Quantization phase coding waveform design method based on alternating direction multiplier method

Abstract

The invention discloses a quantization phase coding waveform design method based on an alternating direction multiplier method, and the method comprises the steps: constructing an optimization model with a minimum integral sidelobe level as a target, and decomposing a complex non-convex optimization problem faced by the optimization model into two sub-problems, namely continuous waveform optimization and quantization phase projection, which can be efficiently solved through an ADMM frame. In the iteration process, an adjacent operator is innovatively introduced, and the obtained phase is accurately quantized into a preset quantized phase set, so that a constant modulus signal with low sidelobe characteristic and high engineering realizability is generated.

Description

Technical Field

[0001] This invention belongs to the field of radar signal processing technology, specifically relating to a quantization phase coding waveform design method based on the alternating direction multiplier method. Background Technology

[0002] With the escalation of electronic warfare, frequency modulation (FM) signals face challenges in maintaining effectiveness in electronic warfare environments due to their high real-time complexity. In contrast, phase modulation techniques offer significant advantages, including stronger noise immunity, higher bandwidth efficiency, robustness to multipath effects, lower power consumption, and simplified hardware complexity. These characteristics make phase modulation techniques particularly suitable for digital communication systems, enabling efficient data transmission within limited bandwidth while maintaining immunity to noise and multipath interference. However, during radar signal pulse compression, high sidelobes inevitably reduce the detection capability of weak targets. Furthermore, some radar systems require phase quantization to simplify signal generation and processing. Therefore, designing low-sidelobe phase-coded waveforms with phase quantization characteristics has become a key research focus.

[0003] Numerous studies have been conducted on the design of low-sidelobe phase coding sequences. Numerical optimization algorithms typically follow a general process: first, a mathematical model is constructed with the goal of minimizing the integrated sidelobe level (ISL) or peak sidelobe level (PSL); then, various optimization methods such as gradient descent and the alternating direction method of multipliers (ADMM) are used to solve the model. With the continuous development of science and technology, deep learning methods have demonstrated superior performance in terms of computational efficiency: first, a training dataset containing randomly initialized waveforms and their performance metrics is generated; then, a neural network is designed and trained by minimizing a preset cost function (such as the desired ISL value); finally, waveforms are generated using the trained model, and its performance is verified through simulation. However, the application of deep learning is often limited by the quality of the training data; in addition, training complex deep learning models requires a large amount of computational resources, including high-performance hardware support. Nevertheless, most existing algorithms still focus on the design of continuous-phase waveforms, i.e., the phase can take any value in the range [0, 2π).

[0004] While continuous phase-coded waveforms offer considerable theoretical design freedom, their implementation is highly complex—requiring not only high-resolution digital-to-analog converters and sophisticated analog circuits, but also susceptible to distortion due to noise and component defects. Quantized phase-coded waveforms effectively mitigate these drawbacks and are therefore widely used in practical radar and communication systems. Although heuristic algorithms can be used to design quantized phase-coded waveforms, their high computational complexity makes them less effective in handling increasingly complex environments. Methods such as Simulated Annealing (SA) and Genetic Algorithms (GA), while capable of phase quantization, also suffer from high computational complexity; while algorithms like Majorization-Minimization (MM) and Extremal Point Pursuit (EXPP), while reducing computational complexity, cannot achieve phase quantization. Summary of the Invention

[0005] To address the aforementioned problems in the existing technology, this invention provides a quantization phase encoding waveform design method based on the alternating direction multiplier method. The technical problem to be solved by this invention is achieved through the following technical solution: This invention provides a quantization phase-encoded waveform design method based on the alternating direction multiplier method, the quantization phase-encoded waveform design method comprising: An optimization model for the quantized phase-coded waveform is established. The optimization objective of the model is to minimize the integral sidelobe level of the phase-coded waveform sequence. The constraints include that each symbol in the phase-coded waveform sequence satisfies the constant modulus condition and that the phase of each symbol belongs to the quantized phase set. By introducing auxiliary variables and Lagrange multipliers, the constrained optimization model of the quantized phase-encoded waveform is reconstructed into an unconstrained optimization model. Solving the unconstrained optimization model and outputting the optimal phase-coded waveform; the process of solving the unconstrained optimization model includes: given initial auxiliary variables and initial Lagrange multipliers, and fixing the initial auxiliary variables and initial Lagrange multipliers, solving the unconstrained optimization model to obtain an intermediate phase-coded waveform sequence; fixing the intermediate phase-coded waveform sequence and the initial Lagrange multipliers, quantizing the phases of the intermediate phase-coded waveform sequence and the initial Lagrange multipliers onto a quantized phase set while simultaneously satisfying the constant modulus condition, obtaining intermediate auxiliary variables; based on the intermediate phase-coded waveform sequence... The process involves determining whether the iteration stopping condition is met using intermediate auxiliary variables. If it is, the intermediate phase-coded waveform sequence of the current iteration is taken as the optimal phase-coded waveform sequence. If it is not, the intermediate Lagrange multiplier is calculated based on the intermediate phase-coded waveform sequence, intermediate auxiliary variables, and initial Lagrange multipliers. The intermediate auxiliary variables are then used as initial auxiliary variables, and the intermediate Lagrange multipliers are used as initial Lagrange multipliers. The process is repeated until the iteration stopping condition is met.

