Networks model
We consider a multi-cell multi-user downlink system with Nc cells as shown in Fig. 1. Each cell consists of one Nt-antenna transmitter, K single-antenna IRs, L single-antenna ERs, and S single-antenna Eves. Nc transmitters transmit signals over a common frequency band, which means that the inter-cell and intra-cell interference co-exist in the system. To prevent the information leakage, the energy-bearing spatially selective AN is embedded in the transmit signals of each transmitter.
For clarity, we use k, l, s, and n to denote the k-th IR, the l-th ER, s-th Eve, and the n-th 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 n-th 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 k-th IR transmitted in the n-th 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 k-th IR and the l-th ER in the n-th cell can be, respectively, given by
$$ {{}{\begin{aligned} {\mathbf{y}^{\text{(IR)}}_{nk}}\left(t \right) &= {\sum\nolimits}_{m = 1}^{{N_{\mathrm{c}}}} {\mathbf{h}_{mnk}^{H}{\mathbf{x}_m}\left(t \right)} + {n_{nk}}\left(t \right)\\ &= \underbrace {\mathbf{h}_{nnk}^{H}{\mathbf{w}_{nk}}{\omega_{nk}}\left(t \right)}_{\text{desired~signal}}+ \underbrace {{\sum\nolimits}_{i \ne k}^{K} {\mathbf{h}_{nnk}^{H}{\mathbf{w}_{ni}}{\omega_{ni}}\left(t \right)} }_{\text{{intra-cell~interference}}} \\ &\quad + \underbrace {\sum\limits_{m \ne n}^{{N_{\mathrm{c}}}} {\sum\limits_{i = 1}^{K} {\mathbf{h}_{mnk}^{H}{\mathbf{w}_{mi}}{\omega_{mi}}\left(t \right)}} }_{\text{{inter-cell~interference}}} + \underbrace {\sum\limits_{m = 1}^{{N_{\mathrm{c}}}} {\mathbf{h}_{mnk}^{H}{\mathbf{z}_m}\left(t \right)} }_{\text{{AN}}} + {n_{nk}}\left(t \right), \end{aligned}}} $$
(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 s-th 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 m-th transmitter to the k-th IR, the l-th ER, and the s-th EveFootnote 1 in the n-th 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 k-th IR, the l-th ER, and the s-th Eve, respectively. Following (2) and (4), the received SINR at the k-th IR in cell n and that at the s-th 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)
Non-linear EH model
Each ER converts the received RF signals into output DC power by its RF-EH circuits. At the l-th ER in the n-th cell, the input power of its RF-EH 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 RF-to-DC conversion efficiency ρ of the RF-EH circuits is regarded as a constant in the interval (0,1], referring to the linear EH model, which indicates that the RF-to-DC conversion efficiency is independent of the input power level. However, in practice, the RF-EH circuits include various non-linearities, such as the diode or diode-connected transistor. As a result, the RF-to-DC conversion efficiency depends on the input power level. To capture the dynamics of the RF-to-DC conversion efficiency for different input power levels, in this paper, the non-linear model is adopted [5–7]. The output DC power (harvested power) of the RF-EH circuits at the s-th 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 RF-EH 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 non-linear 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 γu.
-
To prevent the information interception of each Eve, its received SINR should be lower than a predefined threshold γe.
-
To guarantee the EH requirement of each ER, its output DC power should be larger than a predefined threshold θ.
Then, our considered power-minimization AN-aided 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 Csec= log(1+γu)− log(1+γu). Note that the value of γu and γe depends on the required QoS of IRs. A larger γu and a smaller γe indicate better system performance, but more power is required at transmitters.
Problem (9) is not convex due to the non-convex constraint (9b)–(9d), which cannot be solved directly. Therefore, in Section 3, we will solve it by using SDR methodsFootnote 2.