Joint Beam, Power, and Waveform Allocation Method for Multi-Target Tracking in Centralized MIMO Radar

Abstract

The present invention discloses a joint beam, power and waveform allocation method for multi-target tracking in a centralized MIMO radar. In the initial stage, the MIMO radar uses a single beam to point to the target and allocates all power to this batch of targets to improve the tracking accuracy of this target. When the estimation accuracy of all targets is improved to a certain extent, multiple beams are used to irradiate the targets and the power is optimally allocated. Through the method of the present invention, the tracking accuracy of the beam coverage of the target is significantly improved, and the tracking accuracy of the joint optimization of the beam, power and waveform is significantly better than the results of separate power optimization or waveform scheduling.

Description

Technical Field

[0001] The present invention relates to the technical field of radar target tracking, and particularly relates to a method for jointly allocating beams, power, and waveforms for multi-target tracking of a centralized MIMO radar. Background Art

[0002] A centralized MIMO radar is an extension of a traditional phased array radar, and its system structure has more practical application value. Moreover, the transmitting and receiving antenna elements of a centralized MIMO radar are relatively close to each other, the viewing angles of each antenna element to the target are approximately the same, and each subarray can transmit mutually orthogonal signal waveforms, thereby obtaining waveform diversity, forming a wider low-gain beam different from that of a traditional phased array, and the beams of each subarray can be directed to different directions in the airspace. In the simultaneous multi-beam working mode, the flexibility of multi-target tracking is greater. Waveform selection has a significant impact on the improvement of tracking accuracy. For a centralized MIMO radar, it can truly achieve the simultaneous tracking of multiple targets by virtue of the simultaneous orthogonal multi-beam technology. In this mode, the peak power of the transmitter can be reduced to meet the low intercept requirement in military applications; at the same time, the dwell time for each target is extended, thereby obtaining higher Doppler resolution. However, in an actual system, in the existing multi-beam centralized MIMO radar resource management methods for multi-target tracking, the simultaneous improvement of tracking accuracy is not comprehensively considered in terms of power, waveform selection, and reasonable allocation of beam resources. Therefore, only by comprehensively considering system resources and target tracking accuracy can high-performance tracking of multiple targets by the radar system be achieved. Therefore, studying the joint allocation of power and waveforms under the background of multi-target tracking of a centralized MIMO radar has high application value. Currently, most of the existing literature on adaptive selection of waveform parameters conducts research on phased array radars, and the resource allocation object is limited to waveform parameters, ignoring the change range of parameters such as the chirp rate due to hardware limitations of the radar system. In addition, combined with the cognitive radar closed-loop feedback optimization idea, there is less research on the problem of joint optimization and allocation of power and waveforms of a centralized MIMO radar under the background of multi-target tracking. Summary of the Invention

[0003] In view of the above existing problems, the present invention proposes a method for jointly allocating beams, power, and waveforms for multi-target tracking of a centralized MIMO radar. To achieve the above object, the technical solution adopted by the present invention is as follows:

[0004] A method for jointly allocating beams, power, and waveforms for multi-target tracking of a centralized MIMO radar, characterized by comprising the following steps:

[0005] Step 1: Obtain the measurement information of each batch of targets at the current moment and deduce the state estimation covariance matrix, and deduce the posterior Cramer-Rao lower bound PCRLB of the state estimation lower bound as the optimization index;

[0006] Step 2: Predict the state estimation covariance matrix at the next moment according to the state estimation error covariance matrix, select multiple target positions with the maximum state estimation error to construct an objective function, and establish a resource joint optimization model with the objective function;

[0007] Step 3: Solve the established resource joint optimization model based on Lemma 1, Lemma 2 and Lemma 3 to obtain the optimal allocation vector, so that each target is allocated corresponding resources;

[0008] Step 4: Use the square root cubature Kalman filter SCKF to update the target information;

[0009] Step 5: Estimate and track the target position at the next moment according to the selected array elements, the final power allocation result and the optimal bandwidth allocation vector.

