Optimal Transmit Antenna Selection Algorithm in Massive MIMOME Channels
^{†}^{†}thanks: Supported by BUPT Excellent Ph.D. Students Foundation
Abstract
This paper studies the transmit antenna selection in massive multiple-input multiple-output (MIMO) system over wiretap channel. The transmitter, equipped with a large-scale antenna array whose size is much larger than that of the eavesdropper and legitimate receiver, selects a subset of antennas to transmit messages. An branch-and-bound (BAB) search based algorithm for antenna selection in independent and identical Rayleigh flat fading channel is proposed to maximize the secrecy capacity between the transmitter and the legitimate receiver when the transmit power is equally allocated into the active antennas. Furthermore, the proposed algorithm is separately applied to two scenarios which is based on whether the channel side information of the eavesdropper (CSIE) is available. Simulation results show that the proposed algorithm has the same performance as the exhaustive search under both scenarios but with much lower complexity.
References
I Introduction
Data traffic in wireless networks has experienced an explosive growth as the number of smart devices of all types increases drastically. With the increasing demand for transmission rate, the significance of transmission security and reliability has become increasingly prominent. In this respect, physical layer (PHY) security [1] has gained pivotal attention in recent years for its remarkable performance in information security enhancement.
The basic model for physical layer security is the wiretap channel in which transmitted messages to a legitimate receiver are being overheard by an eavesdropper [1]. Different from the traditional cryptographic techniques [2], physical layer security utilizes the inherent characteristics of wireless channels to ensure reliable transmission. Researches on PHY security has already shifted from single antenna systems [1] to multiple-input multiple-output (MIMO) wiretap channels, also referred to as Multiple-Input Multiple-Output Multiple-Eavesdropper (MIMOME) channels. The works in [3, 4, 5] showed the MIMOME configuration result in a promising performance in light of information theory.
Massive MIMO system drastically improves the spectral efficiency by deploying large-scale antenna arrays at the base station (BS) [6] and thus is believed to be the most promising candidate technology in the 5th generation cellular networks (5G). The work in [7] showed the significant improvements of transmission security and reliability in massive MIMOME channels compared with the small-scale MIMOME system. More specifically, the transmitter can reduce the information disclosure to the eavesdroppers by focusing its main transmit beam to the legitimate receivers[7].
To guarantee communication, each antenna should be connected with a radio-frequency (RF) chain, which results in high hardware cost in Massive MIMO system. Antenna selection (AS) technology [8] is regarded as an alternative to alleviate the requirement on the RF transceivers by selecting a subset of antennas to transceive signals. At the BS, transmit antenna selection (TAS) has been applied into massive MIMOME channels. Conventional researches usually concentrate on the closed-form expressions of secrecy capacity in different scenarios but ignore the algorithm design. Most of them merely consider single-antenna selection and the corresponding closed-form expression of the secrecy outage probability [9, 10, 11, 12, 13, 14]. However, a few researches discussed the performance of multiple-antenna selection and very simplistic algorithms were applied to it, such as the norm-based method [15, 16, 17]. Furthermore, few literatures focused on the algorithm design of multiple-antenna selection in massive MIMOME channels.
This paper concentrates on transmit antenna selection algorithm design in massive MIMOME channels. To the best of our knowledge, this is the first time to propose an optimal multiple transmit antenna selection algorithm in massive MIMOME channels, the complexity of which is much lower than exhaustive search. For simplicity, assume that the total transmit power is uniformly allocated over active antennas. An optimal TAS algorithm is proposed to maximize the secrecy capacity in massive MIMOME channels and discussed in two scenarios : 1) For Scenario A: the eavesdropper’s channel side information (CSIE) is unavailable at the transmitter (NCSIE), and 2) For Scenario B: CSIE is available. In each scenario, simulation results demonstrate that the proposed algorithm obtains an optimal solution at the expense of much lower complexity than exhaustive search.
The remaining parts of this manuscript is structured as follows: Section II describes the system model. In Section III, the optimal TAS algorithm is proposed. The simulation results and corresponding analysis are shown in Section IV. Finally, Section V concludes the paper.
: Throughout the paper, scalars, vectors and matrices are denoted by non-bold, bold lower case, and bold upper letters, respectively. stands for the complex plain. The Hermitian of is indicated with , and is the identity matrix.
