I Introduction
Wireless communication is usually prone to eavesdropping and active malicious attacks due to its broadcast characteristics. Secure spatial modulation (SSM), as enhanced SM [1] with secure capacity, is attracting everincreasing research interest from academic world and industry due to its high energy efficiency. It is suitable for lowpowerconsumption scenarios like internet of things and wireless sensor networks.
Unlike the directional modulation (DM), SSM naturally suits the multipath fading channel and the DM can only be applied in the lineofsight scenarios [2]. In the recent years, the researches on SSM focus on the following several aspects: precoding [3], artificial noise (AN) injection [4], transmit antenna selection (TAS)[5][6] and power allocation [7]. The authors in [3] proposed a precoding scheme to improve secrecy rate (SR) for SM systems. By projecting the AN into the null space of the legitimate channel, a secrecy enhancement scheme was proposed in [4]. In [5], the authors investigated a SSM system, of which two extremely lowcomplexity TAS schemes were proposed. As a powerful way of enhancing the security, some power spitted schemes [7] between the legitimate signal and AN were also devised.
The above literature only concentrates on the transmit techniques for SSM. However, there is very little literature concerning the secure precoding SM (PSM). First, let us review literature about SM without taking security into account. The concept of PSM was proposed in [8] to achieve the goal of the lowdetection complexity. Considering that the number of the receive antennas (RAs) is not a power of two, the authors in [9] proposed two efficient RA selection (RAS) schemes for the purpose of improving the bit error rate performance in PSM networks. Moreover, the authors in [10] investigated the physical layer security performance of PSM schemes via precoding optimization by jointly exploiting the power difference between the desired user and eavesdropper.
To the best of our knowledge, there is no research of investigating how to improve the security of PSM by using RAS. Our main contributions are summarized as follows:

To reduce the computational complexity of optimizing the RAS, two lowcomplexity RAS strategies, called the maximizing SR in the low SNR (MaxSRL) and the maximizing SR in the high SNR (MaxSRH), are proposed. The former and the latter can achieve a perfect SR performances in the low and high SNR regions, respectively. In accordance with simulation results, the proposed two methods can outperform the random method in terms of the SR performance. More importantly, their extremely lowcomplexities are very of importance for practical applications.

To address the problem that there exist a large SR performance loss for the proposed MaxSRL and MaxSRH in the medium SNR region, we propose a novel method maximizing SR at the all SNR region (MaxSRA), which can perform well for all the SNR regions. Simulation results show that, compared to MaxSRL and MaxSRH, the proposed MaxSRA performs better in the medium SNR region in terms of SR. In addition, it can achieve an acceptable SR performance over the whole range of SNR, but at the cost of a slightly higher computational complexity than the previous two methods.
Notations:
Matrices, vectors, and scalars are denoted by letters of bold uppercase, bold lowercase, and lowercase, respectively.
indicates a complex matrix with dimensions. Signs , and denote matrix inverse, conjugate transpose and Frobenius norm. stands for the expectation operation. The operators and indicate the floor function and euclidean norm. is the binominal coefficient.indicates a complex Gaussian distribution with mean
and covariance matrix .Ii System Model
Fig. 1 sketches a secure PSM system with RAS. Here, there is a transmitter (Alice) with transmit antennas, a legitimate receiver (Bob) with RAs, and an eavesdropper (Eve) with RAs. It is assumed that is not a power of two, thus we have to select out of RAs for mapping the bits to antenna index. Notice that there are possible patterns in total, represented as , where the denotes the antenna set of the th pattern.
After Bob chooses one pattern from the pattern set , the transmitted symbol with the aid of AN is given by
(1) 
where is the th column of for , indicating that the th RA is activated at Bob; denotes the input symbol from an ary APM constellation. Besides, is the random AN vector, is the total power constraint, and are the power allocation factors with . Assuming that the channel status information of Bob is available at Alice, the useful signal’s precoding matrix and the AN project matrix can be implemented by Alice.
