A method for integrating radar detection and jamming of a movable antenna

Abstract

This application relates to a mobile antenna-enabled radar detection and jamming integrated method, which includes the following steps: constructing a mobile antenna MA-enabled radar and jamming integrated IRAJ optimization problem for multi-target deceptive range image shaping and radar sensing, proposing an alternating optimization AO algorithm based on block continuous convex approximation BSCA, wherein waveform update adopts an internal approximation strategy and antenna position update adopts backtracking correction, and the performance of the proposed method is evaluated through numerical experiments. This application jointly optimizes the transmitted waveform and antenna position under the constraint of sensing signal-to-interference-plus-noise ratio to achieve deceptive jamming.

Description

Technical Field

[0001] This application relates to the field of radar and jamming integration, and more particularly to a radar detection and jamming integration method empowered by a movable antenna. Background Technology

[0002] With the increasing congestion of the electromagnetic spectrum, the importance of multifunctional integrated radio frequency (RF) platforms is constantly rising. Among them, integrated radar and jamming (IRAJ) has become a promising technical approach in electronic warfare (EW). In early research, the AMRFC Plan achieved the integration of electronic warfare, radar, and communications by sharing RF hardware. More recently, IRAJ-related research has further considered frequency diversity arrays for joint detection and electronic countermeasures, pulse-coded integrated waveforms for simultaneous target localization and countermeasures, joint design of waveforms and mismatched filters, knowledge-aided dual-function waveform design, and waveform-filter co-design for signal-dependent interference.

[0003] Movable antennas (MAs) can provide additional spatial degrees of freedom through adaptive antenna position optimization and have already demonstrated performance gains in beamforming problems. In addition, waveform design problems have been studied from the perspective of range profile (RP) separability, where RP is characterized by the convolution of the target range profile and the transmit waveform (TW). By jointly optimizing the waveform and the geometry of the UAV swarm, RP-based electromagnetic camouflage can be achieved. However, existing MA-related work mainly focuses on beamforming or pure deception design, while existing IRAJ studies generally do not utilize array reconfiguration capabilities. Summary of the Invention

[0004] The purpose of this application is to provide a radar detection and jamming integrated method empowered by a movable antenna, which jointly optimizes the transmitted waveform and antenna position under the constraint of perceived signal-to-interference-plus-noise ratio to achieve deception jamming.

[0005] To achieve the above objectives, this application provides the following technical solution:

[0006] This application provides a radar detection and jamming integrated method empowered by a movable antenna, comprising the following steps:

[0007] Constructing a mobile antenna MA-enabled radar and jamming integrated IRAJ optimization problem for multi-target deceptive range image shaping and radar sensing combined;

[0008] An alternating optimization AO algorithm based on block continuous convex approximation (BSCA) is proposed, in which waveform update adopts an internal approximation strategy and antenna position update adopts backtracking correction.

[0009] The performance of the proposed method was evaluated through numerical experiments.

[0010] Constructing a movable antenna MA-enabled radar and jamming integrated IRAJ optimization problem for multi-target deceptive range image shaping and radar sensing, specifically considering a co-located multi-function radar system whose transmitter uses N... T A MA array with N antenna elements, the receiver uses an N-type antenna array. R This radar system is a fixed array of antenna elements. It needs to simultaneously detect and jam multiple enemy targets. Let x = [x1, ..., x...]. N ] T ∈ R N T × 1 is the transmit antenna position vector APV, where x n Let S = [s1, ..., sn] represent the position of the nth transmitting antenna. N ] ∈ C L×N T is the transmitted waveform matrix, where s n ∈ C L×1 Let L be the waveform radiated by the nth transmitting antenna. · 2 and · F Let represent the Euclidean norm and the Frobenius norm, respectively; ℜ{·}, Tr(·), and diag(·) represent the real part, trace, and diagonalization operations, respectively; (·) T and(·) H These represent transpose and conjugate transpose, respectively.

