Satellite-ground downlink covert communication system and method combining beamforming and cooperative interference

Abstract

The invention discloses a satellite-ground downlink covert communication system and a satellite-ground downlink covert communication method combining beamforming and cooperative interference, which belong to the technical field of wireless communication, and are characterized in that the satellite-ground downlink covert communication is realized through the construction of a covert communication model based on the beamforming and cooperative interference technology in a satellite-ground downlink communication scene, the concealment calculation and the concealment rate calculation. The satellite-ground downlink covert communication system can obtain a covert rate greater than zero, and is composed of a multi-antenna satellite transmitter, a multi-antenna cooperation ground jammer, a single-antenna ground receiver and a single-antenna ground monitor. Transmitter-receiver transmission is performed by means of hidden beam forming, jammer-listener interference is performed by means of null-space beam forming, a hidden communication system can be efficiently supported in a satellite-ground system by means of the scheme of combining beam forming and cooperative interference, and a good effect is achieved.

Description

Technical Field

[0001] This invention relates to the field of wireless communication technology, and more specifically to a satellite-to-ground downlink covert communication system that combines beamforming and cooperative jamming.

[0002] This invention also relates to a satellite-to-ground downlink covert communication method that combines beamforming and cooperative jamming. Background Technology

[0003] Traditional methods for protecting communication content include cryptographic methods and physical layer security methods based on the inherent characteristics of the channel. However, in some scenarios with extremely high security requirements, protecting the communication content alone is insufficient, and it is also necessary to protect the existence of the communication process. Therefore, covert communication technology, which aims to hide the communication process, has attracted widespread attention. Covert communication hides communication signals in background noise to protect the signal transmission itself from detection, and it holds great promise as a robust form of security for modern wireless communication systems.

[0004] Existing research on covert communication in satellite-to-ground communication link systems primarily focuses on scenarios where the satellite transmitter is always equipped with a single antenna. Research results indicate that beamforming jamming and multiple antennas are two attractive techniques for effectively supporting covert communication. However, how to apply these two techniques to achieve more efficient covert communication in satellite-to-ground communication link systems remains an open question. Therefore, it is necessary and feasible to study covert communication systems and methods based on beamforming and cooperative jamming techniques in satellite-to-ground communication links, and to analyze their covertness and covert transmission rates. Summary of the Invention

[0005] The purpose of this invention is to address the aforementioned problems by providing a satellite-to-ground downlink covert communication system and method that combines beamforming and cooperative jamming. Based on the construction of a covert communication model using beamforming and cooperative jamming technologies, covertness calculation and covert rate calculation can achieve a covert rate greater than zero. The proposed beamforming and cooperative jamming combination scheme can efficiently support covert communication systems in satellite-to-ground systems.

[0006] The first technical solution adopted in this invention is as follows: A satellite-to-ground covert communication system combining beamforming and cooperative jamming, the system comprising a satellite transmitter Alice, a ground receiver Bob, a ground monitor Willie, and a ground cooperative jammer Jammer; The satellite transmitter Alice is equipped with a planar array antenna, the ground cooperative jammer Jammer is equipped with multiple isotropic antennas, and the ground receiver Bob and the ground monitor Willie are each equipped with a single antenna. The satellite transmitter Alice transmits covert information to the ground receiver Bob using beamforming technology. The ground monitor Willie detects the covert information within its monitoring range. The ground-based cooperative jammer Jammer sets its beamforming vector to null beamforming of the jammer-receiver channel using null beamforming and then sends jamming signals to the ground receiver Bob to confuse the ground monitor Willie's detection.

[0007] The second technical solution adopted in this invention is a satellite-to-ground downlink covert communication method combining beamforming and cooperative jamming, employing a satellite-to-ground downlink covert communication system combining beamforming and cooperative jamming. The method specifically includes: System link initialization; A monitor detection model is established. The monitor Willie performs a binary hypothesis test based on the signals it receives to test whether the satellite transmitter Alice sends a covert signal to the ground receiver Bob. Then, the monitor Willie detects the test results through an energy detector to obtain the detection error probability, false alarm rate, and false negative rate. Monitor performance analysis is performed to determine the minimum detection error probability, false alarm rate, and false negative rate under perfect / imperfect monitor positioning conditions. With a fixed satellite antenna line of sight under perfect / imperfect monitor location information, and at maximum antenna gain, the covert transmission rate is optimized through beamforming-artificial interference joint design. The satellite antenna line-of-sight direction was optimized, and finally, under perfect / imperfect monitor positions, the covert transmission rate was optimized through a joint design of beamforming, artificial interference, and antenna line-of-sight.

[0008] Furthermore, the binary hypothesis testing process of the monitor detection model is specifically as follows: (1) In the formula, For the detection threshold, and They support the null hypothesis respectively and alternative assumptions The decision, The average value of the signal energy received at the monitor Willie's location is... Greater than Monitor Willie made The decision was made that the transmitter Alice was transmitting a covert signal; conversely, the monitor Willie made... The decision was made that the transmitter Alice was not currently transmitting any covert signals; The probability of detection errors includes the false alarm rate and the false negative rate, which are respectively , , For when Make a decision for the real-time monitor Willie The probability, For when When the result is true, monitor Willie makes a decision. The probability of detection error is thus determined by the probability of [missing information]. The specific formula is as follows: (2).

[0009] Furthermore, the monitor performance analysis is as follows: Utilizing the perfect monitor position , , These represent the minimum false detection probability, false alarm rate, and false negative rate, respectively. The false alarm rate is specifically calculated as follows: (3) In the formula, The variance of the additive white Gaussian noise at the location of the monitor Willie is... The detection threshold; The specific formula for the false negative rate is as follows: (4); The specific probability of detection error is as follows: (5) In the formula: (6) (7) In the formula, , , , , , , It is a first-order Marcum function; Optimal detection threshold under perfect monitor positioning conditions The specific formula is as follows: (8) Minimum detection error probability corresponding to the optimal detection threshold The specific formula is as follows: (9) Due to the minimum detection error probability Since it contains a first-order Marcum function, we use the lower bound of the detection error probability for further analysis. The lower bound of the minimum detection error probability is as follows: (10).

