range. Compared with other representative methods, the present invention has more superior performance in terms of main lobe gain and computational cost. BRIEF DESCRIPTION OF THE DRAWINGS

[0053] Figure 1 This is the power gain pattern generated by the PGPS-based algorithm proposed in the present invention.

[0054] Figure 2 The power gain pattern generated by the Riemann-Newton method based on SBPS.

[0055] Figure 3 The power gain pattern generated by the convex relaxation method based on SBPS.

[0056] Figure 4 The power gain pattern produced by the PGPS-based digital array proposed in “S. Lei, Y. Yang, H. Hu, Z. Zhao, B. Chen, and X. Qiu, “Power gain optimization method for wide-beam array antenna via convex optimization,” IEEE Transactions on Antennas and Propagation, vol. 67, no. 3, pp. 1620-1629, 2018.”

[0057] Figure 5 It is the MPGM obtained by the above four methods in the whole angle area of ​​main lobe scanning. DETAILED DESCRIPTION

[0058] The present invention, namely a method for synthesizing a phased array transmission pattern that maximizes wide main lobe minimum power gain, is further described below with reference to the accompanying drawings and examples.

[0059] A phased array transmit pattern synthesis method that maximizes the minimum power gain of a wide main lobe. Taking the higher minimum power gain in the desired wide main lobe as the design criterion, the transmit pattern synthesis is described as a non-convex constrained optimization problem with an objective function in the form of a maximum and minimum fraction and a constant modulus constraint. This problem can find the optimal solution that maximizes the minimum gain of the main lobe, which can effectively improve power efficiency and reduce overhead without losing beamforming performance. By using ADMM and SCA technology, an effective problem-solving method is designed. Firstly, a model of wide mainlobe minimum power gain maximization problem under phase-only transmitting array is established. Then, the Dinckbach transform is used to process the established non-convex constrained fractional quadratic function minimization problem, and the fractional quadratic objective function is converted into an integer quadratic objective function. Then, auxiliary variables are introduced to express the minimization problem equivalently as a constant modulus plus non-convex quadratic constrained minimization problem. Subsequently, the constant modulus and non-convex quadratic constraints are decoupled based on the alternating direction method of multipliers (ADMM) framework, and the non-convex quadratic constraint sub-problem is solved using the successive convex approximation (SCA) technique and the standard convex optimization algorithm, and the closed-form optimal solution of the constant modulus constraint sub-problem is given.

[0060] The specific implementation steps are as follows:

[0061] Step 1: For a phased array radar system, a flat-top beam with maximum main lobe gain is generated. The narrowband and far-field transmission of a linear array composed of N antenna elements, under the action of the transmission weight w, the far-field radiation pattern in the θ direction can be described as:

[0062] P(θ)=|a H (θ)w| (1)

[0063] in, is the steering vector, θ is the azimuth radiation direction, θ∈[-90°,90°]. n (θ) is the antenna pattern of the nth unit. When the unit is isotropic, g n (θ)=1. When n=1,2,…,N, r n is the distance from the nth unit to the reference unit. λ is the wavelength, is the phase-only emission weight vector, all elements of which satisfy |w n |=p,n=1,2,…,N,where p is determined by the preset transmit power of each element. To simplify the description, p=1. Under the action of w, the directivity D(θ) of the array antenna in the θ direction can be expressed as:

[0064]

[0065] Among them, A θ =a(θ)a H (θ),

[0066] The present invention takes obtaining the maximum minimum power gain in a wide main lobe as the design criterion and describes the transmission pattern synthesis as a non-convex constrained optimization problem with a maximum and minimum fraction objective function and a constant modulus constraint:

[0067]

[0068] Among them, Θ ml is the main lobe pointing angle area, θ m is θ ml The discrete angle in. According to formula (2), problem (3) can be equivalently expressed as:

[0069]