Ii System Model
In this paper, we consider a massive MIMO wiretap channel. The transmitter is equipped with antennas, the legitimate receiver is equipped with antennas and the eavesdropper is equipped with antennas. The received signal vector at the legitimate receiver reads
(1) |
where is the transmitted signal and suppose the transmitted symbols from different antennas are independent; is the Signal to Noise Ratio (SNR) at each receive antenna of the legitimate receiver; is the additive complex Gaussian noise. Considering independent and identical (i.i.d) Rayleigh flat fading channel, the elements in channel matrix are i.i.d. complex Gaussian random variables with zero-mean. The received signal in the eavesdropper reads
(2) |
where is the SNR at the eavesdropper. The channel is still suffering i.i.d Rayleigh flat fading with Gaussian noise. Assume that the channel side information for both the eavesdropper and the legitimate receiver (CSIL) is available at the transmitter. The instantaneous achievable secrecy rate at the legitimate receiver is then written as[5]
(3) |
where , and denote the channel capacity over the eavesdropper channel and the legitimate channel respectively. Suppose that the transmit power is uniformly allocated, and could be written as[18]
(4a) | ||||
(4b) |
Then, consider the TAS at the transmitter and antennas are selected. Let denote the selected subset of transmit antennas, the cardinality of which is . Actually, selecting a subset of antennas from the transmitter is equal to selecting the corresponding columns of channel matrix. Using and to denote the submatrix after TAS, the secrecy capacity is represented as
(5) |
where and are defined as normalized SNR.
Iii TAS Algorithm
In this section, an optimal TAS algorithm with low comlexity in MIMOME channels is formulated. We assume that full CSIL is available at the transmitter.
Most of the TAS algorithms used in MIMOME channels are norm-based [15, 16, 17] i.e., to select antennas corresponding to the largest norms of the column vectors in the channel matrix . The norm-based method is of low complexity but moderately poor performance. Thus, it is important to explore an optimal solution for TAS in massive MIMOME system. Exhausitive search (ES) is definitely an optimal algorithm, but it’s prohibitively complex and even impratical for its huge complexity especially in the large-scale situation. Does an optimal algorithm exist for TAS with much lower complexity in contrast to the ES? Branch-and-bound (BAB) method[19, 20] could answer this question.
BAB was used for receive antenna selection in massive MIMO system[20]. In the followings, consider two scenarios in the former statement depend on whether the CSIE is available or not, and propose corresponding BAB based algorithms for these situations respectively. It’s shown that scenario III-A is the same as [20], but scenario III-B is totally different.
Iii-a Ncsie
In this case, the transimitter knows nothing about the eavesdropper channel and the transmitted power is equally allocated to the selected antennas. Since the CSIE is unavaiable at the transmitter, only the legitimate channel is considered in antenna selection. In this setup, the antenna selection for the eavesdropper channel could be treated as a random selection. And the transmit antenna selection problem could be summarized as
(6) |
where denotes the full set of all the candidate column index subsets with size .
Let denote the submatrix of legitimate channel after antennas are selected and denote the corresponding channel capacity. Assuming that the th row of matrix is selected in the step, , the channel matrix would be and the capacity reads
(7) | ||||
where and . The last equality holds for the Sylvester’s determinant identity[21] . Using the Sherman-Morrison formula[21], we could continue updating the expression to simplify the computation for the matrix inverse . We use to denote the antenna index selected in the th step. Then, could be rewritten as
(8) | ||||
where . Define , which can be updated as
(9) | ||||
where . Therefore, can be written as .
The BAB search is a classical algorithm in integer programming [19, 20] which could achieve the optimal solution but holds much lower complexity than exhaustive search. To use branch-and-bound search, a search tree as decribed in Fig.1 is built to implement the whole search. One can see from Fig.1 that the depth of the search tree is . The exhausitive search is to traverse the whole tree. To apply branch-and-bound into TAS, firstly we have to adjust the object function to which reads
(10) |
where , and , . The index set consists of all the candidate antenna set in the th level. Next, we show the monotonicity of the new object function in detail. According to the definition of , we have
(11) |
Equ. (8) shows that as , then
(12) | ||||
By the definition of , holds. As a result, here comes
(13) | ||||
By Equ. (10), holds, which indicates that is monotonically decreasing with the increase of the number of selected antennas.
As is the maximal value in each level, they are constants once the search tree is fixed. Therefore, maximizing is equivalent to maximizing by Equ. (10). Branch-and-bound search is suitable to find maximum with a monotonically-decreasing object function[19]. Suppose that the depth-first and best-first strategy is used during tree search. Since the object function along a path is decreasing, the function value of a complete path from the root node to the tip node could servr as a lower bound for other nodes. When we visit a node of a path in the th level, all the child nodes produced by can be discard if the real-time object function value is smaller than the lower bound. When we arrive at another tip node, we need to update the lower bound as the larger one if the corresponding object function value for this new complete path is larger than the current lower bound. This procedure will not stop until the whole tree is traversed. We could set minus infinity as the initial global lower bound. The tighter the initial bound is, the lower complexity the BAB algorithm would possess. The branch-and-bound algorithm is summarized in Alg. 1 attached with the complexity analysis, where is the subnode index set in the th level of the th node.
