A transmitting beamforming method for a phased array system based on the Riemann-Newton method

Abstract

The present invention discloses a transmitting beamforming method for a phased array system based on the Riemannian Newton method. This method transforms the problem of minimizing the beam pattern matching error under constant modulus constraints into an unconstrained optimization problem in the complex circular manifold, and uses the second-order Riemannian Newton method to solve it, so as to ensure that the direction updated in each iteration is always the descending direction of the objective function and has a superlinear convergence rate. The present invention has better performance advantages in terms of main lobe fitting accuracy and side lobe level reduction, and has a faster convergence rate.

Description

Technical Field

[0001] The present invention belongs to the technical field of array signal processing, and particularly relates to a transmitting beamforming method for a phased array system based on the Riemannian Newton method. Background Art

[0002] A phased array system can synthesize a desired transmitting pattern by only adjusting the phase of the transmitting weights under the condition of a constant transmitting weight amplitude, and has extremely high transmitting efficiency. Since the phase adjustment only needs to rely on a simple phase shifter to complete, this not only greatly reduces the hardware cost, but also reduces the power consumption and simplifies the structure of the transmitting array. Therefore, the phase-only transmitting beamforming is more and more widely used in the fields of wireless communication, radar, remote sensing systems, etc., especially in the case of large-scale transmitting arrays, and its advantages are fully reflected.

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

[0004] Compared with common algorithms such as the alternating direction method of multipliers (ADMM), the majorization-minimization method (MM), the cyclic coordinate descent (CCD), etc., minimizing the beam pattern matching error under the constant modulus constraint is transformed into an unconstrained optimization problem in the complex circular manifold, and the second-order Riemannian Newton method is used to solve it, which can achieve better performance and has a superlinear convergence rate.

[0005] The phase-only transmitting beamforming technology is currently widely used in radar and communication transmitting systems due to its advantages such as low latency, high flexibility, and simple implementation. In a phased array transmitting system, in order to pursue high cost performance and high power efficiency, a fixed transmitting weight amplitude is usually adopted, and only the phase of the transmitting weight is optimized to form a desired beam pattern. However, due to the existence of the constant modulus constraint, the transmitting 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 for solution, but it will inevitably lead to performance loss, and the computational complexity largely depends on the solver used. Summary of the Invention

[0006] The object of the present invention is to propose a transmitting beamforming method for a phased array system based on the Riemannian Newton method.

[0007] The technical solution for achieving the object of the present invention is as follows: A transmitting beamforming method for a phased array system based on the Riemannian Newton method, comprising:

[0008] Step 1: Construct a transmitting beamforming model for the phased array system that minimizes the pattern matching error according to the steering vector of the antenna linear array;

[0009] Step 2: Successively introduce a preset weight, an auxiliary variable, and a unimodular auxiliary variable to perform an equivalent transformation on the transmitting beamforming model of the phased array system, where the auxiliary variable is determined according to the desired beam pattern;

[0010] Step 3: Convert the transmitting beamforming model of the phased array system after the equivalent transformation in Step 2 into a transmitting beamforming model of the phased array system in the form of Riemannian manifold optimization;

[0011] Step 4: Use the Riemannian Newton method to solve the Riemannian manifold optimization problem, obtain the optimal unique-phase transmission weight vector, and use the optimal unique-phase transmission weight vector to achieve the transmitting beamforming of the phased array system that minimizes the pattern matching error.

[0012] Preferably, the constructed transmitting beamforming problem model for the phased array system that minimizes the pattern matching error is specifically:

[0013] ,

[0014] wherein, is the desired beam pattern, N is the number of radiating elements, is the unique-phase transmission weight vector, K is the number of sampling points in the angular space, is the k-th angle in the space, is the steering vector in the direction, and

[0015] is the unique-phase transmission weight of the n-th element.

[0016] Preferably, the specific process of successively introducing a preset weight, an auxiliary variable, and a unimodular auxiliary variable to perform an equivalent transformation on the transmitting beamforming problem model of the phased array system is as follows: Introduce the preset weight

[0017] ,

[0018] and convert the transmitting beamforming model of the phased array system into: , Introduce the auxiliary variable

[0019] ,

[0020] Introduce unimodular auxiliary variables , and continue to transform the transmit beamforming model of the phased array system into:

[0021] ,

[0022] where , represents a diagonal matrix with the main diagonal filled with the vector filled with , satisfies , represents obtaining the phase operation.