[0006] The beneficial effects of this invention are: This invention proposes a quantization phase-encoded waveform design method based on the alternating direction multiplier method. Using a phase-encoded signal, compared to a linear frequency modulated signal, the core advantage of the phase-encoded signal lies in its sophisticated internal modulation, perfectly resolving the contradiction between "long operating range" and "high range resolution" in radar detection. More importantly, this invention constructs an optimization model aimed at minimizing the integral sidelobe level and utilizes the ADMM framework to decompose the complex non-convex optimization problem into two efficiently solvable sub-problems: continuous waveform optimization and quantized phase projection. During the iteration process, a proximity operator is innovatively introduced to accurately quantize the obtained phase into a preset quantized phase set, thereby generating a constant-mode signal with both low sidelobe characteristics and high engineering feasibility. Experimental results show that the waveform designed in this invention outperforms existing mainstream methods in both sidelobe suppression performance and algorithm computational efficiency.

[0007] The present invention will be further described in detail below with reference to the accompanying drawings and embodiments. Attached Figure Description

[0008] Figure 1 This is a flowchart illustrating a quantization phase encoding waveform design method based on the alternating direction multiplier method provided in an embodiment of the present invention; Figure 2 These are the autocorrelation waveforms corresponding to the phase-encoded waveform sequences output at different stages provided in the embodiments of the present invention. , ) Schematic diagram; Figure 3This is based on the embodiments of the present invention. Figure 2 A schematic diagram of the phase distribution of the optimal phase-coded waveform sequence; Figure 4 These are the normalized ISL values ​​for different iteration numbers provided in the embodiments of the present invention. Comparison diagram; Figure 5 The four algorithms provided in the embodiments of this invention are , A comparative diagram of autocorrelation waveforms is shown below; Figure 6 The four algorithms provided in the embodiments of this invention are , A schematic diagram comparing the autocorrelation waveforms below. Detailed Implementation

[0009] The present invention will be further described in detail below with reference to specific embodiments, but the implementation of the present invention is not limited thereto.

[0010] Please see Figure 1 This invention provides a quantization phase encoding waveform design method based on the alternating direction multiplier method, specifically including the following steps: S10. Establish an optimization model for the quantized phase-coded waveform. The optimization objective of the optimization model is to minimize the integral sidelobe level of the phase-coded waveform sequence. The constraints include that each symbol in the phase-coded waveform sequence satisfies the constant modulus condition and that the phase of each symbol belongs to the quantized phase set.

[0011] Suppose: the phase-coded waveform sequence is denoted as , , Indicates the first phase in the phase-coded waveform sequence Each symbol. To improve transmission efficiency, in this embodiment of the invention, each symbol in the phase-coded waveform sequence is constrained to a constant-modulus signal, that is: (1); in, , express phase, , This represents the set of quantized phases. For phases within the interval... A multi-phase coded signal with a uniform internal distribution, if its quantization phase number is denoted as... The corresponding quantization phase set is defined as follows: (2); Quantizing the phase into a uniformly distributed discrete point helps to better control the uniformity and correlation of the signal, which is particularly advantageous in applications involving nonlinear amplifiers.

[0012] The optimization model for the quantized phase-coded waveform established in this embodiment of the invention aims to minimize the integral sidelobe level of the phase-coded waveform sequence, thereby reducing the sidelobe energy after pulse compression. The autocorrelation coefficient of the phase-coded waveform sequence is denoted as... Then the calculation of the integral sidelobe level can be expressed as: (3); Among them, the autocorrelation coefficient The specific form is as follows: (4); in, express The conjugate operation. To achieve the goal of minimizing ISL and suppressing sidelobes, an optimization model for the final quantization phase-coded waveform is established based on formulas (1), (2), and (3), and is expressed as: (5); in, Represents a phase-coded waveform sequence. , Indicates the first phase in the phase-coded waveform sequence Each code element This indicates the number of symbols in the phase-coded waveform sequence. express The conjugate transpose operation. , , This indicates calculating the square of the modulus. express The constant modulus condition is met. Represents the set of quantized phases. This indicates the number of phase discretizations. This indicates the phase calculation operation. express The phase belongs to .

[0013] The present invention uses a phase-coded signal because a long pulse is used to ensure sufficient transmission energy to achieve long-distance detection. At the same time, a large bandwidth is obtained through precise phase transitions within the pulse, which allows the long pulse energy to be compressed into an extremely narrow main lobe at the receiving end through pulse compression technology. This achieves extremely high range resolution and effectively resists interference and interception.

[0014] S20. By introducing auxiliary variables and Lagrange multipliers, the optimization model of the constrained quantized phase-encoded waveform is reconstructed into an unconstrained optimization model.

[0015] The embodiments of the present invention first introduce auxiliary variables. The optimization model shown in formula (6) is reconstructed and expressed as follows: (6); in, Indicates will Phase projection in The projection operator in , This indicates the phase calculation operation. Indicates the first auxiliary variable One element, express The phase belongs to , express The phase does not belong to , express The constant modulus condition is met.

