A smart reflective surface-assisted electromagnetic stealth method based on Lagrangian duality

Abstract

This invention discloses a Lagrange dual-based intelligent reflector-assisted electromagnetic stealth method, comprising the following steps: S1, inputting the number of sensing units, the number of passive reflective units of the intelligent reflector, the number of units on the untunable material surface, and their absorption efficiency and reflection phase information; S2, using a sensor array to estimate the direction of incoming waves from the enemy detection radar within the current coherent processing time interval, as well as its transmit and receive beam gain information; S3, constructing the intelligent reflector-assisted electromagnetic stealth problem; S4, using the Lagrange dual method to calculate the amplitude and phase of the passive reflective units of the intelligent reflector. The Lagrange dual-based intelligent reflector-assisted electromagnetic stealth method proposed in this invention significantly reduces algorithm complexity and improves the target's stealth performance, thereby meeting the stealth combat requirements of modern battlefields and adapting to complex and ever-changing battlefield situations.

Description

Technical Field

[0001] This invention relates to the field of electromagnetic stealth technology, specifically to a smart reflective surface-assisted electromagnetic stealth method based on Lagrange duality, applicable to aircraft, military equipment, and other stealth applications. Background Technology

[0002] In today's increasingly complex military warfare environment, radar technology plays a crucial role in detecting and tracking targets such as drones, aircraft, and missiles. However, the rapid development of radar technology in recent years has brought serious challenges to military equipment, making electromagnetic stealth technology an important tool for improving its survivability. The main objective of this technology is to achieve stealth by minimizing the electromagnetic wave energy reflected from combat equipment. Traditional electromagnetic stealth methods generally involve changing the shape of the target to reduce its radar cross-section or using absorbing materials to absorb incident electromagnetic waves. However, the effectiveness of traditional electromagnetic stealth technology is limited by multiple factors, including the characteristics of the absorbing materials, the angle of the incident electromagnetic waves, and the operating frequency, especially in high-dynamic electromagnetic environments, where it cannot achieve ideal stealth results. Therefore, traditional "one-step" electromagnetic stealth methods can no longer meet the needs of modern complex electromagnetic warfare environments.

[0003] In recent years, with the rapid development of related disciplines such as metamaterials, electromagnetic information, and interface electromagnetics, smart reflective surfaces, with their high integration, low cost, strong reconfigurability, ease of deployment, and lossless characteristics, have introduced a new technological paradigm into the fields of wireless communication and radar. The advantage of smart reflective surfaces lies in their ability to provide flexible and high-density deployment for future wireless communication and sensing systems through compact, lightweight, and customizable geometries. Furthermore, smart reflective surfaces possess real-time reconfiguration capabilities, enabling them to adapt to rapidly changing and highly dynamic wireless environments. Therefore, smart reflective surfaces can provide effective assistance in highly dynamic wireless environments, compensating for the shortcomings of traditional stealth technologies and opening new paths in the field of electromagnetic stealth. Current artificial electromagnetic metamaterials / metasurfaces can effectively reduce target backscattering over a wide frequency band; however, these materials are still affected by the frequency band, direction, and environment of electromagnetic signals, and low-cost, low-loss, and easily manufactured targets have not yet been achieved, making them unable to cope with complex and ever-changing electromagnetic stealth environments. Summary of the Invention

[0004] In order to at least solve one of the problems existing in the prior art, the present invention provides an intelligent reflective surface-assisted electromagnetic stealth method based on Lagrange duality, which has the advantages of low complexity, intelligent controllability, high flexibility and strong adaptability.

[0005] To achieve the objectives of this invention, a smart reflective surface-assisted electromagnetic stealth method based on Lagrange duality is provided, applicable to evading single-station or multi-station radar detection. The method is implemented based on a stealth system comprising at least one sensor array with L sensing units, at least one smart reflective surface with N1 passive reflective units, and at least one unadjustable material surface with N2 reflective units. The electromagnetic stealth method includes the following steps:

[0006] S1, the number of input sensing units, the number of passive reflection units of the intelligent reflective surface, the number of reflection units on the surface of the non-adjustable material, and the absorption efficiency and reflection phase information of the surface of the non-adjustable material.

[0007] S2. Use the sensor array to estimate the direction of incoming waves from the enemy's detection radar and its transmitted and received beam gains within the current coherent processing time interval.

[0008] S3. The problem of electromagnetic stealth by constructing intelligent reflective surfaces to assist in evading detection by single-station or multi-station radar. The problem of achieving ideal electromagnetic stealth using intelligent reflective surfaces is described as follows:

[0009]

[0010]

[0011] Where, θ n This represents the reflection coefficient of the nth passive reflective element of the intelligent reflector, where K is the number of radars. Let k be the received beam gain of radar k during the coherent processing time interval. Let i be the transmitted beam of radar i during the coherent processing time interval, and the superscript H denotes the conjugate transpose operation. and Let θ represent the cascaded array response vectors on the smart reflective surface and the non-tunable material surface, respectively, where θ is the reflection coefficient of the passive reflective unit of the smart reflective surface and φ is the reflection coefficient of the non-tunable material unit.

[0012] S4. Calculate the amplitude and phase of the passive reflective element of the smart reflective surface using the Lagrange duality method.

[0013] Furthermore, the sensors in the sensor array are arranged in a cross shape.

[0014] Furthermore, the sensor array is located at the center of the stealth target's surface.