[0023] Preferably, the specific method for transforming the transmit beamforming model of the phased array system after the equivalent transformation in step 2 into a Riemannian manifold optimization problem is:

[0024] Introduce auxiliary variables:

[0025] ,

[0026] ,

[0027] In the formula, is the transpose of the unique-phase transmit weight vector , the transpose of the unimodular auxiliary variable , is the conjugate transpose of; is the identity matrix of dimension u is a regularization parameter and ;

[0028] Through vectorization representation, the transmit beamforming model of the phased array system after the equivalent transformation in step 2 is transformed into:

[0029] ,

[0030] In the formula, is the i-th element of.

[0031] Preferably, the specific process of using the Riemannian Newton method to solve the Riemannian manifold optimization problem to achieve the transmit beamforming of the phased array system with minimized pattern matching error is:

[0032] Step 4.1: Set the initial variables , , is the initial value of

[0033] Step 4.2: Calculate the pseudo-Riemannian Newton gradient at the point. The specific formula is as follows:

[0034] ,

[0035] The corresponding complex form is:

[0036] ,

[0037] In the formula, and are respectively the vector composed of the first N elements of and the vector composed of the last N elements of , , , is the Riemannian gradient, is the objective function with respect to the Hessian matrix of

[0038] Step 4.3: Determine the update point at the j-th iteration as

[0039] ,

[0040] In the formula, is the complex Riemannian Newton gradient obtained in the j-th iteration;

[0041] Project the point on the tangent space onto the complex circular manifold, where represents the contraction operation:

[0042] ,

[0043] is the solution obtained by the (j + 1)-th optimization;

[0044] Step 4.4: Calculate the objective function at the point;

[0045] Step 4.5: Determine whether the iteration stop condition is satisfied. If it is satisfied, then is the final optimization result, the solution of the Riemannian manifold optimization problem and are respectively the vectors composed of the first N elements and the last N elements of ;

[0046] Otherwise, let , and return to step 4.2 to continue the iterative solution.

[0047] Preferably, the iteration stop condition is specifically:

[0048] ,

[0049] being a set threshold.

[0050] Compared with the prior art, the significant advantages of the present invention are as follows: The present invention adopts the Riemann-Newton method, has a faster convergence speed, and does not require line search, so it has a better guarantee of convergence speed. In addition to being able to ensure the fitting effect of the main lobe, the present invention can also take into account deep nulls and low sidelobe levels. BRIEF DESCRIPTION OF THE DRAWINGS

[0051] In order to more clearly illustrate the embodiments of the present application or the prior art, the following will briefly introduce the drawings required for the description of the embodiments or the prior art. Obviously, the drawings in the following description are only some embodiments recorded in the present application. For those of ordinary skill in the art, other drawings can be obtained based on these drawings without creative efforts.

[0052] Figure 1 is the flow chart of the present invention.

[0053] Figure 2 is the focused beam pattern without null constraint.

[0054] Figure 3 is Figure 2 the enlarged view of the main lobe region in

[0055] Figure 4 is the focused beam pattern with null constraint.

[0056] Figure 5 is Figure 4 the enlarged view of the main lobe region in

[0057] Figure 6 is the focused beam fitting pattern with the weight of the null region changed.

[0058] Figure 7 is Figure 6 the enlarged view of the main lobe region in DETAILED DESCRIPTION OF THE EMBODIMENTS

[0059] The present invention will be further described below in conjunction with the drawings and examples, that is, a phased array system transmit beamforming method based on the Riemann-Newton method.

[0060] AsFigure 1 As shown in Figure 1 , a transmitting beamforming method for a phased array system based on the Riemannian Newton method, specifically, an algorithm for achieving effective matching between the transmitting beam pattern and the desired beam is proposed based on the Riemannian Newton method. While ensuring unique phase, this algorithm minimizes the deviation between the actually generated beam pattern and the desired beam pattern, thus meeting different requirements for the transmitting beam pattern in different scenarios. To set different requirements for the matching situation within different spatial angle ranges, a preset weight is introduced. Then, the problem with constant modulus constraint is transformed into a Riemannian manifold optimization problem on a complex circular manifold, and the Riemannian Newton method is used to efficiently solve this problem. While ensuring the unique-phase transmitting weights, this algorithm minimizes the matching error between the actual beam pattern and the desired beam pattern, thus meeting the transmitting beamforming requirements in different scenarios. The specific steps of the present invention are as follows:

[0061] Step 1: Assume that a linear array composed of N radiation units realizes transmitting narrowband signals under the premise of far field. The steering vector of the linear array is expressed as:

[0062] (1)

[0063] In the formula, is the azimuth radiation direction, , is the th unit's antenna pattern, , when the unit is isotropic, , is the distance from the th unit to the reference unit, is the wavelength.

