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SWIPT in mMIMO system with non-linear energy-harvesting terminals: protocol design and performance optimization

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Abstract

In this paper, we design the simultaneous wireless information and power transfer (SWIPT) protocol for massive multi-input multi-output (mMIMO) system with non-linear energy-harvesting (EH) terminals. In this system, the base station (BS) serves a set of uplink fixed half-duplex (HD) terminals with non-linear energy harvester. Considering the non-linearity of practical energy-harvesting circuits, we adopt the realistic non-linear 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 two-layer 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 time-switching ratios are optimized. Simulation results show the superiority of the proposed beam-domain 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 real-time traffic information, and so on. A major problem facing is that how to prolong the life of the energy-constrained devices, so that they could be utilized in a sustainable and low-cost way. Currently, a feasible solution is to adopt the radio frequency (RF) signal-based wireless energy transfer (WET) technique [24], 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 fifth-generation (5G) communication system, massive multiple-input multiple-output (mMIMO) [5] technology can obtain the diversity gain and multiplexing gain, and hence can greatly improve the spectrum efficiency (SE) of wireless networks [68]. 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 battery-powered IoT network in 5G systems and hence extend the service lifetime of the IoT network, RF energy-harvesting (EH) technology has been widely studied [1012]. 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 time-switching (TS) and power-splitting (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 amplify-and-forward (AF) one-way single-antenna SWIPT relay networks. In [17], the SWIPT protocol was designed for one-way 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 two-way relaying (TWR) system can further improve the SE of multi-hop relay system [1820], 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 RF-powered 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 wireless-powered 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, 2123], 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 [2429], 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 non-linear function of the received RF signal due to various causes of non-linearity, e.g., diode non-linearity, and saturation non-linearity. 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 energy-harvesting purposes. Therefore, in practice, the amount of harvested energy is clearly a non-linear function of the received power.

In [31], the SWIPT protocol for three-dimensional (3D) mMIMO system is designed, where the matched filter (MF) precoder is applied. In [32], the SWIPT protocol for multi-pair mMIMO TWR system is designed, where linear precoders, i.e., zero-forcing (ZF) and maximal ratio combining (MRC), are employed at the relay. A low-complexity 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 full-dimensional channel state information (CSI) is required to perform transmit and receive precoding [34]. The above four works [3134] 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 beam-domain SWIPT protocol for mMIMO system with fixed half-duplex (HD) non-linear energy-harvesting terminals (TEs) is designed, and the system spectral efficiency performance is maximized by optimizing the TS ratio and transmit power of the BSFootnote 1. The contributions are summarized as follows.

  • We consider the SWIPT protocol design for TEs with non-linear energy harvesters in mMIMO systems. Moreover, a more practical power consumption model is employed at the TE. In particular, after the energy-harvesting 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 beam-domain 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 beam-domain energy beamforming method is employed for wireless energy transmission. In the second phase, the BS forms the two-layer 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 time-switching ratio are optimized. The simulation results show that compared with the traditional mMIMO SWIPT protocol, the proposed beam-domain 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 beam-domain mMIMO SWIPT transmission scheme. Section 1 analyzes the achievable sum-rate 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 B1 and columns of set B2. 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. IV 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 Kt 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 non-linear energy harvester to collect energy and uses the harvested energy for subsequent data transmission.

Fig. 1
figure1

Illustration of mMIMO SWIPT system

In this paper, the frame-based 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 ki 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 cluster-based channel model [35], that is, the signal of TE ki received by the BS is the sum of the contributions from Mu scatterings. Similarly, the signal of the BS received by the TE ki is the sum of the contributions from Md 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 ki can be written as [35]

$$ {\mathbf{h}^{o}_{{k_{i}}}} = \sum\limits_{m = 1}^{{M_{o}}} {\int_{\theta_{{k_{i},m}}^{o,\min }}^{\theta_{{k_{i},m}}^{o,\max }} {\mathbf{a}\left(\theta \right){r^{o}_{{k_{i},m}}}\left(\theta \right)d\theta} } $$
(1)

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,

$$ {\mathbf{h}^{o}_{{k_{i}}}} = \sum\limits_{n = 1}^{N} {{{\tilde h}^{o}_{{k_{i},n}}}{\mathbf{f}_{n}}} = \mathbf{F}{\tilde {\mathbf{h}}^{o}_{{k_{i}}}} $$
(2)

where F=[f1,f2,,fN], and the basis vector fn 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 beam-domain (BD) channel vector [38] and can be obtained by

$$ \widetilde{\mathbf{h}}^{o}_{{k_{i}}} = {\mathbf{F}^{H}}{\mathbf{h}^{o}_{{k_{i}}}} = \sum\limits_{m = 1}^{{M_{o}}} {\int_{\theta_{{k_{i}},m}^{o,\min }}^{\theta_{{k_{i}},m}^{o,\max }} {{\mathbf{F}^{H}}\mathbf{a}\left(\theta \right){r^{o}_{{k_{i}},m}}\left(\theta \right)d\theta} } $$
(3)

Usually, the effective channel length of the mMIMO channel, that is, the number of elements with non-negligible 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]

$$ \begin{aligned} \mathbf{h}_{{k_{i}}}^{o} &= \sum\limits_{m \in {B_{{k_{i}}}}} {\tilde h_{{k_{i}},m}^{o}{\mathbf{f}_{m}}} + \sum\limits_{m \notin {B_{{k_{i}}}}} {\tilde h_{{k_{i}},m}^{o}{\mathbf{f}_{m}}} \\ &\approx \sum\limits_{m \in {B_{{k_{i}}}}} {\tilde h_{{k_{i}},m}^{o}{\mathbf{f}_{m}}} = {\mathbf{F}^{\left\{ {{B_{{k_{i}}}}},: \right\}}}\tilde {\mathbf{h}}_{o,{k_{i}}}^{\left\{ {{B_{{k_{i}}}}} \right\}} \end{aligned} $$
(4)

where \(B_{k_{i}}\) is the index set of the selected active beams of the terminal ki. The approximation in the middle of (4) comes from the assumption that the channel gain of the terminal ki 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 reduced-dimension \(\widetilde {\mathbf {h}}_{o,{k_{i}}}^{\left \{ {{B_{{k_{i}}}}} \right \}} \in {\mathbb {C}^{\left | {{B_{{k_{i}}}}} \right | \times 1}}\) 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

