LPI radar waveform design and resource allocation method based on manifold optimization

Abstract

The invention discloses an LPI radar waveform design and resource allocation method based on manifold optimization, and aims to improve the target tracking precision of a radar. According to the method, the multi-target tracking performance is quantified through the target function based on the posterior Cramer-Rao lower bound. In order to improve accuracy and calculation efficiency, a manifold optimization framework is constructed to carry out optimization post-processing on a constructed objective function: firstly, an accurate penalty function is introduced, an LPI constraint condition is smoothly fused into the objective function, and conversion from a constraint problem to a penalty form is realized; constructing a volume manifold space based on resource constraint characteristics, and reconstructing an original optimization task into an unconstrained optimization problem on a manifold; meanwhile, a parallel conjugate gradient descent algorithm with a self-adaptive step length mechanism is adopted to execute optimization solution. According to the algorithm, parallel computing can be fully utilized to realize rapid convergence, and efficient exploration can be realized in a product manifold space through a self-adaptive step length strategy, so that a better solution and more stable optimization performance are obtained.

Description

Technical Field

[0001] This invention relates to the field of radar technology, and in particular to a method for LPI radar waveform design and resource allocation based on manifold optimization. Background Technology

[0002] In MIMO radar systems, waveform design and resource allocation are crucial for effective multi-target tracking. In practical applications, radar performance is limited by scarce resources; effective management of transmission resources can improve multi-target tracking capabilities. Furthermore, resource management and waveform design can meet the requirements of low probability of intercept (LPI), thereby improving the radar system's survivability in harsh environments. Therefore, waveform design and resource allocation have attracted considerable research attention.

[0003] Existing research methods are mainly divided into two categories: (1) heuristic methods, which may be impractical in MIMO radar systems due to high hardware costs and power consumption; (2) convex relaxation methods, which reduce computational costs but have relaxation losses.

[0004] A typical first-class approach is proposed in the paper "ZHANG Y, PAN M, HAN Q. Joint sensor selection and power allocation algorithm for multiple-target tracking of unmanned cluster based on fuzzy logic reasoning[J]. Sensors, 2020, 20: 5.", which proposes a hybrid intelligent method combining stochastic simulation and genetic algorithms. It employs chromosome encoding and a single-point crossover operator to explore the solution space through random search. However, the random population initialization of this method increases the number of computational iterations, resulting in a slower convergence speed. To address this issue, "SU Y, CHENG T, HE Z, LI X. Joint waveform control and resource optimization for maneuvering targets tracking in netted colocated MIMO radar systems[J]. IEEE Systems Journal, 2022, 16, 3:3960–3971." proposes an improved particle swarm optimization algorithm based on chaotic sequence initialization. It replaces random initialization with chaotic sequences, obtaining high-quality initialization solutions and thus accelerating the convergence speed. Despite these improvements, a large number of gene mutation operations still require significant computational resources, especially when numerous optimization parameters are involved.

[0005] In the second category, the paper "LU X, YI W, Kong L. LPI-based transmit resources cheduling for target tracking with distributed MIMO radar systems[J]. IEEE Transactions on Vehicular Technology[J]. IEEE Transactions on Vehicular Technology, 2023, 72, 11:14230–14244." employs a two-step solution method, combining the relaxation stage with the fine-tuning process. It uses a logarithmic transformation to relax the non-convex problem into a convex one and then uses the spectral gradient projection (SPG) algorithm to solve it. However, its direct approximation of the objective function may introduce errors, causing the final solution to deviate from the true objective of the original problem. In contrast, the paper "LIZ. XIE J, LIU W, ZHANG H, XIANG H. Joint strategy of power and bandwidth allocation for multiple maneuvering target tracking in cognitive MIMO radar with collocated antennas[J]. IEEE Transactions on Vehicular Technology, 2023, 72, 1:190–204." introduces auxiliary variables to transform non-convex constraints into convex constraints and combines Wolf gradient descent and alternating optimization methods to solve the problem. However, its solution cannot avoid relaxation errors and may not satisfy the original constraints, requiring further processing. Summary of the Invention

[0006] To address the shortcomings of existing technologies, this invention provides a manifold-optimized method for LPI radar waveform design and resource allocation to improve radar target tracking accuracy.

