Robust MIMO Harvest-and-Jam Helpers and Relaying for Secret Communications

Cooperative jamming has been demonstrated to be an effective means to provide secret wireless communications and yet this is realized at the expense of the power consumption of the friendly jammers. This paper considers the scenario where friendly jammers harvest energy wirelessly from the signal transmitted by the source, and then use only the harvested energy to transmit the jamming signals for improving the secrecy of the communications of the source to the destination. In particular, we investigate the use of multi-antenna harvest-and-jam (HJ) helpers in a multi-antenna amplify-and-forward (AF) relay wiretap channel assuming that the direct link between the source and destination is broken. Our goal is to maximize the achievable secrecy rate at the destination subject to the transmit power constraints of the AF relay and HJ helpers. In the case of perfect channel state information (CSI), the joint optimization of the artificial noise (AN) covariance for jamming and the AF beamforming matrix is presented as well as a suboptimal solution with lower complexity based on semidefinite relaxation (SDR) which is tight in this case. For the practical case where the CSI is imperfect at both the relay and the HJ helpers, we provide the formulation of the robust optimization for maximizing the worst-case secrecy rate. Using SDR techniques, a near-optimal robust scheme is proposed. Numerical results are provided to validate the effectiveness of the HJ protocol.


I. INTRODUCTION
The pressing demand for high data rate in wireless communications networks coupled with the fact that mobile devices are physically small and power-limited by batteries, has driven the notion of energy harvesting (EH) to become a promising resolution for green communications [1,2]. Among the varied available resources for EH, radio-frequency (RF)-enabled wireless energy transfer (WET) has aroused an upsurge of interest for its long operation range, ubiquitous existence in the electromagnetic radiation, and effective energy multicasting, which motivates the paradigm of simultaneous wireless information and power transfer (SWIPT), e.g., [3][4][5][6]. A typical SWIPT system consists of one access point (AP) that has constant power supply and broadcasts wireless signals to a group of user terminals, amongst which some intend to decode information, referred to as information receivers (IRs), while others scavenge energy from the ambient radio signals, named energy receivers (ERs). This gives rise to a challenging physical (PHY)-layer security issue where the ERs may eavesdrop the information sent to the IRs due to This paper has been presented in part at the IEEE Global Communications Conference (GLOBECOM), Austin, TX, USA, December 8-12, 2014. H. Xing and A. Nallanathan are with the Centre for Telecommunications Research, King's College London (e-mails: hong.xing@kcl.ac.uk; arumugam.nallanathan@kcl.ac.uk).
K.-K. Wong is with the Department of Electronic and Electrical Engineering, University College London (e-mail: kai-kit.wong@ucl.ac.uk).
Z. Chu is with the School of Electrical and Electronic Engineering, Newcastle University (e-mail: z.chu@ncl.ac.uk).
their close proximity to the AP To overcome this problem, in [7][8][9], several researchers presented various approaches to guarantee secret communication to the IRs and maximize the energy simultaneously transferred to the ERs or to satisfy the individual EH requirement for the ERs and maximize the secrecy rate for the IR, by advocating the dual use of the artificial noise (AN) or jamming.
However, previous works all assumed that the ERs in the SWIPT systems attempt to intercept the information for the IR, which is overly protective. On the contrary, it is possible that some ERs are cooperative, especially when they are EHenabled wirelessly. Following the recent advances in wireless powered communications networks [10,11], this paper proposes a self-sustaining harvest-and-jam (HJ) relaying protocol, where in the first transmission phase a single-antenna transmitter transfers confidential information to a multiple-antenna amplify-and-forward (AF) relay and power to a group of multiantenna EH-enabled idle helpers simultaneously, while in the second phase, the relay amplifies and forwards the information to the IR under the protection of the AN generated by the helpers using the energy harvested from their received signals in the first transmission phase.
Physical (PHY)-layer security issues in the rapidly growing cooperative networks have attracted much attention. Cooperative approaches, such as, cooperative jamming, communications have been widely examined [12][13][14][15]. The idea is to assist the transmitter in the secrecy transmission by generating an AN to interfere with the eavesdropper via either multiple antennas or external trusted helpers [16][17][18][19]. However, all of those utilizing ANs require additional supply of power and therefore incur extra system costs. Meanwhile, collaborative use of relays to form effective beams jamming the eavesdropper, i.e., secure collaborative relay beamforming, has been studied for relay-wiretap channels with single eavesdropper in [20], multiple eavesdroppers with AF relays and decodeand-forward (DF) relays in [21] and [22], respectively. All, however, assumed the availability of perfect channel state information (CSI). Though [23] proposed robust AF relay beamforming against the eavesdropper's channel, the solutions were yet suboptimal.
