A method for maximizing confidentiality rate based on smart reflective surface-assisted wireless energy transmission

Abstract

The present invention relates to a method for maximizing the confidentiality rate based on wireless energy transmission assisted by intelligent reflective surfaces, and belongs to the field of communication technology. The method comprises the following steps: an intelligent reflective surface IRS-assisted wireless power supply communication network WPCN security transmission model is proposed, and the transmission is divided into two stages: energy transmission and information transmission; according to the receiving power range of the user in the energy transmission stage, it is determined whether there is an eavesdropper; according to the constraint conditions of the information transmission stage, a system confidentiality rate optimization problem is established, and the user and the eavesdropper are optimized step by step in a step-by-step optimization manner; the user's accessible rate is maximized; according to the eavesdropper's channel state information, a corresponding optimization algorithm is designed to minimize the eavesdropper's accessible rate. The present invention can use intelligent reflective surfaces in wireless power supply communication networks to achieve beam cancellation at the eavesdropping end, and can effectively improve the confidentiality rate of the wireless energy transmission system by jointly optimizing time allocation, power allocation, and the passive reflection phase of the reflective surface.

Description

Technical Field

[0001] The invention belongs to the technical field of communications and relates to a method for maximizing a confidentiality rate based on intelligent reflective surface-assisted wireless energy transmission. Background Art

[0002] The Sixth Generation (6G) wireless communication networks are gaining more and more attention from academia and industry, who hope to achieve ultra-high spectrum / energy efficiency and high-density user connections. Due to these demands, wireless devices are consuming more and more energy, and energy-constrained devices in wireless communication networks have become a bottleneck restricting the quality of network services. Wireless Power Transfer (WPT) technology, which uses RF energy to charge devices, can effectively solve the problem of insufficient energy supply for devices. WPCN is a new framework for WPT network design, where hybrid access points transmit energy to energy-constrained devices through downlinks, while wireless devices use the collected energy to transmit information in uplinks. The emergence of WPCN provides a new way to achieve energy-efficient and durable communications.

[0003] As the communication environment becomes increasingly complex, the security and privacy of user information need to be protected. The use of traditional encryption technology will cause a huge key management burden for the network and consume a lot of network resources. Physical layer security mainly uses the difference between legitimate user channels and eavesdropper channels to achieve communication security from the physical characteristics. The advantages are lower algorithm complexity, stronger confidentiality, no threat from the increased computing power of eavesdroppers, and no additional overhead due to protecting security keys. Considering the importance of energy in WPCN, applying physical layer security in WPCN can effectively solve the problem of secure communication in the network.

[0004] Recently, IRS has attracted wide attention for its potential in achieving secure communications. As a key enabler of intelligent reconfigurable radio environments, it has been shown to effectively improve the spectral efficiency and energy efficiency of wireless systems. The system consists of a large number of reconfigurable reflective elements, which are managed by an intelligent controller. These elements are small, low-cost, and low-energy, and can enhance signal reception by intelligently adjusting the required signal phase without the need for specialized RF processing. By properly adjusting the phase shift of the reflective unit of the intelligent reflective surface, its reflected signal can be combined with the signal of other paths in a consistent manner, reducing the reachability of the link at the eavesdropping end. To further improve security, some methods add artificial noise (AN) to deteriorate the eavesdropper channel, and some methods study cooperative jammer (CJ) technology to interfere with the eavesdropper. Although the auxiliary method of using AN and CJ can further improve the confidentiality rate, it consumes the transmission power other than that used for legitimate users. In the case of limited transmission power, the performance of secure communications has always been a bottleneck. In addition, the existing methods all assume the existence of eavesdroppers, without considering whether the eavesdroppers really exist, which will consume more resources.

[0005] In order to improve the security of WPCN, the present invention uses IRS to implement a dynamic beam cancellation, and detects whether eavesdropping exists while transmitting energy in the first stage. If so, the IRS is used to shield the signal at the eavesdropper in the second stage. In the present invention, the intelligent reflective surface can not only enhance the information transmission rate and energy transmission efficiency, but also shield the signal at the eavesdropper to enhance security. Summary of the invention

[0006] In view of this, an object of the present invention is to provide a method for maximizing the confidentiality rate based on intelligent reflective surface-assisted wireless energy transmission.

