I Introduction
In multipleinput multipleoutput (MIMO) systems, spatial modulation (SM) [7] is a promising technology which exploits both the index of activated antenna and amplitude phase modulation (APM) symbol to transmit messages. Recently, SM has exhibited its benefits of increasing energy efficiency [3] and avoiding the innerchannel interference (ICI). However, due to the broadcasting characteristic of wireless transmission, confidential information might be eavesdropped by illegal receivers. This is a problem of physical layer security [1, 4, 2, 16, 10]. In [16], the author enhance the legitimate security by joint precoding optimization with and without eavesdroppong channel state information (CSI) . In [11, 15], security is enhanced by emitting the artificial noise (AN) onto the nullspace of the desired channel to interfere unknown eavesdropper and the latter derived the closedform approximated expression ESR with the aid of AN in perfect and imperfect CSI, respectively. In [6], the authors emit the AN by a fullduplex desired receiver, where the confidential information is received and at the same time efficiently interfere the eavesdropper. Moreover, the authors in [8] investigate the power allocation between confidential message and AN so that we can get a fixed power allocation factor to achieve higher SR performance.
As for secure SM, the authors in [9, 14] investigate the transmit antenna selection (TAS) schemes, the former proposed leakagebased and MaxSR schemes achieving a good balance between complexity and SR performance, and the latter selected two antennas each time to improve SR performance. In [13], the author proposed three active antennagroup (AAG) selection methods which performed well in the low, medium and high SNR regions with low complexity, respectively. In [12], the author jointly optimized the problem of maximizing SR over TAS and AN projection with joint and separate solutions. Meanwhile, the author in [5] proposed deterministic and random antenna selection methods with the aid of zeroforcing precoding and fast radiofrequency switches to enhance the security of the PSM system.
However, in the aforementioned literature, the illegal receivers are all passive eavesdroppers, which only receive confidential message (CM) and don’t emit malicious jamming to the desired receiver. In such a situation, Bob cannot measure the CSI from Eve to Bob. It is impractical to know the CSIs from Alice to Eve and Eve to Bob in advance. If Eve becomes Mallory, which means it can send a malicious jamming towards Bob, this dilemma will disappear. Why? Due to the fact that Mallory emits jamming, Bob can estimate the CSI from Mallory to Bob. These estimated CSI can be used to suppress the jamming from Mallory. Similarly, Alice also obtain the CSI from Alice to Mallory, and prevent the interception from Eve with the help of AN. To simplify the derivation process, the perfect CSIs from Mallory and Alice to Bob are available in the following.
Considering the worst scenario, we propose a novel secure SM (FDMSSM) system with a FD malicious attacker Mallory having eavesdropping ability. In our FDMSSM system, Mallory can simultaneously send malicious jamming signal and intercept the CMs from Alice. Therefore, how to optimize the receive beamforming (RBF) such that the malicious jamming is reduced efficiently and the performance is improved obviously is a challenging issue. This motivates us to propose several RBF methods, and our main contributions are summarized as follows:

To suppress the malicious interference from Mallory and achieve a high performance, the traditional maximum receive power (MaxRP) at Bob method is first presented via the system. However, consider the noise plus interference at Bob includes three parts: AN from Alice, malicious jamming from Mallory and receiver noise at Bob, which obviously is colored, the conventional MaxRP doesn’t exploit this colored property. Therefore, we propose an improved whiteningfilterbased MaxRP (MaxWFRP). First, we compute the covariance matrix of colored noise plus interference at Bob, multiply the original receive vector at Bob from left by the root of the inverse of covariance matrix to whiten the colored jamming plus noise. By simulation, we find the proposed MaxWFRP harvests substantial performance gains over the conventional MaxRP in terms of SR and BER.

To completely eliminate the malicious jamming signal from Mallory, a MaxRP with zeroforcing constraint (ZFC) is proposed to force the malicious jamming from Mallory to zero and meanwhile maximize the receive power of CMs at Bob. However, due to the strict ZFC, the transmit space of CMs is also reduced dramatically. To remove the ZFC, a maximum signaltojammingplusnoise ratio (MaxSJNR) is proposed to strike a good balance between suppressing jamming and improving performance. From simulation results, the proposed MaxSJNR performs much better than the proposed MaxRP with ZFC in terms of SR and BER performance.
The rest of this paper is organized as follows. Section II presents the FDMSSM system model and gives a definition for its average SR. In Section III, the RBF schemes for MaxSR are proposed and their closedform expression is given. In Section IV, numerical simulation results are presented. Finally, we draw our conclusions in Section V.