[0010] Furthermore, the monitoring performance analysis under the condition of imperfect monitor location includes: assuming that the location information of monitor Willie has estimation errors, firstly, modeling the true position of the monitor including the estimation errors, as follows: (11) In the formula, This represents the estimation error of the location information of the monitor Willie, and should meet the following requirements. and ,in and Given constraints; Given the imperfect location information of the monitor Willie, the minimum detection error probability is determined by setting Willie to the optimal position. The optimization problem is as follows: (12) (13) Solve the above equation to obtain the minimum detection error probability (DEP) for the monitor Willie's optimal and imperfect position information. .

[0011] Furthermore, with the satellite antenna line of sight determined by perfect observer location information, and under the condition of maximum antenna gain, the covert transmission rate is optimized through a joint design of beamforming and artificially assisted interference. The optimization problem is specifically as follows: (14a) (14b) (14c) (14d) (14e) (14f) (14g) (14h) In the formula, and Let Alice be the maximum transmit power and Jammer be the maximum transmit power, respectively. Constraint formula (14b) ensures that there is no signal leakage to the monitor Willie on the direct path of the satellite-to-ground link. Constraint formula (14c) ensures that the jamming signal of Jammer will not affect the receiver Bob. Constraint formula (14h) represents the concealment constraint under the perfect monitor Willie's position information. Solve the optimization problem under perfect monitor location information to obtain the maximum covert transmission rate.

[0012] Furthermore, given imperfect observer location information, the satellite antenna line of sight is determined. Under maximum antenna gain, the covert transmission rate is optimized through a joint design of beamforming and artificially assisted interference. The optimization problem is specifically as follows: (15) (16) (14c) (14d) (14e) (14f) (14g) In the above formula, formula (16) represents the concealment constraint under imperfect monitor location information, let , ,therefore, Follower The value increases monotonically with the increase of , and in order to satisfy the hidden constraint, we can obtain ,in For the Lambert-W function, we can further obtain the range of values ​​for the transmit power that satisfies the concealment constraint: ; Similar to the method for finding the optimal covert transmission rate under perfect monitor location information, the above optimization problem is solved to obtain the optimal covert transmission rate under imperfect monitor location information.

[0013] Furthermore, optimizing the line-of-sight direction of the satellite antenna specifically includes: First, project points S, W, and B in the three-dimensional Cartesian coordinate system onto a unit sphere centered on the satellite transmitter Alice. Their corresponding mapping points on the sphere are: , and spherical triangle The arc length is , and Therefore, the off-axis viewing angles of ground receiver Bob and monitor Willie must meet the following requirements. , To maximize the covert transmission rate (CR), it is necessary to maximize Minimize at the same time ; Let the optimal aiming point of the satellite transmitter Alice on the sphere be denoted as: At this time, the off-axis angle of ground receiver Bob is expressed as... For any , needs to be maximized To achieve the maximum covert transmission rate, according to the spherical cosine theorem, the following formula applies: (17) In the formula, Only when equal To achieve the minimum value, therefore, under the line of sight of the optimal satellite transmitter Alice antenna, the point... , and They are concyclic, and simultaneously, in three-dimensional Cartesian coordinates, the point... It lies on the ray from point W to point B.

[0014] Furthermore, after optimizing the satellite antenna line-of-sight direction, under perfect observer location information, the beamforming-artificial interference-antenna line-of-sight joint design optimization problem for maximizing concealed transmission rate is as follows: (18) (19) exist Under the condition that, the variables in the above formula The off-axis angle of ground receiver Bob can be used. Instead, the optimization problem is further simplified as follows: (20) (twenty one) (twenty two) (14c) (14d) (14e) (14f) (14g) (14h) Solve the above optimization problem formula to obtain the maximum covert transmission rate under perfect monitor location information.

[0015] Furthermore, after optimizing the satellite antenna line-of-sight direction, the joint design problem of beamforming, artificial interference, and antenna line-of-sight for maximizing concealed transmission rate under imperfect observer position information is expressed as follows: (twenty three) (twenty four) (14c) (14d) (14e) (14f) (14g) (twenty one) (twenty two) Similar to the solution to the optimization problem under perfect monitor location information, the maximum covert transmission rate is obtained under imperfect monitor location information.

[0016] In summary, due to the adoption of the above technical solution, the beneficial effects of the present invention are: The system and method of this invention construct a covert communication model based on beamforming and cooperative jamming techniques in a satellite-to-ground downlink communication scenario, calculate the covertness and covert rate, and achieve a covert rate greater than zero. The satellite-to-ground downlink covert communication system consists of a multi-antenna satellite transmitter, a multi-antenna cooperative ground jammer, a single-antenna ground receiver, and a single-antenna ground monitor. By utilizing covert beamforming for transmitter-to-receiver transmission and null-space beamforming for jammer-to-monitor interference, this invention proposes a novel beamforming and cooperative jamming scheme for covert transmission in related covert systems. It provides theoretical models of covert rate and detection error probability to describe the performance of the proposed scheme under scenarios with perfect monitor location information and imperfect eavesdropping location information. Based on these results, this invention presents the optimal beamforming and cooperative jamming design problem for maximizing the covert rate under fixed default settings of the satellite antenna line of sight and perfect / imperfect monitors. By carefully considering the inherent propagation characteristics of the satellite-to-ground downlink communication link, an efficient algorithm based on semi-definite relaxation SDR is developed to solve this optimization problem. A non-convex optimization problem is further proposed for the joint optimization design of beamforming, jamming, and satellite antenna line of sight to achieve the maximum covert rate under perfect / imperfect observer location information. With the help of spherical geometry, semidefinite relaxation, and continuous convex approximation (SCA) techniques, relevant algorithms are designed to solve this optimization problem. Finally, extensive numerical simulation results are presented to verify the theoretical results of this invention and demonstrate that the proposed beamforming and cooperative jamming combination scheme can efficiently support covert communication systems in satellite-to-ground systems, achieving excellent results. Attached Figure Description

