Phased array system transmitting beam forming method based on Riemannian-Newton method

Abstract

The invention discloses a phased array system transmitting beam forming method based on a Riemann-Newton method, which converts a beam pattern matching error minimization problem under a constant modulus constraint into an unconstrained optimization problem under a complex circle manifold, and adopts a second-order Riemann-Newton method to solve. Therefore, it is ensured that the direction of each iteration update is always the descending direction of the target function, and the super-linear convergence speed is achieved. The method has better performance advantages in the aspects of main lobe fitting precision and sidelobe level reduction, and the convergence speed is higher.

Description

Technical Field

[0001] The present invention belongs to the technical field of array signal processing, and in particular relates to a phased array system transmit beamforming method based on the Riemann-Newton method. Background Art

[0002] The phased array system can achieve the desired transmission pattern synthesis by adjusting the phase of the transmission weights at a constant transmission weight amplitude, and the transmission efficiency is extremely high. Since the phase adjustment can be completed only by relying on a simple phase shifter, this not only greatly reduces the hardware cost, but also reduces power consumption and simplifies the structure of the transmission array. Therefore, phase-only transmission beamforming is increasingly used in wireless communications, radar, and remote sensing systems, especially in large-scale transmission arrays, where its advantages are fully reflected.

[0003] In essence, the synthesis of the phased array transmission pattern can be equivalently expressed as a non-convex optimization problem with constant modulus constraints. Under a large number of non-convex constraints, this type of unimodular least squares (ULS) problem is NP-hard. In recent years, many solutions to this type of non-convex optimization problem have been proposed, which can be roughly divided into two categories. One is the convex relaxation method, among which the semi-definite relaxation SDR is a representative algorithm. However, SDR has high requirements for storage resources and computing resources, and the solution efficiency is low when the array scale is large. Moreover, the solution obtained after convex relaxation cannot be guaranteed to be the local optimal solution of the original problem. In some complex cases, it will be difficult to avoid performance loss, making the problem worse. Therefore, many methods have emerged in recent years to directly solve certain specific or well-structured non-convex optimization problems.

[0004] Compared with common algorithms such as alternating direction method of multipliers (ADMM), upper bound function minimization (MM), cyclic coordinate descent (CCD), etc., the beam pattern matching error minimization problem under constant modulus constraint is transformed into an unconstrained optimization problem under complex circular manifold and solved by the second-order Riemann-Newton method, which can achieve better performance and has a superlinear convergence speed.

[0005] Phase-only transmit beamforming technology is currently widely used in radar and communication transmission systems due to its advantages such as low latency, high flexibility and simple implementation. In phased array transmission systems, in order to pursue high cost performance and high power efficiency, a fixed transmit weight amplitude is usually used to only optimize the transmit weight phase to form the desired beam pattern. However, due to the existence of constant modulus constraints, the transmit beamforming of the phased array system is a non-convex and difficult to solve problem. Generally speaking, this problem can be relaxed to a convex problem, but it will inevitably lead to performance loss, and the computational complexity depends largely on the solver used. Summary of the invention

[0006] The purpose of the present invention is to propose a phased array system transmit beamforming method based on the Riemann-Newton method.

[0007] The technical solution to achieve the purpose of the present invention is: a phased array system transmit beamforming method based on the Riemann-Newton method, comprising:

[0008] Step 1: Based on the steering vector of the antenna array, a phased array system transmit beamforming model that minimizes the pattern matching error is constructed; Step 2: Introduce preset weights, auxiliary variables, and unimodular auxiliary variables in sequence to perform equivalent transformation on the phased array system transmit beamforming model, wherein the auxiliary variables are determined according to the desired beam pattern; Step 3: Convert the phased array system transmit beamforming model after equivalent transformation in step 2 into a phased array system transmit beamforming model in Riemann manifold optimization form; Step 4: Use the Riemann-Newton method to solve the Riemann manifold optimization problem to obtain the optimal phase-only transmit weight vector, and use the optimal phase-only transmit weight vector to achieve the phased array system transmit beamforming that minimizes the pattern matching error.

[0009] Preferably, the phased array system transmit beamforming problem model that minimizes the pattern matching error is specifically constructed as follows:

[0010] , In the formula, is the desired beam pattern, N is the number of radiating elements, is the phase-only emission weight vector, K is the number of sampling points in the angle space, is the kth angle in space, for The steering vector in the direction, is the phase-only emission weight of the nth unit.