[0011] Assume the directions of the K enemy targets are respectively Under the assumptions of far field and narrow band, the pointing direction θ k The launch steering vector is,

[0012] (1)

[0013] Where λ is the carrier wavelength, and for the k-th enemy target, let s h,k ∈ C L h,k ×1 represents the transmitted waveform of the enemy target, and it is assumed that this waveform has been accurately intercepted by our detector. Let d ∈ C. LRP is the desired distance image DRP, where L RP = L h,k + L – 1, to simplify the notation, the same symbol L is used for all k. RP This expression can be naturally generalized to different length cases related to the target. Therefore, the range image ORP observed by the enemy receiver is represented as s. h,k ⊛ Sa k (x), where ⊛ represents linear convolution. Since the design goal is to make all enemy targets perceive the same camouflaged target, a uniform DRP d is adopted, and the normalized mean square error (NMS) is used to measure the amplitude-sensitive mismatch between ORP and d.

[0014] (2)

[0015] Using Toeplitz convolution, the term is represented as... Equivalent to , in It is by The constructed Toeplitz matrix, i.e., the matrix representation of linear convolution with the enemy waveform, is rewritten as Equation (2) as follows:

[0016] (3)

[0017] Sensing performance is measured using the detection signal-to-interference-plus-noise ratio (SINR). For the k-th target, the corresponding SINR is:

[0018] (4)

[0019] Where, g k To comprehensively characterize the complex coefficients of the target's radar cross section, R n = σ 2 I LN Given the noise covariance matrix, if the sensing constraint SINR is required to be satisfied... k ⩾ Γ k , where Γ k Given a SINR threshold for the k-th target, it is equivalent to writing as

[0020] (5)

[0021] Where ρ k ≜ Γ k Tr(R n ) / (|g k | 2 N R In addition to perception constraints, the transmitted waveform matrix is ​​also limited by the total energy budget, i.e.

[0022] (6)

[0023] Furthermore, the antenna position is restricted to the interval [0, X]. max Within the [missing information] area, adjacent array elements must satisfy a minimum spacing constraint, and the corresponding feasible region for antenna positions is defined as follows:

[0024] (7)

[0025] To mitigate the mutual coupling effect in the MA array, the minimum spacing is taken as δ = λ / 2. Under the above constraints, the overall deception objective function is defined as follows:

[0026] (8)

[0027] Under the constraints of perception, energy, and antenna position, we jointly design S and x to optimize the deception performance on all enemy targets, thus obtaining the following joint waveform-array optimization problem.

[0028] (9).

[0029] An alternating optimization AO algorithm based on block continuous convex approximation (BSCA) is proposed, in which waveform updates employ an internal approximation strategy and antenna position updates employ backtracking correction. The specific steps are as follows:

[0030] To solve problem (9), the waveform matrix S and the antenna position vector x are updated alternately. Specifically, S is processed by internal approximation combined with the alternating direction multiplier method (ADMM), while x is first obtained based on a quadratic surrogate function to obtain candidate solutions, and then backtracked and corrected using the original objective function and SINR constraints.

[0031] Waveform update: In the t-th outer iteration, the APV is fixed as x = x(t), thus obtaining a subproblem concerning S. For ease of notation, define...

[0032] (10)

[0033] in Indicates the current iteration point The value of is used as a reference point for constructing a feasible internal approximation, and the corresponding... ,

[0034] (11)

[0035] The objective function of equation (11) is convex with respect to S, but the constraints... Because it is an anti-convex constraint, the subproblem remains non-convex even when x is fixed, which can be addressed by using convex functions. At the current point q (t) =S(t)a (t) The lower bound of the first order has

[0036] (12)

[0037] Therefore, constraints A feasible internal approximation can be expressed as

[0038] (13)

[0039] then, The following convex programming problem is obtained by solving it.

[0040] (14)

[0041] The ADMM algorithm is used to solve equation (14), and penalty parameters ϱ1 > 0 and ϱ2 > 0 are introduced to define auxiliary variables.

[0042] (15)

[0043] And record and Let be the corresponding scaling dual variable, then the ℓth ADMM iteration can be written as (16)

[0044] (17)

[0045] (18)

[0046] For each k, define

[0047] (19)

[0048] (20)

[0049] (twenty one)

[0050] like Then there is

[0051] (twenty two)

[0052] (twenty three)

[0053] Otherwise, take ADMM The subproblem can be reduced to: Projected onto the Frobenius norm sphere, that is...