$$ \begin{aligned} \widetilde {\mathbf{h}}_{o,{k_{i}}}^{\left\{ {{B_{{k_{i}}}}} \right\}}& = {\left({{\mathbf{F}^{\left\{ {{B_{{k_{i}}}}},: \right\}}}} \right)^{H}}{\mathbf{h}^{o}_{{k_{i}}}}\\ &= \sum\limits_{m = 1}^{{M_{o}}} {\int_{\theta_{{k_{i}},m}^{o,\min }}^{\theta_{{k_{i}},m}^{o,\max }} {{{\left({{\mathbf{F}^{\left\{ {{B_{{k_{i}}}}},: \right\}}}} \right)}^{H}}\mathbf{a}\left(\theta \right){r^{o}_{{k_{i}},m}}\left(\theta \right)d\theta} } \end{aligned} $$
(5)

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 gt denote the group index and \(B_{g_{t}}\) denote the corresponding AB set of the group gt. Based on the orthogonality of active beam space across disjoint groups, that is, \(B_{g_{t}}\bigcap B_{{g^{\prime }}_{t}} = \emptyset \) when gtgt, the BD effective channel dimensions of TEs within group gt is reduced to \(|B_{g_{t}}|=b_{g_{t}}\).

Energy-harvesting model

At present, the total energy collected by wireless-powered terminals in the wireless charging stage can usually be modeled as two models: the ideal linear EH model, and the practical non-linear EH model.

In linear EH model [2123], the collected energy of the terminal ki can be expressed as

$$ \begin{aligned} E_{k_{i}}^{{\mathrm{L}}} = \eta P^{\text{EH}}_{k_{i}}T_{h}\\ \end{aligned} $$
(6)

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 wireless-powered user ki, and Th denotes the duration of energy harvesting.

In the practical non-linear EH model, the amount of the harvested energy is modeled using the logistic (or sigmoid) function, i.e., S-shaped curve, as follows [28, 29]:

$$ \begin{aligned} E_{k_{i}}^{{\text{NL}}} &= \frac{{\left[ {\frac{{P_{s}}}{{1 + \exp \left({ - {a}\left({ P^{\text{EH}}_{k_{i}} - {b}} \right)} \right)}} - {P_{s}}{\Omega}} \right]}T_{h}}{{1 - {\Omega}}}\\ \end{aligned} $$
(7)

and

$$ {\Omega} = \frac{1}{{1 + \exp \left({{a}{b}} \right)}} $$
(8)

where Ps 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 non-linear EH model given in (7) and the conventional linear EH model given in (6). We assume that all the EH TEs apply the same non-linear energy harvester.

Fig. 2
figure2

A comparison between the linear EH model and the non-linear EH model. For linear EH model, η=0.8, for non-linear EH model [28, 29], Ps=0.024 mW, a = 150, and b = 0.014

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, 2123]. In this work, we employ a more practical power consumption model for the TE. In particular, after the energy-harvesting 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 energyFootnote 2; in the wireless information transmission phase of time period TI, the transmit power of TE ki can be expressed as

$$ {p^{u}_{{k_{i}}}} = \frac{{E_{{k_{i}}}^{{\text{NL}}} - E_{{k_{i}}}^{\mathrm{P}}}}{{{T_{\mathrm{I}}}}} $$
(9)

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 beam-domain SWIPT protocol

In this paper, the frame-based protocol is applied to realize beam-domain 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.

Fig. 3
figure3

Data frame structure of the proposed beam-domain SWIPT protocol

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.

Fig. 4
figure4

Illustration of the signal flow for the proposed beam-domain SWIPT protocol

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 gt, where \(K_{{g_{t}}}\) denotes the number of TEs in group gt, τt 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 Gt 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 gt can be expressed as

$$ \begin{aligned} {\mathbf{Y}^{d}_{g_{t}}}[\!n] &= \mathbf{H}_{d,g_{t}}^{H}[\!n]\sum\limits_{{g^{\prime}}_{t} \in G_{t}} {{\mathbf{F}^{\left\{ {{B_{{g^{\prime}}_{t}}[n]}},: \right\}}}{{\left({\Phi_{{{g^{\prime}}_{t}}}^{\mathrm{E}}} \right)}^{T}}} + {\mathbf{N}^{d}_{g_{t}}}[\!n]\\ \end{aligned} $$
(10)

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 σ2. The received signal at the gt,kth TE can be expressed as

$$ {\mathbf{y}^{d}_{g_{t,k}}}[\!n] = (\mathbf{h}^{d}_{g_{t,k}}[\!n])^{H}\!\!\sum\limits_{{g^{\prime}}_{t} \in G_{t}}\!\! {{\mathbf{F}^{\left\{ {{B_{{g^{\prime}}_{t}}}[n]},: \right\}}}{{\left({\Phi_{{{g^{\prime}}_{t}}}^{\mathrm{E}}} \right)}^{T}}} + {\mathbf{n}^{d}_{g_{t,k}}}[\!n] $$
(11)

where \({\mathbf {n}_{{g_{t,k}}}}[\!n]\sim \mathcal {CN}(0,\sigma ^{2})\) denotes the AWGN at TE gt,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 gt,k during the frame n can be written as

$$ \begin{aligned} {\tilde{E}^{\mathrm{L}}_{{g_{t,k}}}}[\!n] &= {\eta \beta}{\left\| (\mathbf{h}^{d}_{g_{t,k}}[\!n])^{H}\sum\limits_{{g^{\prime}}_{t} \in G_{t}} {{\mathbf{F}^{\left\{ {{B_{{g^{\prime}}_{t}}}[n]},: \right\}}}{{\left({\Phi_{{{g^{\prime}}_{t}}}^{\mathrm{E}}} \right)}^{T}}}\right\|^{2}} \end{aligned} $$
(12)