[0010] Furthermore, the specific steps of Step 1 include:

[0011] Step 11: Under high signal-to-noise ratio conditions, use the CRLB in unbiased parameter estimation to characterize the measurement error covariance matrix R k :

[0012]

[0013] where, T = diag(c / 2, c / (2f c )) is the transformation matrix; J k is the BFIM and:

[0014]

[0015] where, η is the SNR, and η ∝ P t ; τ is the time delay; ν is the Doppler frequency shift; A is the ambiguity function of the transmitted waveform, and the expression is:

[0016]

[0017] where, is the complex envelope of the transmitted waveform;

[0018] Step 12: Characterize the measurement covariance matrix R of target q in the k-th sampling interval with the corresponding transmit power and waveform parameters : k :

[0019]

[0020] where, is a scalar or vector;

[0021] Step 13: In the Kalman filter, the iterative formula for the posterior state estimation error matrix is:

[0022]

[0023]

[0024] Step 14: Make the following approximation to Equation (6):

[0025]

[0026] where, and are the approximate values of the Jacobian matrix of the measurement function and the measurement covariance matrix at the target state prediction point respectively.

[0027] Furthermore, Step 2 specifically includes the following content:

[0028] Step 21: Assume that the utilization of the beams for each target in the k-th sampling interval is and the waveform parameter is the transmit power is Sort the trace of from largest to smallest to obtain:

[0029]

[0030] Step 21: Based on Equation (8), the resource allocation optimization model is obtained as:

[0031]

[0032] where, means that the batch of targets is not assigned a waveform; is the minimum threshold for power allocation; and constitute the optional interval of the waveform parameter; P total is the total power; M is the maximum number of simultaneously orthogonal beams.

[0033] Furthermore, assume that for the feasible number of beam utilizations m k , m k ∈{1,2,…,M}, the corresponding beam allocation vector is and there is Then Lemma 1, Lemma 2, and Lemma 3 are respectively:

[0034] Lemma 1: For a given m k , regardless of P tk and λ kRegardless of the value of the optimal solution of the corresponding beam allocation vector is uniquely determined;

[0035] Lemma 2: For a given and any power allocation vector the waveform allocation problem can be transformed into m k constraint problems that only involve and can be solved quickly;

[0036] Lemma 3: By transforming the multi-objective function into a single-objective function, for a given the joint power and waveform allocation problem is equivalent to a convex optimization problem.

[0037] Furthermore, the specific operation steps of Step 3 include:

[0038] Step 31: According to Lemma 1, for a given m k , can be calculated as:

[0039]

[0040] Step 32: According to Lemma 2, for the determined the optimal waveform allocation vector can be obtained. Then, when enumerating all feasible Equation (9) is equivalent to:

[0041]

[0042] Step 33: For the determined and it can be obtained by solving the following equation:

[0043]

[0044] Step 34: Transform Equation (23) into a single-objective function using the weighted summation method, select 1 / m k as the weight, and at the same time introduce the power allocation vector and the waveform selection vector to rewrite Equation (23) as:

[0045]

[0046] Step 35: Based on Lemma 3, Equation (24) is a convex optimization problem with 2m k variables, and it can be quickly solved using the Zoutendijk feasible direction method. Therefore, when and are obtained, and can be calculated respectively as:

[0047]

[0048]

[0049] Step 36: Through the above variable separation and transformation of the multi-objective function into a single-objective function, the joint allocation problem of beams, waveforms, and power is rewritten as M convex optimization problems. Using the Zoutendijk feasible direction method, the optimal allocation vector can be obtained:

[0050]

[0051] The beneficial effects of the present invention are as follows:

[0052] In the joint allocation method of beams, power, and waveforms for centralized MIMO radar multi-target tracking proposed by the present invention, at the initial stage, the MIMO radar uses a single beam to point to the target and allocates all power to this batch of targets to improve the tracking accuracy of the target; when the estimation accuracy of all targets is improved to a certain extent, multiple beams are used to irradiate the target and the power is optimally allocated; the simulation results show that the present invention is significantly superior to the results of separate power optimization or waveform scheduling in terms of target tracking accuracy, and the tracking accuracy of the beams covering the target is significantly improved, thereby improving the target tracking performance of the centralized MIMO radar and at the same time improving the resource utilization rate. Compared with the prior art, it mainly focuses on (1) the number of beams that the radar system needs to use at each moment; (2) the scheme for the target to allocate specific resources; (3) the resource joint allocation scheme that maximizes the radar working performance under the constraints of the given radar transmit power and the total amount of transmitted waveforms; (4) the analysis and research on the role of the joint allocation scheme in improving the system performance compared with the separate allocation of resources. Description of the Drawings

[0053] Figure 1 It is the cognitive tracking framework diagram of the present invention;

[0054] Figure 2a It is the power allocation result diagram after a single simulation; Figure 2b It is the waveform selection result diagram after a single simulation;

[0055] Figure 3 The trace diagram of the posterior estimation error covariance matrix of each target;

[0056] Figure 4a It is the power allocation result diagram after 100 simulation experiments; Figure 4b It is the waveform selection result diagram after 100 simulation experiments;

[0057] Figure 5 It is the schematic diagram of the performance comparison between the present invention and other three resource allocation schemes;

[0058] Figure 6 Schematic diagram of the power allocation result under the condition of non-waveform scheduling Detailed implementation mode

[0059] In order to enable ordinary technicians in the field to better understand the technical solution of the present invention, the technical solution of the present invention will be further described below in conjunction with the drawings and embodiments

[0060] I. Connection between transmitted wave and tracking filter

[0061] Under high signal-to-noise ratio conditions, the measurement error covariance matrix R k Can be characterized by the CRLB in unbiased parameter estimation, that is

[0062]

[0063] Where T = diag(c / 2, c / (2f c )) is the transformation matrix; J k Is the BFIM and can be obtained by the ambiguity function A of the transmitted waveform

[0064]

[0065] Where η is the SNR, and η ∝ P t ; τ is the time delay; ν is the Doppler frequency shift

[0066] And the ambiguity function A of the transmitted waveform can be calculated as

[0067]

[0068] Where Is the complex envelope of the transmitted waveform

[0069] Therefore, for the measurement covariance matrix of target q within the k-th sampling interval, the corresponding transmitted power And waveform parameters Can be used to represent

[0070]

[0071] Where Is a scalar or vector

[0072] In addition Is closely related to the state estimation error. In Kalman filtering, the iterative formula of the posterior state estimation error matrix is

[0073]

[0074]

[0075] Obviously, the prediction estimation error is only related to the target motion model. During the k-th sampling interval, the selected waveform parameters and the transmit power affect the final estimation error by influencing However, in a non-linear system, H and need to be estimated by the Monte Carlo method. Equation (6) is approximated as follows:

[0076]

[0077] where and are the approximate values of the Jacobian matrix of the measurement function and the measurement covariance matrix at the target state prediction point respectively.

[0078] II. Construction and solution of the resource allocation optimization model

[0079] In fact, the joint allocation problem of beams, power, and waveforms is an optimization problem under the constraints of the total amount of beams, power, and the number of waveforms. Previously, the CRLB was mostly selected as the optimization criterion. However, when waveform scheduling is introduced, the CRLB needs to be solved according to the specific waveform, and its analytical formula is difficult to obtain. The idea of Kalman filtering is widely used in various filtering mechanisms, and its estimation error matrix is predictable; moreover, the filtering equation of Kalman and the expression of PCRLB have potential consistency. Therefore, the present invention selects the state estimation error covariance matrix P k|k of the filter output as the optimization index.

[0080] Assume that the utilization of beams for each target during the k-th sampling interval is and the waveform parameters are the transmit power is To improve the overall performance of the system, appropriate numbers of beams, power, and waveforms need to be selected to minimize the state estimation error of multiple batches of targets. Since the total amount of beams is fixed, only some targets will be irradiated at each moment. Therefore, multiple targets with the worst estimation accuracy are selected to construct the objective function.