[0015] Further, in step S1, the stealth system includes at least one stealth target surface. Assume a stealth target surface comprises N1 passive reflective units and N2 reflective units from unadjustable material surfaces, and the passive reflective units and the reflective units from the unadjustable material surfaces are arranged in a uniform planar array. A sensor array with L sensing units is deployed at the center of the stealth target surface. The reflection coefficient of each passive reflective unit at time t is defined as... in and These represent the modulation amplitude and phase range of the nth passive reflector unit, respectively. e denotes an exponential function, and the superscript T indicates the transpose of a vector; the reflection coefficient of each untunable material unit is defined as... in This represents the absorption efficiency of the m-th untunable material unit. It represents the reflection phase determined by material properties and can be obtained in advance through a large number of actual measurements.

[0016] Furthermore, the estimation step in step S2 includes:

[0017] The one-dimensional steering vector function is defined as: Where φ is defined as the signal phase difference between two adjacent antennas, adjacent passive reflective elements, or adjacent untunable material reflective elements, and N is defined as the size of a uniformly spaced linear array.

[0018] Assuming the wavelength of the enemy's detection radar signal is λ, and the distance between two adjacent sensing units is denoted as Δ... s The number of sensor units in the sensor array along the x-axis and y-axis are L respectively. x and L y At time t, the enemy radar k (denoted as R) k The detection signal reaches the surface of the stealth target (denoted as T). a The azimuth angle is and pitch angle The array response of the sensor array is expressed as follows: its array response along the x-axis and y-axis is expressed as follows:

[0019]

[0020] in, and It is the phase offset.

[0021] The received signal of the sensor array is represented as

[0022]

[0023] in, This indicates that the sensor array receives signals with a mean of 0 and a power of Gaussian white noise, The array response of the sensor array is represented by s(t) = [s1(t), ..., s2(t)]. K (t)] T This represents the transmitted signal vectors of K radars. This represents the transmitted signal of radar k. This represents the radar pulse waveform, where B is the signal bandwidth and t is the signal strength. p The pulse interval;

[0024] First, a multi-signal classification algorithm is used to estimate the direction of the incoming signal. Secondly, the least squares method is used to estimate the signal. The estimated signal is represented as

[0025]

[0026] in, This represents the estimated transmitted signal value of radar K. This represents the estimated noise vector. This represents the noise estimate from radar K. express The conjugate transpose of .

[0027] Furthermore, define and Let be the transmit beam gain and receive beam gain of radar k during the coherent processing time interval, respectively. Based on channel reciprocity, the transmit beam gain and receive beam gain are equal for each monostation radar. K represents the number of radars, and the estimated transmit beam gain is expressed as:

[0028]

[0029] Among them, T p Indicates the pulse repetition interval. Indicates source from radar The noise estimate; This indicates the operation of taking the real part. Indicates to Take conjugate.

[0030] Furthermore, the process of obtaining the electromagnetic stealth problem in step S3 includes:

[0031] The direction of the detection signal for each detection radar at time t is obtained based on the estimation. and transmit beam gain and received beam gain definition This indicates that radar k at time t is related to the intelligent reflector I and the untunable material surface N, respectively. a Surface of stealth target T a The far-field line-of-sight channel, the stealth target surface, and the array response of the radar. and Defined as:

[0032]

[0033]

[0034] in, and These represent the azimuth and elevation angles Δ of the radar transmitting or receiving signals, respectively. e and Δ a N represents the spacing between two adjacent elements on the surface of the stealth target and the radar antenna array, respectively; x and N y Representing the surface of the stealth target along the x-axis and Number of elements along the coordinate axes; M x and M x These represent the number of antennas along the x-axis and y-axis, respectively; the response of the smart reflector array. and array response of untunable material surfaces Similar to the definition of the surface array response of stealth targets, given the positional relationship between the smart reflective surface, the unadjustable material surface, and the stealth target surface, the array response of these three entities has the following relationship:

[0035]

[0036] Therefore, the far-field line-of-sight channel matrix between any two points (denoted by X and Y) at time t is expressed as:

[0037]

[0038] Where X∈{I,N} a T a}and α is defined as the channel gain at a reference distance of 1 meter. Defined as radar Real-time propagation distance to the stealth target Indicates path gain;

[0039] Since the far-field line-of-sight channel between any two points satisfies channel reciprocity during uplink and downlink transmission, then

[0040] Based on the co-location relationship between the stealth target surface, the smart reflector, and the untunable material surface, the channel matrix relationship between radar and these three satisfies:

[0041] After applying intelligent reflective surfaces to assist in stealth targeting, radar The received signal at time t is represented as

[0042]

[0043] in, w i and Let n represent the transmit beam gain of radar i and the receive beam gain of radar k, respectively. k (t) represents a mean of 0 and a variance of σ. 2 Gaussian white noise;

[0044] Assuming the stealth target's position remains constant within each radar coherent processing time interval, and thus negligible changes in the channel and geometric parameters between the stealth target and the radar, and omitting the time index [t] in the calculation, the received echo signal power of radar k within one pulse repetition time interval is expressed as:

[0045]

[0046] The total power of the received signal of the multi-station radar detection system is expressed as:

[0047]

[0048] According to the above formula, by using intelligent reflective surfaces to help reduce or completely eliminate the signal power reflected back to the radar detection system to achieve ideal stealth performance, we first define st as constrained by min x f(x) is defined as minimizing the objective function f(x); therefore, the problem of minimizing the radar received signal energy using a smart reflector is formulated as follows:

[0049]

[0050]

[0051] in, This represents the cascaded array response vector at the overall stealth target. and These represent the cascaded array response vectors on the smart reflective surface and the non-tunable material surface, respectively.

