A method for precise control of the transmitted beam of a phase-only array under colored noise environment
By unifying the optimization of array gain maximization, main lobe ripple suppression, and side lobe level control under colored noise environment into a Rayleigh quotient optimization problem, and by using the Riemannian manifold framework and augmented Lagrangian method, the array gain loss problem of phase-only arrays under colored noise environment is solved, achieving low-cost and high-efficiency beamforming effect.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- NANJING UNIV OF SCI & TECH
- Filing Date
- 2026-05-12
- Publication Date
- 2026-07-31
AI Technical Summary
Existing technologies have failed to effectively address the loss of array gain performance in phase-only arrays under colored noise environments, resulting in complex beamforming that is difficult to control precisely.
Maximizing array gain, suppressing main lobe ripple, and controlling side lobe levels are modeled as a non-convex constraint Rayleigh quotient optimization problem. Adaptation to colored noise environment is achieved by explicitly embedding the noise covariance matrix. The problem is transformed into a solvable manifold optimization problem using the Riemannian manifold framework and augmented Lagrangian method. The weight vector is iteratively updated to satisfy the optimal conditions of the non-convex constraint.
It maximizes array gain in colored noise environments, reduces hardware complexity and cost, and provides a more efficient beamforming solution suitable for large-scale arrays and low-cost phased array radar systems.
Smart Images
Figure CN122174695B_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the field of array signal processing and radar signal processing technology, specifically a method for precise control of the transmitted beam of a phase-only array under colored noise environment. Background Technology
[0002] Antenna arrays are widely used in many modern remote sensing, radar, and wireless communication systems. Beam pattern is one of the most important characteristics for evaluating array performance. Determining the composite weights of array elements to obtain the desired beam pattern, i.e., beam pattern synthesis, is a fundamental problem. It is well known that beam shape is important and has been extensively considered in the literature. However, in practical applications, given a fixed input signal-to-noise ratio (SNR), the array gain is directly proportional to the output SNR, directly affecting system performance.
[0003] Array gain, as an optimization metric, directly quantifies the signal-to-noise ratio (SNR) improvement capability and explicitly embeds the noise covariance matrix, simultaneously maximizing main lobe power and suppressing sidelobe interference. Compared to traditional metrics (SNR, SNR, etc.), array gain balances main lobe gain, sidelobe suppression, and noise robustness in multi-objective optimization, providing an efficient and reliable beamforming solution for mobile platforms, low-cost systems, and high-interference scenarios. Furthermore, phase adjustment in phase-only transmit arrays requires only a simple phase shifter, significantly reducing hardware complexity, power consumption, and cost, which is beneficial for building low-cost communication and radar systems.
[0004] Previous work has addressed the array gain map synthesis problem using semidefinite relaxation and quasi-convex optimization methods, with maximizing array gain as the performance metric and optimization objective. Other studies have addressed the beamforming problem in phase-only arrays by designing different solutions. For example, the main lobe gain maximization problem was studied, with the sidelobe power sum added as a penalty term to the objective function to suppress sidelobe levels, followed by convex relaxation (CR) to solve the problem. For instance, in addition to maximizing main lobe gain, precise control of sidelobe levels was also achieved. This was achieved by introducing auxiliary variables and using the alternating direction multiplier method (ADMM); or by treating the constant mode constraint as a complex circular manifold and using the Riemann-Newton method (RNM). However, none of the above studies considered the impact of colored noise on array gain performance in real-world operating environments. Unlike white noise, colored noise causes additional losses to array gain performance. Currently, research on array gain problems of phase-only arrays in colored noise environments remains lacking, and the precise beamforming problem in this scenario is even more complex and difficult to solve. Summary of the Invention
[0005] The purpose of this invention is to propose a method for precise control of the transmitted beam of a phase-only array under colored noise conditions.
