The explosive demand for new services and data traffic is constantly on the rise, pushing new developments of signal processing and communication technologies [Andrews-14-A]
. It is widely believed that multiple-antenna techniques can offer extra degrees-of-freedom (DoF) to efficiently allocate resources, which helps reduce the bandwidth and/or power while still maintaining the same quality-of-service (QoS) requirements[MietznerCST09]. However, the half-duplex (HD) radio, where downlink (DL) and uplink (UL) transmissions occur orthogonally either in time or in frequency, leads to under-utilization of radio resources, and may no longer provide substantial improvements in system performance even if multiple antennas are employed.
By enabling simultaneous transmission and reception on the same channel, full-duplex (FD) radio, which offers considerable potential of doubling the spectral efficiency compared to its HD counterpart, has arisen as a promising technology for the fifth generation of mobile communications (5G) [ZhangCM15, Wong5Gbook17]. The major challenge in designing an FD radio is to suppress the self-interference (SI) caused by signal leakage from the DL transmission to the UL reception on the same device to a potentially suitable level, such as a few dB above background noise. Fortunately, recent advances in hardware design have allowed the FD radio to be implemented at a reasonable cost while canceling out most of the SI [Riihonen-SP-11, DUPLO, Duarte:TWC:12, Saetal14]. Since then, applying FD radio to a base station (BS) in small cell-based systems or to an access point in WiFi, in which the transmit power is relatively low, has been widely considered. FD for multiple-input multiple-output (MIMO) in single-cell systems has been investigated to achieve a higher spectral efficiency [Dan:TWC:14, SH:TWC:15, Nguyen:Access:17], and an extension to multi-cell scenarios has also been considered in [Tam:TCOM:16, YadavAcess17, AquilinaTCOM17]. Another downside of FD radio in a typical cellular network is that co-channel interference (CCI) caused by the UL transmission of UL users severely impairs the DL reception of DL users. Therefore, it is challenging to fully capitalize on the benefits that FD radios may bring to 5G wireless networks.
Wireless networks have a very wide range of applications, and an unprecedented amount of personal and sensitive information is transmitted over wireless channels. Consequently, wireless network security is a crucial issue due to the unalterable open nature of the wireless medium. Physical-layer (PHY-layer) security can potentially provide information privacy at the PHY-layer by taking advantage of the characteristics of the wireless medium [ChenCST16, WuTCOM17, WuIT16, Nguyen:TIFS:16, ZhengTSP13, LiSPL16, AkgunTCOM17, ZhuTSP14, ZhuTWC16, SunTWC16, Lui:SP:14, Ng-14-A]. An effective means to delivering PHY-layer security is to adopt artificial noise (AN) to degrade the decoding capability of the eavesdropper (Eve) [WuTCOM17, WuIT16, Nguyen:TIFS:16], such that the confidential messages are useless for Eve. Notably, with FD radio, we can exploit AN even more effectively [ChenCST16].
I-a Related Work
In this subsection, we discuss the most recent and relevant works for PHY-layer security that exploit FD radio. Zheng et al. [ZhengTSP13] proposed a self-protection scheme by exploiting FD radio at the desired user to simultaneously receive information and transmit AN in point-to-point transmission. The secrecy rate (SR) optimization was studied in both cases of known and unknown channel state information (CSI) of the Eve. The work in [LiSPL16] considered a MIMO Gaussian wiretap channel with an FD jamming receiver to secure the DoF. The authors in [AkgunTCOM17] extended the prior work to a multiuser scenario, where the transmitter is equipped with multiple antennas to guarantee the individual SR for multiple users. However, these approaches require high hardware complexity of the receivers, which may be difficult to achieve in practice because the end devices should be of very low complexity.
By shifting the computational hardware complexity from the receivers to the BS in cellular networks, FD-BS secure communications are of great interest thanks to providing communication secrecy to both UL and DL transmissions. In [ZhuTSP14], joint information and AN beamforming at the FD-BS was investigated to guarantee PHY-layer security of single-antenna UL and DL users. However, this work assumed that there is no SI and CCI, which is highly idealistic. Therefore, an extension was proposed in [ZhuTWC16] by considering both SI and CCI. The work in [SunTWC16]
analyzed a trade-off between the DL and UL transmit power in FD systems to secure multiple DL and UL users. The common technique used in these works is semi-definite relaxation (SDR) that relaxes nonconvex constraints to arrive at a semi-definite program (SDP). Though in-depth results were presented, the beamforming vectors that are recovered from the covariance matrices of SDR may not perform effectively, and the dimension of covariance matrices is relatively large[PhanTSP]. By leveraging the inner approximation principle, our previous works in [Nguyen:Access:17] and [Nguyen:TCOM:17] have recently proposed path-following algorithms to efficiently address the nonconvex optimization problems in FD multiuser systems by solving a sequence of convex programs. These algorithms completely avoid rank-one constraints and jointly optimize all optimization variables in a single layer.