[0007] The technical solution adopted in this invention is: a method for LPI radar waveform design and resource allocation based on manifold optimization, in... Each equipment In a radar network consisting of MIMO radars with root transmitting antennas, the following steps are performed:

[0008] Step 1: Construct an optimization model for waveform design and resource allocation under radar tracking conditions with low interception in the k-th frame:

[0009]

[0010] Where k represents the frame index. Indicates the target index. Indicates the radar (radar node) index; Represents the trace of a matrix; target set Where Q represents the number of targets; radar number ;radar Position defined ,vector Used to characterize the target In the The state at frame time, where, and These represent the x and y coordinates of the target, respectively. and This indicates its corresponding speed;

[0011] Radar-Target Selection Matrix ,when When, it means that target q is tracked by radar n in the kth frame; Indicates assignment to any target The upper limit of the number of radars; Represents a single The maximum number of targets that can be tracked;

[0012] Given For the number of samples, radar The transmitted signal in the k-th frame is represented as ,in Indicates radar No. The transmitted signals from each antenna are defined, and the radar waveform of the k-th frame is defined as follows:

[0013] This represents the azimuth angle of target q relative to radar n; Indicates radar At angle Power gain;

[0014] For the first The predicted Cramerlow lower bound for each target is an important indicator in radar resource awareness design for measuring target tracking accuracy and guiding resource allocation. Extract the matrix for the location;

[0015] Indicates the effective signal bandwidth. Indicates the minimum available bandwidth for radar n; This represents the minimum available bandwidth for radar n; the bandwidth allocation for the k-th frame. ,in ;

[0016] and This represents the available power and bandwidth resources for each radar n. Indicates the minimum available bandwidth for radar n;

[0017] Each radar point is located at the azimuth angle of the interceptor. The power gain at the location cannot exceed the threshold. Meanwhile, to ensure tracking, the power gain at the target's azimuth angle needs to exceed a threshold. ;

[0018] For radar At angle Power gain. For the first The predicted Cramerlow lower bound for a target is an important indicator in radar resource awareness design for measuring target tracking accuracy and guiding resource allocation. Extract the matrix for the location.

[0019] Step 2: Solve the constructed optimization model using an optimization framework based on Riemannian manifolds to obtain the optimization results for radar beam selection, radar waveform, and bandwidth allocation.

[0020] Furthermore, in step 2, solving the optimization model specifically includes:

[0021] The Continuous Convex Approximation (SCA) method is used to optimize the radar-target selection matrix in the model. The optimization solution is performed to obtain the optimized solution result. ;

[0022] Then optimize the radar waveform in the model. Bandwidth allocation Optimization solutions are performed, including:

[0023] Step 201, Initialization settings, including:

[0024] Initialize radar waveform Bandwidth allocation , recorded as Initialize penalty parameters , recorded as Initialize smoothing parameters , recorded as Initialize convergence precision , recorded as ; and the number of initial iterations ; and initialization penalty update parameters ,in,

[0025] Step 202: The radar waveform is processed using a parallel conjugate gradient descent algorithm with an adaptive step size mechanism. Bandwidth allocation Perform the update to obtain the updated value: ;

[0026] Step 203, update the smoothing parameters and convergence accuracy: ;

[0027] Step 204, if detection constraints and low interception constraints When none of them are satisfied, that is Then the penalty parameter is updated as follows: Otherwise, the penalty parameter remains unchanged, i.e. ;

[0028] Step 205, let Determine whether the preset first iteration convergence condition is met; if so, output the solution results of the radar waveform and bandwidth allocation. Otherwise, return to step 202.

[0029] Furthermore, the convergence condition for the first iteration is set as follows:

[0030] and and

[0031] in, For the minimum step size, For the minimum smoothing parameter, For minimum convergence accuracy, all three values ​​are preset and can be adjusted in the initialization settings.

[0032] (1) Initialize the number of iterations and initialization step size ;

[0033] (2) Calculate the descent direction of the conjugate gradient descent algorithm. ;

[0034] (3) If satisfied Then update the parallel Euclidean gradient. Otherwise, end the update and output the radar waveform. Bandwidth allocation The updated value; where, and These represent the corresponding Riemann gradients. Let represent a smooth optimization objective function with resource constraints. and The objective functions are respectively The 2-norm and F-norm;

[0035] (4) Update the parallel Riemann gradient based on the parallel Euclidean gradient. ;

[0036] (5) Update conjugate parameters And let The initial value of the conjugate parameter is 0.

