Joint Pilot Design and Beamforming Optimization in Massive MIMO Surveillance Systems

Caihong Kai,Xiangru Zhang,Xinyue Hu,Wei Huang,*

1 School of Computer Science and Information Engineering,Hefei University of Technology,Hefei 230601,China

2 Intelligent Interconnected Systems Laboratory of Anhui Province,Hefei University of Technology,Hefei 230601,China

3 School of Electronics and Information Engineering,Anhui University,Hefei 230601,China

Abstract:This paper proposes a novel joint channel estimation and beamforming scheme for the massive multiple-input-multiple-output(MIMO)frequency-division duplexing(FDD)wireless legitimate surveillance system.With the proposed scheme,the monitor with the full duplex capability realizes the proactive eavesdropping of the suspicious link by leveraging the pilot attack approach.Specifically,exploiting the effective eavesdropping rate and the mean square error as performance metrics and setting a total power budget at the training and transmission phases,while guaranteeing the information from suspicious source can be successfully decode,joint pilot design,power allocation and beamforming strategy are formulated as optimization problems for the two objective functions: MSE and effective eavesdropping rate.A closed-form expression of the optimal pilot with the limited length can be obtained via the channel correlation.The optimal power problem at the training phase can be solved by a simple bisection method.Then,based on the obtained imperfect estimated channel,the jamming beamforming at monitor optimization algorithm is proposed by utilizing the convex Semidefinite Programming approach to maximize the effective eavesdropping rate.Numerical results show that the proposed joint pilot design,power allocation and beamforming optimization scheme can improve the surveillance performance of the legitimate monitor as compared to the existing passive eavesdropping and jamming-assisted eavesdropping.

Keywords:proactive eavesdropping;massive MIMO;channel estimation;pilot contamination;beamforming

I.INTRODUCTION

In the past years,the rapid development of wireless communication technology has significantly improved our life,which brings us drastic convenience by providing high-speed wireless connection among the anytime,anywhere,and between any device.However,the ubiquitous accessibility of wireless communication has posed new threats to public security,because it is vulnerable to be misused by criminals or terrorists to commit crimes.Therefore,there is an increasingly urgent need for authorized agencies to implement effectively information surveillance measures so as to achieve various legal pur-pose,such as intelligence gathering,terrorism/crime prevention and investigation,etc.

As an initial work,the authors in[1]and[2]proposed proactive eavesdropping system via cognitive jamming,where the legitimate full-duplex monitor listens to the suspicious communication while sends noise-like jamming signals to intentionally adjust the suspicious communication,so that it can decode suspicious data reliably.After this pioneering work,the researchers have carried out lots of works on proactive eavesdropping[3–6].Ref.[3]proposed a proactive eavesdropping approach via a spoofing relay,where the legitimate full-duplex monitor simultaneous eavesdropping and spoofing relaying to vary the source transmission rate in favor of the eavesdropping performance,and jointly optimizing the power splitting ratios and relay precoding matrix to maximize the achievable eavesdropping rate.The author in[4]designed the optimal receive/transmit beamforming vectors to maximize eavesdropping non-outage probability.Ref.[5]investigated legitimate surveillance of suspicious communication with a battery-aided fullduplex wireless powered monitor.By designing the power splitting ratio and the jamming transmit power,an optimal offline algorithms and online greedy algorithm were proposed to maximizing the number of slots that the monitor can successfully wiretap the suspicious communication.Under the assumptions that the suspicious receiver is capable of detecting the presence of artificial noise,a jamming power optimization algorithm was designed to maximize the surveillance performance[6].Furthermore,surveillance performance for wireless communication in various setups have also been studied,such as dual-hop[7–9],relay-assist[10–13],unmanned aerial vehicle(UAV)communication[14–16],non-orthogonal multiple access[17],wireless powered communication[18],and intelligent reflecting surface-aided wireless systems[19],etc.On the other hand,the deep learning(DL)tool and game theory are exploited in the proactive eavesdropping[20–23].In[20]and[21],the authors considered the cooperative proactive eavesdropping to assist the central monitor,which proposed an efficient deep learning-based method to identify the optimal mode selection for the intermediate nodes.The such cooperative scheme can increase the surveillance performance as well as decease the time complexity.In order to further reduce the algorithm complexity,a learning-based iteration algorithm was developed in[22],while it was able to achieve the same performance as the conventional ergodic search approach.Moreover,an adversarial mechanism between a transmitter and the opposing receiver was studied in[23].To capture their adversarial relationship,a power-domain Bayesian Stackelberg game with incomplete channel state information was proposed to combat the tactical communication link for an opponent receiver.