[0017] Figure 1 This is a schematic diagram of a satellite-to-ground downlink covert communication system combining beamforming and cooperative jamming according to the present invention. Figure 2 This is a geometric diagram showing the relationship between the planar array antenna equipped on the satellite and the ground node in the system of this invention; Figure 3 It refers to the geometric relationship between the satellite and the ground node in the system of this invention; Figure 4 This is a flowchart of a satellite-to-ground downlink covert communication method combining beamforming and cooperative jamming according to the present invention; Figure 5 This is a flowchart of the monitor detection process in the method of the present invention; Figure 6 This is a flowchart of the method for optimizing the line of sight of a satellite antenna in this invention; Figure 7 This describes how the detection error probability changes with the detection threshold in the method of this invention; Figure 8This describes how the minimum detection error probability varies with signal transmission power and cooperative interference power under the condition of perfect / imperfect monitor location information in the method of this invention. Figure 9 This describes the variation of the covert transmission rate with the maximum transmission power of the transmitter and the maximum interference power of the jammer under the condition of perfectly monitoring the location information in the method of this invention. Figure 10 This describes the variation of the covert transmission rate with the maximum transmission power of the transmitter and the maximum interference power of the jammer in the case of imperfect monitor location information in the method of this invention. Figure 11 This describes the change in covert transmission rate with the monitor's location under the condition of perfect / imperfect monitor location information in the method of this invention. Detailed Implementation

[0018] The present invention will now be described in detail with reference to the accompanying drawings.

[0019] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention.

[0020] Example 1 This embodiment provides a covert communication system based on beamforming and cooperative jamming techniques in a satellite-to-ground communication link, such as... Figure 1 As shown, the system includes the satellite transmitter Alice, the ground receiver Bob, the ground monitor Willie, and the ground-based jammer Jammer. Both the ground receiver Bob and the ground monitor Willie are within the coverage area of ​​the satellite transmitter Alice. The transmitter Alice is equipped with [number of antennas missing]. Jammer is equipped with a uniform planar array antenna. Using isotropic antennas, both Bob and the monitor Willie are equipped with a single antenna. This embodiment assumes that the monitor Willie knows the exact locations of the transmitter Alice, the receiver Bob, and the jammer Jammer. Regarding the monitor Willie's location information, this embodiment considers two cases: perfect monitor Willie location information and imperfect monitor Willie location information. In the case of perfect monitor Willie location information, the exact location of the monitor Willie is available to the transmitter Alice, the receiver Bob, and the jammer Jammer; in the case of imperfect monitor Willie location information, only the location information of the monitor Willie with prediction errors is known at the aforementioned nodes.

[0021] In the covert communication system proposed in this embodiment, the transmitter Alice attempts to transmit covert information to the receiver Bob using beamforming technology, while simultaneously being detected by the monitor Willie. To assist in covert transmission, the jammer Jammer applies null-space beamforming, setting its beamforming vector to the null-space beamforming of the jammer-receiver channel. This allows the jamming signal to confuse the monitor Willie's detection without affecting the receiver.

[0022] This embodiment provides a covert communication system involving satellite-to-ground and ground-to-ground links. For the satellite-to-ground links between the satellite transmitter Alice and various ground nodes, the satellite antenna pattern, path loss, and channel fading are modeled as follows: The satellite antenna pattern is modeled as follows: Satellite antenna pattern refers to the gain of a satellite antenna in different directions. This embodiment uses... This indicates the distance from the satellite launcher Alice to the ground node. i The antenna gain is given by the following formula: (25) In the formula, This represents the maximum antenna gain of the planar array antenna. The 3 dB beamwidth of the satellite antenna The level is near the sidelobe. , The distal lateral lobe, , ground nodes The deviation angle from the line of sight relative to the satellite antenna's line of sight is as follows: (26) In the formula, This refers to the antenna line-of-sight vector of the satellite transmitter Alice. The vector pointing from receiver Bob to the intersection of the satellite antenna's line of sight and the Earth's surface. To point from transmitter Alice to ground node The vector.

[0023] The path loss is modeled as follows: Free space loss refers to the signal loss transmitted from the satellite transmitter Alice to the ground node. The attenuation along the path is calculated using the following formula: (27) In the formula, Indicates wavelength. From satellite launcher Alice to node The distance.

[0024] Channel fading is modeled as follows: The satellite-to-ground link follows quasi-static Ricean fading, in which the fading coefficient remains constant within a time slot but varies between different time slots. This indicates the distance from the satellite transmitter Alice to the node. The channel fading vector is given by the following formula: (28) In the formula, Let be the Rice factor, representing the power ratio of the direct component to the scattered component of the channel. and These are the channel fading vectors for the direct component and the multipath scattering component, respectively. It follows a complex Gaussian random variable with zero mean and unit variance, i.e. Because the Alice satellite transmitter uses, Figure 2 The planar array antenna shown, From uniform planar array antenna to node The starting angle is determined as follows: (29) In the formula, For Kronecker product, The steering vector of a uniform planar array antenna. and These are the vertical and horizontal components of the starting angle, respectively. and The corresponding steering vectors of the uniform planar array antennas are respectively Components and The components are as follows: (30) (31) In the formula, , This is the distance between adjacent antenna elements. In this embodiment, let... ,use To represent the satellite launcher Alice and the ground node The channel coefficient vector of the link between them. : (32) In the formula, .

[0025] Assuming the communication link between ground nodes is affected by quasi-static Rayleigh fading, the signal from the jammer Jammer to the node will be affected. Channel fading is defined as , Each component of follows a circularly symmetric complex Gaussian distribution with zero mean and unit variance, i.e. Send the Jammer jammer to the ground node The channel coefficient is expressed as As shown in the following formula: (33) In the formula, Power gain per unit distance Jammer and nodes The distance between them Let be the path loss coefficient. Assume that transmitter Alice and jammer Jammer send orthogonal pilot sequences to Bob, so receiver Bob can obtain accurate channel coefficients. However, monitor Willie cannot decode the pilot signals, so only... and It can be obtained by Willie through long-term observation, and only statistical quantities can be obtained, not instantaneous values.

[0026] This embodiment uses the covert transmission rate to measure the performance of a covert communication system, defined as the maximum achievable rate from Alice to Bob while meeting the covertness requirements. Let represent the transmission rate between Alice and Bob. Then, the covert transmission rate is defined as: (34) (35).

[0027] Example 2 This embodiment provides a satellite-to-ground downlink covert communication method combining beamforming and cooperative jamming, employing a satellite-to-ground downlink covert communication system combining beamforming and cooperative jamming provided in Embodiment 1. The specific steps are as follows: (1) Initialize each link in the system; Transmitter Alice and jammer Jammer simultaneously send orthogonal pilot signals to receiver Bob, who then estimates the channel coefficients. Jammer obtains the location information of the monitor Willie and feeds it back to the receiver Bob.

