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SWIPT in mMIMO system with nonlinear energyharvesting terminals: protocol design and performance optimization
EURASIP Journal on Wireless Communications and Networking volume 2019, Article number: 72 (2019)
Abstract
In this paper, we design the simultaneous wireless information and power transfer (SWIPT) protocol for massive multiinput multioutput (mMIMO) system with nonlinear energyharvesting (EH) terminals. In this system, the base station (BS) serves a set of uplink fixed halfduplex (HD) terminals with nonlinear energy harvester. Considering the nonlinearity of practical energyharvesting circuits, we adopt the realistic nonlinear EH model rather than the idealistic linear EH model. The proposed SWIPT protocol can be divided into two phases. The first phase is designed for terminals EH and downlink training. A beam domain energy beamforming method is employed for the wireless power transmission. In the second phase, the BS forms the twolayer receive beamformers for the reception of signals transmitted by terminals. In order to improve the spectral efficiency (SE) of the system, the BS transmit power and timeswitching ratios are optimized. Simulation results show the superiority of the proposed beamdomain SWIPT protocol on SE performance compared with the conventional mMIMO SWIPT protocols.
Introduction
With the rapid rise of the Internet of Things (IoT) industry [1], 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 [2–4], as RF signal is controllable, stable, and efficient compared to the natural sources such as solar and wind.
As one of the key technologies of fifthgeneration (5G) communication system, massive multipleinput multipleoutput (mMIMO) [5] technology can obtain the diversity gain and multiplexing gain, and hence can greatly improve the spectrum efficiency (SE) of wireless networks [6–8]. Compared with traditional MIMO, the spatial resolution of mMIMO systems is significantly improved [9], which can greatly reduce various types of interference [7].
As a practical solution to improve the energy efficiency (EE) of batterypowered IoT network in 5G systems and hence extend the service lifetime of the IoT network, RF energyharvesting (EH) technology has been widely studied [10–12]. In an actual simultaneous wireless information and power transfer (SWIPT) system [13], the base station (BS) or access point (AP) provides the receiver with concurrent information and energy supply. Currently, the practical SWIPT receiver architecture can be mainly classified as timeswitching (TS) and powersplitting (PS) [14] according to the signal partition method for EH and information decoding (ID). The TS SWIPT receiver alternates between EH and ID according to the TS ratio [15], and as a result, the received RF signal is first sent to the energy harvester and then to the information decoder. For the PS SWIPT receiver, the received RF signal is split into two parts according to the PS ratio [13]; one portion is sent to the energy harvester, and the remainder is sent to the information decoder.
In [16], the TS and PS protocols were designed for the amplifyandforward (AF) oneway singleantenna SWIPT relay networks. In [17], the SWIPT protocol was designed for oneway MIMO relay network, where the MIMO relay can simultaneously harvest energy from the signal sent by the destination and decode the signal sent by the source. Since twoway relaying (TWR) system can further improve the SE of multihop relay system [18–20], recently, SWIPT protocol for TWR systems were studied in many literatures. The authors in [21] studied the SWIPT protocol in the AF TWR system, wherein the RFpowered AF relay node is applied to help the two source nodes exchange information. In [22], the problem of maximizing the sum throughput in the TWR networks with wirelesspowered nodes was studied. The authors in [23] considered the SWIPT protocol design in multiple relay AF TWR networks, where two source nodes harvest energy from multiple relays.
In [16, 17, 21–23], it was assumed that the amount of energy harvested by the EH circuit is linearly proportional to the received RF power, namely, the linear EH model. However, as studied in recent literature [24–29], the linear EH model is too idealistic and impractical because linear assumptions do not hold in practice. The energy conversion efficiency of an actual EH circuit is a nonlinear function of the received RF signal due to various causes of nonlinearity, e.g., diode nonlinearity, and saturation nonlinearity. In addition, the sensitivity of the energy harvester is limited [30], that is, signals with power around communications sensitivity can be decoded at a SWIPT receiver but cannot be exploited for energyharvesting purposes. Therefore, in practice, the amount of harvested energy is clearly a nonlinear function of the received power.
In [31], the SWIPT protocol for threedimensional (3D) mMIMO system is designed, where the matched filter (MF) precoder is applied. In [32], the SWIPT protocol for multipair mMIMO TWR system is designed, where linear precoders, i.e., zeroforcing (ZF) and maximal ratio combining (MRC), are employed at the relay. A lowcomplexity SWIPT scheme with retrodirective maximum ratio transmission (MRT) beamforming is introduced in [33]. In [34], each frame is divided into three phases: the uplink channel estimation phase, the downlink wireless energy transmission phase, and the uplink wireless information transmission phase. ZF and MRC beamforming are applied at the access point. For SWIPT system with mMIMO, instantaneous fulldimensional channel state information (CSI) is required to perform transmit and receive precoding [34]. The above four works [31–34] have considered linear energy harvesters. To obtain full dimensional CSI, the uplink training overhead is linearly proportional to the number of user equipments, thus causing significant bottlenecks on the achievable SE.
Methods
In this paper, the beamdomain SWIPT protocol for mMIMO system with fixed halfduplex (HD) nonlinear energyharvesting terminals (TEs) is designed, and the system spectral efficiency performance is maximized by optimizing the TS ratio and transmit power of the BS^{Footnote 1}. The contributions are summarized as follows.

We consider the SWIPT protocol design for TEs with nonlinear energy harvesters in mMIMO systems. Moreover, a more practical power consumption model is employed at the TE. In particular, after the energyharvesting phase, a portion of the harvested energy is used for necessary processing, such as the energy consumption of channel estimation, channel feedback, and energy consumption of the circuit, to achieve its primary task. The remainder energy is used for the subsequent wireless information transmission.