However,the common assumption in the aforementioned works is that channel state information(CSI)of all links is available at monitor,which is extremely difficult in practice.In massive MIMO systems,acquisition of CSI becomes challenging,due to the plenty of channel parameters[24].To accurately derive the CSI in time-division duplexing(TDD)massive MIMO systems.Ideally,pilot contamination caused by exploiting non-orthogonal uplink pilot in neighboring cells will limit the system performance[25].To address the difficulty,statistical channel properties is leveraging to improve the estimation accuracy[26,27].On the other hand,for the FDD systems,a line of works have studied the acquisition of CSI about the frequency-division duplexing(FDD)massive MIMO to reduce the pilot and feedback overhead by using the channel sparsity in the beamspace[28–31].Therefore,for the proactive eavesdropping FDD systems that the suspicious source and monitor are equipped with massive antenna array,by using the limited pilot overhead,obtaining the CSIs of suspicious link and eavesdropping link at the monitor is a challenge problem.Unfortunately,the most existing research on proactive eavesdropping scheme mainly focuses on the resource allocation,i.e.,subcarrier,power and bandwidth,and beamforming design with the assumption that the CSIs of all link have been available[9–16].To increase the effective eavesdropping rate,a pilot contamination attack mechanism was proposed in[32,33]and the jointly optimize pilot contamination power and jamming power strategy to maximize eavesdropping rate.However,in the practical massive FDD communication systems,in order to reduce the pilot and feedback overhead,the pilot with the limited length(much smaller than the number of transmitting antennas)was utilized to estimate the channel.Moreover,in order to improve the effective surveillance rate,the legitimate monitor should have to be also equipped with the large-scale antenna array.Consequently,for the surveillance system with large-scale antenna array,when the suspicious source obtains the CSI of the suspicious link by using the limited length of pilot,the monitor has to elaborately design the deception pilot so as to decrease the estimation performance of the link between suspicious source and destination as well as improve the effective eavesdropping rate.As a result,the joint pilot design at the training phase and beamforming optimization at the transmission is important,which has not been studied.This thus motivates our current work.

In this paper,we consider a basic setup of the massive MIMO FDD surveillance system,where the monitor transmits a deception pilot with the limited length of the pilot to jam the suspicious link.Our objective is to maximize the effective eavesdropping rate by jointly optimizing the deception pilot,power at the training phase and beamforming vector at the transmission phase.The formulated optimization is difficult to be solved directly due to the following reasons.Firstly,for the massive suspicious system,the limited length of the pilot will impose the pilot contamination attack performance and increase the channel estimation errors.Thus,it is imperative to design the pilot and power allocation scheme that are robust to channel estimation error of suspicious link.Second,the analytical expression of the data rates at monitor and suspicious destination is not tractable to obtained due to the uncertain terms(channel estimation error of suspicious link and eavesdropping link).Third,the optimization variables in the formulated problem is coupled.Thus,the different pilot design and power allocation scheme will affect estimation performance of link between suspicious users and the effective eavesdropping performance.To sum up,the main contributions of this paper can be summarized as follows.

First,we study the massive FDD proactive eavesdropping system,where the suspicious source and monitor equipped with a massive antenna array transmit the pilot with the limited length to obtain the CSIs of suspicious link and eavesdropping link,respectively.In order to improve effective eavesdropping rate,the monitor transmits a deception pilot at the training phase to increase the channel estimation error of suspicious link.Then,the monitor transmits a beamformed jamming symbol to the destination to further decrease the rate of the suspicious link.Due to the non-convex of the formulated optimization problem,we first derive the closed-form expression of the optimal pilot sequence.Then,based on the obtained optimal deception pilot,we transform the optimization problem into the convex power allocation of pilot subproblem and transmit/received beamforming optimization subproblem.

Second,to the uncertain terms in effective eavesdropping rate.i.e.,the channel estimation error of suspicious link and eavesdropping link,the accurate closed-form expressions of data rates at monitor and suspicious destination are extremely difficult to obtain.To circumvent this difficulty,an projection relationships between the effective eavesdropping rate,pilot and transmit/received beamforming vectors over a general correlated channel model is established by exploiting the Jensen’s inequality,which is the lower bound rate expression of the monitor and approximation rate expression of suspicious destination.

