Joint design of beamforming and time switching/power splitting for wirelesspowered multiantenna dualrelay network
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Abstract
In this paper, we consider a wirelesspowered dualrelay network consisting of one multiantenna source, two singleantenna energyconstrained relays and one singleantenna destination without direct source to destination link. In order to establish the communication flow, the energyconstrained relays harvest energy from the radio frequency transmitted by the source firstly, then exploit the harvested energy to forward the source information to the destination based on distributed space time coding (DSTC). Under this network architecture, three decodeandforward (DF) techniquebased relaying protocols, i.e., time switchingbased relaying (TSR) protocol, power splittingbased relaying (PSR) protocol, and hybrid relaying (HR) protocol, are considered to drive the energy transfer and information transmission. To maximize the network throughput, the joint design for the optimal energy and information beamforming vectors employed at the source, the optimal time switching, and power splitting ratios under these three protocols are investigated and solved efficiently by employing simple sequential optimization approach or alternating optimization approach. Simulations are conducted to show the superior performance achieved by our proposed scheme. Moreover, we find that the TSR protocol outperforms the PSR protocol in the low signaltonoise ratio (SNR) region, while the latter outperforms the former in the high SNR region. And the HR protocol achieves the best performance in any SNR region. At the same time, the effect of the relays’ locations on the throughput performance of these three protocols is also investigated.
Keywords
Wireless powered communication network Beamforming vector design TSR PSR HRAbbreviations
 CCI
Cochannel interference
 CSI
Channel state information
 DF
Decodeandforward
 DSTC
Distributed spacetime coding
 FD
Fullduplex
 HD
Halfduplex
 HR
Hybrid relaying
 HTT
Harvestthentransmit
 IOTs
Internet of Things
 MIMO
Multipleinputmultioutput
 NOMA
Nonorthogonal multiple access
 OFDM
Orthogonal frequencydivision multiplexing
 PSR
Power splitting relaying
 QoS
Quality of Service
 RF
Radio frequency
 SDP
Semidefinite programming
 SNR
Signaltonoise ratio
 TB
Transmission block
 TSR
Time switchingbased relaying
 WET
Wireless energy transfer
 WIT
Wireless information transmission
 WPCN
Wireless powered communication network
1 Introduction
With the rapid rise of the Internet of Things (IoT) [1] industry, massive intelligent devices will be deployed in anywhere to monitor engineering structures, diagnose the physical condition of the patients, report the realtime traffic information, and so on. A major problem facing is that how to prolong the life of the energyconstrained devices, so that they could be utilized in a sustainable and lowcost way. Currently, a feasible solution is to adopt the radio frequency (RF) signalbased wireless energy transfer (WET) technique [2], as RF signal is controllable, stable, and efficient compared to the natural sources such as solar and wind.
By integrating WET and wireless data communications, wireless powered communication network (WPCN) [3, 4, 5, 6, 7, 8] has raised as an emerging research topic. Up to now, the most popular research model under WPCN was the threenode relay network [9, 10, 11, 12, 13, 14], in which the “harvestthentransmit" (HTT) scheme was employed, i.e., the singleantenna relay node was energy constrained and had to harvest energy from the singleantenna source firstly, and then exploited the harvested energy to forward the source information to the singleantenna destination under the case of no direct link. Under this scene, [9] firstly proposed the time switchingbased relaying (TSR) protocol and the power splittingbased relaying (PSR) protocol. For the TSR protocol, energy harvesting and information reception at the relay have taken different time slots. For the PSR protocol, the received signal was split into two streams, one stream got into the energy harvesting circuit and the other got into the information receiver. Subsequently, [10] creatively proposed the hybrid relaying (HR) protocol by combining the advantages of the TSR and PSR protocols. By joint optimizing the TS and PS ratios, the HR protocol showed its great gain compared with the TSR and PSR protocols. At the same time, [11] analyzed the outage probability and the ergodic capacity under the HR protocol. Assuming existing direct source to destination link, [12] analyzed the outage probability and optimized the network performance under the PSR protocol and showed that the large diversity gain can be obtained by exploiting the direct link. By adjusting the number of subframes for each link, [13, 14] set the different information transmission rate at the source and the relay, based on which [13, 14] further proposed several generalized HR protocols.