[0028] (2) Establish a monitor detection model. The monitor Willie performs a binary hypothesis test based on the received signal to test whether the satellite transmitter Alice sends a covert signal to the ground receiver Bob. Then, the monitor Willie detects the test results through an energy detector to obtain the detection error probability, false alarm rate, and false negative rate. The specific operation is as follows: like Figure 5As shown, in this embodiment, the monitor Willie uses a radiometer to detect signals. To determine whether the transmitter Alice is sending a covert signal to the receiver Bob, Willie makes a hypothesis including a null hypothesis based on his observations. and alternative assumptions The binary hypothesis test, the signal received by the monitor Willie As shown in the following formula: (36) In the formula, and These represent the transmit power of the transmitter Alice and the jamming power of the jammer Jammer, respectively. and For the first The secondary channel is used to transmit covert signals from the transmitter Alice and jamming signals from the jammer Jammer. , Both signals simultaneously satisfy , , and The beamforming vectors for concealed and jamming signals are assumed to be... , , The variance of the additive white Gaussian noise at the location of the monitor Willie is . The average signal energy received at monitor Willie was positioned as Since this embodiment assumes an unlimited number of channel uses, according to the strong law of numbers, As shown in the following formula: (37) In the formula, , In order to detect covert communications, the monitor Willie uses an energy detector, which is given as follows: (1) In the formula, It is the detection threshold of the energy detector. and To support the hypothesis and Decision, that is: if Greater than Monitor Willie made The decision was made that the transmitter, Alice, was transmitting a covert signal. Conversely, the monitor, Willie, made... The decision was made that the transmitter Alice was not currently transmitting any covert signals.

[0029] However, this can lead to two types of errors: false alarm rate and false negative rate, expressed as: , The former indicates when Make a decision for the real-time monitor Willie The latter represents the probability when; When the result is true, monitor Willie makes a decision. The probability of detection error; therefore, the probability of detection error. It can be defined as (2) From the transmitter Alice's perspective, the detection threshold set by the monitor Willie Since it is difficult to obtain, this embodiment takes into account the worst-case scenario to ensure the concealment of communication. The monitor, Willie, can set an optimal threshold. To achieve the lowest possible detection error probability Therefore, as long as it is implemented for arbitrarily small positive values , to achieve Under the condition of satisfying This will satisfy the requirement for concealment.

[0030] (3) Monitor performance analysis, to solve for the minimum detection error probability, false alarm rate and false negative rate under perfect / imperfect monitor position conditions; With a fixed satellite antenna line of sight based on perfect / imperfect monitor location information, and at maximum antenna gain, the covert transmission rate is optimized through a joint design of beamforming and artificially assisted interference. The specific operation is as follows: Based on the definition of the detection error probability in equation (1), using , , Let $\mathbf$ represent the minimum false detection probability, false alarm rate, and false negative rate under the condition of perfect monitor location information. By definition, the false alarm rate can be expressed as: (38) The probability density function can be given as: (39) Therefore, the false alarm rate is specifically calculated as follows: (3) In the formula, The variance of the additive white Gaussian noise at the location of the monitor Willie is... The detection threshold; Similarly, the false negative rate can be given as: (40) because It concerns two independent random variables and The function first obtains The probability density function is: (41) in: (42) Based on the previous assumptions, , , and It was known to the monitor, Willie, so The uncertainty comes only from ,therefore It is crucial, because Obey Rice factor and The total power follows a Ricean distribution, therefore The probability density function is: (43) In the formula, Denotes the zeroth-order modified Bessel function of the first kind, defined as follows: , The cumulative distribution function can be given as: (44) in: (45) (46) Will Substituting into equation (45), we get: (47) In the formula, For the incomplete Toronto function, the false negative rate is further obtained as: (4) In summary, under the condition of perfect monitor positioning, the probability of monitor detection error is... We can obtain the following: (5) in: (6) (7) In the formula, , , , , , , It is a first-order Marcum function.

[0031] Optimal detection threshold under perfect monitor positioning conditions It can be obtained as: (8) The corresponding minimum detection error probability for: (9) Due to the minimum detection error probability It contains a first-order Marcum function, therefore it is not suitable for implicit constraints. Further analysis and calculations are required, therefore, the lower bound of the detection error probability is used in subsequent steps for further analysis.

[0032] In the case of a perfect monitor position, the lower bound of the detection error probability can be given as: (48) in: (49) According to equation (48), the minimum error detection probability can be seen. Follow The beamforming vector increases and then monotonically decreases; therefore, it is necessary to adjust the beamforming vector accordingly. Designed to meet and The orthogonality of the beam, which guarantees the minimum beam leakage from the transmitter Alice to the eavesdropper Willie, yields the lower bound for the minimum false detection probability as follows: (10) therefore, It is defined as a concealment constraint under the condition of a perfect observer position.

[0033] In real-world scenarios, obtaining the exact location of the monitor, Willie, is difficult. Therefore, only imperfect monitor location information is available. This embodiment assumes that the monitor's information (that the monitor is a child) contains estimation errors. To analyze the minimum false detection probability in this scenario, we first need to model the monitor's true location, which includes estimation errors. This can be modeled as follows: (11) In the formula, This represents the estimation error of the location information of the monitor Willie, and should meet the following requirements. and ,in and Given constraints; Therefore, Willie's four maximum location information estimation errors are given as follows: (50) In the case of imperfect Willie location information, this embodiment considers the worst-case scenario, i.e., Willie's location makes... The value is the largest ( Based on the above assumptions, Design and Related to but It is irrelevant; as discussed above, it can be concluded that... Even with imperfect Willie location information, the distribution still follows a Rice distribution. Therefore, with imperfect Willie location information, the minimum detection error probability can be determined by assuming Willie is in an optimal location. Thus, the optimization problem can be formulated as: (12) (13) By solving equations (12) and (13), the optimal position of Willie and the corresponding minimum detection error probability under imperfect position information can be determined. Through observation of the objective function to be optimized, three parameters are affected by Willie's position: , and At the same time, the objective function (12) can also be obtained as... The increase is monotonically decreasing, and Follower It increases and then monotonically decreases. Therefore, it can be maximized. To obtain the optimal solution of the objective function and further obtain the optimal Willie position, it can be expressed as: (51) In the formula, as well as All are non-convex functions, therefore (51) is a non-convex function and difficult to process. Therefore, this embodiment finds the optimal... Maximize each and This leads to the maximum value of formula (51).