Based on the idea of TS, the entire transmission process of the proposed beamdomain SWIPT protocol can be divided into two phases. The first phase is designed for TEs EH and downlink training. During this phase, the BS transmits energy signals to the TEs. The TEs utilize one portion of the received energy signals for EH and the remainder for downlink channel estimation. After obtaining the channel CSI, TEs feedback it to the BS. The beamdomain energy beamforming method is employed for wireless energy transmission. In the second phase, the BS forms the twolayer receive beamformers for the reception of signals transmitted by TEs.

In order to improve the SE of the system, the base station transmit power and timeswitching ratio are optimized. The simulation results show that compared with the traditional mMIMO SWIPT protocol, the proposed beamdomain SWIPT protocol has superior performance in SE performance.
The rest of the paper is organized as follows. The system and channel model are introduced in Section 1. Section 1 illustrates the proposed beamdomain mMIMO SWIPT transmission scheme. Section 1 analyzes the achievable sumrate performance and proposes the optimization method. Section 1 presents the simulation results. Our conclusions are presented in Section 1.
Notations: In this paper, \(\mathbb {E}(\cdot)\) denotes the expectation. \(\phantom {\dot {i}\!}\mathbf {A}^{\{B_{1}, B_{2}\}}\) is the submatrix of A containing rows of set B_{1} and columns of set B_{2}. A^{{B,:}} and A^{{:,B}} are the submatrix of A containing rows and columns of set B, respectively. δ(·) denotes the dirac delta function. (·)^{T}, (·)^{∗}, (·)^{H}, ·, ∥·∥, and tr(·) denote transpose, conjugate, conjugate transpose, determinant, Frobenius norm, and trace of a matrix, respectively. ⌊·⌋ and ⌈·⌉ denote the floor and ceilings, respectively. I_{V} denotes V×V identity matrix. 〈·,·〉 denotes the inner product operator.
System model
We consider a mMIMO SWIPT system, where the BS serves a set of K_{t} fixed uplink TEs \(\phantom {\dot {i}\!}\mathbf {K}_{T}=\{k_{1},k_{2},\cdots,k_{K_{t}}\}\), as shown in Fig. 1. We assume that all TEs are uniformly distributed in the coverage area of the BS. The BS is equipped with a massive antenna array with N antennas. We assume that all TEs operate in HD mode and are equipped with a single antenna. Each TE employs a nonlinear energy harvester to collect energy and uses the harvested energy for subsequent data transmission.
In this paper, the framebased SWIPT protocol is designed for mMIMO system. The length of one frame is fixed to T symbols, and we assume that T is less than the coherence interval of the fading channel. Let \(\mathbf {h}^{u}_{k_{i}}\in \mathbb {C}^{N \times 1}\) and \(\mathbf {h}^{d}_{k_{i}}\in \mathbb {C}^{N \times 1}\) denote the channel vector of the uplink channel between transmit antenna array of BS and TE k_{i} and corresponding downlink channel vector. We assume that the channel reciprocity between uplink and downlink channels can be exploited, that is, the uplink channel can be obtained by the downlink channel estimation.
Uplink/downlink channel model
In this paper, we apply the general clusterbased channel model [35], that is, the signal of TE k_{i} received by the BS is the sum of the contributions from M_{u} scatterings. Similarly, the signal of the BS received by the TE k_{i} is the sum of the contributions from M_{d} scatterings. We assume that the direction of arrival (DOA) or direction of departure (DOD) of signals from cluster m is within the region \(\left [\theta ^{o,{\text {min}}}_{k_{i},m}, \theta ^{o,{\text {max}}}_{k_{i},m}\right ], o\in \{u,d\}\), then the uplink/downlink channel vector between the BS and TE k_{i} can be written as [35]
where \(\mathbf {a}\!\left (\theta \right)\! \,=\,\! {\left [ {1,\exp \! \!\left (\! { \frac {{j2\pi d\sin \left (\theta \right)}}{\lambda }}\! \right)\!,\! \cdots,\!\exp \!\! \left (\!{ \frac {{j2\pi d\left ({N  1} \right)\sin \left (\theta \right)}}{\lambda }}\! \right)} \right ]^{T}}\) represents the array response vector, d denotes the antenna spacing, and λ denotes the carrier wavelength. \(r^{o}_{k_{i},m}(\theta)\) denotes the complex uplink/downlink channel response gain.
The uplink/downlink channel vector \({\mathbf {h}^{o}_{{k_{i}}}}\) can be expanded by a set of uniform basis vectors \(\{\mathbf {f}_{1},\mathbf {f}_{2},\cdots,\mathbf {f}_{N}\}\in \mathbb {C}^{N\times 1}\) by utilizing the basis expansion model (BEM) [36], that is,
where F=[f_{1},f_{2},⋯,f_{N}], and the basis vector f_{n} is also called a beam [37]. The vector \(\tilde {\mathbf {h}}^{o}_{k_{i}} =\left [\tilde {h}^{o}_{k_{i},1}, \tilde {h}^{o}_{k_{i},2}, \cdots, \tilde {h}^{o}_{k_{i},N}\right ]^{T}\) is called beamdomain (BD) channel vector [38] and can be obtained by