Based on the classic threenode model, researchers put forward kinds of novel ideas, techniques, and scenarios for WPCN. Specifically, [15] proposed the idea of exploiting fullduplex (FD) technique to implement selfenergy recycling at the multiantenna relay, so that an additional energy can be obtained by the relay. Luo et al. and Chen et al. [16, 17] proposed the idea of allowing the source harvesting energy from the relay in the relaying phase, so that the energy efficiency can be improved greatly. Li and Rong [18, 19] proposed the idea of setting different transmit power in the WET phase and wireless information transmission (WIT) phase, then analyzed a generalized TSR protocol and showed its great gain compared with the traditional TSR protocol. In [20], the idea of channel conditionbased adaptive PS at each antenna of the relay was proposed, and the problem of joint PS ratio and power allocation at each transmit antenna of the relay to maximize the throughput was solved. In [21], the HR protocol was applied in the multihop relays network. In [22], the effect of cochannel interference (CCI) at the relay node on the ergodic and outage capacities was investigated. Interestingly, [22] found that CCI may be favorable, as CCI also contributed the larger harvested energy except for causing inconvenience for information decoding at the destination. Liu and Wang [23] considered the orthogonal frequencydivision multiplexing (OFDM)based WPCN, in which all subcarriers were split into two parts at the relay, one for energy harvesting and one for information reception. [24] considered the multipleinputmultioutput (MIMO) FDbased WPCN, then analyzed and optimized the throughput. [25] investigated the nonorthogonal multiple access (NOMA)based WPCN, in which the user near the singleantenna source first decoded its needed information from the received signal, then exploited the harvested energy to help forward the information of the far user. Xu et al. [26] aimed to optimize the joint design of beamforming at the multiantenna source and the PS ratio at the near user in the NOMA based WPCN for the purpose of maximizing the throughput of the near user while satisfying the Quality of Service (QoS) of the far user. Lee et al., Dong et al., Niu et al., and Mei et al. [27, 28, 29, 30] aimed to improve the security of WPCN by exploiting beamforming, cooperative jamming, or artificial noise technique.
In view of the prior works, most assumed that there was only one relay assisting the transmission. This setting was easy and would cause large performance loss, as the channel capacity of the second hop (from the relay to the destination) suffered from twice channel fading compared with the case that the relay was energy rich. By placing multiple relays between the source and the destination and exploiting all relays for information forwarding, the channel capacity of the second hop can be enhanced significantly. Meanwhile, there would be no necessity to consume more power, as the harvested energy of all relays will still come from the same RF signal. In addition, multiple antenna technique, as a powerful tool for improving the energy transfer efficiency and the spatial diversity gain, is expected to play an important role in WPCN. Therefore, under the condition that the source is equipped with multiple antennas and there are multiple relays, to maximize the throughput, the joint design of beamforming and time switching/power splitting needs careful consideration and is not a trivial work, which is the motivation for writing this paper.

In our work, we introduce three relaying protocols, i.e., TSR, PSR, and HR protocols, for energy transfer and information transmission in the considered dualrelay system. Although these three protocols have been reported in prior work [10], we should note that the mentioned paper only considered singlerelay case. When there exist dual relays and each relay adopt distributed spacetime block code for information forwarding, it would become more complex to derive the optimal time switching and power splitting ratios. In our work, after complex derivation, we obtain the optimal closedform solutions.

We extend the case of singleantenna source to the case of multiantenna source. Therefore, we have to accordingly design the energy and information beamforming vectors. After some discussion, we find the favorable structure of the optimal energy and information beamforming vectors, which enable us to just optimize single variable for obtaining the optimal solutions instead of resorting to the convex optimization frame such as semidefinite programming (SDP) [34, 35, 36, 37, 38]. However, the optimization process for the single variable is not trivial and requires numerical analysis. In our work, we provide the detailed analysis process for finding the optimal energy and information beamformers.

We provide numerical insights on each protocol and further give the detailed analysis for the effect of the time switching ratio, the power splitting ratio, and the relays’ locations on the achievable throughput performance, which are omitted in prior works.
Notation: (·)^{∗},(·)^{T} and (·)^{H} denote the complex conjugate, matrix transpose and matrix conjugate transpose, respectively. h_{XY} denotes the absolute value of a scalar h_{XY}. ∥h_{XY}∥ denotes the Frobenius norm of a complex vector h_{XY}. I is the identity matrix, Π_{X}=X(X^{H}X)^{−1}X^{H} denotes the orthogonal projection onto the column space of X, and ΠX⊥=I−Π_{X} denotes the orthogonal projection onto the orthogonal complement of column space X. \({\mathcal {C}}{\mathcal {N}}(0,1)\) denotes a scalar complex Gaussian distribution with zero mean and unit variance.