[0034] First, solve The maximization problem The expression is , Is it following Monotonically increasing, and Is it following The condition is monotonically decreasing; therefore, to determine Willie's optimal position, we need to minimize... The optimization problem can be expressed as: (52) Formula (52) is still difficult to solve due to its non-convexity, and requires further transformation, let , Representing vectors and The unit vector, then, by relaxing Unit modulus constraint of vectors. Equation (52) can be rewritten as: (53) In the formula, Let be the normal vector of the plane formed by two adjacent vectors, and thus we can obtain: (54) Note that equation (53) is a convex optimization problem that can be solved using the standard convex problem solution method, therefore, by obtaining the minimum value... You can then obtain further Represented as .

[0035] solve The maximization problem can be represented as: (55) Formula (55) is also a non-convex problem. Due to the large distance between the satellite and the monitor Willie, the estimation error of the monitor Willie's position information is significant. The distance between the satellite transmitter Alice and the monitor Willie has little impact, that is: To solve the problem (55), we assume... Not following The equation (55) changes with the change of , therefore, formula (55) can be rewritten as: (56) By solving formula (56), we can obtain and The upper bound value of can be defined as follows: and Based on the above theories and results, the minimum DEP under the imperfect monitor Willie's location information is determined. It can be obtained.

[0036] (4) Optimize the line-of-sight direction of the satellite antenna. Finally, under perfect / imperfect observer positions, optimize the covert transmission rate through beamforming, artificial interference, and antenna line-of-sight joint design, as follows: This study investigates the beamforming-artificial-assisted jamming joint design problem to maximize the covert transmission rate under perfect Willie location information and a fixed satellite antenna line of sight (fixed maximum antenna gain). The problem can be expressed as: (14a) (14b) (14c) (14d) (14e) (14f) (14g) (14h) In the formula, and Let be the maximum transmit power of Alice and the maximum transmit power of Jammer, respectively. Constraint formula (14b) ensures that no signal is leaked to the monitor Willie on the direct path of the satellite-to-ground link. Constraint formula (14c) ensures that the jamming signal of Jammer will not affect the receiver Bob. Constraint formula (14h) represents the concealment constraint under the perfect monitor Willie location information.

[0037] Substituting formulas (14c) and (14b) into formulas (14a) and (14h) respectively, the optimization problem can be rewritten as: (57) (58) (14b) (14c) (14d) (14e) (14f) (14g) In theory , , ,make , The optimization problem described above can be rewritten as: (59) (60) (61) (62) (63) (64) (14f) (14g) (58) Note that constraint (64) is still difficult to solve because it is non-convex. Therefore, this study uses the semidefinite relaxation technique SDR to relax constraint (64). Thus, the optimization problem can be transformed into an SDR problem, which can be expressed as: (65) (14f) (14g) (58) (60) (61) (62) (63) The SDR problem described above can be solved using a standard convex optimization solver. The optimal solution to this problem is defined as follows: , , and At the same time, the matrix cannot be determined. and Is it a rank-one matrix? To obtain high-quality beamforming vectors, this study uses a Gaussian stochastic process to decompose... and ,definition For matrix Coordinates are elements, For matrix The A vector of rows.

[0038] Specifically, the following steps are required.

[0039] 1) Input the SDR solutions of the optimization problems (59)~(64), (14f), (14g), (58) ; 2) Set the number of iterations and iteration count index ,matrix The singular value decomposition is ; 3) If If it is a rank-one matrix, execute step 4); otherwise, execute step 5). 4) ; 5) Initialization Looping from 0 Second-rate; 6) Generate random vectors And calculate

[0040] 7) If If the above constraints are satisfied, the objective function value can be obtained. and ; 8) ; 9) Output the optimal beamforming vector Furthermore, an optimization problem (14) is solved under perfect Willie location information to obtain the corresponding maximum covert transmission rate. .

[0041] Specifically, the following steps are required.

[0042] 1) Input channel coefficient vector and transmit power upper limit; 2) Obtained by solving the SDP problem , , and ; 3) Obtained using Gaussian random processes and ; 4) Based on the formula (14a), we obtain ; 5) Output , , , and .

[0043] This study investigates the beamforming-artificial-assisted jamming joint design problem to maximize the covert transmission rate under the condition of a fixed satellite antenna line of sight (fixed maximum antenna gain) given the location information of an imperfect monitor, Willie. The problem can be expressed as: (15) (16) (14c) (14d) (14e) (14f) (14g) Equation (67) represents the concealment constraint under the location information of the imperfect monitor Willie, letting , .therefore, Follower It monotonically increases with the increase of . In order to satisfy the hidden constraint, we can obtain ,in This is the Lambert-W function. Furthermore, the range of values ​​for the transmit power that satisfies the concealment constraint can be obtained: .

[0044] Similar to solving the optimization problem (14), the above optimization problem needs to be simplified first, and then represented using SDR techniques to represent this SDP problem. After the above operations, the optimization problem can be rewritten as: (66) (67) (14f) (14g) (61) (62) (63) The optimal solution to this optimization problem and the corresponding optimal covert transmission rate. The solution method is the same as that used for the location information of the perfect monitor Willie.

[0045] It should be noted that properly setting the satellite's aiming line of sight is beneficial for improving the rate of covert transmission. Therefore, this embodiment first addresses the issue of optimally setting the direction of the satellite antenna's line of sight.

[0046] like Figure 3 As shown, points S, W, and B in the three-dimensional Cartesian coordinate system are first projected onto a unit sphere centered on Alice, and their corresponding mapping points on the sphere are... , and spherical triangle The arc length is , and Therefore, Bob and Willie's off-axis perspective needs to meet the following requirements. , According to formulas (1), (14), and (32), we can obtain the following for a given... and , yes A monotonically decreasing function. yes It is a monotonically increasing function. Therefore, in order to maximize the covert transmission rate CR, it is necessary to maximize as much as possible. Minimize at the same time .