[0052] Furthermore, the cascaded array response vector at the overall stealth target for

[0053]

[0054] Furthermore, step S4 includes:

[0055] Based on the different parameters required for estimating and optimizing the reflection coefficient when evading single-station and multi-station radar detection, the reflection coefficient of the intelligent reflector is designed for two scenarios:

[0056] In the case of single-station radar detection, where there are no cross-linked signals in the radar received signal, the response vector of the cascaded array of the intelligent reflector and the untunable material surface is... and It can be directly expressed as and The superscript H denotes the conjugate transpose operation. Furthermore, only the direction of the incoming signal needs to be estimated, without concern for the radar's transmit and receive beam gains. Therefore, the optimization problem of smart reflector-assisted electromagnetic stealth can be reconstructed as follows:

[0057]

[0058]

[0059] in, The reflection gain of the untunable material surface unit is represented; then the optimal reflection coefficient is solved by applying the Lagrange duality method.

[0060] In the case of multi-station radar detection (K≥2), the optimization problem of smart reflector-assisted electromagnetic stealth can be restructured as follows:

[0061]

[0062]

[0063] in, and The joint gain of the received and transmitted beams between radar i and radar k is given; then the optimal smart reflector reflection coefficient is solved by applying the Lagrange duality method.

[0064] Furthermore, the optimal reflection coefficient of the smart reflector is solved using the Lagrange duality method, where...

[0065] In the context of single-station radar detection, the Lagrangian function of its optimization problem is defined as:

[0066]

[0067] in, Describe a Lagrange multiplier and satisfy

[0068] Then, the Lagrange function Take the partial derivative with respect to θ and set its derivative to 0:

[0069]

[0070] This gives us the expression for θ in terms of the Lagrange multiplier λ:

[0071] θ ＊ =-C(uu) H +diag(λ)) -1 u.

[0072] Secondly, θ ＊ Substitute into the Lagrange function The expression for the dual function is obtained as follows:

[0073]

[0074] Given the strong duality of the primal and dual problems, the optimal dual variables are obtained by solving the dual problem; the dual problem is then transformed into a positive semidefinite optimization problem using Schur complement theory.

[0075]

[0076]

[0077] Finally, the dual problem is solved using a solver, and the desired λ is substituted into θ. ＊ The optimal intelligent reflector coefficient can then be obtained; in the scenario of multi-station radar (K≥2) detection, the Lagrangian function of the above optimization problem is defined as

[0078]

[0079] in, Describe a Lagrange multiplier and satisfy

[0080] Then, the Lagrange function Take the partial derivative with respect to θ and set its derivative to 0:

[0081]

[0082] This gives us the expression for θ in terms of the Lagrange multiplier λ:

[0083]

[0084] Secondly, θ ＊ Substitute into the Lagrange function The expression for the dual function is obtained as follows:

[0085]

[0086] Given the strong duality of the primal and dual problems, the optimal dual variables are obtained by solving the dual problem; the dual problem is then transformed into a positive semidefinite optimization problem using Schur complement theory.

[0087]

[0088]

[0089] Finally, the dual problem is solved using a solver, and the desired λ is substituted into θ. ＊ This allows us to obtain the optimal intelligent reflective surface coefficient.

[0090] Furthermore, the solver is a CVX solver.

[0091] The present invention has the following advantages and effects compared with the prior art:

[0092] 1) The Lagrange duality-based intelligent reflector-assisted electromagnetic stealth method proposed in this invention can effectively reduce or even eliminate radar echo signals. This scheme reduces or cancels electromagnetic wave signals that are not absorbed by the unadjustable material surface by adjusting the reflection coefficient of the intelligent reflector, thereby reducing the echo signal reflected back to the radar system. Compared with traditional stealth technology, this method has the ability to adapt to high-speed flying targets and quickly switch radar operating modes, significantly improving stealth performance in complex electromagnetic environments.

[0093] 2) For both single-station and multi-station radar applications, this invention proposes a low-complexity angle and channel gain estimation scheme. This scheme requires only a small number of sensing devices to effectively estimate the direction and path gain information of the incoming signal, thereby significantly reducing hardware costs and simplifying algorithm complexity.

[0094] 3) The Lagrange dual-based intelligent reflector-assisted electromagnetic stealth method proposed in this invention yields a semi-closed solution for the reflection coefficient of the intelligent reflector. Compared to directly using a convex optimization solver, this stealth method has lower complexity. Moreover, the reflection coefficient can be adjusted in real time to improve the target's stealth performance under both single-station and multi-station radar detection scenarios. Attached Figure Description

[0095] The accompanying drawings, which are included to provide a further understanding of the invention and form part of this application, illustrate exemplary embodiments of the invention and, together with their description, serve to explain the invention and do not constitute an undue limitation thereof. In the drawings:

[0096] Figure 1This is a system model diagram of the intelligent reflective surface-assisted stealth aircraft evading radar detection in an embodiment of the present invention;

[0097] Figure 2 This is a flowchart of an intelligent reflective surface-assisted electromagnetic stealth method based on Lagrange duality provided in an embodiment of the present invention;

[0098] Figure 3 This is a performance comparison simulation diagram in Embodiment 1 of the present invention. (a) The diagram shows the case of a single-station radar, and (b) The diagram shows the case of a multi-station radar.

[0099] Figure 4 This is a performance comparison simulation diagram in Embodiment 2 of the present invention. (a) The diagram shows the case of a single-station radar, and (b) The diagram shows the case of a multi-station radar.