[0081] First, sort in descending order of the trace to obtain:

[0082]

[0083] Second, based on Equation (8), the resource allocation optimization model is constructed as:

[0084] ​

[0085] Among them, means that the batch of targets is not assigned a waveform; is the minimum threshold for power allocation; and constitute the optional interval of waveform parameters; P total is the total power; M is the maximum number of simultaneously generated orthogonal beams.

[0086] As can be seen from the above, Equation (9) is an optimization problem regarding three variables u k , P tk and λ k . For such problems, the common solution method is to separate the variables first and then transform them into a single-objective optimization problem. Before describing the solution method, assume that for the feasible number of beam utilizations m k , m k ∈{1, 2, …, M}, the corresponding beam allocation vector is and there is and the following three lemmas are given:

[0087] Lemma 1 For a given m k , regardless of the values of P tk and λ k , the optimal solution of the corresponding beam allocation vector is uniquely determined.

[0088] Lemma 2 For a given and any power allocation vector , the waveform allocation problem can be transformed into m k constraint problems only regarding and can be quickly solved.

[0089] Lemma 3 By transforming the multi-objective function into a single-objective function, for a given , the power and waveform joint allocation problem is equivalent to a convex optimization problem.

[0090] Among them,

[0091] Lemma 1: If within the (k - 1)-th sampling interval, the state estimation error covariance matrix of target q is then the one-step prediction error covariance matrix is:

[0092] Obviously, if then In the formula, S ++ represents a symmetric positive definite matrix. And for any there is holds. According to Equation (6), it can be obtained that:

[0093]

[0094] To solve First, for each batch of targets perform a descending order arrangement:

[0095]

[0096] In the formula, IX is the sorting vector of

[0097] Then, use mathematical induction to prove that for a given m k , can be uniquely determined by formula (13). For a given m k , can be calculated as:

[0098]

[0099] The first step: Prove that the proposition holds when m k = 1.

[0100] When m k = 1, all feasible beam allocation schemes can be enumerated as:

[0101] In the formula, represents using only one beam to point to target q. At this time, the objective function is:

[0102]

[0103] Similar to formula (13), when q = IX(1), for any and there is

[0104]

[0105] and hold. Therefore, to minimize formula (14), the optimal beam allocation result must be is a unit vector containing Q elements and the IX(1)-th element is 1. Thus, the proposition holds when m k = 1.

[0106] The second step: When m k = L holds, prove that the proposition still holds when m k = L + 1.

[0107] When m k = L holds, there is the following formula:

[0108]

[0109] At this time, when m k = L + 1, the feasible beam directions can be enumerated as:

[0110] In the formula, is defined as

[0111]

[0112] Therefore, the objective function can be written as:

[0113]

[0114]

[0115] In the formula, and respectively satisfy and c L is:

[0116]

[0117] Similar to before, for q = IX(L + 1), there is

[0118]

[0119] and Thus, to minimize the objective function, the optimal solution must be:

[0120]

[0121] Up to this point, it is proved that the proposition still holds when m k = L + 1.

[0122] In addition, using Lemma 2, for a given the optimal waveform allocation vector can be obtained, which means that P tk and λ k can be decoupled. Therefore, when enumerating all feasible (m k = 1, 2, … M), equation (9) is equivalent to:

[0123]

[0124] That is, for a given and it can be obtained by solving the following formula:

[0125]

[0126] Equation (23) is a multi-objective optimization problem. Although it can be solved using multi-objective optimization algorithms, the following two points should still be noted during the solution process:

[0127] (1) To meet the real-time requirements of the system, the algorithm should be as simple and efficient as possible;

[0128] (2) After solving Equation (23) m k times, it is also necessary to compare the objective functions. However, the number of objective functions corresponding to each m k is different, making the subsequent problems still complex.