We use instead of as the object function because the former is monotonically decreasing. By using tree search, many nodes could be pruned during the search procedure. If we use , the whole procedure would degrade into an exhausitive search with huge computational complexity. If the tree search stops at the first level, it would degrade into the norm-based method. During the procedure of branch-and-bound algorithm, the total number of visited node is uncertain, the computational complexity could be calculated by where denotes the visited node’s total number. Because many of the branches are pruned during the algorithm, it could achieve the optimal solution with much lower computation cost than exhaustive search.
Iii-B Csie
By now, we have dicussed the scenario without CSIE. Then, we consider the situation when the transmitter have both full CSIL and full CSIE. The following demonstration shows that the BAB still works well when CSIE is available.
Since the transmitter knows the channel matrix of the eavesdropper, the antenna selection for the eavesdropper channel can’t be treated as random selection any more. The channel side information must be taken into consideration. Let and denote the submatrix of legitimate channel and eavesdropper channel after antennas are selected and denote the corresponding security capacity. Assuming that the th row of matrix is selected in the step, the channel matrix is denoted by . For the eavesdropper, the th row of matrix is selected.
Following the same derivations in Equ. (7), the capacity reads
(14) |
where and . These matrix inverses could be still updated using Sherman-Morrison formula which read
(15a) | ||||
(15b) |
in which and hold the similiar expressions as before . Define
(16a) | ||||
(16b) |
where and .
Then, . To maximize , we firstly make some adjustments to convert the object function to be a monotonically-decreasing one as follows
(17) |
where , in which and is defined as . The index set consists of all the candidate antenna set in the th level, and . Suppose that the th antenna has been selected in the th step, we have
(18) |
Since , we could derivate that once all the antennas the by its definition, which shows that the final would be fixed in whatever selection order. Thus we have
(19) | ||||
where the step () holds for that we could treat as the result of any selection order which includes the first antennas index that results in .
According to the definition of , we have ; thus . In addition, it has been proved in Equ. (12) and (13) that . Consequently, here comes
(20) | ||||
Thus holds by Equ. (18), which indicates that is monotonically decreasing. It’s clear that the branch-and-bound for the situation with full CSIE is different from the one with NCSIE. Actually, in the last section, we adjust a monotonically-increasing function to a monotonically-decreasing function . Nevertheless, the original function isn’t monotonic, which makes it even harder to do the construction. The branch-and-bound with full CSIE is summarized in Alg. 2, where is the subnode index set of the th node in the th level. The complexity of Alg. 2 is where denotes the total number of the visited nodes,
Iv Simulation Results
This part gives the simulation results followed by the computation complexity analysis of all the proposed algorithms.
Fix , , and . Fig.2 shows the ergodic secrecy capacity versus when CSIE is unclear at the transmitter with increasing form 32 to 128. Compared with the norm-based search, BAB has superior performance in all conditions.
Fig.3 shows the ergodic secrecy capacity for BAB and norm-based method when CSIE is available. BAB still outperforms the norm-based method. To verify that the branch-and-bound search has the same optimal peoformance as exhausitive search, Fig.4 shows the ergodic secrecy capacity of BAB and ES in both NCSIE and CSIE case. From Fig.4, it is truely that BAB can find the opimal antenna index subset to maximize the secrecy capacity.
Fig.5 shows the complexity of BAB, norm-based method and ES. As they can all be treated as tree search and the complexity of BAB is related with the number of visited nodes, it makes sense to use the number of visited nodes asking for updating operations during the tree search to measure the complexity of these algorithms[20]. It’s shown that norm-based method has the lowest complexity and ES’s complexity is nearlly ten times than that of the BAB. Basd on these simulation results, the advantages of BAB are apparent for its optimality and low-complexity.
V Conclusion
This paper studies transmit antenna selection in massive MIMOME channels. An optimal algorithm based on branch-and-bound search is proposed and discussed in the situations when the CSIE is available or unavailable. Simulation shows that branch-and-bound search can guarantee optimal performance with much lower complexity compared with exhaustive search. The proposed algorithm could serve as a benchmark in the future work on TAS algorithm design in massive MIMOME channels.
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