On optimization of cooperative MIMO for underlaid secrecy Industrial Internet of Things*

Xinyao WANG ,Xuyan BAO ,Yuzhen HUANG ,Zhong ZHENG ,Zesong FEI

1School of Information and Electronics,Beijing Institute of Technology,Beijing 100081,China

2China Academy of Information and Communications Technology,Beijing 100191,China

3Academy of Military Sciences of PLA,Beijing 100091,China

Abstract:In this paper,physical layer security techniques are investigated for cooperative multi-input multi-output(C-MIMO),which operates as an underlaid cognitive radio system that coexists with a primary user (PU).The underlaid secrecy paradigm is enabled by improving the secrecy rate towards the C-MIMO receiver and reducing the interference towards the PU.Such a communication model is especially suitable for implementing Industrial Internet of Things (IIoT) systems in the unlicensed spectrum,which can trade offspectral efficiency and information secrecy.To this end,we propose an eigenspace-adaptive precoding(EAP)method and formulate the secrecy rate optimization problem,which is subject to both the single device power constraint and the interference power constraint.This precoder design is enabled by decomposing the original optimization problem into eigenspace selection and power allocation sub-problems.Herein,the eigenvectors are adaptively selected by the transmitter according to the channel conditions of the underlaid users and the PUs.In addition,a simplified EAP method is proposed for large-dimensional C-MIMO transmission,exploiting the additional spatial degree of freedom for a low-complexity secrecy precoder design.Numerical results show that by transmitting signal and artificial noise in the properly selected eigenspace,C-MIMO can eliminate the secrecy outage and outperforms the fixed eigenspace precoding methods.Moreover,the proposed simplified EAP method for the large-dimensional C-MIMO can significantly improve the secrecy rate.

Key words: Cognitive radio network;Physical layer security;Cooperative multi-input multi-output (C-MIMO);Eigenspace-adaptive precoding;Difference convex programming

1 Introduction

1.1 Background

The Industrial Internet of Things (IIoT) is one of the most important applications of beyond 5thgeneration/6thgeneration (B5G/6G) mobile communication technology (Chettri and Bera,2020;Nguyen et al.,2022).It supports critical data harvesting from machinery sensors as well as control signaling delivery to actuators for the smart factory(Ho?ej?í et al.,2020;Hussain et al.,2020).As an example,high-definition real-time video transmission in the IIoT provides a wide range of sensing capabilities for the smart factory,including physical-space security surveillance (Borges and Izquierdo,2010),vision-based quality inspection(Akhyar et al.,2019),and high-precision video-based localization (Chen et al.,2017).In these cases,high-volume videos need to be transmitted from the sensors to the collector,which requires high-speed and secure communication links to guarantee video quality and industrial information security.In addition to the rigorous confidentiality requirement,one of the major issues that have limited massive deployments of IIoT systems is the high cost of acquiring or leasing the spectrum license.

To acquire the operation spectrum with reduced cost,the unlicensed spectrum has been introduced to IIoT systems,opportunistically exploiting airtime among other spectrum users (Hampel et al.,2019;Lu et al.,2019).Traditionally,there exist two sharing mechanisms in the unlicensed spectrum,i.e.,the listen-before-talk (LBT) mode and the underlying cognitive radio (CR) mode.LBT is contentionbased spectrum access and is not suitable for delaysensitive IIoT applications.Although the underlying CR mode avoids the delay due to spectrum contention,it requires IIoT devices to restrict the radiated interference power towards the primary users(PUs).

To address both the information security and the interference mitigation issues,we resort to the cooperative multi-input multi-output(C-MIMO)transmission technique for underlaid IIoT systems.As shown in Fig.1,several IIoT sensors inside the factory building form a cooperative cluster and jointly transmit to the remote data collector.The transmissions are subject to secrecy constraints that prevent eavesdroppers outside the building from intercepting confidential messages,and also to the interference constraint that avoids excessive interference toward PUs.

Fig.1 Diagram of the underlaid secrecy C-MIMO with the location-constrained multiple eavesdroppers in an IIoT scenario (C-MIMO: cooperative multi-input multi-output;IIoT: Industrial Internet of Things)

1.2 Related works

Physical layer security (PLS) techniques guarantee information security from an informationtheoretic perspective,where the wireless transmission is between the legitimate transmitter (Alice)and receiver(Bob),but overheard by the eavesdropper(Eve).PLS is a classical model,where communication security is realized by proper coset coding at Alice,so that the mutual information between Alice and Eve is zero (Wyner,1975).With this secrecy constraint,the limited information rate between Alice and Bob is represented by the secrecy capacity,which is given by the difference between the Shannon rates of wireless channels from Alice to Bob and from Alice to Eve(Csiszar and K?rner,1978).To this end,Alice has to obtain the knowledge of both channels to adapt its coding scheme and coding rate.Therefore,the knowledge of channel state information (CSI) is crucial in designing secrecy transmissions.However,in realistic systems,a covert eavesdropper behaves as a receiver and does not transmit any signal.Alice cannot obtain the realizations of the eavesdropping channel by measuring the transmission from Eve.