At last,numerical simulation results are presented,which demonstrate that the effective eavesdropping rate achievable by the proposed proactive eavesdropping scheme is significantly improved than that of the two benchmark schemes,namely,passive eavesdropping and jamming-assisted proactive eavesdropping.

The rest of this paper is organized as follows.Section II discusses the FDD-based massive MIMO surveillance system with a spatially correlated channel.In Section III introduces the problem formulation for the eavesdropping system and proposes a jointly pilot and beamforming design scheme.Section IV evaluates the performance of the proposed scheme by simulations and finally Section V concludes the paper.

Notations:Bold uppercase and lowercase letters and italic letters represent matrices,vectors,and scalars,respectively.The superscripts(·)H,(·)T,and(·)*represent the conjugate transpose operator,transpose operator,and conjugate operator,respectively.|·|and‖·‖denote the absolute value and the spectral norm,respectively.E{·}denotes the expectation operation.diag{x}stands for a diagonal matrix with the vector x on its diagonal.CN(0,R)stands for the circular symmetric complex Gaussian distribution with mean 0 and covariance matrix R.

II.SYSTEM MODEL

As shown in Figure 1,we consider a three-node point-to-point FDD massive MIMO surveillance system,where a legitimate monitor E with 2Nantennas is intended to eavesdrop a dubious communication link from the suspicious source S withMantennas to a single-antenna destination D.Generally,due to the limitation of physical size,the destination is only equipped with the single-antenna.To enable eavesdropping and jamming simultaneously,the legitimate monitor uses a full-duplex mode and is equipped with two sets of antennas,where the number of the transmitted and received antennas are denoted asNTandNR,respectively.For simplicity,we assume thatNT=NR=N.Furthermore,the perfect selfinterference is technique is used to reduce the complexity of the surveillance system[34,35].

Figure 1.Information surveillance scenario.

Figure 2.Approximation expression and Monte-Carlo evaluated values of the suspicious data rate for different numbers of monitor.

Figure 3.Lower bound and Monte-Carlo evaluated values of the eavesdropping data rate for different numbers of monitor.

Figure 4.Average effective eavesdropping data rate under different power consumption of monitor,M =100 and N =50.

Figure 5.Average effective eavesdropping data rate under different number of monitor antennas,M = 50 and PE =20dBm.

2.1 Channel Training Procedure

In general,the suspicious source needs to obtain the CSI between S and D before the information transmission.However,due to the existence of the legitimate monitor,the estimation accuracy of the suspicious channel can be affected by transmitting the same pilots to the D during the training procedure.In such case,the overall training procedure corresponding to suspicious link and eavesdropping link will be discussed in the following section.

2.1.1 Channel Estimation of Suspicious Link

In the FDD massive MIMO systems,since the transmitter is equipped with massive antenna array,the length of pilotLtransmitted by suspicious source is no larger than the number of the transmit antennasMso as to reduce training overhead.In order to increase the effective eavesdropping rate in the surveillance system,legitimate monitor operating in a full-duplex mode usually transmits the deception pilotto reduce the accuracy of channel estimation corresponding to the suspicious link.

2.1.2 Eavesdropping Channel Estimation

2.2 Data Transmission Procedure with Jamming Attack

When suspicious source and monitor obtain the individual CSI,they will transmit the beamformed symbolsxSandxEto the suspicious destination,where the symbolxEtransmitted by monitor is usually called as the jamming symbol.After propagating via the suspicious channel and jamming channel,the received signal at the suspicious destination can be written as

To successfully eavesdrop the suspicious link,ifRE≥ RD,monitor E can decode the information of suspicious link reliably with arbitrarily small error probability.As a result,the eavesdropping rate is equal to the data rate at suspicious destination.On the other hand,ifRE＜RD,it is impossible for monitor E to decode the suspicious information without any error.Therefore,we define the effective eavesdropping rate asRev,which can be expressed as[3]

Through the above analysis,it is observed that improving the surveillance performance requires a joint optimization over the four variables: pilot sequence Ψ,power allocated factorα,the transmit beamforming vector wE,and the received beamforming vector vEdirectly impact on surveillance performance.Unfortunately,finding an optimization solution for the constrained joint optimization problem is often intractable[49,50].To decouple the joint power,pilot and beamforming optimization problem,we consider the training and transmission phase recessively in next section.