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

$$ \begin{aligned} {\tilde{E}^{\mathrm{L}}_{{g_{t,k}}}}[n] &\approx {\eta \beta}{\left\| (\mathbf{h}^{d}_{g_{t,k}}[n])^{H}{{\mathbf{F}^{\left\{ {{B_{{g}_{t}}}[n]},: \right\}}}{{\left({\Phi_{{{g}_{t}}}^{\mathrm{E}}} \right)}^{T}}}\right\|^{2}}\\ &= {\eta \beta\tau_{t}}{p^{d}_{g_{t,k}}}[\!n]{\left\| {\tilde{\mathbf{h}}_{d,g_{t,k}}^{\left\{ {{B_{g_{t}}}[n]} \right\}}} [\!n]\right\|^{2}}\triangleq \eta\beta{\alpha T} {\hat{p}^{\mathrm{H}}_{d,{g_{t,k}}}} \end{aligned} $$
(13)

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 non-linear model, the amount of the harvested energy at gt,k is modeled using the logistic (or sigmoid) function, i.e., S-shaped curve, as follows [28]:

$$ \begin{aligned} \tilde E_{{g_{t,k}}}^{{\text{NL}}}[n] &\approx \frac{{{P_{s}}\alpha T\left[ {\frac{1}{{1 + \exp \left({ - {a}\left({\beta {\hat{p}^{\mathrm{H}}_{d,{g_{t,k}}}} - {b}} \right)} \right)}} - {\Omega}} \right]}}{{1 - {\Omega}}}\\ \end{aligned} $$
(14)

Another portion of the TE gt,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

$$ \begin{aligned} \tilde {\mathbf{h}}_{d,g_{t,k},LS}^{\left\{ {{B_{g_{t}}}[n]} \right\}}[\!n] &= \frac{1}{{{\tau_{t}}{p^{d}_{t,k}}}[n]}{\left({{\mathbf{Y}^{d}_{g_{t}}}[\!n]{{\left({\Phi_{{g_{t}}}^{\mathrm{E}}} \right)}^{*} }} \right)^{H}}{\mathbf{e}^{k}}\\ &= \tilde {\mathbf{h}}_{d,g_{t,k}}^{\left\{ {{B_{g_{t}}}[n]} \right\}}[\!n] + \sum\limits_{{g^{\prime}}_{t} \in {G_{t}} \left/ \right. {g_{t}}} {\tilde{\mathbf{h}}_{d,g_{t,k}}^{\left\{ {{B_{{g'}_{t}}}[n]} \right\}}}[\!n] \\ & + \frac{1}{{{\tau_{t}}{p^{d}_{t,k}}[\!n]}}{\left({{\mathbf{N}_{g_{t}}}[\!n]{{\left({\Phi_{{g_{t}}}^{\mathrm{E}}} \right)}^{*} }} \right)^{H}}{\mathbf{e}^{k}} \end{aligned} $$
(15)

where ek 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

$$ {\begin{aligned} \tilde {\mathbf{ h}}_{d,g_{t,k},LM}^{\left\{ {{B_{g_{t}}}}[n] \right\}}[\!n] &=\! \!\!\sum\limits_{{g^{\prime}}_{t} \in G_{t}} {\sum\limits_{m = 1}^{{M_{d}}} {\int_{\theta_{g_{t,k,m}}^{d,\min }}^{\theta_{g_{t,k,m}}^{d,\max }} {{{\left({{\mathbf{F}^{\left\{ {{B_{g_{t}}}[n]},: \right\}}}} \right)}^{H}}\!\mathbf{a}\left(\theta \right)\!{\mathbf{a}^{H}}\left(\theta \right)}} } {\mathbf{F}^{\left\{ {{B_{{g^{\prime}}_{t}}}[n]},: \right\}}}{S^{d}_{g_{t,k,m}}}\left(\theta \right)d\theta \\ &\times \left({\sum\limits_{{g^{\prime}}_{t},{g^{\prime\prime}}_{t} \in G_{t}} \!{\sum\limits_{m = 1}^{{M_{d}}}\! {\int_{\theta_{g_{t,k,m}}^{d,\min }}^{\theta_{g_{t,k,m}}^{d,\max }} \!\!{{{\left(\! {{\mathbf{F}^{\left\{ {{B_{{g^{\prime}}_{t}}}[n]},: \right\}}}}\! \right)}^{H}}\!\!\mathbf{a}\left(\theta \right)\!}} }} \right.\\ &\times {\mathbf{a}^{H}}\!\!\left(\theta \right){\left. {{\mathbf{F}^{\left\{ {{B_{{g^{\prime\prime}}_{t}}}[n]},: \right\}}}{S^{d}_{g_{t,k,m}}}\left(\theta \right)d\theta + \frac{\sigma^{2} }{{{\tau_{t}}{p^{d}_{t,k}}}}} \!\!\right)^{- 1}} \tilde {\mathbf{ h}}_{d,g_{t,k},LS}^{\left\{ {{B_{g_{t}}}[n]} \right\}}[\!n] \end{aligned}} $$
(16)

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 feedbackFootnote 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 KT TEs. The transmit power of TE gt,k can be expressed as

$$ p^{u}_{g_{t,k}}[\!n] = \frac{{{{\breve{{E}}^{\text{NL}}_{{g_{t,k}}}}}[\!n] - E_{g_{t,k}}^{\mathrm{P}}[\!n]}}{(1-\alpha)T} $$
(17)

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,

$$ \begin{aligned} {\breve{E}^{\text{NL}}_{{g_{t,k}}}} [\!n] &= \sum_{i=1}^{n} {\tilde{E}^{\text{NL}}_{{g_{t,k}}}} [\!i] \,-\,\sum_{i=1}^{n-1} E_{g_{t,k}}^{\mathrm{P}}[\!i]- \sum_{i=1}^{n-1} \left(1\,-\,\alpha[\!i]\right) T p^{u}_{g_{t,k}} [\!i] \end{aligned} $$
(18)

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 gt,k, and the transmit power of TE gt,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

$$ \mathbf{y}^{u}_{t}[\!n] = \sum\limits_{{g^{\prime}}_{t} \in {G_{t}}} {{\mathbf{H}_{u,{g^{\prime}}_{t}}}[\!n]{\mathbf{s}^{u}_{{g^{\prime}}_{t}}}}[\!n] + \mathbf{n}^{u}_{t}[\!n] $$
(19)