The assumption of perfect CSI of the eavesdroppers appears to be too ideal because the eavesdroppers, despite being legitimate users, wish to hide from the transmitter without being cooperative in the stage of channel estimation. Even if they are registered users and bound to help the transmitter in obtaining their CSIs to facilitate their own communication, the CSIs at the transmitter side will change due to mobility and Doppler effect, and may be outdated. Moreover, even for the legitimate users, the estimated CSIs may also be subject to quantization errors due to the limited capacity of the feedback channel, although the inaccuracy is reasonably assumed less severe than that for the eavesdroppers. To tackle this issue, state-of-art schemes have been developed ( [24] and the references therein), among which the worst-case secrecy rate is commonly employed to formulate the robust secrecy rate maximization problem [8,19,[25][26][27]. The robust transmit covariance design for the secrecy rate maximization in a multiple-input-single-output (MISO) channel overheard by multi-antenna eavesdroppers was considered in [25,28] while the enhanced secrecy performance was achieved by introducing a friendly jammer in the same scenario in [26], in which a joint optimization of the robust transmit covariance and power allocation between the source and the helper was studied via geometric programming. More recently, [8] studied a joint robust design of the information beams, the AN and the energy signals for SWIPT networks with quality-of-service (QoS) constraints.
The contribution of this paper is threefold. First, with perfect CSI, in addition to the joint optimal solutions, we propose two near-optimal schemes with much reduced complexity by exploiting the optimal structure of the relay weight matrix, and providing a semi-closed form solution for the relay weight matrix given fixed null-space jamming, respectively. Second, besides the imperfect eavesdropper's channel, legitimate channels such as those from the K HJ helpers (the transmitter) to the legitimate receiver (K HJ helpers), and from the AF relay to the receiver are jointly modeled with imperfect estimation, and multiple semi-indefinite non-convex constraints have been judiciously replaced by linear matrix inequalities (LMIs) to fit the semi-definite programming (SDP). Third, a rank-one reconstruction algorithm exploiting the structure of the semidefinite relaxation (SDR)-based solutions has been proposed to provide promising performance at low computational cost.
Of particular relevance to our work is [29] which jointly optimizes the AF matrices and AN covariances in a relay wiretap channel with multiple multi-antenna AF relays and multiple multi-antenna eavesdroppers via a worst-case robust formulation. While our network model is similar, the difference of our work from [29] is twofold. On one hand, in this paper, the AN generated by the friendly jammers are subject to their respective channels from the transmitter during WET in the first transmission phase. On the other hand, the technique in [29,Proposition 1] cannot be applied to our problem since the AN beams and the forwarded information are transmitted via different channels in ours. As a consequence, to the best of authors' knowledge, our proposed worst-case based robust optimization scheme that incorporates imperfect CSIs into all the HJ helpers, has not been addressed in the literature.
It is worth noting that devising a wireless-powered friendly jammer to enhance PHY-layer security for a direct transmission protocol was studied in [30], in which the "harvesting" blocks and "jamming" blocks were well exploited to compose four different types of harvesting-jamming cycles. Compared to [30], which focused on the dedicated scheduling of "harvest" and "jam" operations and its long-term performance, ours are concerned with adaptive rate/power optimization with multiple HJ helpers to achieve higher worst-case secrecy rate. Moreover, instead of assuming perfect channels to/from the HJ helpers, our robust optimization algorithm takes imperfect legitimate channels into account to provide robustness.
Note that in this paper, as in [23,29], we assume that the channel between the transmitter and the AF relay is perfectly known and there is no direct link between the transmitter and the receiver or the eavesdropper, a common assumption in the concerned AF relay wiretap channel [20,21].
Notations-Throughout, we use the upper case boldface letters for matrices and lower case boldface letters for vectors. The superscripts (·) T , (·) † and (·) H represent the transpose, conjugate and conjugate transpose, respectively. Also, tr(·) and E[·] stand for the trace of a matrix and the statistical expectation for random variables, respectively. Likewise, vec(A) is defined as a column vector obtained by stacking the rows of A on top of one another. vec (−1) is the inverse operation of vec. null(A) denotes the null space of A. ⊗ represents the Kronecker product of two matrices. In addition, the notation A 0 indicates that A is a positive semi-definite matrix and I (0) denotes an identity (all-zero) matrix with appropriate size. Furthermore, · represents the Euclidean norm of a vector, while P r (·) stands for the probability of an input random event. Finally, [x] + denotes max(0, x) and (·) * stands for an optimal solution.