[0016] Due to auxiliary variables Constrained By restricting and fixing the variables, this problem is equivalent to the original problem. However, the variables in the original problem are subject to phase quantization constraints. The variables are transformed into unconstrained variables, while the constraints are transferred to auxiliary variables. Then, the ADMM algorithm can be used to solve the problem.

[0017] Next, by introducing Lagrange multipliers Formula (6) is then used to construct the final unconstrained optimization model, which is expressed as: (7); in, These represent the phase-coded waveform sequence, auxiliary variables, and Lagrange multipliers, respectively. Indicates about An unconstrained optimization model. This indicates the number of symbols in the phase-coded waveform sequence. express The conjugate transpose operation. , , This indicates calculating the square of the modulus. , Represents the set of quantized phases. This indicates the number of quantized phases in the quantized phase set. It is a penalty parameter. This indicates calculating the square of the L2 norm. (Penalty parameter) A value that is too small will lead to xUnable to converge; conversely, penalty parameter An excessively large value for the penalty parameter may cause the optimization model to fail. Extensive numerical experiments have shown that adjusting the penalty parameter... The value is set to 5 N Up to 7 N Within the range, it can effectively guarantee x Towards z Convergence is achieved while ensuring numerical stability throughout the optimization process.

[0018] S30. Solve the unconstrained optimization model and output the optimal phase-coded waveform. The process of solving the unconstrained optimization model includes: given initial auxiliary variables and initial Lagrange multipliers, and fixing the initial auxiliary variables and initial Lagrange multipliers, solving the unconstrained optimization model to obtain an intermediate phase-coded waveform sequence; fixing the intermediate phase-coded waveform sequence and initial Lagrange multipliers, quantizing the phases of the intermediate phase-coded waveform sequence and initial Lagrange multipliers onto a quantized phase set while simultaneously satisfying the constant modulus condition, to obtain intermediate auxiliary variables; based on the intermediate phase-coded waveform... The sequence and intermediate auxiliary variables are used to determine whether the iteration stopping condition is met. If it is met, the intermediate phase-coded waveform sequence of the current iteration is taken as the optimal phase-coded waveform sequence. If it is not met, the intermediate Lagrange multiplier is calculated based on the intermediate phase-coded waveform sequence, intermediate auxiliary variables, and initial Lagrange multipliers. The intermediate auxiliary variables are used as initial auxiliary variables, and the intermediate Lagrange multipliers are used as initial Lagrange multipliers. The process of fixing the initial auxiliary variables and initial Lagrange multipliers and solving the unconstrained optimization model to obtain the intermediate phase-coded waveform sequence continues until the iteration stopping condition is met.

[0019] For the unconstrained optimization model shown in formula (7), this invention proposes a method for solving the unconstrained optimization model, specifically including: given initial auxiliary variables and initial Lagrange multipliers, and fixing the initial auxiliary variables and initial Lagrange multipliers, solving the unconstrained optimization model to obtain the intermediate phase encoded waveform sequence; fixing the intermediate phase encoded waveform sequence and the initial Lagrange multipliers, quantizing the phases of the intermediate phase encoded waveform sequence and the initial Lagrange multipliers onto the quantized phase set while simultaneously satisfying the constant modulus condition, to obtain the intermediate auxiliary variables; based on the intermediate phase encoded waveform... The sequence and intermediate auxiliary variables are used to determine whether the iteration stopping condition is met. If it is, the intermediate phase-coded waveform sequence of the current iteration is taken as the optimal phase-coded waveform sequence. If not, the intermediate Lagrange multipliers are calculated based on the intermediate phase-coded waveform sequence, intermediate auxiliary variables, and initial Lagrange multipliers. The intermediate auxiliary variables are then used as initial auxiliary variables, and the intermediate Lagrange multipliers are used as initial Lagrange multipliers. The process of fixing the initial auxiliary variables and initial Lagrange multipliers and solving the unconstrained optimization model to obtain the intermediate phase-coded waveform sequence continues until the iteration stopping condition is met. Specifically: Step 1: Solve the phase-coded waveform sequence

[0020] Given initial auxiliary variables and initial Lagrange multipliers, for example: initialize the phase-coded waveform sequence as follows: Initial auxiliary variables Initial Lagrange multipliers .

[0021] By fixing the initial auxiliary variables and initial Lagrange multipliers, the unconstrained optimization model is solved to obtain an intermediate phase-coded waveform sequence. This process includes: fixing the initial auxiliary variables and initial Lagrange multipliers, transforming the unconstrained optimization model into a first unconstrained sub-optimization model; converting the solution of the first unconstrained sub-optimization model from complex form to real-valued form to obtain a second unconstrained sub-optimization model; using the NAG (Nesterov Accelerated Gradient) algorithm to solve the second unconstrained sub-optimization model to obtain the real-valued form of the phase-coded waveform sequence; and converting the real-valued form of the phase-coded waveform sequence into a complex form as the intermediate phase-coded waveform sequence. By fixing the initial auxiliary variables and the initial Lagrange multipliers, the unconstrained optimization model shown in formula (7) can be converted into a first unconstrained sub-optimization model, expressed as follows: (8); in, Indicates the first The intermediate phase-coded waveform sequence obtained from the next iteration. Indicates the first The initial auxiliary variables obtained in the next iteration Indicates the first The initial Lagrange multipliers obtained in the next iteration Represents a phase-coded waveform sequence. This indicates the number of symbols in the phase-coded waveform sequence. express The conjugate transpose operation. , , It is a penalty parameter. This indicates that the square of the L2 norm is being calculated.