[0064] Define as the unique-phase transmitting weight vector, and the elements of the unique-phase transmitting weight vector satisfy , where is the modulus constraint of each element in the waveform , which is determined by the preset transmitting power of each element. Under the action of the transmitting weight , the far-field radiation pattern in the direction is described as:

[0065] (2)

[0066] Divide the entire spatial region into two parts: the main lobe and the side lobe , is the desired beam pattern. To ensure the working performance of the phased array radar in different scenarios, it is necessary to minimize the deviation between the actual transmitted beam pattern and the desired pattern in the current application scenario. Based on this consideration, a model of the transmitting beamforming problem for the phased array system that minimizes the pattern matching error is constructed, which is specifically described as follows:

[0067] (3)

[0068] where, is the number of sampling points in the angular space, is the k-th angle in the space.

[0069] Step 2: To set different requirements for the matching in different spatial angular ranges, a preset weight is introduced. Therefore, formula (3) is equivalently expressed as:

[0070] (4)

[0071] An auxiliary variable , is introduced. The original problem of minimizing the pattern matching error is transformed into the form of a least squares problem:

[0072] (5)

[0073] A unimodular auxiliary variable is introduced to remove the modulo operation, where satisfies ( represents the phase acquisition operation) and , , is the n-th element of

[0074] (6)

[0075] where , represents the diagonal matrix with the vector filled on the main diagonal of .

[0076] Step 3: Since formula (6) is a non-convex constrained optimization problem and is difficult to solve directly. Therefore, a method based on manifold optimization is adopted to transform the problem with the constant modulus constraint of the complex circular manifold into a Riemannian manifold optimization problem on the complex circular manifold. For this purpose, auxiliary variables

[0077] (7)

[0078] and

[0079] ,

[0080] where is the identity matrix of dimension is a regularization parameter. Since each element of satisfies the unimodular constraint,

[0081] (8)

[0082] Step 4: Equation (8) is a unimodular constrained least squares problem, which can be solved with high performance using the Riemannian Newton method to achieve the transmit beamforming of the phased array system that minimizes the pattern matching error.

[0083] The basic principle of the Riemannian Newton method is to search for the update point along the Riemannian Newton direction (Riemannian Newton gradient) to ensure that the update point is always close to the complex circular manifold, and then project the update point back to the complex circular manifold. The Riemannian Newton direction needs to satisfy

[0084] (9)

[0085] wherein, represents the complex circular manifold, where is the Hadamard product, is a vector of all ones of dimension is the tangent space of the manifold at the point where is a vector of all zeros of dimension is the Riemannian gradient, is the orthogonal projection on is the Euclidean gradient. is the Riemannian Hessian matrix, , , denotes the diagonal matrix with filled on its main diagonal.

[0086] Convert the Newton equation (9) into a real form:

[0087] (10)

[0088] wherein, , , , and , are respectively represented as:

[0089] , ,

[0090] To make positive definite, transform Equation (10) into the following form:

[0091] (11)

[0092] where is the diagonal loading factor. By appropriately selecting , the pseudo-Riemannian Newton gradient at the

[0093] (12)

[0094] Its complex form is

[0095] (13)

[0096] where and are respectively the vectors composed of the first N elements and the last N elements of .

[0097] Let be the search point at the -th iteration, and be the corresponding complex Riemannian Newton gradient obtained by the above method. Determine the update point at the -th iteration as

[0098] (14)

[0099] Project the point on the tangent space onto the complex circular manifold, where represents the contraction operation.

[0100] (15)

[0101] is projected onto the manifold to generate the iterative output, and there is

[0102] (16)

[0103] The specific solution steps of Equation (8) are as follows:

[0104] 1) Input parameters: , , ,

[0105] 2) Initial settings: ,

[0106] 3) Start iteration:

[0107] i. Calculate the Riemannian Newton gradient through formulas (12) and (13)

[0108] ii. Obtain through formulas (14) and (15):

[0109] iii.