[0054] (twenty four)

[0055] Subsequently, the dual variable behaves more...

[0056] (25)

[0057] When both the original residual and the dual residual are less than The inner loop terminates when the number of iterations reaches the upper limit.

[0058] Antenna position update, under the condition of fixed S = S(t+1), defines

[0059] (26)

[0060] right exist Performing a first-order Taylor expansion, we get...

[0061] (27)

[0062] in, Let x(t) be the Jacobian matrix calculated at x(t). Further

[0063] (28)

[0064] Then there is

[0065] (29)

[0066] Based on this approximation, for any Constructing a proxy function

[0067] (30)

[0068] Using the linear approximation in equation (29) and ignoring constant terms independent of x, we can obtain

[0069] (31)

[0070] in

[0071] (32)

[0072] (33)

[0073] Therefore, the candidate position update is given by the following formula.

[0074] (34)

[0075] Similarly, equation (34) is solved using ADMM, where the penalty parameter is denoted as... After completing the variable transformation Then, project onto the set. The process can be achieved through bounded equal-order projection. Let... Output to ADMM and define the search direction as...

[0076] (35)

[0077] To ensure the descent property of the original problem, further along the direction Backtracking search, i.e.

[0078] (36)

[0079] The initial step size is set to 1, and then... Decrease until satisfied.

[0080] (37)

[0081] in To fix the backtracking factor, For Armijo parameters, accept updates .

[0082] Compared with the prior art, the beneficial effects of the present invention are as follows: For the obtained waveform-position coupling optimization problem, a BSCA-AO algorithm is proposed for solving it. Numerical results show that, while meeting the perception requirements, the proposed scheme can achieve better range image deception performance than the FPA benchmark scheme, and the performance gain is more significant in more complex multi-target scenarios. Attached Figure Description

[0083] To more clearly illustrate the technical solutions of the embodiments of this application, the accompanying drawings used in the embodiments of this application will be briefly introduced below. It should be understood that the following drawings only show some embodiments of this application and should not be regarded as a limitation of the scope. For those skilled in the art, other related drawings can be obtained based on these drawings without creative effort.

[0084] Figure 1 This is a diagram of a radar and jamming integrated system powered by a movable antenna provided in an embodiment of the present invention.

[0085] Figure 2 (a) shows the variation of the average NMSE with Γ; (b) shows the convergence performance comparison.

[0086] Figure 3 This is a comparison chart of the DRP and its corresponding ORP obtained by the method provided in this embodiment of the invention and the FPA benchmark scheme.

[0087] Figure 4 This is a comparison diagram of MA and FPA beammaps under different NT conditions provided in the embodiments of the present invention. Detailed Implementation

[0088] The technical solutions of the embodiments of this application will now be described with reference to the accompanying drawings. It should be noted that similar reference numerals and letters in the following drawings indicate similar items; therefore, once an item is defined in one drawing, it does not need to be further defined and explained in subsequent drawings.

[0089] The terms “comprising,” “including,” or any other variations thereof are intended to cover a non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such a process, method, article, or apparatus. Without further limitation, an element defined by the phrase “comprising one…” does not exclude the presence of other identical elements in the process, method, article, or apparatus that includes said element.

[0090] The terms “first,” “second,” etc., are used only to distinguish one entity or operation from another, and should not be construed as indicating or implying relative importance, nor as requiring or implying any such actual relationship or order between these entities or operations.

[0091] This embodiment is achieved through the following technical solution: a radar detection and jamming integrated method empowered by a movable antenna, comprising the following steps:

[0092] Constructing a mobile antenna MA-enabled radar and jamming integrated IRAJ optimization problem for multi-target deceptive range image shaping and radar sensing combined;

[0093] An alternating optimization AO algorithm based on block continuous convex approximation (BSCA) is proposed, in which waveform update adopts an internal approximation strategy and antenna position update adopts backtracking correction.

[0094] The performance of the proposed method was evaluated through numerical experiments.