[0100] Figure 5 This is a performance comparison simulation diagram in Embodiment 3 of the present invention. (a) The diagram shows the case of a single-station radar, and (b) The diagram shows the case of a multi-station radar. Detailed Implementation

[0101] 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 clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0102] Please see Figure 1 , Figure 1 This is a system model diagram of the intelligent reflective surface-assisted electromagnetic stealth technology in all embodiments of the present invention. The stealth system includes at least one sensor array with L sensing units, at least one intelligent reflective surface with N1 passive reflective units, and at least one non-tunable material surface with N2 reflective units (hereinafter also referred to as non-tunable material units). The sensor array is used to estimate the direction of the radar detection signal and its transmission and reception beam gain. The intelligent reflective surface is used to control the amplitude and phase of the incident electromagnetic wave signal. The non-tunable material surface is used to absorb the radar detection signal. The unabsorbed signal is reflected and controlled by the intelligent reflective surface, thereby effectively reducing or even eliminating the signal returning to the radar.

[0103] To illustrate the technological advancements of this invention, the radar signal energy of the proposed Lagrange duality-based intelligent reflector-assisted electromagnetic stealth method was compared with other methods under different implementations on the MATLAB platform. These other methods included: 1) an intelligent reflector random phase shift method: the reflection phase is randomly generated within [0, 2π). 2) a DFT-based codebook search method: the codebook based on the Discrete Fourier Transform (DFT) is searched using... The reflection coefficient vector of the smart reflector is searched on the radar to minimize the sum of the signal power received by the radar.

[0104] Example 1

[0105] In this embodiment, the specific parameter settings are as follows:

[0106] Assuming the stealth target and the radar are located in the same two-dimensional plane, such that Only the elevation angle of arrival of the incoming signal needs to be considered. Other parameter settings are as follows: the number of radar transceiver antennas is M = 8 × 8 = 64, and the intelligent reflector and the non-adjustable material surface each have passive reflective elements N1 = N 1，x ×2(N 1，x The number of passive reflective units of the smart reflective surface along the x-axis and the number of unadjustable reflective units N2 = 100 × 2 = 200, the number of sensing units L = 9, the sensor array is deployed at the center of the stealth target surface, and the absorption efficiency of the unadjustable material (such as radar absorbing material or other materials on the aircraft surface) is... The reflection phase of the reflective elements on the surface of the untunable material is randomly generated and follows a uniform distribution in [0, 2π). The maximum reflection amplitude of the smart reflector is 1. Assume the radar system operates in the 150 GHz ultra-high frequency band, the wavelength of the transmitted signal is λ = 0.05 m, and the distance between two adjacent antennas in the radar system is Δ. a =λ / 2 = 0.025m, the distance between two adjacent passive reflective units is Δ e =λ / 4 = 0.0125m. The radar transmission waveform is... The signal bandwidth is B = 100MHz, and the pulse repetition interval and pulse interval are respectively set as T. p =100μs and t p =30μs, the shortest distance between the radar and the stealth target is set to 100 meters, the channel gain at a reference distance of 1 meter is β0 = -30dB, and the transmit power of each radar is set to P = 15dBm.

[0107] The following combination Figure 1 and Figure 2 The following describes the process steps of an intelligent reflective surface-assisted electromagnetic stealth method based on Lagrange duality disclosed in Example 1.

[0108] In this embodiment, step S1 is specifically implemented as follows:

[0109] Input information includes the number of sensing units, the number of passive reflective units on the smart reflective surface, the number of reflective units on the untunable material surface, their absorption efficiency, and reflection phase information.

[0110] The stealth system includes at least one stealth target surface. Assume that a stealth target surface comprises N1 passive reflective units and N2 reflective units from unadjustable material surfaces, and that the passive reflective units and the reflective units from the unadjustable material surfaces are arranged in a uniform planar array. A cross-shaped sensor array with L sensing units is deployed at the center of the stealth target surface. The reflection coefficient of each passive reflective unit at time t is defined as... in and These represent the modulation amplitude and phase range of the nth passive reflector unit, respectively. e denotes an exponential function, and the superscript T indicates the transpose of a vector; the reflection coefficient of each untunable material unit is defined as... in This represents the absorption efficiency of the m-th untunable material unit. This represents the reflection phase, which is determined by material properties and can be obtained in advance through numerous actual measurements. is the reflection coefficient of the N1th passive reflective unit.

[0111] In this embodiment, step S2 uses a sensor array to estimate the direction of incoming waves from the opposing detection radar within the current coherent processing time interval, and is specifically implemented as follows:

[0112] The one-dimensional steering vector function is defined as: Wherein, φ is defined as the signal phase difference between two adjacent antennas, adjacent passive reflective elements, or adjacent untunable material reflective elements. Defined as the size of a uniformly spaced linear array;

[0113] Assuming the wavelength of the enemy's detection radar signal is λ, and the distance between two adjacent sensing units is denoted as Δ... s The number of sensing units in the cross-shaped sensor array along the x-axis and y-axis are L respectively. x and L y At time t, the enemy radar k (denoted as R) k The detection signal of ) reaches the stealth target (denoted as T) a The azimuth angle is and pitch angle The array response of the sensor array is expressed as follows: its array response along the x-axis and y-axis is expressed as follows:

[0114]

[0115] in, and It is the phase offset.

[0116] The received signal of the sensor array is represented as

[0117]

[0118] in, This indicates that the sensor array receives signals with a mean of 0 and a power of Gaussian white noise, The array response of the sensor array is represented by s(t) = [s1(t), ..., s2(t)]. K (t)] T This represents the transmitted signal vectors of K radars. This represents the transmitted signal of radar k. This represents the radar pulse waveform, where B is the signal bandwidth and t is the signal strength. p The pulse interval is used. The direction of arrival of the signal is estimated using the Multiple Signal Classification (MUSIC) algorithm. Then, the least squares (LS) method is used to estimate the signal s(t), and the estimated signal is expressed as follows:

[0119]

[0120] in, This represents the estimated transmitted signal value of radar K. This represents the noise vector after LS estimation. This represents the noise estimate from radar K. express The conjugate transpose of . Definition and Radar During the coherent processing time interval, the transmit and receive beam gains are equal for each monostation radar due to channel reciprocity. K represents the number of radars. Finally, the estimated transmit beam gain can be expressed as:

[0121]

[0122] Among them, T p Indicates the pulse repetition interval. Indicates source from radar The noise estimate; This indicates the operation of taking the real part. Indicates to Take conjugate.

