Synthesis method of wide mainlobe transmission power pattern of phase-only array based on rank-one matrix optimization

Abstract

The present invention discloses a method for synthesizing a wide mainlobe transmission power pattern of a phase-only array based on rank-one matrix optimization, comprising: constructing a wide beam power pattern synthesis problem model of minimum power gain in a maximum mainlobe region, wherein the objective function in the wide beam power pattern synthesis problem model is a maximum minimization fractional objective function; introducing a first auxiliary variable, and using generalized fractional programming to convert the maximum minimization fractional objective function form into a maximum minimization integer objective function form; introducing a second auxiliary variable to equivalently express the maximum minimization problem in a minimization form, and solving it using an equivalent rank-one method, a continuous convex approximation and a penalty method to obtain an optimal phase-only transmission weight.

Description

Technical Field

[0001] The invention belongs to the technical field of array signal processing and radar signal processing, and specifically is a method for synthesizing a wide main lobe transmission power pattern of a phase-only array based on rank-one matrix optimization. Background Art

[0002] When using an array antenna mounted on top of a mobile platform to receive satellite signals, the random motion of the platform requires the array antenna to have a wide beam main lobe above a certain gain threshold. When a phased array radar uses phase-only transmission to obtain maximum transmit power, a wide main lobe expands the radar's monitoring range and increases the radar's data rate. Therefore, wide beam pattern synthesis has important engineering application value.

[0003] The measurement standard of traditional beamforming pattern synthesis (SBPS) is the radiation pattern, while the power pattern synthesis (PGPS) mentioned in the present invention directly uses power gain as the measurement standard, which can form high power gain in a wide main lobe area, which is beneficial to improving the performance of satellite communication receivers and radar target detection. On the other hand, phase adjustment in the phase-only transmit array can be achieved with only a simple phase shifter, which significantly reduces hardware complexity, power consumption and cost, and is conducive to building low-cost communication and radar systems.

[0004] The prior art uses convex optimization methods to solve the problem of wide beam power pattern synthesis under digital arrays. At the same time, different solutions are designed for the SBPS wide beam problem under phase-only arrays. Specifically, the main lobe gain maximization problem is studied, and the power sum of the side lobes is added to the objective function as a penalty term to suppress the side lobe level, and then the convex relaxation (CR) method is used to solve the corresponding problem. On the basis of maximizing the main lobe gain, precise control of the side lobe level is achieved. However, the research on the power pattern synthesis problem of phase-only arrays is still blank, so that the power gain cannot be directly optimized, resulting in a certain loss in the performance of the power gain. Summary of the invention

[0005] The purpose of the present invention is to propose a method for synthesizing wide mainlobe transmission power patterns of phase-only arrays based on rank-one matrix optimization.

[0006] The technical solution adopted by the present invention is: a method for synthesizing a wide main lobe transmission power pattern of a phase-only array based on rank-one matrix optimization, comprising:

[0007] Step 1: constructing a wide beam power pattern synthesis problem model with minimum power gain in the maximum main lobe region, wherein the objective function in the wide beam power pattern synthesis problem model is a maximum minimization fractional objective function;

[0008] Step 2: Introduce the first auxiliary variable and use generalized fractional programming to transform the maximum-minimization fractional objective function form into the maximum-minimization integral objective function form;

[0009] Step 3: Introduce the second auxiliary variable to equivalently express the maximum minimization problem in a minimization form, and use the equivalent rank-one method, continuous convex approximation and penalty method to solve it and obtain the optimal phase-only emission weight.

[0010] Preferably, the wide beam power pattern synthesis problem model is specifically:

[0011] ,

[0012] In the formula, is the main lobe area, for The discrete directions in , is the discrete angle of the main lobe area, for The steering vector at is the phase-only emission weight vector, for Middle elements, , For the entire angle space, Angle The steering vector at .

[0013] Preferably, an auxiliary variable is introduced , use generalized fractional programming to transform the maximum-minimization fractional objective function form into the maximum-minimization integral objective function form, specifically:

[0014]

[0015] In the formula, .