2 Methods
The rest of this paper is organized as follows: Section 3 describes the system model. Section 4 gives the optimization process for the TSR protocol. Section 5 gives the optimization process for the PSR protocol. Section 6 gives the optimization process for the HR protocol. Section 7 presents the simulation results and provides some insightful details. Finally, conclusion is given in Section 8.
3 System models

There is no direct link between S and D, which may be resulted from the deep shadowing or shielding effect caused by the obstacles [41].

It is assumed that all channel gains keep constant during each transmission block (TB) but change independently from one TB to another.

The relay nodes R_{1} and R_{2} have no external power supply and would harvest energy from the source RF signal firstly, then exhaust the harvested energy to help the source information transmission based on decodeandforward (DF) technique.

As the necessary conditions for designing optimal scheduling strategies (such as the design of energy and information beamforming vectors at S, the TS, and PS ratios) under different relaying protocols, the global channel state information (CSI) are assumed to be available at all nodes, which can be realized through various methods, e.g., the pilotassisted reverselink channel training [42].
Based on the network model and above assumptions, we introduce three different relaying protocols, i.e., TSR, PSR, and HR to drive the energy harvesting at the relays and information transmission from the source to the destination for each TB. More importantly, in order to maximize the network throughput, we present a detailed investigation on the optimal design of beamformers at the source, the TS ratio, the PS ratio, and the joint TS&PS ratio under these protocols in the following sections.
4 Time switchingbased relaying (TSR) protocol
It is worthy noted that in the relays’ transmission phase, the two relays would forward the source information to D with the help of distributed spacetime block code (DSTC) [43].
where i∈{1,2},0<η<1 denotes the energy conversion efficiency, m denotes the path loss factor, and w_{1} is the N×1 energy beamforming vector employed at S.
where N_{0} denotes the noise power.
where \({A_{1}} = \frac {{{P_{S}}}}{{d_{{SR}}^{m}{N_{0}}}}\).
where \({A_{2}} = \frac {{2\eta {P_{S}}{{\left  {{f_{{R_{1}}D}}} \right }^{2}}}}{{d_{{SR}}^{m}d_{{RD}}^{m}{N_{0}}}}\) and \({A_{3}} = \frac {{2\eta {P_{S}}{{\left  {{f_{{R_{2}}D}}} \right }^{2}}}}{{d_{{SR}}^{m}d_{{RD}}^{m}{N_{0}}}}\).
To proceed, we first have the following Proposition.
Proposition 1
where x_{i} is a real number in [0,1].
Proof
Same with the objective function shown in Eq. (2) of [44], R_{DF} is nondecreasing with respect to \({\left  {\mathbf {h}_{S{R_{1}}}^{T}{\mathbf {w}_{1}}} \right ^{2}}, {\left  {\mathbf {h}_{S{R_{2}}}^{T}{\mathbf {w}_{1}}} \right ^{2}}, {\left  {\mathbf {h}_{S{R_{1}}}^{T}{\mathbf {w}_{2}}} \right ^{2}}\), and \({\left  {\mathbf {h}_{S{R_{2}}}^{T}{\mathbf {w}_{2}}} \right ^{2}}\). And the beamforming vectors w_{i} in fact should play the role for balancing the values of \({\left  {\mathbf {h}_{S{R_{1}}}^{T}{\mathbf {w}_{i}}} \right ^{2}}\) and \({\left  {\mathbf {h}_{S{R_{2}}}^{T}{\mathbf {w}_{i}}} \right ^{2}}\) as far as possible. Therefore, the process for proving Proposition 1 can be obtained by resorting to the proof of Lemma 1 presented in [44]. □
where \({f_{1}}({x_{1}}) = \min \left ({A_{1}}{a^{2}}{x_{1}^{2}},{A_{1}}{\left (b{x_{1}} + c\sqrt {1  x_{1}^{2}} \right)^{2}}\right), {f_{2}}({x_{2}}) = {A_{2}}{a^{2}}x_{2}^{2} + {A_{3}}{\left (b{x_{2}} + c\sqrt {1  x_{2}^{2}} \right)^{2}}\), \(a = \frac {{\left  {\mathbf {h}_{S{R_{1}}}^{T}{\prod _{\mathbf {h}_{S{R_{1}}}^{*}}}\mathbf {h}_{S{R_{2}}}^{*}} \right }}{{\left \ {{\prod _{{\mathbf {h}_{S{R_{1}}}}}}{\mathbf {h}_{S{R_{2}}}}} \right \}}, b = \left \ {{\prod _{{\mathbf {h}_{S{R_{1}}}}}}{\mathbf {h}_{S{R_{2}}}}} \right \\) and \(c = \left \ {\prod _{{\mathbf {h}_{S{R_{1}}}}}^ \bot {\mathbf {h}_{S{R_{2}}}}} \right \\).