[0095] like Figure 1As shown, the optimization problem of constructing a mobile antenna MA-enabled radar and jamming integrated IRAJ system for multi-target deceptive range image shaping and radar sensing is specifically considered. This optimization problem involves a co-located multi-function radar system whose transmitter uses N... T A MA array with N antenna elements, the receiver uses an N-type antenna array. R This radar system is a fixed array of antenna elements. It needs to simultaneously detect and jam multiple enemy targets. Let x = [x1, ..., x...]. N ] T ∈ R N T × 1 is the transmit antenna position vector APV, where x n Let S = [s1, ..., sn] represent the position of the nth transmitting antenna. N ] ∈ C L×N T is the transmitted waveform matrix, where s n ∈ C L×1 Let L be the waveform radiated by the nth transmitting antenna. · 2 and · F Let represent the Euclidean norm and the Frobenius norm, respectively; ℜ{·}, Tr(·), and diag(·) represent the real part, trace, and diagonalization operations, respectively; (·) T and(·) H These represent transpose and conjugate transpose, respectively.

[0096] Assume the directions of the K enemy targets are respectively Under the assumptions of far field and narrow band, the pointing direction θ k The launch steering vector is,

[0097] (1)

[0098] Where λ is the carrier wavelength, and for the k-th enemy target, let s h,k ∈ C L h,k ×1 represents the transmitted waveform of the enemy target, and it is assumed that this waveform has been accurately intercepted by our detector. Let d ∈ C. L RP is the desired distance image DRP, where L RP = L h,k + L – 1, to simplify the notation, the same symbol L is used for all k. RP This expression can be naturally generalized to different length cases related to the target. Therefore, the range image ORP observed by the enemy receiver is represented as s.h,k ⊛ Sa k (x), where ⊛ represents linear convolution. Since the design goal is to make all enemy targets perceive the same camouflaged target, a uniform DRP d is adopted, and the normalized mean square error (NMS) is used to measure the amplitude-sensitive mismatch between ORP and d.

[0099] (2)

[0100] Using Toeplitz convolution, the term is represented as... Equivalent to , in It is by The constructed Toeplitz matrix, i.e., the matrix representation of linear convolution with the enemy waveform, is rewritten as Equation (2) as follows:

[0101] (3)

[0102] Sensing performance is measured using the detection signal-to-interference-plus-noise ratio (SINR). For the k-th target, the corresponding SINR is:

[0103] (4)

[0104] Where, g k To comprehensively characterize the complex coefficients of the target's radar cross section, R n = σ 2 I LN Given the noise covariance matrix, if the sensing constraint SINR is required to be satisfied... k ⩾ Γ k , where Γ k Given a SINR threshold for the k-th target, it is equivalent to writing as

[0105] (5)

[0106] Where ρ k ≜ Γ k Tr(R n ) / (|g k | 2 N R In addition to perception constraints, the transmitted waveform matrix is ​​also limited by the total energy budget, i.e.

[0107] (6)

[0108] Furthermore, the antenna position is restricted to the interval [0, X]. max Within the [missing information] area, adjacent array elements must satisfy a minimum spacing constraint, and the corresponding feasible region for antenna positions is defined as follows:

[0109] (7)

[0110] To mitigate the mutual coupling effect in the MA array, the minimum spacing is taken as δ = λ / 2. Under the above constraints, the overall deception objective function is defined as follows:

[0111] (8)

[0112] Under the constraints of perception, energy, and antenna position, we jointly design S and x to optimize the deception performance on all enemy targets, thus obtaining the following joint waveform-array optimization problem.

[0113] (9).

[0114] An alternating optimization AO algorithm based on block continuous convex approximation (BSCA) is proposed, in which waveform updates employ an internal approximation strategy and antenna position updates employ backtracking correction. The specific steps are as follows:

[0115] To solve problem (9), the waveform matrix S and the antenna position vector x are updated alternately. Specifically, S is processed by internal approximation combined with the alternating direction multiplier method (ADMM), while x is first obtained based on a quadratic surrogate function to obtain candidate solutions, and then backtracked and corrected using the original objective function and SINR constraints.