[0129] Therefore, it can be seen that to convert the multi-objective function in Equation (23) into a single-objective function using the weighted summation method, 1 / m k is selected as the weight. At the same time, the power allocation vector and the waveform selection vector are introduced to rewrite Equation (23) as:

[0130]

[0131] Based on Lemma 3, Equation (24) is a convex optimization problem with 2m k variables. The present invention uses the Zoutendijk feasible direction method to quickly solve it, thereby obtaining and When obtaining and after that, and can be calculated respectively as:

[0132]

[0133]

[0134] To sum up, through variable separation and converting the multi-objective function into a single-objective function, the joint allocation problem of beam, waveform, and power is rewritten as M convex optimization problems. Using the Zoutendijk feasible direction method, the optimal allocation vector can be obtained:

[0135]

[0136] III. Cognitive Tracking Based on the Resource Allocation Optimization Model

[0137] Here, we use the SCKF for state filtering to solve the target tracking problem under non-linear measurements. After obtaining the tracking results of each batch of targets, the appropriate transmission parameters for the next moment are selected by predicting the posterior estimation error matrix. After the radar performs beam, power, and waveform optimization allocation, higher target estimation accuracy is obtained, and a closed-loop feedback loop is formed. As shown in the appendix Figure 1As shown in the figure, the entire process of cognitive tracking can be described as follows:

[0138] Step 1: Obtain the measurement information and state estimation covariance matrix of each batch of targets at the current moment;

[0139] Step 2: Rely on the obtained information to predict the state estimation covariance matrix at the next moment, and call the beam and power waveform joint allocation algorithm for resource allocation;

[0140] Step 3: For targets q = 1, 2,..., m k,opt Allocate corresponding resources and use SCKF to update target information; for other targets q = m k,opt +1,..., Q, update relevant information using Equation (6).

[0141] Embodiment

[0142] To further verify the effectiveness of the algorithm proposed by the present invention, the resource allocation scheme proposed by the present invention is compared with other resource allocation schemes. Consider a centralized MIMO radar system located at the origin, which can transmit a maximum number of simultaneous orthogonal beams of M = 4. Each transmitting unit transmits an OFDM-LFM signal with an effective bandwidth of 1 MHz and an effective time width of 1 ms. The frequency modulation slope is [0.1λ max , λ max ; the carrier frequency is 1 GHz; the lower bound of the transmission power is 0.1P total . The number of targets to be tracked simultaneously is 8; the sampling interval is 1 s. The benchmark measurement error is at a distance of 500 km from the radar σ r = 50 m; σ θ = 0.1 rad.

[0143] In the experiment, three benchmark allocation schemes are considered:

[0144] Scheme 1 (benchmark 1, B1) The MIMO radar uses (u k,opt , P tk,opt , λ0) for resource allocation. That is, the MIMO radar allocates beams and power using the proposed algorithm, but randomly selects waveforms.

[0145] Scheme 2 (benchmark 2, B2) The MIMO radar uses for resource allocation, m k = 1, 2... M. Wherein,

[0146]

[0147]

[0148] That is, for a given number of beam utilisations, the radar evenly distributes power among batches of targets and randomly selects waveforms.

[0149] Scheme 3 (benchmark 3, B3) MIMO radar uses for resource allocation. That is, for a given number of beam utilisations, the radar evenly distributes power among batches of targets and selects the optimal waveform.

[0150] Select the trace of the average posterior estimation error covariance matrix as the performance evaluation metric:

[0151]

[0152] To explore the influence of range and motion models on the resource allocation algorithm, the targets are divided into 4 groups, with each group of targets having the same range but different model errors, namely targets 1 and 2, targets 3 and 4, targets 5 and 6, targets 7 and 8. The specific parameters are shown in Table 1.

[0153] Table 1 Parameters of each target

[0154]

[0155] Appendix Figure 2a is the power allocation result of the proposed algorithm in a single simulation. Among them, the horizontal axis indicates the frame number, and the vertical axis indicates the target number. Rectangles of different colours represent different power allocation ratios, which are defined as:

[0156]

[0157] Among them, the dark blue area represents the power allocation ratio and the beam utilisation situation Other coloured areas represent From Figure 2a it can be seen that in the initial stage, the radar only uses one beam to irradiate the targets and allocates all the power to this batch of targets. Subsequently, all available beams are used to irradiate the targets and the power is optimally allocated. This is mainly because in the initial stage, the state estimation accuracy of all targets is relatively poor, and concentrated power is needed to improve the estimation accuracy to a relatively high level. When this goal is achieved, the radar then uses all available beams to improve the tracking performance of the targets.