[0022] It is evident that formula (8) is an unconstrained optimization problem, which can be solved using various optimization methods. This embodiment of the invention proposes converting the complex-form optimization model shown in formula (8) into a real-valued optimization model, namely, the second unconstrained sub-optimization model, expressed as: (9); in, Indicates the first The real-valued form of the intermediate phase-coded waveform sequence obtained in the next iteration. Represents the real-valued form of the phase-coded waveform sequence. express The transpose operation, , , This indicates the operation of taking the real part. This indicates the operation of taking the imaginary part. , , Indicates the first The real-valued form of the initial auxiliary variable obtained in the next iteration. Indicates the first The real-valued form of the initial Lagrange multipliers obtained in the next iteration. It is a penalty parameter. This indicates that the square of the L2 norm is being calculated.

[0023] To improve efficiency, this invention utilizes the NAG algorithm to solve formula (9). The NAG algorithm can predict the next position using momentum and correct the gradient direction in advance, thereby accelerating the standard gradient descent process. The specific process of this algorithm is shown in Algorithm 1.

[0024]

[0025] In Algorithm 1 That is, in formula (9) Solving using Algorithm 1 yields... After that, it is also necessary to Convert from real-valued form to response number form to obtain the intermediate phase-coded waveform sequence of the current iteration, for example, if it is the th... In the next iteration, the intermediate phase-coded waveform sequence is denoted as... .

[0026] Step 2: Solve for auxiliary variables

[0027] In this embodiment of the invention, the intermediate phase encoded waveform sequence and the initial Lagrange multiplier are fixed. The phases of the intermediate phase encoded waveform sequence and the initial Lagrange multiplier are quantized onto a set of quantized phases using a neighbor operator, simultaneously satisfying the constant modulus condition, to obtain intermediate auxiliary variables. These include: fixing the intermediate phase encoded waveform sequence and the initial Lagrange multiplier; converting the unconstrained optimization model into a third unconstrained sub-optimization model; solving the third unconstrained sub-optimization model using a neighbor operator to map the phases of the intermediate phase encoded waveform sequence and the initial Lagrange multiplier to the nearest quantized phase; and solving for the intermediate auxiliary variables based on the nearest quantized phase. By fixing the intermediate phase encoded waveform sequence obtained in the first step and the given initial Lagrange multipliers, the unconstrained optimization model shown in Equation (7) is transformed into a third unconstrained sub-optimization model, expressed as follows: (10); in, Indicates the first The intermediate auxiliary variables obtained in the iteration are the first One element, The value can be 1~ , This indicates the number of symbols in the phase-coded waveform sequence. Indicates the first auxiliary variable One element, express The constant modulus condition is met. Represents the set of quantized phases. This indicates the number of quantized phases in the quantized phase set. Indicates will Phase projection in The projection operator in Indicates the first The intermediate phase encoded waveform sequence obtained in the nth iteration is the first Each code element Indicates the first The initial Lagrange multipliers obtained in the second iteration are the first... One element, It is a penalty parameter. This indicates that the square of the L2 norm is being calculated.

[0028] The most recent quantization phase of the mapping is expressed by the formula: (11); in, Indicates the index of the most recent quantized phase of the mapping. Indicates the first The intermediate phase encoded waveform sequence obtained in the nth iteration is the first Each code element Indicates the first The initial Lagrange multipliers obtained in the second iteration are the first... One element, This indicates the index of the quantized phase in the quantized phase set. , This indicates the number of quantized phases in the quantized phase set. Indicates the th phase in the most recent quantization phase Each element.

[0029] Finally, the most recent quantization phase is calculated. Then, the current intermediate auxiliary variable can be calculated, and the formula is expressed as: (12); in, Indicates the first Intermediate auxiliary variables obtained from the next iteration.

[0030] The results were obtained through the first and second steps respectively. , ,Will As an iteration stopping condition, if the iteration stopping condition is met, the intermediate phase encoded waveform sequence of the current iteration is used. If the optimal phase-coded waveform sequence does not meet the iteration stopping condition, then proceed to the third step.

[0031] Step 3: Solve for the Lagrange multipliers

[0032] In this embodiment of the invention, the intermediate Lagrange multiplier is calculated using the following formula: (13); in, Indicates the first The intermediate Lagrange multipliers obtained in the next iteration Indicates the first The initial Lagrange multipliers obtained in the next iteration Indicates the first The intermediate phase-coded waveform sequence obtained in the next iteration is the result of the first step. Indicates the first The intermediate auxiliary variables obtained in the second iteration are the results of the second solution.