[0011] Preferably, the specific process of sequentially introducing preset weights, auxiliary variables, and unimodular auxiliary variables to perform equivalent transformation on the phased array system transmit beamforming problem model is as follows:

[0012] Importing preset weights , the phased array system transmit beamforming model is converted into: , Introducing auxiliary variables , , the phased array system transmit beamforming model is further converted into: , Introducing unimodular auxiliary variables , the phased array system transmit beamforming model is further converted into: , in , Represented by vector filling The diagonal matrix of the main diagonal of satisfy , Indicates the acquisition phase operation.

[0013] Preferably, the specific method of converting the phased array system transmit beamforming model after the equivalent transformation in step 2 into a Riemann manifold optimization problem is: Introduce auxiliary variables: , , In the formula, is the phase-only emission weight vector The transpose of Unimodular auxiliary variables The transpose of for The conjugate transpose of ; yes dimensional identity matrix, u is a regularization parameter and ; Through vectorization, the phased array system transmit beamforming model after equivalent transformation in step 2 is converted into: , In the formula, for The ith element of .

[0014] Preferably, the Riemann-Newton method is used to solve the Riemann manifold optimization problem, so as to achieve the specific process of minimizing the pattern matching error of the phased array system transmit beamforming as follows: Step 4.1: Set initial variables , , yes Initial value of Step 4.2: Calculate The quasi-Riemann-Newton gradient at the point is as follows: , The corresponding plural form is: , In the formula, and They are The vector consisting of the first N elements and the vector consisting of the last N elements, is the diagonal loading factor, , , , is the Riemann gradient, is the objective function about The Hessian matrix of Step 4.3: Update point at iteration Determined as , In the formula, is the complex Riemann-Newton gradient obtained at the jth iteration; Project the points on the tangent space onto the complex circular manifold, where Represents a shrink operation: , is the solution obtained by the j+1th optimization; Step 4.4: Calculation The objective function at the point ; Step 4.5: Determine whether the iteration stop condition is met. If so, This is the final optimization result, the solution to the Riemann manifold optimization problem and Respectively by A vector consisting of the first N elements and the last N elements of ; Otherwise, let , return to step 4.2 to continue iterative solution.

[0015] Preferably, the iteration stopping condition is specifically: , is the set threshold.

[0016] Compared with the prior art, the present invention has the following significant advantages: the present invention adopts the Riemann-Newton method, has a faster convergence speed, and does not require line search, so it has a better convergence speed guarantee. In addition to ensuring the fitting effect of the main lobe, the present invention can also take into account deep nulling and low sidelobe levels. BRIEF DESCRIPTION OF THE DRAWINGS

[0017] In order to more clearly illustrate the embodiments of the present application or the existing technical solutions, the drawings required for use in the embodiments or the description of the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments recorded in the present application. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative labor.

[0018] Figure 1 It is a flow chart of the present invention.

[0019] Figure 2 is the focused beam pattern without null-steering constraint.

[0020] Figure 3 for Figure 2 A magnified view of the main lobe region in .

[0021] Figure 4 is the focused beam pattern with null-steering constraint.

[0022] Figure 5 for Figure 4 A magnified view of the main lobe region in .

[0023] Figure 6 Fitting pattern of focused beam for varying null region weights.

[0024] Figure 7 for Figure 6 A magnified view of the main lobe region in . DETAILED DESCRIPTION

[0025] The present invention, namely a transmit beamforming method for a phased array system based on the Riemann-Newton method, is further described below with reference to the accompanying drawings and examples.

[0026] like Figure 1 As shown, a method for transmitting beamforming of a phased array system based on the Riemann-Newton method is specifically an algorithm for realizing effective matching of a transmitting beam pattern with a desired beam based on the Riemann-Newton method. While ensuring phase-only transmission weights, the algorithm minimizes the deviation between the actual beam pattern and the desired beam pattern, thereby meeting different requirements for the transmitting beam pattern in different scenarios. In order to set different requirements for matching conditions within different spatial angle ranges, preset weights are introduced. Then, the problem with constant modulus constraints is transformed into a Riemann manifold optimization problem on a complex circular manifold, and the Riemann-Newton method is used to efficiently solve the problem. While ensuring phase-only transmission weights, the algorithm minimizes the matching error between the actual beam pattern and the desired beam pattern, thereby meeting the transmitting beamforming requirements in different scenarios. The specific steps of the present invention are as follows:

[0027] Step 1: Assume that a linear array consisting of N radiating elements transmits a narrowband signal in the far field. The steering vector of the linear array is expressed as:

[0028] (1)

[0029] In the formula, is the azimuth radiation direction, , It is The antenna pattern of each unit, , when the unit is isotropic, , It is The distance from the unit to the reference unit, is the wavelength.