[0037] (6) Update descent direction ;

[0038] (7) Update radar waveform Bandwidth allocation To obtain the value of the next iteration And update the step size to obtain the step size for the next iteration. ;

[0039] (8) Number of update iterations Return to step (2).

[0040] 5. The method as described in claim 4, characterized in that step (7) further includes: adaptively adjusting the step size of the next iteration:

[0041]

[0042] in, The parameter represents the step size of the next iteration after adaptive adjustment. , It is the preset value, and , , where n represents the number of searches for the step size.

[0043] The technical solution provided by this invention brings at least the following beneficial effects:

[0044] In this invention, multi-target tracking performance is quantified using an objective function based on a posterior Cramero lower bound. To improve accuracy and computational efficiency, a manifold optimization framework is constructed to perform post-processing on the constructed objective function: First, an exact penalty function is introduced to smoothly integrate LPI constraints into the objective function, transforming the constrained problem into a penalty form; based on resource constraint characteristics, a product manifold space is constructed, reconstructing the original optimization task into an unconstrained optimization problem on the manifold; simultaneously, a parallel conjugate gradient descent algorithm with an adaptive step-size mechanism is employed to perform the optimization solution. This solution method not only fully utilizes parallel computing to achieve fast convergence but also achieves efficient exploration in the product manifold space through an adaptive step-size strategy, thereby obtaining a better solution and more stable optimization performance. Attached Figure Description

[0045] To more clearly illustrate the technical solutions in the embodiments of the present invention, the accompanying drawings used in the description of the embodiments will be briefly introduced below. Obviously, the accompanying drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0046] Figure 1 This is a schematic diagram of a radar network resource allocation model;

[0047] Figure 2 It is the result of normalized bandwidth resource allocation;

[0048] Figure 3 These are the emission patterns of different tracking frames;

[0049] Figure 4 This is a comparison of the multi-target tracking performance of the method proposed in this invention, CRM, FNTA, and FBDB;

[0050] Figure 5 It is the time taken per frame. Detailed Implementation

[0051] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be described in detail and completely below with reference to the accompanying drawings. Obviously, the described embodiments are only a part of the embodiments of this application, and not all of them. Generally, the components of the embodiments of the present invention described and shown in the accompanying drawings can be arranged and designed using different configurations. Therefore, the following detailed description of the embodiments of the present invention provided in the accompanying drawings is not intended to limit the scope of the claimed application, but merely represents selected embodiments of the present invention.

[0052] Inspired by the paper "ZHONG K, HU J, ZHAO Z, et al. MIMO radar unimodular waveform design with learned complex circle manifold network[J]. IEEE Transactions on Aerospace and Electronic Systems, 2024, . 60, 2: 1798–1807.", it can be noted that complex spherical manifolds naturally satisfy transmit power constraints, while real oblique manifolds satisfy bandwidth constraints. LPI constraints can be transformed into penalty functions to supplement the objective function. Based on these characteristics, this invention proposes an LPI radar waveform design and resource allocation method based on manifold optimization. First, the LPI constraints are transformed into penalty functions using the exact penalty function technique. Then, a product manifold space that satisfies resource constraints can be constructed, transforming the problem into an unconstrained problem on the product manifold. In addition, a parallel conjugate gradient descent (PCGD) algorithm with adaptive step size is also disclosed. The parallel conjugate gradient descent algorithm can achieve fast convergence, while the adaptive step size can thoroughly explore the product manifold space, thereby obtaining better results.

[0053] For ease of description, let's define it as follows:

[0054] Consider a A cooperative multiple-input multiple-output (MIMO) radar system consisting of several radar nodes. The position is determined by Given, among which Each node is equipped with One transmitting antenna and There are 1 receiving antenna, with the spacing between adjacent elements being half a wavelength. Assume a target set... The targets are sufficiently dispersed within the monitored area, where Q represents the number of targets. The multi-target tracking problem can be decomposed into a series of independent single-target tracking tasks. Figure 1 This invention illustrates radar network resource allocation modes considered in embodiments of the present invention.

[0055] definition The resulting matrix is ​​the radar-target assignment matrix. When When, it means that target q is tracked by radar n in the kth frame; otherwise... Given For the number of samples, radar In the The transmit signal matrix at frame time can be represented as: ,in, Indicates radar No. The transmitted signal from each antenna.