[0070] Step 2: Problem (4) can be classified as generalized fractional programming. Using the Dinkelbach transformation, the fractional quadratic objective function is transformed into an equivalent integral form, and the equivalent expression of problem (4) is obtained as follows:

[0071]

[0072] in,

[0073] Step 3: Introduce auxiliary variables The objective function of problem (5) is simplified, and the mini-max problem is equivalently expressed as a constant modulus plus non-convex quadratic constraint minimization problem:

[0074]

[0075] Introducing auxiliary variables Constructing the augmented Lagrangian function

[0076]

[0077] Among them, β>0 is the penalty factor, r=[r1,r2,...,r N ] T is the scaled dual variable. Then, problem (6) is transformed into the following form:

[0078]

[0079] Step 4: Use an ADMM-based method to decompose problem (8) into several easily solvable sub-problems, and solve these sub-problems by alternating iterations to achieve the phased array transmit beam design that maximizes the minimum mainlobe gain. For the sub-problems with non-convex power constraints, the SCA technology is used to transform the non-convex constraints into convex constraints, and then the standard convex optimization algorithm is used to solve them; for the sub-problems with non-convex constant modulus constraints, the closed-form solution is directly derived.

[0080] Under the ADMM framework, the following iterative process is used to find The optimal solution is:

[0081]

[0082]

[0083]

[0084] The original optimization problem (6) is transformed into solving subproblems (9), (10) and (11).

[0085] By ignoring the constant term, subproblem (9) can be expressed as

[0086]

[0087] definition Since the positive definiteness of R cannot be guaranteed, the constraints in problem (12) may be non-convex, so it is difficult to solve them using existing standard convex optimization methods. To this end, the present invention uses SCA technology to perform a second-order Taylor expansion on the constraints and converts them into the following form:

[0088]

[0089] Among them, w t is w obtained in the previous iteration. Therefore, problem (12) can be transformed into the following series of sub-problems:

[0090]

[0091] Problem (14) is a standard convex quadratic constrained quadratic programming problem, which can be solved directly using existing convex optimization algorithms, such as the interior point method.

[0092] For subproblem (10), by ignoring the constant term, it can be equivalently expressed as:

[0093]

[0094] Its closed-form solution is:

[0095]

[0096] In summary, the specific solution steps are summarized as Algorithm 1:

[0097] 1) Input parameters:

[0098] 2) Initialization setting: t = 0

[0099] 3) Start iteration:

[0100] i. Solve problem (14) using the convex optimization algorithm to obtain: w t+1 ,q t+1

[0101] ii. By formula (16), we can obtain:

[0102] iii. Update by formula (11): r t+1

[0103] iv.t=t+1

[0104] 4) If Then stop the loop

[0105] 5) Output: Optimized phase-only emission weight w *

[0106] Example

[0107] The specific implementation scheme of the phased array transmission pattern synthesis method for maximizing the wide main lobe minimum power gain of the present invention is further explained through Matlab simulation.

[0108] 1) Simulation system parameter settings

[0109] The simulation is based on N = 41 interval non-uniform linear arrays, and the specific array element position d = [-10, -9.6065, -8.8098, -8.2995, -7.8973, -7.3497, -6.8494, -6.5302, -5.6299, -5.3749, -5, -4.6065, -3.8098, -3.2995, -2.8973, -2.3497, -1.8494, The spatial angle domain [0°,180°] is divided into 181 angular grids with an interval of 1°. The main lobe beamwidth is 20°, and ξ=10 -3 and T max = 1000. The dual and auxiliary variables are initialized randomly.

[0110] 2) Beam diagram drawing

[0111] In order to intuitively display the effect of the phased array transmission pattern synthesis that maximizes the minimum power gain of the wide main lobe, the weight vector solved by Algorithm 1 is used to weighted combine the signal, and finally the beam pattern under different parameters is drawn. The horizontal axis of the beam pattern represents the angle range of [0°, 180°]. The amplitude value after the weighted combination of the signal is modulo and normalized to the maximum value, so the unit of the vertical axis of the beam pattern is dB.