Let and be the channel matrices from Alice to Bob and Alice to Eve, respectively. Each element of and obeys the circularly symmetric complexvalued Gaussian distribution. Accordingly, the received signals at Bob and Eve can be expressed by
(2) 
(3) 
where is RAS matrix, constituted by the specifically selected rows of . Assume that . , where . , where . It is obvious that , so that the AN has no effect on Bob. Additionally, and are additive white gaussian noise. As such, the average SR is defined as
(4) 
where . and represent the average mutual information of the AlicetoBob and AlicetoEve channels, respectively.
Considering the discreteinput continuousoutput memoryless channel, the mutual information for Bob is expressed as
(5) 
where .
Due to the effect of the injected AN, the received noise at Eve is a colored noise. Then, a whitening filter is required and given by . Therefore, the mutual information for Eve is
(6) 
where and . Thus, . Herein, we assume that is also available at Alice [11], then the SR maximization problem over TAS is cast as
(7) 
Note that evaluating the SR in (7) requires an extremely large computational amount and is computationally prohibitive due to the use of the montecarlo method.
Iii Proposed RAS strategies
In this section, we first aim to reduce the computational complexity of evaluating in (7). To further mitigate the computational burden, three lowcomplexity RAS schemes, namely MaxSRL, MaxSRH and MaxSRA, are proposed.
To reduce the computational complexity of the exact SR value, a closedform approximation to the mutual information can be employed, namely the cutoff rate [6][12], is given by
(8) 
(9) 
with the approximate SR as . Upon replacing by , it will save a large computational amount. Although is not the achievable SR, its efficiency has been repeatedly demonstrated in [6]. Then, the optimization problem in (7) is reduced to
(10) 
where has a closed form. However, (10) is still an intractable optimization problem due to its prohibitive computational complexity. Below, we propose some methods to further mitigate this computational burden.
Iiia Proposed MaxSRL RAS Method
Firstly, the nature of will be investigated as the SNR tends to infinitely small, i,e., . It can be noted that the value of expression is closed to , with the aid of the firstorder Taylor expansion, we have and . As such, an approximation to Eve’s mutual information in the low SNR can be expressed as
(11)  
where with . Similarly, the mutual information term can be approximated as
(12) 
Upon using the difference of (11) and (12), an approximation to the SR is
(13) 
Replacing the objective function in (10) by (13) yields
(14) 
It is apparent that the objective function in (14) has a much simpler form than that in (10). Actually, the twolayer summation over all possible transmit symbols has been removed from the derived objective function in (14). Therefore, the computational complexity is significantly reduced.
IiiB Proposed MaxSRH RAS Method
It should be pointed out that the proposed scheme in (14) has a low complexity, but it merely suits for the lowSNR region. As we will show in the next section, its SR performance will be seriously deteriorated in the high SNR region. Here, we propose a new scheme suiting for the highSNR region. When , i.e., in the high SNR region, we have
Thus, the achievable rate obtained by Bob is close to , which can be observed from (5). From this perspective, choosing different RA combinations will result in little effect on the achievable rate of Bob. Considering that the SR is calculated from the difference between the achievable rates of Bob and Eve. Because Bob’s rate is almost unchanged, the SR can be increased by reducing the achievable rate of Eve. From the perspective of information theory, reducing Eve’s achievable rate can be achieved by minimizing its received SNR of the useful signals at Eve. Consider that the AN and thermal noise are whitened after the received signal passes by taking the benefit of the whitening filter . The corresponding optimization problem can be described as
(15) 
As such, a large number of the computational load will be saved by taking the benefit of the simplified objective function of (15). Actually, the MaxSRH scheme may coordinate with the MaxSRL together to avoid the disadvantage of the SR performance degradation in the high SNR region.
