IRS-Aided WPCN

Abstract

This chapter presents three paradigms of IRS-aided WPCN to show the potential gain achieved by the IRS. In the first paradigm, we first study three special cases of the dynamic IRS beamforming, namely user-adaptive IRS beamforming, uplink (UL) adaptive IRS beamforming, and static IRS beamforming, and then a general optimization framework is proposed for dynamic IRS beamforming in half-duplex (HD) WPCN where any given number of IRS phase-shift vectors are available during one channel coherence. Moreover, we further extend the UL multi-use orthogonal multiple access (OMA) scheme to the NOMA scheme. In the second paradigm, we study the full-duplex (FD) system where the HAP is able to transmit signal and receive signal over the same frequency simultaneously and three IRS beamforming designs, are presented in FD-WCPN with linear EH model. In the third paradigm, we consider a more general case with the practical non-linear EH model and the multi-antenna system in the FD-WPCN, i.e, IRS-aided MIMO FD-WPCN. Several efficient optimization approaches are proposed to cater to different types of optimization problems. In addition, simulation results validate the theoretical findings, illustrate the practical significance of IRS with dynamic beamforming for spectral efficient WPCNs, and demonstrate the effectiveness of our proposed designs over various benchmark schemes.

Appendices

Appendix 1: Proof of Proposition 4.1

To prove Proposition 1, we only need to prove $R^*_{\mathrm {User-adp}} \geq R^*_{\mathrm {UL-adp}}$ and $R^*_{\mathrm {UL-adp}}=R^*_{\mathrm {Static}}$, respectively. Note that (P2) is a special case of (P1) with $\boldsymbol v_k=\boldsymbol v_m, \forall k\neq m, k \geq 1, m\geq 1$. As such, the optimal solution to (P2) is also a feasible solution to (P1), which yields $R^*_{\mathrm {User-adp}} \geq R^*_{\mathrm {UL-adp}}$.

We next prove $R^*_{\mathrm {UL-adp}}=R^*_{\mathrm {Static}}$. First, it can be readily shown that in the optimal solution to (P2), constraint (5b) is met with equality. Then, the objective function of (P2) can be written as

$$\displaystyle \begin{aligned} {} \sum_{k=1}^{K} \tau_k \log_2\left(1+\frac{\eta_kP_{\mathrm{A}} \tau_0|h^H_{d,k} + \boldsymbol q^H_k \boldsymbol v_0|{}^2 |h^H_{d,k} + \boldsymbol q^H_k \boldsymbol v_1|{}^2}{\sigma^2\tau_k}\right). \end{aligned} $$

(4.132)

The key to proving $R^*_{\mathrm {UL-adp}}=R^*_{\mathrm {Static}}$ lies in splitting the upper bound of the objective function in (P2) into two independent terms, which are the functions of $\boldsymbol v_0$ and $\boldsymbol v_1$, respectively. To this end, we provide the following lemma to facilitate the proof, which can be obtained with simple algebraic operations.

Lemma 4.2

For any arbitrary numbers $a\geq 0$ and $b\geq 0$, it follows that $1+ab\leq \sqrt {(1+a^2)(1+b^2)}$ and the equality holds if and only if $a=b$.

Let $f(\boldsymbol v)\triangleq |h^H_{d,k} + \boldsymbol q^H_k \boldsymbol v|{ }^2/\sqrt { \eta _kP_{\mathrm {A}}\tau _0/(\sigma ^2\tau _k)}$ and denote by $\boldsymbol v^*$ the vector maximizing $f(\boldsymbol v)$ subject to constraints $|[\boldsymbol v]_n|=1, \forall n$. Then, we can establish the following inequalities for the objective function in (4.132)

$$\displaystyle \begin{aligned} {} & \sum_{k=1}^{K} \tau_k\log_2\left(1+ {f(\boldsymbol v_0)f(\boldsymbol v_1) } \right) \\ \overset{(a)}{\leq} & \sum_{k=1}^{K} \tau_k\log_2\left(\sqrt{(1+{f^2(\boldsymbol v_0) } ) (1+ f^2(\boldsymbol v_1) ) } \right) \\ \!\!\!\! = & \! \sum_{k=1}^{K} \tau_k\log_2\left(\sqrt{ 1 \! +\! {f^2(\boldsymbol v_0) } } \right) \! + \! \sum_{k=1}^{K} \tau_k\log_2\left(\sqrt{ 1 \! +\! {f^2(\boldsymbol v_1) } } \right) \\ \overset{(b)}{\leq} & \sum_{k=1}^{K} \tau_k\log_2\left({ 1+{f^2(\boldsymbol v^*) } } \right), \end{aligned} $$

(4.133)

where $(a)$ is based on Lemma 4.2 and the equality holds when $\boldsymbol v_0=\boldsymbol v_1$, and $(b)$ holds due to the optimality of $\boldsymbol v^*$ in maximizing $f(\boldsymbol v)$ and the equality holds when $\boldsymbol v_0=\boldsymbol v_1=\boldsymbol v^*$. Based on (4.133), we have $\boldsymbol v_0=\boldsymbol v_1$ holds in the optimal solution to (P2), which means that (P2) is simplified to (P3) with $R^*_{\mathrm {UL-adp}}=R^*_{\mathrm {Static}}$. This thus completes the proof.

