Multi-objective robust secure beamforming for cognitive satellite and UAV networks

WANG Zining ,LIN Min,* ,TANG Xiaogang ,GUO Kefeng ,HUANG Shuo ,and CHENG Ming

1.School of Telecommunications and Information Engineering,Nanjing University of Posts and Telecommunications,Nanjing 210003,China;2.School of Space Information,Space Engineering University,Beijing 101407,China;3.Shanghai Aerospace Electronic Technology Institute,Shanghai 201109,China

Abstract:A multi-objective optimization based robust beamforming (BF) scheme is proposed to realize secure transmission in a cognitive satellite and unmanned aerial vehicle (UAV) network.Since the satellite network coexists with the UAV network,we first consider both achievable secrecy rate maximization and total transmit power minimization,and formulate a multi-objective optimization problem (MOOP) using the weighted Tchebycheff approach.Then,by supposing that only imperfect channel state information based on the angular information is available,we propose a method combining angular discretization with Taylor approximation to transform the non-convex objective function and constraints to the convex ones.Next,we adopt semi-definite programming together with randomization technology to solve the original MOOP and obtain the BF weight vector.Finally,simulation results illustrate that the Pareto optimal trade-off can be achieved,and the superiority of our proposed scheme is confirmed by comparing with the existing BF schemes.

Keywords:cognitive satellite and unmanned aerial vehicle network (CSUN),multi-objective optimization,robust secure beamforming (BF),weighted Tchebycheff approach.

1.Introduction

Recently,application of satellite communication (SatCom)in future wireless systems has attracted significant attention,since it can provide ubiquitous connections and high quality service for users all over the world [1−4].Meanwhile,compared with SatCom,unmanned aerial vehicles(UAVs) has the advantages of adjusting their altitudes conveniently and avoiding obstacles to establish line-ofsight (LoS) communication links,so that the quality of communication links can be improved efficiently. As such,UAV technology is developing rapidly,which has been widely used in many fields,such as surveillance and monitoring,and emergency communications [5−9].

To take advantages of both SatCom and UAV communications,the 3rd Generation Partnership Project (3GPP)has proposed to use both satellites and UAVs as integrated access platforms for the next generation wireless communications [10].However,the scarcity of spectrum resources has become a bottleneck to further enhance the capacity of wireless systems.To address this issue,the concept of cognitive satellite and UAV networks (CSUNs)that utilizes the cognitive radio (CR) to realize the spectrum coexistence of both SatCom and UAV networks has been introduced in [11−13] and attracted increasing attention.Specifically,the authors of [11] investigated the multi-domain resource allocation to improve the efficiency of massive access for CSUNs.In [13],the authors proposed a cooperative beamforming (BF) scheme to achieve energy-efficient communication in CSUNs.

Besides realizing the spectrum coexistence in CSUNs,another urgent issue is the privacy and security,which is due to the broadcasting nature of wireless communications.In this regard,the cryptographic encryption method is commonly used at the upper layer to achieve secure transmission.However,this security scheme faces great challenges with the rapid development of computational capability,especially the application of quantum computing.By exploiting signal processing and the randomness of wireless channels,the physical layer security (PLS) is considered as a promising approach to achieve secure communication.Since BF technology is capable of both enhancing the received signal at the legitimate users and suppressing the signal to eavesdroppers (Eves),the employment of BF in PLS to improve the secrecy performance of the system has become a new research hotspot[14−16].For example,in [14],the authors proposed a BF method using semi-definite programming (SDP) combined with one-dimension search to maximize the sum secrecy rate in multibeam SatCom.Considering the scenarios of single Eve and multiple Eves,respectively,the authors of [15] proposed two joint BF schemes to improve the security of integrated satellite terrestrial networks.It is worth mentioning that the previous work in[14] and [15] has assumed that perfect channel state information (CSI) is available at the transmitter,which is unrealistic in some cases due to the estimation error,mobility of terminals and limited feedback channels [17].Thereby,the authors of [18] proposed a robust BF scheme to maximize the secrecy rate for a two-tier heterogeneous cellular network,where theS-procedure is adopted to transform quadratic matrix inequality constraints into linear matrix inequality.In [19],the authors investigated the robust secure BF and power splitting for cognitive satellite and terrestrial networks (CSTNs).