[0116] Waveform update: In the t-th outer iteration, the APV is fixed as x = x(t), thus obtaining a subproblem concerning S. For ease of notation, define...

[0117] (10)

[0118] in Indicates the current iteration point The value of is used as a reference point for constructing a feasible internal approximation, and the corresponding... ,

[0119] (11)

[0120] The objective function of equation (11) is convex with respect to S, but the constraints... Because it is an anti-convex constraint, the subproblem remains non-convex even when x is fixed, which can be addressed by using convex functions. At the current point q (t) =S(t)a (t) The lower bound of the first order has

[0121] (12)

[0122] Therefore, constraints A feasible internal approximation can be expressed as

[0123] (13)

[0124] then, The following convex programming problem is obtained by solving it.

[0125] (14)

[0126] The ADMM algorithm is used to solve equation (14), and penalty parameters ϱ1 > 0 and ϱ2 > 0 are introduced to define auxiliary variables.

[0127] (15)

[0128] And record and Let be the corresponding scaling dual variable, then the ℓth ADMM iteration can be written as (16)

[0129] (17)

[0130] (18)

[0131] For each k, define

[0132] (19)

[0133] (20)

[0134] (twenty one)

[0135] like Then there is

[0136] (twenty two)

[0137] (twenty three)

[0138] Otherwise, take ADMM The subproblem can be reduced to: Projected onto the Frobenius norm sphere, that is...

[0139] (twenty four)

[0140] Subsequently, the dual variable behaves more...

[0141] (25)

[0142] When both the original residual and the dual residual are less than The inner loop terminates when the number of iterations reaches the upper limit.

[0143] Antenna position update, under the condition of fixed S = S(t+1), defines

[0144] (26)

[0145] right exist Performing a first-order Taylor expansion, we get...

[0146] (27)

[0147] in, Let x(t) be the Jacobian matrix calculated at x(t). Further

[0148] (28)

[0149] Then there is

[0150] (29)

[0151] Based on this approximation, for any Constructing a proxy function

[0152] (30)

[0153] Using the linear approximation in equation (29) and ignoring constant terms independent of x, we can obtain

[0154] (31)

[0155] in

[0156] (32)

[0157] (33)

[0158] Therefore, the candidate position update is given by the following formula.

[0159] (34)

[0160] Similarly, equation (34) is solved using ADMM, where the penalty parameter is denoted as... After completing the variable transformation Then, project onto the set. The process can be achieved through bounded equal-order projection. Let... Output to ADMM and define the search direction as...

[0161] (35)

[0162] To ensure the descent property of the original problem, further along the direction Backtracking search, i.e.

[0163] (36)

[0164] The initial step size is set to 1, and then... Decrease until satisfied.

[0165] (37)

[0166] in To fix the backtracking factor, For Armijo parameters, accept updates .

[0167] The performance of the proposed method was evaluated through numerical experiments. Unless otherwise specified, g is taken as... k = 1, λ = 1 m, L = 32, X max = 10 m, and N R = 10 2 Among them, s h,k Randomly generated, the expected distance image d was generated by FEKO software. All experiments were performed on a personal computer equipped with an Intel Ultra 9 285H, 2.9GHz CPU, and 64 GB RAM. As the baseline scheme, a ULA-type FPA scheme with half-wavelength spacing and an MA baseline scheme modified from the literature were used, with the initial antenna position uniformly set to x. (0) = nX max / (N T + 1), n ​​= 1, ..., N T The average NMSE is defined as F / K. Since the literature shows that the gain brought by optimizing only the antenna position is relatively limited, the results of position optimization alone will not be given separately in this paper.

[0168] Unless otherwise specified, take σ. 2 = −10 dB, in the case of three targets, let θ1 = 30 ◦ θ2 = 120 ◦ θ3 = 150 ◦, Figure 2 (a) By comparing the average NMSE of the proposed method, the FPA baseline scheme, and the rewritten MA baseline scheme in the literature under different common Γ, it can be seen that the proposed method consistently achieves the lowest average NMSE throughout the entire range of Γ variation. Furthermore, as the number of enemy targets increases, the advantage of the proposed method over the baseline scheme becomes more obvious. Figure 2 (b) gives a representative scenario where K = 3 and Γ 1,2,3 Convergence performance at 15 dB. The objective function value converged by the proposed algorithm is lower than that of the rewritten MA baseline scheme in the literature (0.06556 vs. 0.10299), with corresponding running times of 0.27 s and 34.04 s, respectively.