[0123] In this embodiment, step S3 constructs an intelligent reflective surface to assist in avoiding detection by single-station or multi-station radar, specifically implemented as follows:

[0124] Based on step S2, the signal direction of each detection radar at time t is estimated. and its transmitted beam gain and received beam gain definition This indicates that radar k at time t is associated with a smart reflector (represented by I) and an untunable material surface (represented by N). a The far-field line-of-sight channel of the stealth target surface is represented by N; N represents the number of elements on the stealth target surface and satisfies N = N1 + N2; the array response of the stealth target surface and radar k. and Defined as:

[0125]

[0126]

[0127] in, and These represent the azimuth and elevation angles of the radar's transmitted or received signals, respectively; Δ e and Δ a N represents the spacing between two adjacent elements on the surface of the stealth target and the radar antenna array, respectively; x and N y M represents the number of elements on the surface of the stealth target along the x-axis and y-axis, respectively; x and M x These represent the number of antennas along the x-axis and y-axis, respectively; the response of the smart reflector array. and array response of untunable material surfaces The definition method is the same as that for the array response of a stealth target surface. Given the positional relationship between the smart reflective surface, the unadjustable material surface, and the stealth target surface, the array response of these three components has the following relationship:

[0128]

[0129] Therefore, the far-field line-of-sight channel matrix between any two points (denoted by X and Y) at time t is expressed as:

[0130]

[0131] Where X∈{I,N} a T a}and α is defined as the channel gain at a reference distance of 1 meter. Defined as the real-time propagation distance between radar k and the stealth target. This represents the path gain; since the far-field line-of-sight channel between any two points satisfies channel reciprocity during uplink and downlink transmission, then... Based on the co-positional relationship between the target surface, the smart reflector, and the untunable material surface, the channel matrix relationship between the radar and these three satisfies:

[0132] After applying a smart reflector to assist in stealth targeting, the received signal of radar k at time t is expressed as follows:

[0133]

[0134] in, w represents the reflectance of the target surface. i and Let n represent the transmit beam gain of radar i and the receive beam gain of radar k, respectively. k (t) represents a mean of 0 and a variance of σ. 2 Gaussian white noise.

[0135] Assuming the stealth target's position remains constant within each radar coherent processing time interval, changes in the channel and geometric parameters between the stealth target and the radar can be ignored, and the time index [t] can be omitted in the calculation. Therefore, the received echo signal power of radar k within one pulse repetition time interval can be expressed as:

[0136]

[0137] in, This represents the channel matrix from the k-th radar to the stealth target; the superscript H indicates the conjugate transpose operation; The cascaded array response vector at the overall target is represented as

[0138]

[0139] in, and Let represent the cascaded array response vectors on the smart reflector and the non-tunable material surface, respectively; then the total received signal power of the multi-station radar detection system is expressed as:

[0140]

[0141] According to the above formula, ideal stealth performance can be achieved by using intelligent reflective surfaces to help reduce or completely eliminate the signal power reflected back to the radar detection system. First, let st be defined as constrained by min... x f(x) is defined as minimizing the objective function f(x); therefore, the problem of minimizing radar received signal energy (achieving ideal electromagnetic stealth) using intelligent reflective surfaces is formulated as follows:

[0142]

[0143]

[0144] Where, θ n This represents the reflection coefficient of the nth passive reflective unit of the intelligent reflective surface.

[0145] In this embodiment 1, step S4 uses the Lagrange duality method to calculate the amplitude and phase of the passive reflective element of the smart reflective surface, which is specifically implemented as follows:

[0146] Based on the different parameters required for estimating and optimizing the reflection coefficient when evading single-station and multi-station radar detection, the reflection coefficient of the intelligent reflector is designed for two scenarios:

[0147] S4.1. In the case of single-station radar (K=1) detection, if there is no cross-link signal in the radar received signal, then the response vector of the cascaded array of the smart reflector and the untunable material surface is... and It can be directly expressed as and The superscript H denotes the conjugate transpose operation. Furthermore, only the direction of the incoming signal needs to be estimated, without concern for the radar's transmit and receive beam gains. Therefore, the optimization problem of smart reflector-assisted electromagnetic stealth can be reconstructed as follows:

[0148]

[0149]

[0150] in, The reflection gain of the untunable material surface unit is represented; then the optimal reflection coefficient is solved by applying the Lagrange duality method.

[0151] First, the Lagrangian function of the above optimization problem is defined as follows:

[0152]

[0153] in, Describe a Lagrange multiplier and satisfy diag(λ) represents a diagonal matrix whose diagonal elements are elements of the vector λ.

[0154] Then, the Lagrange function Take the partial derivative with respect to θ and set its derivative to 0:

[0155]

[0156] This gives us the expression for θ in terms of the Lagrange multiplier λ:

[0157] θ ＊ =-C(uu) H +diag(λ)) -1 u.

[0158] Secondly, θ ＊ Substitute into the Lagrange function The expression for the dual function is obtained as follows:

[0159]

[0160] in, Given the strong duality of the primal and dual problems, the optimal dual variable is obtained by solving the dual problem. Based on Schur complement theory, an auxiliary variable q is introduced to transform the dual problem into a positive semidefinite optimization problem.