As we can see, when α is fixed, maximizing R_{DF} is equivalent to maximizing \(\min \left ({f_{1}}({x_{1}}),\frac {\alpha }{{1  \alpha }}{f_{2}}({x_{2}})\right)\). Observing that f_{1}(x_{1}) is independent of f_{2}(x_{2}), hence next we aim to find the optimal x_{1} and x_{2} to maximize f_{1}(x_{1}) and f_{2}(x_{2}), respectively.
 The process of finding \(x_{1}^{*}\): based on the conclusion shown in the Proposition 2 of [45], the optimal \(x_{1}^{*}\) is given as follows:$$ x_{1}^{*} = \left\{ {\begin{array}{*{20}{c}} 1&{a < b}\\ {\frac{c}{{\sqrt {{c^{2}} + {{(a  b)}^{2}}} }}}&{b \le a < \frac{{{b^{2}} + {c^{2}}}}{b}}\\ {\frac{b}{{\sqrt {{b^{2}} + {c^{2}}} }}}&{a \ge \frac{{{b^{2}} + {c^{2}}}}{b}} \end{array}} \right. $$(9)
 The process of finding \(x_{2}^{*}\): f_{2}(x_{2}) can be further written as \({f_{2}}({x_{2}}) = \left ({A_{2}}{a^{2}} + {A_{3}}{b^{2}}  {A_{3}}{c^{2}}\right)x_{2}^{2} + 2{A_{3}}bc{x_{2}}\sqrt {1  x_{2}^{2}} + {A_{3}}{c^{2}}\). When A_{2}a^{2}+A_{3}b^{2}−A_{3}c^{2}≥0, by letting the first derivative of f_{2}(x_{2}) equaling 0, we have the unique root \({\widehat x_{2}} = \sqrt {\frac {{1 + \sqrt {\frac {{{u^{2}}}}{{{u^{2}} + 4{v^{2}}}}} }}{2}} \), where u=2(A_{2}a^{2}+A_{3}b^{2}−A_{3}c^{2}) and v=2A_{3}bc. Moreover, because that f_{2}^{′}(x_{2})>0 and f_{2}^{′}(x_{2})<0 when \(x \in [0,{\widehat x_{2}})\) and \(x \in ({\widehat x_{2}},1]\), we can claim that f_{2}(x_{2}) achieves its maximum at the point \({\widehat x_{2}}\). When A_{2}a^{2}+A_{3}b^{2}−A_{3}c^{2}<0, due to the fact that \(\left ({A_{2}}{a^{2}} + {A_{3}}{b^{2}}  {A_{3}}{c^{2}}\right)x_{2}^{2}\) and \(2{A_{3}}bc{x_{2}}\sqrt {1  x_{2}^{2}} \) are two concave functions, f_{2}(x_{2}) must be a concave function and achieves its maximum value at the point \({\widehat x_{2}} = \sqrt {\frac {{1  \sqrt {\frac {{{u^{2}}}}{{{u^{2}} + 4{v^{2}}}}} }}{2}} \). Therefore, the optimal x_{2} is given as follows:$$ x_{2}^{*} = \left\{ {\begin{array}{*{20}{c}} {\sqrt {\frac{{1 + \sqrt {\frac{{{u^{2}}}}{{{u^{2}} + 4{v^{2}}}}} }}{2}} }&{{A_{2}}{a^{2}} + {A_{3}}{b^{2}}  {A_{3}}{c^{2}} \ge 0}\\ {\sqrt {\frac{{1  \sqrt {\frac{{{u^{2}}}}{{{u^{2}} + 4{v^{2}}}}} }}{2}} }&{{A_{2}}{a^{2}} + {A_{3}}{b^{2}}  {A_{3}}{c^{2}} < 0} \end{array}} \right. $$(10)
Remark 1
With α increasing in the interval \([0,\overline \alpha ]\), the efficient information transmission time would decrease linearly and the SNR of the second hop would increase at the same time. This tradeoff makes the optimal α exist. However, γ_{DF,2} always does not exceed γ_{DF,1} when the throughput is maximized. In the HR protocol as introduced in Section 6, we can see that in the second phase, each relay could adaptively split some fraction of the received signal as an additional harvested energy (expected the harvested energy in the first phase) to further improve the SNR of the second hop and reduce the SNR of the first hop. By selecting suitable splitting ratio, γ_{DF,1} and γ_{DF,2} would be balanced as far as possible and the efficient SNR would be improved greatly. Hence, without decreasing the information transmission time, the HR protocol offsets the shortcoming of the TSR protocol efficiently and achieves the superior throughput performance, as shown in the simulations.