II. NETWORK MODEL
We consider a cooperative relay wiretap channel for SWIPT over a given frequency band as shown in Fig. 1(a). We assume that there is a transmitter, named Alice, sending confidential messages to the IR, Bob, in the presence of an eavesdropper [31], Eve, with the aid of a multi-antenna AF relay and K ERs willing to act as HJ helpers, H helper = {H 1 , . . . , H K }. The transmitter, ERs, and the AF relay are deployed in a same cluster that is relatively far away from the destination and Eve, such that there is no direct link from the transmitter to the receiver or Eve, respectively. Moreover, the ERs are assumed to be located closer to the transmitter than the AF relay in order that they can harvest sufficient amount of energy for jamming. Alice, Bob and Eve are all assumed to be equipped with single antenna, while the AF relay and each of the K helpers are assumed to have the same N t antennas.
Using two equal slots for the HJ relaying protocol, as shown in Fig. 1(b), for the first phase, Alice sends a confidential message to the relay while simultaneously transferring energy to the K helpers; for the second phase, the relay amplifies and forwards the message to Bob while the K helpers perform cooperative jamming using their respective harvested energy from the first transmission phase, to compromise Eve. In this paper, we assume a quasi-static fading environment and for convenience denote h 0 ∈ C Nt×1 as the complex channel from the transmitter to the relay and h k ∈ C Nt×1 , k = 1, . . . , K, as that from the transmitter to the kth helper;h 0 as the transpose of the complex channel from the relay to Bob andh k ∈ C Nt×1 , k = 1, . . . , K, as that from H k to Bob; g 0 ∈ C Nt×1 and g k ∈ C Nt×1 , k = 1, . . . , K, as those from the relay and H k to Eve, respectively.  In the first transmission phase, the baseband received signal at the AF relay can be expressed as where s is a circularly symmetric complex Gaussian (CSCG) random variable, denoted by s ∼ CN (0, 1) and n r is the additive complex noise vector, denoted by n r ∼ CN (0, σ 2 r I). Also, P s denotes the given transmit power at Alice. Further, the received signal at each helper H k is expressed as where n ′ k is the additive noise, denoted by n ′ k ∼ CN (0, σ 2 h I). On the other hand, for WET, the harvested energy of H k in each unit slot is given by where 0 < η ≤ 1 denotes the EH efficiency.
In the second transmission phase, the linear operation at the AF relay can be represented by where x ′ ∈ C Nt×1 is the retransmit signal at the AF relay and W ∈ C Nt×Nt is the beamforming matrix. Note that the transmit power of the AF relay can be shown as which is constrained by the maximum available power at the AF relay, i.e., P r , which is given by In the meantime, each H k will help generate an AN n k ∈ C Nt×1 to interfere with Eve. Similar to [16], we assume that n k 's are independent CSCG vectors denoted by n k ∼ CN (0, Q k ), ∀k, since the worst-case noise for Eve is known to be Gaussian. In addition, each H k has a transmit power constraint due to its harvested energy in the previous transmission phase, i.e., tr(Q k ) ≤ ηP s h k 2 (c.f. (3)), ∀k. The received signal at Bob can thus be expressed as where n b ∼ CN (0, σ 2 b I) is the additive noise at Bob. Similarly, the received signal at Eve can be expressed as where n e ∼ CN (0, σ 2 e I). According to (7) and (8), the signalto-interference-plus-nose-ratio (SINR) at Bob and Eve can be, respectively, expressed as and As such, the achievable secrecy rate at Bob is [16] III. JOINT AN-AF BEAMFORMING WITH PERFECT CSI

A. Problem Formulation for Perfect CSI
We aim to maximize the secrecy rate at Bob subject to the transmit power constraints at the AF relay and each individual helper H k , k = 1, . . . , K. Thus, our problem is to solve Next, we define a new functionF ({Q k }, W ) as It can be easily shown that the optimal solution {Q * k }, W * to (P1), is also optimal for (P1 ′ ) given by Hence, we focus on solving problem (P1 ′ ) in the rest of the paper. However, since (P1 ′ ) is in general a non-convex problem that is hard to solve, we will reformulate it into a two-stage optimization problem. First, we constrain the SINR at Eve to beγ e , it thus follows from (13) is maximized when γ b is maximized, which can be obtained by solving the following problem: (12a) − (12c). (15) Let H(γ e ) denote the optimal value of (P1 ′ .1) givenγ e . Then (P1 ′ ) can be equivalently solved by Lemma 3.1: Problem (P1 ′ ) has the same optimal value as (P1 ′ .2), and the same optimal solution as (P1 ′ .1) whenγ e takes the optimal solution for (P1 ′ .2).