[0006] The technical solution adopted in this invention is: a method for precise control of the transmitted beam of a phase-only array under colored noise environment, comprising:
[0007] Step 1: The array gain maximization, main lobe ripple suppression and side lobe level control are uniformly modeled as a Rayleigh quotient optimization problem with non-convex constraints. By explicitly embedding the noise covariance matrix in the objective function, the spatial correlation of the colored noise environment is adaptively achieved.
[0008] Step 2: Relax the main and side lobe constraints into penalty terms of the objective function to obtain the augmented Lagrangian function, and map the remaining phase constraints to the complex circular manifold space, thereby transforming the original non-convex optimization problem into a solvable manifold optimization problem;
[0009] Step 3: Design an augmented Lagrangian solver based on the Riemannian manifold framework, and iteratively update the weight vector using manifold gradient descent until the iteration converges to the optimal phase-only weight vector that satisfies the non-convex constraint.
[0010] Compared with existing technologies, the significant advantages of this invention are as follows: This invention, through the Riemann Augmented Lagrange Method (RALM) framework, theoretically solves key problems in traditional methods such as large constraint relaxation errors and weak multi-objective optimization capabilities; at the engineering level, it achieves breakthroughs in low-power hardware adaptation and color noise suppression capabilities, providing a beamforming solution with better performance and lower cost for large-scale arrays, low-cost phased array radars and other systems. Attached Figure Description
[0011] To more clearly illustrate the embodiments of this application or existing technical solutions, the accompanying drawings used in the description of the embodiments or existing technology will be briefly introduced below. Obviously, the drawings described below are merely some embodiments recorded in this application. For those skilled in the art, other drawings can be obtained based on these drawings without any creative effort.
[0012] Figure 1 The spatial signal-to-noise ratio beam pattern of the method proposed in this invention under white noise environment.
[0013] Figure 2 This is the spatial signal-to-noise ratio beam pattern obtained by the RNM method in a white noise environment.
[0014] Figure 3 This is the spatial signal-to-noise ratio beam pattern of the method of the present invention in a colored noise environment.
[0015] Figure 4 This is the spatial signal-to-noise ratio beam pattern of the RNM method in a colored noise environment.
[0016] Figure 5This is a flowchart for obtaining the optimal phase-only array transmit weights using RALM.
[0017] Figure 6 This is a flowchart for solving unconstrained subproblems using the Riemann gradient descent method (RGD). Detailed Implementation
[0018] The present invention will be further described below with reference to the accompanying drawings and examples, which is a method for precise control of the transmission beam of a phase-only array under colored noise environment.
[0019] This invention presents a method for precise control of the transmit beam of a phase-only array in a colored noise environment based on RALM (Radar Evolution of the Lagrangian Model). This method proposes an efficient solution algorithm that integrates array gain maximization, Riemannian manifold optimization, augmented Lagrangian function (ARLM) method, and phase-only array related problems. This algorithm is applicable to solving the multi-constraint phase-only array gain pattern synthesis problem of maximizing array gain. In this algorithm, array gain maximization, main lobe ripple suppression, and side lobe level control are uniformly modeled as a Rayleigh quotient optimization problem with non-convex constraints. By explicitly embedding the noise covariance matrix into the objective function, spatial correlation adaptation to the colored noise environment is achieved. The augmented Lagrangian function transforms the main and side lobe constraints into penalty terms of the objective function, and the remaining phase-only constraints are mapped to the complex circular manifold space, thus transforming the original non-convex optimization problem into a solvable manifold optimization problem. An augmented Lagrangian solver is designed based on the Riemannian manifold framework, and the weight vector is iteratively updated using manifold gradient descent, ultimately converging to the optimal phase-only weight vector satisfying the non-convex constraints.