[0007] In order to achieve the above object, the present invention provides the following technical solutions:

[0008] A method for maximizing confidentiality rate based on intelligent reflective surface-assisted wireless energy transmission, the method comprising the following steps:

[0009] Step 1: The IRS-assisted WPCN secure transmission model is proposed, and the transmission is divided into two stages: energy transmission and information transmission;

[0010] Step 2: Determine whether there is an eavesdropper based on the user's receiving power range during the energy transmission phase;

[0011] Step 3: Establish the system confidentiality rate optimization problem according to the constraints in the information transmission stage, and optimize the users and eavesdroppers step by step by adopting a step-by-step optimization method;

[0012] Step 4: Maximize the user's achievable rate by optimizing the algorithm design;

[0013] Step 5: According to the eavesdropper's channel state information, design a corresponding optimization algorithm to minimize the eavesdropper's achievable rate.

[0014] Optionally, in the step one, in the IRS-assisted WPCN secure transmission model, the two stages of energy transmission and information transmission are: in the first stage, i.e., time period T1, energy transmission is performed, and energy signals are transmitted with a fixed power P; in the second stage, i.e., time T2, confidential information is transmitted, and in this stage, users use the energy collected from downlink to transmit information.

[0015] Optionally, in step 2, the process of determining whether there is an eavesdropper according to the receiving power range of the user in the energy transmission stage is as follows: energy transmission is performed in time period T1, assuming that the signal powers transmitted to the legitimate user and the eavesdropper are P1 and P2 respectively, and P1+P2=P; then the reachable rates of the legitimate user and the eavesdropper are:

[0016]

[0017]

[0018] If there is eavesdropping, then R UE -R Eve ≤0, from the above formula, we can get that the power range received by the legal receiver is P1≤σ.

[0019] Optionally, in step 3, the process of establishing the system confidentiality rate optimization problem according to the constraint conditions of the information transmission stage is: maximizing the confidentiality rate of the system by optimizing the reflection phase shift, power allocation and time allocation of the intelligent reflection surface; the confidentiality rate achieved by the system is:

[0020] R s =R' UE -R' Eve

[0021] Based on the derived objective function and constraints, the optimization problem is finally established; to maximize the confidentiality rate, it is obviously necessary to maximize the achievable rate of legitimate users; the constraints include user energy collection constraints, total transmission time constraints, and passive reflection phase shift constraints of the smart reflective surface.

[0022] Optionally, in step 4, for the achievable rate optimization problem, it is obviously necessary to make the legitimate user rate of the objective function as large as possible and the eavesdropping user rate as small as possible; a step-by-step optimization method is adopted to first optimize the achievable rate of the legitimate user, and by jointly optimizing the reflection phase shift Φ1 and Φ2, the power allocation P′=[P3, P4] and the time allocation t=[T1, T2] optimization problem is expressed as:

[0023]

[0024] stC1:E1≥(P3+P4)T2

[0025] C2:T1+T2≤T,T1>0,T2>0

[0026] C3:|φ 1,n |=1,n=1,...,N

[0027] C4:|φ 2,n |=1,n=1,...,N

[0028] By designing an optimization algorithm, the non-convex problem is transformed into a semidefinite programming problem, and CVX is used to optimize the solution.

[0029] Optionally, in the step 5, a corresponding optimization algorithm is designed according to the eavesdropper's channel state information to minimize the eavesdropper's achievable rate; since the eavesdropper usually tries to hide its existence from the base station, it is difficult to obtain Eve's perfect channel state information, and therefore, according to the perfect and imperfect channel state information of the eavesdropper's channel state information CSI, corresponding optimization algorithms are designed respectively to minimize the eavesdropper's achievable rate, that is, to achieve the second step of the step-by-step optimization to maximize the confidentiality rate, so as to minimize the information rate at the eavesdropper as much as possible;

[0030] The optimization problem of beam cancellation optimization at the eavesdropper under perfect CSI conditions is established as:

[0031]

[0032] stC5:|φ 3,n |=1,n=1,...,N

[0033] Solve the above problem using CVX and get the optimal solution, from which the eavesdropper's information accessibility rate can be easily obtained;

[0034] The optimization problem of beam cancellation optimization at the eavesdropper under non-perfect CSI conditions is established as:

[0035]

[0036] stC5:|φ 3,n |=1,n=1,...,N

[0037] C13:Δh2∈K B,E

[0038] Where Δh2 represents the estimation error. The problem is first simplified and transformed into a standard semidefinite program using the S-Lemma lemma, and then optimized and solved using CVX. The eavesdropper's signal-to-noise ratio obtained by optimization can be used to easily obtain the eavesdropper's achievable rate.