[0158] Figure 2b is the corresponding waveform selection result. Each colour block is a subplot. In each subplot, each differently coloured block represents each frame (sampling time). The abscissa represents time, and the ordinate indicates the beam situation allocated to each target. Its colour represents the allocation situation of waveform parameters represented by given stripe colours according to the ratio. From Figure 2bIt can be seen that the radar uses the same waveform for illumination at the initial moment. As the target moves, the radar uses different waveforms for illumination of different targets.

[0159] Attached Figure 3 The changes of the covariance matrix trace of the posterior estimation error of each target are given. Figure 2a and Figure 2b It can be seen that the radar system adaptively distributes power and waveform in real time to minimize the tracking error of the covered target. Figure 3 It can be seen that after each resource allocation, the estimation error of the unilluminated targets increases. This is because the state estimation error of these targets comes from information prediction, that is, Equation (6). However, after 10 frames, the estimation error of all targets is reduced to a low level.

[0160] Figure 4 shows the resource allocation results after 100 simulation experiments. Figure 4a It can be seen that the targets with poor model accuracy obtain more power resources, specifically target 2>target 4>target 6>target 8. This is mainly because target 2 has a larger model error and is also the farthest away, resulting in the largest measurement error and requiring more power resources to improve tracking accuracy. In addition, Figure 4b It can be seen that MIMO radar optimizes tracking performance by adjusting the transmission waveform for each batch of targets in real time.

[0161] Attached Figure 5 The performance comparison results of the algorithm proposed in this invention and the other three resource allocation schemes are shown in Figure 2. Intuitively speaking, the more beams are used, the earlier the first inflection point appears in the figure. The value of the inflection point represents the radar system's use of the beam. The first inflection point appears only after all targets are illuminated. In the initial tracking stage, the proposed algorithm only uses one beam to illuminate the target and allocates all power and waveform resources to the target. Therefore, the first inflection point in the proposed resource allocation scheme appears relatively late. However, after all targets enter stable tracking, the proposed algorithm achieves the optimal E metric , demonstrating its superiority. Furthermore, in Schemes 2 and 3, the greater the number of beams utilized, the better the performance. While Scheme 1 utilized the optimal power allocation algorithm, its performance was poor due to the random selection of waveforms. Therefore, the combined allocation of power and waveforms is more conducive to improving the system's tracking performance.

[0162] Since the resources of the present invention include two variables, power and waveform, under the condition of keeping the waveform unchanged, the simulation experiment is repeated and the power distribution results are obtained as shown in the attached figure. Figure 6 As shown. Figure 6It can be seen that the power allocation result generally remains unchanged, but the power allocation ratio for Target 2 has increased, which is because this target has the farthest distance and the largest model error.

[0163] From the above simulations, the following conclusions can be drawn:

[0164] (1) Regardless of whether the RCS of the target follows a Swerling I or Swerling II distribution, the performance gap between the dynamic antenna configuration and the random antenna configuration will decrease as the SNR increases;

[0165] (2) Compared with the random configuration and the heuristic search algorithm, the proposed antenna selection algorithm has better performance;

[0166] (3) Although the exhaustive search algorithm can provide the optimal result, due to the limitation of computational complexity, it is difficult to be applied in practice. The proposed algorithm has lower computational complexity and higher computational accuracy, and is more suitable for engineering requirements.

[0167] The above shows and describes the basic principles, main features and advantages of the present invention. Those skilled in the art should understand that the present invention is not limited by the above embodiments. What is described in the above embodiments and the specification only illustrates the principles of the present invention. Without departing from the spirit and scope of the present invention, the present invention will have various changes and improvements, and these changes and improvements fall within the scope of the present invention claimed. The scope of protection of the present invention is defined by the appended claims and their equivalents.