[0161]

[0162]

[0163] Finally, the dual problem is solved using the CVX solver, and the desired λ is substituted into θ. ＊ This allows us to obtain the optimal intelligent reflective surface coefficient.

[0164] S4.2. In the case of multi-station radar detection (K≥2), the optimization problem of intelligent reflector-assisted electromagnetic stealth can be reconstructed as follows:

[0165]

[0166]

[0167] in, and For radar i and radar The combined gain formed by the received and transmitted beams.

[0168] The optimal reflection coefficient of the smart reflector is solved using the Lagrange duality method. First, the Lagrange function for the above optimization problem is defined as...

[0169]

[0170] in, Describe a Lagrange multiplier and satisfy

[0171] Then, the Lagrange function Take the partial derivative with respect to θ and set its derivative to 0:

[0172]

[0173] This gives us the expression for θ in terms of the Lagrange multiplier λ:

[0174]

[0175] Secondly, θ ＊ Substitute into the Lagrange function The expression for the dual function is obtained as follows:

[0176]

[0177] Given the strong duality of the primal and dual problems, the optimal dual variable is obtained by solving the dual problem. Based on Schur complement theory, an auxiliary variable q is introduced to transform the dual problem into a positive semidefinite optimization problem.

[0178]

[0179]

[0180] Finally, the dual problem is solved using the CVX solver, and the desired λ is substituted into θ. ＊ The optimal intelligent reflective surface coefficient can then be obtained. It is understandable that other convex optimization solvers can also be used.

[0181] like Figure 3 As shown, Figure 3 The relationship between radar received signal power and angle of arrival estimation error was plotted. The method of this invention, within an estimation error range of no more than 2 degrees for both single-station and multi-station radars, shows significantly lower received signal power compared to the aforementioned intelligent reflector random phase shift method and DFT codebook search method. Furthermore, in single-station radar scenarios, the received signal power is only about 6.3% of that of the DFT codebook search method and approximately 4.7% of that of the random phase shift method; in multi-station radar scenarios, the total received signal power is only about 6.4% of that of the DFT codebook search method and approximately 3.5% of that of the random phase shift method.

[0182] Example 2

[0183] In this embodiment 2, the specific parameter settings are as follows:

[0184] Assuming the number of passive reflective elements in the intelligent reflector is N1 = 4 × 2 = 8, the shortest distance between the radar and the target... The distances are 0 meters, 30 meters, 40 meters, 50 meters, 60 meters, 70 meters, 80 meters, 90 meters, and 100 meters, respectively. For the settings of other parameters, please refer to Example 1.

[0185] In this embodiment 2, the distance between the radar and the target is measured each time. After selection, change the distance between the radar and the target. The relevant variables are determined, and steps S1-S4 are performed once, wherein steps S1-S4 are described in Example 1.

[0186] like Figure 4 As shown, Figure 4 The relationship between radar received signal power and the shortest distance between the radar and the target is demonstrated. First, the method of this invention, which deploys a smart reflector-assisted electromagnetic stealth system on the target, can significantly reduce the signal power received by the radar. Furthermore, the stealth performance of the Lagrange dual-based smart reflector-assisted electromagnetic stealth method is significantly superior to that of random phase-shift design and DFT codebook search-based methods. This also verifies the effectiveness of the smart reflector design scheme in reducing or eliminating reflected signal power, thereby preventing detection by enemy radar.

[0187] Example 3

[0188] In this embodiment 3, the specific parameter settings are as follows:

[0189] Assuming the distance parameters between the radar and the target The number of passive reflective units N1 of the intelligent reflective surface is fixed at 10, 20, 30, 40, 50, 60, 70, 80, 90, and 100 respectively. For the setting of other parameters, please refer to Example 1.

[0190] In this embodiment 3, each time the number of passive reflective units N1 of the smart reflective surface is taken, the variable related to the number of passive reflective units N1 of the smart reflective surface is changed, and steps S1-S4 are implemented once. For steps S1-S4, please refer to steps S1-S4 in embodiment 1.

[0191] like Figure 5 As shown, Figure 5A graph showing the relationship between radar received signal power and passive reflector elements was plotted. First, as the number of reflector elements on the smart reflector increases, the total received signal power of the radar decreases significantly. This is because the enhanced signal cancellation capability of the smart reflector eliminates or mitigates reflected signals to a single radar system. Second, even in the case of interconnected multi-station radars, the stealth method designed in this example can still achieve complete electromagnetic stealth when the number of reflector elements is greater than 90, meaning the signal power received by each radar is zero. This is because, compared to a single radar system, the minimum number of reflector elements required only needs to cancel the detection signal from a single radar, while achieving ideal electromagnetic stealth in a multi-station radar scenario requires a larger number of reflector elements to cancel the superimposed detection signals from different detection radars. The electromagnetic stealth system without the application of smart reflectors remains unchanged, and the electromagnetic stealth method based on random phase smart reflectors increases with the number of reflector elements N1, because the lack of suitable signal cancellation leads to an increase in the signal power reflected from the target.

[0192] In summary, the intelligent reflector-assisted electromagnetic stealth method based on Lagrange duality proposed in this invention can significantly reduce the signal power reflected from the target to the enemy's radar system, and improve the flexibility and adaptability of target stealth.

[0193] The above embodiments are preferred implementation modes of the present invention, but the implementation modes of the present invention are not limited to the above embodiments. Any other changes, modifications, substitutions, combinations, and simplifications that do not deviate from the spirit and principles of the present invention should be considered as equivalent replacement methods and are included in the scope of protection of the present invention.