[0039] The beneficial effects of the present invention are:

[0040] 1) Hybrid access points can transfer energy to energy-constrained devices through downlink, while wireless devices use the collected energy to transmit information in uplink. The considered WPCN scenario provides a new way to achieve energy-efficient and durable communications;

[0041] 2) Smart reflective surface, as a key driver of intelligent reconfigurable radio environment, the reflective element has the characteristics of small size, low cost and low energy consumption, and does not require special RF processing. By properly adjusting the phase shift of the reflective unit of the smart reflective surface, its reflected signal can be consistently combined with the signals of other paths, which is beneficial to enhance the information transmission rate and energy transmission efficiency.

[0042] 3) The IRS considered implements dynamic beam cancellation, not only detecting whether eavesdropping exists while transmitting energy in the first stage, but also using IRS to shield the signal at the eavesdropper in the second stage, taking into account the difficulty of obtaining the CSI of the eavesdropping user in the actual scenario, and jointly optimizing the established problem. In the present invention, the intelligent reflective surface can not only enhance the information transmission rate and energy transmission efficiency, but also shield the signal at the eavesdropper to enhance security.

[0043] Other advantages, objectives and features of the present invention will be described in the following description to some extent, and to some extent, will be obvious to those skilled in the art based on the following examination and study, or can be taught from the practice of the present invention. The objectives and other advantages of the present invention can be realized and obtained through the following description. BRIEF DESCRIPTION OF THE DRAWINGS

[0044] In order to make the purpose, technical solutions and advantages of the present invention more clear, the present invention will be described in detail below in conjunction with the accompanying drawings, wherein:

[0045] Figure 1 A process for maximizing the confidentiality rate of wireless energy transfer based on smart reflective surface-assisted wireless energy transfer;

[0046] Figure 2 This is a system model diagram of the confidentiality rate maximization method based on smart reflective surface-assisted wireless energy transmission;

[0047] Figure 3 This is the two-stage transmission diagram of the system. DETAILED DESCRIPTION

[0048] The following describes the embodiments of the present invention by specific examples, and those skilled in the art can easily understand other advantages and effects of the present invention from the contents disclosed in this specification. The present invention can also be implemented or applied through other different specific embodiments, and the details in this specification can also be modified or changed in various ways based on different viewpoints and applications without departing from the spirit of the present invention. It should be noted that the illustrations provided in the following embodiments only illustrate the basic concept of the present invention in a schematic manner, and the following embodiments and features in the embodiments can be combined with each other without conflict.

[0049] Among them, the drawings are only used for illustrative explanations, and they only represent schematic diagrams rather than actual pictures, and should not be understood as limitations on the present invention. In order to better illustrate the embodiments of the present invention, some parts of the drawings may be omitted, enlarged or reduced, and do not represent the size of actual products. For those skilled in the art, it is understandable that some well-known structures and their descriptions in the drawings may be omitted.

[0050] The same or similar numbers in the drawings of the embodiments of the present invention correspond to the same or similar parts; in the description of the present invention, it should be understood that if the terms "upper", "lower", "left", "right", "front", "rear", etc. indicate the orientation or position relationship, they are based on the orientation or position relationship shown in the drawings, which is only for the convenience of describing the present invention and simplifying the description, rather than indicating or implying that the device or element referred to must have a specific orientation, be constructed and operate in a specific orientation. Therefore, the terms describing the position relationship in the drawings are only used for illustrative purposes and cannot be understood as limiting the present invention. For ordinary technicians in this field, the specific meanings of the above terms can be understood according to specific circumstances.

[0051] like Figure 1 As shown, the present invention provides a method for maximizing the confidentiality rate based on intelligent reflective surface-assisted wireless energy transmission.

[0052] like Figure 2 As shown, it is a system model diagram of the present invention, which is described below in conjunction with the accompanying drawings:

[0053] The secure transmission model of the smart reflective surface assisted wireless power supply communication network considered in the present invention is as follows: Figure 1 As shown, it includes a base station (BS), a legitimate user (UE), and an eavesdropper (Eve), where the UE is equipped with an energy harvesting receiver, the BS has M antennas, the UE is set to a single antenna, and it is assumed that Eve also has a single antenna. Communication is carried out through a smart reflective surface with N reflective units.