[0030] definition is the phase-only emission weight vector, the elements of which satisfy ,in It is a waveform The modulus constraint of each element in is determined by the preset transmit power of each element. is the emission weight Under the effect of The far-field radiation pattern of the direction is described as:

[0031] (2)

[0032] The entire space area Main lobe , side lobe Two parts, is the desired beam pattern. In order for the phased array radar to ensure working performance in different scenarios, it is necessary to minimize the deviation between the actual transmit beam pattern and the desired pattern in the current application scenario. Based on this consideration, a phased array system transmit beamforming problem model that minimizes the pattern matching error is constructed, which is described as follows:

[0033] (3)

[0034] In the formula, is the number of sampling points in the angle space, is the kth angle in the space.

[0035] Step 2: In order to set different requirements for matching in different spatial angle ranges, introduce preset weights Therefore, formula (3) can be expressed as:

[0036] (4)

[0037] Introducing auxiliary variables , . Convert the original directional pattern matching error minimization problem into the form of a least squares problem:

[0038] (5)

[0039] Introducing unimodular auxiliary variables Removal Modulo upper operation, where satisfy ( represents the acquisition phase operation) and , , for The nth element of . So formula (5) is equivalent to formula (6):

[0040] (6)

[0041] in , Represents the vector filling Diagonal matrix with main diagonal.

[0042] Step 3: Since formula (6) is a non-convex constrained optimization problem, it is difficult to solve it directly. Therefore, a manifold optimization method is used to transform the problem with constant modulus constraints on complex circular manifolds into a Riemann manifold optimization problem on complex circular manifolds. To this end, an auxiliary variable is introduced.

[0043] (7)

[0044] and

[0045] ,

[0046] in yes dimensional identity matrix, is a regularization parameter. Since Every element of satisfies the unimodular constraint, is a constant. Therefore, by vectorization, formula (6) can be equivalently described as:

[0047] (8)

[0048] Step 4: Formula (8) is a unimodular constrained least squares problem, which can be solved with high performance using the Riemann-Newton method to achieve the transmit beamforming of the phased array system that minimizes the pattern matching error.

[0049] The basic principle of the Riemann-Newton method is to search for update points along the Riemann-Newton direction (Riemann-Newton gradient), ensure that the update points are always close to the complex circular manifold, and then project the update points back to the circular manifold. Needs to be satisfied

[0050] (9)

[0051] in, represents a complex circular manifold, where It is Hadamard. yes -dimensional all-1 vector. For manifold exist The tangent space at the point, where yes dimensional all-zero vector. is the Riemann gradient, yes Orthogonal projection on is the Euclidean gradient. is the Riemann Hessian matrix, , , Indicates A diagonal matrix with its main diagonal filled.

[0052] Convert Newton's equation (9) into real form:

[0053] (10)

[0054] in, , , , , , Respectively expressed as:

[0055] , ,

[0056] In order to Positive definite, transform formula (10) into the following form:

[0057] (11)

[0058] in is the diagonal loading factor. Appropriate selection , can be obtained in Quasi-Riemann-Newton gradient at a point

[0059] (12)

[0060] Its plural form is

[0061] (13)

[0062] in, and They are A vector consisting of the first N elements and the last N elements of .

[0063] make For the The search point for the iteration, is the corresponding complex Riemann-Newton gradient obtained according to the above method. The update point at the iteration is determined as

[0064] (14)

[0065] Project the points on the tangent space onto the complex circular manifold, where Represents a shrink operation.

[0066] (15)

[0067] for Projection onto the manifold The iterative output generated by

[0068] (16)

[0069] The specific steps for solving formula (8) are:

[0070] 1) Input parameters: , , ,

[0071] 2) Initialization settings: ,

[0072] 3) Start iteration:

[0073] i. Calculate the Riemann-Newton gradient using formula (12) and formula (13)

[0074] ii. Using formula (14) and formula (15), we can obtain:

[0075] iii.

[0076] 4) If Select the best Stop the loop, otherwise go back to 3).

[0077] 5) Output: , the solution of formula (6) and Respectively by A vector consisting of the first N elements and the last N elements of .

