# Secrecy-Capacity-Optimization-Artificial-Noise in Large-Scale MIMO: Ergodic Lower Bound on Secrecy Capacity and Optimal Power Allocation

Authors: Yebo Gu; Zhilu Wu; Zhendong Yin
Journal: Wireless Communications and Mobile Computing
 (2021)

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## Abstract

The security of wireless information transmission in large-scale multi-input and multioutput (MIMO) is the focus of research in wireless communication. Recently, a new artificial noise—SCO-AN which shows no orthogonality to the channel, is proposed to overcome the shortcomings of traditional artificial noise. In the previous research, the optimization function of SCO-AN is not convex, and its extremum cannot be obtained. Usually, nonconvex optimization algorithms or iterative relaxation algorithms are used to get the maximum value of the optimization objective function. Nonconvex optimization algorithms or iterative relaxation algorithms are greatly affected by the initial value, and the extremum cannot be obtained by a nonconvex optimization algorithm or iterative relaxation algorithm. In this paper, we creatively apply the strong law of large numbers to obtain the optimal value of the optimization function of SCO-AN under the condition of large-scale MIMO: the strong law of large numbers is applied to obtain the ergodic lower bound (ELB) expression of SC for SCO-AN. The power allocation (PA) problem of the SCO-AN system is discussed. We use a statistical method to get the formula for calculating the optimal power distribution coefficient of the SCO-AN system. The transmitter can use the optimal power ratio of PA to distribute the transmitted power without using the PA algorithm. The effect of imperfect channel state information is discussed. Through simulation, we found that more power should be generated for SCO-AN if the channel estimation is imperfect and the proposed method can achieve better security performance in the large-scale MIMO system.


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## Body

## 1. Introduction

Security is the core issue of wireless communication [1]. In this paper, the security of wireless information transmission is discussed. Since the wireless communication signal is transmitted in broadcast ways, the wireless communication signal can be received by eavesdroppers without restriction. Since the eavesdropper cannot be prevented from obtaining the transmitted information, previous research on secure communication is focused on using the principles of encryption to prevent eavesdroppers from deciphering the obtained information. However, encryption cannot fundamentally solve secure transmission problem if the eavesdropper obtains the key of the information or has enough computing power to crack the information. The secure transmission of information based on cryptography will fail. Therefore, reducing the amount of information obtained by eavesdroppers is the correct way to achieve secure communication. In recent years, physical layer security (PLS) technology has become a new research focus.The PLS technology was proposed in [2]. In [2], a classic model for secure communication model is proposed. In [3], the secure communication model is extended to the field of wireless communication, the channels in the secure communication models are all broadcast channels.The application of MIMO provides an opportunity for the development of PLS. In [4], the concept of secrecy capacity (SC) is outlined by Barros and Rodrigues. The numerical expression of SC is studied in [5]. In [6], Zhu studies the SC of wiretap fading channels in a massive MIMO communication system. Zeng proposes the SC of downlink massive MIMO-NOMA networks [7]. A novel method to select the proper antenna and PA in massive MIMO is proposed in [8], the algorithm in [8] makes the wireless communication system to adjust the PA coefficient, and the communication quality of eavesdropper is reduced effectively.SC is determined by the capacity of legitimate and eavesdropping channels. Generally, increasing the capacity of the legitimate channel or decreasing the capacity of the eavesdropping channel could increase SC [9–11]. However, for a particular communication system, the capacity of the legitimate channel is challenging to change. Therefore, previous studies focus on reducing the capacity of eavesdropping channels [12–14].Artificial noise (AN) is a specific application for PLS technology [15]. AN is orthogonal to the receiver’s channel, so AN will not affect the legitimate channel. However, AN is not orthogonal to the eavesdropping channel; AN will affect the eavesdropping channel. Adding AN to the transmitted signals will raise the SC of secure transmission.However, the design of AN is under strict restriction.H denotes the legitimate channel state information (CSI), and wk denotes the AN. By definition of AN, Hwk=0. wk is the zero solution of homogeneous linear equation group HX=0. The dimension of H is m×n. If X is nonzero, the rank (H) must be guaranteed that rank (H)<minm,n. It means that the number of transmitting and receiving antennas has to be different. However, transmitting antennas have the same number as receiving antennas in most MIMO real-life communication systems. We have to reduce the transmitting or receiving antennas. Meanwhile, the capacity of legitimate channel is reduce.SCO-AN is proposed in [16] to solve the problem. SCO-AN does not show orthogonality to the legitimate and eavesdropping channels. Therefore, SCO-AN is achievable in any situation. Although this method has a certain degree of negative impacts on the legitimate channel, SCO-AN has a greater degree of negative impacts on the eavesdropping channel. SCO-AN is considered to increase the SC to a greater degree when compared to the traditional AN. SCO-AN is a tool that converts the receiver’s antinoise ability into SC.Recently, intelligent reflecting surface (IRS) technology has become a research hotspot in communication technology [17, 18]. Because the channel of IRS is divided into uplink and downlink. Artificial noise is only valid in the uplink or downlink so that AN cannot be applied to IRS. The nonorthogonal characteristic of SCO-AN makes it a perfect application in IRS. In 2020, [19] applied nonorthogonal artificial noise to improve the security performance of IRS systems. It should be pointed out that the idea of nonorthogonal artificial noise was first put forward in [16].We should consider the transmission power limits. It is important to allocate the power of the SCO-AN system [20]. The objective function of SCO-AN PA is nonconvex. It has been proved that we cannot get the optimal PA coefficient of the SCO-AN system. In general, nonconvex optimization algorithm or relaxation iteration algorithm is used to optimize nonconvex functions [21–24]. The simulation result has proved that these two algorithms are greatly affected by initial states of the communication models. We can only adopt an iterative approach to calculate the maximum of the objective function in a certain range. However, in the massive MIMO communication system, we can use strong law of large numbers to approximate the ELB of SC. Based on the ELB of SC, the optimal PA coefficient of SCO-AN is obtained as well. The expression for ELB on SC in fading channels which adds SCO-AN is derived. The optimal ratio of PA of the SCO-AN system is then deduced.Base on the optimal PA coefficient, in a large-scale MIMO system, we can allocate the power without calculating. Furthermore, the effects of the perfect CSI and the imperfect CSI on the ELB of SC are discussed. The study of this paper is highly compatible with the features of 6G networks which equip massive MIMO communication system [25]. Therefore, the application introduced in this paper is a promising solution in addressing security threats for 6G networks.The contributions of this paper are as follows:(i)
The nonconvexity of the SCO-AN PA function makes it impossible to obtain the maximum value. In this paper, the SCO-AN PA function is reconstructed creatively using statistical theory, and the reconstructed PA function approximates the theoretical limit. From the reconstructed PA function, we can deduce the channel capacity of the legitimate channel and eavesdropping channel in large-scale MIMO. Thereby, the secrecy capacity in large-scale MIMO is deduced. We reach a crucial conclusion: the PA objective function of the channel between transmitter and receiver depends mainly on the partition of power and the number of transmitting antennas(ii)
Based on the reconstructed PA function, we derive the optimal PA coefficient. In large-scale MIMO, PA can be carried out according to this coefficient without calculation, and the system can achieve excellent security performance. To simplify the calculation, the precision of the traditional PA algorithm based on statistics is very low. We improve the traditional PA algorithm based on statistics and propose a new PA algorithm based on statistics. The accuracy of the PA algorithm is improved(iii)
We consider the effect of the imperfect CSI. An expression for ELB of SC with imperfect CSI is derived. Based on the expression, we conclude that we should generate more power for SCO-AN once the level of imperfect CSI increases. The negative effect of imperfect CSI on PA is proved by simulationThe rest of the paper is organized as follows: Section2 introduces the related work. Section 3 defines the system, channel, and signal model. Section 4 investigates the ELB of SC. The optimal PA for the proposed system model is discussed in Section 5. The effect of the imperfect is studied in Section 6. Simulation results are given in Section 7. Finally, concluding remarks are given in Section 8.