I-B Motivation and Contributions
Even with recent advances in hardware design for SI cancellation techniques, the harmful effect of residual SI cannot be neglected if it is not properly controlled, and is proportional to the DL transmission power. The CCI may become strong and uncontrolled whenever an UL user is located near DL users. These shortcomings limit the performance of FD systems [Dan:TWC:14, SH:TWC:15, Nguyen:Access:17, Tam:TCOM:16, YadavAcess17, AquilinaTCOM17, SunTWC16]. In addition, a major part of the FD-BS power budget allocated to the DL users with poor channel conditions to improve their QoS will significantly reduce the QoS for other users due to an increase in both the SI and multiuser interference (MUI). Recently, non-orthogonal multiple access (NOMA) [Wong5Gbook17, Islam:CST17, DSP16, Nguyen:JSAC:17] has been introduced to improve the far users’ (i.e., the cell-edge users) throughput by allowing near users (i.e., the cell-center users) to access and decode their intended signals. In other words, far users must sacrifice their own information privacy in NOMA [Islam:CST17, DSP16, Nguyen:JSAC:17, Choi15, NguyenCLFT17]. In all aforementioned work in FD security, the number of transmit antennas is usually required to be larger than the number of users to efficiently manage the network interference. Otherwise, the AN and MUI will impair the channel quality of the desired users, leading to a significant loss in system performance.
In this paper, we propose a new transmission design to further resolve the practical restrictions given above. Specifically, the near DL users and far UL users are served in a fraction of the time block, and then FD-BS uses the remaining fractional time to serve far DL users and near UL users. It is worth noting that the effects of SI, CCI and MUI are clearly reduced while the information privacy for far DL users is preserved (all DL users are allowed to access and decode their intended signals only). On the other hand, FD-BS can effectively perform transmit beamforming even if the number of DL users exceeds the number of transmit antennas because the number of users that are served at the same time is effectively reduced. There are multiple-antenna Eves that overhear the information signals from both DL and UL channels. We are concerned with the problem of jointly optimizing linear precoders/beamformers at the FD-BS, UL transmit power allocation and fractional time (FT) to maximize the minimum SR among all legitimate users subject to power constraints. In general, such a design problem involves optimization of nonconcave and nonsmooth objective functions subject to nonconvex constraints, for which the optimal solution is computationally difficult to find. Note that SDR cannot be directly applied to such a challenging problem since the optimization problem resulting from the SDR is still highly nonconvex. This paper is centered on the inner approximation framework to directly tackle the nonconvexity of the considered optimization problem. The main contributions of this paper are summarized as follows:
We propose a new transmission model for FD security to optimize simultaneous DL and UL information privacy by exploring user grouping-based FT model, which helps manage the network interference more effectively than aiming to focus the interference at Eves.
We first assume perfect CSI to realize the potential benefits of our new model, for which a path-following computational procedure is proposed. The core idea behind our approach is to develop a new inner approximation of the nonconvex problem, which guarantees convergence at least to local optima. The convex program solved at each iteration is of moderate dimension since it does not require rank-constrained optimization, and thus, its computation is very efficient.
When only the statistics of CSI (SCSI) for Eves are known, we reformulate the optimization problem by replacing a nonconvex probabilistic constraint with a tractable nonconvex constraint which can be further shaped to have a set of convex constraints.
We determine the optimal solution for a worst-case scenario (WCS) of secure communications, where Eves adopt a more advanced linear decoder to cancel all multiuser interference.
Extensive numerical results show that the proposed algorithms converge quite quickly in a few iterations and greatly improve the SR performance over existing schemes, i.e., HD, conventional FD and FD-NOMA. It also confirms the robustness of the proposed approach against the significant effects of SI and DoF bottleneck.
I-C Paper Organization and Notation
The rest of the paper is organized as follows. The system model and problem statement are given in Section II. Path-following algorithms based on a convex approximation for the SR maximization (SRM) problem with known CSI and statistical CSI of Eves are developed in Section III and Section IV, respectively. Section V is devoted to the computation for the SRM-WCS problem. Numerical results are illustrated in Section VI, and Section VII concludes the paper.
Notation: Lower-case letters, bold lower-case letters and bold upper-case letters represent scalars, vectors and matrices, respectively. , and are the Hermitian transpose, normal transpose and conjugate of a matrix , respectively. The trace of a matrix is denoted by . , and denote a matrix’s Frobenius norm, a vector’s Euclidean norm and absolute value of a complex scalar, respectively. represents an identity matrix. means that
is a random vector following a complex circularly symmetric Gaussian distribution with meanand covariance matrix . denotes the statistical expectation. The notation () means that matrix is positive semi-definite (definite). represents the real part of the argument. is the gradient of .
