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

This paper proposes a hybrid-relaying scheme empowered by a self-sustainable intelligent reflecting surface (IRS) in a wireless powered communication network (WPCN), to simultaneously improve the performance of downlink energy transfer (ET) from a hybrid access point (HAP) to multiple users and uplink information transmission (IT) from users to the HAP. We propose time-switching (TS) and power-splitting (PS) schemes for the IRS, where the IRS can harvest energy from the HAP's signals by switching between energy harvesting and signal reflection in the TS scheme or adjusting its reflection amplitude in the PS scheme. For both the TS and PS schemes, we formulate the sum-rate maximization problems by jointly optimizing the IRS's phase shifts for both ET and IT and network resource allocation. To address each problem's non-convexity, we propose a two-step algorithm, where the formulated problem is decoupled into two sub-problems and each sub-problem can be solved separately in an efficient way. To show the structure of resource allocation, we also investigate sub-optimal solutions for the schemes with random phase shifts. Through numerical results, we show that our proposed schemes can achieve over hundred times sum-rate gain compared to the baseline scheme without IRS.

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I Introduction

With nearly 50 billion Internet of Things (IoT) devices by 2020 and even 500 billion by 2030 [1], we have already stepped into the new era of IoT. Having the vision of being self-sustainable, IoT has observed the energy limitation as a major issue for its widespread development. Recent advances in energy harvesting (EH) technologies, especially radio frequency (RF) EH [2], opened a new approach for self-sustainable IoT devices to harvest energy from dedicated or ambient RF sources. This leads to an emerging topic of wireless powered communication networks (WPCNs), in which low-cost IoT devices can harvest energy from a dedicated hybrid access point (HAP) and then use the harvested energy to transmit data to the HAP [3]. As a result, the development of WPCNs has been a promising step toward the future self-sustainable IoT networks [4].

Although possessing significant benefits and attractive features for low-cost IoT networks, e.g., multi-device and long-distance charging, WPCNs are facing some challenges which need to be addressed before they can be widely deployed in practice. In particular, the uplink information transmission (IT) of IoT devices in WPCNs relies on their harvested energy from downlink energy transfer (ET) of the HAP. However, the IoT devices typically suffer from doubly attenuations of RF signal power over distance [3], which severely limits the network performance, e.g., the amount of energy harvested by the IoT devices and achievable rates at the HAP. Hence, solutions to enhance the downlink ET efficiency and improve the uplink transmission rates for WPCNs are urgent needs. Reducing the distances between the HAP and IoT devices is one solution to enhance EH efficiency and achieve greater transmission rates. However, this is not a viable option because IoT devices are randomly deployed in practice, and thus we may not be able to control all of them over their locations. Hence, more efficient and cost-effective solutions are required to guarantee that WPCN can be seamlessly fitted into the IoT environment with satisfying performance.

Relay cooperation is an efficient way to enhance the performances of WPCNs, which can be classified into two categories of active relaying and passive relaying. Active relaying refers to scenarios in which the communication between a transmitter and its destined receiver is assisted by a relay which forwards the user’s information to the destination via active RF transmission

[5]-[7]. However, active relaying schemes have several limitations. Considering that the relays are energy-constrained, they need to harvest sufficient energy from the RF sources and use the harvested energy to actively forward information to the receiver. Due to the higher circuit power consumption of active relays, it may take long for these relay to harvest enough energy. This reduces the IT time of the network. Besides, most active relays operate in half-duplex mode, which further shortens the effective IT time, resulting in network performance degradation. Full-duplex (FD) relays can relax this issue; however, complex self-interference (SI) cancellation techniques are needed at FD relays to ensure that SI is effectively mitigated [8].

Passive relaying exploits the idea of backscatter communication (BackCom) for assisting the source-destination communication [9]-[11]. Specifically, BackCom relay nodes do not need any RF components as they passively backscatter the source’s signal to strengthen the received signal at the receiver. Accordingly, the power consumption of BackCom relay nodes is extremely low and no dedicated time is needed for the relays’ EH [12]. Nonetheless, as no active signal generation is involved and the passive relays simply reflect the received signal from the source, passive relaying schemes suffer from poor performance.

