Intelligent reflecting surface assisted transceiver design optimization in non-linear SWIPT network with heterogeneous users

Abstract

This paper studies the simultaneous wireless information and power transfer multi-antenna network with heterogeneous users and intelligent reflecting surface (IRS). Information decoding (ID) users, energy harvesting (EH) users, and power-splitting (PS) hybrid users require data and energy at the same time from the base station (BS) in the assistance of IRS. In the system model, the IRS reflects the incoming signal at each element by collaboratively adjusting phase shift to enhance the received signal at the users. In addition, the PS users exploit the PS circuit structure to simultaneously decode information and harvest energy via PS ratios. The target is to maximize the total PS-EH harvested power at PS and EH users with the non-linear EH model and limitation of BS transmit power while satisfying the signal-to-interference-plus-noise ratio requirements at ID and PS users. The information vectors at the BS, phase-shift vector at the IRS, and PS ratios at the hybrid users are jointly optimized by alternating optimization, semidefinite relaxation, sequential parametric convex approximation, and Gaussian randomization methods in the challenging non-convex problem. The numerical experiments present the convergence and effectiveness of the proposed algorithm. For certain parameters, the proposed IRS-assisted method enhances 17.65% total harvested power than the No-IRS one.

Appendix: Obtaining the equivalent optimization problem (15)

To simplify the complex form of the objective function (14a), we utilize the new auxiliary variables $\left\{ {{o_l},{{{\tilde{o}}}_j}} \right\}$ for replacing $\left\{ {\mathrm{{HP}}_{\mathrm{{NL,}}l}^{\mathrm{{PS}}},\mathrm{{HP}}_{\mathrm{{NL,}}j}^{\mathrm{{EH}}}} \right\}$, and express the equivalent form as:

$$\begin{aligned} \mathop {\mathrm{{maximize}}}\limits _{\left\{ {{{\mathbf{y}}_l},{{\mathbf{z}}_k},{\theta _l}} \right\} } \,\,\,&\sum \limits _{l = 1}^L {{o_l}} + \sum \limits _{j = 1}^J {{{{\tilde{o}}}_j}} \end{aligned}$$

(30a)

$$\begin{aligned} \text {s.t.:}\,\,\,\,&{{\left( {\frac{{{M_l}}}{{1 + {e^{ - {a_l}\left( {\mathrm{{HP}}_l^{\mathrm{{PS}}} - {b_l}} \right) }}}} - \frac{{{M_l}}}{{1 + {e^{{a_l}{b_l}}}}}} \right) } \Bigg / {\left( {1 - \frac{1}{{1 + {e^{{a_l}{b_l}}}}}} \right) }} \ge {o_l},\forall l \end{aligned}$$

(30b)

$$\begin{aligned}&{{\left( {\frac{{{{{\tilde{M}}}_j}}}{{1 + {e^{ - {{{\tilde{a}}}_j}\left( {\mathrm{{HP}}_j^{\mathrm{{EH}}} - {{{\tilde{b}}}_j}} \right) }}}} - \frac{{{{{\tilde{M}}}_j}}}{{1 + {e^{{{{\tilde{a}}}_j}{{{\tilde{b}}}_j}}}}}} \right) } \Bigg / {\left( {1 - \frac{1}{{1 + {e^{{{{\tilde{a}}}_j}{{{\tilde{b}}}_j}}}}}} \right) }} \ge {{\tilde{o}}_j},\forall j \end{aligned}$$

(30c)

$$\begin{aligned}&{o_l} \ge 0,{{\tilde{o}}_j} \ge 0,\forall l,j. \end{aligned}$$

(30d)

Then, we continue the simplified form of the new constraints (30b) as follows. By introducing new auxiliary variables, ${u_l} \ge 0,{v_l} \ge 0$, we replace ${e^{ - {a_l}\left( {\mathrm{{HP}}_l^{\mathrm{{PS}}} - {b_l}} \right) }}$ by ${u_l}$ and $\mathrm{{HP}}_l^{\mathrm{{PS}}}$ by $v_l^2$ in (30b). This results the equivalent form as:

$$\begin{aligned}&0 \ge \left( {1 - \frac{1}{{1 + {e^{{a_l}{b_l}}}}}} \right) {o_l} + \frac{{{M_l}}}{{1 + {e^{{a_l}{b_l}}}}} - \frac{{{M_l}}}{{1 + {u_l}}},\forall l \end{aligned}$$

(31a)

$$\begin{aligned}&{u_l} \ge {e^{ - {a_l}\left( {v_l^2 - {b_l}} \right) }},\forall l \end{aligned}$$

(31b)

$$\begin{aligned}&\mathrm{{HP}}_l^{\mathrm{{PS}}} \ge v_l^2,\forall l \end{aligned}$$

(31c)

$$\begin{aligned}&{u_l} \ge 0,{v_l} \ge 0,\forall l. \end{aligned}$$

(31d)

Similarly, the equivalent form of the new constraint (30c) is expressed as:

$$\begin{aligned}&0 \ge \left( {1 - \frac{1}{{1 + {e^{{{{\tilde{a}}}_j}{{{\tilde{b}}}_j}}}}}} \right) {{\tilde{o}}_j} + \frac{{{{{\tilde{M}}}_j}}}{{1 + {e^{{{{\tilde{a}}}_j}{{{\tilde{b}}}_j}}}}} - \frac{{{{{\tilde{M}}}_j}}}{{1 + {{{\tilde{u}}}_j}}},\forall j \end{aligned}$$

(32a)

$$\begin{aligned}&{{\tilde{u}}_j} \ge {e^{ - {a_l}\left( {{{{\tilde{v}}}_j} - {b_l}} \right) }},\forall j \end{aligned}$$

(32b)

$$\begin{aligned}&\mathrm{{HP}}_j^{\mathrm{{EH}}} \ge {{\tilde{v}}_j},\forall j \end{aligned}$$

(32c)

$$\begin{aligned}&{{\tilde{u}}_j} \ge 0,{{\tilde{v}}_j} \ge 0,\forall j. \end{aligned}$$

(32d)

Moreover, we take the function $\ln \left( \cdot \right)$ for both sides in (31b) and (32b), thus achieve the equivalent optimization problem (15).