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 gt. \(\mathbf {n}^{u}_{t}[\!n]\) denotes the AWGN vector with variance σ2. In this paper, we apply the two-layer beamforming scheme. The first-layer 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 gt can be expressed as

$$ \begin{aligned} {\tilde {\mathbf{y}}^{u}_{g_{t}}}[\!n] &= { {{\mathbf{F}^{^{\left\{ {{B_{{g_{_{t}}}}}[n]},: \right\}}}}}}\mathbf{y}^{u}_{t}[\!n]\\ &= \tilde {\mathbf{ H}}_{u,g_{t}}^{\left\{ {{B_{{g_{_{t}}}}}[n]},: \right\}}[\!n]{\mathbf{s}^{u}_{g_{t}}}[\!n]\!\! +\!\!\! \sum\limits_{{g^{\prime}}_{t} \in {{G_{t}} \left/\right. {\left\{ {g_{t}} \right\}}}}\!\!\! {\tilde{\mathbf{H}}_{u,{g^{\prime}}_{t}}^{\left\{ {{B_{{g_{_{t}}}}}}[n],: \right\}}}[\!n] {\mathbf{s}^{u}_{{g^{\prime}}_{t}}[\!n]}\,+\, {\tilde {\mathbf{n}}^{u}_{g_{t}}}[\!n] \end{aligned} $$
(20)

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 gt, the BS employs the second-layer 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

$$ \begin{aligned} {{\dot y}^{u}_{g_{t,k}}}[\!n] &= \mathbf{w}_{g_{t,k}}^{H}[\!n]\tilde {\mathbf{ h}}_{u,g_{t,k}}^{\left\{ {{B_{g_{t}}}}[n] \right\}}[\!n]{s^{u}_{g_{t,k}}}[\!n] + \mathbf{w}_{g_{t},k_{{g_{t}}}}^{H}[\!n]\sum\limits_{k^{\prime} = 1,k^{\prime} \ne k}^{K_{{g_{t}}}} {\tilde{\mathbf{h}}_{g_{t,k^{\prime}}}^{\left\{ {{B_{g_{t}}}}[n] \right\}}[\!n]{s^{u}_{g_{t,k^{\prime}}}}}[\!n] \\ &\quad+ \mathbf{w}_{g_{t,k}}^{H}[\!n]\!\!\sum\limits_{{g^{\prime}}_{t} \in {{G_{t}^{e}} \left/\right. {\left\{ {g_{t}} \right\}}}}\!\!\! {\tilde{\mathbf{H}}_{u,{g^{\prime}}_{t}}^{\left\{ {{B_{{g_{_{t}}}}}}[n],: \right\}}}[n]{\mathbf{s}^{u}_{{g^{\prime}}_{u}}}[\!n] + {{{{\tilde n}}}^{u}_{g_{t}}}[\!n] \end{aligned} $$
(21)

The signal-to-interference-and-noise ratio (SINR) of the BS received signal from TE gt,k can be expressed as

$$ \text{SINR}{_{g_{t,k}}}[\!n] = \frac{{{p^{u}_{g_{t,k}}}[\!n]{{\left| {\mathbf{w}_{g_{t,k}}^{H}[\!n]\tilde{\mathbf{ h}}_{u,g_{t,k}}^{\left\{ {{B_{g_{t}}}} [n]\right\}}}[\!n] \right|}^{2}}}}{{{\sigma^{2}}\|\mathbf{w}_{g_{t,k}}^{}[\!n]\|^{2} + {\Theta_{g_{t,k}}}[\!n]}} $$
(22)

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

$$ {\begin{aligned} {\Theta_{{g_{t,k}}}}[\!n] &= \underbrace {p_{{g_{t,k}}}^{u}[n]{{\left| {\mathbf{w}_{{g_{t,k}}}^{H}[\!n]\Delta \tilde {\mathbf{ h}}_{t,{g_{t,k}}}^{\left\{ {{B_{{g_{t}}}}[n]} \right\}}[\!n]} \right|}^{2}}}_{{\text{CE}}{{\mathrm{E}}_{{g_{t,k}}}}\left[ n \right]}\\ &\quad+ \underbrace {\sum\limits_{k^{\prime} = 1,k^{\prime} \ne k}^{{K_{{g_{t}}}}} p_{{g_{t,k^{\prime}}}}^{u}[\!n]{{\left| {\mathbf{w}_{{g_{t,k}}}^{H}[\!n]\Delta \tilde {\mathbf{ h}}_{u,{g_{t,k^{\prime}}},LM}^{\left\{ {{B_{{g_{t}}}}[n]} \right\}}[\!n]} \right|}^{2}}}_{{\text{IU}}{{\mathrm{I}}_{{g_{t,k}}}}\left[ n \right]}\\ &\quad+ \underbrace {\sum\limits_{{{g^{\prime}}_{t}} \in {{{G_{t}}} \left/\right. {\left\{ {{g_{t}}} \right\}}}} {{{\left\| {\mathbf{w}_{{g_{t,k}}}^{H}[\!n]\tilde {\mathbf{H}}_{u,{{g^{\prime}}_{t}}}^{\left\{ {{B_{{g_{t}}}}[n],:} \right\}}[\!n]\Lambda_{{{g^{\prime}}_{t}}}^{{{\mathrm{1}} \left/\right. {\mathrm{2}}}}[\!n]} \right\|}^{2}}} }_{{\text{IG}}{{\mathrm{I}}_{{g_{t,k}}}}\left[ n \right]} \end{aligned}} $$
(23)

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), inter-user interference (IUI), and inter-group 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 gt.