Although the aforementioned works have conducted indepth investigations on PLS in wireless communications,they only focused on the single performance criterion,such as the transmit power or the achievable secrecy rate(ASR).Actually,the next generation wireless network aims at providing a variety of services with high transmission rates,high reliability,low energy consumption and low latency.In this context,the multi-objective optimization problem (MOOP) implementing the trade-off between multiple objectives has attracted much interest in recent years [20−22].In [20],the MOOP about the tradeoff between the total transmit power and the energy harvesting efficiency was investigated for a multiuser multiple-input single-output downlink system.The authors of[21] investigated the MOOP through the modified weighted Tchebycheff approach.In [22],the authors adopted semidefinite programming relaxation to solve the MOOP about total downlink and uplink transmit power minimization.Besides,the MOOP for secure beamforming design of CSTNs was studied in [23],where the perfect CSI was used to conduct the optimization design.The most existing multi-objective research focuses on the system design in terrestrial networks or CSTNs,whereas the application of the multiple objective problem in CSUNs is still an open yet challenging topic that needs to be studied further.

Motivated by these observations,we propose a multiobjective optimization based robust BF scheme under the condition of imperfect CSI to achieve the trade-off between the ASR and the total transmit power in CSUNs.In summary,the contributions of this paper can be listed as follows:

(i) We present a framework to implement PLS in the CSUNs,where the satellite is equipped with array fed reflector antennas and the UAV employs uniform planar arrays (UPAs).With the help of CR technology,the satellite network termed as the primary network shares spectrum resources with the UAV network termed as the secondary network.In this context,it is urgent to investigate new BF schemes that should consider multiple objectives in wireless communications.

(ii) We adopt the weighted Tchebycheff approach to implement the trade-off between the ASR and the total transmit power,while the total transmit power budget and the quality-of-service requirements of secondary users(SUs) and primary users (PUs) are chosen as the constraints.Compared with the existing work in [18] and[19],which only takes single objective optimization into account,our proposed scheme can achieve the Pareto optimal trade-off between the multiple objectives [24].

(iii) Considering that only the imperfect CSI based on angular information of PUs and Eves is known,we propose a method combining angular discretization with Taylor approximation to transform the non-convex objective function and constraints to the convex ones and obtain the BF weight vector through SDP joint with a randomization technology.Compared with other existing methods,our proposed method can offer a global optimal solution with a fast convergence.We utilize the imperfect CSI to perform robust BF design,which is more realistic than BF schemes based on perfect CSI in previous works,such as [21] and [23].

The rest of this paper is organized as follows:Section 2 introduces the system model and formulates the MOOP.In Section 3,the robust BF scheme is proposed to solve the MOOP.Section 4 provides simulation results and discussions.Finally,we conclude our paper in Section 5.

2.System model and problem formulation

As shown inFig.1,we investigate the downlink transmission of a CSUN,where the satellite network acting as the primary network shares the spectrum resources with the UAV network termed as the secondary network.The geostationary orbit (GEO) satellite exploits multicast transmission to provide wireless services toLPUs,while the UAV transmits the private signal to an SU that will be wiretapped byKEves within its coverage.The satellite employs array fed reflector antennas withNsfeeds and the UAV is equipped with a UPA havingNuantennas.

Fig.1 System model

Similar to the related work [25],we assume that the interference from satellite to SU and Eves can be ignored owing to the large path loss of satellite links.The UAV performs transmit BF with weight vectorwand transmits messages(t) satisfying E[js(t)j2]=1 to the SU,where E[·] represents the expectation,denotes the complex space ofNu×1 and j·j represents the absolute value.After passing through the wireless channel,the received signals at the SU and thekth Eve can be,respectively,expressed as

where (·)Hdenotes the Hermitian transpose,hi(ifs,(e,k)g) denotes the channel vector from UAV to the SU or thekth Eve.ns(t) andne,k(t) denote independent and identically distributed (i.i.d) additive white Gaussian noise (AWGN) with zero mean and variancesκBT(ifs,(e,k)g),whereTdenotes the receive noise temperature,Bis the noise bandwidth and κ ≈1.38×10−23J/K represents the Boltzmanns constant.According to (1) and (2),the output signal-to-noise ratio(SNR) at the SU and thekth Eve are,respectively,given by

According to [26],the ASR of the SU can be described as

where [x]+=maxfx,0g .The equation implies that the secure communication is realized whenR>0,and the greaterR,the better secrecy performance is achieved[27].