To guarantee perfect secrecy with partial or no channel knowledge of Eve,artificial noise (AN) injection by Alice has been considered(Goel and Negi,2008;Zhou and McKay,2010;Zhu Y et al.,2013).With AN injection,the total transmit power is split between the confidential signal and the AN,and AN is transmitted in the null-space of the channel between Alice and Bob.He and Yener(2014)presented codebook construction and the related secrecy rate,where the signal and AN were superimposed in the same vector space.In Goel and Negi (2008),Zhou and McKay(2010),Zhu Y et al.(2013),and He and Yener (2014),the lower bound of the secrecy rate was obtained by ignoring the thermal noise at Eve,which is equivalent to the case in which Eve can be anywhere in the network.In some communication scenarios,this is an overly conservative assumption that leads to a pessimistic estimation of the secrecy rate.Although information security can be guaranteed by deducing the previously referenced lower bound of the achievable secrecy rate,it sacrifices the efficiency of scarce spectrum resources in the IIoT scenario.In Zhang et al.(2013),Zheng TX et al.(2015),and Deng et al.(2016),a certain secrecy outage probability was allowed to improve the information rate at the expense of the compromised secrecy constraint.In Wang et al.(2016),the secrecy rate was evaluated for a large distributed antenna system,where Alice was composed of distributed antennas and Eve had a single fixed location known a priori.Therein,the secrecy rate maximization was recast into a max-min problem using results from the random matrix theory and solved by the iterative block coordinate descent algorithm.However,the adopted asymptotic analysis was valid only when the number of antennas at Alice,Bob,and Eve approached infinity.

In Sibomana et al.(2015),Zhu FC and Yao(2016),and Hu et al.(2018),PLS techniques were investigated for CR systems,assuming a pair of PU transceivers and a pair of secondary user transceivers coexisting with a malicious eavesdropper.Therein,the transmitters sent their own confidential messages to intended receivers in case of being overheard by eavesdroppers,and kept the mutual interference below a tolerable threshold.To this end,different interference metrics were chosen as the subjection for this secrecy rate optimization problem,e.g.,the outage probability constraint for the PUs (Sibomana et al.,2015),interference power constraint for the PUs (Hu et al.,2018),and lower-bound signal-tointerference-plus-noise ratio (SINR) constraint for the PUs along with the upper-bound SINR constraint for eavesdroppers (Zhu FC and Yao,2016).In addition,prior knowledge of CSI is crucial for PLS technology.In Pei et al.(2010),the underlaid secrecy rate for a multi-input single-output singleeavesdropper (MISOSE) wiretap channel was deduced assuming that perfect CSI of all users was known at Alice.The worst-case secrecy rate for a multi-input multi-output multi-eavesdropper(MIMOME)wiretap channel,assuming that perfect CSI of all users was known at Alice,was derived in Fang et al.(2016) and Hu et al.(2018).Previous works have focused mainly on the secrecy capacity under the perfect assumption of the eavesdropping channels,which would lead to an optimistic estimation of the secrecy rate and cause an information leakage issue in practical IIoT systems.

1.3 Our contributions

Due to the passive nature of the eavesdropping devices,only partial or no channel knowledge can be harnessed to design and optimize the considered secrecy communications of the underlaid CR systems.Specifically,a novel PLS optimization framework is investigated assuming the constrained eavesdropper’s location,which potentially improves the achieved secrecy rate while leveraging the practical location constraint.To further disrupt the information received at Eve,the confidential signal and AN are jointly precoded in the eigenspace of the channel matrix and randomly superimposed at Eve,while satisfying the interference constraint at the PU.Our main contributions are summarized as follows:

1.We consider a realistic underlaid secrecy CR system,where the C-MIMO transceivers coexist with a PU and the eavesdroppers can appear at multiple possible locations in the network.This is useful to relax the “anywhere Eve” assumption adopted in other works(Goel and Negi,2008;Zhou and McKay,2010;Zhu Y et al.,2013;He and Yener,2014),and the considered framework can take into account the practical constraints of the system topology.Specifically,if there exists an area around Alice guaranteed to be free of an eavesdropper,the signal and AN can be optimized assuming that Eve is outside the eavesdropper-free region and is located at an arbitrary finite number of sampled locations.