Usually, the effective channel length of the mMIMO channel, that is, the number of elements with nonnegligible channel gains in vector \(\tilde {\mathbf {h}}^{o}_{k_{i}}\), is far less than the number of antennas. Therefore, applying BEM can greatly reduce the size of the channel that needs to be estimated and the number of CSIs that need feedback. Thus, for a given set of beams, we can approximate the channel vector \({\mathbf {h}^{o}_{{k_{i}}}}\) as [38]
where \(B_{k_{i}}\) is the index set of the selected active beams of the terminal k_{i}. The approximation in the middle of (4) comes from the assumption that the channel gain of the terminal k_{i} on the beams corresponding to the index set \(B_{k_{i}}\) is not negligible, and the channel gain on other beams is close to zero and can be neglected, that is, \(\sum \limits _{m \notin {B_{{k_{i}}}}} {\tilde h_{{k_{i}},m}^{o}{\mathbf {f}_{m}}} \approx 0\). \(\mathbf {F}^{\left \{B_{k_{i}},:\right \}}\) denotes the corresponding active beam (AB) space with columns are indexed in the set \(B_{k_{i}}\). The channel vector with reduceddimension \(\widetilde {\mathbf {h}}_{o,{k_{i}}}^{\left \{ {{B_{{k_{i}}}}} \right \}} \in {\mathbb {C}^{\left  {{B_{{k_{i}}}}} \right  \times 1}}\) is named as the BD effective channel and (4) holds with equality as N→∞.
Based on (1) and (4), \(\widetilde {\mathbf {h}}_{o,{k_{i}}}^{\left \{ {{B_{{k_{i}}}}} \right \}}\) can be written as
Hence, in order to obtain \({\mathbf {h}^{o}_{{k_{i}}}}\) by channel estimation, \(\left  {{B_{{k_{i}}}}} \right \)dimension BD effective channel estimation is enough. As a result, the consumption of training resources will be significantly reduced.
As introduced in [11], all TEs can be partitioned into groups according to their AB sets. Let g_{t} denote the group index and \(B_{g_{t}}\) denote the corresponding AB set of the group g_{t}. Based on the orthogonality of active beam space across disjoint groups, that is, \(B_{g_{t}}\bigcap B_{{g^{\prime }}_{t}} = \emptyset \) when g_{t}≠g^{′}_{t}, the BD effective channel dimensions of TEs within group g_{t} is reduced to \(B_{g_{t}}=b_{g_{t}}\).
Energyharvesting model
At present, the total energy collected by wirelesspowered terminals in the wireless charging stage can usually be modeled as two models: the ideal linear EH model, and the practical nonlinear EH model.
In linear EH model [21–23], the collected energy of the terminal k_{i} can be expressed as
where 0<η<1 denotes the constant energy conversion efficiency for converting RF energy to electrical energy, \(P^{\text {EH}}_{k_{i}}\) denotes the total received RF power at wirelesspowered user k_{i}, and T_{h} denotes the duration of energy harvesting.
In the practical nonlinear EH model, the amount of the harvested energy is modeled using the logistic (or sigmoid) function, i.e., Sshaped curve, as follows [28, 29]:
and
where P_{s} denotes the maximum amount of harvested power when the EH circuit is saturated, and a and b are positive constants related to the circuit specification [28, 29]. Figure 2 illustrates the practical nonlinear EH model given in (7) and the conventional linear EH model given in (6). We assume that all the EH TEs apply the same nonlinear energy harvester.
Energy consumption model
Most of the previous works assume that all harvested energy is used for power amplifier during the wireless information transmission phase [11, 21–23]. In this work, we employ a more practical power consumption model for the TE. In particular, after the energyharvesting phase, one portion of the harvested energy is used for necessary processing at TE to achieve its main task, and the remainder energy is used for uplink transmission during the wireless information transmission phase. Note that the similar model has also been applied in [39]. Let \(E^{\mathrm {P}}_{k_{i}}\) denote the processing energy^{Footnote 2}; in the wireless information transmission phase of time period T_{I}, the transmit power of TE k_{i} can be expressed as
Note that (9) has assumed that the harvest energy is larger than the processing energy. Otherwise, \({p^{u}_{{k_{i}}}}\) should be set to zero and an outage occurs.
Proposed beamdomain SWIPT protocol
In this paper, the framebased protocol is applied to realize beamdomain SWIPT in the cellular system with mMIMO BS. The whole transmission process is divided into N frames. Each frame can be divided into two phases based on the idea of TS, i.e., phase I and phase II, as shown in Fig. 3.
In phase I of time period αT (0≤α≤1), the BS transmits energy signals to all the TEs, where α denotes TS ratio. We assume that all TEs turn on the receive circuit during the interval αT. TEs utilize one portion of the received energy signals for EH and the remainder for downlink channel estimation. In this phase, the main purpose of the protocol design is to enable TEs to collect as much energy as possible.