## 2. Related Work

SCO-AN was first proposed in [16]. In this paper, the author proposes an SCO-AN to overcome the defects caused by the orthogonality of traditional artificial noise. For a long time, SCO-AN is not drawing attention. In recent years, the intelligent reflecting surface is a hot spot in the field of communication technology. IRS has two channels—uplink and downlink channels, and artificial noise cannot be orthogonal with two channels simultaneously. The nonorthogonal SCO-AN is perfectly suitable for intelligent reflecting surface technology; it also brought new prospects for the application of SCO-AN. Hong et al. have started to add nonorthogonal artificial noise to the intelligent reflection surface and achieved excellent security performance [19].Here, we briefly introduce the principle of AN. LetH be the CSI matrix. H is a Nh×Nh matrix. AN is a Nh×1 vector. AN is orthogonal to channel H. AN should satisfy:(1)Hwk=0,wk denotes AN. We find that AN can hardly be designed in the simulation. AN does not exist, especially when the number of transmitting antennas equals the number of receiving antennas. According to matrix theory, for a linear equation Hwk=0, if H is not full rank, that is, H=0, Hwk=0 has a nonzero solution. However, in wireless communication, H=0 almost does not exist. Therefore, for real-life communication, AN has only zero solutions, and AN cannot be designed.SCO-AN is designed to solve the orthogonality defect of AN. SCO-AN is not orthogonal to channelH; that is,(2)Hwm=α,wm denotes SCO-AN; α is a Nh×1 vector, and each elements in α is arbitrarily small. When designing SCO-AN, α satisfies condition rankH=rankH,α. According to matrix theory, SCO-AN must have a nonzero design scheme. The design of SCO-AN does not meet any restrictions.We have performed a thousand simulation experiments to design AN and SCO-AN. From Table1, it can be seen that when rank (H) is equal to Nh, in 1000 experiments, AN is not designed successfully in any experiments. Only when rank (H) is less than Nh, AN can be designed successfully. The design of SCO-AN is not affected by the rank(H).Table 1
Design feasibility of AN and SCO-AN.
ANSCO-ANRankH=Nh01000RankH<Nh10001000Table2 shows the number of solution vectors for AN and SCO-AN under different channel conditions, N1 is the number of solution vectors for AN, and N2 is the number of solution vectors for SCO-AN. When rankH is equal to Nh, the number of solution vectors for AN is 0, that is, AN cannot be designed, and the number of solution vectors of SCO-AN is 1, that is, under a specific power limit, there is only one design scheme of SCO-AN. When rankH−Nh=1, under a specific power limit, the number of AN solution vectors is 1; that is, under a specific power limit, there is only one design solution for AN. Moreover, there are countless kinds of SCO-AN design schemes. Only when rankH−Nh>1, the solution vectors of AN and SCO-AN are both infinite, which also shows that SCO-AN is a more designable than artificial noise.Table 2
Solution space size of AN and SCO-AN.
ANSCO-ANRankH=NhN1=0N2=1RankH−Nh=1N1=1N2=∞RankH−Nh>1N1=∞N2=∞A comparison of AN and SCO-AN secrecy performance is shown in Figure1. The previous analysis shows that, in most cases, the design of AN is very few or even unique, so for a particular channel, the improvement of AN for SC is limited; while the number of design scheme for SCO-AN is infinite, SCO-AN has no theoretical upper limit for the improvement of SC. Therefore, the SC of SCO-AN is always higher than that of AN.Figure 1
The SC of AN and SCO-AN versus different SNR.In the real-life communication environment, Alice must consider the influence of power when transmitting signals. A reasonable power allocation will effectively increase the SC of the system. After considering the transmission power, the SC of the system is as follows:(3)Csec=IyB;S−IyE;S=log1+H2σu2σn2+H2P−σu2−log1+G2σu2G2P−σu2+σe2.In (3), H and G are legitimate channel CSI and eavesdropping channel CSI, respectively. σn2 and σe2 are the variance of noise in legitimate channel and eavesdropping channel, respectively. σu2 is the power of the transmitted signal, and the power of SCO-AN is σv2. σu2+σv2=P; therefore, σv2=P−σu2. If the variables in (3) are σv2 and σu2, the Hessian matrix of (3) is nonpositive definite so that (3) is nonconvex.Due to the nonconvex property of the SCO-AN PA function, the extremum of the function and the optimal power allocation cannot be obtained. The iterative algorithms in [26–29] are applied to derive the maximum for PA function within a specific range. The iterative algorithms are affected by the initial value seriously and have a limited optimization effect.Figure2 shows a comparison of the different nonconvex optimization algorithms.Figure 2
The comparison of the different nonconvex optimization algorithms.From Table3, we can see that the different initial states of the transmitted signal and SCO-AN can lead to different optimization results, which is caused by the nonconvexity of the PA function. Therefore, traditional nonconvex optimization algorithms cannot get excellent optimization results. SQP is a classic iterative optimization algorithm. In the following sections, we continue to study the optimal PA and lower bound of secrecy capacity of SCO-AN under ideal conditions.Table 3
The initial state influence on SC.
AlgorithmInitial (σu2,σv2)Initial SCOptimized SCSQP(5,5)0.56120.6375SQP(3,7)0.47850.5364SQP(1,9)0.38210.4357