2.We propose an eigenspace-adaptive precoding (EAP) method for the underlaid C-MIMO system by jointly designing the signal and the AN.The signal and the AN are adaptively transmitted in the eigenspace of the main channel or the null-space of the interference channel according to the interference power constraint at the PU and the CSI of the legitimate channel.Depending on the selected eigenspace,the secrecy rate maximization is re-formulated as two canonical difference convex (CDC) problems,which are then solved by an iterative outer approximation algorithm,where the required average mutual information between Alice and Eve is given by closedform approximation.

3.To adapt to the low-powered and massively populated device scenario,we simplify the EAP method by adopting uniform power allocation.Specifically,when the null-space AN injection is considered,the original power allocation problem is simplified into a two-variable optimization problem.It is solved by an iterative power allocation algorithm from Lin et al.(2013),which achieves almost the same performance as a brute-force search with much lower complexity.When the null-space of the PU’s channel is considered,the orthogonal subspace projection method is adopted to improve the information rate of the legitimate channel by aligning the sub-null-space of the PU channel with the eigenspace of the main channel,which is also solved by the iterative power allocation algorithm.

2 System model

As shown in Fig.2,we consider a secondary C-MIMO system betweenKclustered transmit nodes and a receive node,where the transmit cluster has a head node at the center and the cluster radius isr.The transmit cluster is referred to as Alice and the receiver is referred to as Bob,where each transmit node is equipped with a single antenna and the receive node is equipped withNBantennas.The distance between the head of Alice and Bob is denoted byb0.The secret messages from the transmit cluster can be intercepted by Eve,which is equipped withNEantenna elements(NE≥1)and is located outside a predefined eavesdropper-free region.In addition,we assume that a PU (PU refers to the PU receiver mentioned in the introduction,that is,the receiver of the primary user) equipped withNPreceive antennas at a distance ofp0from the head of Alice can overhear the underlaid transmissions,which is regarded as an unnecessary interference toward the PU.

Fig.2 Mathematical model of a C-MIMO underlaid CR system with a location-restricted region for Eve around transmitters (C-MIMO: cooperative multiinput multi-output;CR: cognitive radio).The possible locations of Eve are along the contour of the region

The transmit head node is responsible for precoding the information and conveys the encoded signals to the transmit cluster members.Similar to Zheng Z and Haas (2017),we assume that the necessary cooperation signaling is perfectly exchanged among transmitters without delay.The fundamental limit of the information rate between the transmit cluster and the receiver is given by the Shannon rate of the distributed MIMO channel(Ozgur et al.,2013).In addition,we focus on the physical layer security of the communication between Alice and Bob,while we assume that the security of the intracluster cooperation within Alice can be guaranteed relatively easily.This is because the communication links have a shorter range and therefore could provide higher channel capacity.

2.1 Channel model

Consider a transmit vectorx=[x1,x2,...,xK]T,wherexk(k=1,2,...,K) denotes the transmitted symbol from thekthtransmit node of Alice.The receive vectors of Bob,PU,and Eve at theithlocation are denoted asy=[y1,y2,...,yNB]T,u=respectively,whereyn,un,andzi,ndenote the receive symbols at thenthreceive antenna of Bob,PU,and theithEve,respectively.The vectorsy,u,andziare expressed as

whereLis the number of possible locations of Eve.The matrix entryHn,kon thenthrow andkthcolumn ofHdenotes the channel coefficient between thekthtransmitter and thenthreceiver,i.e.,similarly,As the antenna elements at Bob are co-located,the legitimate channel between Alice and Bob can be written as

whereLdenotes the fast fading coefficients,modeled as an independent and identically distributed(i.i.d.) standard complex Gaussian random matrix,i.e.,L～CN(0,I).TheK × Kdiagonal matrixΩ=diag(ω1,ω2,...,ωK)denotes the average channel gains with thekthdiagonal entry being

wherebkdenotes the distance between thekthtransmitter of Alice and Bob,αis the path-loss exponent,andcpis the path loss at the unit distance.Similarly,the channel between Alice and Eve at theithlocation can be written as

whereWi～CN(0,I).TheK×Kdiagonal matrixΣi=diag(σi,1,σi,2,...,σi,K) denotes the average channel gains with thekthdiagonal entry being

whereei,kdenotes the distance between thekthtransmitter and theithEve.The channel matrix between Alice and the PU is represented as