[0047] Assume Alice's optimal aiming point on the sphere is represented as At this point, Bob's off-axis angle can be expressed as... For any , needs to be maximized To achieve the maximum covert transmission rate, according to the spherical cosine theorem, we can obtain: (17) From formula (17), we can obtain that Only when equal To achieve the minimum value, therefore, under the optimal Alice antenna line of sight, the point... , and They are concyclic. Furthermore, in three-dimensional Cartesian coordinates, the point... It lies on the ray from point W to point B.

[0048] After optimizing the satellite antenna line-of-sight direction, under perfect observer location information, the beamforming-artificial interference-antenna line-of-sight joint design optimization problem for maximizing concealed transmission rate is as follows: (18) (19) exist Under the condition that, the variables in the above formula The off-axis angle of ground receiver Bob can be used. Instead, the optimization problem is further simplified as follows: (20) (twenty one) (twenty two) (14c) (14d) (14e) (14f) (14g) (14h) The aforementioned optimization problem remains difficult to solve due to its complexity, so it is transformed into two sub-problems. The first sub-problem involves beamforming vectors. and The joint optimization, the second subproblem involves finding the optimal beamforming vector. and Down , and Joint optimization.

[0049] The first subproblem can be expressed as: (68) (14c) (14d) (14e) The aforementioned optimization problems can also be solved using SDR technology.

[0050] For the second subproblem, substituting (14c) into the objective function (20) and substituting (14b) into the implicit constraint (14h), the subproblem can be expressed as: (69) (70) (14f) (14g) (twenty one) (twenty two) Note Follow Monotonically increasing and the objective function (69) and Irrelevant. Therefore, when At that time, the objective function (69) reaches its maximum value. Based on this conclusion, The value is set to And take the logarithm of (69) and (70). Therefore, the above optimization problem can be rewritten as: (71) (72) (14f) (twenty one) (twenty two) In the formula, the equation This makes the objective functions (71) and (72) nonconvex; therefore, a first-order restricted approximation is used here. Convert to convex form. For a given feasible point... and , It can be approximated as: in: (73) After the above approximation, a convex problem can be obtained, as follows: (74) (75) (14f) (twenty one) Based on this convex problem, the optimal solution to the rewritten optimization problem is obtained iteratively using the SCA technique.

[0051] In summary, the detailed solution to the above optimization problem involves the following steps: 1) Input channel coefficient vector and transmit power upper limit; 2) By solving the SDP problem, i.e., the first subproblem, we obtain... and ; 3) Obtained using Gaussian random processes and ; 4) Given and Set the number of iterations and iteration count index Initialize feasibility points ; 5) Update by solving optimization problems , If the following conditions are met: or Execute step 6); otherwise, execute step 4). 6) , ; 7) According to (20), we get ; 8) Output , , , and ; Under imperfect Willie location information, the joint design problem of beamforming-artificial interference-antenna line-of-sight for maximizing concealed transmission rate is as follows: (76) (77) (14c) (14d) (14e) (14f) (14g) (twenty one) (twenty two) Similar to the optimization problem under perfect Willie location information, the above optimization problem needs to be transformed into two sub-problems. First, the SDR technique is used to solve... , The optimization subproblem. Utilizing the optimal beamforming vector. , Using the first-order restricted approximation method to and The optimization subproblem is transformed into the following convex optimization problem: (78) (79) (80) (14f) (twenty one) In the formula, For a detailed solution to the optimization problem, please refer to the solution to the optimization problem under perfect Willie location information.

[0052] Verification Example To verify the performance of the proposed method, the following simulation experiments were conducted: The performance of the proposed optimal joint beamforming and jamming strategy (Opt-JBJ) and the optimal joint beamforming, jamming, and antenna pointing strategy (Opt-JBJ-B) under perfect / imperfect Willie location information was compared and analyzed. The experiments also compared the covert communication performance of the proposed scheme with existing MRT-NS and ZF-NS schemes: the MRT-NS scheme uses maximum ratio transmission beamforming for covert transmission, combined with null-space beamforming for jamming; while the ZF-NS scheme utilizes zero-forced beamforming to prevent signal leakage to the monitoring Willie, and also employs a null-space jamming strategy. Unless otherwise specified, system parameters were set according to Table 1.

[0053] Table 1. Simulation Experiment System Parameter Table

[0054] like Figure 7 As shown, the detection threshold is given. False detection probability The influence of the experimental parameters was set to... and ,Depend on Figure 7 As can be seen, the theoretical calculation results and simulation results are in high agreement, effectively verifying the proposed method. Further observation revealed that the correctness of the theoretical model increased with the detection threshold. The increase, It shows a trend of first decreasing and then increasing. This phenomenon can be explained as: when As the value increases, the false alarm probability at the Willie end decreases while the false negative probability increases. When the value is low, the reduction in false alarm probability plays a dominant role, making It decreases as τ increases; when Once the probability of false negatives increases to a certain extent, it becomes the dominant factor, leading to... It has turned into an upward trend. Figure 7 Simulation results prove the existence of... Minimize the optimal detection threshold This is completely consistent with the previous theoretical analysis conclusions.

[0055] like Figure 8 a and Figure 8 As shown in b, the covert signal transmission power is given. With interference signal power The impact of different Willie location information (WLI) conditions on the minimum false detection probability. Figure 8 a and Figure 8 b shows the respective W, and Under the parameter settings, the optimal covert detection error probability Perfect WLI Nether and the imperfect WLI lower bound Follow and The changing pattern. From... Figure 8 It can be seen that, with the increase in the power of the concealed signal... The increase, Both its theoretical lower bound and its lower bound exhibit a monotonically decreasing trend. This is because... Increasing the value of enhances the signal power at the Willie receiver, reducing its detection uncertainty and thus decreasing the probability of detection errors. However, Figure 8 The result of b shows that when the interference signal power When increasing, Its lower bound, however, increases significantly. This is because... The increase in the channel uncertainty will increase the difficulty of Willie's detection, which in turn increases the probability of false detection of covert communication.

[0056] Further observation Figure 8 a and Figure 8 b can be observed that when Decrease and When the value increases, the optimal probability of detecting concealed errors is... Its theoretical lower bound The gradually narrowing gap indicates that under conditions of low concealment power and high interference power, the theoretical model can more accurately approximate the actual performance. These results demonstrate that concealment signal power and interference power have a bidirectional regulatory effect on system concealment, requiring reasonable configuration of power parameters to optimize concealment performance. Furthermore, the applicability of the theoretical lower bound across different power ranges is verified.