[0016] Preferably, the second auxiliary variable is introduced to equivalently express the maximum minimization problem in a minimization form, and is solved using an equivalent rank-one method, a continuous convex approximation and a penalty method. The specific method for obtaining the optimal phase-only emission weight is:

[0017] Step 3.1: Randomly initialize the emission weights , ,definition ;

[0018] Step 3.2, The optimization problem of this iteration is expressed as:

[0019]

[0020] Step 3.3: Update auxiliary variables The expression is:

[0021]

[0022] Step 3.4, define auxiliary variables , transform the objective function in the maximum minimization form of step 3.2 into the objective function in the minimization form, specifically:

[0023]

[0024] Step 3.5: Use the penalty method and the equivalent rank-one transformation method to transform the problem model in step 3.4 into a convex problem for solution, and obtain the The weight of the iteration ;

[0025] Step 3.6: Determine whether the iteration termination condition is reached , The number of iterations set, if reached, will be The weight of the iteration As the optimal phase-only emission weight, otherwise, let , return to step 3.3.

[0026] Preferably, the specific method of using the penalty method and the equivalent rank-one transformation method to transform the problem model in step 3.4 into a convex problem for solving is:

[0027] Step 3.5.1. Define the penalty factor , No. t The emission weight of the iteration , define auxiliary variables ,in , number of iterations ;

[0028] Step 3.5.2: The problem model of step 3.4 is equivalently expressed as:

[0029] ,

[0030] In the formula, refers to the matrix rank;

[0031] Step 3.5.3: Express the rank-one constraint equivalently as:

[0032] ,

[0033] in, and Respectively represent matrices The nuclear norm and spectral norm of ;

[0034] Step 3.5.4: Use the penalty method to add the expression of the equivalent rank-one constraint as a penalty term to the objective function of the optimization problem and transform it into the following form:

[0035] ,

[0036] Step 3.5.5: Use the continuous convex approximation method to obtain the local optimal solution of step 3.5.4. In each continuous convex approximation iteration, the objective function is first-order Taylor approximated, specifically:

[0037] ,

[0038] in, Expressed as The solution of the iteration, for The eigenvector corresponding to the maximum eigenvalue of ;

[0039] No. The iterations are expressed as:

[0040] ,

[0041] Solved by the standard convex optimization toolbox, we get ;

[0042] Step 3.5.6: Determine whether the iteration termination condition is reached , is the set number of iterations. If it is satisfied, Perform eigenvalue decomposition to obtain , the phase-only array transmit weight , if not satisfied , return to step 3.5.3.

[0043] Compared with the prior art, the present invention has the following significant advantages: the present invention adopts an equivalent rank-one method, a continuous convex approximation and a penalty method to deal with rank-one constraints, transforms a non-convex problem into a convex problem and directly solves it using a standard convex optimization toolbox, so that there is no relaxation in each transformation, and the solution obtained does not cause any loss in system performance. BRIEF DESCRIPTION OF THE DRAWINGS

[0044] In order to more clearly illustrate the embodiments of the present application or the existing technical solutions, the following is a brief introduction to the drawings required for use in the embodiments or the description of the prior art. Obviously, the drawings described below are only some embodiments recorded in the present application, and for ordinary technicians in this field, other drawings can be obtained based on these drawings without creative labor.

[0045] Figure 1 is the power pattern obtained using the convex relaxation method.

[0046] Figure 2 is the power pattern obtained using the alternating direction multiplier method.

[0047] Figure 3 is the power pattern obtained using the Riemann-Newton method.

[0048] Figure 4 The power pattern is obtained by using the method of the present invention.

[0049] Figure 5 It is the power pattern of digital array.

[0050] Figure 6 The main lobe power gain of the convex relaxation method, the alternating direction multiplier method, the Riemann-Newton method, the method proposed by the present invention, and the method under the digital array varies with the main lobe center angle.

[0051] Figure 7 The main lobe power gain of the convex relaxation method, the alternating direction multiplier method, the Riemann-Newton method, the method proposed by the present invention, and the method under the digital array varies with the main lobe width.

[0052] Figure 8 Flowchart for obtaining the optimal phase-only emission weights using the equivalent rank-one method, continuous convex approximation, and penalty method.

[0053] Fig. 9 A flowchart for solving the problem model by using penalty method, continuous convex approximation and equivalent rank-one transformation to transform the problem model into a convex problem. DETAILED DESCRIPTION

[0054] The present invention is further described below with reference to the accompanying drawings and examples, which is a method for synthesizing wide mainlobe transmission power patterns of phase-only arrays based on rank-one matrix optimization.

