3.6 Appendix
3.6.1 Proof of Lemma 1
By dropping the rank-one constraint, (3.11) is a convex SDP problem. It is easy to verify that this problem satisfies the Slater’s condition [101] and the corresponding duality gap is zero. Therefore, the Lagrangian of (3.11) can be expressed as
L(Q,Cj,λj,µj) = K
∑
j=1 λj( δd2 ρk +σk2) +µj( ek 1−ρk −σk2) + K∑
j=1 Tr(CjQ), (3.18) whereCj=PsH+σs2,rI−∑Kj=1( λj γk+µj)PsBk+∑ K j=1((λj−µj)σs2,rGk1)andλj≥0,µj≥0are the dual variables associated withkth SNR and EH constraints, respectively. The dual problem of (3.11) is then defined as
max λj,µj,ρk K
∑
j=1 λj( δd2 ρk +σk2) +µj( ek 1−ρk −σk2) s.t. Cj≽0,∀j. (3.19) It is assumed thatnλ∗j o andnµ∗j oare the optimal dual variables for problem (3.19). The resultingCj becomesC∗j. Then by complementary slackness condition, it obtains
Tr(C∗jQ∗) =0,∀j, (3.20)
whereQ∗is the optimal solution of (3.11) and the equation (3.20) is equivalentC∗jQ∗=0,∀j.
Due to 0<ρk<1, condition, two cases of the dual variables: λ∗j >0,µ∗j =0 and λ∗j =
0,µ∗j >0 is not possible. Ifλj∗=0 andµ∗j =0, thenC∗j=σs2,rI+PsH≻0and Rank(C∗j) =M
3.6 Appendix 57 cannot be the optimal solution for problem (3.11). Therefore, bothλ∗j,µ∗j =0 must be great than zero.
Due to PsH+σs2,rI≻0, it has Rank(C∗j) =Rank(M−(∑Kj=1(
λj
γk +µj)PsBk)), then it gets Rank(C∗j)≥M−1. Note that if Rank(C∗j)is full rank matrix, it hasQ∗=0, which cannot be the optimal solution. Hence, it follows that Rank(C∗j) =M−1,∀j. According toC∗jQ∗=0,
Chapter 4
Secure Beamforming and Power Splitting
Designs in a Multi-antenna Relay Network
This chapter is organized as follows. The main contributions of this work is first introduced in 4.1. The system model is described in Section 4.2. The PM optimization problems based on perfect CSI is solved in Section 4.3, whereas the robust PM optimization problem is considered in Section 4.4. Section 4.5 presents simulation results to validate the performance of the proposed algorithms for both perfect and imperfect CSI cases. Finally, conclusions are presented in Section 4.6.4.1
Introduction
In this chapter, a secure transmission in a multi-antenna AF relay network based on perfect CSI and imperfect CSI assumption is investigated. In the considered scenario, multiple idle receivers attempt to overheard the transmission message. PS receiver architecture is employed at each receiver. The main objective is to jointly design secure beamforming and the PS ratio by a proposed PM problem subject to the constraints on secrecy rate and EH under the assumption of perfect and imperfect CSI at the transmitter and the relay. The
4.2 Network Model 59 proposed PM problem for both channel model is non-convex and challenge to solve due to the coupled variables [101]. Two methods are proposed to tackle the difficulties. One is based on 1-D search and the other one is based on SPCA [110]. The primary contributions of this chapter are summarized as follows.
• For the model with perfect CSI at the transmitter and the relay, the PM problem is transformed into a two-level optimization problem. In the inner level, the PM problem can be relaxed as a SDP problem by using SDR method [108]. In the outer level, the problem can be solved by using 1-D line search respect to a single variable. In addition, a rank reduction algorithm is proposed to guarantee the tightness of the relaxation based on the concept in [111]. On the other hand, a low-complexity algorithm based on the SPCA method is proposed to iteratively solve the PM problem.
• For the imperfect CSI model with bounded CSI error, SDR method, 1-D linear search, and the rank reduction method can be still adopted to solve the difficulties caused by the quadratic terms, the coupled variables, and rank constraint. Furthermore, S-procedure method is used to convert the semi-infinite constraints into LMI. In order to reduce the computational complexity, an SPCA-based iterative algorithm is also proposed.
4.2
Network Model
In this chapter, a cooperative relay network is considered as shown in Figure 4.1 which consists of a transmitter,(K+1)th receiver, and a half duplex (HD) AF relay. All nodes are deployed in a same cluster. It is assumed that there is no direct link from the transmitter to all the receivers since the transmitter is relative far away from the receivers. The transmitter and all receivers are equipped with single antenna as well as the relay is equipped withMantennas (M>1). All receivers are able to decode information and harvest energy from radio signals.
4.2 Network Model 60 Two types of receiver are considered in the system; one desired receiver andK idle receivers. In addition, it is assumed that only the desired receiver is active to decode information and harvest energy simultaneously. The other idle receivers are supposed to harvest energy from the RF signal when they are inactive. However, these idle receivers may attempt to eavesdrop the information for the desired receiver rather than to harvest energy since all receivers are deployed in the range of service coverage. Hence, the secure communication should be provided by taking into account for the idle receivers as the potential eavesdroppers.
Fig. 4.1 System model.
In the first transmission phase, the transmitter sends coded confidential informationsto the AF relay. Thus, the received signal at the relay is given by
yr=
√
Pshrs+nr, (4.1)
wherePs denotes the average transmit power at the transmitter;s∼CN(0,1)is a transmit signal andnr∼CN(0,σr2I)is an additive complex noise at the relay;hr∈CM×1represents
4.2 Network Model 61 In the second transmission phase, the relay employs the AF transformation matrix W∈
CNt×Nt to amplify the received signal and then forwards to the DR. Thus, the transmitted
signal at the relay can be expressed as
xr=Wyr. (4.2)
Note that the transmit power at the relay is given by
Pr =E{xrxHr }=Ps∥Whr∥2+σr2∥W∥2F. (4.3)
Then, the received signal at the DR and thek-th idle receiver is given by
yd=√PshHdWhrs+hHdWnr+nD, (4.4a)
ye,k=
√
PsgHkWhrs+gHk Wnr+nk, (4.4b)
wherehd∈CM×1andgk∈CM×1denote the channel vector from the relay to the DR and the
k-th idle receiver, respectively;nD∼CN(0,σD2)andnk∼CN(0,σk2)represent the additive
complex noise at the DR and thek-th idle receiver, respectively.
It assumes that power splitter is employed at the DR to perform ID and EH simultaneously. Hence, the received signal for ID can be expressed as
yIDd =√ρ
√
PshHdWhrs+hHdWnr+nD
+nP, (4.5)
whereρ(0≤ρ≤1)is PS ratio andnP∼CN(0,σP2)is the signal processing noise power at
the receiver. On the other hand, the total harvested energy at the DR can be expressed as
4.3 Problem Design Based on Perfect CSI 62