B. Optimal Solution to
Here, we consider solving problem (P1 ′ .1) by jointly optimizing the covariance matrix for the AN at each of the HJ helper, Q k 's, and the beamforming matrix, W . To facilitate the analysis in the sequel, we rewrite the following equations in line with our definition of vec(·) [32, Chapter 13]: In addition, Then by ignoring the rank-one constraint on X, (P1 ′ .1-RW) is modified as , via Charnes-Cooper transformation [33], can be equivalently recast as

Lemma 3.2:
The constraints in (23a) and (23b) can be replaced by , respectively, where both inequalities will be activated when problem (P1 ′ ) obtains its optimum value.
Proof: See [34, Appendix A]. Since problem (P1 ′ .1-RW-SDR) is a standard convex optimization problem and satisfies the Slater's condition, its gap with its dual problem is zero [35]. Now, let λ denote the dual variable associated with the equality constraint in (23a), α associated with the other equality constraint in (23b), β 0 associated with the transmit power constraint for the AF relay in (23c), {β k } associated with the transmit power constraints for each H k in (23d), and ζ associated with τ . Then the Lagrangian of problem (P1 ′ .1-RW-SDR) is given by (24) where Ω denotes the set of all primal and dual variables, 2) X * can be expressed as where a n ≥ 0 ∀n, b > 0, r c = rank(C * ) (c.f. (88)) and , which consists of orthonormal basis for null(C * ); 3) According to (28), if rank(X * ) > 1, then we have the following sufficient condition to yield an optimal solution of X with rank-one: is also optimal to problem ( Proof: See Appendix A. Note from Proposition 3.1 that if rank(X * ) = 1, then the optimal w * to (P1 ′ .1-RW) can be found directly from the eigenvalue decomposition (EVD) of X * , where X * = X * /τ * . Namely, the upper-bound optimum value obtained by solving (P1 ′ .1-RW-SDR) is tight in this case; otherwise, (X * , {Q * k }, τ * ) only serves as an upper-bound solution. Now, we show that this upper-bound is always achievable by a rank-one X * . When rank(X * ) > 1, firstly, we check whether the sufficient condition proposed in (32) is satisfied. If it is met, then a direct reconstruction of (X * , {Q * k },τ * ) with rank(X * ) = 1 follows according to (29)-(31); otherwise, assume that any optimal solution to problem (P1 ′ .1-RW-SDR) has no zero component, i.e., (X * = 0, {Q * k = 0}, τ * = 0). In addition, the number of optimization variables and the number of shaping constraints are denoted by L and M , respectively. Since L = K + 2 and M = K + 3 for (P1 ′ .1-RW-SDR), we have M ≤ L + 2 satisfied. Thus, according to [36,Proposition 3.5], (P1 ′ .1-RW-SDR) has an optimal solution ofX * that is rank-one. Also, the detailed rank reduction procedure based on an arbitrary-rank solution has been given in [  • Initializeγ e search = 0 : α :γe max and i = 0 (29)- (31) and set w * = √ bξ; else constructX * using the procedure in [36,Algorithm 1].

Lemma 3.3:
The optimal relay beamforming matrix W for problem (P1 ′ .1) is of the form: where B ∈ C 2×2 and C ∈ C 2×(Nt−2) are two unknown matrices, and respectively. Since C has 2(N t − 2) complex variables, we devise a suboptimal design for C to reduce the size of variables by suboptimal design for problem (P1 ′ .1) by ignoring the rank constraints on Z and V is thus given by , are of much reduced size. Further, the reconstruction of v * from V can be briefly explained as follows. Given the Lagrangian of (P1 ′ .1-sub1-SDR), the KKT conditions with respect to (w.r.t.) V * are given by Post-multiplying (37) is an arbitrary vector with unit norm. With V solved, (P1 ′ .1-sub1-SDR) reduces to a problem with similar structure as (P1 ′ .1-RW-SDR), and the proof for existence of a rank-one Z can be referred to Proposition 3.1.
2) Zero-forcing: We propose a low-complexity ZF scheme for (P1 ′ .1), in which the jamming signal places a null at Bob, and then a semi-closed form solution for W is derived. In line with the principle of ZF jamming [17], the jamming signal n k is designed as Thus, given any W ,ñ k 's can be optimized to maximize the effect of jamming at Eve by max whereg k =Ṽ T k g k , and ζ k = √ ηP s h k / g k is determined by (23d), ∀k. As such, K k=1 g T kṼ kQ * kṼ H k g † k turns out to be K k=1 ηP s h k 2 g k 2 , which is denoted by q.
Proof: See Appendix C. The only problem in Proposition 3.2 is the dual problem of (P1 ′ .1-sub2-SDR), which admits a much simpler structure to solve than the primal one.