Notations: Boldface lower case and upper case letters denote vectors and matrices, respectively. denotes conjugate and transpose operation. represents expectation operation. denotes 2norm.
Ii System Model
As shown in Fig. 1, the FDMSSM system considered consists of a legal transmitter (Alice) with transmit antennas (TAs),a receiver (Bob) with receive antennas (RAs), and a FD malicious (Mallory) with antennas. Here, Mallory works on FD model. In other words, he intercepts the CMs from Alice and transmits a malicious jamming towards Bob.
In general, it is noted that is not a power of 2. For a SM system, the number of active antennas are chosen from , where is equal to . And Alice activates one of TAs to emit ary APM symbol and uses the index of activated antenna to convey spatial bits. As a result, bits per channel use can be transmitted. The transmit signal with the aid of AN from Alice and the jamming signal sent from Mallory can be expressed as
(1) 
and
(2) 
respectively, where is the power allocation factor and denotes the total transmit power of Alice. Accordingly, is the th column of , indicating that the th antenna is chosen to transmit symbol , which is equiprobably drawn from discrete ary APM constellation for with . Moreover, matrix is the AN projection matrix and is the AN vector. In (2), is the transmit power of Mallory and is the transmit beamforming matrix with of jamming vector .
Let , , and denote the channel gain matrices from Alice to Bob, from Alice to Mallory, from Mallory to Bob, and Mallory’s selfinterference, respectively. Then the receive signal at Bob and Mallory after receive beamforming can be respectively expressed as
(3) 
and
(4) 
where and are the receive beamforming vectors at Bob and Mallory. In addition, and denote the complex additive white Gaussian noise (AWGN) vectors at Bob and Mallory, respectively. Meanwhile, is the transmit antennas selection matrix constituted by the specially selected columns of , which is determined by the leakagebased method in [9]. In (4), is the selfinterference observed at Mallory. Consider Mallory design the matrix by nullspace projection method, i.e., is the nullspace of the RBF with . As a result, Mallory can eliminate the selfinterference. The average SR is given as
(5) 
where
(6) 
and
(7) 
where . , , , and . Herein, , , , and are the possible transmit vectors. Besides, , and . and are the covariance matrices of interference plus noise of Bob and Mallory respectively, where
(8) 
and
(9) 
According to [11], it is known that premultiplying by will not change the mutual information, which results in and .
Iii Proposed Three Receive Beamforming Schemes at Bob
For the newly proposed FDMSSM system, the design of receive beamforming at Bob is very important to improve the system performance. In this section, the conventional MaxRP is presented, and three highperformance RBF schemes are proposed as follows: MaxWFRP, MaxRP with ZFC, and MaxSJNR. Finally, complexity comparison is made among them.
Iiia Conventional MaxRP
In this subsection, we derive the MaxRP from MaxSJNR rule. According to the definition of SJNR, we first write the SJNR at Bob as follows
SJNR  
(10) 
As mentioned in Section II, , and is the th column of . To simplify the above optimization, w
is approximated as a white noise, i.e.,
, (IIIA) can be rewritten asSJNR  (11) 
Hence, the optimization problem of MaxSJNR reduces to
(12) 
which actually is a MaxRP. The Lagrangian function associated with the above optimization is defined as
(13) 
whose firstorder derivative with respect to is set to be zero as follows
(14) 
which means the fact that
is the eigenvector corresponding to the largest eigenvalue of the matrix
.IiiB Proposed MaxWFRP
In fact, the malicious jamming signal plus interference plus noise w is colored not white. Therefore, the conventional MaxRP method in Subsection A may not perform optimally in terms of SNR. In what follows, we propose the MaxWFRP method, which first whitens the colored noise plus interference at Bob using their covariance matrix, and then apply MaxRP method to maximize the average SNR.
As , and are the independent and identically distributed i.e.,i.i.d, random vectors, w has a mean vector of allzeros and covariance matrix
(15) 
Observing (IIIB), it is evident that the covariance matrix
is a positive definite matrix, its eigenvalue decomposition (EVD) is in the form
, where is an unitary matrix, and is a diagonal matrix with being the th eigenvalue of matrix . Let us define the WF matrix satisfying , which yields(16) 
where . Applying the WF matrix in (16) on yields a new system model,
(17) 
where has covariance matrix . Similar to (12), we have the form of MaxWFRP
(18) 
Hence, can be set to be the eigenvector to the largest eigenvalue of the matrix .