[0169] Figure 3 By comparing the DRP and its corresponding ORP obtained by the proposed method with those obtained by the FPA baseline scheme, it can be seen that the ORP generated by the proposed method matches the expected distance image more accurately, while the FPA baseline scheme exhibits a more obvious mismatch phenomenon.

[0170] Figure 4 Comparison of different N T Under the same conditions, the beam diagrams of the proposed method and the FPA benchmark scheme show that, at the same N... T Below, the proposed MA design maintains the response in the target direction better than the corresponding FPA benchmark scheme, when N T When N is increased from 4 to 8, the proposed MA design maintains a good response in the target direction, while the sidelobes become more balanced overall. Specifically, when N... T When N = 6, the suppression in the middle corner region is stronger, but residual peaks still exist near the edge direction; while when N = 6, the suppression in the middle corner region is stronger, but residual peaks still exist near the edge direction; T When the value is 8, the overall side lobe contour is more uniform.

[0171] The above description is merely an embodiment of this application and is not intended to limit the scope of protection of this application. Various modifications and variations can be made to this application by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of this application should be included within the scope of protection of this application.

Claims

1. A method for integrating radar detection and jamming powered by a movable antenna, characterized in that, Includes the following steps: Constructing a mobile antenna MA-enabled radar and jamming integrated IRAJ optimization problem for multi-target deceptive range image shaping and radar sensing combined; An alternating optimization AO algorithm based on block continuous convex approximation (BSCA) is proposed, in which waveform update adopts an internal approximation strategy and antenna position update adopts backtracking correction. The performance of the proposed method was evaluated through numerical experiments.

2. The integrated radar detection and jamming method empowered by a movable antenna according to claim 1, characterized in that, Constructing a movable antenna MA-enabled radar and jamming integrated IRAJ optimization problem for multi-target deceptive range image shaping and radar sensing, specifically considering a co-located multi-function radar system whose transmitter uses N... T A MA array with N antenna elements, the receiver employs an N-type antenna array. R A fixed array of antenna elements. This radar system needs to simultaneously detect and deceive multiple enemy targets, denoted as x = [x1, ..., x]. N ] T ∈ R N T × 1 is the transmit antenna position vector APV, where x n Let S = [s1, ..., sn] represent the position of the nth transmitting antenna. N ]∈ C L×N T is the transmitted waveform matrix, where s n ∈ C L×1 Let L be the waveform radiated by the nth transmitting antenna. · 2 and · F Let represent the Euclidean norm and the Frobenius norm, respectively; ℜ{·}, Tr(·), and diag(·) represent the real part, trace, and diagonalization operations, respectively; (·) T and(·) H These represent transpose and conjugate transpose, respectively.