whereMis a standard complex Gaussian random matrix and the diagonal matrixΘ=diag(θ1,θ2,...,θK) denotes the average channel gains with thekthdiagonal entry being

wherepkdenotes the distance between thekthtransmitter and the PU.In addition,cpandαare assumed to be the same as those of Eve’s channel.The additive noisesnB,nP,andnEat Bob,PU,and Eve are modeled as i.i.d.complex Gaussian vectors with normalized power,i.e.,nB～CN(0,),nP～CN(0,),andnE～I(xiàn)n this study,we assume thatH,Fi,andGall follow block fading processes.In addition,the instantaneous CSI ofHis known by Alice and Bob,while the instantaneous CSI ofGis known by Alice and the PU.However,only the statistical CSI ofFi,depending on the number of receive antennas at Eve and the location of Eve,could be acquired by Alice by evaluating the worst-case scenario;i.e.,NEis set to the maximum number of antennas allowed at Eve and Eve’s locations are along the contour of the eavesdropper-free region,as shown in Fig.2.

2.2 Signal model with artificial noise injection

The wiretap channel (Eqs.(1)–(3)) with multiple possible locations of Eve can be modeled as the compound wiretap channel(Bloch and Laneman,2013).With CSI available at the receivers,the following secrecy rate is achievable:

where [x]+=xforx ≥0 and 0 otherwise.We denoteI(a;b1,b2)as the mutual information between the random variableaand the random variablesb1,b2.The outer optimization in Eq.(10) is taken overp(x),the probability distribution ofx.By following similar arguments in Lin et al.(2013),the mutual informationI(x;zi,Fi)is calculated as

where the first equality is due to the chain rule of mutual information,and the second equality is obtained because the block fading channelFiis independent ofx.

To achieve rate maximization through optimizing the probability distributionp(x) is still an open problem for generic MIMO wiretap channels.To proceed,we adopt the widely used Gaussian signaling applied in other works (Goel and Negi,2008;Zhou and McKay,2010;Zhang et al.,2013;Zhu Y et al.,2013;He and Yener,2014;Zheng TX et al.,2015;Wang et al.,2016),wherexis a multivariate Gaussian vector.In addition,to obscure the information reception by Eve,AN is injected by Alice along with the information symbols.The transmitted symbolxis the sum of the precoded information and AN,i.e.,

wheres～CN(0,IK) anda～CN(0,IK) are the information symbols and AN,respectively.The nonnegative diagonal matrixΨx0 (x ∈{s,a}) denotes the power allocation,where thekthdiagonal elementψx,k ≥0 is the power allocated to thekthtransmit antenna.Moreover,to reflect the lowpower characteristic of the considered IIoT scene,we set 0≤≤Γs(k=1,2,...,K),whereΨp=VsΨsV?s +VaΨaV?a (?denotes the conjugate transpose operation for a matrix)andΓsdenotes the maximum transmit power of a single antenna.The spatial precoding matrixVx(x ∈{s,a}) maps the symbols to the transmit antennas.To improve the worst-case secrecy rate under the constraint of the total power at Alice and the interference power at the PU,we design the transmit signal to be able to flexibly choose the underlying eigenspace according to the problem constraints,i.e.,the EAP method,where the precoder is selected from the following two sets of eigenvectors:

1.Eigenspace precodingVH

LetVs=Va=VHand the columns ofVHare the right singular vectors of the main channelH;i.e.,Hhas the singular value decompositionH=Note that the null-space AN injection in other works (Goel and Negi,2008;Zhou and McKay,2010;Zhang et al.,2013;Zhu Y et al.,2013;Zheng TX et al.,2015)can be viewed as a special case of Eq.(12),whereVsandVaare chosen as the orthogonal sub-spaces ofVH,i.e.,=0.Therefore,the precoder structure(Eq.(12))is more general than the null-space AN injection,which can be realized by Eq.(12)by properly setting the power allocation variablesψx,k=0.

2.Null-space precodingVG

To avoid the interference toward the PU,the signal and AN can be precoded in the null-space of the interference channelG,i.e.,Vs=Va=VG,whereGVG=0.The columns ofVGcan be selected as the right singular vectors of the interference channel corresponding to the zero singular values.Note that this precoding scheme further requires that the number of transmitters be larger than the number of antennas at the PU,i.e.,K ≥NP.

Accordingly,following the above two eigenvector precoding schemes,the transmit signal of the proposed EAP can be written as

whererdenotes the number of zero singular values of the interference channelG.According to Eq.(13),we can improve the secrecy rate of the underlaid secrecy CR system by jointly optimizing the power allocation vectors and the eigenspace selector.The eigenspace selection will be specifically discussed in Section 3.2.