Specifically, the BS transmits the energy signals to all the TEs. The signal received by ith TE can be expressed as \(y_{r,i}^{S}\). Part of the received energy signal, \(\sqrt {\beta }y_{r,i}^{S}\), is used for EH and part of the received signal, \(\sqrt {1\beta }y_{r,i}^{S}\), is used for downlink channel estimation, where β denotes the PS ratio, as shown in Fig. 4. The detailed training signals design will introduced in the next subsection.
In phase II of time period (1−α)T, the TEs feedback the uplink CSI to the BS and transmit their data during the interval (1−α)T using the harvested energy. Specifically, the ith TE transmits signal \(\sqrt {\beta \eta }y_{t,i}^{S}\) to the BS. Note that the resource consumption for CSI feedback of TEs are omitted in this paper.
Energy signal design
Let \({\Phi _{g_{t}}^{\mathrm {E}}} \in {{\mathbb {C}}^{{\tau _{t}}\times K_{g_{t}}}}\) be the orthogonal energy sequence set transmit by the BS to the TE group g_{t}, where \(K_{{g_{t}}}\) denotes the number of TEs in group g_{t}, τ_{t} denotes the length of energy sequence and satisfies \(\phantom {\dot {i}\!}{\tau _{t}} \ge b_{g_{t}}\), and \({\left  {{{\left [ {{\Phi ^{\mathrm {E}}_{{g_{t}}}}} \right ]}_{j,k}}} \right ^{2}} = \tau _{t}{p^{d}_{g_{t,k}}}\). \({p^{d}_{g_{t,k}}}\) denotes the power of each energy symbol. Let G_{t} denote the terminal group set, the time allocated for the TEs EH is \( \alpha T \geq \max \{b_{g_{t}}g_{t} \in G_{t}\}\).
Energy transfer and TE downlink channel estimation
During phase I of frame n, the received signal of TE group g_{t} can be expressed as
where \(\mathbf {H}_{d,g_{t}}[\!n] = \left [ {\mathbf {h}^{d}_{g_{t,1}}[\!n], \cdots,\mathbf {h}^{d}_{g_{t,{K_{g_{t}}}}}}[\!n] \right ]\) and \({\mathbf {N}^{d}_{g_{t}}}[n]\) denote additive white Gaussian noise (AWGN) with variance σ^{2}. The received signal at the g_{t,k}th TE can be expressed as
where \({\mathbf {n}_{{g_{t,k}}}}[\!n]\sim \mathcal {CN}(0,\sigma ^{2})\) denotes the AWGN at TE g_{t,k}.
The TE harvests energy from one portion of the whole received signal, that is, \(\sqrt {\beta }{\mathbf {y}^{d}_{g_{t,k}}}[\!n]\). For signal with large transmit power, the energy of noise is negligible in the harvested energy [16]. In the linear EH model, the harvested energy of TE g_{t,k} during the frame n can be written as
Based on the orthogonality of active beam space across disjoint groups, that is, \(\left \langle {{\mathbf {F}^{\left \{ {{B_{{g_{t}}}}\left [ n \right ]},: \right \}}},{\mathbf {F}^{\left \{ {{B_{g{^{\prime }_{t}}}}\left [ n \right ]},: \right \}}}} \right \rangle = 0\) when \({B_{{g_{t}}}}\left [ n \right ] \ne {B_{g{^{\prime }_{t}}}}\left [ n \right ]\), we have the following approximation
where \({\hat {p}^{\mathrm {H}}_{d,{g_{t,k}}}} = q_{g_{t,k}} {p^{d}_{g_{t,k}}}[\!n]\) and \(q_{g_{t,k}}={\left \ {\tilde {\mathbf {h}}_{d,g_{t,k}}^{\left \{ {{B_{g_{t}}}[n]} \right \}}}[n] \right \^{2}}\).
In the adopted nonlinear model, the amount of the harvested energy at g_{t,k} is modeled using the logistic (or sigmoid) function, i.e., Sshaped curve, as follows [28]:
Another portion of the TE g_{t,k} received signal, that is, \(\sqrt {1\beta }{\mathbf {y}^{d}_{g_{t,k}}}[\!n]\), is used for the channel estimation. Since the TE knows the values of β and hence can accurately compensate it, we omit it in the following channel estimation. The least square (LS) estimator [38] of the downlink channel can be expressed as
where e^{k} denotes the unit vector with the kth element is 1 and the rest of the elements are 0. The linear minimum mean square error (LMMSE) estimate [38] of the downlink channel can be expressed as
The uplink channel CSI \(\tilde {\mathbf { h}}_{u,g_{t,k},LM}^{\left \{ {{B_{g_{t}}}[n]} \right \}}[\!n]\) can be obtained at the BS by using the CSI feedback^{Footnote 3}.
TE uplink information transmission
In phase II of time period (1−α)T, the TEs send signal using one portion of the harvested energy during phase I, and the BS uses the estimated effective channels to perform receive beamforming and receive data sent by K_{T} TEs. The transmit power of TE g_{t,k} can be expressed as
where \({\breve {E}^{\text {NL}}_{{g_{t,k}}}} [\!n]\) denotes the currently available energy in the nth frame, which can be expressed as the difference between the cumulative harvested energy and the consumed energy during the {1,2,⋯,n−1}th frame, that is,
where \({\tilde {E}^{\text {NL}}_{{g_{t,k}}}} [\!i]\), \(E^{\mathrm {P}}_{k_{i}}[\!i]\), α[ i], and \(p^{u}_{g_{t,k}} [\!i]\) denote the harvested energy, processing energy, TS ratio, and transmit power in the ith frame, respectively.
Let \({s}^{u}_{g_{t,k}}[\!n]\) denote the transmit signal of TE g_{t,k}, and the transmit power of TE g_{t,k} is set to \(p^{u}_{g_{t,k}}[\!n]={\left  {s}^{u}_{g_{t,k}} [\!n] \right ^{2}} \). The received signal at the BS can be expressed as
