Networks model
We consider a multicell multiuser downlink system with N_{c} cells as shown in Fig. 1. Each cell consists of one N_{t}antenna transmitter, K singleantenna IRs, L singleantenna ERs, and S singleantenna Eves. N_{c} transmitters transmit signals over a common frequency band, which means that the intercell and intracell interference coexist in the system. To prevent the information leakage, the energybearing spatially selective AN is embedded in the transmit signals of each transmitter.
For clarity, we use k, l, s, and n to denote the kth IR, the lth ER, sth Eve, and the nth cell/transmitter, respectively, where \(k \in \mathcal K\), \(l \in \mathcal L\), \(s \in \mathcal S\), and \(n \in \mathcal N_{\mathrm {c}}\). \(\mathcal K\buildrel \Delta \over = \left [1,2,\ldots,K\right ]\), \(\mathcal L\buildrel \Delta \over = \left [1,2,\ldots,L\right ]\), \(\mathcal S\buildrel \Delta \over =\left [1,2,\ldots,S\right ]\), and \(\mathcal N_{\mathrm {c}}\buildrel \Delta \over = \left [1,2,\ldots,N_{\mathrm {c}}\right ]\) denote the sets of IRs, ERs, Eves, and cells/transmitters, respectively. Thus, the transmit signal from the nth transmitter is
$$\begin{array}{*{20}l} \mathbf{x}_{n}\left(t \right) = {\sum\nolimits}_{k = 1}^{K} {{\mathbf{w}_{nk}} {\omega_{nk}}\left(t \right)} + \mathbf{z}_{n}(t) \in \mathbb{C}^{N_{\mathrm{t}} \times 1}, \end{array} $$
(1)
where \(\omega _{nk}\in \mathbb {C}\) with \({\mathbb E}\{\omega _{nk}^{2}\}=1\) is the data symbol for the kth IR transmitted in the nth cell. \(\mathbf {w}_{nk}\in \mathbb {C}^{{N_{\mathrm {t}}} \times 1}\) is the corresponding transmit beamforming vector. \(\mathbf {z}_{n} \in \mathbb {C}^{{N_{\mathrm {t}}} \times 1}\) denotes the AN vector which follows Gaussian distribution, i.e., \(\mathbf {z}_{n}\sim \mathcal {CN}(\mathbf {0},\mathbf {\Sigma }_{n})\) and Σ_{
n
}≽0. Thus, the total required power of the system can be given by \( {\sum \nolimits }_{n = 1}^{{N_{\mathrm {c}}}} {\left ({{\sum \nolimits }_{k = 1}^{K} {\left \ {{\mathbf {w}_{nk}}} \right \_{2}^{2}} + {\text {Tr}}\left ({{\boldsymbol {\Sigma }_{n}}} \right)} \right)}\).
Then, the received signal at the kth IR and the lth ER in the nth cell can be, respectively, given by
$$ {{}{\begin{aligned} {\mathbf{y}^{\text{(IR)}}_{nk}}\left(t \right) &= {\sum\nolimits}_{m = 1}^{{N_{\mathrm{c}}}} {\mathbf{h}_{mnk}^{H}{\mathbf{x}_m}\left(t \right)} + {n_{nk}}\left(t \right)\\ &= \underbrace {\mathbf{h}_{nnk}^{H}{\mathbf{w}_{nk}}{\omega_{nk}}\left(t \right)}_{\text{desired~signal}}+ \underbrace {{\sum\nolimits}_{i \ne k}^{K} {\mathbf{h}_{nnk}^{H}{\mathbf{w}_{ni}}{\omega_{ni}}\left(t \right)} }_{\text{{intracell~interference}}} \\ &\quad + \underbrace {\sum\limits_{m \ne n}^{{N_{\mathrm{c}}}} {\sum\limits_{i = 1}^{K} {\mathbf{h}_{mnk}^{H}{\mathbf{w}_{mi}}{\omega_{mi}}\left(t \right)}} }_{\text{{intercell~interference}}} + \underbrace {\sum\limits_{m = 1}^{{N_{\mathrm{c}}}} {\mathbf{h}_{mnk}^{H}{\mathbf{z}_m}\left(t \right)} }_{\text{{AN}}} + {n_{nk}}\left(t \right), \end{aligned}}} $$
(2)
$$ {\begin{aligned} &{\mathbf{y}^{\text{(ER)}}_{nl}}\left(t \right) = {\sum\nolimits}_{m = 1}^{{N_{\mathrm{c}}}} {\mathbf{h}_{mnl}^{H}{\mathbf{x}_m}\left(t \right)} + {n_{nl}}\left(t \right) \end{aligned}} $$
(3)
and that at the sth Eve in cell n is
$$ \begin{aligned} {\mathbf{y}^{\text{(Eve)}}_{ns}}\left(t \right) &= {\sum\nolimits}_{m = 1}^{{N_{\mathrm{c}}}} {\mathbf{g}_{mns}^{H}{\mathbf{x}_m}\left(t \right)} + {v_{ns}}\left(t \right), \end{aligned} $$