2.3 Problem formulation

According to Eq.(10),the secrecy rate under a pair of fixedΨsandΨacan be explicitly written as

whereRBandRE,iare the information rates of the main and theitheavesdropping channel,respectively.As the main channel (Eq.(1)) and the eavesdropping channel(Eq.(3))are Gaussian MIMO channels,these information rates with AN injection follow the well-known MIMO channel capacity in Chiurtu et al.(2001)as

By denotingXas a multi-column selection matrix andYas aK ×Kpositive semi-definite Hermitian matrix,the information rate of Bob(fB)and that of Eve (fE,i)are separately defined as

To balance the underlaid secrecy rate and the interference at the PU,the objective function is designed to maximize the worst-case secrecy rate in the underlaid communication system while subject to both the total power constraint at Alice and the interference power constraint at the PU.We optimize the worst-case underlaid secrecy rate by jointly optimizing the power allocation vectors of information symbols and AN,and the eigenspace selector of the precoder,i.e.,{S,Ψs,Ψa}.Accordingly,the optimization problem in our proposed underlaid secrecy CR network can be written as

whereΨp=and tr(·) denotes the trace of a matrix.Constraint (20b) is due to the maximum transmit powerΓsof a single antenna imposed by Alice,and constraint(20c)is due to the fact that the maximum interference received at the PU should be below a given thresholdΓI.

Compared with Eq.(10),the secrecy rate(Eq.(20))is suboptimal due to the Gaussian signaling and the specific precoder structure (Eq.(13)).However,optimization problem (20) can be solved relatively easily and the numerical results in Section 6 will show that substantial secrecy rates can still be achieved.In the next section,we present an approximate closed-form expression for the average ratesRE,iand the optimization framework to solve problem(20).

3 Secrecy rate maximization under eigenspace-adaptive precoding

3.1 Approximation of the ergodic information rate for eavesdropping channels

In the literature,the functionfE(Ψ) is the ergodic capacity of Rayleigh MIMO channels with transmitter-side correlationT=Note that we ignore the subindexiwhenever it is clear from the context.The expression offE(Ψ) (e.g.,Eq.(123) in Simon et al.(2006)) depends on the eigenvalues ofT.For a matrix with arbitrary dimension,there does not exist a closed-form expression of its eigenvalues,so one must resort to numerical and iterative routines.WhenfE(·) is used in optimization problem (20),the relevant algorithm typically starts from an initial valueT0and approaches the optimal solution via a sequence of iterations,sayingT1,T2,...,Ta.Therefore,one must calculate all the eigenvalues ofT0,T1,...,Ta,which requires prohibitively more computational resources.

To address this issue,we approximate the information rate between Alice and Eve as

Here,Zis a diagonal matrix,Φis a random unitary matrix,and each instance ofΦis drawn uniformly and randomly from the Haar measure (Sternberg,1995).The expectation in Eq.(22)is taken over bothWandΦ.ComparingwithfE(·) in Eq.(19),we replace the unitary matrixwith a random Haar unitary matrixΦ,and apply Jensen’s inequality.

Fig.3 compares the approximate mutual informationbetween Alice and theithEve calculated in Eq.(21) with the exact mutual informationRE,iin Eq.(17) withK=N=4 andM=2,using randomly chosen power allocationsΨsandΨa.The locations of transmitters are as shown in Fig.2 and the location of theithEve is set to(ei,0).By increasing the cluster radiusr,the radial distances of the transmitters are proportionally increased.We generate 104realizations of the precoding matrixVH,and plot one standard deviation above and below the expectation ofRE,iaveraged overVH.Fig.3 shows that approximation (21) tends to over-estimate the exact rateRE,i,which is favorable in the context of secrecy communications because it prevents setting a code rate higher than the achievable rate obtained by Eq.(20).Additionally,because the legitimate channel given by Eq.(4) and the PU channel given by Eq.(8)have the same formulation,the numerical simulations of the exact rateRE,iby using the precoderVGare therefore identical to those ofRE,iby using the precodingVH,if these two channels have the same configuration.Therefore,we omit the comparison betweenRE,iachieved by theVGprecoding and

Fig.3 Comparison of approximation (21) with Eq.(17) using randomly generated power allocations Ψs and Ψa

3.2 Eigenspace-adaptive precoding architecture and the decomposed problem formulation

According to Eq.(22),the information rate of the wiretap channel is approximated by substituting the channel eigenspace vectorVxby a random unitary matrixΦ;thus,the precodersVHandVGcan achieve the same approximated information rate at the eavesdroppers.Accordingly,substitutingRE,iin Eq.(15) by Eq.(21),it is observed that the secrecy rateRsof the C-MIMO system achieved by precodersVHandVGcan be simply analyzed by only comparing theRBachieved by theVHandVGprecoding methods.