[0055] The present invention is a method for synthesizing wide mainlobe transmission power patterns of phase-only arrays based on rank-one matrix optimization. This method combines the advantages of generalized fractional programming, equivalent rank-one method, continuous convex approximation, and penalty method, and proposes an efficient solution algorithm. The algorithm is suitable for solving the gain pattern synthesis problem of wide beam phased arrays that maximizes the mainlobe gain. In the algorithm, generalized fractional programming and equivalent rank-one are first used to convert the original problem into a series of non-convex constrained sub-problems, and then the equivalent rank-one method, continuous convex approximation, and penalty method are used to solve the corresponding sub-problems. The present invention is suitable for solving the power pattern synthesis problem of wide beam phased arrays that maximizes the mainlobe gain. In the present invention, in order to convert the objective function in fractional form into a polynomial form, generalized fractional programming is used to introduce auxiliary variables for implementation. Then, the equivalent rank-one method, continuous convex approximation, and penalty method are used to handle constant modulus constraints, and the problem is converted into a convex problem, which is directly solved using the standard convex optimization toolbox. Figure 8 As shown, the specific implementation steps of the present invention are as follows:

[0056] Step 1: Assumptions A linear array consisting of array elements, . At an angle The steering vector at is expressed as:

[0057] (1)

[0058] in, It is Antenna pattern of array elements, when the array elements are isotropic , It is The distance from the array element to the reference array element, It is about wavelength The wave number, is the center direction of the preset main lobe.

[0059] definition is the phase-only emission weight vector, Each of All satisfied , is the preset transmit power of each antenna element. For convenience, assume . At an angle The far-field radiation pattern at is expressed as:

[0060] (2)

[0061] In the entire angular space In the radiation pattern , at an angle The directivity of the array antenna at is expressed as:

[0062] (3)

[0063] Substituting equation (2) into equation (3), the directionality It is expressed as:

[0064] (4)

[0065] in, , .

[0066] For any array, the power pattern is equal to the product of the array directivity and its radiation efficiency. Once the array antenna structure is determined, its radiation efficiency is basically unchanged. That is, the power pattern depends entirely on the directivity. In radar and navigation applications, wide-beam power patterns with maximum power gain play an indispensable role in search mode, and for phase-only arrays, they can significantly reduce costs and hardware complexity. Therefore, the problem of synthesizing wide-beam power patterns with minimum power gain in the maximum main lobe region can be expressed as:

[0067] (5)

[0068] in, is the main lobe area, for The discrete directions in , is the discrete angle of the main lobe area;

[0069] Step 2: The fractional objective function under the maximum minimization in problem (5) is difficult to handle, so an auxiliary variable is introduced , use generalized fractional programming to transform the maximum-minimization fractional objective function form into the maximum-minimization integer objective function form. Specifically, define for:

[0070] (6)

[0071] Through generalized fractional programming, the original problem can be transformed into a maximum-minimum problem as follows:

[0072] (7)

[0073] Step 3: Introduce the second auxiliary variable to equivalently express the maximum minimization problem in a minimization form, and use the equivalent rank-one method, continuous convex approximation and penalty method to solve it and obtain the optimal phase-only emission weight.

[0074] Step 3.1: Randomly initialize the emission weights , ,definition ;

[0075] Step 3.2, The optimization problem of this iteration is expressed as:

[0076] (8)

[0077] Step 3.3: Update auxiliary variables The expression is:

[0078] (9)

[0079] Step 3.4: Then, in order to solve problem (9) more efficiently, the objective function in the form of maximum minimization is transformed into the objective function in the form of minimization. Define auxiliary variables , problem (8) can be equivalent to the following form:

[0080] (10)

[0081] Step 3.5: Fig. 9 As shown, the penalty method and the equivalent rank-one transformation method are used to transform the problem (10) in step 5 into a convex problem for solution, specifically:

[0082] Step 3.5.1. Define the penalty factor , No. t The emission weight of the iteration , define auxiliary variables ,in , number of iterations .

[0083] Step 3.5.2, Problem (10) is equivalently expressed as:

[0084] (11)

[0085] At this time, problem (11) is a constrained optimization problem, and only the rank-one constraint is non-convex. Usually, the semi-positive definite relaxation method is used to discard the rank-one constraint for solving, but it will inevitably lead to system performance loss. The present invention uses the equivalent rank-one method to deal with non-convex rank-one constraints to avoid performance loss.