(4)
where h_{
mnk
}, h_{
mnl
}, and \(\mathbf {g}_{mns} \in \mathbb {C}^{{N_{\mathrm {t}}} \times 1}\) denote the channel vectors from the mth transmitter to the kth IR, the lth ER, and the sth Eve^{Footnote 1} in the nth cell, respectively. n_{
nk
}(t), n_{
nl
}(t), and v_{
ns
}(t) are the Gaussian noises with variance \(\sigma _{nk}^{2}\), \(\sigma _{nl}^{2}\), and \(\sigma _{ns}^{2}\) at the kth IR, the lth ER, and the sth Eve, respectively. Following (2) and (4), the received SINR at the kth IR in cell n and that at the sth ER in cell n are given by (5) and (6), respectively. [!t]
$$ {}\begin{aligned} &\text{SINR}_{nk}\left({\left\{ {{\mathbf{w}_{m1,}} \cdots {\mathbf{w}_{mK}},{{\boldsymbol{\Sigma}} _m}} \right\}_{m = 1}^{{N_{\mathrm{c}}}}} \right) = \frac{{{{\left {\mathbf{h}_{nnk}^{H}{\mathbf{w}_{nk}}} \right}^{2}}}}{{{\sum\nolimits}_{i \ne k}^{K} {{{\left {\mathbf{h}_{nnk}^{H}{\mathbf{w}_{ni}}} \right}^{2}} + {\sum\nolimits}_{m \ne n}^{{N_{\mathrm{c}}}} {{\sum\nolimits}_{i = 1}^{K} {{{\left {\mathbf{h}_{mnk}^{H}{\mathbf{w}_{mi}}} \right}^{2}} + {\sum\nolimits}_{m = 1}^{{N_{\mathrm{c}}}} {\mathbf{h}_{mnk}^{H}\mathbf{\Sigma}_m \mathbf{h}_{mnk}+ \sigma_{nk}^{2}}}}}}} \end{aligned} $$
(5)
[!t]
$$ {}\begin{aligned} \text{SINR}_{ns}^{e}\left({\left\{ {{\mathbf{w}_{m1,}} \cdots {\mathbf{w}_{mK}},{{\boldsymbol{\Sigma}} _m}} \right\}_{m = 1}^{{N_{\mathrm{c}}}}} \right) = {\underset{k \in \mathcal K}{\max}} \left({\frac{{{{\left {\mathbf{g}_{nns}^{H}{\mathbf{w}_{nk}}} \right}^{2}}}}{{\sum\limits_{i \ne k}^{K} {{{\left {\mathbf{g}_{nnk}^{H}{\mathbf{w}_{ni}}} \right}^{2}} + {\sum\nolimits}_{m \ne n}^{{N_{\mathrm{c}}}} {\sum\limits_{i = 1}^{K} {{{\left {\mathbf{g}_{mns}^{H}{\mathbf{w}_{mi}}} \right}^{2}} + \sum\limits_{m = 1}^{{N_{\mathrm{c}}}} {\mathbf{g}_{mns}^{H}{{\boldsymbol{\Sigma}} _m}{\mathbf{g}_{mns}} + \sigma_{ns}^{2}}} }} }}} \right) \end{aligned} $$
(6)
Nonlinear EH model
Each ER converts the received RF signals into output DC power by its RFEH circuits. At the lth ER in the nth cell, the input power of its RFEH circuits from the received RF signals is
$$\begin{array}{*{20}l} &P_{nl}^{{\left(\text{ER}\right)}} =\\ &{{\mathbf{h}}_{mnk}^{H}\left({\sum\nolimits}_{n = 1}^{N}\left({\sum\nolimits}_{k = 1}^{K} {{{\mathbf{w}}_{nk}}{\mathbf{w}}_{nk}^{H}} + {{\boldsymbol{\Sigma}}_n}\right) \right){\mathbf{h}}_{mnk}^{}}. \end{array} $$
(7)
In most existing works, the RFtoDC conversion efficiency ρ of the RFEH circuits is regarded as a constant in the interval (0,1], referring to the linear EH model, which indicates that the RFtoDC conversion efficiency is independent of the input power level. However, in practice, the RFEH circuits include various nonlinearities, such as the diode or diodeconnected transistor. As a result, the RFtoDC conversion efficiency depends on the input power level. To capture the dynamics of the RFtoDC conversion efficiency for different input power levels, in this paper, the nonlinear model is adopted [5–7]. The output DC power (harvested power) of the RFEH circuits at the sth ER is
$$\begin{array}{*{20}l} {\Phi_{nl}}\left({\left\{ {{\mathbf{w}_{m1,}} \cdots {\mathbf{w}_{mK}},{{\boldsymbol{\Sigma}} _m}} \right\}_{m = 1}^{{N_{\mathrm{c}}}}} \right) = \frac{{{\Psi_{nl}}}}{{{X_{nl}}}}  {Y_{nl}} \end{array} $$
(8)
with
$$\begin{array}{*{20}l} {\Psi_{nl}} = \frac{{{M_{nl}}}}{{1 + \exp \left({  {a_{nl}}\left({P_{nl}^{{\left(\text{ER}\right)}} {b_{nl}}} \right)} \right)}}, \end{array} $$