## 3. Problem State

### 3.1. System Model

In Figure3, the system model of MIMO communication with eavesdroppers is shown. The noncolluding and colluding eavesdropper cases are discussed in this paper. Therefore, the system model has two scenarios: noncolluding and colluding eavesdroppers.Figure 3
System model of MIMO communication with eavesdroppers.Alice is the transmitter of information. The number of Alice’s antennas isNK. Bob is the receiver of information. The number of Bob’s antennas is NP. Eve is the eavesdropper of information.There areNL eavesdroppers in the system. For colluding scenario, the NL eavesdroppers can eavesdrop on information, which means that the eavesdropped information will be exchanged and processed jointly. The noncolluding eavesdropper is a special scenario that these NL eavesdroppers independently eavesdrop on information. When studying this scenario, we conducted research on each independent eavesdropper, which can be considered NL=1. Similar settings can be found in [15, 30]. For the facilitation of discussion, we assumed NK>NL and NK=NP.Here we have the following:(4)yB=Hx+n,(5)yE=Gx+e,yB is Bob’s received signal, and yE is Eve’s eavesdropping signal. H is the CSI matrix of the legitimate channel. The dimension of H is NK×NP. G is the CSI of eavesdropping channel. For the colluding scenario, G is a matrix and the dimension of G is NK×NL; for the noncolluding scenario, G is a vector and the dimension of G is NK×1. Each element in H and G is zero-mean complex Gaussian random variables and independent identically distributed (i.i.d). x is signal transmitted by Alice. n and e are the additive Gaussian white noise (AWGN) in legitimate and eavesdropping channels. The variance of n is σn2. The variance of e is σe2.Because AN is orthogonal toH, according to the theory of matrices, AN cannot be designed when NK=NP. The key idea of SCO-AN is to design a signal which does not show orthogonality to legitimate and eavesdropping channels. SCO-AN effectly reduces eavesdropping channel capacity while slightly reducing the capacity of legitimate channels [16]. Therefore, SCO-AN will improve the SC of the communication system.We design the transmitted signal as below:x is a vector, the dimension of x is NK×1. x=r1u+r2v, where r1=H†/H. u is the signal which bears the information. The power of u is σu2. v is the SCO-AN. The elements of v are i.i.d. In v, each power of the NK−1 column vectors is σv2. ∗† is the conjugate transpose operation of ∗. ∗ is the norm of ∗. So the legitimate receiving and eavesdropping signal becomes the following:(6)yB=Hr1u+Hr2v+n=Hu+H1v+n,(7)yE=Gr1u+Gr2v+e=G1u+G2v+e,where H1=Hr2, G1=Gr1, and G2=Gr2.Let the power for each transmission beP, P=σu2+NK−1σv2. The proportion of SCO-AN to P is ϕ. Here,(8)σu2=ϕP,(9)σv2=1−ϕPNK−1.From (8) and (9), we can come to the conclusion that Alice could adjust the PA strategy by adjusting ϕ.