The optimal \(\mathbf {w}_{g_{t,k}}^{}[\!n]\) is the vector that maximizes the \(\text {SINR}_{g_{t,k}}[n]\), and, thus, the second-layer beamformer can be obtained by

$$ {\underset{\left\| {\mathbf{w}_{g_{t,k}}^{}}[\!n] \right\| = 1}{\max}} \!\!\!\!\! \frac{{{p^{u}_{g_{t,k}}}[\!n]\mathbf{w}_{g_{t,k}}^{H}[\!n]\tilde {\mathbf{ h}}_{u,g_{t,k}}^{\left\{ {{B_{g_{t}}}}[n] \right\}}[n]\!{{\left(\! {\tilde{\mathbf{h}}_{u,g_{t,k}}^{\left\{ {{B_{g_{t}}}}[n] \right\}}}[\!n]\!\! \right)}^{H}}\!\!\!\!\!\mathbf{w}_{g_{t,k}}^{}}\![\!n]}{{\mathbf{w}_{g_{t,k}}^{H}[\!n]\left({{\mathbf{\Xi }_{g_{t,k}}}[\!n]+ {\sigma^{2}}{\mathbf{I}_{b_{t}[n]}}} \right)\mathbf{w}_{g_{t,k}}}[\!n]} $$
(24)

where \({{\mathbf {\Xi }_{g_{t,k}}}}[\!n]\) can be written as

$$ {\begin{aligned} {{\mathbf{\Xi }_{g_{t,k}}}}[n] &= {p^{u}_{g_{t,k}}}[n]\Delta \tilde {\mathbf{ h}}_{u,g_{t,k}}^{\left\{ {{B_{g_{t}}}}[n] \right\}}[n]{\left({\Delta \tilde {\mathbf{ h}}_{u,g_{t,k}}^{\left\{ {{B_{g_{t}}}}[n] \right\}}}[n] \right)^{H}}\\ &\quad+ \sum\limits_{k^{\prime} = 1,k^{\prime} \ne k}^{{K_{g_{t}}}} {{p^{u}_{g_{t,k^{\prime}}}}[\!n]\Delta \tilde{\mathbf{h}}_{u,g_{t,k^{\prime}},LM}^{\left\{ {{B_{g_{t}}}}[n] \right\}}[\!n]{{\left({\Delta \tilde{\mathbf{h}}_{u,g_{t,k^{\prime}},LM}^{\left\{ {{B_{g_{t}}}} [n]\right\}}}[n] \right)}^{H}}} \\ &\quad+ \sum\limits_{{g^{\prime}}_{t} \in {{{G_{t}}} \left/\right. {\left\{ {g_{t}} \right\}}}} \tilde{\mathbf{H}}_{u,{g^{\prime}}_{t}}^{\left\{ {{B_{g_{t}}}}[n],: \right\}}[\!n]\Lambda_{{g^{\prime}}_{t}}^{{1 \left/\right. 2}}[\!n]{{\left(\tilde{\mathbf{H}}_{u,{g^{\prime}}_{t}}^{\left\{ {{B_{g_{t}}}}[n],: \right\}}[\!n]\Lambda_{{g^{\prime}}_{t}}^{{1 \left/\right. 2}} [\!n]\right)}^{H}} \end{aligned}} $$
(25)

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

$$ \mathbf{w}_{g_{t,k}}[n]=\frac{{{{\left({{\mathbf{\Xi }_{g_{t,k}}}[\!n] + {\sigma^{2}}{\mathbf{I}_{b_{t}[n]}}} \right)}^{- 1}}\tilde {\mathbf{ h}}_{u,g_{t,k},LM}^{\left\{ {{B_{g_{t}}}}[n] \right\}}}[n]}{{\left\| {{{\left({{\mathbf{\Xi }_{g_{t,k}}}[\!n] + {\sigma^{2}}{\mathbf{I}_{b_{t}[n]}}} \right)}^{- 1}}\tilde {\mathbf{ h}}_{u,g_{t,k},LM}^{\left\{ {{B_{g_{t}}}} [n]\right\}}}[n] \right\|}} $$
(26)

Achievable sum-rate analysis and optimization

When optimizing the system achievable sum-rate, 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 power-limited 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,

$$ {\begin{aligned} {\breve{E}^{\text{NL}}_{{g_{t,k}}}} [\!n] &= \sum_{i=1}^{n} {\tilde{E}^{\text{NL}}_{{g_{t,k}}}} [\!i] -\sum_{i=1}^{n-1} \left(1-\alpha[\!i]\right) T p^{u}_{g_{t,k}} [i]\\ &\quad- \sum_{i=1}^{n-1} E_{g_{t,k}}^{\mathrm{P}}[\!i]\geq \eth, \forall g_{t,k}\in \mathbf{K}_{U}, n\in \{1,2,\cdots,n-1\} \end{aligned}} $$
(27)

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 gt,k can be expressed as

$$ {\begin{aligned} \begin{array}{c} {R_{{g_{t,k}}}} (\mathcal{P}[\!n],\alpha[\!n])= \left({1 - \alpha[\!n]} \right){\log_{2}}\left({1 + {p^{u}_{g_{t,k}}}[\!n]\|\tilde {\mathbf{ h}}_{u,g_{t,k}}^{\left\{ {{B_{g_{t}}}}[n] \right\}}[\!n]\|^{2}} \right.\\ \,\quad\quad\times\left({\underbrace {{p^{u}_{g_{t,k}}}[\!n]{{\left| {\mathbf{w}_{{g_{t,k}}}^{H}[\!n]\Delta \tilde {\mathbf{ h}}_{u,{g_{t,k}}}^{\left\{ {{B_{{g_{t}}}}} [n]\right\}}} [n] \right|}^{2}}}_{{\text{CEE}_{g_{t,k}}}[n]}} + \|\mathbf{w}_{g_{t,k}}^{}[n]\|^{2}\right.\\ \quad+ \underbrace {\sum\limits_{k^{\prime} = 1,k^{\prime} \ne k}^{{K_{{g_{t}}}}} {{p^{u}_{{g_{t,k^{\prime}}}}}[\!n]{{\left| {\mathbf{w}_{{g_{t,k}}}^{H}[\!n]\Delta \tilde {\mathbf{ h}}_{u,{g_{t,k^{\prime}}},LM}^{\left\{ {{B_{{g_{t}}}}}[n] \right\}}}[\!n] \right|}^{2}}} }_{\text{IUI}_{g_{t,k}}[n]}\\ \quad\left. {{{\left. + {\underbrace {\sum\limits_{{g^{\prime}}_{t} \in {{{G_{t}}} \left/\right. {\left\{ {{g_{t}}} \right\}}}}^{} {{{\left\| {\mathbf{w}_{{g_{t,k}}}^{H}[\!n]\tilde {\mathbf{ H}}_{u,{g^{\prime}}{_{t}}}^{\left\{ {{B_{{g_{t}}}}}[n],: \right\}}[\!n]\Lambda_{{g^{\prime}}{_{t}}}^{{1 \left/\right. 2}}}[\!n] \right\|}^{2}}} }_{\text{IGI}_{g_{t,k}}[n]}} \right)}^{- 1}}} \right) \end{array} \end{aligned}} $$
(28)