[0056] Radar at frame k At the far field angle The expression for the synthesized signal at point is:

[0057]

[0058] in, It is the launch steering vector.

[0059] The launch pattern is as follows

[0060]

[0061] in, Yes, waveform correlation matrix, transmit power. ,in Represents the trace of a matrix. This represents the signal duration. The receive steering vector is defined as... Radar at frame k The composite pattern can be described as follows:

[0062]

[0063] For simplicity, the target motion model adopts the CV model (constant velocity model), then the target... The state transition equation is as follows.

[0064]

[0065] Among them, the target In the The state at frame time is determined by vectors express, and These represent the coordinates of the target. and This indicates its corresponding speed. The noise is Gaussian process noise with a mean of 0 and a covariance matrix of... State transition matrix Represented as:

[0066]

[0067] and for,

[0068]

[0069] in, Indicates the tracking interval. Indicates the intensity of process noise. yes The identifier matrix, It is the Kronecker product operator. Frame-time radar For the goal The nonlinear measurement model is for

[0070]

[0071] in, It is a measurement function. and These represent distance and azimuth, respectively. Noise Following a zero Gaussian distribution, the covariance matrix ,in and satisfy

[0072]

[0073] in, Indicates the effective signal bandwidth. Indicates the radar's transmission wavelength. This is equivalent to the aperture size of an antenna. Radar n for the target The signal-to-noise ratio (SNR) is derived from the following expression:

[0074]

[0075] in, Indicates the duration of stay. and These correspond to the transmit and receive antenna gains, respectively. Indicates radar Observed target Radar cross section (RCS). It is radar and target The distance between them. A constant. This refers to Boltzmann's constant. It is the noise temperature of the receiving system. This indicates the filter bandwidth used by the radar receiver. Furthermore, It is the noise figure of the receiver. Used to calculate the total loss of the system. This indicates the system processing gain.

[0076] Due to the need to optimize the transmit beam and bandwidth allocation, for convenience, the measurement error covariance matrix can be rewritten as follows:

[0077]

[0078] matrix It is derived from other relevant parameters.

[0079] The posterior Cramero lower bound (PCRLB) characterizes the theoretical lower bound of the target state estimation error and is an important benchmark for evaluating tracking accuracy. This lower bound is derived from the Bayesian Fisher Information Matrix (BFIM), which quantifies the amount of available information about the target state. In the The Bayesian information matrix of the frame is:

[0080]

[0081] in, It is a measurement function Relative to the target state The Jacobian matrix. For ease of representation, , ,in .

[0082] PCRLB is the inverse of BFIM and can be written as Since position error plays a major role in estimating the target state, the position submatrix of PCRLB is extracted as a metric, and the sum of PCRLBs is used as the objective function:

[0083]

[0084] in, .

[0085] When assuming the potential location of an interceptor based on prior information and auxiliary information obtained through electronic reconnaissance, the power gain of the interceptor's possible azimuth angle must be kept below a predetermined threshold to ensure a low probability of interception. In summary, the first... The LPI problem model when the interceptor azimuth angle is known at frame time is as follows:

[0086]

[0087] in, It can be assigned to any target. The upper limit of the number of radar nodes, and Represents any single radar node The maximum number of targets that can be tracked. and This indicates the available power and bandwidth resources for each radar node. To improve tracking efficiency, the radar transmits a beam at the target azimuth angle. The power gain at the point must exceed the power gain threshold. Since the precise angle of the interceptor cannot be obtained, it is assumed that the interceptor is located within a specific azimuth range, denoted as... ,in and This is supplementary knowledge obtained through electronic reconnaissance methods. Radar is used for anti-interception purposes. In the The launch mode on each tracking frame must be The set forms a depth of The notch. That is This indicates the power gain threshold.