5 Power splittingbased relaying (PSR) protocol
where \({B_{1}} = \frac {{\eta {P_{S}}{{\left  {{f_{{R_{1}}D}}} \right }^{2}}}}{{d_{{SR}}^{m}d_{{RD}}^{m}{N_{0}}}}\) and \({B_{2}} = \frac {{\eta {P_{S}}{{\left  {{f_{{R_{2}}D}}} \right }^{2}}}}{{d_{{SR}}^{m}d_{{RD}}^{m}{N_{0}}}}\).
where \({B_{3}} = \frac {{{B_{1}}}}{{1 + \frac {{{B_{1}}}}{{{A_{1}}}} + \frac {{{B_{2}}}}{{{A_{1}}}}}}\) and \({B_{4}} = \frac {{{B_{2}}}}{{1 + \frac {{{B_{1}}}}{{{A_{1}}}} + \frac {{{B_{2}}}}{{{A_{1}}}}}}\).
where \({g_{1}}(x) = {B_{3}}{a^{2}}{x^{2}} + {B_{4}}{\left (bx + c\sqrt {1  {x^{2}}} \right)^{2}}\) and \({g_{2}}(x) = \min \left ({A_{1}}{a^{2}}{x^{2}},{A_{1}}{\left (bx + c\sqrt {1  {x^{2}}} \right)^{2}}\right)\).
To obtain the optimal x that maximizes min(g_{1}(x),g_{2}(x)), next we discuss three cases in term of different solution with respect to x that maximizes g_{2}(x):
5.1 Case 1: a<b
where u_{1}=2(B_{3}a^{2}+B_{4}b^{2}−B_{4}c^{2}) and v_{1}=2B_{4}bc.
Lemma 1
where w_{1}=B_{4}c^{2} and z_{1}=A_{1}a^{2}−B_{3}a^{2}−B_{4}(b^{2}−c^{2}).
Proof
 1.
In this scenario, g_{2}(1)≤g_{1}(1). There is no cross point between g_{1}(x) and g_{2}(x), and g_{1}(x) is always greater than g_{2}(x). Hence, min(g_{1}(x),g_{2}(x)) achieves its maximal value when g_{2}(x) is maximized, i.e., x^{∗}=1.
 2.
In this scenario, g_{2}(1)>g_{1}(1) and \({g_{2}}(\widehat x) \le {g_{1}}(\widehat x)\). There exists one cross point between g_{1}(x) and g_{2}(x), which occurs after the point at which g_{1}(x) achieves its maximal value. Hence, min(g_{1}(x),g_{2}(x)) is maximized when g_{1}(x)=g_{2}(x), i.e., \({x^{*}} = \sqrt {\frac {{2{z_{1}}{w_{1}} + v_{1}^{2} + \sqrt {v_{1}^{4} + 4{z_{1}}{w_{1}}v_{1}^{2}  4v_{1}^{2}w_{1}^{2}} }}{{2\left (z_{1}^{2} + v_{1}^{2}\right)}}} \).
 3.
In this scenario, \({g_{2}}(\widehat x) > {g_{1}}(\widehat x)\). There also exists one cross point between g_{1}(x) and g_{2}(x), which occurs before the point at which g_{1}(x) is maximized. Hence, min(g_{1}(x),g_{2}(x)) achieves its maximal value at the point when g_{1}(x) is maximized, i.e., \({x^{*}} = \widehat x\).
5.2 Case 2: \(b \le a < \frac {{{b^{2}} + {c^{2}}}}{b}\)
In this case, g_{2}(x) is a piecewise function, as shown by the blue line in Fig. 4b. Obviously, the maximum of g_{2}(x) is obtained at \(\overline x = \frac {c}{{\sqrt {{{(a  b)}^{2}} + {c^{2}}} }}\), i.e., the cross point of the functions A_{1}a^{2}x^{2} and \({A_{1}}{(bx + c\sqrt {1  {x^{2}}})^{2}}\).
Lemma 2
where w_{2}=(B_{4}−A_{1})c^{2},z_{2}=(A_{1}−B_{4})(b^{2}−c^{2})−B_{3}a^{2}, and v_{2}=2(B_{4}−A_{1})bc.
Proof
 1.
