Optimized Energy and Information Relaying in Self-Sustainable IRS-Empowered WPCN

IRS has recently been applied to various wireless communication networks and demonstrated promising results for improving the performance in terms of spectrum efficiency (SE) and energy efficiency (EE). References [16] and [17] consider the EE maximization problem in an IRS-assisted multi-user downlink communication network. The authors jointly optimize the power allocation at the AP and the phase shifts at the IRS and show that the proposed IRS-assisted communication remarkably outperforms the conventional relay-assisted communication in terms of EE. The authors in [18] propose an architecture where a few IRS elements are assumed to be active. Based on the proposed architecture, the authors develop two solutions using compressive sensing and deep learning for designing IRS’s reflection matrices. The authors in [19] exploit deep reinforcement learning based algorithms to jointly design the transmit beamforming at the base station and phase shifts at the IRS to maximize the sum-rate of downlink multiple-input single-output (MISO) systems. A low-complexity channel estimation protocol is proposed in [20], which does not require any prior knowledge on channel state information (CSI) or any active participation from IRS. The authors design the near-optimal active beamforming at the power beacon and passive beamforming at the IRS in order to maximize the received power at an EH user. In [21], the authors propose two efficient algorithms for finding the optimal beamformer at the AP and phase shifts at the IRS in an IRS-assisted MISO communication system. The authors in [22] study the problem of transmit power minimization in a multi-user downlink communication network by jointly optimizing the active transmit beamforming at the AP and passive reflect beamforming at the IRS subject to the users’ individual signal-to-interference-plus-noise ratio (SINR) constraints. Compared to the conventional massive MIMO system, the proposed IRS-enhanced model in [22] can considerably reduce the required transmit power. [23] and [24] study the integration of IRS with simultaneous wireless information and power transfer (SWIPT) technology, where the transmit precoders at the AP and the passive beamforming at the IRS are jointly optimized for maximizing the weighted sum-power at EH users [23] and weighted sum-rate at information receiving users [24]. Physical layer security and outage probability analysis in IRS-assisted MISO networks are investigated in [25] and [26], respectively. Signal-to-noise-ratio (SNR) maximization problem in a self-sustainable single-user IRS-assisted MISO communication system is studied in [27], where IRS elements use part of the downlink information signal for harvesting their required energy.

A survey on recent research efforts in the area of IRS can be found in [28].

Although IRS has lately received significant interests from the research community, it is still at the very early stage of development and more investigations are needed to fully capture the potentials of IRS and make it applicable to practical scenarios. Specifically, the integration of IRS technology with WPCN is a great step toward the realization of efficient and self-sustainable IoT networks, which has not been well investigated in the literature. Recently, a few research works have investigated the application of IRS for improving the performance of WPCNs [29, 30]. In [29], the authors study the application of IRS for WPCN performance enhancement, where IRS elements assist in downlink ET from the HAP to the users and uplink IT from users to the HAP. The authors in [30] propose a similar idea to use the IRS as a hybrid energy and information relay, where the user cooperation is also investigated for a two-user WPCN scenario. These preliminary works on the integration of IRS with WPCN provide some insights on the performance enhancements offered by using IRS in WPCNs. However, this integration needs to be studied more deeply with practical considerations for the network setup and network elements.

One of the most important points that is often overlooked in the studies on IRS is the IRS’s power consumption. Although IRS elements passively reflect the incident signals, the power consumption of the IRS cannot be neglected [16, 17, 27]. However, the majority of the works in this area (e.g., [18]-[26]) assume that the IRS’s power consumption is negligible because it does not perform complex signal processing tasks. In practice, the power consumption of IRS depends on the type and characteristics of its reflecting elements [16, 17]. For example, the values of each reflecting element’s circuit power consumption are 1.5 and 6 mW for 3- and 5-bit resolution phase shifting, respectively [17]. As the number of IRS elements is typically large, the circuit power consumption of the IRS can be even comparable to its power supply and cannot be neglected.

In the self-sustainable IoT networks, devices are expected to operate in an uninterrupted manner and have theoretically perpetual lifespans. Considering the non-negligible power consumption of IRS elements, it is important to propose efficient strategies which can keep the IRS operational for very long periods. Although embedded batteries can power the IRS temporarily, they cannot be relied on for the long-term functionality and uninterrupted operation of the IRS. Wired charging may also be unavailable if the IRS is deployed in inaccessible places. Thus, equipping IRS elements with EH modules can resolve these issues and make the IRS energy-neutral [15, 31]. This is our main motivation for studying a self-sustainable IRS-empowered WPCN, where the EH-enabled IRS, powered by energy transmission of the HAP, can act as a hybrid energy and information relay assisting in both downlink ET and uplink IT.