Claims

1. A method for intelligent reflective surface-assisted electromagnetic stealth based on Lagrange duality, characterized in that, This method, applied to evade single-station or multi-station radar detection, is based on a stealth system. The stealth system comprises at least one sensor array with L sensing units, at least one smart reflective surface with N1 passive reflective units, and at least one unadjustable material surface with N2 reflective units. The electromagnetic stealth method includes the following steps: S1, the number of input sensing units, the number of passive reflection units of the intelligent reflective surface, the number of reflection units on the surface of the non-adjustable material, and the absorption efficiency and reflection phase information of the surface of the non-adjustable material. S2. Use the sensor array to estimate the direction of incoming waves from the enemy's detection radar and its transmit and receive beam gains within the current coherent processing time interval. S3. The problem of electromagnetic stealth by constructing intelligent reflective surfaces to assist in evading detection by single-station or multi-station radar. The problem of achieving ideal electromagnetic stealth using intelligent reflective surfaces is described as follows: Where, θ n This represents the reflection coefficient of the nth passive reflective element of the intelligent reflector, where K is the number of radars. Let k be the received beam gain of radar k during the coherent processing time interval. Let i be the transmitted beam of radar i during the coherent processing time interval, and the superscript H denotes the conjugate transpose operation. and Let θ represent the cascaded array response vectors on the smart reflective surface and the non-tunable material surface, respectively, where θ is the reflection coefficient of the passive reflective unit of the smart reflective surface and φ is the reflection coefficient of the non-tunable material unit. S4. Calculate the amplitude and phase of the passive reflective element of the smart reflective surface using the Lagrange duality method.

2. The method for intelligent reflective surface-assisted electromagnetic stealth based on Lagrange duality according to claim 1, characterized in that, The sensors in the sensor array are arranged in a cross shape.

3. The method for intelligent reflective surface-assisted electromagnetic stealth based on Lagrange duality according to claim 1, characterized in that, The sensor array is located at the center of the stealth target's surface.

4. The method for intelligent reflective surface-assisted electromagnetic stealth based on Lagrange duality according to claim 1, characterized in that, In step S1, the stealth system includes at least one stealth target surface. It is assumed that a stealth target surface includes N1 passive reflection units and N2 reflection units of non-adjustable material surfaces, and the passive reflection units and the reflection units in the non-adjustable material surfaces are arranged in the form of a uniform planar array. A sensor array with L sensing units is deployed at the center of the stealth target's surface; the reflection coefficient of each passive reflective unit at time t is defined as... in and These represent the modulation amplitude and phase range of the nth passive reflector unit, respectively. e denotes an exponential function, and the superscript T indicates the transpose of a vector; the reflection coefficient of each untunable material unit is defined as... in This represents the absorption efficiency of the m-th untunable material unit. This indicates the reflection phase, which is determined by the material properties.

5. The method for intelligent reflective surface-assisted electromagnetic stealth based on Lagrange duality according to claim 1, characterized in that, The estimation steps in step S2 include: The one-dimensional steering vector function is defined as: Wherein, φ is defined as the signal phase difference between two adjacent antennas, adjacent passive reflective elements, or adjacent untunable material reflective elements. Defined as the size of a uniformly spaced linear array; Assuming the wavelength of the enemy's detection radar signal is λ, and the distance between two adjacent sensing units is denoted as Δ... s The number of sensor units in the sensor array along the x-axis and y-axis are L respectively. x and L y At time t, the azimuth angle at which the detection signal from enemy radar k reaches the stealth target is... and pitch angle R k Indicates enemy radar, T a Indicates a stealth target. The array response of the sensor array is expressed as follows: its array response along the x-axis and y-axis is expressed as follows: in, and It is the phase offset; The received signal of the sensor array is represented as in, This indicates that the sensor array receives signals with a mean of 0 and a power of Gaussian white noise, The array response of the sensor array is represented by s(t) = [s1(t),...,s...]. K (t)] T This represents the transmitted signal vectors of K radars. This represents the transmitted signal of radar k. This represents the radar pulse waveform, where B is the signal bandwidth and t is the signal strength. p The pulse repetition interval; First, a multi-signal classification algorithm is used to estimate the direction of the incoming signal. Secondly, the least squares method is used to estimate the signal s(t), and the estimated signal is expressed as... in, This represents the estimated transmitted signal value of radar K. This represents the estimated noise vector. This represents the noise estimate from radar k. express The conjugate transpose operation.

6. The method for intelligent reflective surface-assisted electromagnetic stealth based on Lagrange duality according to claim 5, characterized in that, definition and These represent the transmit and receive beam gains of radar k within the coherent processing time interval, respectively. Based on channel reciprocity, the transmit and receive beam gains of each monostation radar are equal, i.e. K represents the number of radars; correspondingly, the estimated value of the transmit beam gain is expressed as... Among them, T p Indicates the pulse repetition interval. This represents the noise estimate from radar k; This indicates the operation of taking the real part. Indicates to Take conjugate.