[0057] Figure 9 Figure a shows Alice's maximum transmit power under perfect WLI conditions. For maximum covert transmission rate The influence of the experimental parameters was set to... and .from Figure 9 As can be seen from a, when When the size increases, the three methods, Opt-JBJ, Opt-JBJ-B, and MRT-NS, are used to obtain... All of them first increase to a certain constant value and then remain unchanged. This phenomenon can be explained as follows: when When the value is small, the concealment constraint is satisfied. The maximum feasible transmit power is ,at this time Follow Increases linearly; while when After exceeding the critical value, the concealment constraint is satisfied. Will make at this time It reached saturation. Further observation revealed that the Opt-JBJ and Opt-JBJ-B methods obtained... The performance of Opt-JBJ-B is significantly better than that of the MRT-NS method, and Opt-JBJ-B outperforms Opt-JBJ, indicating that joint optimization of beamforming, jamming strategies and satellite antenna pointing can effectively improve covert communication performance.

[0058] Figure 9 b gives the maximum interference power under perfect WLI conditions. For maximum covert transmission rate The impact is shown in the figure. hour Follow The changing pattern. From... Figure 5 As can be seen, when When the size increases, the Opt-JBJ and Opt-JBJ-B schemes obtain It first grows and then tends to stabilize, while the MRT-NS scheme... It continues to rise. The reasons are as follows: When When the power is low, Alice needs to use a lower transmission power. To meet the concealment constraint conditions ,lead to Lower. Following In addition, Alice can improve her performance while satisfying the concealment constraint. Thus promoting Growth. Once the threshold is exceeded, Alice uses the maximum transmission power. And still satisfied ,at this time It reaches saturation. Furthermore, when... When the value is large, the Opt-JBJ and Opt-JBJ-B schemes The two schemes tend to be consistent because the concealment constraint is easily satisfied under high interference power. In both schemes, Bob uses the maximum transmit power and the minimum off-axis angle (i.e., the maximum antenna gain) for transmission, resulting in the same concealment rate.

[0059] Figure 10 a gives the maximum transmit power of Alice under imperfect WLI conditions. For maximum covert transmission rate The effect is shown in the parameters. , hour, Follow The changing pattern. From... Figure 10 As can be seen in a, those who follow The increase in size, the three schemes Opt-JBJ, Opt-JBJ-B and MRT-NS obtained All of them first increase to a maximum value and then tend to stabilize, similar to the principle in the case of perfect WLI. It is worth noting that the Opt-JBJ and MRT-NS schemes obtained... The difficulty in accurately obtaining and satisfying the constraints of the optimization problem leads to problems in the beamforming vector optimization of the Opt-JBJ scheme. The beamforming strategy is similar to that of MRT. Therefore, given a given transmit power, the two schemes have the same covert transmission rate performance.

[0060] Figure 10 b shows the maximum interference power under imperfect WLI conditions. For maximum covert transmission rate The impact is specifically shown in the parameters. , , hour, Follow The changing pattern. From... Figure 10 As can be seen in b, all three schemes All follow The increase first reaches a maximum value and then tends to saturate. The mechanism of this phenomenon is consistent with the law under perfect WLI conditions: when When the signal strength is low, Alice needs to increase the interference power to compensate for the loss of stealth caused by channel information errors, thus allowing for higher transmit power. To increase the concealment rate; and when After exceeding the critical value, the concealment constraint The optimal power allocation has been fully satisfied. It reaches the theoretical upper limit determined by the system parameters.

[0061] Figure 11 Willie's location was shown. The impact of perfect WLI and imperfect WLI conditions on the maximum covert transmission rate is demonstrated, showing the effect of parameters. , hour, and Follow The changing pattern can be seen from the graph, as... As the value increases, the concealment rate under both conditions first decreases to a minimum and then rises back to a constant value. The reason is as follows: Willie's change in position has a dual effect on the concealment rate: on the one hand, The increased distance between Jammer, the interference source, and Willie reduced the channel uncertainty at Willie's location, forcing Alice to use a lower transmit power. This suppresses the concealment rate; on the other hand, Increasing this value will weaken the satellite antenna gain of the Alice-Willie link, reduce the power of the covert signal received by Willie, and lower the probability of detection error. Ascend, allowing Alice to rise To improve the concealment speed When the size is small, the former plays a dominant role, and the concealment rate increases with... Increase and decrease; when Once the critical value is exceeded, the latter takes over and drives the covert transmission rate back to saturation.

[0062] Further analysis shows that under imperfect WLI conditions, optimizing the satellite antenna line-of-sight pointing (Opt-JBJ-B) significantly improves the concealment rate compared to the default pointing scheme. Furthermore, when the position error range expands to... At that time, the three options All results are lower than those obtained when the error range is 300m. This is because a larger positional error allows Willie to reduce the error by selecting a better location. This forced Alice to further reduce her transmission power to meet stealth constraints, ultimately leading to a decrease in stealth rate.

Claims

1. A satellite downlink covert communication system combining beamforming and cooperative jamming, characterized in that, The system comprises a satellite transmitter Alice, a ground receiver Bob, a ground monitor Willie and a ground cooperative jammer Jammer; The satellite transmitter Alice is equipped with a planar array antenna, the ground cooperative jammer Jammer is equipped with a plurality of isotropic antennas, and the ground receiver Bob and the ground monitor Willie are each equipped with a single antenna; The satellite transmitter Alice transmits covert information to the ground receiver Bob through beamforming technology, the ground monitor Willie detects the covert information within its monitoring range, and the ground cooperative jammer Jammer sets its beamforming vector to the zero space beamforming of the jammer-receiver channel through zero space beamforming, and then sends an interference signal to the ground receiver Bob to confuse the detection of the ground monitor Willie. 2.A method for satellite downlink covert communication with joint beamforming and cooperative jamming, using the satellite downlink covert communication system with joint beamforming and cooperative jamming according to claim 1, characterized in that, The method specifically comprises: System link initialization; Establishing a monitor detection model, the ground monitor Willie performs binary hypothesis testing on the received signal, and tests whether the satellite transmitter Alice sends a covert signal to the ground receiver Bob; then the ground monitor Willie detects the test result through an energy detector to obtain a detection error probability, a false alarm rate and a missed detection rate; Monitor performance analysis, solving the minimum detection error probability, false alarm rate and missed detection rate under the condition of perfect / imperfect monitor position; Fixing the satellite antenna boresight under the perfect / imperfect monitor position information, optimizing the covert transmission rate through beamforming-artificial auxiliary interference joint design under the condition of maximum antenna gain; Optimizing the satellite antenna boresight direction, and finally optimizing the covert transmission rate through beamforming-artificial auxiliary interference-antenna boresight joint design under the perfect / imperfect monitor position.