## 3.1. System Model

In Figure3, the system model of MIMO communication with eavesdroppers is shown. The noncolluding and colluding eavesdropper cases are discussed in this paper. Therefore, the system model has two scenarios: noncolluding and colluding eavesdroppers.Figure 3
System model of MIMO communication with eavesdroppers.Alice is the transmitter of information. The number of Alice’s antennas isNK. Bob is the receiver of information. The number of Bob’s antennas is NP. Eve is the eavesdropper of information.There areNL eavesdroppers in the system. For colluding scenario, the NL eavesdroppers can eavesdrop on information, which means that the eavesdropped information will be exchanged and processed jointly. The noncolluding eavesdropper is a special scenario that these NL eavesdroppers independently eavesdrop on information. When studying this scenario, we conducted research on each independent eavesdropper, which can be considered NL=1. Similar settings can be found in [15, 30]. For the facilitation of discussion, we assumed NK>NL and NK=NP.Here we have the following:(4)yB=Hx+n,(5)yE=Gx+e,yB is Bob’s received signal, and yE is Eve’s eavesdropping signal. H is the CSI matrix of the legitimate channel. The dimension of H is NK×NP. G is the CSI of eavesdropping channel. For the colluding scenario, G is a matrix and the dimension of G is NK×NL; for the noncolluding scenario, G is a vector and the dimension of G is NK×1. Each element in H and G is zero-mean complex Gaussian random variables and independent identically distributed (i.i.d). x is signal transmitted by Alice. n and e are the additive Gaussian white noise (AWGN) in legitimate and eavesdropping channels. The variance of n is σn2. The variance of e is σe2.Because AN is orthogonal toH, according to the theory of matrices, AN cannot be designed when NK=NP. The key idea of SCO-AN is to design a signal which does not show orthogonality to legitimate and eavesdropping channels. SCO-AN effectly reduces eavesdropping channel capacity while slightly reducing the capacity of legitimate channels [16]. Therefore, SCO-AN will improve the SC of the communication system.We design the transmitted signal as below:x is a vector, the dimension of x is NK×1. x=r1u+r2v, where r1=H†/H. u is the signal which bears the information. The power of u is σu2. v is the SCO-AN. The elements of v are i.i.d. In v, each power of the NK−1 column vectors is σv2. ∗† is the conjugate transpose operation of ∗. ∗ is the norm of ∗. So the legitimate receiving and eavesdropping signal becomes the following:(6)yB=Hr1u+Hr2v+n=Hu+H1v+n,(7)yE=Gr1u+Gr2v+e=G1u+G2v+e,where H1=Hr2, G1=Gr1, and G2=Gr2.Let the power for each transmission beP, P=σu2+NK−1σv2. The proportion of SCO-AN to P is ϕ. Here,(8)σu2=ϕP,(9)σv2=1−ϕPNK−1.From (8) and (9), we can come to the conclusion that Alice could adjust the PA strategy by adjusting ϕ.


## 4. ELB of SC for SCO-AN

The ELB of SC for SCO-AN is derived in this section. The legitimate channel capacity is as follows:(10)C1=EHlog21+σu2H2σv2H2+σn2=EHlog21+σu2σv2+σn2/H2=EHlog21+ϕP1−ϕP/NK−1+σn2/H2=EHlog21+ϕ1−ϕ/NK−1+σn2/H2P.In (10), for the facilitation of discussion, we hypothesize the signal-noise-ratio (SNR) of communication is high enough, so P≫1. Therefore, the H2P is much larger than σn2 so σn2/H2P is an arbitrarily small number which can be ignored. (10) is written as follows:(11)C1=log21+ϕP1−ϕP/NK−1=log21+ϕ1−ϕ/NK−1.From (11), we can conclude that the capacity of the legitimate channel is mainly related to Nk and ϕ in a high SNR regime. Alice can adjust the capacity of the legitimate channel by adjusting ϕ and NK.Alice has no knowledge of SNR ofG. In order to enable secure communication, we assume that σe2 is arbitrarily tiny, which means the eavesdropper has the best communication environment.Therefore, the upper bound on the eavesdropping channel capacity without consideringσe2 is as follows:(12)C2=Eh,G1,G2log2I+σu2G1G1†σv2G2G2†−1=Eh,G1,G2log21+NK−1r−1G1†G2G2†−1G1.r is the inverse of ϕ, r=ϕ−1.R=r1r2 is a unitary matrix. GR=G1G2 is a complex Gaussian matrix, and G1 and G2 are independent. Therefore, G1†G2G2†G1 is equivalent to the signal-to-interference ratio of a NL−branch mininum mean square error (MMSE) diversity combiner with NK−1 interferers. According to [30, 31], the complementary cumulative distribution function of X=G1†G2G2†−1G1 is as follows:(13)MXx=∑n=0NL−1NK−1nxn1+xNK−1.According to (13), we can rewrite (12) as follows:(14)C2=Eh1ln2∫0∞NK−1r−11+NK−1r−1x−1Mxxdx=Eh1ln2∑n=0NL−1NK−1kNK−1r−1Betk+1,NK−1−k×K11,k+1;NK;r−NKr−1.In (14), Bet· is the function of beta distribution and Beta,b=ΓaΓb/Γa+b. K1· is Gaussian hypergeometric function. Formula (14) uses conclusion in [31].From [5], the ELB of SC is C=C1−C2+ and a+ denotes max0,α. The ELB of SC is as follows:(15)C=1ln2log21+NK−1r−1−∑n=0NL−1NK−1kNK−1r−1×Betk+1,NK−1−k×K11,k+1;NK;r−NKr−1+.In the following sections, the simplified and approximated expressions on ELB of SC will be discussed.In (15), from [32], here we have(16)K11,1;M+1;y=1+yM−1K1M,M;M+1;y,(17)dM−1dyM−1K11,1;2;y=1M−11M−12M−1K1M,M;M+1;y,where mn is the rising factorial. So,(18)K1M,M;M+1;y=2M−11M−11M−1dM−1dyM−1K11,1;2;y=−MM−1!∑t=0M−1M−1tdtdytln1−ydM−1−tdyM−1−ty−1.In [32], we know K11,1;2;y=−ln1−y/y. Therefore, here, we have(19)dtdytln1−y=−t−1!1−yt,t=1,2,3⋯,dtdyty−1=−1tt!yt+1,t=0,1,2,3⋯.Substituting (19) into (18), we can obtain the following:(20)K1M,M;M+1;y=−MM−1!×ln1−y−1M−1M−1!yM−∑t=1M−1M−1!t!M−1−t!t−1!1−yt−1M−1−tM−1−t!yM−t=−1MMyMln1−y−∑t=1M−11tyty−1t.From (20), it is easy to obtain(21)K11,1;M+1;y=−1MM1−yM−1yMln1−y−∑t=1M−11tyty−1t.