[0033] Furthermore, As the initial Lagrange multiplier, as The initial auxiliary variables are returned to the steps in the first step of fixing the initial auxiliary variables and initial Lagrange multipliers, and solving the unconstrained optimization model to obtain the intermediate phase-coded waveform sequence. The next iteration is then performed to solve the intermediate phase-coded waveform sequence. The first, second, and third steps are repeated until the iteration stopping condition is met. The intermediate phase-coded waveform sequence obtained at this point is taken as the optimal phase-coded waveform sequence.

[0034] To verify the effectiveness of the quantization phase encoding waveform design method based on the alternating direction multiplier method provided in this embodiment of the invention, the following experiments were conducted.

[0035] I. Algorithm Iteration Analysis The initial phase-coded waveform sequence is generated randomly. The method proposed in this invention is denoted as the PO-ADMM algorithm, and the iterative process of the PO-ADMM algorithm is described and analyzed. The momentum coefficient of the NAG algorithm is set as... The learning rate is set to .

[0036] Figure 2 The experimental results are shown when the number of symbols in the phase-coded waveform sequence is N=60. Figure 2 The horizontal axis represents the number of symbols in the phase-coded waveform sequence, and the vertical axis represents the autocorrelation coefficient of the phase-coded waveform sequence. Figure 2 In the diagram, the blue curve represents the autocorrelation waveform corresponding to the initial phase-coded waveform sequence, the black curve represents the autocorrelation waveform corresponding to the phase-coded waveform sequence after one iteration of the PO-ADMM algorithm proposed in this invention, and the red curve represents the autocorrelation waveform corresponding to the optimal phase-coded waveform sequence output after multiple iterations.

[0037] Figure 3 The phase distribution of the optimal phase-coded waveform sequence is presented. Figure 3 The horizontal axis represents the number of symbols in the optimal phase-coded waveform sequence, and the vertical axis represents the phase of the optimal phase-coded waveform sequence. Figure 3 It is evident that the phase of the optimal phase-coded waveform sequence is constrained within a specific set of quantized phases. These visualizations clearly demonstrate the iterative optimization process and verify that the PO-ADMM algorithm proposed in this invention can satisfy the preset phase constraint conditions while reducing the integral sidelobe level.

[0038] Figure 4 The normalized integral sidelobe level (ISL) values ​​are given for different iteration numbers. Figure 4 The horizontal axis represents the number of iterations, and the vertical axis represents the normalized ISL value. Figure 4 It can be seen that the number of different symbols in the phase-coded waveform sequence The corresponding normalized ISL curves all show a trend of first decreasing rapidly and then stabilizing, and the number of symbols... The larger the value, the smoother the curve. In the iterative process of the Alternating Direction Multiplier Method (ADMM), the neighbor operator can cause a temporary mismatch between local corrections and the global direction, leading to oscillations; and as the number of symbols increases... The increase in the number of symbols improves the dimensionality and degrees of freedom of the optimization problem, thus improving the problem conditions. It can make fuller use of statistical properties to suppress fluctuations, and can also achieve better coordination between the neighbor operator and the global objective in the iteration of ADMM, making the variable coupling more consistent. Therefore, the oscillation intensity is reduced and the convergence curve is more stable.

[0039] II. Differences and Optimization result analysis The algorithms PO-ADMM, p-MM (Penalized Majorization-Minimization), DPM (Discrete Phase Modulation), and GA are compared under different conditions. and The optimization performance of the p-MM algorithm is compared. Specifically, the core of the p-MM algorithm is waveform design based on norm minimax optimization, the DPM algorithm uses coordinate descent for waveform design, and the GA algorithm is a search-based heuristic method. The p-MM algorithm also incorporates the nearest neighbor operator to achieve phase quantization. The experimental setup is as follows: (1) The number of quantized phases in the quantized phase set The number of symbols in the phase-encoded waveform sequence is fixed at 2, and is set to 2 respectively. For the NAG algorithm, the momentum coefficient is... Learning rate .

[0040] (2) Number of symbols in the phase-coded waveform sequence The value is fixed at 30, and the number of quantization phases in the quantization phase set is respectively taken as... K ={4,8,16}. At this point, the momentum coefficient is... The learning rate is .

[0041] Figure 5 A comparison was made between PO-ADMM, p-MM, DPM, and GA. ( The optimization results under ) Figure 5 The horizontal axis represents the number of symbols in the phase-coded waveform sequence, and the vertical axis represents the autocorrelation coefficient of the phase-coded waveform sequence. The red curve corresponds to the results of the PO-ADMM algorithm proposed in this invention. Table 1 shows the results of the four algorithms under different conditions. The ISL value below.

