Secure Full-Duplex Two-Way Relaying for SWIPT

This letter studies bi-directional secure information exchange in a simultaneous wireless information and power transfer (SWIPT) system enabled by a full-duplex (FD) multiple-input multiple-output (MIMO) amplify-and-forward (AF) relay. The AF relay injects artificial noise (AN) in order to confuse the eavesdropper. Specifically, we assume a zeroforcing (ZF) solution constraint to eliminate the residual self-interference (RSI). As a consequence, we address the optimal joint design of the ZF matrix and the AN covariance matrix at the relay node as well as the transmit power at the sources. We propose an alternating algorithm utilizing semi-definite programming (SDP) technique and one-dimensional searching to achieve the optimal solution. Simulation results are provided to demonstrate the effectiveness of the proposed algorithm.


II. SYSTEM MODEL
We consider SWIPT in a three-node MIMO relay network with sources S A and S B , consisting of one transmit and receive antenna for information transmission and reception, respectively, exchanging confidential information with the aid of a multiantenna AF relay R, in the presence of a single antenna eavesdropper E. The relay harvests energy to complete the information exchange as also assumes in [3]. We assume that: i) S A , S B and R all operate in FD mode, ii) there is no direct link between S A and S B , and iii) the source nodes are not aware of any eavesdropper thus, no direct link exist between the source nodes and the eavesdropper [16]. The relay however, is aware of the eavesdropper. As a result, the relay injects AN signals to confuse the eavesdropper.
In the first phase, the relay receives confidential information from S A and S B , while in the next phase, R amplifies and forwards the processed information to both sources with the AN signal being superimposed to jam the eavesdropper [16]. The harvested energy at the relay is used to complete the bidirectional information exchange between the source nodes. Using the transmit power P A and P B , respectively, S A and S B transmit their confidential messages simultaneously to R. On the other hand, R employs linear processing with amplification matrix W to process the received signal and broadcasts the processed signal to the nodes with harvested power U.
The antennas at R are separated for transmission and reception with M T transmit antennas and M R receive antennas. Also, we denote h XR ∈ C MR×1 and h RX ∈ C MT ×1 to, respectively, represent the directional channel vectors between the source node X ∈ {A, B} and R. Similarly, we use h RE to denote the channel between E and R. To achieve FD communication, self-interference (SI) must be significantly suppressed, as total cancellation is not possible as a result of imperfect channel estimation [13]. Therefore, we adopt the use of existing SI cancellation mechanisms (e.g., antenna isolation, digital and analog cancellation, etc.), to reduce the effect of SI. For convenience, we denote h AA , h BB , and H RR ∈ C MR×MT as the residual SI (RSI) channels at the respective nodes [13]. Also, the RSI channel is represented as a Gaussian distribution random variable with zero mean and variance σ 2 X , for X ∈ {A, B, R} [13]. Furthermore, the relay, assumed to be equipped with a PS device, coordinates information decoding and energy harvesting. Specifically, the relay splits the received signal power such that a ρ ∈ (0, 1) portion of the received signal power is fed to the information receiver (IR) and the remaining (1 − ρ) portion of the power is fed to the energy receiver (ER) at the relay.

A. Signal Model
The received signal y r [n] and the transmit signal x R [n] at R at time instant n, can be written, respectively, as where τ is the processing delay to implement FD operation and assumed short enough to be neglected as far as the achievable rate computation is concerned, n R ∽ CN (0, σ 2 R I) is the additive white Gaussain noise (AWGN) at R, z[n] ∽ CN (0, Q), with Q 0, is the AN used for interfering E, and y IR R [n] is the signal split to the IR at R given by (3) Thus, the signal transmitted by R can then be expressed as As shown in [17], the relay output can be further written as where j denotes the index of the delayed symbols. We define the covariance matrix of (5) as Clearly, the relay's transmit covariance is indeed a complicated function of W. In this letter, we adopt the zeroforcing (ZF) solution constraint to cancel the RSI from the relay output to the relay input via the optimization of W [13]. In particular, the ZF constraints may take the following forms [16] For convenience, we only consider the case M T > M R as the other case can be handled similarly. Thus, (5) becomes with the relay output power expressed as In the second time slot after cancelling the SI signal s A [n−τ ], the received signal at S A is given as where n A [n] is the AWGN at source node S A . From this, we can work out the rates at S A and S B as where . (13) The signal received at E can be expressed as where n E is the AWGN at E. Also, the achievable sum-rate at E is upper bounded as The achievable secrecy sum-rate is then defined as [16] where [x] + represents max(x, 0). Meanwhile, the signal split to the ER at R is given by The harvested energy at the relay is thus given as [13] and β denotes the energy conversion efficiency of the ER at the relay which is assumed unity.