### 4.1. The Noncolluding Eavesdropper Case for SC

In the noncolluding eavesdroppers case, there is no cooperation between each eavesdropper; therefore,NL=1. Then,(22)C2=Eh1ln2×1r−1K11,1;NK;r−NKr−1=Eh1ln2NK−1NK−rNK−1×lnNK−1r−1−∑l=1NK−2NK−rNK−1l.(22) is derived by applying the Gaussian hypergeometric function. The ELB of SC for noncolluding eavesdroppers is given by the following:(23)C=1ln2log21+NK−1r−1−EhNK−1NK−rNK−1×lnNK−1r−1−∑l=1NK−2NK−rNK−1l+.


### 4.2. The Colluding Eavesdropper Case for SC

In this subsection, the ELB of SC is discussed whenNK is large enough. We know that limNK⟶∞G2G2†/NK−1=I when we use the strong law of large numbers. So (12) can be written as follows:(24)limNK⟶∞C2=limNK⟶∞EG1,G2log21+1r−1G1†G2G2†NK−1−1G1=EG1log21+1r−1G12=1ln2expr−1∑n=1NLEkr−1,where Ek is the generalized exponential integral. The channel response of Rayleigh fading channel is an exponential function. So G12 is a Gamma function. The paramater of G12 is NL,1.WhenNK is large enough, the ELB of SC is(25)C=log21+NK−1r−1−1ln2expr−1∑n=1NLEkr−1+.From (25), we can conclude that the ELB of SC relates to Nk and r. The optimal PA can be obtained by deriving the ELB of SC. The optimal PA is a function of Nk and r.


## 4.1. The Noncolluding Eavesdropper Case for SC

In the noncolluding eavesdroppers case, there is no cooperation between each eavesdropper; therefore,NL=1. Then,(22)C2=Eh1ln2×1r−1K11,1;NK;r−NKr−1=Eh1ln2NK−1NK−rNK−1×lnNK−1r−1−∑l=1NK−2NK−rNK−1l.(22) is derived by applying the Gaussian hypergeometric function. The ELB of SC for noncolluding eavesdroppers is given by the following:(23)C=1ln2log21+NK−1r−1−EhNK−1NK−rNK−1×lnNK−1r−1−∑l=1NK−2NK−rNK−1l+.


## 4.2. The Colluding Eavesdropper Case for SC

In this subsection, the ELB of SC is discussed whenNK is large enough. We know that limNK⟶∞G2G2†/NK−1=I when we use the strong law of large numbers. So (12) can be written as follows:(24)limNK⟶∞C2=limNK⟶∞EG1,G2log21+1r−1G1†G2G2†NK−1−1G1=EG1log21+1r−1G12=1ln2expr−1∑n=1NLEkr−1,where Ek is the generalized exponential integral. The channel response of Rayleigh fading channel is an exponential function. So G12 is a Gamma function. The paramater of G12 is NL,1.WhenNK is large enough, the ELB of SC is(25)C=log21+NK−1r−1−1ln2expr−1∑n=1NLEkr−1+.From (25), we can conclude that the ELB of SC relates to Nk and r. The optimal PA can be obtained by deriving the ELB of SC. The optimal PA is a function of Nk and r.


## 5. Optimal PA of SCO-AN

The ELB of SC is discussed in the last section. In this section, the optimal PA of the SCO-AN system is studied. The ELB of SC for noncolluding eavesdroppers and infiniteNK are obtained by (23) and (25). For the facilitation of calculation and discussion, the expressions obtained in the last sections will be applied in this section.


### 5.1. Optimal PA: Noncolluding Eavesdropper

For the noncolluding case, theNL=1, and the ELB of SC C=C1−C2+ in last sections helps us to find the optimal values for the PA coefficients r and ϕ.The first derivatives of (23)—the ELB of SC—concerning the PA coefficient r will help us to find the optimal PA coefficient. Therefore, the first derivative of (23) satisfies the following:(26)dCdr=dC1dr−dC2dr=0.From (26), the optimal PA coefficient should satisfy(27)1ln2r−12+NK−1r−1−NK−1×NK−1NK−rNK−2×1−NKNK−r2×lnNK−1r−1−NK−rNK−1+NK−1NK−rNK−1×r−1NK−1×1−NKr−12+1NK−1+=0.(27) is a higher-order equation of r on NK. Its closed-form solution is challenging to find or even nonexistent. Moreover, according to the difference of NK, its root-finding formula is also different. Therefore, in the following research content, we provide some of the optimal PA coefficients with a particular number of commonly used antennas.


### 5.2. Optimal PA: Colluding Eavesdroppers

In this subsection, the colluding eavesdropper’s case is studied. We improve the optimal PA coefficient algorithm under colluding eavesdroppers in the classic paper [30].In [30], for the convenience of calculation, expr−1∑n=1NLEkr−1 is simplified to expr−1Ekr−1. The first NL−1 items in expr−1∑n=1NLEkr−1 have been omitted. This will cause large errors in the calculation results. So we will improve this formula in order to get better calculation results.Considering the expression ofC in (25) and taking its derivative concerning r, the optimal r satisfies the following:(28)dCdr=dC1dr−dC2dr=ln2r−NK−2−1−ln2r−1−1−er−1NLNL−1+2r−12r−1∑n=1NLEkr−1−NLr−1.Hence, we have shown that for sufficiently large NK, the optimal PA is given by (28). The solution of Equation (28) is the optimal PA coefficient. The optimal ratio of PA depends on NL and NK.In Table4, we provide the optimal PA coefficients for some commonly used transmitting and eavesdropping antennas. Alice can allocate power to the transmitted signal by looking up the table instead of using an algorithm for power allocation. From Table 4, we can also see that when the number of eavesdropping antennas increases, it is wise to allocate more power to the SCO-AN to confuse the eavesdropper. At the same time, when NK increases, the capacity of the legitimate channel increases, which can reduce the power allocated for SCO-AN.Table 4
Optimalϕ for different NK and NL.
Number ofNKNumber ofNLOptimalϕ210.292220.313240.327280.339410.271420.282440.312480.327810.255820.262840.272880.2841610.2321620.2381640.2521680.2673210.2073220.2123240.2323280.2426410.1856420.2026440.2316480.25212810.17112820.19012840.20112880.224