[0042] Table 1 Differences Comparison of ISL values ​​for the following four algorithms

[0043] Table 2 Differences K Comparison of ISL values ​​for the following four algorithms

[0044] Compared to the other three algorithms (DPM, p-MM, GA), in all In this scenario, the ISL value of the PO-ADMM algorithm is relatively lower. For example, when At that time, the ISL value of PO-ADMM was 3.74 dB, while the ISL values ​​of DPM, p-MM, and GA were 3.89 dB, 3.96 dB, and 4.12 dB, respectively. This indicates that the PO-ADMM algorithm has a certain advantage in suppressing sidelobes of autocorrelation waveforms, and can more effectively reduce sidelobe levels, thereby improving radar signal performance. The superior performance of the PO-ADMM algorithm, which achieves the lowest ISL value, can be attributed to its alternating iterative framework: this framework decomposes the optimization problem into multiple independent subproblems that can be solved efficiently; in addition, the NAG algorithm incorporated into PO-ADMM effectively reduces the risk of the algorithm getting trapped in local optima. In contrast, the other three algorithms are more likely to converge to local minima when dealing with constrained optimization problems.

[0045] Figure 6 Four algorithms were demonstrated for Comparison of the designed autocorrelation waveforms Figure 6 The horizontal axis represents the number of symbols in the phase-coded waveform sequence, and the vertical axis represents the autocorrelation coefficient of the phase-coded waveform sequence. The red curve corresponds to the results of the PO-ADMM algorithm proposed in this invention. Table 2 shows the results of the four algorithms under different conditions. The ISL value under [condition]. For the PO-ADMM algorithm, when [condition]... The ISL value was 3.74 dB when When the value is increased to 16, it drops to 3.36 dB. This trend indicates that the sidelobe suppression capability of the PO-ADMM algorithm gradually improves, and the integral sidelobe level continues to decrease as the number of phase quantization points increases.

[0046] Compared to the other three algorithms, in all tests At this value, the PO-ADMM algorithm consistently exhibits a lower ISL value. For example, when At that time, the ISL value of PO-ADMM was 3.67 dB, while the corresponding ISL values ​​of DPM, p-MM, and GA were 3.73 dB, 3.83 dB, and 4.03 dB, respectively; and Similar advantages to PO-ADMM can also be observed in different scenarios. These results indicate that, in different... Under these conditions, the PO-ADMM algorithm outperforms the algorithm in reducing the integral sidelobe level, highlighting its effectiveness in sidelobe suppression.

[0047] Table 3 Differences Comparison of PSLR values ​​for the following four algorithms

[0048] Table 4 Differences K Comparison of PSLR values ​​for the following four algorithms

[0049] Tables 3 and 4 compare the four algorithms—PO-ADMM, DPM, p-MM, and GA—in different... Different Peak Sidelobe Ratio (PSLR) at this level. When Increase from 30 to 80 As the value increases from 2 to 16, the PSLR decrease trend of the PO-ADMM algorithm is more significant than that of the other three algorithms, and it consistently achieves lower PSLR values. In summary, at different... and different Under these conditions, the PSLR performance of the PO-ADMM algorithm is superior to that of the DPM, p-MM, and GA algorithms.

[0050] Under the premise that phase quantization is used in all cases, the four algorithms PO-ADMM, P-MM, DPM and GA are compared under different conditions. Different The optimized performance was evaluated. Experimental results show that: 1. In terms of suppressing autocorrelation waveform sidelobes, the PO-ADMM algorithm consistently outperforms the other three algorithms; 2. With As the value increases, the average integral sidelobe level of all algorithms decreases, while the PO-ADMM algorithm consistently achieves the lowest integral sidelobe level. 3. In , The PO-ADMM algorithm maintains a significant advantage across different scenarios, and its sidelobe suppression performance increases with the number of values. The increase is due to the growth of; 4. A comparison of ISL values ​​further confirms that, in different... and Under these conditions, the PO-ADMM algorithm can consistently achieve a lower ISL value.

[0051] III. Algorithm Computational Complexity Analysis Tables 5 and 6 show the four algorithms at different code lengths. With different phase modulation numbers The computation time required. This varies with code length. and phase modulation number As the value increases, the computation time of all four algorithms shows an upward trend. Among them, the PO-ADMM algorithm is significantly faster than the other three algorithms. This is because the Nesterov Accelerated Gradient (NAG) algorithm used by PO-ADMM is more computationally efficient than the max-min optimization algorithm used by the p-MM algorithm and the coordinate descent method used by the DPM algorithm. These research results will be explained in more detail below.

[0052] Table 5 Differences Comparison of computation time for the following four algorithms

[0053] Table 6 Differences K Comparison of computation time for the following four algorithms

[0054] As shown in Tables 5 and 6, the NAG algorithm has high computational efficiency. It shortens the running time by combining synchronous updates and parallel processing mechanisms for global optimization using full gradients. In contrast, both the max-min optimization algorithm and the coordinate descent method rely on iterative processes: the former requires nested loops, while the latter uses a successive update method. These two characteristics result in higher computational complexity and longer running time for both.

[0055] Four algorithms were studied at different code lengths. and phase modulation number The computation time is as follows. The results show that: 1. The computation time of all algorithms varies with... and Increases with the increase of; 2. The PO-ADMM algorithm proposed in this invention has the fastest computation speed because it uses the NAG algorithm, which supports parallel processing and efficient global optimization. 3. In contrast, p-MM, DPM and GA algorithms involve nested loops, successive updates or complex evolutionary processes, resulting in higher computational complexity.