[0020] The specific implementation steps of this invention are as follows:
[0021] Step 1: Model the array gain maximization, main lobe ripple suppression, and side lobe level control into a unified Rayleigh quotient optimization problem with non-convex constraints. By explicitly embedding the noise covariance matrix into the objective function, we achieve spatial correlation adaptation to the colored noise environment. The specific process is as follows:
[0022] First, define a linear array containing N elements. For simplicity, consider it as an isotropic one-dimensional uniform linear array, and the orientation-dependent steering vector can be given as follows:
[0023]
[0024] In the formula The phase delay of the nth element is given, followed by the power response of the normalized array:
[0025]
[0026] In the formula, For weight vectors, For the conjugate transpose operator. The beam axis direction is the incident angle of the desired signal.
[0027] In the case of a single desired signal, the array observations can be expressed as:
[0028]
[0029] in and These represent the signal component and the noise component, respectively. The covariance matrices of the signal and noise are as follows:
[0030]
[0031]
[0032] in, and These are the power of the signal and the power of the noise, respectively. It is positive semi-definite and used to characterize the structure of noise; it is called the normalized noise covariance matrix. Clearly, the input signal-to-noise ratio is:
[0033]
[0034] For a given weight vector The array output can be written as And the output power satisfies:
[0035]
[0036] in and Let represent the output components of signal and noise, respectively. Then the output signal-to-noise ratio can be expressed as:
[0037]
[0038] Array beamforming improves the output signal-to-noise ratio by coherently introducing a signal and incoherently introducing noise; its performance is measured by the array gain. The specific definition of array gain is:
[0039]
[0040] To ensure a high output signal-to-noise ratio for the array system, under the specific shape constraints of the beam pattern, it is desirable to minimize the main lobe loss, or equivalently maximize the array gain. More specifically, the upper bound of the beam pattern is typically the envelope over some sidelobe regions, while in the main lobe region, it usually requires upper and lower bound constraints on the beam pattern. Based on these beam direction constraints, the optimization problem can be formulated as:
[0041]
[0042] in, The level is the sidelobe region level. and These are the upper and lower boundaries of the ripples in the main lobe region, respectively. These three parameters are preset constants. and These are the angle sets for the main lobe and the side lobe regions, respectively.
[0043] Step 2: Transform the main lobe and side lobe constraints into penalty terms of the objective function by using the augmented Lagrangian function, and map the remaining phase-only constraints to the complex circular manifold space, thereby transforming the original non-convex optimization problem into a solvable manifold optimization problem.
[0044] For the optimization problem described above, we first relax the main and side lobe constraints into penalty terms of the objective function to obtain the augmented Lagrangian function. Then, we map the remaining phase-only constraints to the complex circular manifold space, thus transforming the original non-convex optimization problem into a solvable manifold optimization problem. Augmented Lagrangian function The specific expression is:
[0045]
[0046] In the formula, , , ; , and These are the penalty factors; , and These are the corresponding Lagrange multipliers; and For the corresponding , The number of unit angles in the angle set; This indicates the search for the maximum value of two real numbers.
[0047] Apply the characteristic-only constraint to a complex circular manifold. Indicates, that is The original constrained optimization problem is transformed into the following unconstrained optimization problem of the manifold:
[0048]
[0049] Step 3: Design an augmented Lagrangian solver based on the Riemannian manifold framework, and iteratively update the weight vector using manifold gradient descent until the iteration converges to the optimal phase-only weight vector that satisfies the non-convex constraint.
[0050] Step 3.1: Randomly generate initial weights Set the initial Lagrange multipliers , and and punishment factors , and Define the iteration index ;
[0051] Step 3.2: Solve the unconstrained optimization problem of the manifold using the Riemann gradient descent method to obtain the... The optimal phase-only weight vector in the next iteration ;
[0052] Step 3.3: Update each factor , and The iterative expression is:
[0053]
[0054]
[0055]
[0056] in, It is a truncation function that makes the variable The output value is at the specified minimum value and maximum value between, This indicates the search for the minimum value of two real numbers.
[0057]
[0058]
[0059]
[0060] in, This refers to the constraint violation degree at the (k+1)th iteration, used to quantify the degree of deviation of the current solution from the sidelobe suppression and main lobe ripple constraints.