IiiC Proposed MaxRP with ZFC
In order to completely eliminate the malicious jamming from Mallory, in that follows, we propose the ZF method of maximizing the receive power of confidential messages at Bob by projecting the malicious jamming signal onto the nullspace of channel matrix from Alice to Bob. The corresponding MaxRP with ZFC optimization problem is formalized as
(19)  
To simplify the above optimization, let us define a new matrix
, and its singular value decomposition (SVD) is
(20) 
where spans the nullspace of the column space of matrix . Now, to remove the first constraint, let us introduce a new beamforming vector as follows , which satisfies . Substituting the above equation in forms the following unstrained optimization
(21) 
Using the generalized RayleighRitz theorem, the new beamforming vector is directly equal to the eigenvector corresponding to the largest eigenvalue of the matrix
(22) 
i.e., . Accordingly, the optimal value of is given by
(23) 
IiiD Proposed MaxSJNR
The ZF constraint in (19
) is extremely strict and reduces the degrees of freedom of transmit space of CMs. Below, after this constraint is removed, a MaxSJNR is proposed to maximize the average SJNR,
(24) 
which can make a good balance between reducing the malicious jamming and improving the performance, where SJNR is defined as
(25) 
Using the generalized RayleighRitz theorem, the beamforming vector of maximizing the SJNR can be directly obtained from the eigenvector corresponding to the largest eigenvalue of the matrix
(26) 
Actually, the MaxSJNR is equivalent to the MaxWFRP proposed by us, which will be verified in the next section.
IiiE Computational Complexity Analysis
Now, the approximate computational complexities of the four RBF methods are analyzed. First, the complexity of the traditional MaxRP method is about floatingpoint operations (FLOPs). For the proposed MaxWFRP, its computational complexity is approximated as . Accordingly, the complexity of MaxRP with ZFC method is . Besides, the computational complexity of the proposed MaxSJNR is approximated as, which is slightly higher than proposed MaxWFRP method. Generally, their complexities have an increasing order as follows: MaxRP, MaxRP with ZFC, MaxWFRP, MaxSJNR.
Iv Simulation and Discussion
In this section, numerical simulations are presented to make a performance comparison among three proposed methods and conventional MaxRP from three different aspects: average SR, cumulative density function (CDF) of SR and the BER. Simulation parameters are set as follows: , , W, , and modulation scheme being QPSK.
Fig. 2 plots the curves of SR versus SNR for two different interference power: W and W. From Fig. 2, it is seen that the proposed MaxWFRP and MaxSJNR methods perform much better than MaxRP with ZFC and MaxRP. The conventional MaxRP is the worst one among all beamformers in terms of SR due to the fact that it omits the colored property of noise plus interference at Bob. Conversely, MaxWFRP achieves the best one among four methods. Additionally, the SR performance of the proposed MaxRP with ZFC method approaches those of MaxSJNR, MaxWFRP, and conventional MaxRP in the high SNR region. In summary, they have an increasing order in SR performance: MaxRP, MaxRP with ZFC, and MaxWFRP MaxSJNR.
Fig. 3 illustrates the curves of average BER versus SNR for the above four methods. From Fig. 3, it is seen that the proposed MaxSJNR and the proposed MaxWFRP still obviously exceed the remaining two methods MaxRP with ZFC and conventional MaxRP in terms of BER performance. Similarly, MaxSJNR is still overlapped with MaxWFRP and the performance difference between them is trivial. In particular, as SNR increases, the BER performance gains achieved by MaxWFRP and MaxSJNR over MaxRP with ZFC and conventional MaxRP become more significant. Similar to Fig. 2, there is the same increasing order in BER performance: MaxRP, MaxRP with ZFC, and MaxWFRP MaxSJNR. Fig. 4 shows the CDF curves of SR for the four methods with two different value of SNRs: 5dB and 5dB. From Fig. 4 , we find the same performance tendency as shown in Fig. 3.
V Conclusion
In this paper, we have made an investigation of RBF methods in the newly proposed SSM system with a malicious fullduplex attacker having an eavedropping ability. First, the conventional MaxRP method was derived to design RBF, after that, three highperformance RBF design methods, MaxWFRP, MaxRP with ZF constraint, and MaxSJNR have been proposed to improve the SR performance of the system. From simulation results, the proposed MaxWFRP achieves the same performance as MaxSJNR. They are much better than conventional MaxRP and MaxRP with ZFC in terms of SR and BER performance and have the same order computational complexity as MaxRP and MaxRP with ZFC. Interestingly, the proposed MaxWFRP has a slightly lower complexity compared to the proposed MaxSJNR.
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