[0112] 3) Metrics

[0113] In the present invention, the effect of the final beam synthesis needs to be measured. In addition to drawing the beam pattern, the minimum mainlobe power gain (MPGM) is also used. After using the beam pattern synthesis algorithm to obtain the corresponding beamforming weight vector, substitute it into the solution formula,

[0114]

[0115] Under the same initial conditions, the larger the minimum main lobe gain, the better the beamforming effect.

[0116] 4) Result analysis

[0117] The present invention has carried out 5 example simulations in total. Figure 1The power patterns generated by the proposed method, the Riemannian Newton Method (RNM) based on SBPS, the Convex Relaxation (CR) method based on SBPS, and the digital array beamforming algorithm (Digital) based on PGPS proposed in the literature [1] under the same parameter settings when the main lobe has different angle ranges. The main lobe center pointing angles are 50°, 70° and 90° respectively, and the main lobe ranges are [40°, 60°], [60°, 80°] and [80°, 100°] respectively. To ensure the fairness of the method comparison, all sidelobe constraints in the SBPS-based and PGPS-based algorithms are removed, and the rest remain unchanged. In order to further explore the scanning performance, we consider the main lobe beam scanning from [0°, 20°] to [160°, 180°], that is, the central main lobe direction changes from 10° to 170° at intervals of 20°. Figure 2 It is the change curve of MPGM of each method during the process of main lobe beam scanning the entire spatial angle.

[0118] Figure 1 The beam patterns generated by the four methods are given when the main lobe center is pointed at 50°, 70° and 90° respectively. Compared with the basic PGPS problem using a digital array, the main lobe gain loss of the algorithm proposed in the present invention is about 1dBi, which is within an acceptable range, while the introduction of phase-only constraints will inevitably lead to an increase in sidelobe power gain. For the phased array transmission pattern synthesis problem with phase-only constraints, the algorithm proposed in the present invention has the highest main lobe power gain because it directly sets the power gain function as the cost function. At the same time, the sidelobe power gain of the proposed algorithm is also significantly lower than that of the two SBPS-based phased array beamforming algorithms RNM and CR. In all cases, compared with the SBPS-based algorithm, the MPGM of the PGPS-based algorithm proposed in the present invention is improved by 0.62 to 3.38dBi. For example, when the main lobe region is [60°, 80°], the MPGM of the PGPS-based algorithm is 2.86dB higher than that of RNM. Therefore, the PGPS-based algorithm proposed in the present invention has the best performance in both main lobe and sidelobe power gain.

[0119] Figure 2The MPGM variation curves obtained by scanning the entire angle area of ​​the main lobe are plotted. The MPGM obtained by these algorithms show nearly symmetrical behavior around 90°, with the lowest value occurring at 90° and the highest value occurring at both ends. Compared with the two SBPS-based algorithms, the algorithm proposed in the present invention obtains the highest MPGM in all directions. As the main lobe constraint gradually increases from CR to RNM, the MPGM gradually decreases. When the main lobe center is pointed at 70° to 110°, the MPGM of the CR method is significantly higher than that of the RNM method. Due to the introduction of phase-only constraints, the MPGM of the algorithm proposed in the present invention has a certain performance loss compared with the unconstrained digital array transmission pattern synthesis algorithm based on PGPS.

[0120] In summary, compared with the existing typical SBPS-based algorithm, the PGPS-based algorithm proposed in the present invention provides a higher minimum power gain in the main lobe, and the performance loss introduced by the phase constraint is also within an acceptable range. It can not only obtain robust beamforming performance, but also reduce the hardware cost and computational complexity of the system, and has a high application value.