[0054] The baseband equivalent channels from the base station to IRS, base station to user, and base station to Eve are recorded as: i=1,2, all the items inside the channel vector are modeled as zero-mean independent and identically distributed complex Gaussian random variables, whose variance depends on the path loss of the respective wireless links. The phase shift matrix of IRS is expressed as: Φ i =diag(φ i,1 ,...,φ i,n ,...,φ i,N ), θ n ∈[0,2π]. Eve may be a legitimate user in the past, but cannot access confidential information during the current communication process, or the base station does not want to send confidential information to these users. In order to ensure the security of communication, the system must regard these receivers as potential eavesdroppers. Therefore, Eve's perfect CSI can be obtained. Since eavesdroppers usually try to hide their existence from the base station, it is difficult to obtain Eve's perfect CSI. In practice, the knowledge of the CSI between IRS and Eve is not accurate, which may also be caused by channel estimation and quantization errors. This paper considers the worst-case channel uncertainty model, and the bounded CSI error model of the channel vector h2 is:

[0055]

[0056] in is the estimated value of the channel vector h2, K B,E represents the uncertainty region of h2, Δh2 represents the estimation error, Denotes the uncertainty region K B,E The radius of .

[0057] like Figure 3 As shown, it is a two-stage transmission diagram (energy transmission and information transmission) of the system of the present invention, which is explained below in conjunction with the accompanying drawings:

[0058] In the first phase (time period T1), energy transmission is performed to transmit energy to the UE while eavesdropping detection can be performed. The IRS can determine whether to adopt the anti-eavesdropping mode in the second phase (time period T2) according to the energy collection status of the UE. Assuming that there is eavesdropping, the UE uses the smart reflective surface to shield the signal at the eavesdropper while transmitting confidential information in the second phase.

[0059] In the first stage, downlink transmission is performed, and the energy signal is transmitted with a fixed power P. The energy signal is represented as x0(t), where E||x0(t)| 2 ]=1, assuming that the signal power transmitted to the legitimate user and the eavesdropper are: P1 and P2 (P1+P2=P). Then the received signals are:

[0060]

[0061]

[0062] Where Φ1=diag(φ 1,1 ,...,φ 1,n ,...,φ 1,N ),|φ 1,N |=1,n=1,...,N, n1(t) and n2(t) represent the power respectively. and Gaussian noise.

[0063] The achievable rates of legitimate users and eavesdroppers are:

[0064]

[0065]

[0066] If there is eavesdropping, then R Bob -R Eve ≤0, the power range received by the legitimate receiver can be obtained from the above formula as P1≤σ. Assuming that the received power range is found to be eavesdropping during the downlink transmission phase, a response must be made during the uplink information transmission phase. While transmitting confidential information, a signal shielding area is formed at the eavesdropper using the intelligent reflective surface.

[0067] In the second stage (time T2), confidential information is transmitted and x1(t) is represented as the transmission signal from the BS, where E[|x1(t)| 2 ]=1. In this stage, the energy collected by downlink is used for transmission. The energy collected by the legitimate user is:

[0068]

[0069] Where ξ(0<ξ<1) represents the energy collection efficiency. The received signals of the legitimate user and the eavesdropper are:

[0070]

[0071]

[0072] Where Φ2=diag(φ 2,1 ,...,φ 2,n ,...,φ 2,N ),|φ 2,N |=1,n=1,...,N;Φ3=diag(φ 3,1 ,...,φ 3,n ,...,φ 3,N),|φ 3,N |=1,n=1,...,N.

[0073] The user uses the energy collected by the downlink to transmit confidential information, and we can get:

[0074] E1≥(P3+P4)T2

[0075] The achievable rates of legitimate users and eavesdroppers are:

[0076]

[0077]

[0078] Theoretically, it can be set That is, in the second stage, the information received by the eavesdropper is offset by using the intelligent reflective surface.

[0079] The confidentiality rate of the system is maximized by optimizing the reflection phase shifts Φ1, Φ2 and Φ3 of the smart reflective surface, the power allocation P′=[P3, P4] and the time allocation t=[T1, T2].