3. The integrated radar detection and jamming method empowered by a movable antenna according to claim 2, characterized in that, Assume the directions of the K enemy targets are respectively Under the assumptions of far field and narrow band, the pointing direction θ k The launch steering vector is, （1） Where λ is the carrier wavelength, and for the k-th enemy target, let s h,k ∈ C L h,k ×1 represents the transmitted waveform of the enemy target, and it is assumed that this waveform has been accurately intercepted by our detector. Let d ∈ C. L RP is the desired distance image DRP, where L RP =L h,k + L – 1, to simplify the notation, the same symbol L is used for all k. RP This expression can be naturally generalized to different length cases related to the target. Therefore, the range image ORP observed by the enemy receiver is represented as s. h,k ⊛ Sa k (x), where ⊛ represents linear convolution. Since the design goal is to make all enemy targets perceive the same camouflaged target, a uniform DRP d is adopted, and the normalized mean square error (NMS) is used to measure the amplitude-sensitive mismatch between ORP and d. （2） Using Toeplitz convolution, the term is represented as... Equivalent to , in It is by The constructed Toeplitz matrix, i.e., the matrix representation of linear convolution with the enemy waveform, is rewritten as Equation (2) as follows: （3） Sensing performance is measured using the detection signal-to-interference-plus-noise ratio (SINR). For the k-th target, the corresponding SINR is: （4） Where, g k To comprehensively characterize the complex coefficients of the target's radar cross section, R n = σ 2 I LN Given the noise covariance matrix, if the sensing constraint SINR is required to be satisfied... k ⩾ Γ k , where Γ k Given a SINR threshold for the k-th target, it is equivalent to writing as （5） Where ρ k ≜ Γ k Tr(R n ) / (|g k | 2 N R In addition to perception constraints, the transmitted waveform matrix is ​​also limited by the total energy budget, i.e. （6） Furthermore, the antenna position is restricted to the interval [0, X]. max Within the [missing information] area, adjacent array elements must satisfy a minimum spacing constraint, and the corresponding feasible region for antenna positions is defined as follows: （7） To mitigate the mutual coupling effect in the MA array, the minimum spacing is taken as δ = λ / 2. Under the above constraints, the overall deception objective function is defined as follows: （8） Under the constraints of perception, energy, and antenna position, we jointly design S and x to optimize the deception performance on all enemy targets, thus obtaining the following joint waveform-array optimization problem. （9）。 4. The integrated radar detection and jamming method empowered by a movable antenna according to claim 2, characterized in that, An alternating optimization AO algorithm based on block continuous convex approximation (BSCA) is proposed, in which waveform updates employ an internal approximation strategy and antenna position updates employ backtracking correction. The specific steps are as follows: To solve problem (9), the waveform matrix S and the antenna position vector x are updated alternately. Specifically, S is processed by internal approximation combined with the alternating direction multiplier method (ADMM), while x is first obtained based on a quadratic surrogate function to obtain candidate solutions, and then backtracked and corrected using the original objective function and SINR constraints. Waveform update: In the t-th outer iteration, the APV is fixed as x = x(t), thus obtaining a subproblem concerning S. For ease of notation, define... （10） in Indicates the current iteration point The value of is used as a reference point for constructing a feasible internal approximation, and the corresponding... , （11） The objective function of equation (11) is convex with respect to S, but the constraints... Because it is an anti-convex constraint, the subproblem remains non-convex even when x is fixed, which can be addressed by using convex functions. At the current point q (t) = S(t)a (t) The lower bound of the first order has （12） Therefore, constraints A feasible internal approximation can be expressed as （13） then, The following convex programming problem is obtained by solving it. （14） The ADMM algorithm is used to solve equation (14), and penalty parameters ϱ1 > 0 and ϱ2 > 0 are introduced to define auxiliary variables. （15） And record and Let be the corresponding scaling dual variable, then the ℓth ADMM iteration can be written as (16) （17） （18） For each k, define （19） （20） （21） like Then there is （22） （23） Otherwise, take ADMM The subproblem can be reduced to: Projected onto the Frobenius norm sphere, that is... （24） Subsequently, the dual variable behaves more... （25） When both the original residual and the dual residual are less than The inner loop terminates when the number of iterations reaches the upper limit. Antenna position update, under the condition of fixed S = S(t+1), defines （26） right exist Performing a first-order Taylor expansion, we get... （27） in, Let x(t) be the Jacobian matrix calculated at x(t). Further （28） Then there is （29） Based on this approximation, for any Constructing a proxy function （30） Using the linear approximation in equation (29) and ignoring constant terms independent of x, we can obtain （31） in （32） （33） Therefore, the candidate position update is given by the following formula. （34） Similarly, equation (34) is solved using ADMM, where the penalty parameter is denoted as... After completing the variable transformation Then, project onto the set. The process can be achieved through bounded equal-order projection. Let... Output to ADMM and define the search direction as... （35） To ensure the descent property of the original problem, further along the direction Backtracking search, i.e. （36） The initial step size is set to 1, and then... Decrease until satisfied. （37） in To fix the backtracking factor, For Armijo parameters, accept updates .

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