We study a self-sustainable IRS-empowered multi-user WPCN, where the IRS is equipped with an EH circuit to harvest RF energy from the HAP to power its operations. Inspired by the conventional wireless-powered active relays [32], time-switching (TS) and power-splitting (PS) schemes are proposed to enable the IRS to harvest energy from the RF signals transmitted by the HAP. In the TS scheme, the ET phase is split into two sub-slots, where the IRS harvests energy in the first sub-slot and assists in the downlink ET to the users in the second sub-slot. Compared to the conventional TS scheme [32], the proposed TS scheme can efficiently improve the amount of harvested energy at the users. In the PS scheme, the IRS harvests energy from the HAP’s signal and assists in the downlink ET to the users by adjusting its amplitude reflection coefficients in the ET phase. Compared to the conventional PS scheme [32], the proposed PS scheme can enhance both ET and IT efficiency and is more spectrum-efficient. To make our study applicable to practical systems, we consider a piece-wise linear EH model for the IRS and the users to account for the saturation behavior of practical EH systems [33]-[36]. We investigate the problem of sum-rate maximization for both TS and PS schemes and optimize the IRS phase shift design and network resource allocation jointly with EH time and amplitude reflection coefficients of the IRS.

The main contributions of this paper are summarized as follows:

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  We propose a self-sustainable IRS-empowered WPCN, where a wireless-powered IRS acts as a hybrid relay to improve the performance of WPCN in both downlink ET from the HAP to the users and uplink IT from users to the HAP.

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  To enable energy collection and hybrid relaying functionalities at the IRS, we propose more efficient TS and PS schemes, which can enhance the ET efficiency from the HAP to the users and assist in the uplink information transmission. We consider a piece-wise linear EH model for the IRS and the users, which is mathematically tractable and is able to capture the saturation effect of practical energy harvesters.

• •

  We study the system sum-rate maximization problem for the TS scheme by jointly optimizing the IRS’s phase shift designs in both ET and IT phases, time allocation for the IRS and users’ EH, time allocation for each user’s IT, and the users’ power allocation. To deal with the non-convexity of the formulated problem, we propose a two-step algorithm to achieve the near-optimal solution: in the first step the phase shifts for the IT are obtained in closed-form, while an efficient method by using one-dimensional search, semidefinite relaxation (SDR) and Gaussian randomization is designed for optimizing the IRS phase shifts in the ET phase, time allocation and power allocation in the second step. In particular, we obtain a closed-form solution for the optimal IRS’s EH time and discuss its implications.

• •

  We then investigate the sum-rate maximization problem for the PS scheme and jointly optimize the IRS’s phase shift design in both ET and IT phases, time allocation for the EH and IT phases, power allocation at the users, and the amplitude reflection coefficient in the EH phase, using a similar two-step algorithm as for the TS scheme. In particular, we analyze the condition for activating the IRS in the PS scheme and obtain the optimal amplitude reflection coefficient as a function of the EH time, from which some interesting observations are revealed.

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  Finally, we evaluate the performance of our proposed schemes via numerical simulations which show that our proposed schemes can achieve significant system sum-rate gain compared to the baseline WPCN protocol.

This paper is organized as follows. Section II describes the system model of the proposed IRS-empowered WPCN for both TS and PS schemes. Sections III and IV investigate the sum-rate maximization problems for TS and PS schemes, respectively. Section V evaluates the performance of the presented algorithms by conducting numerical simulations and Section VI concludes the paper.

As mentioned earlier, the IRS is assumed to be energy-constrained, which needs to harvest energy from the HAP for powering its relaying operations. In this regard, we design efficient TS and PS schemes for the IRS.

In the IT phase, the users transmit information to the HAP via time division multiple access, using the harvested energy in the ET phase. Denote the duration of IT for $U_{i}$ as $t_{i}$. Let $s_{u,i}$ be the information-carrying signal of $U_{i}$ with unit power. The transmit signal of $U_{i}$ during $t_{i}$ is then expressed as $x_{u,i}=\sqrt{P_{u,i}}s_{u,i}$, where $P_{u,i}$ is $U_{i}$’s transmit power and satisfies

with $P_{c,i}$ being the circuit power consumption of $U_{i}$. As the amplitude reflection coefficients are set to be the same, the IRS’s circuit power consumption is mainly caused by performing each element’s phase shifting [16, 17]. The other power consumptions, such as powering the EH circuit and signaling overhead, can be considered to be negligible [27, 32, 43]. By denoting the power consumption of each element as $\mu$, the circuit power consumption of the IRS is thus expressed as $K\mu$. To power its operations, IRS needs to harvest sufficient energy in the ET phase. We assume that all the harvested energy stored in the energy storage can be used to power the IRS’ circuits, the following constraints are thus held:

for TS and PS schemes, respectively. Note that the power consumption of the IRS in the first sub-slot of the TS scheme is neglected because the IRS’s power consumption is mainly determined by the reflection operation [16, 17], which do not take place during $\tau_{0}$.