[0086] Step 3.5.3: Express the rank-one constraint equivalently as:

[0087] (12)

[0088] in, and Respectively represent matrices The nuclear norm and spectral norm of . For any matrix ,inequality Always holds, where express The singular values. When is rank one, the equality in equation (12) holds.

[0089] Step 3.5.4, then, use the penalty method to add the expression of the equivalent rank-one constraint as a penalty term into the objective function of the optimization problem. Thus, problem (11) is transformed into the following form:

[0090] (13)

[0091] Among them, the penalty factor To penalize the violation of constraint (11) in the objective function. When is small enough, Problem (11) and Problem (13) are equivalent. After the rank-one solution of is obtained by solving problem (13), the emission weights It can be decomposed from the eigenvalue is recovered from the rank-one solution.

[0092] Step 3.5.5: In problem (13), due to the difference convex objective function The problem is still a non-convex optimization problem. The present invention adopts a continuous convex approximation method to obtain the local optimal solution of problem (13).

[0093] In each consecutive convex approximation iteration, the first-order Taylor approximation is performed on the objective function, specifically:

[0094] (14)

[0095] Among them, among them, Expressed as The solution of the iterations, and for The eigenvector corresponding to the maximum eigenvalue of ;

[0096] Therefore, the The iterations are expressed as:

[0097] (15)

[0098] This problem is a convex problem and can be solved directly by the standard convex optimization toolbox, obtaining ;

[0099] Step 3.5.6: Determine whether the iteration termination condition is reached , if satisfied, then Perform eigenvalue decomposition to obtain , the phase-only array transmit weight , if not satisfied , return to step 3.5.3;

[0100] Step 3.6: Determine whether the iteration termination condition is reached , if reached, then the The weight of the iteration As the optimal phase-only emission weight, otherwise, let , return to step 3.3.

[0101] Example

[0102] Through Matlab simulation, the present invention further illustrates the phase-only array wide mainlobe transmission power pattern synthesis method based on rank-one matrix optimization.

[0103] 1) Simulation parameter settings

[0104] use The specific position of the array elements is set to: 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,-1.5302,-0.6299,-0.3749,0,0.3749,0.6299, 1.5302,1.8494,2.3497,2.8973,3.2995,3.8098,4.6065,5,5.3749,5.6299,6.5302,6.8494, 7.3497,7.8973,8.2995,8.8098,9.6065,10]. by The interval is discretized into 181 angles, and set , the auxiliary variables are randomly initialized. In addition, for a fair comparison, the control of the sidelobe level in the three methods, namely, the convex relaxation method, the alternating direction multiplier method and the Riemann-Newton method, is omitted.

[0105] 2) Beam diagram drawing

[0106] In order to intuitively show the effect of the power pattern, the obtained weight vector is used to draw the pattern at different main lobe center angles, and compared with the power patterns generated by three typical phase-only array methods and one digital array method. , the unit of the vertical coordinate of the directional diagram is .

[0107] 3) Metrics

[0108] In the present invention, in addition to the beam pattern, the minimum power gain (MPG) in the main lobe area is also used as a measurement indicator. The minimum power gain MPG in the main lobe area is defined as follows:

[0109]

[0110] in, is the main lobe area. Under the same initial conditions, the larger the minimum power gain in the main lobe area, the better the beamforming effect.

[0111] 4) Result analysis

[0112] The present invention has carried out seven example simulations in total. Figure 1-4 They are respectively the power patterns generated by the convex relaxation method, the alternating direction multiplier method, the Riemann-Newton method and the method proposed by the present invention under the phase-only array; Figure 5 It is the power pattern formed by the PGPS method under the digital array, which is used as the performance measurement benchmark; Figure 6 is the power gain of the five methods at different main lobe center angles; Figure 7 is the power gain of the five methods at different main lobe widths.

[0113] pass Figure 1-Figure 4 It can be seen that among the four phase-only array transmission pattern synthesis methods, the minimum power gain in the main lobe area formed by the method proposed in the present invention is the highest, and the power gain in the side lobe area is also the lowest. Figure 5 In comparison, the phase-only array inevitably brings the main lobe power gain In practical applications, the use of large-scale phase-only arrays can reduce this performance loss.