[0088] Mathematically, the LPI problem is non-convex due to the presence of both binary and continuous variables. To effectively address this problem, this invention proposes a two-stage solution combining radar beam selection and resource allocation. This is based on the isotropic transmission waveform... and uniform bandwidth allocation First determine This guides subsequent joint transmit waveform design and bandwidth allocation. Then, based on the overall performance of the MMT, the transmit waveform design and bandwidth allocation are optimized in parallel using the MO framework. For example, the specific stages are as follows:

[0089] Phase 1: Node Allocation via SCA - Target

[0090] Due to the initial minimization problem Aimed at optimizing MTT performance, inequality constraints and This is transformed into an equation. Based on the Continuous Convex Approximation (SCA) method, a set of convex optimization problems can be solved step by step to find a feasible solution to the original problem. In SCA... During this iteration, the following subproblems were solved:

[0091]

[0092] in, Objective function.

[0093]

[0094]

[0095] in, and This means stacking the matrices column by column into a column vector. Indicates the first The solution at the next iteration. It is a very large constant (preset value). These are dual variables to ensure that the binary constraints are satisfied. The results are obtained using SCA. As a result, the design of the transmission waveform and bandwidth allocation can continue.

[0096] Phase Two: Waveform Design and Bandwidth Allocation Using the MO (Multi-Objective) Framework

[0097] Transmit waveform design and bandwidth allocation are non-convex problems, making them difficult to solve. Existing methods mostly employ computationally expensive heuristic evolutionary approaches or convex relaxation methods, lacking the real-time adaptability required for dynamic scenarios. To overcome this problem, an MO framework for transmit signal waveform design and optimized bandwidth resource allocation is proposed. It was found that a product manifold can be constructed to satisfy transmit power and bandwidth constraints, and that LPI constraints can be combined with a smoothing function. Leveraging these characteristics, a product manifold-based MO framework is proposed for relaxation-free transmit waveform design and bandwidth allocation. Specifically, the LPI constraints are transformed into a penalty function through smoothing techniques, thus reformulating the problem as an unconstrained problem on a product manifold. To address this issue, a parallel conjugate gradient descent (PCGD) algorithm is derived, employing an adaptive step size to avoid relaxation.

[0098] An adaptive precise penalty step is employed to transform the LPI constraints into precise penalty terms, thereby reformulating the problem as a multivariate optimization. First, the precise penalty terms are constructed: specifically, the difference between the power gain and the threshold at the azimuth angles of the opposing interceptor and the target is calculated.

[0099]

[0100]

[0101] Then, by introducing the maximum value function This allows us to obtain the precise penalty item.

[0102]

[0103] in, This represents the penalty parameter. The exact penalty term is not smooth and is difficult to solve directly. However, since the penalty function term has a structure that takes the maximum of two terms, the LSE function can be used. Smooth it, where It is a smoothing parameter. Therefore, the penalty cost function can be obtained:

[0104]

[0105] Based on the above formula, the comprehensive function weighted by the adaptive penalty factor can be derived as follows:

[0106]

[0107] Finally, the problem is transformed into a smooth optimization problem with resource constraints.

[0108]

[0109] The original problem is solved using the exact penalty method, and the penalty parameters are continuously adjusted. This is to ensure that, after reaching a certain critical value, the solution to the penalty problem remains consistent with that of the original problem. Considering that if... Setting the value too high will increase the cost of obtaining a feasible solution; therefore, a smaller value is usually used first. The solution is then obtained. When the penalty parameter is relatively small, the solution lies outside the feasible region. In this case, it can be determined according to... Adjustments were made, including It is a constant factor used to update the penalty parameter, subscript Indicates the number of iterations. A threshold parameter is introduced. This is to terminate the iteration of the parallel conjugate gradient descent algorithm. As the iteration progresses and a more accurate solution is desired, the threshold parameter must be reduced. Therefore, it is achieved by... Update, among which It is the lower limit. It is used for adjustment The constant. Introducing a smoothing parameter. This is to relax the inequality constraints, such as To progressively improve the approximation quality during iteration, a threshold is set. It will gradually decrease.

[0110] At the same time, bandwidth auxiliary variables are introduced. To integrate the constraints. The matrix composed of bandwidth allocation auxiliary variables is as follows: and This represents the bandwidth allocation auxiliary vector for radar n.

[0111]

[0112] Therefore, the problem can be restated as follows:

[0113]

[0114] It is noted that It satisfies the characteristics of a real skew manifold, and It satisfies the characteristics of a complex spherical manifold. Therefore, a real skew manifold and For a complex spherical manifold, there will be a constrained optimization problem. This is transformed into an unconstrained optimization problem on a manifold. The manifold is constructed as follows:

[0115]

[0116] in, represent A one-dimensional matrix. The product manifold is... Cartesian product of manifolds.