Ii System Model and Problem Formulation
Ii-a Signal Processing Model
Consider a multiuser communication system illustrated in Fig. 1, where the FD-BS is equipped with transmit antennas and receive antennas to simultaneously serve DL users and UL users, respectively, over the same radio frequency band. Each legitimate user is equipped with a single antenna and operates in the HD mode to ensure low hardware complexity. Both DL and UL transmissions are overheard by non-colluding Eves, where Eve has antennas.111The scenario can be easily extended to colluding Eves by incorporating Eves into one with antennas. Herein, we use the most natural and efficient divisions of the coverage area [DSP16, Nguyen:JSAC:17]. In particular, users are randomly placed into two zones, such that there are DL users and UL users located in a zone nearer the FD-BS (referred to as zone-1 of near users), and DL users and UL users are located in a zone farther from the FD-BS (called zone-2 of far users) [DSP16]. Note that our proposed algorithms can be further adjusted to the case of different numbers of users located in each zone.
In this paper, we split each communication time block, denoted by , into two sub-time blocks orthogonally, as shown in Fig. 2. As previously mentioned, in order to mitigate the harmful effects of SI and CCI, far DL users (near DL users) and far UL users (near UL users) should be scheduled in a different time slot. The FD-BS serves the DL users (UL users) with similar channel conditions, which helps reduce the MUI. As a consequence, near DL users and far UL users are grouped into group-1 and are served in the first duration , while far DL users and near UL users are grouped into group-2 and are served in the remaining duration . Although each group still operates in the FD mode, the inter-group interference, i.e., interference across groups 1 and 2, is perfectly eliminated thanks to the FT allocation. This is in contrast with the conventional FD systems which simultaneously serves all UL and DL users. Without loss of generality, the communication time block is normalized to 1. Upon denoting and , the sets of DL and UL users are and for , respectively. Thus, the -th DL user and the -th UL user in the -th group are referred to as DL user and UL user , respectively.
1) Received Signal Model at the FD-BS and DL Users: We consider that the FD-BS deploys a transmit beamformer to transfer the information bearing signal with to DL user . Since all UL users have only a single antenna, they are unable to generate AN for jamming. To guarantee secure communication in both DL and UL channels, FD-BS also injects AN signals to interfere with the reception of the Eves. Hence, the DL transmit signals at the FD-BS intended for DL users in zone- can be expressed as
is the AN vector whose elements are zero-mean complex Gaussian random variables, i.e.,with . All channels are assumed to follow frequency-flat fading, which accounts for the effects of both large-scale path loss and small-scale fading. The received signal at DL user can be expressed as
where is the transmit channel vector from the FD-BS to DL user . In (II-A), the term represents the CCI from UL users to DL user , where , and with are the complex channel coefficient from UL user to DL user , transmit power and message of UL user , respectively. denotes the additive white Gaussian noise (AWGN) at DL user . By defining and , the information rate decoded by DL user in nats/sec/Hz is given by [Nguyen:TCOM:17]
where with and are the matrix encompassing the beamformers/precoders in the DL and transmit power allocation of all users in the UL in the -th group, and
The received signal at the FD-BS for reception of UL users in the -th group can be expressed as
where is the receive channel vector from UL user to the FD-BS. The term in (II-A) represents the residual SI after all real-time cancellation in analog and digital domains [Saetal14]; denotes a fading loop channel which impairs the UL signal detection at the FD-BS due to the concurrent DL transmission and is used to model the degree of residual SI propagation [Riihonen-SP-11]. denotes the AWGN at the FD-BS. To maximize the information rates of UL users, we adopt the minimum mean square error and successive interference cancellation (MMSE-SIC) decoder at the FD-BS [Tse:book:05]. For UL users in each group, we assume that the decoding order follows the UL users’ index, i.e., , with the Foschini ordering. In other words, the strongest signal is decoded first, while weakest signal is decoded last to support the most vulnerable UL users. Hence, the information rate in decoding UL user ’s message is given by [Nguyen:TCOM:17]
2) Received Signal Model at Eves: After performing handshaking with the FD-BS, we assume that the Eves are also aware of the FT . The information signals of group- leaked out to the -th Eve during the FT can be expressed as222Note that if the Eves are not aware of the FT , the received signals at Eve in (II-A) will include the inter-group interference, which leads to a lower bound on the information rate of the Eves. Such a design would be unfair to the Eves, and therefore, we do not pursue this here.
where and are the wiretap channel matrix and vector from the FD-BS and UL user () to the -th Eve, respectively. denotes the AWGN at Eve . The worst-case information (WCI) rates at the -th Eve, corresponding to the signals targeted for DL user and UL user , are given by
Ii-B Optimization Problem Formulation
The channel of each legitimate user together with Eves form a compound wiretap channel for which the SR expressions of DL user and UL user can be expressed as [Lian:EUR:09, SunTWC16]
respectively, where and .