As shown inFig.2,we assume that UAV employs UPA of dimensionNu=N1×N2to achieve a high gain with a compact size,and the UAV downlink channel consists of a predominant LoS component and multiple single-bounce non-LoS (NLoS) components due to the highly directional and quasi-optical properties of the radio wave propagation [28].

Fig.2 Geometric relation between UAV and any user

Taking the effect of path loss into account,we obtain the UAV downlink channel vectorhas

where ⊗ denotes the Kronecker product,andLndenotes the number of NLoS paths. ρ0and ρm(m=1,···,Ln) are the pass loss associated with the LoS path and themth NLoS path,respectively.The pass loss amplitude of the LoS component can be calculated as

where λ is the wavelength andd0is the transmission distance.It should be pointed out that the value of jρmj2is typically 5 dB to 10 dB lower than jρ0j2[29].In addition,ge(θ,φ) in (6) denotes the element pattern with θ and φ being the horizontal and vertical degree of arrival (DoA),respectively.According to the model introduced by International Telecommunication Union (ITU),10log10(ge(θ,φ)) with the element pattern in dB [30] can be expressed as

whereEmaxrepresents the maximum antenna gain,Srepresents the side-lobe level,andgx(θ,φ) andgy(θ,φ) denote the relative patterns inXandYplanes,which can be,respectively,expressed as

Moreover,due to spectral sharing between the satellite and the UAV,the interference from UAV to PUs is also an important factor to be considered in the system design.The interference-to-noise ratio (INR) at thelth PU can be expressed as

wherehp,ldenotes the channel vector between the UAV and thelth PU,andis the AWGN variance at thelth PU.

In addition,we assume the imperfect CSI of PUs and Eves is unknown,but the imperfect CSI based on angular information is available at the UAV,which can be expressed as

In the considered CSUN,transmit power minimization and ASR maximization are both significant design criteria.In the following,we will first investigate the single objective problem under these two criteria separately.Then,we propose a trade-off design to balance two objectives through MOOP.By using (14),we formulate the worst-case ASR maximization problem subject to the quality-of-service (QoS) requirements of SU,the INR threshold of each PU and the total transmit power budget.The first problem can be expressed as

P1 (worst-case ASR maximization):

whereRmindenotes the ASR threshold of SU,γthis the INR threshold of each PU andPmaxrepresents the total transmit power budget.

Similarly,the second problem is designed to minimize the total transmit power,which can be expressed asP2 (total transmit power minimization):

Since a high achievable ASR leads to a high transmit power consumption and vice versa,the objectives ofP1 andP2 are in conflict [31,32].In order to obtain the Pareto optimal solution between these two single-objectives,we adopt the weighted Tchebycheff approach in[33] to formulate the MOOP as

P3 (multi-objective optimization):

In the following,we will propose a robust BF scheme to solve the MOOP in (17).

3.Proposed multi-objective optimization based robust BF scheme

It can be observed that due to the continuity of CSI uncertainty sets,the constraints involve infinite inequality constraints that make problems difficult to be solved effectively.In order to overcome this predicament,we recastP1,P2 andP3 by introducing several auxiliary variables.Then,a method combining angular discretization with Taylor approximation is proposed to transform the intractable constraints into solvable forms. Finally,the MOOP can be solved efficiently.

First of all,by substituting (3)−(5) into (15),P1 can be derived as

where Tr(·) is the trace and rank(·) represents the rank of a matrix.Since the objective function of (19) is nonconvex,we introduce slack variable to replace the objective function,and remove the nonconvex rank-one constraint which can be solved effectively by randomization technology [35].Finally,the problem (19) can be converted into

wheretis an auxiliary variable.In order to address the non-convexity of constraints in (20),we introduce a set of auxiliary variables fx,ykg,8kand define the following equations:

Substituting (21) and (22) into problem (20),we obtain

It can be found that both the constraints C4 and C5 are mathematically intractable due to the channel uncertainty region ∆i.Therefore,we adopt a method combining angular discretization with Taylor approximation to address the channel uncertain region in (23).