[0078] Example

[0079] The phased array system transmit beamforming method based on the Riemann-Newton method of the present invention is further illustrated through Matlab simulation.

[0080] 1) Simulation system parameter settings

[0081] In the simulation process, taking the uniform linear array as an example, the number of array elements is set to 32 and the array element spacing is half a wavelength. It is divided into 181 points with an interval of 1°. The upper limit of the number of algorithm iterations is set to , and set The auxiliary variables are generated by random initial values. For a focused beam without null sinking, the expected beam pattern is Set to:

[0082] (17)

[0083] The weight coefficients in all directions are set to ; For a focused beam with null, the expected beam pattern is and preset weight coefficients They are respectively set as:

[0084] (18)

[0085] (19)

[0086] 2) Transmit beam pattern drawing

[0087] In order to intuitively demonstrate the actual effect of beam pattern matching, the solved weight vector can be used to perform weighted combination processing on the signal. The beam pattern generated by the present invention is plotted, and the results are compared with three typical phase-only pattern synthesis methods. Subsequently, the performance advantages of the method of the present invention are further explored by changing the parameter settings. In the beam pattern, the horizontal axis represents an angle range covering from -90° to 90°, and the vertical axis represents the amplitude value after weighted combination of the signal. These amplitude values ​​are presented in dB after modulo and maximum value normalization processing, so as to more clearly reflect the characteristics and advantages and disadvantages of beamforming.

[0088] 3) Metrics

[0089] In addition to generating the beam pattern, the present invention also uses the peak sidelobe level (PSL), the integrated sidelobe level (ISL) and the integrated null depth (INL) as metrics to measure the performance of the beam pattern, which are specifically defined as follows:

[0090] (20)

[0091] in, and They are the sidelobe area and the null area respectively.

[0092] 4) Result analysis

[0093] The present invention carries out three example simulations, and compares the Riemann-Newton method of the present invention with three other representative methods: alternating direction method of multipliers (ADMM), upper bound function minimization (MM), and cyclic coordinate descent (CCD). Figures 2~5 These are the focused beam patterns synthesized by the four methods with and without null-steering requirements. Figure 4 and Figure 5 The beam pattern matching of the Riemann-Newton method is shown in Figure 2 when the weight coefficient applied to the null region is changed. The setting of the desired beam pattern of the third simulation experiment remains unchanged, and the preset weight coefficient is changed to

[0094] (twenty one)

[0095] Figures 2~5 The specific index parameters of the transmit beam synthesized in are shown in Tables 1 and 2. As shown in Table 1, in the case of a focused beam without nulling, the Riemann-Newton method can obtain the smallest fitting error. Although the main lobe gain generated by all methods is equivalent, the Riemann-Newton method has the lowest peak sidelobe level. As shown in Table 2, in the case of a focused beam with nulling, the fitting error of the Riemann-Newton method increases, but it still has the lowest peak sidelobe level, and the integrated nulling depth in the nulling area is significantly lower than that of the other three methods.

[0096] Table 1 Simulation results of focused beam without nulling

[0097] method Riemann-Newton method Alternating Direction Multiplier Method Upper bound function minimization method Cyclic Coordinate Descent Fitting error 0.76 1.19 0.83 0.77 Peak sidelobe level (dB) 14.51 16.78 15.09 15.89 Main lobe gain (dB) 29.04 28.31 29.07 29.08

[0098] Table 2 Simulation results of focused beam with nulling

[0099] method Riemann-Newton method Alternating Direction Multiplier Method Upper bound function minimization method Cyclic Coordinate Descent Fitting error 0.94 0.87 0.82 0.82 Integrated null depth (dB) -14.05 -7.28 -10.34 -9.34 Peak sidelobe level (dB) 16.05 16.10 16.14 16.42 Main lobe gain (dB) 29.21 29.18 29.11 29.23

[0100] Figure 6 and Figure 7The influence of the null region weight coefficient on the transmit beam performance when N=32 is given, and the specific index parameters are shown in Table 3. It can be seen that the integral depth of the null region gradually decreases with the increase of the weight coefficient, and the gain in the main lobe pointing direction gradually increases; however, Then continue to grow , the effect of deepening the null is very limited, and the peak sidelobe level will increase. Therefore, in practical applications, the appropriate .