## 5.1. Optimal PA: Noncolluding Eavesdropper

For the noncolluding case, theNL=1, and the ELB of SC C=C1−C2+ in last sections helps us to find the optimal values for the PA coefficients r and ϕ.The first derivatives of (23)—the ELB of SC—concerning the PA coefficient r will help us to find the optimal PA coefficient. Therefore, the first derivative of (23) satisfies the following:(26)dCdr=dC1dr−dC2dr=0.From (26), the optimal PA coefficient should satisfy(27)1ln2r−12+NK−1r−1−NK−1×NK−1NK−rNK−2×1−NKNK−r2×lnNK−1r−1−NK−rNK−1+NK−1NK−rNK−1×r−1NK−1×1−NKr−12+1NK−1+=0.(27) is a higher-order equation of r on NK. Its closed-form solution is challenging to find or even nonexistent. Moreover, according to the difference of NK, its root-finding formula is also different. Therefore, in the following research content, we provide some of the optimal PA coefficients with a particular number of commonly used antennas.


## 5.2. Optimal PA: Colluding Eavesdroppers

In this subsection, the colluding eavesdropper’s case is studied. We improve the optimal PA coefficient algorithm under colluding eavesdroppers in the classic paper [30].In [30], for the convenience of calculation, expr−1∑n=1NLEkr−1 is simplified to expr−1Ekr−1. The first NL−1 items in expr−1∑n=1NLEkr−1 have been omitted. This will cause large errors in the calculation results. So we will improve this formula in order to get better calculation results.Considering the expression ofC in (25) and taking its derivative concerning r, the optimal r satisfies the following:(28)dCdr=dC1dr−dC2dr=ln2r−NK−2−1−ln2r−1−1−er−1NLNL−1+2r−12r−1∑n=1NLEkr−1−NLr−1.Hence, we have shown that for sufficiently large NK, the optimal PA is given by (28). The solution of Equation (28) is the optimal PA coefficient. The optimal ratio of PA depends on NL and NK.In Table4, we provide the optimal PA coefficients for some commonly used transmitting and eavesdropping antennas. Alice can allocate power to the transmitted signal by looking up the table instead of using an algorithm for power allocation. From Table 4, we can also see that when the number of eavesdropping antennas increases, it is wise to allocate more power to the SCO-AN to confuse the eavesdropper. At the same time, when NK increases, the capacity of the legitimate channel increases, which can reduce the power allocated for SCO-AN.Table 4
Optimalϕ for different NK and NL.
Number ofNKNumber ofNLOptimalϕ210.292220.313240.327280.339410.271420.282440.312480.327810.255820.262840.272880.2841610.2321620.2381640.2521680.2673210.2073220.2123240.2323280.2426410.1856420.2026440.2316480.25212810.17112820.19012840.20112880.224


## 6. Secrecy Capacity: Imperfect Channel State Information

The previous section explored the ELB of SC obtained with the perfect CSI. We will continue to discuss the effect of the imperfect CSI.The CSI estimation error introduces extra noise into the legitimate channel. Bob uses the channel estimation algorithm to get the CSI. The CSI is imperfect due to the Gaussian noise. Therefore,(29)H=H^+H~,where H^ denotes the channel state information obtained by MMSE channel estimate and H~ denotes the channel estimation error. H^ and H~ are uncorrelated due to the property of the MMSE estimation. Therefore,(30)σH2=σH^2+σH~2.r1 is designed as r1=H∧†/H^, and the symbol received by Bob is(31)yB=H^x+H~x+Hr2v+n=H^u+H~RuvTT+H1v+n,where H1=Hr2. R is a unitary matrix, so the distribution of h~R and h~ is the same. Therefore, the ELB of SC for the legitimate channel is(32)C^1=Ehlog21+σu2H∧2σv2H∧2+σH~2P+1.With σH2 normalized to unity, we have σH^2+σH~2=1. Each subchannel between Alice and Bod obeys exponential distribution. Therefore, H∧2 follows a Gamma distribution with parameter (NK,1−σH~2). Here, we have(33)C^1=∫0∞gx1−σH~2NK−1ΓNKHxNK−1e−1−σH~2hxdx.In (33), hx is the inverse function (H∧2=x) of (32), and gx is the first derivative of hx. Therefore,(34)hx=ln22x2x−11+σH^2Pσv2σu2−2x−1σv22+2x1+σH^2Pσu2−2x−1σv2,(35)gx=2x−11+σH^2Pσu2−2x−1σv2.The error of channel estimation does not affect Eve’s channel. Therefore, the ELB of SC is as follows:(36)C=∫0∞hx1−σH~2NK−1ΓNKgxNK−1e−1−σH~2gxdx−1ln2ψ0NL+r−ψ0r+,where Γ denotes the Gamma function and Ψn denotes the nth derivative of the polygamma function.