Let first discuss the uncertainty region ∆e,k.According to the concept of convex hull,any CSI in the uncertainty region ∆e,kcan be effectively expressed as a convex combination of a finite number of CSI sampling points.Thus,we uniformly selectM1andM2samples from vertical DoA setand horizontal DoA setas

According to (26),the convex hull Ωe,kof ∆e,kcan be expressed as

which is equivalent to

In order to achieve the maximum value on the left side of (29),we adopt Cauchy-Schwarz inequality to calculate the weighting factorSpecifically,we have

where the equality is satisfied when

Note that the constraint (33) is still non-convex due to the exponential function,we adopt first-order Taylor expansion and obtain

Substituting (34) and (36) into optimization problem(23),we obtain

Similarly,P2 andP3 can be,respectively,transformed as

(i) TransformedP2 (total transmit power minimization):

(ii) TransformedP3 (multi-objective optimization):

4.Numerical results and discussion

In this section,computer simulations are conducted to evaluate the performance of the proposed robust BF scheme for a CSUN in the presence ofL=2 andK=2.All of the plots are obtained by performing 10 000 channel realizations.We set the tolerance of accuracy ε=10−3,the maximum number of iterationNmax=10.Other parameters are listed inTable 1[6].Moreover,we compare the proposed robust BF scheme with three different BF schemes,namely,zero forcing (ZF) BF scheme,maximum ratio transmission (MRT) BF scheme and non-robust BF scheme.

Table 1 Main simulation parameters

Fig.3andFig.4plot the beampatterns of the BF weight vector obtained by the proposed robust BF scheme from 3D and vertical vision,respectively.It can be observed that the maximum direction of beam points to the SU,and four nulls are generated with at least−50 dB at the uncertainty region of PUs and Eves,respectively,demonstrating that our proposed robust BF scheme can effectively guarantee the signal quality at the intended user and suppress the signal leaked to unintended users among the uncertainty regions.

Fig.3 3D beampattern of w

Fig.4 Beampattern from vertical vision

Fig.5depicts trade-off regions for the ASR and total transmit power of the four schemes.The curve is obtained fromP3 by var∑ying the values of λj≥0 by the step of 0.1 subject toIt can be observed that the ASR of SU is a monotonically increasing function with respect to the total transmit power.It can be observed that the ASR of the nonrobust BF scheme,MRT BF scheme and ZF BF scheme would converge gradually as the transmit power grows.Moreover,our proposed robust BF scheme has the superior performance to the other three schemes and the difference becomes larger as the total transmit power increases.The reason is that the proposed robust BF scheme takes the channel uncertainty region into consideration,having good robustness,and can well deal with the performance loss caused by channel errors.The curves verify the superiority and security of our proposed robust BF scheme.

Fig.5 ASR versus total transmit power

Fig.6plots the trade-off region for the INR and the total transmit power.It can be seen that the INR of our proposed scheme increases slowly with the total transmit power and is below the threshold γth=−50 dB.Moreover,other three schemes cannot satisfy the QoS requirement of PUs.As the increase of power,they will produce serious interference to PUs.Therefore,our proposed robust scheme outperforms the other schemes significantly.

Fig.6 INR versus total transmit power

Fig.7depicts the relationship between the total transmit power,INR and ASR.The weighted MOOP is set as λ1=λ2=0.5.It can be observed that the total transmit power increases as the ASR threshold increases because more total transmit power is needed to satisfy the increasing requirement of ASR.However,the INR threshold has a small influence on the total transmit power.The main reason is that our proposed robust BF scheme can effectively suppress the interference to the PU.

Fig.7 Total transmit power versus INR and ASR

Fig.8illustrates the ASR against the antenna number of UAV.It can be observed that the difference between the proposed scheme and the other schemes becomes larger as the antenna number increases.The reason is that employing more antennas can enhance the advantage of the proposed scheme,but will both simultaneously increase the signal strength of SU and Eves in the other schemes,thus eventually increase the performance gap.The curves verify the superiority of our proposed robust BF scheme.

Fig.8 ASR versus antenna number of UAV

5.Conclusions

By using the weighted Tchebycheff approach,we propose a multi-objective robust secure BF scheme in the CSUN to jointly considering the ASR and the total transmit power,where only imperfect CSI of PUs and Eves is available at the UAV.Specifically,we formulate a MOOP under the constraints of minimum ASR of SUs,QoS requirement of PUs and total transmit power budget.Then,we propose a method combining angular discretization with Taylor approximation to transform the MOOP into convex one.Next,we adopt SDP along with randomization technology to solve the MOOP and obtain the BF weight vector.Finally,Monte Carlo simulation results demonstrate the effectiveness and superiority of the proposed robust BF scheme.