A. Problem Formulation for Imperfect CSI
We use a deterministic spherical model [25,26] to characterize the resulting CSIs' uncertainties such that whereĝ 0 ,ĝ k 's,ĥ 0 ,ĥ k 's andĥ k 's are the estimates of the corresponding channels; ∆g 0 , ∆g k 's, ∆h 0 , ∆h k 's and ∆h k 's are their respective channel errors; the matrices W 0 , W k 's, W ′ 0 , W ′′ k 's and W ′ k 's determine the shape of each error region. Without loss of generality (w.l.o.g.), we set and ǫ ′′ k represent the respective size of the bounded error regions, k = 1, . . . , K.
As stated in Lemma 3.1, similarly, (P2 ′ ) can be proved to have the same optimal value as (P2 ′ .2) and the same optimal solution as (P2 ′ .1) whenγ e takes its optimal value. As a result, (P2 ′ ) can be solved in a two-stage fashion as well. Specifically, given anyγ e , we first solve (P2 ′ .1) to obtain H(γ e ) and then search for the optimalγ e to (P2 ′ .2).
B. Solutions to (P2 ′ .1) By ignoring (44d), (P2 ′ .1) is recast as It is worth noting that due to the rank-one relaxation of (P2 ′ .1-RW-SDR-Eqv), solution provided by (P2 ′ .1-RW-SDR-Eqv) in general yields an upper-bound for H(γ e ), which may not be achievable. However, in the sequel we insist on solving (P2 ′ .1-RW-SDR-Eqv) that is regarded as an upper-bound benchmark for our proposed problem detailed later in this subsection.
In the following, we equivalently transform these constraints to tractable ones using S-Procedure and a generalized S-Procedure given in Lemmas 4.1 and 4.2, respectively.
where A m = A H m ∈ C N ×N , b m ∈ C N ×1 and c m ∈ R, and ℜ gives the real part of the input entity. Then the implication provided there exists a pointx such that f m (x) > 0, m = 1, 2.
is equivalent to First, by rearranging terms, (46a) can be equivalently transformed into the following linear form: Recalling the following matrix equalities in line with our definition of vec(·) operation: it follows that (41)). By introducing X ′′ = (h 0 ⊗ I)X(h H 0 ⊗ I) and X ′ = I ⊗ X, (51) can thus be recast as Hence, according to Lemma 4.1, the implication ∆h 2 ≤ N t ǫ ′ 0 ⇒ (57) holds if and only if there exists w (0) ≥ 0 such that the following LMI holds: where . Now, (46a) has been equivalently reformulated as (58). To further cope with channel uncertainties with regards toh k 's such that (58) holds for h k ∈H k , k = 1, . . . , K, we need the following proposition.
Proof: See Appendix D. Next, (44a) is rewritten as where g ∈ C N 2 t ×1 = vec(g T 0 ⊗I) and the equivalent imperfect channel model is given by g =ĝ + ∆g such that ∆g 2 ≤ N t ǫ 0 .

Proposition 4.2:
The semi-indefinite constraint of (62) is satisfied if and only if there exists v (k) ≥ 0, k = 1, . . . , K, such that the following block matrix inequality holds: where H  Proof: See Appendix E. Last, we rewrite (44c) to facilitate the robust optimization against the errors introduced by ∆h k 's. By applying Lemma 4.1, (44c) holds if and only if there exists µ k ≥ 0, k = 1, . . . , K, such that the following LMI constraint is met: Because of the non-convex term such as δX ′ in (59), problem (P2 ′ .1-RW-SDR-Eqv) remains very hard to solve. We thus use the bisection method [35] w.r.t. δ to solve it. However, using bisection in addition to solving (P2 ′ .2) by one-dimension search overγ e may lead to very high complexity. As a result, we propose an alternative problem to approximateĤ(γ e ).

Remark 4.1:
It is worth noting that as the numerator and the denominator of the objective function in (P2 ′ .1) are coupled by common uncertaintyh 0 , Charnes-Cooper transformation, in general, cannot be applied to realize equivalent decoupling. As a result, (P2 ′ .1-RW-SDR) yields a more conservative approximation forĤ(γ e ) than (P2 ′ .1-RW-SDR-Eqv). However, considering that (P2 ′ .1-RW-SDR) needs to be solved only once for givenγ e in contrast with (P2 ′ .1-RW-SDR-Eqv) requring isection over δ, we exploit it in the sequel. The effectiveness of this approximation will be evaluated in Section V-B.