In this scenario, \(\widehat x \le \overline x \) and \({g_{1}}(\widehat x) \ge {g_{2}}(\widehat x)\). There is only one cross point between g_{1}(x) and A_{1}a^{2}x^{2}, which occurs after the point \(\widehat x\). Hence, min(g_{1}(x),g_{2}(x)) achieves its maximal value when \({B_{3}}{a^{2}}{x^{2}} + {B_{4}}{\left (bx + c\sqrt {1  {x^{2}}} \right)^{2}} = {{A_{1}}{a^{2}}{x^{2}}}\), i.e., \({x^{*}} = \sqrt {\frac {{2{z_{1}}{w_{1}} + v_{1}^{2} + \sqrt {v_{1}^{4} + 4{z_{1}}{w_{1}}v_{1}^{2}  4v_{1}^{2}w_{1}^{2}} }}{{2\left (z_{1}^{2} + v_{1}^{2}\right)}}} \).
 2.
In this scenario, \(\widehat x \le \overline x\) and \({g_{1}}(\widehat x) < {g_{2}}(\widehat x)\). And g_{1}(x) and A_{1}a^{2}x^{2} also has one cross point, which occurs before the point at which g_{1}(x) is maximized. Hence, min(g_{1}(x),g_{2}(x)) achieves its maximal value at the point when g_{1}(x) is maximized, i.e., \({x^{*}} = \widehat x\).
 3.
In this scenario, \(\widehat x > \overline x\) and \({g_{1}}(\widehat x) \ge {g_{2}}(\widehat x)\). There may be one or two cross points between g_{1}(x) and \({{A_{1}}{{\left (bx + c\sqrt {1  {x^{2}}} \right)}^{2}}}\), and the bigger one would be located in the interval \((\overline x,\widehat x)\). Hence, the optimal x is the bigger root of \({B_{3}}{a^{2}}{x^{2}} + {B_{4}}{\left (bx + c\sqrt {1  {x^{2}}} \right)^{2}} = {{A_{1}}{{\left (bx + c\sqrt {1  {x^{2}}} \right)}^{2}}}\), i.e., \({x^{*}} = \sqrt {\frac {{2{z_{2}}{w_{2}} + v_{2}^{2} + \sqrt {v_{2}^{4} + 4{z_{2}}{w_{2}}v_{2}^{2}  4v_{2}^{2}w_{2}^{2}} }}{{2\left (z_{2}^{2} + v_{2}^{2}\right)}}} \).
 4.
In this scenario, \(\widehat x > \overline x\) and \({g_{1}}(\widehat x) < {g_{2}}(\widehat x)\). Obviously, the optimal x is \(\widehat x\).
5.3 Case 3: \(a \ge \frac {{{b^{2}} + {c^{2}}}}{b}\)
In this case, g_{2}(x) is also a piecewise function, as shown by the blue line in Fig. 4c. However, the maximum of g_{2}(x) is obtained at \(\overline x = \frac {b}{{\sqrt {{b^{2}} + {c^{2}}} }}\), i.e., the maximum point of the function \({A_{1}}{\left (bx + c\sqrt {1  {x^{2}}} \right)^{2}}\).
Lemma 3
Proof
 1.
In this scenario, \({g_{1}}(\overline x) \ge {g_{2}}(\overline x)\). There may be one or two cross points between g_{1}(x) and \({{A_{1}}{{\left (bx + c\sqrt {1  {x^{2}}} \right)}^{2}}}\), but which is(are) before the point at which g_{2}(x) is maximized. Hence, the optimal x is \({\overline x }\).
 2.
In this scenario, \({g_{1}}(\overline x) < {g_{2}}(\overline x)\) and \({g_{1}}(\widehat x) \ge {g_{2}}(\widehat x)\). The optimal solution is obtained at the point making \({B_{3}}{a^{2}}{x^{2}} + {B_{4}}{\left (bx + c\sqrt {1  {x^{2}}} \right)^{2}} = {{A_{1}}{{\left (bx + c\sqrt {1  {x^{2}}} \right)}^{2}}}\) and is the bigger root of this equation, i.e., \({x^{*}} = \sqrt {\frac {{2{z_{2}}{w_{2}} + v_{2}^{2} + \sqrt {v_{2}^{4} + 4{z_{2}}{w_{2}}v_{2}^{2}  4v_{2}^{2}w_{2}^{2}} }}{{2\left (z_{2}^{2} + v_{2}^{2}\right)}}} \).
 3.