B. Problem Statement
Due to the inherent SI at each FD node, the source nodes may not use the maximum available transmit power in order not to increase the level of SI. Thus, there is a need to transmit at optimum values. Furthermore, it is known that optimal values of system parameters guarantees that the secrecy rate is as large as possible [16]. Thus, in this letter, our aim is to maximize the secrecy sum-rate for SWIPT by ensuring system parameters are optimal. We achieve this by jointly optimizing the transmit power at the source nodes (P A , P B ), the relaying matrix (W) and the AN covariance matrix (Q) at the relay. Thus, we have min ρ∈(0,1),W,Q 0 0<P A ≤Pmax ,0<P B ≤Pmax As (18) is nonconvex, we solve (18) in an alternating manner.

III. PROPOSED SCHEME
A. Optimization of W and Q at the Relay Here, we study the optimal beamforming matrix and the AN covariance matrix assuming the source power (P A , P B ) and the PS ratio (ρ) all being fixed. For convenience, we define W = N t V, where N t ∈ C MT ×MT represents the null space of H RR , and V ∈ C MT ×MT is the new optimization variable. As a consequence, the optimization of W reduces to optimizing V. Hence, we remove the ZF constraint in (18) and obtain the equivalent optimization problem: (19) is a nonconvex problem due to the coupled optimization variables in the constraints. However, by rearranging the terms in the constraints, (19) can be re-expressed as Problem (20) can be efficiently solved by existing solvers such as CVX [18]. Once the optimal Σ is obtained, optimal V can be constructed through matrix decomposition.

B. Optimization of the PS Coefficient (ρ)
For fixed values of the relay beamforming matrix (W), AN covariance (Q) and for given values of the transmit power (P A , P B ) at the sources, (18) can be reformulated as min ρ∈(0,1) which can be expressed in a form solvable by existing solvers by rearranging the terms in the constraints as min ρ∈{0,1} C. Optimization of the Source Power (P A , P B ) For given values of the relay beamforming matrix (W), AN covariance matrix (Q) and the relay PS ratio, problem (18) can be written as min PA,PB It is worth noting that full-duplexity in communication systems is preceded by successful SI cancellation. In our model, the source nodes are equipped with a single transmitter-receiver pair for signal transmission and reception, respectively. As a result, it is impossible to cancel the SI in the spatial domain [13]. The relay, in contrast, equipped with at least two transmitter-receiver pairs, can cancel the generated SI in the spatial domain. We proceed to investigate the optimal power solution (P A , P B ) assuming W, Q and ρ all being fixed. Evidently, it is easy to check that at the optimum, at least one source will be transmitting with maximum power [13] i.e., P A = P max or P B = P max . As a consequence, we can relax (23) into two sub-problems with: (i) P A = P max , (ii) P B = P max . Considering the symmetric nature of case (i) and case (ii), we study case (i) as an example and solve problem (23) analytically. Problem (23) is thus reformulated as we can obtain the feasible range [P min B , P max B ] for P B . Also, the constraints in (24) can be analysed with respect to P B : 1) A continuous increase in P B should guarantee that (24b) remains satisfied. As a consequence, we can set the minimum of P B as P min 2) Constraint (24c) is a decreasing function of P B . Therefore, the maximum P B satisfying (24c) to equality is defined as P max 3) An upper bound of the eavesdropping constraint in (24d) is satisfied when P B ≤ P max B . 4) A lower bound of the energy harvesting constraint in (24e) is guaranteed to be satisfied when P B ≥ P min B . The optimal P * B is chosen between P min B and P max B which satisfies (24b)-(24e). Accordingly, to obtain the optimal P B , we perform a 1-D search over P B starting from P min

IV. SIMULATION RESULTS
In this section, we present numerical results to investigate the performance of the proposed scheme through computer simulations. We consider a Rayleigh flat fading channel. The results are averaged over 1000 independent realizations and SINR at node A, node B and the eavesdropper is given, respectively, as γ A = −5(dB), γ B = −5(dB), γ E = −15(dB). We also assume that 60% of the SI at node A and node B has been eliminated through digital cancellation [13].
In Fig. 1, we investigate the secrecy sum-rate for SWIPT in FD systems versus the transmit power budget P max (dB) for different values of the harvested power constraint. We study the performance of the proposed scheme (denoted 'Joint Opt.' in the figure) in comparison with the relay-only optimization scheme (denoted 'Relay Only Opt.'). We see that the proposed scheme yields a higher secrecy sum-rate than the relay-only optimization scheme. Also, it can be observed that the secrecy sum-rate increases with P max to a certain level after which it begins to experience a decrease with a continuous increase in P max , due to the increase of RSI [19]. In Fig. 2, we investigate further the secrecy sum-rate performance against the RSI for different values of the harvested power constraints. Evidently, as RSI increases, a corresponding decrease in the secrecy sum-rate is observed. However, the proposed scheme yields higher secrecy sum-rate compared to the secrecy sum-rate of the relay-only optimization scheme for different values of the harvested energy constraint. Hence, the need for joint optimization is justified.

V. CONCLUSION
This letter investigated the joint optimization of the source transmit power, AN covariance matrix, and the relay beamforming matrix for SWIPT in FD AF relaying system in the presence of an eavesdropper. Specifically, using SDP and 1-D searching, we proposed an algorithm that minimizes the total transmit power for secure SWIPT in a FD MIMO AF relay system.