Intelligent reflecting surface (IRS), consisting of a large number of low-cost reflecting elements, has recently emerged as a promising solution to improve the performance of wireless communication networks [13, 14]. IRS elements smartly induce phase shifts and amplitude change to the incident signals and passively reflect them such that the signals are constructively combined at the receiver. In this way, IRS can adjust the communication environment and create favorable conditions for energy and information transmission without using energy-hungry RF chains. This technology has lately been utilized for total transmit power minimization in wireless communication networks [15], ET enhancement in multiple-input single-output (MISO) systems [16], weighted sum-power and weighted sum-rate maximization in simultaneous wireless information and power transfer (SWIPT) systems [17, 18], spectrum efficiency maximization in MISO communication systems [19], secure transmit power allocation [20]

, outage probability minimization

[21], etc., and demonstrated promising results and significant performance gains as compared to the conventional wireless networks without IRS.

Having the capability of cooperating in downlink ET and uplink IT, IRS has several advantages over the conventional active and passive relaying techniques [13]. First of all, IRS is a cost-effective technology and it can be readily integrated into existing wireless communication networks without incurring high implementation costs. Furthermore, IRS is more energy- and spectrum-efficient as compared to conventional relaying methods because it consumes very low power and at the same time, helps using the limited spectrum resources more efficiently. IRS essentially works in the full-duplex mode without causing any interference and adding thermal noise, which further improves the spectral efficiency. Moreover, it is easy to increase the number of IRS elements to achieve higher performance gains. Motivated by its numerous benefits, IRS offers a promising green solution to improve the performance for WPCNs. Recently, a few research works have investigated the application of IRS for improving the performance of WPCNs [22, 23] In our preliminary work [22], we proposed a solution using the IRS as a hybrid relay to simultaneously improve energy and data communications efficiency for WPCNs, where the IRS first operates as an energy relay to assist the downlink ET from the HAP to a number of users and then works as an information relay to help the uplink IT from the users to the HAP. Zheng et al. proposed a similar idea to use the IRS as a hybrid relay, which is not only used to enhance both ET and IT efficiency but also involved in user cooperation [23]. However, [23] only considered a two-user scenario and the proposed solution is not applicable to the general scenario with multiple users. More importantly, [15]-[23] assume that the energy consumption of IRS is negligible because it does not need any active RF chains but only reflects incident signals passively. However, as mentioned in [24]-[26], the energy consumption of IRS is in fact proportional to the number of IRS elements. Hence, the energy consumption of IRS cannot be ignored since the number of IRS elements is relatively large to achieve better performance. Thus, it is challenging to power IRS and keep its hybrid-relaying operations in WPCNs.

To address the above issue effectively, we propose a self-sustainable IRS-empowered WPCN, where the IRS is equipped with EH circuit to harvest RF energy from the HAP to power its operations. Similar to the conventional wireless-powered active relays [27], time-switching (TS) and power-splitting (PS) schemes are proposed to enable 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 downlink ET to the users in the second sub-slot. In the PS scheme, the IRS harvests energy from the HAP’s signal by adjusting its amplitude reflection coefficients. To maximize the system sum-rate for both TS and PS schemes, we thus need to optimize IRS phase shift design and network resource allocation jointly with EH time and amplitude reflection coefficients of the IRS. This optimization is thus much more challenging to address than the ones studied in other works on IRS and conventional wireless-powered relays. Therefore, we propose efficient algorithms to find the optimal solutions to the sum-rate maximization problems for both TS and PS schemes. Numerical simulations endorse the effectiveness of our proposed schemes for improving the performance of WPCN and show remarkable performance gain as compared to the baseline WPCN without IRS in [3].

The main contributions of this paper are summarized as follows:

  • 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.

  • To enable energy collection and hybrid relaying functionalities at the IRS, we consider TS and PS schemes, where the IRS uses a portion of the ET phase for its own EH in the TS scheme or adjusts its amplitude reflection coefficients to harvest energy from a part of the received energy signal in the PS scheme.

  • We study the system sum-rate maximization problem for the TS scheme by jointly optimizing the IRS’s phase shift design in both ET and IT phase, time allocation for the IRS and users’ EH, time allocation for each user’ IT, and the users’ power allocation. To deal with the non-convexity of the formulated problem, we propose a two-step algorithm, where the problem is decoupled into two sub-problems. For the first sub-problem, showing that the optimal phase shift design in the IT phase is independent of other variables, we find the optimal phase shifts at the IRS in each time-slot of the IT phase by applying the semidefinite relaxation (SDR) [28] and Gaussian randomization methods. We then solve the second sub-problem to find the optimal values of other optimization variables. In particular, we obtain the IRS’s EH time in a closed-form and discuss its implications. We finally obtain the optimal solution to our problem by sequentially solving the two sub-problems.