Proposition 4.3:
The semi-indefinite constraints in (72) can be replaced by the following LMI constraint:   H where H (k) and F (k) are recursively given by , k = 1, . . . , K, and {s (k) ≥ 0} denote the auxiliary variables.
Proof: It is observed that (68c) differs from (44a) in the only respect that σ 2 e is replaced by τ σ 2 e . Hence the proof for Proposition 4.2 can be directly applied herein by substituting τ σ 2 e for σ 2 e . Last, by replacing "ηP s " in (44c) with "τ ηP s " in (68e), (68e) can be replaced by a similar LMI as (67), denoted by (68e ′ ), in which the pertinent auxiliary variables are denoted by {µ k ≥ 0}.
Consequently, the equivalent reformulation for problem (P2 ′ .1-RW-SDR) can be summarized as is convex and can be solved efficiently by convex optimization tools such as CVX. Next, we derive the Lagrangian of (P2 ′ .1-RW-SDR). Note that in the following expression, we only consider the uncertainties regardingh 0 , h k 's,h k 's, g 0 and g k 's when K = 1 for the purpose of simplicity and the results can be easily extended to the case of K > 1. Denote the dual variables associated with (68d), (70), (73) and (75) by β 0 , W , V and Y , respectively. Then the partial Lagrangian of (P2 ′ .1-RW-SDR) w.r.t. X is where Ω is the set of all primal and dual variables, and In (79),F 1 =f † 1f T 1 ; W i,j , i, j = 1, 2, V i,j , i, j = 1, . . . , 3 and Y i,j , i, j = 1, . . . , 3 are the block submatrices of W ∈ C (N 2 and Y (i,i) 2,2 are the ith block diagonal submatrices of Y 2,1 ∈ C N 3 t ×N 3 t , and Y 2,2 ∈ C N 3 t ×N 3 t , respectively. Proposition 4.5: 1) The optimal X * to (P2 ′ .1-RW-SDR) is expressed as whereā n ≥ 0, ∀n,b > 0, andξ ∈ C N 2 t ×1 is a unit-norm vector orthogonal toΞ (c.f. (28)).

V. NUMERICAL RESULTS
Here we provide numerical examples to validate our results. We assume a typical scenario where the K helpers are evenly distributed around Alice with a radius of ρ k = 2m and θ k = 2π(k−1) K (radian by default), where θ k is the angle of direction (w.r.t. the Alice-relay link by default) of the kth helper, k = 1, . . . , K. Alice, Bob and Eve are, w.l.o.g., assumed to have the same distance away from the AF relay with their angle of direction π, π/6 and 11π/6, respectively. We also assume channel models with both large-scale fading, i.e., path loss and shadowing, and small-scale fading, i.e., multi-path fading. The simplified large-scale fading model is given by [39] where z is a log-normal random variable capturing the effect of shadowing with the standard derivation σ = 4dB, A 0 = 10 −3 , d is the distance, d 0 is a reference distance set to be 1m, and α = 2 is the path loss exponent. Specifically, the channels including h k 's, h 0 ,h 0 and g 0 , are assumed to suffer from Rician fading while the channels from the HJ helpers to Bob (h k 's) and Eve (g k 's) follow Rayleigh distribution due to the missing of line-of-sight (LOS) components with their respective average gain specified by (84). Take h k , ∀k, as 84)),ȟ k is the Rayleigh fading component denoted byȟ k ∼ CN (0, DI), and K R is the Rician factor set to be 3. Note that for the involved LOS component, we use the far-field uniform linear antenna array to model the channels [40]. In addition, unless otherwise specified, the number of HJ helpers, K is set to be 5; the AF relay is assumed to be 5m away from Alice; the EH efficiency, η = 0.5 and σ 2 r = σ 2 b = σ 2 e = −50dBm. The results presented in Section V-A are obtained by averaging over 500 times of independent trials.

A. The Perfect CSI Case
We compare the proposed optimal solutions with three suboptimal schemes in the case of perfect CSI. One suboptimal scheme, denoted by "Suboptimal 1", is introduced in Section III-C1 by exploiting the optimal structure of W . The other described in Section III-C2 is known as optimal null-space ZF, denoted by "Suboptimal 2". Specifically, each jamming beam n k is restricted to lie in the orthogonal space ofh † k such that n k 's cause no interference to the IR while maximizing its effect of jamming at the eavesdropper. As a benchmark, we also present the well-known isotropic jamming that is particularly useful when there is no Eve's CSI known at each HJ helper, H k , ∀k [41], denoted by "Suboptimal 3". Note that the difference between "Suboptimal 2" and "Suboptimal 3" only lies in the design of jamming noise, for which the former also aligns the jamming noise to an equivalent Eve's channel to confront Eve with most interference, while the latter transmits isotropic jamming withñ k ∼ CN (0, ηP s h k 2 /(N t − 1)), k = 1, . . . , K, in directions orthogonal toh k 's, due to lack of knowledge of Eve's channel and thus is expected to be less efficient than "Suboptimal 2" with perfect CSI.
First, we study the secrecy rate at the receiver versus the transmit power of the transmitter, P s with P r = 10dBm. Fig. 2 demonstrates that for both cases of N t = 3 and N t = 5, the average secrecy rate increases and tends to be saturated as P s goes to 30dBm. It also illustrates that "suboptimal 1" and "suboptimal 2" closely approach the optimal solutions while "Suboptimal 3" is outperformed more succinctly with larger number of antennas at the AF relay and the HJ helpers. Moreover, with N t increasing, the average secrecy rate gets larger as a result of the higher array gain of the AF relay and more available power transferred to the HJ helpers.
In addition, we show in Fig. 3 the secrecy rate achieved by different schemes versus the transmit power of the AF relay, P r with P s = 30dBm. It is seen that the average secrecy rate first grows faster and then slower, since when P r increases, not only the desired signal but also the noise yielded from the first transmission phase is amplified to a larger extent. In addition, the performance gap between the optimal scheme and suboptimal schemes is almost negligible. Similar to Fig. 2, "Suboptimal 3" appears to have certain performance loss from the optimality but is considered as a promising scheme when no Eve's CSI is available at the HJ helpers.