Denote the phase shift of the $k$-th element for $U_{i}$’s IT as $\theta_{d,i,k}\in\mathcal{R}$. Then, the phase shift matrix during $t_{i}$ is denoted by $\bm{\Theta}_{d,i}$, where $\bm{\Theta}_{d,i}=\sqrt{\rho}\text{diag}\{v_{d,i,1},\ldots,v_{d,i,K}\}$, $v_{d,i,k}=e^{j\theta_{d,i,k}}$, and $|v_{d,i.k}|=1$. Note that we have set the amplitude reflection coefficients to be 1 to maximize the signal reflection in the IT phase [13]. The received signal at the HAP from $U_{i}$, denoted by $y_{h,i}$, is thus given by

where $n_{h}\sim\mathcal{CN}(0,\sigma_{h}^{2})$ is the AWGN at the HAP. The SNR at the HAP during $t_{i}$, denoted by $\gamma_{i}$, is expressed as $\gamma_{i}=\frac{P_{u,i}|\bm{g}_{r}^{H}\bm{\Theta}_{d,i}\bm{g}_{u,i}+g_{h,i}|^{2}}{\sigma_{h}^{2}}$. The achievable rate from $U_{i}$ to the HAP in bits/second/Hz is then formulated as

It is obvious that P1 is a non-convex optimization problem due to the coupling of variables in the objective function and the constraints, and convex optimization techniques cannot be used to solve it directly. In the following, we propose a two-step algorithm to solve the sum-rate maximization problem in P1. Specifically, we first obtain the optimal phase shifts for the IT in closed-form and then propose an efficient algorithm to solve the simplified problem.

To reduce the computational complexity and show more insights about resource allocation, we consider a special case with random design of phase shifts and focus on the time and power allocation optimization in the IRS-assisted WPCN. As will be shown in Section V, using IRS is beneficial for improving the performance of WPCN even with randomly designed phase shifts [22, 48]. Letting $e_{u,i}=P_{u,i}t_{i}$, we have

and the sum-rate maximization problem with random phase shifts is formulated as

where $\gamma_{d,i}=|\bm{g}_{r}^{H}\bm{\Theta}_{d,i}\bm{g}_{u,i}+g_{h,i}|^{2}$. The constraint C15 is an equality at the optimal solution as we discussed in Section III-A2. Hence, we have

Substituting (III-B) into $R_{i}$, we have

where $a_{i}=\min\{\eta P_{ts,r,i,0},P_{u,i,sat}\}\tau_{0}\gamma_{d,i}/\sigma_{h}^{2}$, $b_{i}=\min\{\eta P_{ts,r,i,1},P_{u,i,sat}\}\gamma_{d,i}/\sigma_{h}^{2}$ and $c_{i}=P_{c,i}\gamma_{d,i}/\sigma_{h}^{2}$. Proposition 2 holds here as well. Hence, P3 is modified as

It can be verified that P3.1 is a convex optimization problem [45], which can be solved by standard convex optimization techniques, e.g., Lagrange duality method. The Lagrangian of P3.1 is given by

where $\xi\geq 0$ is the Lagrange multiplier.

Using (III-B) and Proposition 3, the optimal energy allocation at each user can be easily obtained.

Similar to P1, P4 is a non-convex optimization problem due to the coupled variables in the objective function and the constraints. It is straightforward to observe that Proposition 1 also holds for P4. Accordingly, P4 can be equivalently reformulated as

In the following, we investigate P4.1 under the condition that (27) is satisfied, because otherwise the IRS is not able to improve the performance of WPCN. Following the same steps as in Section III-A2, the sum-rate maximization problem is formulated as

where

Similarly, solving P4.2 is equivalent to solving P4. From P4.2, we first obtain the following proposition about the optimal amplitude reflection coefficient.

For solving P4.2, we first fix $t_{0}$ and optimize other variables. The optimal value of $t_{0}$ can then be obtained by a one-dimensional search over $\big{(}\max\{K\mu/(\eta P_{h}||\bm{h}_{r}||^{2}),K\mu/P_{irs,sat}\},1\big{)}$. Given $t_{0}$, the optimal value of $\beta_{e}$ can be found from Proposition 4 and we will have the following optimization problem:

After the relaxation of the rank-one constraint in C14, P4.3 is similar to P2.2 in Section III-A2 and can be solved following the same procedure. For brevity and to avoid repetition, we do not explain the details of solving P4.3 here.

Algorithm 2 describes the process of solving the sum-rate maximization problem for the PS scheme, which is implemented at the HAP. Similar to what has been mentioned for Algorithm 1, the computational complexity of Algorithm 2 is $\mathcal{O}(\bar{M}\max(K,N)^{4}K^{0.5}\log(1/\epsilon)+\bar{M}DN)$, where $\bar{M}$ is the number of iterations for updating $t_{0}$. Again, by setting the appropriate step size for updating $t_{0}$ and relatively large number of randomizations for the Gaussian randomization method, we can obtain the near-optimal solution to P4.