[0056] As can be seen, the PO-ADMM algorithm proposed in this invention exhibits superior computational efficiency under all test conditions.

[0057] In summary, the quantized phase-coded waveform design method based on the alternating direction multiplier method proposed in this invention uses a phase-coded signal. Compared with linear frequency modulated signals, the core advantage of the phase-coded signal lies in its ingenious internal modulation, which perfectly resolves the contradiction between "long range" and "high range resolution" in radar detection. More importantly, this invention constructs an optimization model with the goal of minimizing the integral sidelobe level and uses the ADMM framework to decompose the complex non-convex optimization problem faced by the optimization model into two efficiently solvable sub-problems: continuous waveform optimization and quantized phase projection. During the iteration process, a proximity operator is innovatively introduced to accurately quantize the obtained phase into a preset quantized phase set, thereby generating a constant-mode signal with both low sidelobe characteristics and high engineering feasibility. Experimental results show that the waveform designed in this invention outperforms existing mainstream methods in both sidelobe suppression performance and algorithm computational efficiency.

[0058] In the description of this invention, it should be understood that the terms "first" and "second" are used for descriptive purposes only and should not be construed as indicating or implying relative importance or implicitly specifying the number of indicated technical features. Therefore, a feature defined as "first" or "second" may explicitly or implicitly include one or more of that feature. In the description of this invention, "a plurality of" means two or more, unless otherwise explicitly specified.

[0059] Although the invention has been described herein in conjunction with various embodiments, those skilled in the art, by reviewing the specification and accompanying drawings, will understand and implement other variations of the disclosed embodiments in carrying out the claimed invention. In the specification, the word "comprising" does not exclude other components or steps, and "a" or "an" does not exclude a plurality. While certain measures are described in different embodiments, this does not mean that these measures cannot be combined to produce good results.

[0060] The above description, in conjunction with specific preferred embodiments, provides a further detailed explanation of the present invention. It should not be construed that the specific implementation of the present invention is limited to these descriptions. For those skilled in the art, various simple deductions or substitutions can be made without departing from the concept of the present invention, and all such modifications and substitutions should be considered within the scope of protection of the present invention.

Claims

1. A quantization phase encoding waveform design method based on the alternating direction multiplier method, characterized in that, The quantization phase encoding waveform design method includes: An optimization model for the quantized phase-coded waveform is established. The optimization objective of the model is to minimize the integral sidelobe level of the phase-coded waveform sequence. The constraints include that each symbol in the phase-coded waveform sequence satisfies the constant modulus condition and that the phase of each symbol belongs to the quantized phase set. By introducing auxiliary variables and Lagrange multipliers, the constrained optimization model of the quantized phase-encoded waveform is reconstructed into an unconstrained optimization model. Solving the unconstrained optimization model and outputting the optimal phase-coded waveform; the process of solving the unconstrained optimization model includes: given initial auxiliary variables and initial Lagrange multipliers, and fixing the initial auxiliary variables and initial Lagrange multipliers, solving the unconstrained optimization model to obtain an intermediate phase-coded waveform sequence; fixing the intermediate phase-coded waveform sequence and the initial Lagrange multipliers, quantizing the phases of the intermediate phase-coded waveform sequence and the initial Lagrange multipliers onto a quantized phase set using the proximity operator while simultaneously satisfying the constant modulus condition, obtaining intermediate auxiliary variables; based on the intermediate phase-coded waveform... The iteration stops if the intermediate phase-coded waveform sequence and intermediate auxiliary variables are satisfied. If satisfied, the intermediate phase-coded waveform sequence of the current iteration is taken as the optimal phase-coded waveform sequence. If not satisfied, the intermediate Lagrange multiplier is calculated based on the intermediate phase-coded waveform sequence, intermediate auxiliary variables, and initial Lagrange multipliers. The intermediate auxiliary variables are used as initial auxiliary variables, and the intermediate Lagrange multipliers are used as initial Lagrange multipliers. The process of fixing the initial auxiliary variables and initial Lagrange multipliers and solving the unconstrained optimization model to obtain the intermediate phase-coded waveform sequence continues until the iteration stops.

2. The quantization phase encoding waveform design method based on the alternating direction multiplier method according to claim 1, characterized in that, The constrained optimization model for the quantized phase-coded waveform is expressed as: ； in, Represents a phase-coded waveform sequence. , Indicates the first phase in the phase-coded waveform sequence Each code element This indicates the number of symbols in the phase-coded waveform sequence. express The conjugate transpose operation. , , This indicates calculating the square of the modulus. express The constant modulus condition is met. Represents the set of quantized phases. This indicates the number of phase discretizations. This indicates the phase calculation operation. express The phase belongs to .

3. The quantization phase encoding waveform design method based on the alternating direction multiplier method according to claim 1, characterized in that, The unconstrained optimization model is expressed by the following formula: in, These represent the phase-coded waveform sequence, auxiliary variables, and Lagrange multipliers, respectively. Indicates about An unconstrained optimization model. This indicates the number of symbols in the phase-coded waveform sequence. express The conjugate transpose operation. , , This indicates calculating the square of the modulus. , Represents the set of quantized phases. This indicates the number of quantized phases in the quantized phase set. Indicates will Phase projection in The projection operator in , This indicates the phase calculation operation. Indicates the first auxiliary variable One element, express The phase belongs to , express The phase does not belong to , It is a penalty parameter. This indicates that the square of the L2 norm is being calculated.