[0080] The achievable confidentiality rate of the system is:

[0081] R s =R' UE -R' Eve

[0082] According to the objective function and constraints derived above, the following optimization problem is finally established:

[0083]

[0084] stC1:E1≥(P3+P4)T2

[0085] C2:T1+T2≤T,T1>0,T2>0

[0086] C3:|φ 1,n |=1,n=1,...,N

[0087] C4:|φ 2,n |=1,n=1,...,N

[0088] C5:|φ 3,n |=1,n=1,...,N

[0089] Among them, C1 is the user energy collection constraint, C2 is the total transmission time constraint, and C3, C4, and C5 are reflection phase shift constraints.

[0090] Finally, it should be noted that the above embodiments are only used to illustrate the technical solution of the present invention rather than to limit it. Although the present invention has been described in detail with reference to the preferred embodiments, those skilled in the art should understand that the technical solution of the present invention can be modified or replaced by equivalents without departing from the purpose and scope of the technical solution, which should be included in the scope of the claims of the present invention.

Claims

1. A method for maximizing confidentiality rate based on intelligent reflective surface-assisted wireless energy transmission, characterized in that: The method comprises the following steps: Step 1: Establish a secure transmission model of the wireless power supply communication network WPCN assisted by the intelligent reflective surface IRS. The model includes a base station, a legitimate user, an eavesdropper, and an intelligent reflective surface. The transmission model includes two stages: energy transmission and information transmission. In the first stage, energy transmission and eavesdropping detection are performed, and energy signals are transmitted with a fixed power P. In the second stage, i.e., time T2, confidential information is transmitted. In this stage, users use the energy collected from the downlink to transmit information, perform wireless information transmission, and adopt an anti-eavesdropping mode. Step 2: In the first stage, the wireless transmission surface determines whether there is an eavesdropper based on the receiving power range of the user in the energy transmission stage, and enters the second stage when there is an eavesdropper. In the second stage, the user transmits confidential information and uses the intelligent reflective surface to shield the signal at the eavesdropper; Step 3: Establish the system confidentiality optimization problem according to the constraints of the information transmission stage, and optimize the user and the eavesdropper step by step by adopting a step-by-step optimization method; maximize the confidentiality of the system by optimizing the reflection phase shift, power allocation and time allocation of the smart reflective surface; Step 4: Convert the non-convex problem into a semi-definite programming problem by designing an optimization algorithm, and use CVX to optimize and solve it to maximize the achievable rate of legal users. Step 5: According to the eavesdropper's channel state information, design a corresponding optimization algorithm to minimize the eavesdropper's achievable rate.

2. The method for maximizing confidentiality rate based on intelligent reflective surface-assisted wireless energy transmission according to claim 1, characterized in that: In step 2, according to the receiving power range of the user in the energy transmission stage, the process of determining whether there is an eavesdropper is as follows: energy transmission is performed in time period T1, assuming that the signal powers transmitted to the legitimate user and the eavesdropper are P1 and P2 respectively, P1+P2=P; then the reachable rates of the legitimate user and the eavesdropper are: h1 represents the UE-IRS and IRS-UE segment channels; h2 represents the IRS-Eve segment channel; Φ1 represents the IRS phase shift matrix in the first phase of wireless energy transmission eavesdropping; g represents BS-IRS and IRS-BS segment channels; Indicates power Gaussian noise; Indicates power Gaussian noise; δ b represents the noise power of legitimate users; δ e represents the noise power of the eavesdropper; H1 represents the UE-BS and BS-UE segment channels; H3 represents the BS-Eve segment channel; H represents the baseband equivalent channel; If there is eavesdropping, then R UE -R Eve ≤0, and the power range received by the legal receiver is obtained from the above formula as P1≤σ.