Inspired by the observations above,we propose an EAP architecture,in which the original optimization problem (20) is decomposed into two sub-problems,i.e.,the optimization of eigenspace selectorSand the optimization of the power allocation vectors{Ψs,Ψa}.First,we solve theSoptimizing sub-problem by fixingΨsandΨa.Herein,Ψa=0 andΨs=Ψwfare given,whereΨwfis the solution of the water-filling power allocation algorithm by optimizingΨsinRBonly,i.e.,Ψwf=arg maxΨs0RB(S,Ψs,0),subject to tr(Ψs)≤Γs.Therefore,theS-optimizing sub-problem can be written as

4 Difference convex program and iterative outer approximation method

The first term is independent of the indexiand can be pulled out of the maximization in Eq.(39),thus becoming

As maximization and summation of a finite number of functions are convex-preserving,bothp(x)(·) andq(x)(·) are convex.Next,we introduce the auxiliary variablestandssuch that

wheret ≤0 andsis real.Inequality (44) can be rewritten as the following system of inequalities:

Comparing inequality (43) with Eq.(40),instead of maximizingdirectly,we can alternatively minimize the auxiliary variablet.Combining the inequality constraints(45)with the power constraint in inequality (32b) and the interference constraint in inequality (32c),while assuming eigenspace precoderVH,the secrecy rate maximization (32),withreplaced by its approximationis equivalent to the CDC program(Horst and Tuy,1996)given as follows:

On the other hand,using the null-space precoderVG,the secrecy rate optimization problem becomes

where the tuplew={Ψa,Ψs,s,t}and we denotetw ≡t.

Both problems (46) and (47) can be solved by the iterative outer approximation method (Horst and Tuy,1996),as collectively outlined in Algorithm 1.For notational convenience,we denote the setsH(x)={w:h(x)(w)≤0},G(x)={w:g(x)(w)≥0},and?G(x)={w:g(x)(w)=0}as the boundary ofG(x).The boundary?G(x)can be determined by linear interpolation between an inner pointx(x ∈G(x)) and an outer pointv(vG(x)),whereg(x)(x)＞0,g(x)(v)＜0,andtv ＜min{tw:w ∈H(x)∩G(x)}.An example of such an outer pointvcan be found by settingΨs=Ψwf,Ψa=0,and

Denoteπ(x)=vx+(1-v)v(0＜v ＜1)as the intersection point between the line segment [x,v] and the boundary?G(x),i.e.,g(x)(π(x))=0.Becauseg(x)(·) is convex,π(x) can be uniquely obtained by an univariate convex minimization min{v:π(x)∈G(x)}.In the initialization of Algorithm 1,the inner pointw0can be determined by running an algorithm to solve the convex maximization problem max{g(x)(x) :x ∈H(x)}until a feasible pointx0is found,if it exists.We then set the initial statew0=π(x0).

Because the monomialt ≤0 andg(x)(w)≥0(Horst and Tuy,1996,Lemma X.2),the CDC problems(46)and(47)are stable in the sense of Def.X.1 in Horst and Tuy(1996).Therefore,the convergence of Algorithm 1 can be easily guaranteed(Horst and Tuy,1996,Prop.X.3).Note that Algorithm 1 may require infinite iterations to converge to the global optimal solution.Therefore,a relaxation parameter∈≥0 is introduced (line 5),which provides a tradeoffbetween optimality and complexity.Specifically,∈=0 corresponds to the global optimal solution of problem(46)or(47).

In addition,we present the computational complexity analysis of the presented iterative outer approximation method.The complexity is composed of two parts,i.e.,the complexity of solving the subproblem(line 4)and the complexity of acquiring the optimal interpolation factorv(line 6).Because the sub-problem (line 4) has been verified to be convex,it is solved using the inner point method (IPM) in the CVX toolbox.According to the arithmetic complexity of the linear programming by IPM shown in Ben-Tal and Nemirovski (2001),the complexity of the power allocation sub-problem (line 4) is scaled asO（(3K+6)3/2(2K+2)2）,whereKis the number of transmitting nodes of Alice.Similarly,the complexity of solving the optimal interpolation subproblem(line 6)isO（(m+n)3/2n2）withn=1 andm=1,which can be neglected.We denoteTas the number of iterations of Algorithm 1 under the relaxation parameter∈.Therefore,while retaining only the highest-order term,the overall complexity of solving this CDC problem can be scaled asO（T K3.5）.