[0061]
[0062]
[0063]
[0064] in, A proportional threshold is used to determine whether the decay of sidelobe constraint violations is sufficient. When the rate of decrease in the violation is insufficient, the penalty coefficient is adjusted. It controls the amplification factor of the penalty coefficient, enhancing the effectiveness of the constraint when the amount of violation is insufficient to decay. For the first The first of the Lagrange multipliers Each element For the first The first of the Lagrange multipliers Each element For the first The first of the Lagrange multipliers Each element For the first The degree of violation For the first The degree of violation For the first The degree of violation For the first The penalty factor, For the first The penalty factor, For the first The penalty factor, For a preset vector The One element, For a preset vector The One element, For a preset vector The One element, For a preset vector The One element, For a preset vector The One element, For a preset vector The Each element.
[0065] Step 3.3: Update all Lagrange multipliers, constraint violation degree, and penalty factor if the conditions are met. and , No. The optimal phase-only weight vector in the next iteration This is the final optimal phase-only weight vector; otherwise, return to step 3.2. For the first The value of the augmented Lagrange function in the next iteration. For the first The value of the augmented Lagrange function in the next iteration. Tolerance factor, This represents the maximum number of iterations.
[0066] In a further embodiment, for step 3.2, the Riemann gradient descent method is used to solve the unconstrained optimization problem of the manifold to obtain the first... The optimal phase-only weight vector in the next iteration The specific method is as follows:
[0067] Step 3.2.1: Define Riemannian manifold constraint parameters Initialize the iteration index j=0 and the weights Set the maximum number of iterations. ;
[0068] Step 3.2.2: Construct the Riemann gradient operator and implement manifold constraints through orthogonal projection:
[0069]
[0070] in, It is the Riemann gradient. It is a Euclidean gradient. To perform operations on the real part of a complex number, This represents the dot product operation. This indicates the conjugate operation.
[0071] Step 3.2.3: Design an adaptive step size selection mechanism, using Armijo backtracking search to determine the step size. The descent condition is met:
[0072]
[0073] in, For line search coefficients, In the direction of descent, This indicates finding the 2-norm of a vector.
[0074] Step 3.2.4: Perform manifold projection update. The formula for the (k+1)th iteration is:
[0075]
[0076] Step 3.2.5: Determine if the iteration termination condition has been met. or Then output The optimal weight is determined; otherwise, the optimal weight is determined. Return to step 3.2.2.
[0077] Example
[0078] The invention's method for phase-only array multi-constraint gain beamforming in colored noise environments, based on RALM, is further illustrated through MATLAB simulation.
[0079] 1) Simulation parameter settings
[0080] A uniform linear array consisting of N=32 elements is used to represent the spatial angle. by The intervals are discrete into 181 angles, and set... Auxiliary variables are all initialized randomly.
[0081] The noise model here can be replaced with colored noise, which better reflects the actual environment. For ease of explanation, the normalized noise covariance matrix in the objective function is modeled as an exponential decay model, and its matrix elements satisfy: ,in This is represented as the correlation of noise. , This represents Gaussian white noise. The larger the value, the stronger the correlation.
[0082] The experimental constraints were set as follows: main lobe ripple constraint of 0.5dB, main lobe width of 10°, side lobe suppression constraint of -10dB, and Gaussian white noise.
[0083] 2) Beam plotting
[0084] Spatial signal-to-noise ratio (SNR) is a metric used in array signal processing to measure the ratio of signal power to noise power in a specific direction. It is defined as follows:
[0085]
[0086] In the formula, the molecule It is a weight vector In direction The array response power (i.e., signal power); denominator The unit power of the array output noise, including the colored noise structure, reflects the signal-to-noise ratio per unit noise power. Its value depends only on the spatial structure of the signal and noise, not the absolute noise power. This formula incorporates the influence of noise structure and can intuitively reflect the algorithm's ability to suppress noise in all directions.