B. The Imperfect CSI Case
Now, we consider the imperfect CSI case and compare the proposed scheme Robust SDR with HJ, which is obtained by solving (P2 ′ . solving the rank constraint relaxed problem (P2 ′ .1-RW-SDR) ((P2 ′ .1-RW-SDR-Eqv)). On the other hand, for Robust w/o HJ, we solve (P2 ′ .1-RW-SDR) by setting Q k = 0, ∀k while for Non-robust with HJ, (11) is evaluated by applying the optimal solutions to (P1 ′ .1) assuming perfect CSI, to the actual channels including errors that are generated from the sets defined in (41).
To assess the worst-case secrecy performance, we use the metric, namely, secrecy outage probability, defined as [42]: where r * 0 obtained by solving (P2 ′ ) is termed as the 100p%secrecy outage rate.
The parameters are set identical to those in the perfect CSI case. Regarding the uncertainty model in (41), we introduce the uncertainty ratios associated with ǫ 0 , ǫ ′ 0 , ǫ k , ǫ ′ k and ǫ ′′ k as α 0 , α ′ 0 , α k , α ′ k and α ′′ k , respectively. For instance, α 0 is while α ′ 0 , α k 's, α ′ k 's and α ′′ k 's are similarly defined and thus omitted here for brevity. Besides, it is reasonable to assume that the channel estimates w.r.t Eve suffer from more errors than those for Alice and Bob. Hence, we set α ′2 0 = α ′2 k = α ′′2 k = 1% while α 2 0 = α 2 k = 10%, k = 1, . . . , K, unless otherwise specified. Fig. 4 demonstrates the cumulative density function (CDF) of the achievable secrecy rate from 1000 samples of random channel errors uniformly distributed over the sets defined by (41) given fixed actual channel realization. We set P r = 20dBm, P s = 30dBm, N t = 3, K = 5 and α ′2 0 = α ′2 k = α ′′2 k = 2%, k = 1, . . . , K. Despite being suboptimal to the upper-bound schemes of "Robust SDR with HJ" and "Robust-eqv with HJ", the proposed "Robust with HJ" scheme outperforms its non-robust counterpart "Non-robust with HJ" particularly in the low range of probability, and overwhelmingly surpasses the "Robust w/o HJ". For example, "Robust with HJ" can achieve a secrecy rate of around 3.5bps/Hz in the 3% worst case versus that of 3.3bps/Hz and 1.0bps/Hz for the "Non-robust with HJ" and "Robust w/o HJ", respectively. The solutions for "Robust SDR with HJ" is also seen to admit very little gap from those for "Robust-eqv with HJ", which suggests that approximatingĤ(γ e ) by solving the  complexity reduced "Robust SDR with HJ " leads almost no performance loss. Fig. 5 illustrates the CDF of the achievable secrecy rate from 1000 samples of random channel errors generated in the same way as Fig. 4, with P r = 20dBm, P s = 30dBm and N t = 3. It is observed that proposed solutions to "Robust with HJ" nearly achieve their upper-bound rank constraint relaxed solutions, i.e., SDR, to "Robust upper SDR with HJ" throughout the whole range of outage probability. Moreover, the "Robust w/o HJ" yields the worst performance. In particular, when the outage probability falls to 3%, the "Robust w/o HJ" achieves a worst-case secrecy rate of less than 1bps/Hz while the proposed scheme can still guarantee an outage rate of rough 1.64bps/Hz and 2.07bps/Hz for K = 5 and K = 6, respectively. Also, it is observed that increasing the number of HJ helpers will improve the secrecy performance, but we do not draw conclusions on the extent to which the secrecy rate can increase, since it also depends on the level of channel estimation inaccuracy. For example, more HJ helpers may also yield larger interference to the legitimate receiver if the channels from HJ helpers to Bob are not as well estimated as this instance of α ′′2 k = 1%, ∀k. Hence we suggest that in practice, a mild number of HJ helpers are sufficient in view of the trade-off between complexity and performance. Fig. 6 shows two different levels (p = 0.20 and p = 0.30) of secrecy outage rate versus the channel uncertainty ratios (assuming α 0 = α k , k = 1, . . . , K), in which P r = 30dBm, P s = 30dBm, N t = 3 and K = 5. It is observed that the secrecy outage rate by the proposed schemes decreases slowly with the eavesdropper's CSI error ratios, which validates the motivation of the worst-case robust optimization. It is worth noting that the advantage of the HJ protocol is more significant when the normalized channel uncertainty of Eve's channels surpasses 10%, since the HJ scheme provides more degree of freedom for robust design and thus capable of guaranteeing larger worst-case secrecy rate against worse channel conditions compared to that without HJ. The reasonably suboptimal performance of the proposed "Robust with HJ" is also seen as from Figs. 4 and 5. Fig. 7 studies the 100p%-secrecy outage rate for p = 0.05 and p = 0.20, respectively, versus the transmit power of the AF relay. Specifically, we set P s = 30dBm, N t = 3, and K = 5. As observed similarly from Fig. 6, the robust schemes with the assistance of HJ helpers perform considerably better than solutions without HJ helpers. Furthermore, when the transmit power is set relatively large, i.e., P s = 30dBm, it is seen that continuously increasing P r does not contribute much to the secrecy performance, because in this situation the increased amplified noise at the AF relay compromises the performance, which provides useful insight for practical setting of P r . In addition, the proposed "Robust with HJ" is observed striking a good trade-off between optimality and complexity compared with the two upper-bound solutions.