III.DESIGN OF PROACTIVE EAVESDROPPING SCHEME

The pilot contamination attack is an efficient approach to improve the eavesdropping performance by sending the same pilot signals at the eavesdropper to fool the transmitter over the correct channel estimation.This changes the precoder used by the legitimate transmitter in a controlled manner to strengthen the signal reception at the eavesdropper during data transmission.Therefore,we propose a joint pilot design and beamforming design scheme in the section.

3.1 Joint Pilot Design and Power Allocation Scheme in the Training Phase

For wireless information surveillance,the legitimate monitor usually needs to be deployed far away from the suspicious transmitter so as to avoid exposure.Hence,the quality of suspicious link is better than that of eavesdropping link.In order to successful decode the information from S to E,the rate of eavesdropping link generally is better than that of suspicious link.To this end,we can actively reduce the accuracy of suspicious link via pilot contamination approach.In this paper,we first guarantee the performance of the channel estimation by allocating the power,and then the residual power is allocated to the beamforming at the monitor to maximize the effective surveillance rate.In this section,we aim for designing and optimizing deception pilot Ψ and power allocation factor to maximize the MSE,while satisfying the MSE constraints to prevent exposure.Specifically,we formulate the following optimization problem

Note that in(21),(21b)ensures the monitor E do not be detected by S,whereρdenotes maximum tolerance of estimation error,(21c)is the pilot length constraints,(21d)is the power constraints,which means the part of power is allocated to the pilot.Moreover,for the fixed power allocated factorα,we can obtain the optimal length of the deception pilot and have the following proposition:

Proposition 1.For any fixed α,the optimal deception training sequence at monitor that maximize the MSE ofhSDis given by

Then,for the given pilot Ψ,we further optimize the power allocated factorα.It is observed that MSE is the monotonic increasing function regarding with the variableα,and the constraints(21b)limits he maximum of the objective function.Therefore,if MSE(α)＜ρ,the optimization problem can be transformed into the power allocation problem,which is written as

Algorithm 1.Bisection-based search approach for problem(26).1: Initialization 2: threshold τ ＞0, αmin = 0,and αmax to a large value.Let αlow =αmin;3: Repeat 4: Let αup =αmax;5: Repeat 6: Let α=(αlow+αup)/2;7: With given Ψ,solve the problem(26)and denote the optimal solution as α⋆;8: if MSE(α)＜ρ 9:set αlow =α;10: else 11:αup =αmax;12: Until αup-αlow ≤τ;13: Output: solution α⋆.

Note that problem(26)can be solved by the simple bisection approach which is presented in Algorithm 1[51].Moreover,in the case with MSE(α)≥ρ,the the optimal power splittingα⋆=1 from the monotone increasing property of the objective function.

3.2 Transmit Beamforming And Received Beamforming Optimization

Although pilot contamination strategy can effectively increase the estimation error of suspicious channel,this strategy should decrease the rate of eavesdropping destination.In our proposed surveillance scheme,the monitor sends artificial noise jamming signals to intentionally further degrade the suspicious communication and improve surveillance performance.Therefore,the problem of maximizing the surveillance rate by optimizing receive and transmit beamforming can be formulated as

Plugging(32)and(33)into(29)yields relevant conclusion.

Due to multiple uncertain terms,it is also found that the accurate closed-form expression of eavesdropping link’s rate is mathematically intractable.Motivated by the results based on Jensen’s inequality in[48,54],the low bound of achievable rate of eavesdropping link can be expressed as

IV.PERFORMANCE EVALUATION

In this section,we carry out numerical simulations to evaluate the performance of the proposed proactive eavesdropping scheme.We set the pilot lengthL=5,the coherence interval isT= 20,and average pilot transit power of suspicious source isPS=8dBm,the maximum tolerance of estimation errorρ= 10-2.In addition,for the suspicious channel and eavesdropping channel,we adopt the exponential correlation model to characterize channel correlation matrix[42].Furthermore,the following schemes are compared:

Scheme 1(Passive eavesdropping): the monitor only is equipped with half-duplex radios and can not transmit deception pilot and jamming signals.

Scheme 2(Jamming-assisted):the full-duplex monitor receive the suspicious data meanwhile transmit jamming signals to intentionally adjust the suspicious communication in data transmission[4].