## 7. Results and Discussion

### 7.1. Simulation Parameters

In the simulation, the legitimate and eavesdropping channels are Rayleigh fading channels. The relevant simulation parameters are shown in Table5.Table 5
Simulation environment.
ParametersValuesNkFrom 1 to 256NLFrom 1 to 256Power for each transmission10 (W)Mean of legitimate and eavesdropping channels0Variances of legitimate and eavesdropping channels0.5SNR of legitimate and eavesdropping channelsFrom -5 to 30 (dB)


### 7.2. Simulation Results and Discussion

Figure4 shows how ELB of SC changes with Nk and Nl. The ELB of SC is calculated by (15) with the optimal ϕ. It can be seen from Figure 4 that when the number of transmitting antennas Nk increases, the SC also increases. It can also be seen from (15) that Nk is located on the molecule, and Nk plays a positive role in the promotion of SC. Similarly, the increase in the number of eavesdropping antennas will reduce the SC of the system. The above conclusions are easy to understand. The increase in the number of transmitting antennas represents an increase in the capacity of the legitimate channel. Therefore, the SC will increase. Similarly, the increase in the number of eavesdropping antennas will result in a decrease in the SC.Figure 4
The ELB of SC versus SNR with differentNK and NL.Figure5 shows the effect of the number of eavesdropping antennas on the SC. It can be seen that with the increase in the number of eavesdropping antennas, the SC of the wireless communication system decreases under the condition that the legitimate channel remains unchanged. At the same time, Figure 5 shows the SC of the wireless communication system for AN and SCO-AN when N approaches infinity. It can be seen that the ELB on SC of SCO-AN is higher than that of AN.Figure 5
The lower bound of SC versus SNR with different numbers ofNL and infinite NK.Figure6 shows the ELB of SC in (15) with the optimal ϕ. For comparison, the ELB of SC for AN with equal power allocation is indicated by solid lines. ELB of SC for SCO-AN is always greater than that for AN. The SC increases as NK grows. The optimal ϕ of SCO-AN is much smaller than that of SC for AN, which means SCO-AN has a better secrecy performance with less power for extra artificial noise. It proves that SCO-AN is a better artificial noise when compared to AN. The SC of the wireless communication system decreases, which shows that the channel estimation error has a counterproductive effect on the security performance of the wireless communication system, which is also in line with reality. The imperfect CSI will cause Alice to error in the distribution of transmitting power, which will leads to a decrease in the SC.Figure 6
The lower bound of SC versus ratio of powerϕ with different NK.The AWGN produces the CSI estimation error. So the CSI estimation error decreases with SNR. The deviation of the PA ratio between the practical system and the perfect CSI system becomes smaller with the increase of SNR. We should consider the channel estimation error when SNR is low in a practical system.Figure7 shows the SC with optimal ϕ and different types of σH~2 (LS and MMSE) [31]. MMSE and LS are classic channel estimation algorithms. It can be seen that for the influence of different levels of imperfect CSI on SC, the influence of wireless communication systems using MMSE channel estimation is always less than that of LS, which is because the MMSE channel estimation accuracy is higher than that of the LS, so the MMSE channel estimation has less influence on Alice’s power allocation. At the same time, the increase of the channel estimation error level will cause the decrease of SC.Figure 7
ϕ versus SNR for different NK and channel estimation errors σH~2.Figure8 shows the ELB of SC with different PA methods. We use SQP and BPA algorithms [26] to compare with the optimal PA coefficient presented in this paper. We can see from Figure 8 that under the condition that transmitting antennas’ number is small, the ELB of SC using the optimal power distribution coefficient is smaller than that of SQP; when the number of antennas increases, the SC using the optimal power distribution coefficient increases significantly, which is consistent with the theory in this paper. Since the conclusions derived in this paper are derived using the strong law of large numbers, which is a statistical theory, the larger the number of samples, the more accurate the results of the theoretical derivation. Therefore, in the case of a small number of antennas, the algorithm in this paper is not as good as the traditional iterative optimization algorithm. But when the number of antennas is large enough, the algorithm in this paper is far better than the traditional iterative optimization algorithm. Since the optimal power distribution coefficient is based on statistics, the optimal power distribution coefficient can be used well for massive MIMO.Figure 8
The ELB of SC versusNK with different PA algorithm.Figures9 and 10 show optimal power distribution coefficient in (28) with different NK and NL. Figures 9 and 10 show optimal ϕ reduces as NL grows; we should allocate more power to SCO-AN as NL increases. On the contrary, when NK increases, less power should be allocated to SCO-AN. Figures 9 and 10 also show that we should generate more power to the information-bearing signal with the SNR increase.Figure 9
The optimal power distribution coefficient versus SNR with differentNK.Figure 10
The optimal power distribution coefficient versus SNR with differentNL.Figure11 shows the comparison between the algorithm in this paper and the algorithm in the classic paper [30]. The solid line SC in Figure 11 is calculated by the optimal PA coefficient obtained by (28). The dotted SC is calculated by the best PA coefficient obtained in the classic paper [30]. Under the same conditions of Nk and NL, the algorithm in this paper has a better improvement on SC, while the algorithm in the classic paper [30] discards too many items, resulting in a significant error in the optimal PA coefficient, and the effect of improving the SC is reduced.Figure 11
Comparison of SC between the algorithm in this paper and the algorithm in the classic papers [30].