IiiC Proposed MaxSRA RAS Method
In the previous two subsections, two simple approximations to the SR expression considered only the low and high SNR regions, respectively. How about the medium SNR region? In this subsection, a new approximation is presented to be independent of SNR regions. Then, a new algorithm, called MaxSRA, is proposed to conquer this disadvantage.
Assume that the set of all possible values of is replaced by and
where the former denotes the values and the latter denotes its probability mass function. For the determined transmission mode,
and are calculated in advance just once. The cutoff rate in (8) for Bob can be simplified as(16) 
Due to the SM nature, the number of can be greatly reduced because of only a unique ’1’ component. Thus the computational complexity of the cutoff rate can be decreased. According to the definition of the matrix norm induced by vector norms, we get , and the inequality can be derived. Thus, the rate is given by
(17) 
Taking the difference between (16) and (17) as the new alternative, the SR approximation can be further reduced to
(18) 
which gives the following optimization problem
(19) 
Clearly, the computational complexity of (19) is much higher than those of MaxSRL and MaxSRH. This is mainly owing to the fact that calculating the objective function requires larger computational load than those of MaxSRL and MaxSRH. However, this method is capable of harvesting stable SR gains over MaxSRL and MaxSRH for the medium SNR region, which will be shown in the simulation section.
IiiD Complexity Analysis and Comparison
Now, we make a complexity comparison concerning the above methods. Notice that there are totally possible RAS patterns, and the complexity of exponential and logarithm operations are omitted. According to the floatingpoint operations (FLOPs) in matrixvector calculus [13], the original MaxSR method using MonteCarlo simulation costs about FLOPs, where is the number of noise samples for evaluating the SR. The proposed MaxSRL doesn’t require exhaustive search over all symbols, and its complexity is reduced to . Similarly, the computation complexity of MaxSRH is FLOPs. The complexity of MaxSRA is FLOPs, where the value of is only related to the specific modulation constellation. Generally, is always a large number and , . Therefore, we have the complexity order as: MaxSR MaxSRA MaxSRL MaxSRH.
Iv Numerical Results and Discussions
In this section, we present simulation results to evaluate the performance of the above schemes. For a fair comparison, it is assumed that . In addition, the total transmit power is set as , and quadrature phase shift keying (QPSK) modulation is used. The metric for quantifying the secrecy performance is the ergodic SR obtained by averaging over 2000 random channel realizations.
Fig. 2 plots the average SR performance versus SNR for the proposed three methods, where the random selection method is invoked as the benchmark. It is seen from Fig. 2 that the proposed three methods have significant performance improvements over the random one. Particularly, the proposed MaxSRL is close to the optimal MaxSR in the low SNR region. However, its performance degrades severely in the high SNR region. By contrast, the SR performance of the proposed MaxSRH is close to that of the MaxSR scheme in the high SNR region, while its performance gradually decreases as SNR moves from medium to low. The proposed MaxSRA is close to the MaxSR for all SNRs whereas it is much better than both the MaxSRL and MaxSRH in the medium SNR region, and slightly worse than MaxSRL and MaxSRH in the low and high SNR regions, respectively.
Fig. 3
shows the cumulative distribution function (CDF) of SR for the proposed three RAS methods with three different values of SNR: 3dB, 3dB and 12dB. From Fig.
3, we find the consistent SR performance tendency as Fig. 2.V Conclusion
In this paper, we have investigated the RAS strategies of the secure PSM systems. To reduce the computational complexity and improve the SR performance, three high performance TAS strategies, called MaxSRL, MaxSRH and MaxSRA, have been proposed. Simulation results showed that the proposed MaxSRL and MaxSRH can attain substantial SR gains over the random method in the low and high SNR regions with extremely lowcomplexities, respectively. The proposed MaxSRA is capable of achieving the nearoptimal SR performance over almost all SNR regions and avoiding the SR performance losses of the MaxSRL and MaxSRH in the high and low SNR regions, respectively.
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