We aim to jointly optimize the transmit information vectors, AN matrices ) and the FT to maximize the minimum (max-min) SR among all legitimate users. The SRM problem with Eves’ WCI rate, referred to as SRM-EWCI for short, can be mathematically formulated as
where and . Constraint (12a) means that the total transmit power at the FD-BS, which is allocated across different time fractions, does not exceed the power budget, , while constraints (13a) and (14a) are individual transmit power at UL user () in its service time, with being the power budget (see e.g., [Nguyen:TCOM:17, Nguyen:Access:17, Yang:TVT:17] for these realistic power constraints). Finding an optimal solution to the SRM-EWCI problem (11a) is challenging because the objective (11a) is nonconcave and constraints (12a)-(14a) are also nonconvex due to coupling between and .
In a practical scenario, the DL and UL traffic demands in current generation wireless networks are typically asymmetric. Another optimization problem of interest is to maximize the minimum SR of DL users subject to the SR constraints of UL users as follows:
where the QoS constraints in (19a) set a minimum SR requirement at UL user . It should be emphasized that the systematic approach in this paper is expected to be applicable for such a problem (this will be elaborated in Section VI).
Iii Proposed Method with Known CSI
In this section, the CSI of the users (including Eves) is assumed to be perfectly known at the transmitters. Channel reciprocity of UL and DL channels in time division duplex (TDD) mode can be adopted for small cell systems as those considered in this paper. The channels for all users (with a low degree of mobility) can be acquired at the FD-BS by requesting them to send pilot signals to the FD-BS, and thus these estimated channels can be assumed to be perfectly available[Nguyen:TCOM:17, Dan:TWC:14]. Likewise, the CSI between an UL user and the receivers (DL users and Eves) can be estimated through TDD since any transmitted signal includes short-training and long-training sequences (e.g., a part of preamble). This way, any UL user can overhear and estimate the channels from DL users and Eves, and then these estimated channels can be acquired at the FD-BS by polling each UL user. After CSI acquisition, we assume that only DL users and UL users are scheduled to be simultaneously served as in IEEE 802.11ac. Herein, unscheduled users are not necessarily malicious, but are untrusted users. Thus, unscheduled users are treated as eavesdroppers but with perfectly known CSI.
Iii-a Equivalent Transformations for (11a)
To solve the max-min SR problem in (11a), we present a path-following algorithm under which each iteration invokes only a simple convex program of low computational complexity. Toward a tractable form, several proper transformations need to be invoked. Let us start by expressing (11a) equivalently as
where is an additional variable to achieve the SR fairness among all DL and UL users. Note that the equivalence between (11a) and (20a) can be readily verified by checking that constraints (23a)-(23a) must hold with equality at optimum. We now provide a sketch of the proof to verify (23a), and other constraints follow immediately. Suppose that for some . Then, there may exist a positive constant to satisfy . As a consequence, is also feasible for (20a) but yielding a strictly larger objective, and thus, this is a contradiction with the optimality assumption. By observing that the objective function (20a) is monotonic in its argument, the main difficulty in solving (20a) is due to the nonconvex constraints (23a) and (23a). To provide a minorant of the SR, we further rewrite (20a) as follows:
where are newly introduced variables to tackle the maximum allowable rate of the Eves [Lui:SP:14].
However, problem (24a) still remains intractable since it is not amendable to a direct application of the inner approximation method. To this end, we make the variable change:
to equivalently rewrite (16a) by the following convex constraint
Iii-B Proposed Convex Approximation-Based Path-Following Method
We are now in a position to approximate the equivalent formulation in (36a). Note that except for (15a), (31) and (40a), the rest of the constraints are nonconvex. The proposed algorithm is mainly based on an inner approximation framework [Marks:78] under which the nonconvex parts are completely exposed.
Approximation of Constraints (32a): To develop a convex approximation, we first introduce the following approximation of function at a feasible point :
The proof of (41) is given in Appendix A. In the spirit of [WES06], for with , it follows that and for all . Thus, can be equivalently replaced by
with the condition
We make use of the following inequality
with due to the convexity of the function to further expose the hidden convexity of the right-hand side (RHS) of (44) as
over the trust region
Note that is a lower bounding concave function of , which also satisfies
As a result, (32a) can be iteratively replaced by the following inequality:
By defining , the left-hand side (LHS) of (33a) is lower bounded at the feasible point as
It follows from (50) that is a concave function, which agrees with at the feasible point as
Thus, constraint (33a) can be iteratively replaced by
which is a result of the concavity of the function . In the sequel, we develop an upper bound of the LHSs of (34a). Let us consider (34a) first. For at a feasible point and using (53), its LHS is upper bounded by