[0114] pass Figure 6 and Figure 7 It can be seen that under different main lobe center angles and main lobe widths, the minimum power gain performance in the main lobe area of ​​the power pattern synthesis algorithm proposed in the present invention is always better than the other three beamforming pattern synthesis algorithms under phase-only arrays, but is inevitably slightly worse than the power pattern synthesis algorithm under digital arrays.

[0115] In summary, the method proposed in the present invention has good power pattern synthesis performance. Compared with other representative methods, the proposed method has the best minimum power gain performance in the main lobe area under different main lobe center angles and different main lobe widths. When applied in radar and satellite communication systems, it can greatly reduce hardware costs and reduce the complexity of hardware systems, and has high application value.

Claims

1. A method for synthesizing wide mainlobe transmission power patterns of phase-only arrays based on rank-one matrix optimization, characterized in that: include: Step 1: construct a wide beam power pattern synthesis problem model with minimum power gain in the maximum main lobe area, wherein the objective function in the wide beam power pattern synthesis problem model is a maximum minimization fractional objective function, and the wide beam power pattern synthesis problem model is specifically: ， In the formula, is the main lobe area, , is the discrete angle of the main lobe area, for The steering vector at is the phase-only emission weight vector, for Middle elements, , For the entire angle space, Angle The steering vector at ; Step 2: Introduce the first auxiliary variable and use generalized fractional programming to transform the maximum-minimization fractional objective function form into the maximum-minimization integral objective function form; Step 3: Introduce the second auxiliary variable to equivalently express the maximum minimization problem in a minimization form, and use the equivalent rank-one method, continuous convex approximation and penalty method to solve it and obtain the optimal phase-only emission weight.

2. The method for synthesizing wide mainlobe transmission power patterns of phase-only arrays based on rank-one matrix optimization according to claim 1, characterized in that: Introducing auxiliary variables , use generalized fractional programming to transform the maximum-minimization fractional objective function form into the maximum-minimization integral objective function form, specifically: ， In the formula, .

3. The method for synthesizing wide main lobe transmission power patterns of phase-only arrays based on rank-one matrix optimization according to claim 2, characterized in that: The second auxiliary variable is introduced to equivalently express the maximum minimization problem as a minimization form, and the equivalent rank-one method, continuous convex approximation and penalty method are used to solve it. The specific method for obtaining the optimal phase-only emission weight is: Step 3.1: Randomly initialize the emission weights , ,definition ; Step 3.2, The optimization problem of this iteration is expressed as: ， Step 3.3: Update auxiliary variables The expression is: ， Step 3.4, define auxiliary variables , transform the objective function in the maximum minimization form of step 3.2 into the objective function in the minimization form, specifically: ， Step 3.5: Use the penalty method and the equivalent rank-one transformation method to transform the problem model in step 3.4 into a convex problem for solution, and obtain the The weight of the iteration ; Step 3.6: Determine whether the iteration termination condition is reached , The number of iterations set, if reached, will be The weight of the iteration As the optimal phase-only emission weight, otherwise, let , return to step 3.

3.

4. The method for synthesizing wide main lobe transmission power patterns of phase-only arrays based on rank-one matrix optimization according to claim 3, characterized in that: The specific method of using the penalty method and the equivalent rank-one transformation method to transform the problem model in step 3.4 into a convex problem for solution is: Step 3.5.

1. Define the penalty factor , No. t The emission weight of the iteration , define auxiliary variables ,in , number of iterations ; Step 3.5.2: The problem model of step 3.4 is equivalently expressed as: ， In the formula, refers to the matrix rank; Step 3.5.3: Express the rank-one constraint equivalently as: ， in, and Respectively represent matrices The nuclear norm and spectral norm of ; Step 3.5.4: Use the penalty method to add the expression of the equivalent rank-one constraint as a penalty term to the objective function of the optimization problem and transform it into the following form: ， Step 3.5.5: Use the continuous convex approximation method to obtain the local optimal solution of step 3.5.

4. In each continuous convex approximation iteration, the objective function is first-order Taylor approximated, specifically: ， in, Expressed as The solution of the iteration, for The eigenvector corresponding to the maximum eigenvalue of ; No. The iterations are expressed as: ， Solved by the standard convex optimization toolbox, we get ; Step 3.5.6: Determine whether the iteration termination condition is reached , is the set number of iterations. If it is satisfied, Perform eigenvalue decomposition to obtain , the phase-only array transmit weight , if not satisfied , return to step 3.5.3.

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