[0087] 3) Measurement indicators
[0088] In this invention, in addition to the beammap, the minimum power gain (MPG) of the main lobe region is also used as a metric. The definition of the minimum power gain (MPG) of the main lobe region is as follows:
[0089]
[0090] in, The main lobe region is considered. Under the same initial conditions, the higher the minimum spatial signal-to-noise ratio of the main lobe region, the better the beamforming effect.
[0091] 4) Results Analysis
[0092] The invention has been simulated in four instances. Figure 1This is the spatial signal-to-noise ratio beam pattern obtained by this method in a white noise environment; Figure 2 This is the spatial signal-to-noise ratio beam pattern obtained by the RNM method under white noise conditions; Figure 3 This method is applicable to colored noise environments. The resulting spatial signal-to-noise ratio beam pattern; Figure 4 It is a spatial signal-to-noise ratio beam pattern obtained by RNM under colored noise conditions.
[0093] pass Figures 1 to 4 In comparison, the method of the present invention performs comparably to RNM under white noise; under colored noise, the method of the present invention exhibits better adaptability, with the main lobe width and gain remaining stable. This indicates that the method of the present invention has greater practical value in complex noise environments, and is particularly suitable for practical applications requiring high robustness.
[0094] In summary, the method proposed in this invention exhibits excellent power pattern synthesis performance. Compared with other representative methods, the proposed method demonstrates superior gain performance even in relatively complex environments. Its application in systems such as radar and satellite communications can significantly reduce hardware costs and system complexity, demonstrating high application value.
Claims
1. A method for accurate control of phase-only array transmit beam in a colored noise environment, characterized in that, include: Step 1: The array gain maximization, main lobe ripple suppression, and side lobe level control are uniformly modeled as a Rayleigh quotient optimization problem with non-convex constraints. By explicitly embedding the noise covariance matrix into the objective function, spatial correlation adaptation to colored noise environments is achieved. The Rayleigh quotient optimization problem with non-convex constraints is specifically as follows: ， In the formula, The phase-only array weight vector; For an N-dimensional uniform linear array at an angle The guide vector in the direction, where Let the incident angle of the desired signal be... For the phase delay of the nth unit, From the angle of the main lobe region, For the angle of the side lobe region; It is the conjugate transpose operator; This is the normalized noise covariance matrix; The level is the sidelobe region level. and These are the upper and lower boundaries of the ripples in the main lobe region, respectively. and These are the sets of angles for the main lobe and the side lobe regions, respectively. The phase-only array weight vector The nth element, where N is the number of elements in the linear array. This is the normalized noise covariance matrix; Step 2: Relax the main and side lobe constraints into penalty terms of the objective function to obtain the augmented Lagrangian function, and map the remaining phase constraints to the complex circular manifold space, thereby transforming the original non-convex optimization problem into a solvable manifold optimization problem; Step 3: Design an augmented Lagrangian solver based on the Riemannian manifold framework, and iteratively update the weight vector using manifold gradient descent until the iteration converges to the optimal phase-only weight vector that satisfies the non-convex constraint.
2. The method for precise control of the phase-only array transmitted beam under colored noise environment according to claim 1, characterized in that, The augmented Lagrangian function obtained by relaxing the main and side lobe constraints into a penalty term of the objective function is as follows: ， In the formula, , , ; , and These are the penalty factors; , and These are the corresponding Lagrange multipliers; and These represent the number of unit angles in the angle sets of the main lobe and side lobe regions, respectively. This indicates the search for the maximum value of two real numbers.
3. The method for precise control of the phase-only array transmitted beam under colored noise environment according to claim 1, characterized in that, The specific method for mapping the remaining phase-only constraints to the complex circular manifold space, thereby transforming the original non-convex optimization problem into a solvable manifold optimization problem, is as follows: The unique constraint in the Rayleigh quotient optimization problem is represented by a complex circular manifold. Indicates, that is The original constrained optimization problem is transformed into the following unconstrained optimization problem of the manifold: ， In the formula, The augmented Lagrangian function obtained by relaxing the main and side lobe constraints into a penalty term of the objective function is... It is an N-dimensional complex vector. This means that it holds true for any value of n, where n is from 1 to n. Natural numbers between.