Claims

1. A joint beam, power, and waveform allocation method for multi-target tracking in a centralized MIMO radar, characterized in that It includes the following steps: Step 1: Obtain the measurement information of each batch of targets at the current moment and derive the state estimation covariance matrix, and derive the posterior Cramer-Rao lower bound PCRLB of the state estimation lower bound as the optimization index; Among them, the specific steps of Step 1 include: Step 11: Under high signal-to-noise ratio conditions, use the CRLB in unbiased parameter estimation to characterize the measurement error covariance matrix R k : where, T = diag(c / 2, c / (2f c )) is the transformation matrix; J k is the BFIM and: where η is the SNR and η ∝ P t ; τ is the time delay; ν is the Doppler shift; A is the ambiguity function of the transmitted waveform, and the expression is: Among them, is the complex envelope of the transmitted waveform; Step 12: With the corresponding transmission power and waveform parameters to characterize the measurement covariance matrix R of target q within the k-th sampling interval k : Among them, is a scalar or a vector; Step 13: In the Kalman filter, the iterative formula of the posterior state estimation error matrix is: Step 14: Make the following approximation to Equation (6): wherein, and are respectively the approximate values of the Jacobian matrix of the measurement function and the measurement covariance matrix at the target state prediction point ; Step 2: Predict the state estimation covariance matrix at the next moment according to the state estimation error covariance matrix, select multiple target positions with the maximum state estimation error to construct the objective function, and establish a resource joint optimization model with the objective function; Among them, Step 2 specifically includes the following content: Step 21: Assume that the utilization of the beams for each target at the k-th sampling interval is and The waveform parameter is The transmit power is Arrange in descending order of trace to obtain: Step 22: Based on Equation (8), the resource allocation optimization model is obtained as: Among them, means that this batch of targets is not assigned waveforms; is the minimum threshold for power allocation; and constitute an optional range of waveform parameters; P total is the total power; M is the maximum number of simultaneous orthogonal beams generated; Step 3: Solve the established resource joint optimization model based on Lemma 1, Lemma 2 and Lemma 3 to obtain the optimal allocation vector, so that each target is allocated corresponding resources; Among them, it is assumed that for the number of feasible beam utilisations \(m\) k , \(m\) k \(\in \{1, 2, \ldots, M\}\), and the corresponding beam allocation vector is and there is Then Lemma 1, Lemma 2 and Lemma 3 are respectively: Lemma 1: For a given m k regardless of the values of P tk and λ k the corresponding optimal solution of the beam allocation vector is uniquely determined; Lemma 2: For a given and any power allocation vector the waveform allocation problem can be transformed into m k constraint problems that only concern and can be solved quickly; Lemma 3: By transforming the multi-objective function into a single-objective function, for a given the joint power and waveform allocation problem is equivalent to a convex optimization problem; The specific operation steps of Step 3 include: Step 31: According to Lemma 1, for a given m k , calculate as follows: Step 32: According to Lemma 2, for the determined the optimal waveform selection vector can be obtained Then when enumerating all feasible after that, Equation (9) is equivalent to: Step 33: For the determined and obtained by solving the following equation: Step 34: Convert Equation (23) into a single-objective function by weighted summation, and select 1 / m k as the weight, and at the same time introduce the power allocation vector and the waveform selection vector to rewrite Equation (23) as: Step 35: Based on Lemma 3, Equation (24) is a convex optimization problem with 2m k variables, which can be quickly solved using the Zoutendijk feasible direction method. Therefore, when and are obtained, and are calculated respectively as: Step 36: Through the above variable separation and transformation of the multi-objective function into a single-objective function, the joint allocation problem of the beam, waveform and power is rewritten as M convex optimization problems, and the optimal allocation vector can be obtained by using the Zoutendijk feasible direction method: Step 4: Use the square root cubature Kalman filter SCKF to update the target information; Step 5: Estimate and track the target position at the next moment according to the selected array elements, the final power allocation result and the optimal bandwidth allocation vector.

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