4. The quantization phase encoding waveform design method based on the alternating direction multiplier method according to claim 1, characterized in that, By fixing the initial auxiliary variables and initial Lagrange multipliers, solving the unconstrained optimization model yields the intermediate phase-coded waveform sequence, including: By fixing the initial auxiliary variables and initial Lagrange multipliers, the unconstrained optimization model is transformed into a first unconstrained sub-optimization model; The solution of the first unconstrained sub-optimization model is transformed from complex form to real form to obtain the second unconstrained sub-optimization model. Using the NAG algorithm, the real-valued form of the phase-coded waveform sequence is obtained by solving the second unconstrained sub-optimization model. The real-valued form of the phase-coded waveform sequence is converted into the complex form of the phase-coded waveform sequence and used as an intermediate phase-coded waveform sequence.

5. The quantization phase encoding waveform design method based on the alternating direction multiplier method according to claim 4, characterized in that, The first unconstrained sub-optimization model is expressed by the following formula: ； in, Indicates the first The intermediate phase-coded waveform sequence obtained from the next iteration. Indicates the first The initial auxiliary variables obtained in the next iteration Indicates the first The initial Lagrange multipliers obtained in the next iteration Represents a phase-coded waveform sequence. This indicates the number of symbols in the phase-coded waveform sequence. express The conjugate transpose operation. , , It is a penalty parameter. This indicates that the square of the L2 norm is being calculated.

6. The quantization phase encoding waveform design method based on the alternating direction multiplier method according to claim 4, characterized in that, The second unconstrained sub-optimization model is expressed as follows: ； in, Indicates the first The real-valued form of the intermediate phase-coded waveform sequence obtained in the next iteration. Represents the real-valued form of the phase-coded waveform sequence. express The transpose operation, , , This indicates the operation of taking the real part. This indicates the operation of taking the imaginary part. , , Indicates the first The real-valued form of the initial auxiliary variable obtained in the next iteration. Indicates the first The real-valued form of the initial Lagrange multipliers obtained in the next iteration. It is a penalty parameter. This indicates that the square of the L2 norm is being calculated.

7. The quantization phase encoding waveform design method based on the alternating direction multiplier method according to claim 1, characterized in that, By fixing the intermediate phase encoded waveform sequence and the initial Lagrange multipliers, the phases of the intermediate phase encoded waveform sequence and the initial Lagrange multipliers are quantized onto the quantized phase set using the proximity operator, while simultaneously satisfying the constant modulus condition, resulting in intermediate auxiliary variables, including: By fixing the intermediate phase encoded waveform sequence and the initial Lagrange multipliers, the unconstrained optimization model is transformed into a third unconstrained sub-optimization model; The third unconstrained sub-optimization model is solved by the proximity operator to map the phase of the intermediate phase-encoded waveform sequence and the initial Lagrange multiplier to the nearest quantized phase. The intermediate auxiliary variables are solved based on the most recent quantized phase.

8. The quantization phase encoding waveform design method based on the alternating direction multiplier method according to claim 7, characterized in that, The third unconstrained sub-optimization model is expressed as follows: ； in, Indicates the first The intermediate auxiliary variables obtained in the iteration are the first One element, The value can be 1~ , This indicates the number of symbols in the phase-coded waveform sequence. Indicates the first auxiliary variable One element, express The constant modulus condition is met. Represents the set of quantized phases. This indicates the number of quantized phases in the quantized phase set. Indicates will Phase projection in The projection operator in Indicates the first The intermediate phase encoded waveform sequence obtained in the nth iteration is the first Each code element Indicates the first The initial Lagrange multipliers obtained in the second iteration are the first... One element, It is a penalty parameter. This indicates that the square of the L2 norm is being calculated.

9. The quantization phase encoding waveform design method based on the alternating direction multiplier method according to claim 7, characterized in that, The most recent quantization phase of the mapping is expressed by the formula: ； in, Indicates the index of the most recent quantized phase of the mapping. Indicates the first The intermediate phase encoded waveform sequence obtained in the nth iteration is the first Each code element Indicates the first The initial Lagrange multipliers obtained in the second iteration are the first... One element, This indicates the index of the quantized phase in the quantized phase set. , This indicates the number of quantized phases in the quantized phase set. Indicates the th phase in the most recent quantization phase Each element.

10. The quantization phase encoding waveform design method based on the alternating direction multiplier method according to claim 1, characterized in that, The formula for calculating the intermediate Lagrange multipliers is as follows: ； in, Indicates the first The intermediate Lagrange multipliers obtained in the next iteration Indicates the first The initial Lagrange multipliers obtained in the next iteration Indicates the first The intermediate phase-coded waveform sequence obtained from the next iteration. Indicates the first Intermediate auxiliary variables obtained from the next iteration.

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