In this scenario, \({g_{1}}(\widehat x) < {g_{2}}(\widehat x)\). Obviously, the optimal x is \(\widehat x\).
Remark 2
For the PSR protocol, γ_{DF,1} and γ_{DF,2} could be balanced as far as possible, but the efficient information transmission time is set to be a constant. The drawback of this setting is explained as follows: the SNR of the second hop is subjected to twice channel fading and the fixed information transmission time (T/2) would result in a small efficient SNR, which is the main factor affecting the network throughput. However, as we will see, in the HR protocol, the information transmission time can be adjusted flexibly.
6 Hybrid relaying (HR) protocol
Hence, the received SNR at R_{i} after the second phase is given in (13) and the SNR of the first hop is given in (14).
Interestingly, (32) can be regarded as the sum of (4) and (21), i.e., the SNR of the second hop under the HR protocol can be understood as the sum of the SNR of the second hop under the TSR and the PSR protocols.
where \({B_{5}} = \frac {{{A_{2}}}}{{1 + \frac {{{B_{1}}}}{{{A_{1}}}} + \frac {{{B_{2}}}}{{{A_{1}}}}}}\) and \({B_{6}} = \frac {{{A_{3}}}}{{1 + \frac {{{B_{1}}}}{{{A_{1}}}} + \frac {{{B_{2}}}}{{{A_{1}}}}}}\).
Due to the fact that α is coupled with x and problem P4 is nonconvex, finding the optimal solution of α and x is difficult. However, we can pursue a suboptimal design by adopting the idea of alternating optimization, which has been shown to achieve a decent performance [48].
The proposed alternating optimization approach is summarized in Algorithm 1. In particular, since the objective function has a finite value and the alternating optimization design is nondecreasing in each step, Algorithm 1 is guaranteed to be convergent. In Section 7, we will show the proposed alternating optimization approach achieves nearoptimal performance. Moreover, given the required precision of the solution σ, Algorithm 1 only requires \(\log \left (\frac {1}{\sigma }\right)\) iterations, while conventional one dimensional search would require \(\frac {1}{\sigma }\) iterations.
Remark 3
The HR protocol combines the advantages of the TSR and PSR protocols and achieves the best tradeoff relationship between the information transmission time and the efficient SNR. Hence, under the proposed network architecture, it can be regarded as the optimal protocol in term of maximizing the network throughput.
7 Simulation results
In this section, numerical results are presented to compare the throughput performance of the TSR, PSR, and HR protocols in terms of some key parameters. And insights on the optimal TS and PS ratios are also provided. In all simulations, the distance between the source and the destination is set to 10 meters (m), the path loss factor is equal to 3, and the energy conversion efficiency is equal to 0.6. For all figures, each point is obtained by averaging 2×10^{4} independent and random simulations. The channel model has been introduced in Section 3. The number of antennas equipped at the source is set to N=3 and N=8.
7.1 Throughput performance comparison

One scheme is called “Beamforming towards the best relay" (BTBR1). In particular, for BTBR1, the beamforming vector w_{i} (i∈{1,2}) would be set to \(\frac {{\mathbf {h}_{S{R_{b}}}^{*}}}{{\left \ {{\mathbf {h}_{S{R_{b}}}}} \right \}}\), where \(b = \arg \mathop {\max }\limits _{i \in \left \{ {1,2} \right \}} \min \left \ {{\mathbf {h}_{S{R_{i}}}}} \right \\left  {{f_{{R_{i}}D}}} \right \).

The other scheme is called BTBR2. In particular, for BTBR2, w_{i} (∀i∈{1,2}) is also set to \(\frac {{\mathbf {h}_{S{R_{b}}}^{*}}}{{\left \ {{\mathbf {h}_{S{R_{b}}}}} \right \}}\) under these three protocols. However, the relay R_{{1,2}/b} does not participant in the transmission process.
Now, let us formally compare these three protocols. As we can see, (i) the TSR protocol outperforms the PSR protocol in the low SNR region (such as 14–24dB and 14–20dB in our proposed scheme when N=3 and N=8), and the latter outperforms the former in the high SNR region (such as 26–30dB and 22–30dB in our proposed scheme when N=3 and N=8); and (ii) the HR protocol owns the best performance in any SNR region. The corresponding reasons are given as follows:
(i) The values of the throughput shown in (5), (16), and (33) depend on the information transmission time and the channel capacities of the two hops. In the low SNR region, the main factor affecting the throughput is the channel capacities of the two hops, the values of which would increase rapidly by expanding the time length of the first phase in the TSR protocol. This is because that the logarithmic function has large slope when the independent variable is in a small range. However, in the PSR protocol, the channel capacities cannot be further enhanced after implementing optimal power splitting scheme at each relay. In the high SNR region, as the logarithmic function grows slowly, the channel capacities would change more stably and the throughput is mainly related to the information transmission time, the value of which is maximized in the PSR protocol.