[0117]

[0118] The unconstrained optimization problem on the integral manifold can be expressed as:

[0119]

[0120] To better integrate optimization methods in Euclidean space, a tangent space is constructed. The tangent space provides a local linear approximation of a point on the manifold, making the derivation and implementation of algorithms on the manifold more direct and efficient. tangent space yes tangent space and tangent space The product of.

[0121]

[0122] in, This represents the tangent vector of the point in the tangent space.

[0123] By introducing the tangent space, a parallel conjugate gradient algorithm based on the product manifold can be derived. First, the Riemann gradient is calculated: the Riemann gradient is projected from the Euclidean gradient onto the tangent space. The conclusion is as follows. exist The Euclidean gradient at point is denoted as And derived from the following formula:

[0124]

[0125] in,

[0126]

[0127] and

[0128]

[0129] exist The Euclidean gradient at point is denoted as .

[0130]

[0131] in

[0132]

[0133]

[0134] Compared to The Euclidean gradient can be obtained by applying the matrix. The derivative is calculated for each element.

[0135]

[0136] objective function Relative to element The derivative is

[0137]

[0138] in,

[0139]

[0140] Associative functions and Parallel Euclidean gradient for

[0141]

[0142] The Riemann gradient is defined as the orthogonal projection of the Euclidean gradient onto the tangent space. Therefore, exist and The Riemann gradients at the points are respectively

[0143]

[0144]

[0145] Merge function and Parallel Riemann gradient for

[0146]

[0147] The descent direction in the conjugate gradient method can be calculated after obtaining the Riemann gradient, which is a combination of the momentum direction and the negative gradient direction. The parallel descent direction in the j-th iteration is...

[0148]

[0149] These are conjugate parameters, used to obtain the descent direction. Then, in the inner space Update .

[0150]

[0151] This represents the adaptive update step size determined by the Armijo linear search algorithm.

[0152] Traditional gradient descent algorithms use a constant step size, which can lead to convergence problems. To overcome this, a linear search strategy is employed to dynamically update the step size, thus accelerating convergence. The linear search strategy can be expressed as follows:

[0153]

[0154] in, , It is the initial step size for the next iteration. This represents the number of iterations in the linear search. (Iteration) The direction of descent at that time is used express:

[0155]

[0156] To further accelerate convergence, the step size of the next iteration... According to satisfy Make appropriate adjustments based on the changes. If satisfy This indicates that the current step size is too small and should be increased to a larger value in the next iteration. ,in .if satisfy This indicates a reasonable initial step size, therefore subsequent iterations must be kept within [the appropriate range]. The steps. If satisfy This indicates that more than three linear searches have been performed, resulting in a small step size at the current point. The step size should be increased in the next iteration. ,in In summary, the step size for the next iteration. The update rules are summarized as follows:

[0157]

[0158] In tangent space The update does not guarantee that the obtained points will still be on the manifold. Above. To ensure feasibility, the point will be moved from... Map back Therefore, manifold The next feasible solution is obtained in the following way:

[0159]

[0160] In one embodiment, the specific solution process for the constructed optimization model (Equation 23) is as follows: First, an exact penalty function mechanism is introduced to smoothly integrate the low interception constraint into the objective function, realizing the transformation of the constrained problem into a penalty form; then, based on the resource constraint characteristics, a product manifold structure space is constructed, reconstructing the original optimization task into an unconstrained optimization problem on the manifold. Furthermore, the solution can also be obtained using the Parallel Conjugate Gradient Descent (PCGD) algorithm with an adaptive step-size mechanism. This algorithm not only fully utilizes parallel computing to achieve fast convergence but also achieves efficient exploration in the product manifold space through an adaptive step-size strategy, thereby obtaining a better solution and more stable optimization performance.

[0161] In one embodiment, the above solution process can be specifically described as follows:

[0162] Algorithm 1: MO framework

[0163] Input: Initialize input Initialize penalty parameters Initialize smoothing parameters Minimum smoothing parameter Initialize convergence accuracy , precision tolerance Penalty update parameters Smoothly update parameters Convergence accuracy update parameters minimum step size .