The transmit power of TE gt,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

$$ {{R}_{{\text{SUM}}}}(\mathcal{P}[\!n],\alpha[\!n]){{=}} \sum\limits_{g_{t,k}[n] \in {\mathbf{K}_{T}}}^{} {{{{R}}_{g_{t,k}}}}(\mathcal{P}[\!n],\alpha[\!n]) $$
(29)

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 gt,k are given in (41). Let Rmin denote the minimum throughput of the TEs, and the system achievable rate maximization problem can be expressed as

$$ \begin{array}{l} {\underset{\mathcal{P}[n],\alpha[n]}{\max}} \,\,\sum\limits_{{g_{t,k}} \in {\mathbf{K}_{T}}}{{{{{{R}}}_{g_{t,k}}}}}(\mathcal{P}[\!n],\alpha[\!n])\\ {\mathrm{s}}{\mathrm{.t}}{{.}}\left\{ {\begin{array}{*{20}{l}} {p1:T>\alpha[\!n] T \geq \max\{b_{g_{t}}|g_{t} \in G_{t}\}}\\ {p2:\sum\limits_{k = 1}^{K_{t}} {{p^{d}_{{g_{t,k}}}}} [\!n] \le {P^{\mathrm{E}}_{d}}}\\ {p3:{p^{d}_{{g_{t,k}}}}[\!n]> 0, k \in {{\mathbf{K}_{T}}}}\\ {p4:{\tilde{E}^{\text{NL}}_{{g_{t,k}}}} [\!n]\leq P_{s}\alpha T, k\in \mathbf{K}_{T}}\\ {p5:{{\breve{E}}^{\text{NL}}_{{g_{t,k}}}} [\!n]\geq \eth, k\in \mathbf{K}_{T}}\\ {p6:{R_{{g_{t,k}}}} (\mathcal{P},\alpha) [\!n]\geq R_{\text{min}}, k\in \mathbf{K}_{T}} \end{array}} \right. \end{array} $$
(30)

where \(P^{\mathrm {E}}_{d}\) denotes the maximum transmit power constraint at the BS during phase I. One can check that (30) is non-convex due to the constraints in p2p6, 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 PsαT; 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

$$ \begin{aligned} \alpha \left[ n \right] & > \frac{E_{g_{t,k}}^{\mathrm{P}}[\!n] + \eth}{P_{s} T} \triangleq \ddot{\alpha}_{g_{t,k}}[\!n] \end{aligned} $$
(31)

where Ω is given in (8). Hence, the SE optimization problem in (30) can be rewritten as

$$ \begin{array}{l} {\underset{\mathcal{P}[n],\alpha[n]}{\max}} \,\,\sum\limits_{{g_{t,k}} \in {\mathbf{K}_{T}}}^{}{{{{{{R}}}_{g_{t,k}}}}}(\mathcal{P}[\!n],\alpha[\!n])\\ {\mathrm{s}}{\mathrm{.t}}{\mathrm{. }}\left\{ {\begin{array}{*{20}{l}} {\hat{p}1:T > \alpha [\!n]T \ge }\\ {\max \left\{ {\max \{ {b_{{g_{t}}}}|{g_{t}} \in {G_{t}}\},\max \{ {{\ddot \alpha }_{{g_{t,k}}}}[\!n]|k \in {\mathbf{K}_{T}}\}} \right\}}\\ {\hat{p}2:\sum\limits_{k = 1}^{K_{t}} {{p^{d}_{{g_{t,k}}}}} [\!n] \le {P^{\mathrm{E}}_{d}}}\\ {\hat{p}3:{p^{d}_{{g_{t,k}}}}[\!n]> 0, k \in {{\mathbf{K}_{T}}}}\\ {\hat{p}4:{R_{{g_{t,k}}}} (\mathcal{P}[\!n],\alpha[\!n])\geq R_{\text{min}}, k\in \mathbf{K}_{T}} \end{array}} \right. \end{array} $$
(32)

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=α0. 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 gt,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

$$ \begin{array}{l} {\underset{\mathcal{P}}{\max}} \prod\limits_{k = 1}^{{K_{t}}} {\left({1 + {\gamma_{t,k}}} \right)} = {\underset{\mathcal{P}}{\min}} {\left[ {\prod\limits_{k = 1}^{{K_{t}}} {\left({1 + {\gamma_{t,k}}} \right)}} \right]^{- 1}}\\ {\mathrm{s}}{\mathrm{.t}}{{.}}\left\{ {\begin{array}{*{20}{l}} {c1:{\gamma_{{g_{t,k}}}} \le p^{u}_{g_{t,k}}{b_{{g_{t,k}}}}{{\left({{c_{{g_{t,k}}}} + {d_{{g_{t,k}}}}} \right)}^{- 1}},}\\ \quad k = 1, \cdots,{K_{u}}\\ {\hat{p}2,\hat{p}3,\hat{p}4,\hat{p}5} \end{array}} \right. \end{array} $$
(33)

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

$$ \hat{p}5 : p^{u}_{g_{t,k}}\!\! \ge \!\! \frac{{{P_{s}}\alpha_{0} T \!\! \left[ \!{\frac{1}{{1 + \exp \left({\! - a\left(\! {\beta \hat p_{d,{g_{t,k}}}^{\mathrm{H}}\! \!- b} \right)}\! \right)}}\! -\! \Omega }\! \right] \!\,-\,\! E_{{g_{t,k}}}^{\mathrm{P}}\!\!\left({1\! -\! \Omega} \right)}}{{\left({1 - \Omega} \right)\left({1 - \alpha_{0}} \right)T}} $$
(34)