5 Underlaid secure precoding for the large-dimensional C-MIMO system

As we all know,the large number of connections is one of the crucial characteristics of IIoT.However,the complexity of the proposed EAP method exponentially increases with the increase of the transmit nodes,which is not adaptive to the large-dimensional scenario.Therefore,we further consider designing a simplified version of the EAP method for the largedimensional underlaid C-MIMO system by adopting uniform power allocation,where the precoder is selected from the two sets of eigenvectors:

1.Null-space AN injection precoding

Therefore,optimization problem (32) can be reformulated as

Therefore,optimization problem (38) can be reformulated as

Although we have simplified optimization problems (32) and (38) by adopting the fixed precoding and uniform power allocation method,it is observed that optimization problems (51) and (55) are still non-convex stochastic ones,for which it is difficult to determine an optimal analytical solution.Instead,we adopt an iterative power allocation algorithm from Lin et al.(2013),which achieves almost the same performance as brute-force search with much lower complexity.

6 Numerical results

In this section,we study the achievable average secrecy rate of the C-MIMO system under both the single antenna power constraint and the interference power constraint toward the PU.A guaranteed minimum distancee0is introduced between Alice’s head node and any possible location of Eve.To simplify this system,let all the possible locations of Eve be evenly placed on a circle with radiuse0centered at the transmit head node;i.e.,in Eq.(7) we sete1,1=e1,2=···=eL,1=e0.We first simulate the EAP method in a scenario where the legitimate channel does not have the null-space(case 1),i.e.,K=4,NB=4,NE=2,andNP=2.Then we assume that the legitimate channel has the orthogonal null-space(case 2),i.e.,K=4,NB=2,NE=2,andNP=2.In cases 1 and 2,the single antenna power constraint isΓs=23 dBm.Additionally,the performance of the low-powered and large-dimensional C-MIMO system is evaluated (case 3),i.e.,K=64,NB=4,NE=2,andNP=2.In case 3,the single antenna power constraint isΓs=13 dBm.

Fig.4 shows the average secrecy rate of the CMIMO system achieved by theVH,VG,and EAP methods with the AN injection in case 1 whene0=10 m,r=3 m.It is observed that the proposed EAP method outperforms both theVHandVGmethods.Specifically,when the interference constraint dominates the problem,e.g.,ΓI=-25 dBm,the EAP method can improve the secrecy rate by about 0.25 nats/(s·Hz)compared to theVHmethod.Meanwhile,when the single antenna power constraintΓsdominates the problem,e.g.,Γs=23 dBm andΓI=-15 dBm,the EAP method can improve the secrecy rate by about 1.2 nats/(s·Hz) compared to theVGmethod.Therefore,the proposed EAP method can adaptively choose an eigenspace that achieves a better secrecy rate than theVHandVGmethods.

Fig.4 Average secrecy rate of the C-MIMO system via the VH, VG,and EAP schemes with AN injection under different interference power constraints ΓI when K=4, NB=4, NE=2, NP=2, r=3 m, e0=10 m,and Γs=23 dBm (AN: artificial noise;C-MIMO: cooperative multi-input multi-output;EAP:eigenspaceadaptive precoding)

Fig.5 shows the cumulative distribution functions (CDFs) of the secrecy rates of the C-MIMO system via the EAP method in case 1 whenr=3 m ande0=10 m.The interference power constraint isΓI=-25,-20,and-15 dBm.Results show that the EAP method can eliminate the secrecy outage under different interference constraints,and that the gain benefits from the AN injection are limited when the interference power is rigorously constrained.Additionally,the fluctuation caused by random propagation fading increases with the relaxation of the interference power constraint.

Fig.5 CDFs of the C-MIMO system secrecy rate via EAP when K=4, NB=4, NE=2, NP=2, r=3 m,e0=10 m,and Γs=23 dBm (CDFs: cumulative distribution functions;C-MIMO: cooperative multi-input multi-output;EAP: eigenspace-adaptive precoding)

Fig.6 shows the CDFs of the secrecy rates via the EAP method in case 1 whenr=3 m,ΓI=-15 dBm,ande0=6,8,and 10 m.Results show that with the increase ofe0,the secrecy rate increases and the performance gain of AN gradually decreases.Moreover,the EAP method with AN injection can fully eliminate the secrecy outage even when the eavesdroppers are distributed on a circle around the transmit cluster with a radius ofe0=6 m.