[0164] Step 1: Initialization

[0165] Step 2: Repeat the operation:

[0166] Step 201: Update according to Algorithm 2 Furthermore, the termination rule is parallel Riemann gradient. ;

[0167] Step 202: Calculation

[0168] Step 203: If detection constraints and low interception constraints When all conditions are met, that is ,So Otherwise, the penalty parameter remains unchanged;

[0169] Step 204: Calculation ,until , ,

[0170] Output:

[0171] Algorithm 2: PCGD Algorithm

[0172] enter:

[0173] Step 1: Initialize the number of iterations Step length

[0174] Step 2: Calculate the descent direction Calculated according to formula (44);

[0175] Step 3: If satisfied Update parallel Euclidean gradients Otherwise, directly output the current updated value, i.e.

[0176] Step 4: Update the parallel Riemann gradient based on the parallel Euclidean gradient. ;

[0177] Step 5: Update conjugate parameters And let ;

[0178] Step 6: Update descent direction ;

[0179] Step 7: Update ;in, The calculation and update are performed according to formulas (45) and (49). Use formula (48) to perform the calculation update;

[0180] Step 8:

[0181] Output:

[0182] Example

[0183] The simulation scenario uses data consisting of 20 frames, with a frame interval of 2 seconds. Each radar node is equipped with a uniform linear array, with the elements in the array spaced at half a wavelength. The total bandwidth of each radar node is... 1MHz , The minimum power gain in the target direction is set to... dB, the maximum power gain in the interception direction is -35 dB. , The normalization of the launch pattern is as follows: Simulation results validate the effectiveness of the proposed method in tracking and identifying multiple targets in complex environments.

[0184] Existing solutions:

[0185] (1) Two-step solution method: relax the non-convex problem into a convex problem and solve it using the spectral gradient projection algorithm: Reference: "LU X, YI W, Kong L. LPI-based transmit resource scheduling for targettracking with distributed MIMO radar systems[J]. IEEE Transactions on Vehicular Technology[J]. IEEE Transactions on Vehicular Technology, 2023, 72,11:14230–14244."

[0186] (2) Relax the non-convex problem into a convex problem and solve it by combining the Wolf gradient descent method and the alternating optimization method: Reference: "LI Z. XIE J, LIU W, ZHANG H, XIANG H. Joint strategy of power and bandwidth allocation for multiple maneuvering target tracking in cognitive MIMO radar with collocated antennas[J]. IEEE Transactions on Vehicular Technology, 2023, 72, 1:190–204."

[0187] from Figure 2 As can be seen, adaptive node selection has been achieved (the maximum number of nodes for each target is...). And bandwidth allocation. In the initial stage of target tracking, each radar node tends to allocate more bandwidth to targets that are closer to it. As tracking time increases and tracking errors decrease, radar nodes tend to allocate bandwidth more evenly.

[0188] like Figure 3 As shown in (c), (d), (e), and (f), the power gain distribution of the proposed method in the transmit pattern of nodes 2 and 3 is basically consistent with that of the CRM method. However, there are significant differences between the two in the transmit pattern of node 1. Figure 3 As shown in (a) and (b) in the present invention, the method proposed in the embodiment of the invention focuses on improving the energy focusing of target 3 in the beam of radar 1, while the CRM method tends to improve the power gain of target 1.

[0189] Figure 4 The results show that, in terms of multi-target tracking accuracy, the method proposed in this invention consistently outperforms the Fixed Node Target Assignment (FNTA) method, the Fixed Beam Pattern Design and Bandwidth Allocation (FBDB) method, and the Convex Relaxation Algorithm (CRM), achieving a lower posterior Cramer-Rayet (PCRLB). Due to the adoption of adaptive radar-target assignment and optimized bandwidth allocation, both the proposed method and CRM significantly improve tracking performance compared to FNTA. Specifically, during tracking, the proposed method improves tracking accuracy by 6.8% compared to CRM.

[0190] Figure 5The comparison between the method proposed in this embodiment and CRM in terms of computation time is shown. The method proposed in this embodiment has higher computational efficiency, with an average computation time of approximately 60% that of CRM.

[0191] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of the present invention.

[0192] The above descriptions are merely some embodiments of the present invention. Those skilled in the art can make various modifications and improvements without departing from the inventive concept of the present invention, and these all fall within the scope of protection of the present invention.