(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=α0 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+γt,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

$$ \begin{array}{l} {\underset{\mathcal{P}^{u}}{\min}} \prod\limits_{k = 1}^{{K_{t}}} {{{\left({{\lambda_{t,k}}} \right)}^{- 1}}(\gamma_{t,k})^{- {\mu_{t,k}}}} \\ {\mathrm{s}}{\mathrm{.t}}{{.}}\left\{ {\begin{array}{*{20}{l}} {c1:{\gamma_{{g_{t,k}}}}\left({{c_{{g_{t,k}}}} + {d_{{g_{t,k}}}}} \right){{\left({{p^{u}_{{g_{t,k}}}}{b_{{g_{t,k}}}}} \right)}^{- 1}} \le 1,}\\ \,k = 1, \cdots,{K_{t}}\\ {\hat{p}2,\hat{p}3,\hat{p}4,\hat{p}5} \end{array}} \right. \end{array} $$
(35)

Constraints c1 can be re-expressed as

$$ {\begin{aligned} &c1:\frac{1}{{{{\left\| \tilde{{\mathbf{h}}_{u,{g_{t,k}}}^{\left\{ {{B_{{g_{t}}}}} \right\}}} \right\|}^{2}}}}\left({\vphantom{{{{{\left\| {\tilde{\mathbf{h}}_{u,{g_{t,k}}}^{\left\{ {{B_{{g_{t}}}}} \right\}}} \right\|}^{2}}}}}} {\gamma_{{g_{t,k}}}}{{\left| {\mathbf{w}_{{g_{t,k}}}^{H}\Delta \tilde {\mathbf{ h}}_{u,{g_{t,k}},LM}^{\left\{ {{B_{{g_{t}}}}} \right\}}} \right|}^{2}}\right.\\ & + {\gamma_{{g_{t,k}}}}\left(p^{u}_{{g_{t,k}}}\right)^{- 1}\sum\limits_{k^{\prime} = 1,k^{\prime} \ne k}^{{K_{{g_{t}}}}} {{p^{u}_{{g_{t,k^{\prime}}}}}{{\left| {\mathbf{w}_{{g_{t,k}}}^{H}\Delta \tilde {\mathbf{ h}}_{u,{g_{t,k^{\prime}}},LM}^{\left\{ {{B_{{g_{t}}}}} \right\}}} \right|}^{2}}} \\ & + {\gamma_{{g_{t,k}}}}\left(p^{u}_{{g_{t,k}}}\right)^{- 1}\sum\limits_{g{^{\prime}_{t}} \in {{{G_{t}}} \left/\right. {\left\{ {{g_{t}}} \right\}}}}^{} {{{\!\sum\limits_{k^{\prime} = 1}^{{K_{g{^{\prime}_{t}}}}} {p^{u}_{{g^{\prime}_{t,k^{\prime}}}}\left|{\mathbf{w}_{{g_{t,k}}}^{H}\tilde{\mathbf{h}}_{u,g{^{\prime}_{t,k^{\prime}}}}^{\left\{ {{B_{{g_{t}}}}} \right\}}} \right|} }^{2}}}\\ &\left. + {\gamma_{{g_{t,k}}}}\left(p^{u}_{{g_{t,k}}}\right)^{- 1}\|\mathbf{w}_{{g_{t,k}}}^{H}\|^{2} {\vphantom{{{{{\left\| \tilde{\mathbf{h}_{u,{g_{t,k}}}^{\left\{ {{B_{{g_{t}}}}} \right\}}} \right\|}^{2}}}}}}\right) \le 1, k=1,\cdots,K_{t} \end{aligned}} $$
(36)

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

$$ {R_{g_{t,k}}}(\alpha)|_{\mathcal{P}^{\ast}} = \left({1 - \alpha} \right){\log_{2}}\left({1 + \gamma_{g_{t,k}}(\alpha)} \right) $$
(37)

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

$$ \begin{aligned} p_{{g_{t,k}}}^{u} &\ge \frac{{{P_{s}}\alpha T\left[ {\frac{1}{{2 - a\left({\beta \breve{p}_{d,{g_{t,k}}}^{\mathrm{H}} - b} \right)}} - \Omega} \right] - E_{{g_{t,k}}}^{\mathrm{P}}\left({1 - \Omega} \right)}}{{\left({1 - \Omega} \right)\left({1 - \alpha} \right)T}}\\ &= \frac{\alpha }{{\left({1 - \alpha} \right)}}\frac{{{P_{s}}\left[ {\frac{1}{{2 - a\left({\beta \breve{p}_{d,{g_{t,k}}}^{\mathrm{H}} - b} \right)}} - \Omega} \right]}}{{\left({1 - \Omega} \right)}} \,-\, \frac{{E_{{g_{t,k}}}^{\mathrm{P}}}}{{\left({1 - \alpha} \right)T}}\\ &= \frac{\alpha }{{\left({1 - \alpha} \right)}}{{\tilde d}_{{g_{t,k}}}} - \frac{{E_{{g_{t,k}}}^{\mathrm{P}}}}{{\left({1 - \alpha} \right)T}} \end{aligned} $$
(38)

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:

$$ \begin{array}{l} {\underset{\alpha}{\max}} \,\, {\vec{{R}}_{{\text{SUM}}}}(\alpha)|_{\mathcal{P}^{\ast}}=\sum\limits_{g_{t,k} \in {\mathbf{K}_{T}}}^{}\left({1 - \alpha} \right){\log_{2}}\left({1 + \vec{\gamma}_{g_{t,k}}}(\alpha) \right) \\ {\mathrm{s}}{\mathrm{.t}}{{.}}\,\,\,\,\,1 > \alpha \ge \frac{{\max \left\{ {\max \{ {b_{{g_{t}}}}|{g_{t}} \in {G_{t}}\},\max \{ {{\ddot \alpha }_{{g_{t,k}}}}|k \in {\mathbf{K}_{T}}\}} \right\}}}{T} \end{array} $$
(39)

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

$$ {\begin{aligned} &\frac{\partial {\vec{{R}}_{{\text{SUM}}}}(\alpha)}{\partial \alpha}\,=\, \,-\,\!\sum\limits_{g_{t,k} \in {\mathbf{K}_{T}}}^{}{\log_{2}}\left({1 + \vec{\gamma}_{g_{t,k}}}(\alpha) \right){{\,+\,}}\!\!\!\sum\limits_{g_{t,k} \in {\mathbf{K}_{T}}}^{} {\frac{{\left({{{\tilde d}_{{g_{t,k}}}} - {{E_{{g_{t,k}}}^{\mathrm{P}}} \left/\right. T}} \right){{\tilde b}_{{g_{t,k}}}}{{\left({{{\widetilde c}_{{g_{t,k}}}} + 1} \right)}^{- 1}}}}{{\ln \left(2 \right)\left({1 + {\vec{\gamma}_{g_{t,k}}}(\alpha)} \right){{\left({1 - \alpha} \right)}}}}} \end{aligned}} $$
(40)