3. The method according to claim 2, wherein, The binary hypothesis testing process of the monitor detection model is specifically as follows: （1） where is the detection threshold, and are the decisions in favor of the null hypothesis and the alternative hypothesis respectively, is the average value of the signal energy received at the monitor Willie, if is greater than , the monitor Willie makes a decision that transmitter Alice is transmitting a covert signal, otherwise, the monitor Willie makes a decision that transmitter Alice is not currently transmitting a covert signal; The probability of detection errors includes the false alarm rate and the false negative rate, which are respectively , , For when Make a decision for the real-time monitor Willie The probability, For when When the result is true, monitor Willie makes a decision. The probability of detection error is thus determined by the probability of [missing information]. The specific formula is as follows: （2）。 4. The method according to claim 2, wherein, The monitor performance analysis is specifically as follows: In the case of a perfect monitor position , , respectively denote the minimum probability of error detection, the false alarm rate and the miss detection rate; The false alarm rate is specifically as follows: （3） wherein is the additive white Gaussian noise at the monitor Willie with variance is the detection threshold The missed detection rate is specifically as follows: （4）； The detection error probability is specifically as follows: （5） In the case of imperfect monitor position, the monitor detection performance analysis specifically comprises: assuming that the position information of the monitor Willie has an estimation error, first modeling the real position of the monitor Willie containing the estimation error, as follows: （6） （7） wherein , , , , , , is a first order Marcum function; Best detection threshold in perfect monitor position case As follows: （8） Minimum detection error probability corresponding to optimal detection threshold Specifically as follows: （9） Since the minimum detection error probability Since the minimum detection error probability The first-order Marcum function is included in the lower bound of the minimum detection error probability, so the lower bound of the minimum detection error probability is used for further analysis, and the lower bound of the minimum detection error probability is as follows: （10）。 5. The method of claim 2, wherein, Under the condition of imperfect monitor Willie position information, the minimum detection error probability is determined by setting the monitor Willie at the best position, and the optimization problem is as follows: （11） wherein represents the position information estimation error of the monitor Willie and should satisfy and wherein and are known constraints; Under the perfect monitor position information, fix the satellite antenna boresight, optimize the covert transmission rate through beamforming-artificial auxiliary interference joint design under the condition of maximum antenna gain, and the optimization problem is specifically as follows: （12） （13） Solving the above equation obtains the best position of the monitor Willie and the corresponding minimum detection error probability DEP under the condition of imperfect position information .

6. The method of claim 2, wherein, Solving the optimization problem under the perfect monitor position information to obtain the maximum covert transmission rate under the perfect monitor position information. （14a） （14b） （14c） （14d） （14e） （14f） （14g） （14h） wherein and respectively, the maximum transmit power of Alice and the maximum transmit power of the jammer Jammer, constraint equation (14b) ensures that no signal leaks from the direct path of the space-to-ground link to the monitor Willie, constraint equation (14c) ensures that the jamming signal of the jammer Jammer does not affect the receiver Bob, and constraint equation (14h) represents the concealment constraint under the perfect monitor Willie location information; Under the imperfect monitor position information, fix the satellite antenna boresight, optimize the covert transmission rate through beamforming-artificial auxiliary interference joint design under the condition of maximum antenna gain, and the optimization problem is specifically as follows:

7. The method according to claim 6, wherein, ​ （15） （16） （14c） （14d） （14e） （14f） （14g） In the above formula, formula (16) is a concealment constraint under imperfect monitor position information, and let , Therefore, With the increase of , in order to meet the concealment constraint, the following can be obtained , wherein is the Lambert-W function, and further, the range of the value of the transmission power that meets the concealment constraint can be obtained as follows: ; The optimal covert transmission rate under imperfect monitor location information is obtained by solving the above optimization problem. 8.The method of claim 2, wherein, The optimization of the satellite antenna visual axis direction specifically includes: Firstly, the points S, W and B in three-dimensional Cartesian coordinate system are projected onto the unit sphere centered at the satellite transmitter Alice, and their mapping points on the sphere are , and , the arc lengths of the spherical triangle are , and , therefore, the off-axis viewing angles of the ground receiver Bob and the monitor Willie should satisfy , In order to maximize the covert transmission rate CR, it is necessary to maximize while minimizing ; Let the optimal pointing point of satellite transmitter Alice on the sphere be denoted as and the off-axis angle of ground receiver Bob be denoted as For any , the maximum transmission rate is achieved by maximizing According to the spherical cosine theorem, the following equation is obtained: （17） wherein only if is equal to is the minimum, thus, under the best satellite transmitter Alice's antenna boresight, the point , and are concyclic, while, in three-dimensional Cartesian coordinates, the point lies on the ray from the point W to the point B.

9. The method according to claim 1 or 8, wherein, After optimizing the satellite antenna visual axis direction, the beamforming-artificial auxiliary interference-antenna visual axis joint design optimization problem for maximizing the covert transmission rate under perfect monitor location information is as follows: （18） （19） Under the condition that the variables in the above equation can be replaced by the off-axis angle of the ground receiver Bob, the optimization problem is further simplified as follows: （20） （21） （22） （14c） （14d） （14e） （14f） （14g） （14h） The maximum covert transmission rate under perfect monitor location information is obtained by solving the above optimization problem formula.

10. The method of claim 1 or 8, wherein, After optimizing the satellite antenna visual axis direction, the beamforming-artificial auxiliary interference-antenna visual axis joint design problem for maximizing the covert transmission rate under imperfect monitor location information is represented as follows: （23） （24） （14c） （14d） （14e） （14f） （14g） （21） （22） The maximum covert transmission rate under imperfect monitor location information is obtained in the same way as the solution to the optimization problem under perfect monitor location information.

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