(ii) The reasons are presented in Remarks 1, 2,and 3; here, we omit the explaining process.
7.2 Insights on the optimal TS, PS, and TS&PS ratios
In this subsection, we provide some insights on the optimal TS ratio α and PS ratio λ of these three protocols under our proposed scheme. Specially, the optimal λ is specified by the average value of \(\lambda _{1}^{*}\) and \(\lambda _{2}^{*}\) in the PSR and HR protocols.
(i) As mentioned in Subsection 7.1, with the increasing transmit power, the channel capacities of the two hops would change slowly. Then, a larger information transmission time would contribute the larger throughput.
(ii) The purpose of power splitting in the PSR protocol is to balance the capacities of the two hops as much as possible. No matter what the transmit power is, once other parameters are fixed, the ratio allocation would not be changed. And the large \(\lambda _{{PSR}}^{*}\) is the result of the twice fading suffered by the SNR of the second hop.
(iii) For the HR protocol, \(\alpha _{{HR}}^{*}\) must be smaller than \(\alpha _{{TSR}}^{*}\), as each relay can split some fraction of the received signal at the second phase for additional harvested energy to increase the channel capacity of the second hop. Obviously, as \(\alpha _{{HR}}^{*}\) decreases, \(\lambda _{{HR}}^{*}\) would increase accordingly so that the best balance can be achieved. Finally, the HR protocol would completely evolve into the PSR protocol when the transmit SNR approaches infinity.
(i) With the increasing d_{SR}, the channel capacities of the first hop and the second hop would reduce and increase, respectively. Hence, although the capacity of the second hop is suffered from twice channel fading, the relays only need smaller power to compensate it.
(ii) For the TSR protocol and the HR protocol, the channel capacity of the second hop would get its minimum when the relays are located at the middle of the source and the destination, which needs to be enhanced by increasing α.
8 Conclusion and future work
In this paper, we have investigated the throughput performance of three relaying protocols, i.e., the TSR protocol, the PSR protocol, and the HR protocol for the wirelesspowered multiantenna dualrelay network. To maximize the throughput of each protocol, we have provided the detailed design processes for the optimal energy and information beamforming vectors at the multiantenna source, the optimal time switching, and power splitting ratios. Based on theoretical analysis and simulation results, we have found that the TSR protocol and the PSR protocol have their respective advantages in the low and high SNR regions. And the best performance can be achieved by employing the HR protocol and setting the relays closer to the source or the destination concurrently.
For future works, we will consider a more challenging scenario such that there exist any number of relays and each relay works in the FD mode. Specifically, the FD strategy of each relay enables the simultaneous transmission/reception of information and energy signals. Then, it is conceivable that these would significantly change the methods of designing energy and information beamformers at the source and the detailed optimization process for finding the optimal time switching and power splitting ratios.
Footnotes
 1.
In our work, we considered the dualrelay case because of the following reasons: the dualrelay network, i.e., diamond network, has been discussed in numerical classical works, such as [38, 39, 40], which all demonstrated that the dualrelay network was worth studying and also provided valuable guidance for us. In addition, to the best knowledge of the authors, there are no related works that have employed the energy harvesting technique in the dualrelay network, where the source is equipped with multiple antennas. This motivates us to carry out detailed research for the energy harvesting based dualrelay network.
 2.
To keep consistency, in the following sections, we also use w_{1} and w_{2} to represent the energy beamforming vector and information beamforming vector, respectively.
 3.
Because that the two relays all participant in the transmission process, the efficient SNR of the first hop must be determined by the minimal received SNR by the two relays.
 4.
Notes
Acknowledgements
This research was supported by National Natural Science Foundation of China (No. 61371122, 61471393, 61771487, and 61501507), Jiangsu Provincial Natural Science Foundation of China (BK20150719).
Authors’ contributions
GH is the main writer of this paper. He proposed the main idea, conducted the simulations, and analyzed it. YC, LA, and XW assisted the review of this manuscript. All authors read and approved the final manuscript.
Funding
This research was supported by National Natural Science Foundation of China (No. 61371122, 61471393, 61771487, and 61501507), Jiangsu Provincial Natural Science Foundation of China (BK20150719).
Competing interests
The authors declare that they have no competing interests.
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