Appendix 2: Proof of Proposition 4.2

We show this proof by contradiction. Suppose that $S^* \,{=}\,\Big \{ {\tau ^*_{0},\{t^*_{k,j}\},\{p^*_{k,j}\}, \boldsymbol v^*_0,\{\boldsymbol v^*_{j}\} } \Big \}$ achieves the optimal solution to (P4) and there exists a device m who performs its UL WIT employing two IRS phase-shift vectors indexed by $m'$ and $\ell$, $\ell \neq m'$, i.e., $t^*_{m, m'}>0$, $p^*_{m,m'}>0$ and $t^*_{m, \ell }>0$, $p^*_{m, \ell }>0$, with $\psi _{m,m'}\triangleq |h^H_{d,m} + \boldsymbol q^H_m \boldsymbol v^*_{m'}|{ }^2 > \psi _{m,\ell } \triangleq |h^H_{d,m} + \boldsymbol q^H_m \boldsymbol v^*_{\ell }|{ }^2$. Then, we construct a different solution ${S}^{\star } = \Big \{ {\tau ^{\star }_{0},\{t^{\star }_{k,j}\},\{p^{\star }_{k,j}\}, \boldsymbol v^{\star }_0,\{\boldsymbol v^{\star }_{j}\} } \Big \}$ where $\tau ^{\star }_{0}= \tau ^{*}_{0}$, $\boldsymbol v^{\star }_0 = \boldsymbol v^{*}_0$, $\boldsymbol v^{\star }_{j} = \boldsymbol v^{*}_{j}$, and

$$\displaystyle \begin{aligned} {} {t}^{\star}_{k,j} = \left\{ \begin{aligned} &t^*_{m, m'} + t^*_{m,\ell}, && k=m, j = m',\\ &0,&& k=m, j\neq m',\\ &t^*_{k,j}, &&k\neq m, 0\leq j\le J, \end{aligned} \right. \end{aligned} $$

(4.134)

(4.135)

It can be verified that the newly constructed solution ${S}^{\star }$ is also a feasible solution to (P4) as it satisfies all the constraints therein. Since the time and transmit power solutions in UL WIT for any device $k\neq m$ remain unchanged in (4.134) and (4.135), the UL throughput of device m achieved by $S^{\star }$ is the same as that achieved by $S^{*}$. Thus, we only focus on the UL throughput of device m via phase-shift vector $m'$, which satisfies the following inequalities

$$\displaystyle \begin{aligned} & {t}^{\star}_{m,m'}\log_2\!\left(\!1\!+\!\frac{ {p}^{\star}_{m,m'}\psi_{m,m'}}{ \sigma^2}\!\right)\! =(t^*_{m,m'}+t^*_{m,\ell})\log_2\left(1+\frac{(p^*_{m,m'}+p^*_{m,\ell})\psi_{m,m'}}{(t^*_{m,m'}+t^*_{m,\ell}) \sigma^2 }\right) \\ & \quad \overset{(a)}{\geq}{t}^*_{m,m'}\log_2 \!\left(\!1\!+\!\frac{{p}^*_{m,m'}\psi_{m,m'}}{{t}^*_{m,m'} \sigma^2 } \!\right)\!+\!{t}^*_{m,\ell}\log_2\left(\!1\!+\!\frac{{p}^*_{m,\ell}\psi_{m,m'}}{{t}^*_{m,\ell} \sigma^2 }\!\right)\\ & \quad \overset{(b)}>{t}^*_{m,m'}\log_2\!\left(\!1\!+\!\frac{{p}^*_{m,m'}\psi_{m,m'}}{{t}^*_{m,m'} \sigma^2 \! }\right)\!+\!{t}^*_{m,\ell}\log_2\left(\!1\!+\!\frac{{p}^*_{m,\ell}\psi_{m,\ell}}{{t}^*_{m,\ell} \sigma^2 }\! \right), \end{aligned} $$

(4.136)

where inequality $(a)$ holds due to the concavity of $x \log _2(1+\frac {y}{x})$ and strict inequality $(b)$ holds due to $\psi _{m,m'}>\psi _{m,\ell }$, $\ell \neq m'$. This means that the constructed solution $S^{\star }$ achieves a higher sum throughput than $S^*$ which contradicts the assumption that $S^*$ is optimal. This thus completes the proof.