VI. CONCLUSION
This paper considered improving the secret wireless communications in a multi-antenna AF relay wiretap channel via a novel harvest-and-jam (HJ) relaying protocol. The AN covariance matrices at HJ helpers and the AF relay beamforming matrix have been jointly optimized to maximize the achievable secrecy rate and/or worst-case secrecy rate at the legitimate receiver subject to the transmit power constraints of the AF relay as well as the HJ helpers, on perfect and imperfect CSI occasions, respectively, using the technique of semi-definite relaxation (SDR). The SDR was shown tight for the perfect CSI case while suboptimal rank-one reconstruction algorithms for the robust formulation under imperfect CSIs were presented achieving promising tradeoffs between complexity and performance. The effectiveness of the proposed schemes were also verified by numerical results. APPENDIX a n η n η H n + bξξ H . Assume that (X * , {Q * k }, τ * ) is the optimal solution to (P1 ′ .1-RW-SDR) with rank(X * ) > 1. Then construct a new solution {X * ,Q * k ,τ * } according to (29)- (31). Now, we check if the reconstructed solution is feasible if (32) holds. First, Moreover, In addition, (23c)-(23e) are easily shown to satisfy. In the above, (a) and (b) hold due to the feasibility in (23a) and (23b), respectively. Further, P s tr(F 1X * ) = P s tr(F 1 X * ) shows that the reconstructed solution achieves the same optimum value as that of (P1 ′ .1-RW-SDR). Hence, an optimal solution to (P1 ′ .1-RW-SDR) with rank-one X is ensured.
Next, by substituting (92) for W in (5), we have P r ≥ P s ( BU H 2 h 0 2 + DU H 2 h 0 2 ) + σ 2 r tr(B H B + C H C + D H D + E H E). Since (P1 ′ ) is a secrecy rate maximization problem subject to the given P r , it turns out that given the optimum secrecy rate, P r is the minimized required power by taking D = 0 and E = 0, while B and C cannot be determined directly. Thus, reformulated into (104). Then, given g k , k = 2, . . . , K, fixed, applying similar procedure to that in Appendix D, it follows that there exists v (1) ≥ 0 such that the following LMI holds: Since v (1) I ǫ1 + A 2 (1) ≻ 0 always holds, (105) is equivalent to the following LMI: Next, devising the method of mathematical induction again as that for (99), (104) holds for g k ∈ G k , ∀k, if and only if there exists {v(k) ≥ 0}, such that (63) is satisfied, which completes the proof.

F. Proof of Proposition 4.3
We only sketch the proof herein since it is quite similar to that of Proposition 4.1. First, apply Lemma 4.2 to (71) givenh k 's, k = 2, . . . , K, fixed and obtain an initial LMI. Next, manipulate the resulting LMI according to the property of Schur-Complements to facilitate using Lemma 4.2. Then, repeat this procedure until all the semi-indefinite constraints w.r.t.h k 's have been incorporated into an equivalent LMI.

G. Proof of Proposition 4.5
According to the KKT conditions of (P2 ′ .1-RW-SDR), we haveĀ * X * = 0, whereĀ * is given by (79). DefineC * = A * − w * 2,2 P sF 1 with rank(C * ) denoted byr c . Then take the similar procedure as Case I and Case II in Appendix A, it can be obtained that X * = N 2 t −rc n=1ā nηnη H n +bξξ H .