4. The method for precise control of the phase-only array transmitted beam under colored noise environment according to claim 1, characterized in that, The specific method for designing an augmented Lagrangian solver based on the Riemannian manifold framework, iteratively updating the weight vector using manifold gradient descent, and finally iteratively converging to the optimal phase-only weight vector satisfying the non-convex constraint is as follows: Step 3.1: Randomly generate initial weights Set the initial Lagrange multipliers , and and punishment factors , and Define the iteration index ,in, It is a complex circular manifold; Step 3.2: Solve the unconstrained optimization problem of the manifold using the Riemann gradient descent method to obtain the... The optimal phase-only weight vector in the next iteration ; Step 3.3: Update each Lagrange multiplier, constraint violation degree, and penalty factor, if satisfied. and , No. The optimal phase-only weight vector in the next iteration This is the final optimal phase-only weight vector; otherwise, return to step 3.
2. For the first The value of the augmented Lagrange function in the next iteration. For the first The value of the augmented Lagrange function in the next iteration. Tolerance factor, This represents the maximum number of iterations.
5. The method for precise control of the phase-only array transmitted beam under colored noise environment according to claim 4, characterized in that, The Riemann gradient descent method is used to solve the unconstrained optimization problem of the manifold, and the result is obtained. The optimal phase-only weight vector in the next iteration The specific method is as follows: Step 3.2.1: Define the complex circular manifold corresponding to the phase-only constraint as follows: Initialize the iteration index j=0 and the weights Set the maximum number of iterations. ; Step 3.2.2: Construct the Riemann gradient operator and implement manifold constraints through orthogonal projection: ， in, It is the Riemann gradient. It is a Euclidean gradient. To perform the operation of taking the real part, This represents the Hadamard product operation. This indicates the conjugate operation; Step 3.2.3: Design an adaptive step size selection mechanism, using Armijo backtracking search to determine the step size. The descent condition is met: ， in, For line search coefficients, In the direction of descent, This indicates finding the 2-norm of a vector. Let be the phase-only weight vector for the j-th iteration; Step 3.2.4: Perform manifold projection update to obtain the phase-only weight vector for the (j+1)th iteration, with the specific formula as follows: ， Step 3.2.5: Determine if the iteration termination condition has been met. or If satisfied, output This is the optimal phase-only weight vector for this iteration; if it does not satisfy this condition, then let... Return to step 3.2.
2.
6. The method for precise control of the phase-only array transmitted beam under colored noise environment according to claim 5, characterized in that, The specific formulas for updating each Lagrange multiplier, constraint violation degree, and penalty factor are as follows: ， ， ， ， ， ， ， ， ， in, It is a truncation function that makes the variable The output value is in and Between, among Both b and are given values. Always true This indicates finding the minimum value of two real numbers; It is the constraint violation degree of the (k+1)th iteration, used to quantify the degree of deviation of the current solution from the sidelobe suppression and main lobe ripple constraints; This is a proportional threshold used to determine whether the decay of sidelobe constraint violations is sufficient. When the rate of decrease in the violation is insufficient, the penalty coefficient is adjusted. It controls the amplification factor of the penalty coefficient, enhancing the effectiveness of the constraint when the attenuation of the violation is insufficient. and These represent the number of unit angles in the angle sets of the main lobe and side lobe regions, respectively. , , ; For the first The first of the Lagrange multipliers Each element For the first The first of the Lagrange multipliers Each element For the first The first of the Lagrange multipliers Each element For the first The degree of violation For the first The degree of violation For the first The degree of violation For the first The penalty factor, For the first The penalty factor, For the first The penalty factor, For a preset vector The One element, For a preset vector The One element, For a preset vector The One element, For a preset vector The One element, For a preset vector The One element, For a preset vector The Each element.