We first conduct an experiment to validate the tightness of proposed approximation expression ofRSand LB ofRE.In this experiment,the beamforming direction of monitor matches the corresponding channel vector,PE= 40dBm andM= 10.Figure 2 shows the suspicious data rate of approximation expression and Monte-Carlo simulation with the number of monitor antenna,whererEis channel correlation of RED.This figure reveals the data rate at suspicious destination becomes worsen the channel correlation between monitor and suspicious destination gets stronger.That is because the channel correlation matrix REDimpacts the accuracy of suspicious channel estimation,which further decrease the accuracy of the MRT beamforming at the suspicious source.

Figure 3 characterizes the tightness of lower bound ofRE,whererSis channel correlation of RS.It is observed that the LB closed-form expressioncurve coincides well with the Monte-Carlo simulation curve,which verifies the correctness of our analysis results.The gap of two curves decreases with the number of transmit antenna of monitor.Especially in the dense area of the antenna,the LB is very tight,and almost equal to the Monte-Carlo numerical results.Moreover,it is shown that the data rate of monitor is improved as the channel correlation of the eavesdropping link gets stronger.

Figure 4 shows the surveillance performance for various schemes are plotted with different transmit power,where the number of antennas at monitor and suspicious source is set toM= 100 andN= 50.It is observed that the proposed proactive eavesdropping schemes is significantly outperforms the other schemes.The reason is the proactive eavesdropping strategy reduces the accuracy of suspicious channel estimation by performing pilot contamination attack.Thus,the data rate at suspicious destination decreases and surveillance performance further is improved.

In Figure 5,the effective eavesdropping rate for various schemes are plotted with the different number of antennas at monitor,where the number of antennas at suspicious is set withM= 50,and the power constraint of monitor is set toPE= 20dBm.As we can see,the surveillance performance of all schemes is improved with the number of monitor antennas,due to the array gains.In addition,the gaps decreases with the number of monitor antennas.This is because the potential gain of legitimate eavesdropping channel becomes better than that of the suspicious channel,when number of monitor antennas increases.For the proposed jointly power allocation and beamforming scheme,it has the better performance than that of the fixed power allocated factorα.Hence,the monitor also can successfully intercept suspicious information without any proactive measures.

Figure 6.average effective eavesdropping data rate under different number of suspicious destination antennas,N =50 and PE =20dBm.

Figure 6 shows the effect of antenna number at suspicious source on the effective eavesdropping rate,where the number of antennas at monitor is set withN= 50,and the power of monitor is set withPE= 20dBm.It is observed that the surveillance performances of the proposed scheme and jammingassisted scheme increases with the number of suspicious source antennas whenM ≤100.The reason is that the number of suspicious source antennas increase,the more eavesdropping channel gain is obtained,resulting in the improved of surveillance performances.On the contrary,the effective eavesdropping rate of such two schemes is visibly deteriorated whenM ≥125.This is because it is becomes extremely difficult for the monitor with massive array antenna to achieve an effective eavesdropping with limited power overhead.On the other hand,since the quality of the legitimate eavesdropping channel is a degraded version of the suspicious channel,the gain of eavesdropping channel is much less than that of suspicious channel with the small number of antennas.Similar with Figure 5,it is not difficult to see that the average eavesdropping rate with the optimal power allocated factorαhas the preferred surveillance performance.Therefore,the surveillance performances of the passive eavesdropping scheme is degraded when the number of suspicious source increases without the proactive measures.

V.CONCLUSION

In this paper,the proactive eavesdropping problem was investigated in a FDD-based massive MIMO surveillance systems.In such system,a full-duplex multi-antenna legitimate monitor aimed to eavesdrop on a dubious link by using the pilot contamination and jamming attacks method.To realize the eavesdrop scheme,a pilot optimization algorithm was developed to maximize the MSE of suspicious channel estimation by joint optimizing deception pilot and power of monitor.Further,in the transmission phase,we jointly designed the beamforming vector at the monitor to maximize the effective eavesdropping rate.Numerical results demonstrated the effectiveness and superiority of the proposed proactive eavesdropping scheme compared with the baseline schemes.

ACKNOWLEDGEMENT

This work was supported in part by the National Natural Science Foundation of China under Grants 61971176 and 61901156;in part by the Anhui Provincial Natural Science Foundation under Grant 2008085QF281;in part by the Fundamental Research Fund for the Central Universities of China under Grant JZ2021HGTB0081.