where
$$\begin{array}{*{20}l} &{X_{nl}} = \frac{{\exp \left({{a_{nl}}{b_{nl}}} \right)}}{{1 + \exp \left({{a_{nl}}{b_{nl}}} \right)}},~{Y_{nl}} = \frac{M_{nl}}{{\exp \left({{a_{nl}}{b_{nl}}} \right)}}. \end{array} $$
Ψ_{
nl
} is a logistic function of \(P_{nl}^{{\left (\text {ER}\right)}}\), M_{
nl
} is a constant denoting the maximum output DC power, which indicates the saturation limitation of the RFEH circuits. a_{
nl
} and b_{
nl
} are constants representing some properties of the EH system, e.g., the resistance, the capacitance and the circuit sensitivity. In general, M_{
nl
}, a_{
nl
}, and b_{
nl
} depend on the choice of hardware components for assembling the EH system and can be estimated through a standard curve fitting algorithm. Figure 2 provides an example of the nonlinear EH model, where the maximum output DC power M_{
nl
} is set as 20 mW. One can observe that the output DC power increases with the increment of the input power at first, and then when it reaches the saturation region, the output DC power cannot surpass this saturation limitation, which is much different from the linear EH model, where the output DC power can always increases with the increment of the input power.
Problem formulation
Our goal is to minimize the total required power of the system by jointly optimizing the transmit beamforming vectors and covariance matrixes of the AN to meet the following two system requirements.

To guarantee the information rate requirement of each IR, its received SINR should be larger than a predefined threshold γ_{u}.

To prevent the information interception of each Eve, its received SINR should be lower than a predefined threshold γ_{e}.

To guarantee the EH requirement of each ER, its output DC power should be larger than a predefined threshold θ.
Then, our considered powerminimization ANaided MCBF design is mathematically formulated as
$$\begin{array}{*{20}l} & {\underset{\left\{ {{\mathbf{w}_{nk}}} \right\}\left\{ {{\mathbf{\Sigma}_{n}}} \right\}}{\min}} \sum\limits_{n = 1}^{{N_{\mathrm{c}}}} {\left({\sum\limits_{k = 1}^{K} {\left\ {{\mathbf{w}_{nk}}} \right\_{2}^{2}} + {\text{Tr}}\left({{\mathbf{\Sigma}_{n}}} \right)} \right)} \end{array} $$
(9a)
$$\begin{array}{*{20}l} \text{s.t.}~&\text{SINR}_{nk}\left({\left\{ {{\mathbf{w}_{m1,}} \cdots {\mathbf{w}_{mK}},{{\boldsymbol\Sigma}_{m}}} \right\}_{m = 1}^{{N_{\mathrm{c}}}}} \right)\ge {\gamma_{\mathrm{u}}} \end{array} $$
(9b)
$$\begin{array}{*{20}l} &\text{SINR}_{ns}^{e}\left({\left\{ {{\mathbf{w}_{m1,}} \cdots {\mathbf{w}_{mK}},{{\boldsymbol{\Sigma}}_{m}}} \right\}_{m = 1}^{{N_{\mathrm{c}}}}} \right)\le {\gamma_{\mathrm{e}}} \end{array} $$
(9c)
$$\begin{array}{*{20}l} &{\Phi_{nl}}\left({\left\{ {{\mathbf{w}_{m1,}} \cdots {\mathbf{w}_{mK}},{{\boldsymbol{\Sigma}}_{m}}} \right\}_{m = 1}^{{N_{\mathrm{c}}}}} \right)\ge\theta\\ &{\boldsymbol{\Sigma}_{n}}\succeq \mathbf{0},~\forall n,m \in \mathcal N_{\mathrm{c}}, \end{array} $$
(9d)
With (9b) and (9c), the secrecy capacity between each IR and its serving transmitter is guaranteed bounded below C_{sec}= log(1+γ_{u})− log(1+γ_{u}). Note that the value of γ_{u} and γ_{e} depends on the required QoS of IRs. A larger γ_{u} and a smaller γ_{e} indicate better system performance, but more power is required at transmitters.
Problem (9) is not convex due to the nonconvex constraint (9b)–(9d), which cannot be solved directly. Therefore, in Section 3, we will solve it by using SDR methods^{Footnote 2}.