where \({{\mathbf {s}^{u}_{g_{t}}}}[\!n]=[s^{u}_{g_{t,1}}[\!n], s^{u}_{g_{t,2}}[\!n],\cdots,s^{u}_{g_{t,K_{g_{t}}}}[\!n]]^{T} \in \mathbb {C}^{K_{g_{t}}\times 1}\) denotes the signal sent by TE group g_{t}. \(\mathbf {n}^{u}_{t}[\!n]\) denotes the AWGN vector with variance σ^{2}. In this paper, we apply the twolayer beamforming scheme. The firstlayer beamformer is a matrix composed of active beam sets \({\mathbf {F}^{\left \{ {{B_{{g_{_{t}}}}}[n]},: \right \}}}\). Multiplying both sides of (19) with \({\mathbf {F}^{\left \{ {{B_{{g_{_{t}}}}}[n]},: \right \}}}\), the received signal of the BS from group g_{t} can be expressed as
where \({\tilde {\mathbf {n}}^{u}_{g_{t}}}[\!n]= { {{\mathbf {F}^{^{\left \{ {{B_{{g_{_{t}}}}}} [n],:\right \}}}}}} \mathbf {n}^{u}_{t}[\!n]\). To detect the signal of group g_{t}, the BS employs the secondlayer receive beamforming, i.e., \({\dot {\mathbf {y}}^{u}_{g_{t}}}[\!t] = \mathbf {W}_{g_{t}}^{H}[\!n]{\tilde {\mathbf {y}}^{u}_{g_{t}}}[\!n]\) and \({\mathbf {W}_{g_{t}}}[\!n] = \left [ {{\mathbf {w}_{g_{t,1}}}[\!n],{\mathbf {w}_{g_{t,2}}}\!n], \cdots,{\mathbf {w}_{g_{t},K_{{g_{t}}}}}}[\!n] \right ] \in {\mathbb {C}^{b_{t}[n] \times K_{{g_{t}}}[n]}}\). The kth element of \({\dot {\mathbf {y}}^{u}_{g_{t}}}[\!n]\) can be expressed as
The signaltointerferenceandnoise ratio (SINR) of the BS received signal from TE g_{t,k} can be expressed as
Let \(\Delta \tilde {\mathbf { h}}_{t,{g_{t,k}}}^{\left \{ {{B_{{g_{t}}}}[n]} \right \}}\) denote the channel estimation error of \(\tilde {\mathbf { h}}_{t,{g_{t,k}}}^{\left \{ {{B_{{g_{t}}}}[n]} \right \}}\), and \({{{\Theta }}_{g_{t,k}}}[\!n]\) can be expressed as
where \({\text {CE}}{{\mathrm {E}}_{{g_{t,k}}}}\left [ n \right ]\), \({\text {IU}}{{\mathrm {I}}_{{g_{t,k}}}}\left [ n \right ]\), and \({\text {IG}}{{\mathrm {I}}_{{g_{t,k}}}}\left [ n \right ]\) denote the total power introduced by channel estimation error (CEE), interuser interference (IUI), and intergroup interference (IGI), respectively. \(\Lambda _{g_{t}}[\!n] = \mathbb {E}\left [ {{\mathbf {s}^{u}_{g_{t}}}[\!n](\mathbf {s}^{u}_{g_{t}}[\!n])^{H}} \right ]\) denotes the diagonal covariance matrix of group g_{t}.
The optimal \(\mathbf {w}_{g_{t,k}}^{}[\!n]\) is the vector that maximizes the \(\text {SINR}_{g_{t,k}}[n]\), and, thus, the secondlayer beamformer can be obtained by
where \({{\mathbf {\Xi }_{g_{t,k}}}}[\!n]\) can be written as
The problem in (24) is in generalized Rayleigh quotient form. The maximum value in (24) can be obtained when \(\mathbf {w}_{g_{t,k}}^{}[n]\) equals to the generalized eigenvector corresponding to the maximum generalized eigenvalue [11], which can be written as
Achievable sumrate analysis and optimization
When optimizing the system achievable sumrate, the following two constraints should be considered: Firstly, the harvested energy at the TEs should be enough for the signal transmission in the subsequent information transmission. Secondly, the minimum transmission rate of TEs should be guaranteed.
To guarantee the effective transmission of the powerlimited devices, the currently available energy \({\breve {E}^{\text {NL}}_{{g_{t,k}}}} [\!n]\) in the nth frame should be enough for the signal transmission at devices in the subsequent information transmission, that is,
where ∂ denotes the minimum transmission energy constraint of TEs.
With the help of the bounding technique [40] and according to (21), the average achievable rates at TE g_{t,k} can be expressed as
The transmit power of TE g_{t,k} is set to \(p^{u}_{g_{t,k}}\). The powers of CEE, IUI, and IGI are introduced in (41).
The system achievable rate can be expressed as
where \(\mathcal {P}[n] = \left \{{p^{d}_{{g_{t,1}}}}[\!n], \cdots,{p^{d}_{{g_{t,{K_{t}}}}}[\!n]} \right \}\) denotes the power set transmitted by the BS to each TE and the uplink rate \({{R}}_{g_{t,k}}\) of TE g_{t,k} are given in (41). Let R_{min} denote the minimum throughput of the TEs, and the system achievable rate maximization problem can be expressed as
where \(P^{\mathrm {E}}_{d}\) denotes the maximum transmit power constraint at the BS during phase I. One can check that (30) is nonconvex due to the constraints in p2∼p6, and therefore it is not straightforward to obtain the globally optimal solution. To tackle this difficulty, we apply the lower bound of the available energy in the frame n, that is, \({{\breve {E}}^{\text {NL}}_{{g_{t,k}}}} [\!n]\geq {\tilde {E}^{\text {NL}}_{{g_{t,k}}}} [\!n]E_{g_{t,k}}^{\mathrm {P}}\!n] \geq \eth \). Since (14) is a transcendental function, this inequality cannot be solved directly. Fortunately, we know that the collected energy \({\tilde {E}^{\text {NL}}_{{g_{t,k}}}} [\!n]\) is bounded by P_{s}αT; hence, to ensure the subsequent uplink transmission of TEs, we have \(P_{s} \alpha T > E_{g_{t,k}}^{\mathrm {P}}[\!n] + \eth \), and if this condition can be met, we have
where Ω is given in (8). Hence, the SE optimization problem in (30) can be rewritten as
In order to simplify the representation, we omit the frame index n in the following derivations.
Optimize \(\mathcal {P}\) for fixed α