Fig.6 CDFs of the C-MIMO system secrecy rate via EAP when K=4, NB=4, NE=2, NP=2, r=3 m,Γs=23 dBm,and ΓI=-15 dBm (CDFs: cumulative distribution functions;C-MIMO: cooperative multi-input multi-output;EAP: eigenspace-adaptive precoding)

Fig.7 shows the CDFs of the secrecy rates via the EAP method in case 1 whene0=10 m,ΓI=-15 dBm andr=1,3,and 5 m.Results indicate that the EAP method can eliminate the secrecy outage even whenr=1 m;i.e.,the cooperative nodes are densely packed together.To further clarify the effect of the distributed node topology and Eve’s distribution on the secrecy rate,Fig.8 shows the average secrecy rates under different configurations ofrande0,showing that the secrecy rate increases with the increase of the cluster radiusr.Furthermore,it is observed that AN injection achieves the maximum gain atr=5 m ande0=8 m,which demonstrates that AN injection can achieve better secrecy gain when the eavesdropper is closer to the distributed nodes.

Fig.7 CDFs of the C-MIMO system secrecy rate via EAP when K=4, NB=4, NE=2, NP=2, e0=10 m,Γs=23 dBm,and ΓI=-15 dBm (CDFs: cumulative distribution functions;C-MIMO: cooperative multi-input multi-output;EAP: eigenspace-adaptive precoding)

Fig.8 Average secrecy rate of the C-MIMO system via EAP under different distributed radius r and different eavesdropper-free region e0,when K=4, NB=4, NE=2, NP=2, Γs=23 dBm,and ΓI=-15 dBm (C-MIMO: cooperative multi-input multi-output;EAP: eigenspace-adaptive precoding)

Fig.9 shows the average secrecy rates via theVH,VG,EAP,and generalized AN-aided precoding(Lin et al.,2013) methods in case 2.In Lin et al.(2013),assuming that the perfect CSI of the legitimate channel and only the statistics of the eavesdropper’s channel are known at the transmitter,the optimal structure of the precoding is derived.Therein,the power of AN is divided into two parts;one is injected into the same eigenspace as the signal and the other is uniformly injected into the null-space of the legitimate channelH.Results show that the proposed EAP method outperforms the other methods when the interference power constraint dominates the problem and achieves almost the same performance as the optimally structured AN-aided precoding method(Lin et al.,2013)when the single antenna power constraint dominates the problem.

Fig.9 Average secrecy rate of C-MIMO system via different AN-aided precoding methods when K=4,NB=2, NE=2, NP=2, r=3 m, e0=10 m, Γs=23 dBm,and ΓI=-15 dBm (AN: artificial noise;C-MIMO: cooperative multi-input multi-output)

Fig.10 shows the average secrecy rates via theand EAP methods in case 3,i.e.,the low-powered and large-dimensional C-MIMO system.Therein,the single antenna power constraint is set toΓs=13 dBm and the interference constraintΓIis set from-15 to-5 dBm.Results indicate that the large-dimensional C-MIMO can significantly increase the secrecy rate in this underlaid CR system.Similarly,the proposed simplified EAP method for the large-dimensional C-MIMO system can adapt to the more superior eigenspace and achieve a better secrecy rate than both themethods.

Fig.10 Average secrecy rate of the large-dimensional C-MIMO system via , ,and the simplifeid EAP methods when K=64, NB=4, NE=2, NP=2,r=3 m, e0=10 m,and Γs=13 dBm (C-MIMO: cooperative multi-input multi-output;EAP: eigenspaceadaptive precoding)

7 Conclusions

EAP together with AN-assisted secrecy transmission is considered for a C-MIMO system coexisting with a PU.The design of underlaid secrecy communications exploits the geographical location constraint of the eavesdropper as well as the eigenspace of the channels.Specifically,the eigenvectors are adaptively selected by the transmitter according to the channel conditions.Also,a simplified EAP method is proposed for the large-dimensional C-MIMO system.Numerical results show that the proposed EAP method outperforms the fixed eigenvector precoding method.Moreover,EAP can eliminate the secrecy outage even when the eavesdroppers are located closer to the transmitter.In addition,the simplified EAP method for large-dimensional CMIMO transmission can significantly improve the secrecy rate with low complexity.

Contributors

Xinyao WANG and Zhong ZHENG designed the research.Xuyan BAO and Yuzhen HUANG processed the data.Xinyao WANG and Zhong ZHENG drafted the paper.Zesong FEI helped organize the paper.Xuyan BAO,Yuzhen HUANG,and Zesong FEI revised and finalized the paper.

Compliance with ethics guidelines

Xinyao WANG,Xuyan BAO,Yuzhen HUANG,Zhong ZHENG,and Zesong FEI declare that they have no conflict of interest.

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.