Claims

1. A method for LPI radar waveform design and resource allocation based on manifold optimization, characterized in that, exist Each equipment In a radar network consisting of MIMO radars with root transmitting antennas, the following steps are performed: Step 1: Construct an optimization model for waveform design and resource allocation under radar tracking conditions with low interception in the k-th frame: ； Where k represents the frame index. Indicates the target index. Indicates the radar (radar node) index; Represents the trace of a matrix; target set Where Q represents the number of targets; radar number ;radar Position defined ,vector Used to characterize the target In the The state at frame time, where, and These represent the x and y coordinates of the target, respectively. and This indicates its corresponding speed; Radar-Target Selection Matrix ,when When, it means that target q is tracked by radar n in the kth frame; Indicates assignment to any target The upper limit of the number of radars; Represents a single The maximum number of targets that can be tracked; Given For the number of samples, radar The transmitted signal in the k-th frame is represented as ,in Indicates radar No. The transmitted signals from each antenna are defined, and the radar waveform of the k-th frame is defined as follows: ; This represents the azimuth angle of target q relative to radar n; Indicates radar At angle Power gain; Indicates the effective signal bandwidth. This represents the minimum available bandwidth for radar n; This represents the minimum available bandwidth for radar n; the bandwidth allocation for the k-th frame. ,in ; and This represents the available power and bandwidth resources for each radar n. This represents the minimum available bandwidth for radar n; Indicates the azimuth angle of the interceptor. Indicates azimuth angle The power gain threshold at that point, The threshold representing the power gain at the azimuth angle of the target. ; For radar At angle Power gain, For the first The predicted lower bound of Cramerlow for each target. Extract the matrix for the location; Step 2: Solve the constructed optimization model using an optimization framework based on Riemannian manifolds to obtain the optimization results for radar beam selection, radar waveform, and bandwidth allocation.

2. The method as described in claim 1, characterized in that, Step 2, solving the optimization model specifically includes: The radar-target selection matrix in the optimization model is obtained by using a continuous convex approximation method. The optimization solution is performed to obtain the optimized solution result. ; Then optimize the radar waveform in the model. Bandwidth allocation Optimization solutions are performed, including: Step 201, Initialization settings, including: Initialize radar waveform Bandwidth allocation , recorded as Initialize penalty parameters , recorded as Initialize smoothing parameters , recorded as Initialize convergence precision , recorded as ; and the number of initial iterations ; and initialization penalty update parameters ,in, ; Step 202: The radar waveform is processed using a parallel conjugate gradient descent algorithm with an adaptive step size mechanism. Bandwidth allocation Perform the update to obtain the updated value: ; Step 203, update the smoothing parameters and convergence accuracy: ; Step 204, if detection constraints and low interception constraints When none of them are satisfied, that is Then the penalty parameter is updated as follows: Otherwise, the penalty parameter remains unchanged, i.e. ; Step 205, let Determine whether the preset first iteration convergence condition is met; if so, output the solution results of the radar waveform and bandwidth allocation. Otherwise, return to step 202.

3. The method as described in claim 2, characterized in that, The convergence condition for the first iteration is set as follows: and and ; in, The preset minimum step size, This is the preset minimum smoothing parameter. This is the preset minimum convergence accuracy.

4. The method as described in claim 2, characterized in that, In step 202, the parallel conjugate gradient descent algorithm with an adaptive step size mechanism specifically includes: (1) Initialize the number of iterations and initialization step size ; (2) Calculate the descent direction of the conjugate gradient descent algorithm. ; (3) If satisfied Then update the parallel Euclidean gradient. Otherwise, end the update and output the radar waveform. Bandwidth allocation The updated value; where, and These represent the corresponding Riemann gradients. Let represent a smooth optimization objective function with resource constraints. and The objective functions are respectively The 2-norm and F-norm; (4) Update the parallel Riemann gradient based on the parallel Euclidean gradient. ; (5) Update conjugate parameters And let The initial value of the conjugate parameter is 0. (6) Update descent direction ; (7) Update radar waveform Bandwidth allocation To obtain the value of the next iteration And update the step size to obtain the step size for the next iteration. ; (8) Number of update iterations Return to step (2).

5. The method as described in claim 4, characterized in that, Step (7) also includes: adaptively adjusting the step size for the next iteration: ； in, The parameter represents the step size of the next iteration after adaptive adjustment. , It is the preset value, and , , where n represents the number of searches for the step size.

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