We fix α to α=α_{0}. The uplink rate \({{R_{g_{t,k}}}}\) given in (41) can be expressed as \({R_{g_{t,k}}} = (1\alpha _{0}){\log _{2}}\left ({1 + \gamma _{g_{t,k}}} \right) \) and the SINR of TE g_{t,k} received signal can be expressed as \(\gamma _{g_{t,k}}=p^{u}_{g_{t,k}}{b_{g_{t,k}}}\left ({{c_{g_{t,k}} + d_{g_{t,k}}}}\right)^{1}\), wherein \(b_{g_{t,k}},c_{g_{t,k}},d_{g_{t,k}}\) are positive variables. Specifically, \(b_{g_{t,k}} = {\left \ \tilde {\mathbf {h}_{g_{t,k}}^{\left \{ {{B_{g_{t}}}} \right \}}} \right \^{2}}\), \(c_{g_{t,k}} = {\text {CE}}{{\mathrm {E}}_{g_{t,k}}} + {\text {IU}}{{\mathrm {I}}_{g_{t,k}}} + {\text {IG}}{{\mathrm {I}}_{g_{t,k}}} \), and \(d_{g_{t,k}} = \\mathbf {w}_{{g_{t,k}}}^{H}\^{2}=1\). Based on (41) and omit the group index, we can rewrite the SE optimization problem as
where \(\hat {p}1\sim \hat {p}4\) are constraints given by (32). Since \(p^{u}_{g_{t,k}}= \frac {{{{\breve {{E}}^{\text {NL}}_{{g_{t,k}}}}}[n]  E_{g_{t,k}}^{\mathrm {P}}}}{(1\alpha _{0})T}\geq \frac {{{{\tilde {E}^{\text {NL}}_{{g_{t,k}}}}}  E_{g_{t,k}}^{\mathrm {P}}}}{(1\alpha _{0})T}\), \(p^{u}_{g_{t,k}}\) is lower bound by
(34) is the constraint \(\hat {p}5\). We can see from Fig. 2 that \(p^{u}_{g_{t,k}}\) is a monotonic function of \(p^{d}_{g_{t,k}}\), the solution \(\mathcal {P}\) of problem (33) is equivalent to the optimization of \(\mathcal {P}^{u}= \left \{{p^{u}_{{g_{t,1}}}}, \cdots,{p^{u}_{{g_{t,{K_{t}}}}}} \right \}\) when α=α_{0} is fixed.
The form of SE optimization problem (33) is close to a geometric programming (GP) except that the target function is not in the posynomial form [41]. The convex optimization tools can be applied to solve (33) by using the technique in [42] to approximate the target. Specifically, 1+γ_{t,k} can be approximated by \({\lambda _{t,k}}\gamma _{t,k}^{{\mu _{t,k}}}\) close to a point \({{\hat \gamma }_{t,k}}\), where \({\mu _{t,k}} = {{\hat \gamma }_{t,k}}{\left ({1 + {{\hat \gamma }_{t,k}}} \right)^{ 1}}\)and \({\lambda _{t,k}} = \hat \gamma _{t,k}^{ {\mu _{t,k}}}\left ({1 + {{\hat \gamma }_{t,k}}} \right)\). Thus, (33) can be rewritten as
Constraints c1 can be reexpressed as
The SE optimization problem (35) becomes a standard GP and can be solved by Algorithm 1.
When the solution \({\widetilde {\mathcal {P}}^{u}}\) of (35) is obtained, we can get the solution \(\widetilde {{\mathcal {P}}}\) according to (14).
Optimize α for fixed \(\mathcal {P}\)
We fix the transmit power of BS to \(\mathcal {P}={\mathcal {P}}^{\ast }\). Since the ratio used for information transmission can be adjusted by changing the duration of phase II, that is, (1−α)T. The uplink rate \({{R_{g_{t,k}}}}(\alpha)\) can be expressed as
where \(\gamma _{g_{t,k}}(\alpha)=p_{{g_{t,k}}}^{u} \tilde {b}_{g_{t,k}} {\left (\tilde {{c}}_{g_{t,k}} +1 \right)^{1}}\), and \(\tilde {b}_{g_{t,k}} = \\tilde {\mathbf { h}}_{u,g_{t,k}}^{\left \{ {{B_{g_{t}}}}[n] \right \}}\^{2}\), \(\tilde {{c}}_{g_{t,k}} = {\left ({{\text {CEE}}_{g_{t,k}}} + {\text {IU}}{{\mathrm {I}}_{g_{t,k}}} + {\text {IG}}{{\mathrm {I}}_{g_{t,k}}} \right)} \). Due to the tremendous gain of antenna array applied in mMIMO system, it is reasonable to assume that the uplink received SINR given by (1) is much greater than 1. Hence, we can treat \(\tilde {b}_{g_{t,k}},\tilde {{c}}_{g_{t,k}}\) as positive variables that are irrelevant to α. From (34) we know that
where \(\breve {p}_{d,{g_{t,k}}} = \hat {p}_{d,{g_{t,k}}}_{{\mathcal {P}}^{\ast }}\). In order to design a computationally efficient resource allocation algorithm, we focus on a lower bound of the objective function:
where \(\vec {\gamma }_{g_{t,k}}(\alpha)=\left (\frac {\alpha }{{\left ({1  \alpha } \right)}}{{\tilde d}_{{g_{t,k}}}}  \frac {{E_{{g_{t,k}}}^{\mathrm {P}}}}{{\left ({1  \alpha } \right)T}}\right)\tilde {b}_{g_{t,k}}{\left (\tilde {{c}}_{g_{t,k}} +1 \right)^{1}}\) denotes the lower bound of the SINR \({\gamma }_{g_{t,k}}(\alpha)\). For the sake of simplification, we ignore \(\mathcal {P}^{\ast }\) in the following derivation. The partial derivative of \({{\vec {R}}_{{\text {SUM}}}}(\alpha)\) can be expressed as
The two order derivative of \({{\vec {R}}_{{\text {SUM}}}}(\alpha)\) can be expressed as
In this paper, we assume that the power of CEE, IUI, and IGI is much larger than AWGN. Hence, we have \(\frac {\partial ^{2}{{R}_{{\text {SUM}}}}(\alpha)}{\partial ^{2} \alpha }<0\) for 0<α<1, and we can obtain the optimal α of R_{SUM}(α) when \(\frac {\partial {{R}_{{\text {SUM}}}}(\alpha)}{\partial \alpha }=0\), that is,
(42) is a transcendental equation, we can resort to Newton iterative method to solve this problem, that is, \(\hat {\alpha }_{k+1}=\hat {\alpha }_{k}\frac {\partial {{\vec {R}}_{{\text {SUM}}}}(\alpha)}{\partial \alpha }/\frac {\partial ^{2}{{\vec {R}}_{{\text {SUM}}}}(\alpha)}{\partial ^{2} \alpha }_{\alpha =\hat {\alpha }_{k}}\). Let \(\hat {\alpha }^{{\text {Opt}}}\) denote the solution of (42), and if \(1>\hat {\alpha }^{{\text {Opt}}} \ge \frac {{\max \left \{ {\max \{ {b_{{g_{t}}}}{g_{t}} \in {G_{t}}\},\max \{ {{\ddot \alpha }_{{g_{t,k}}}}k \in {\mathbf {K}_{T}}\}} \right \}}}{T}\), then the solution of (42) is \({\alpha }^{{\text {Opt}}}=\hat {\alpha }^{{\text {Opt}}}\). Moreover, we can see that the maximum sumrate is \({{R}^{\text {OPT}}_{{\text {SUM}}}}=B({\alpha }^{{\text {Opt}}})\). Otherwise, the solution of (42) is \({\alpha }^{{\text {Opt}}}=\frac {{\max \left \{ {\max \{ {b_{{g_{t}}}}{g_{t}} \in {G_{t}}\},\max \{ {{\ddot \alpha }_{{g_{t,k}}}}k \in {\mathbf {K}_{T}}\}} \right \}}}{T}\), and the maximum sumrate is