3. The method for maximizing confidentiality rate based on intelligent reflective surface-assisted wireless energy transmission according to claim 1, characterized in that: In the step 3, the process of establishing the system confidentiality rate optimization problem according to the constraint conditions of the information transmission stage is: maximizing the confidentiality rate of the system by optimizing the reflection phase shift, power allocation and time allocation of the intelligent reflection surface; the confidentiality rate achieved by the system is: R s =R' UE -R' Eve R' UE It represents the achievable rate of legitimate users in the second phase of wireless energy transmission and anti-eavesdropping phase; R' Eve It represents the achievable rate of the eavesdropper in the second phase of wireless energy transmission and anti-eavesdropping phase; Propose an optimization problem based on the objective function and constraints to maximize the achievable rate for legitimate users; The constraints include,user energy collection constraint, total transmission time constraint, and passive reflection phase shift constraint of smart reflective surface; In step 4, the optimization of the achievable rate includes: by jointly optimizing the reflection phase shifts Φ1 and Φ2, the power allocation P′=[P3, P4] and the time allocation t=[T1, T2], the optimization problem is expressed as: stC1:E1≥(P3+P4)T2 C2:T1+T2≤T,T1>0,T2>0 C3:|φ 1,n |=1,n=1,...,N C4:|φ 2,n |=1,n=1,...,N R UE Indicates the achievable rate of legitimate users in the second phase; T2 represents the transmission time of the second stage wireless information transmission and the eavesdropping mode; T represents the total transmission time in the first phase of unlimited energy transmission eavesdropping detection and the second phase of wireless information transmission and eavesdropping mode; P3 represents the signal power of the legitimate user in the second phase of wireless information transmission and eavesdropping mode; Indicates power Gaussian noise; h1 represents the UE-IRS and IRS-UE segment channels; H1 represents the UE-BS and BS-UE segment channels; Indicates power Gaussian noise; H represents the baseband equivalent channel; st means constraint; C1 represents the first optimization constraint, the energy collected by the legitimate users, i.e., the energy collected in the downlink is greater than the energy consumption in the second stage; C2 represents the second optimization constraint, the sum of the first and second stage times is less than the total time and both the first and second stage times are greater than 0; C3 represents the third optimization constraint. In the first stage, the IRS phase shift matrix satisfies the modulus value of 1 during BS-IRS-UE and BS-IRS-Eve transmission. C4 represents the fourth optimization constraint, the IRS phase shift matrix satisfies the modulus value of 1 during the second stage UE-IRS-BS transmission; CVX stands for Convex Optimization Toolbox, a toolbox for convex optimization problems; By designing an optimization algorithm, the non-convex problem is transformed into a semi-definite programming problem, and CVX is used to optimize the solution. In the step 5, according to the perfect and imperfect channel state information CSI of the eavesdropper Eve, corresponding optimization algorithms are designed to minimize the achievable rate of the eavesdropper; The optimization problem of beam cancellation optimization at the eavesdropper under perfect CSI conditions is established as: Use CVX to solve the above problem and get the optimal solution, thus obtaining the eavesdropper's information accessibility rate; The optimization problem of beam cancellation optimization at the eavesdropper under non-perfect CSI conditions is established as: Φ3 means that the IRS phase shift matrix satisfies the modulus value of 1 during the second phase UE-IRS-Eve transmission; R Eve It represents the information rate of the eavesdropper in the second stage of wireless information transmission and the eavesdropping mode. Considering that Eve may be a legitimate user in the past, under perfect channel state information, the information beam cancellation at the eavesdropper in this case is solved by optimizing the phase shift matrix Φ3. T2 represents the transmission time of the second stage wireless information transmission and the eavesdropping mode; T represents the total transmission time in the first phase of unlimited energy transmission eavesdropping detection and the second phase of wireless information transmission and eavesdropping mode; P4 represents the signal power of Eve in the second stage of wireless information transmission and eavesdropping mode; h2 represents the IRS-Eve segment channel; h1 represents the UE-IRS and IRS-UE segment channels; H2 represents the UE-Eve segment channel; Indicates power Gaussian noise; stC5 represents the fifth optimization constraint, the IRS phase shift matrix satisfies the modulus value of 1 during the second phase of wireless information transmission and UE-IRS-Eve transmission in the wiretap mode; φ 3,n It represents the IRS phase shift matrix during the second phase of wireless information transmission and UE-IRS-Eve transmission in the wiretap mode; N represents the number of phase-shift elements in the diagonal matrix; C13 represents the thirteenth optimization constraint; K B,E The uncertainty region K of h2 B,E ; represents the signal power, represents the noise power; Δh2 represents the estimation error. The optimization problem is first simplified and converted into a standard semidefinite program using the S-Lemma lemma, and then optimized and solved using CVX. The eavesdropper's signal-to-noise ratio obtained from the optimization is used to obtain the eavesdropper's achievable rate.

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