The two order derivative of \({{\vec {R}}_{{\text {SUM}}}}(\alpha)\) can be expressed as

$$ {\begin{aligned} &\frac{{{\partial^{2}}{\vec{R}_{{\text{SUM}}}}(\alpha)}}{{\partial {\alpha^{2}}}}= - \sum\limits_{{g_{t,k}} \in {\mathbf{K}_{T}}}^{} {\frac{{{{\left({{{\tilde d}_{{g_{t,k}}}} - {{E_{{g_{t,k}}}^{\mathrm{P}}} \left/\right. T}} \right)}^{2}}{{\left({{{\tilde b}_{{g_{t,k}}}}} \right)}^{2}}}}{{{{\left({\ln \left(2 \right)\left({1 + {{\vec \gamma }_{{g_{t,k}}}}(\alpha)} \right)\left({{{\widetilde c}_{{g_{t,k}}}} + 1} \right)} \right)}^{2}}{{\left({1 - \alpha} \right)}^{3}}}}} \end{aligned}} $$
(41)

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 RSUM(α) when \(\frac {\partial {{R}_{{\text {SUM}}}}(\alpha)}{\partial \alpha }=0\), that is,

$$ \begin{array}{l} \underbrace {\sum\limits_{g_{t,k} \in {\mathbf{K}_{T}}}^{} {\frac{{\left({{{\tilde d}_{{g_{t,k}}}} - {{E_{{g_{t,k}}}^{\mathrm{P}}} \left/\right. T}} \right){{\tilde b}_{{g_{t,k}}}}{{\left({{{\widetilde c}_{{g_{t,k}}}} + 1} \right)}^{- 1}}}}{{\ln \left(2 \right)\left({1 + {\vec{\gamma}_{g_{t,k}}}(\alpha)} \right)}}} }_{B\left(\alpha \right)}\\ = \left({{\mathrm{1}} - \alpha} \right)\sum\limits_{g_{t,k} \in {\mathbf{K}_{U}}}^{} {{{\log }_{2}}} \left({1 + {\vec{\gamma}_{g_{t,k}}}(\alpha)} \right)\\ ={\vec{{R}}_{{\text{SUM}}}}(\alpha)|_{\mathcal{P}^{\ast}} \end{array} $$
(42)

(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 sum-rate 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 sum-rate is

$$ {{R}_{{\text{SUM}}}^{\text{Opt}}}=\sum\limits_{g_{t,k} \in {\mathbf{K}_{T}}}^{} {{{R}_{g_{t,k}}}}(\alpha^{{\text{Opt}}}) $$
(43)

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 first-order condition numerically, whose complexity is similar with the approach in [43].

Results and discussion

In this section, the performance of the proposed beam-domain SWIPT protocol is evaluated using the 3GPP LTE simulation model in macro-cell 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 non-linear EH [28, 29], Ps 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 non-linear 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 [3133] is also harvested from the BS downlink transmit signal. In simulations, the CSI obtained by beam-domain channel estimation is adopted in the above schemes, respectively.

Figure 5 illustrates the SE of the proposed beam-domain 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 signal-to-noise ratio (SNR) is 10 dB. We can see from Fig. 5 that with estimated effective beam-domain CSI, the required length of pilot sequence can be reduced significantly; hence, significant SE gain can be achieved over the traditional mMIMO SWIPT protocols [3133].

Fig. 5
figure5

Spectral efficiency of the proposed SWIPT protocol, the channel coherent time is 300 symbols, N = 256. The processing energy of TE is set to 10−8 mJ

Figure 6 depicts the SE of the proposed beam-domain 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 beam-domain 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 beam-domain 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 [3133] will increase as the number of antennas increases.

Fig. 6
figure6

Spectral efficiency of the proposed SWIPT protocol with the varying of the processing energy of TE, the channel coherent time is 300 symbols, N = 256. The transmit power of the BS is set to 46 dBm

Fig. 7
figure7

Spectral efficiency of the proposed SWIPT protocol with the varying of the number of antennas, the channel coherent time is 300 symbols. The transmit power of the BS is set to 46 dBm

Conclusion

In this paper, we propose a beam-domain SWIPT protocol for mMIMO system with non-linear 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 two-layer 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 time-switching ratio are optimized. Simulation results show the superiority of the proposed protocol on spectral efficiency compared with the conventional mMIMO SWIPT protocols.

Notes

  1. 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 sum-rate performance optimization under new constraints.

  2. 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. 3.

    This scenario is currently widely studied [43, 47]. In this paper, we use beam domain downlink channel estimation. Since the effective channel length in the beam domain is greatly compressed, the feedback overhead for downlink channel estimation will be greatly reduced.

Abbreviations

3D:

Three-dimensional

5G:

5th-generation

AB:

Active beam

AF:

Amplify-and-forward

AP:

Access point

AWGN:

Additive white Gaussian noise

BD:

Beam-domain

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:

Half-duplex

ID:

Information decoding

IGI:

Inter-group interference

IoT:

Internet of Things

IUI:

Inter-user interference

LMMSE:

Linear minimum mean square error

LS:

Least square

MF:

Matched filter

MIMO:

Multi-input multi-output

MRC:

Maximal ratio combining

MRT:

Maximum ratio transmission

PS:

Power splitting

RF:

Radio frequency

SE:

Spectral efficiency

SNR:

Signal-to-noise ratio

SI:

Self-interference

SINR:

Signal-to-interference-and-noise ratio

SWIPT:

Simultaneous wireless information and power transfer

TE:

Terminal

TS:

Time switching

TWR:

Two-way relaying

WET:

Wireless energy transfer

ZF:

Zero-forcing

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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.

Correspondence to Ming Zhang.

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Authors’ information

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, full-duplex 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, full-duplex 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, full-duplex 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
  • Beam-domain channel representation
  • Simultaneous wireless information and power transfer
  • Beamforming
  • Achievable sum-rate