Joint optimization of α and \(\mathcal {P}\)
In this subsection, we optimize \(\mathcal {P}\) and α alternatively to maximize \({{R}_{{\text {SUM}}}}(\mathcal {P},\alpha)\). The detailed algorithm is summarized in Algorithm 2.
In each iteration of Algorithm 2, the update of α is performed by evaluating the firstorder condition numerically, whose complexity is similar with the approach in [43].
Results and discussion
In this section, the performance of the proposed beamdomain SWIPT protocol is evaluated using the 3GPP LTE simulation model in macrocell environment [44]. The center frequency and bandwidth are set to 2.4 GHz and 20 MHz, respectively. The path loss between BS and TEs is modeled as 2.7+42.8 log10(R) [dB] and the path loss between terminals are modeled as 55.78+40 log10(R) [dB], where R denotes the distance. Thermal noise density is set to − 174 dBm/Hz. The power ratio used for energy harvesting β is set to 0.8. The energy conversion efficiency η for linear harvester is set to 0.8. For the nonlinear EH [28, 29], P_{s} in this paper is set to 0.024 mW, a = 150, and b = 0.014, as shown in Fig. 2. Hence, the energy conversion efficiency for nonlinear EH is not a fixed number. For comparison, the performance of traditional mMIMO SWIPT protocol with MF precoder [31], with ZF precoder [32], and with MRT precoder [33] are also simulated. In the simulations, the uplink transmit power of TEs in [31–33] is also harvested from the BS downlink transmit signal. In simulations, the CSI obtained by beamdomain channel estimation is adopted in the above schemes, respectively.
Figure 5 illustrates the SE of the proposed beamdomain SWIPT protocol for mMIMO system. All the TEs are gathered into three groups, respectively. Each group contains four TEs, and the DOA regions of TEs in each group are identical. The detailed DOA regions of three groups are [− 45^{∘},− 35^{∘}], [12^{∘},22^{∘}], and [42^{∘},52^{∘}], respectively. The resulting AB sets for all the groups satisfy the TE grouping criteria in [11]. The transmit power of the BS is set to 20.2 dBm, and the distance between TEs and BS is set to 500 m. As a result, the average BS receive signaltonoise ratio (SNR) is 10 dB. We can see from Fig. 5 that with estimated effective beamdomain CSI, the required length of pilot sequence can be reduced significantly; hence, significant SE gain can be achieved over the traditional mMIMO SWIPT protocols [31–33].
Figure 6 depicts the SE of the proposed beamdomain SWIPT protocol with the varying of the processing energy of TE. In Fig. 6, the number of antennas at the BS is N = 256 antennas. We can see from Fig. 6 that by optimizing the TS ratio and the transmit powers of the BS for different terminals, the proposed beamdomain SWIPT protocol obtains the best performance. When the processing energy of TE is grater than 2.6×10^{−8} mJ, the SE performance of all the schemes is close to 0. Figure 7 depicts the SE of the proposed beamdomain SWIPT protocol with the varying of the processing energy of TE. We can conclude from Fig. 7 that the SE performance gap between the proposed scheme and traditional mMIMO SWIPT protocols [31–33] will increase as the number of antennas increases.
Conclusion
In this paper, we propose a beamdomain SWIPT protocol for mMIMO system with nonlinear energy harvesting. In order to reduce the pilot and feedback resource cost used for channel estimation, we resort BEM to represent mMIMO channel. In the wireless energy transmission phase, the beam domain energy beamforming is applied. In the information transmission phase, the twolayer receive beamforming is applied for the reception of signals transmitted by TEs. In order to improve the SE of the system, the BS transmit power and timeswitching ratio are optimized. Simulation results show the superiority of the proposed protocol on spectral efficiency compared with the conventional mMIMO SWIPT protocols.
Notes
 1.
Different from the previous works in [45, 46], in this paper, the nonlinear energy harvesting model is applied, while in [45, 46], the linear energy harvesting model is applied. A practical energy consumption model is applied in this paper. Hence, this paper will meet new problems such as more practical harvested energy and available power at TEs, the sumrate performance optimization under new constraints.
 2.
Processing energy includes all energy consumption except for TE uplink transmission, such as the energy consumption of the channel estimate, the energy consumption of the circuit, etc.
 3.
Abbreviations
 3D:

Threedimensional
 5G:

5thgeneration
 AB:

Active beam
 AF:

Amplifyandforward
 AP:

Access point
 AWGN:

Additive white Gaussian noise
 BD:

Beamdomain
 BEM:

Basis expansion model
 BS:

Base station
 CEE:

Channel estimation error
 CSI:

Channel state information
 DOA:

Direction of arrival
 DOD:

Direction of departure
 EE:

Energy efficiency
 EH:

Energy harvesting
 GP:

Geometric programming
 HD:

Halfduplex
 ID:

Information decoding
 IGI:

Intergroup interference
 IoT:

Internet of Things
 IUI:

Interuser interference
 LMMSE:

Linear minimum mean square error
 LS:

Least square
 MF:

Matched filter
 MIMO:

Multiinput multioutput
 MRC:

Maximal ratio combining
 MRT:

Maximum ratio transmission
 PS:

Power splitting
 RF:

Radio frequency
 SE:

Spectral efficiency
 SNR:

Signaltonoise ratio
 SI:

Selfinterference
 SINR:

Signaltointerferenceandnoise ratio
 SWIPT:

Simultaneous wireless information and power transfer
 TE:

Terminal
 TS:

Time switching
 TWR:

Twoway relaying
 WET:

Wireless energy transfer
 ZF:

Zeroforcing
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Acknowledgements
The authors would like to thank the reviewers for their careful readings and valuable comments. This work is supported by Jiangsu Province Natural Science Foundation under Grant BK20160079, National Natural Science Foundation of China (No. 61671472,61771486).
Funding
This work is supported by Jiangsu Province Natural Science Foundation under Grant (BK20160079, BK20181335), National Natural Science Foundation of China (No. 61671472,61771486).
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KX is the main writer of this paper and proposed the main idea. MZ, ZS, NS, WX, and LC assisted in the simulations and analysis. All authors read and approved the final manuscript.
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Correspondence to Ming Zhang.
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Kui Xu was born in 1982. He received the B.S. degree in wireless communications and the Ph.D. degree in software defined radio from the PLA University of Science and Technology, Nanjing, China, in 2004 and 2009, respectively. He is currently an Associate Professor with the College of Communications Engineering, Army Engineering University of PLA, Nanjing, China. Since 2013, he has been a Postdoctoral Fellow with the PLA University of Science and Technology. His research interests include broadband wireless communications, signal processing for communications, network coding, and wireless communication networks. He has authored about 50 papers in refereed journals and conference proceedings and holds five patents in China. He is currently serving on the Technical Program Committee of the IEEE WCSP 2014. He received the URSI Young Scientists Award in 2014 and the 2010 Ten Excellent Doctor Degree Dissertation Award of PLAUST. He also serves as the Reviewer of the IEEE TRANSACTIONS ON WIRELESS COMMUNICATION, the IEEE TRANSACTION VEHICLE TECHNOLOGY, the IEEE COMMUNICATIONS LETTER, the IEEE SIGNAL PROCESSING LETTERS.
Ming Zhang was born in 1992. He received the B.E. degree in communications engineering from the Xidian University in 2015, he is currently working toward the Ph.D. degree with the Institution of Communications Engineering in Army Engineering University of PLA. His research interests include MIMO techniques, heterogeneous network, fullduplex communication, and network coding.
Jie Liu was born in 1984. He received the B.E. and M.S. degrees in communications engineering from the PLA University of Science and Technology in 2006 and 2009. His research interests include MIMO techniques, heterogeneous network, fullduplex communication, and network coding.
Nan Sha was born in China, 1981. He received the B.S. degree in wireless communications and the Ph.D. degree in software defined radio from the PLA University of Science and Technology, Nanjing, China, in 2003 and 2013, respectively. He is currently an Associate Professor with the College of Communications Engineering, Army Engineering University of PLA, Nanjing, China. His research interests include broadband wireless communications, signal processing for communications, network coding, and wireless communication networks.
Wei Xie was born in 1972. He received the B.S. and Ph.D degrees in communication engineering in 1999 and 2015, respectively, from the PLA University of Science and Technology, Nanjing, China. His research interests include MIMO techniques, heterogeneous network, fullduplex communication, and network coding.
Lihua Chen was born in 1972. She received the B.S. and M.S. degrees in communication engineering in 1993 and 2005, respectively, and the Ph.D. degree in communications and information system from the PLA University of Science and Technology, Nanjing, China, in 2013. She is currently a Professor in the College of Communications Engineering, Army Engineering University of PLA, Nanjing, China. Her research interests include new generation wireless mobile communication system, radio resource management, and network coding in wireless communication.
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Keywords
 Massive MIMO
 Beamdomain channel representation
 Simultaneous wireless information and power transfer
 Beamforming
 Achievable sumrate