## 7.1. Simulation Parameters

In the simulation, the legitimate and eavesdropping channels are Rayleigh fading channels. The relevant simulation parameters are shown in Table5.Table 5
Simulation environment.
ParametersValuesNkFrom 1 to 256NLFrom 1 to 256Power for each transmission10 (W)Mean of legitimate and eavesdropping channels0Variances of legitimate and eavesdropping channels0.5SNR of legitimate and eavesdropping channelsFrom -5 to 30 (dB)


## 7.2. Simulation Results and Discussion

Figure4 shows how ELB of SC changes with Nk and Nl. The ELB of SC is calculated by (15) with the optimal ϕ. It can be seen from Figure 4 that when the number of transmitting antennas Nk increases, the SC also increases. It can also be seen from (15) that Nk is located on the molecule, and Nk plays a positive role in the promotion of SC. Similarly, the increase in the number of eavesdropping antennas will reduce the SC of the system. The above conclusions are easy to understand. The increase in the number of transmitting antennas represents an increase in the capacity of the legitimate channel. Therefore, the SC will increase. Similarly, the increase in the number of eavesdropping antennas will result in a decrease in the SC.Figure 4
The ELB of SC versus SNR with differentNK and NL.Figure5 shows the effect of the number of eavesdropping antennas on the SC. It can be seen that with the increase in the number of eavesdropping antennas, the SC of the wireless communication system decreases under the condition that the legitimate channel remains unchanged. At the same time, Figure 5 shows the SC of the wireless communication system for AN and SCO-AN when N approaches infinity. It can be seen that the ELB on SC of SCO-AN is higher than that of AN.Figure 5
The lower bound of SC versus SNR with different numbers ofNL and infinite NK.Figure6 shows the ELB of SC in (15) with the optimal ϕ. For comparison, the ELB of SC for AN with equal power allocation is indicated by solid lines. ELB of SC for SCO-AN is always greater than that for AN. The SC increases as NK grows. The optimal ϕ of SCO-AN is much smaller than that of SC for AN, which means SCO-AN has a better secrecy performance with less power for extra artificial noise. It proves that SCO-AN is a better artificial noise when compared to AN. The SC of the wireless communication system decreases, which shows that the channel estimation error has a counterproductive effect on the security performance of the wireless communication system, which is also in line with reality. The imperfect CSI will cause Alice to error in the distribution of transmitting power, which will leads to a decrease in the SC.Figure 6
The lower bound of SC versus ratio of powerϕ with different NK.The AWGN produces the CSI estimation error. So the CSI estimation error decreases with SNR. The deviation of the PA ratio between the practical system and the perfect CSI system becomes smaller with the increase of SNR. We should consider the channel estimation error when SNR is low in a practical system.Figure7 shows the SC with optimal ϕ and different types of σH~2 (LS and MMSE) [31]. MMSE and LS are classic channel estimation algorithms. It can be seen that for the influence of different levels of imperfect CSI on SC, the influence of wireless communication systems using MMSE channel estimation is always less than that of LS, which is because the MMSE channel estimation accuracy is higher than that of the LS, so the MMSE channel estimation has less influence on Alice’s power allocation. At the same time, the increase of the channel estimation error level will cause the decrease of SC.Figure 7
ϕ versus SNR for different NK and channel estimation errors σH~2.Figure8 shows the ELB of SC with different PA methods. We use SQP and BPA algorithms [26] to compare with the optimal PA coefficient presented in this paper. We can see from Figure 8 that under the condition that transmitting antennas’ number is small, the ELB of SC using the optimal power distribution coefficient is smaller than that of SQP; when the number of antennas increases, the SC using the optimal power distribution coefficient increases significantly, which is consistent with the theory in this paper. Since the conclusions derived in this paper are derived using the strong law of large numbers, which is a statistical theory, the larger the number of samples, the more accurate the results of the theoretical derivation. Therefore, in the case of a small number of antennas, the algorithm in this paper is not as good as the traditional iterative optimization algorithm. But when the number of antennas is large enough, the algorithm in this paper is far better than the traditional iterative optimization algorithm. Since the optimal power distribution coefficient is based on statistics, the optimal power distribution coefficient can be used well for massive MIMO.Figure 8
The ELB of SC versusNK with different PA algorithm.Figures9 and 10 show optimal power distribution coefficient in (28) with different NK and NL. Figures 9 and 10 show optimal ϕ reduces as NL grows; we should allocate more power to SCO-AN as NL increases. On the contrary, when NK increases, less power should be allocated to SCO-AN. Figures 9 and 10 also show that we should generate more power to the information-bearing signal with the SNR increase.Figure 9
The optimal power distribution coefficient versus SNR with differentNK.Figure 10
The optimal power distribution coefficient versus SNR with differentNL.Figure11 shows the comparison between the algorithm in this paper and the algorithm in the classic paper [30]. The solid line SC in Figure 11 is calculated by the optimal PA coefficient obtained by (28). The dotted SC is calculated by the best PA coefficient obtained in the classic paper [30]. Under the same conditions of Nk and NL, the algorithm in this paper has a better improvement on SC, while the algorithm in the classic paper [30] discards too many items, resulting in a significant error in the optimal PA coefficient, and the effect of improving the SC is reduced.Figure 11
Comparison of SC between the algorithm in this paper and the algorithm in the classic papers [30].


## 8. Conclusion

This paper researches ELB of SC under large-scale MIMO. Aiming at the problem that traditional optimization algorithms cannot find the maximum value of nonconvex PA function, this paper uses statistical methods to study the PA technology of SCO-AN in large-scale MIMO systems. First, this paper derives the expression of ELB of SC. Then, the best PA coefficient function based on this expression of ELB of SC is derived. According to the optimal PA coefficient function, the optimal PA coefficient under the conditions of different transmitting antennas and the number of eavesdropping antennas can be deduced. The application of the PA coefficient can significantly improve the SC without the optimization algorithm. This paper also considers the impact of imperfect CSI on PA; whenNK increases or NL decreases, the power of SCO-AN should be reduced. At the same time, when the level of imperfect CSI increases, more power should be allocated to SCO-AN.



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*Source: 9420982-2021-08-19.xml*