[0101] Table 3 Effect of changing the weight coefficient of the null region on the beam under the focused beam with null

[0102] Null sink weight Integrated null depth (dB) Peak sidelobe level (dB) Main lobe gain (dB) w=1 13.16 17.72 29.35 w=10 -8.94 16.87 29.46 w=100 -21.19 17.22 29.50 w=1000 -21.96 18.00 29.52

[0103] In summary, the method of the present invention has good comprehensive performance. Compared with other representative methods, the proposed method has almost the best main lobe fitting accuracy in directional beam fitting, ensures low side lobe level and meets the requirements of deep nulling, and has high practical value in phased array radar systems.

Claims

1. A method for transmitting beamforming of a phased array system based on the Riemann-Newton method, characterized in that: include: Step 1: Based on the steering vector of the antenna array, a phased array system transmit beamforming model that minimizes the pattern matching error is constructed; Step 2: Introduce preset weights, auxiliary variables, and unimodular auxiliary variables in sequence to perform equivalent transformation on the phased array system transmit beamforming model, wherein the auxiliary variables are determined according to the desired beam pattern; Step 3: Convert the phased array system transmit beamforming model after equivalent transformation in step 2 into a phased array system transmit beamforming model in Riemann manifold optimization form; Step 4: Use the Riemann-Newton method to solve the Riemann manifold optimization problem to obtain the optimal phase-only transmit weight vector, and use the optimal phase-only transmit weight vector to achieve the phased array system transmit beamforming that minimizes the pattern matching error.

2. The method for transmitting beamforming of a phased array system based on the Riemann-Newton method according to claim 1, characterized in that: The constructed phased array system transmit beamforming model that minimizes the pattern matching error is as follows: ， In the formula, is the desired beam pattern, N is the number of radiating elements, is the phase-only emission weight vector, K is the number of sampling points in the angle space, is the kth angle in space, for The steering vector in the direction, is the phase-only emission weight of the nth unit.

3. The method for transmitting beamforming of a phased array system based on the Riemann-Newton method according to claim 2, characterized in that: The specific process of introducing preset weights, auxiliary variables, and unimodular auxiliary variables in turn to perform equivalent transformation on the phased array system transmit beamforming model is as follows: Importing preset weights , the phased array system transmit beamforming model is converted into: ， Introducing auxiliary variables , , the phased array system transmit beamforming model is further converted into: ， Introducing unimodular auxiliary variables , the phased array system transmit beamforming model is further converted into: ， in , Represented by vector filling The diagonal matrix of the main diagonal of satisfy , Indicates the acquisition phase operation.

4. The method for transmitting beamforming of a phased array system based on the Riemann-Newton method according to claim 3, characterized in that: The specific method of converting the phased array system transmit beamforming model after the equivalent transformation in step 2 into a Riemann manifold optimization problem is as follows: Introduce auxiliary variables: ， ， In the formula, is the phase-only emission weight vector The transpose of Unimodular auxiliary variables The transpose of for The conjugate transpose of ; yes dimensional identity matrix, u is a regularization parameter and ; Through vectorization, the phased array system transmit beamforming model after equivalent transformation in step 2 is converted into: ， In the formula, for The ith element of .

5. The method for transmitting beamforming of a phased array system based on the Riemann-Newton method according to claim 4, characterized in that: The Riemann-Newton method is used to solve the Riemann manifold optimization problem, so as to achieve the specific process of the phased array system transmit beamforming that minimizes the pattern matching error: Step 4.1: Set initial variables , , yes Initial value of Step 4.2: Calculate The quasi-Riemann-Newton gradient at the point is as follows: ， The corresponding plural form is: ， In the formula, and They are The vector consisting of the first N elements and the vector consisting of the last N elements, is the diagonal loading factor, , , , is the Riemann gradient, is the objective function about The Hessian matrix of Step 4.3: Update point at iteration Determined as ， In the formula, is the complex Riemann-Newton gradient obtained at the jth iteration; Project the points on the tangent space onto the complex circular manifold, where Represents a shrink operation: ， is the solution obtained by the j+1th optimization; Step 4.4: Calculation The objective function at the point ; Step 4.5: Determine whether the iteration stop condition is met. If so, This is the final optimization result, the solution to the Riemann manifold optimization problem and Respectively by A vector consisting of the first N elements and the last N elements of ; Otherwise, let , return to step 4.2 to continue iterative solution.

6. The method for transmitting beamforming of a phased array system based on the Riemann-Newton method according to claim 5, characterized in that: The specific conditions for stopping the iteration are: ， is the set threshold.

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