Claims

1. A phased array transmission pattern synthesis method for maximizing wide main lobe minimum power gain, characterized in that: include: Establish a model for maximizing the minimum power gain of a wide main lobe in a phase-only transmitting array; The Dinkelbach transformation is used to process the non-convex constrained fractional quadratic function maximization and minimization problem, and the fractional quadratic objective function is converted into an integer quadratic objective function. Auxiliary variables are introduced to express the mini-max problem equivalently as a constant modulus plus non-convex quadratic constraint minimization problem; Based on the framework of alternating direction multiplier method, the constant modulus and non-convex quadratic constraints are decoupled, and the non-convex quadratic constraint subproblem is solved using successive convex approximation technology and standard convex optimization algorithm. Finally, the closed-form optimal solution of the constant modulus constraint subproblem with respect to the phase-only emission weight vector is given.

2. The phased array transmission pattern synthesis method for maximizing wide main lobe minimum power gain according to claim 1, characterized in that: The specific model for maximizing the minimum power gain of the wide main lobe under the phase-only transmitting array is established as follows: Where w is the phase-only emission weight vector, w n is the phase-only transmission weight vector of the nth unit, N is the total number of antenna array elements; D(θ m ) is θ m Directivity of the array antenna, Θ ml The main lobe points to the angle area.

3. The phased array transmission pattern synthesis method for maximizing wide main lobe minimum power gain according to claim 2, characterized in that: θ m The directivity of the array antenna in the direction is specifically expressed as: 3、 In the formula, a(θ m ) is the steering vector, θ m is θ ml The discrete angle in .

4. The phased array transmission pattern synthesis method for maximizing wide main lobe minimum power gain according to claim 1, characterized in that: The integral quadratic objective function is specifically: In the formula, in the formula, a(θ m ) is the steering vector, Θ ml is the main lobe pointing angle area, θ m is θ ml The discrete angles in the image are: w is the phase-only transmit weight vector, N is the total number of antenna elements, and w is the phase-only transmit weight vector. n is the phase-only emission weight vector of the nth unit.

5. The method for synthesizing a phased array transmission pattern with a maximized wide main lobe minimum power gain according to claim 4, characterized in that: Auxiliary variables can be used to represent the mini-max problem as a constant modulus plus non-convex quadratic constraint minimization problem. The specific method is: Introducing auxiliary variables The integral quadratic objective function is equivalently expressed as a constant modulus plus a non-convex quadratic constraint minimization objective function, specifically expressed as: Where M is the main lobe pointing angle area Θ ml The number of sampling points Introducing auxiliary variables Constructing the augmented Lagrangian function Among them, β>0 is the penalty factor, r=[r1,r2,...,r N ] T is the scaled dual variable; The constant modulus plus non-convex quadratic constraint minimization objective function is transformed into the following form:

6. The method for synthesizing a phased array transmission pattern for maximizing wide main lobe minimum power gain according to claim 5, characterized in that: The specific method for decoupling constant modulus and non-convex quadratic constraints based on the framework of alternating direction multiplier method is as follows: The transformed constant modulus plus non-convex quadratic constraint minimization objective function is transformed into a sub-problem with non-convex power constraints, specifically: By neglecting the constant term, The equivalent expression is: definition Using SCA technology to constrain Perform a second-order Taylor expansion and transform it into the following form: Among them, w t is the phase-only emission weight vector obtained in the previous iteration; Transformed into a series of sub-problems as follows: for By neglecting the constant term, it can be expressed equivalently as: Its closed-form solution is:

7. The phased array transmission pattern synthesis method for maximizing wide main lobe minimum power gain according to claim 6, characterized in that: The specific method of solving the non-convex quadratic constraint subproblem by using the successive convex approximation technique and the standard convex optimization algorithm and giving the closed-form optimal solution of the constant modulus constraint subproblem is as follows: S1: Solve the following equation using a convex optimization algorithm: Get w t+1 ,q t+1 S2: By formula Solution S3： S4: Determine whether the iteration stop condition is met Let t = t + 1. If it is not satisfied, return to S1. If it is satisfied, then the last obtained w t+1 As the optimized phase-only emission weight w * .

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