[0110] 4) If i.e., select the optimal Stop the loop, otherwise go back to 3).

[0111] 5) Output: , the solution of formula (6) and respectively are vectors composed of the first N elements and the last N elements of .

[0112] Embodiment

[0113] Through Matlab simulation, the phased array system transmitting beamforming method based on the Riemannian Newton method of the present invention is further illustrated.

[0114] 1) Simulation system parameter settings

[0115] During the simulation, taking the uniform linear array as an example, the number of array elements is set to 32, and the element spacing is half wavelength. The spatial region is divided into 181 points with an interval of 1°. The upper limit of the algorithm iteration times is set to , and let . The auxiliary variable is generated by randomizing the initial value. For the zero-nulling focused beam, the desired beam pattern is set to:

[0116] (17)

[0117] The weight coefficients in all directions are set to ; for the focused beam with zero nulls, the desired beam pattern and the preset weight coefficient are respectively set to:

[0118] (18)

[0119] (19)

[0120] 2) Transmit beam pattern plotting

[0121] To visually display the actual effect of beam pattern matching, the solved weight vector can be used to perform weighted combination processing on the signal. Plot the beam pattern generated by the present invention and compare the results with three typical phase-only pattern synthesis methods. Subsequently, by changing the parameter settings, the performance advantages of the method of the present invention are further explored. In the beam pattern, the horizontal axis represents the angular 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 modulus and maximum normalization processing, so as to more clearly reflect the characteristics and advantages and disadvantages of beamforming.

[0122] 3) Measurement metrics

[0123] 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. The specific definitions are as follows:

[0124] (20)

[0125] Among them, and are the sidelobe region and the null region respectively.

[0126] 4) Result analysis

[0127] A total of three instance simulations were carried out in the present invention. The Riemannian Newton method of the present invention was compared with three other representative methods: the alternating direction method of multipliers (ADMM), the majorization-minimization (MM), and the cyclic coordinate descent (CCD). Figures 2 - 5 They are the focused beam patterns synthesized by the four methods without null requirement and with null requirement respectively. Figure 4 and Figure 5 are the beam pattern matching situations of the Riemannian Newton method when changing the weight coefficient applied to the null region. Among them, the setting of the desired beam pattern in the third simulation experiment remains unchanged, and the preset weight coefficient is changed to

[0128] (21)

[0129] Figures 2 - 5The specific index parameters of the synthesized transmit beam are shown in Table 1 and Table 2. As can be seen from Table 1, in the case of a focused beam without nulls, the Riemann-Newton method can obtain the minimum fitting error. Although the main lobe gains produced by all methods are comparable, the Riemann-Newton method has the lowest peak sidelobe level. As can be seen from Table 2, in the case of a focused beam with nulls, the fitting error of the Riemann-Newton method increases, but it still has the lowest peak sidelobe level, and the integrated null depth in the null region is significantly lower than the other three methods.

[0130] Table 1 Simulation results for focused beam without nulls

[0131] Method Riemann - Newton method Alternating direction method of multipliers Upper - bound function minimization method Cyclic coordinate descent method 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

[0132] Table 2 Simulation results for focused beam with nulls

[0133] Method Riemann - Newton method Alternating direction method of multipliers Upper - bound function minimization method Cyclic coordinate descent method 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

[0134] Figure 6 and Figure 7 shows the influence of the weight coefficients in the null region on the performance of the transmit beam when N = 32. The specific index parameters are shown in Table 3. It can be seen that the integrated depth in the null region gradually decreases as the weight coefficient increases, and the gain in the main lobe pointing direction gradually increases. However, after continuing to increase , the effect of deepening the null is very limited, and the peak sidelobe level will increase. Therefore, in practical applications, an appropriate can be selected according to the requirements.

[0135] Table 3 Influence of changing the weight coefficients in the null region on the beam for a focused beam with nulls

[0136] Null 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

[0137] In summary, the method described in the present invention has good comprehensive performance. Compared with other representative methods, the proposed method has almost the best main lobe fitting accuracy in the directional beam fitting, ensures a low sidelobe level, and meets the requirement of a deep null, 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. Specifically: ， 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; 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. The auxiliary variables are determined according to the desired beam pattern. The specific process 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 of phase operation; 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 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 the equivalent transformation in step 2 is converted into: ， In the formula, for The ith element of .

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 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.

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 conditions for stopping the iteration are: ， is the set threshold.

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