7. The method for intelligent reflective surface-assisted electromagnetic stealth based on Lagrange duality according to claim 1, characterized in that, The process of obtaining the electromagnetic stealth problem in step S3 includes: Based on the estimation, the direction of the detection signal of each detection radar at time t can be obtained. and its transmitted beam gain and received beam gain definition Let radar k be at time t and interact with the intelligent reflective surface I and the untunable material surface N, respectively. a Surface of stealth target T a The far-field line-of-sight channel between the stealth target surface and the array response of radar k. and Defined as: in, and These represent the azimuth and elevation angles of the radar transmitting or receiving signals, respectively. and Let λ be the azimuth and elevation angles at which the enemy radar k's detection signal reaches the stealth target at time t, and let λ be the wavelength of the enemy detection radar signal. e and Δ a N represents the spacing between two adjacent elements on the surface of the stealth target and the radar antenna array, respectively; x and N y M represents the number of elements on the surface of the stealth target along the x-axis and y-axis, respectively; x and M x These represent the number of antennas along the x-axis and y-axis, respectively; the response of the smart reflector array. and array response of untunable material surfaces The definition method is the same as that for the surface array response of stealth targets; given the positional relationship between the smart reflective surface, the uncontrollable material surface, and the stealth target surface, the array response of these three has the following relationship: Let X and Y represent any two points. The far-field line-of-sight channel matrix between any two points at time t is expressed as: Where X∈{I,N a ,T a }and α is defined as the channel gain at a reference distance of 1 meter. Defined as the real-time propagation distance between radar k and the stealth target. Indicates real-time path gain; Since the far-field line-of-sight channel between any two points satisfies channel reciprocity during uplink and downlink transmission, then Based on the co-location relationship of the stealth target surface, the smart reflector surface, and the unadjustable material surface, the channel matrix relationship between the radar transmitted signal reaching these three surfaces satisfies: After applying a smart reflector to assist in stealth targeting, the received signal of radar k at time t is expressed as follows: in, w i and Let n represent the transmit beam gain of radar i and the receive beam gain of radar k, respectively. k (t) represents a mean of 0 and a variance of σ. 2 Gaussian white noise; Assuming the stealth target's position remains constant within each radar coherent processing time interval, and thus ignoring changes in the channel and geometric parameters between the stealth target and the radar, and omitting the time index [t] in the calculation; therefore, the received echo signal power of radar k within one pulse repetition time interval is expressed as: The total power of the received signal of the multi-station radar detection system is expressed as: In the formula, T p This indicates the pulse repetition interval, and P is the radar's transmit power. Let be the power of the received echo signal from radar k; According to the above formula, ideal stealth performance can be achieved by using intelligent reflective surfaces to help reduce or completely eliminate the signal power reflected back to the radar detection system; firstly, st is defined as constrained by min x f(x) is defined as minimizing the objective function f(x); therefore, the problem of minimizing the radar received signal energy using a smart reflector is formulated as follows: in, This represents the cascaded array response vector at the overall stealth target. and These represent the cascaded array response vectors on the smart reflective surface and the non-tunable material surface, respectively.

8. The method for intelligent reflective surface-assisted electromagnetic stealth based on Lagrange duality according to claim 7, characterized in that, Cascaded array response vector at the overall stealth target for and These represent the cascaded array response vectors of the intelligent reflective surface and the non-tunable material surface, respectively.

9. A method for intelligent reflective surface-assisted electromagnetic stealth based on Lagrange duality according to any one of claims 1-8, characterized in that, Step S4 includes the following steps: Based on the different parameters required for estimating and optimizing the reflection coefficient when evading single-station and multi-station radar detection, the reflection coefficient of the intelligent reflector is designed for two scenarios: In the case of single-station radar detection, where there are no cross-linked signals in the radar received signal, the response vector of the cascaded array of the intelligent reflector and the untunable material surface is... and Directly represented as u H and The superscript H denotes the conjugate transpose operation; furthermore, only the direction of the incoming signal needs to be estimated, without concern for the radar's transmit and receive beam gains; therefore, the optimization problem of smart reflector-assisted electromagnetic stealth can be reconstructed as follows: in, The reflection gain of the untunable material surface unit is represented; then, the optimal reflection coefficient is solved by applying the Lagrange duality method. In the case of multi-station radar detection, i.e., when K≥2, the optimization problem of intelligent reflector-assisted electromagnetic stealth is restructured as follows: in, and The joint gain of the received and transmitted beams between radar i and radar k is given; then, the optimal smart reflector reflection coefficient is solved by applying the Lagrange duality method.

10. The method for intelligent reflective surface-assisted electromagnetic stealth based on Lagrange duality according to claim 9, characterized in that, The optimal reflection coefficient of the smart reflector is solved using the Lagrange duality method, where... In the context of single-station radar detection, the Lagrangian function of its optimization problem is defined as: in, Describe a Lagrange multiplier and satisfy Then, the Lagrange function Take the partial derivative with respect to θ and set its derivative to 0: We can obtain the expression for θ in terms of the Lagrange multiplier λ: i ＊ =-C(uu H +diag(λ)) -1 you Secondly, θ ＊ Substitute into the Lagrange function The expression for the dual function is obtained as follows: Given the strong duality of the primal and dual problems, the optimal dual variables are obtained by solving the dual problem; the dual problem is then transformed into a positive semidefinite optimization problem using Schur complement theory. q is an auxiliary variable; Finally, the dual problem is solved using a solver, and the desired λ is substituted into θ. ＊ This allows us to obtain the optimal intelligent reflective surface coefficient; In the scenario of multi-station radar detection, i.e., when K≥2, the Lagrangian function of the above optimization problem is defined as: in, Describe a Lagrange multiplier and satisfy Then, the Lagrange function Take the partial derivative with respect to θ and set its derivative to 0: We can obtain the expression for θ in terms of the Lagrange multiplier λ: Secondly, θ ＊ Substitute into the Lagrange function The expression for the dual function is obtained as follows: Given the strong duality of the primal and dual problems, the optimal dual variables are obtained by solving the dual problem; the dual problem is then transformed into a positive semidefinite optimization problem using Schur complement theory. Finally, the dual problem is solved using a solver, and the desired λ is substituted into θ. ＊ This allows us to obtain the optimal intelligent reflective surface coefficient.

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