  • 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 phase, time allocation for the EH and IT phases, power allocation at the users, and the amplitude reflection coefficients in the EH phase, using a similar two-step algorithm as for the TS scheme. We obtain the optimal amplitude reflection coefficient as a function of the EH time, from which some interesting observations are revealed.

  • To shed more light on the structure of resource allocation for our proposed schemes, we also investigate special problems of optimizing time and power allocation with random phase shifts for both TS and PS schemes.

  • Finally, we evaluate the performance of our proposed schemes via numerical results, which show that our proposed schemes can achieve over hundred times sum-rate gain as compared to the baseline WPCN protocol in [3]. In addition, we show that the PS scheme usually achieves a better performance than the TS scheme. However, the PS scheme is not applicable when the HAP’s transmit power is low and/or the channel between the HAP and IRS is weak.

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

Ii System Model

As illustrated in Fig. 1, we consider an IRS-assisted WPCN, consisting of an HAP with stable power supply, energy-constrained users (denoted by ), and an energy-constrained IRS. HAP and users have single antenna each. The IRS is composed of passive reflecting elements, which can be configured to direct the incident signals to desired directions. The IRS assists in both downlink ET from the HAP to the users and uplink IT from the users to the HAP.

The downlink channels from the HAP to , from the HAP to the IRS, and from the IRS to are denoted by , , and , respectively. The counterpart uplink channels are denoted by , , and , respectively. All channels are assumed to be quasi-static flat fading, which remain constant during one block but may change from one block to another [17]. We assume that the channel state information (CSI) of all links is perfectly known111

The CSI of all links can be precisely estimated by the advanced channel estimation techniques. Even if there exist channel estimation errors in realistic scenarios, the sum-rate derived under the perfect CSI condition can serve as an upper-bound for the system performance.

.

Fig. 1: System model for an IRS-assisted WPCN.
(a) Time-switching scheme.
(b) Power-splitting scheme.
Fig. 2: Transmission block structure.

The transmission block, normalized to one, is divided into two phases, i.e., ET phase and IT phase. In the ET phase, the HAP transfers energy to the users and IRS in the downlink. The IRS uses the harvested energy from HAP’s signals for its own EH and energy relaying to the users. In the IT phase, the users use the harvested energy to transmit data to the HAP with the assistance of the IRS. The details of the ET and IT phases are shown in Fig. 2 and elaborated in the following subsections.

Ii-a Energy Transfer Phase

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 consider TS and PS schemes which have been widely used in conventional wireless-powered relay communications [27].

Ii-A1 Time-switching scheme

For the TS scheme, the ET phase with the duration of is divided into two sub-slots, having the duration of and , respectively, which satisfy . In this scheme, the users can harvest energy over the entire ET phase. For the IRS, it will spend the first sub-slot in the ET phase for harvesting energy and the second sub-slot for improving the EH efficiency at the users. In particular, in the second sub-slot, the IRS can cooperate with the HAP by adjusting its elements’ phase shits in order to enhance the total received signal power at the users. The transmission block structure for the TS scheme is illustrated in Fig. 2 (a). Denote the transmit signal in the ET phase as

(1)

where is the transmit power and is the energy-carrying signal with .

The received signals at the IRS and in the first sub-slot are expressed as

(2)
(3)

where and denote the additive white Gaussian noises (AWGNs) at the IRS and , respectively. Note that the noise power is usually very small and ineffective for EH and can be thus neglected. Hence, the harvested energy by the IRS, denoted by , is expressed as

(4)

In the second sub-slot, the IRS assists in the downlink ET. The phase shift matrix of the IRS during is denoted by , where and are the amplitude reflection coefficient and the phase shift of the -th element, respectively. For the TS scheme, since the IRS only harvests energy during , all incident signals at the IRS during can be reflected to enhance the EH efficiency, i.e., [13]. Let . During , the received signal at for the TS scheme is given by

(5)

The harvested energy of for the TS scheme, denoted by , is thus obtained as

(6)

where the first term denotes the energy harvested by from the HAP directly during , and the second term is the harvested energy with the aid of the IRS during .

Ii-A2 Power-splitting scheme

Different from the TS scheme, the dedicated EH time is not required in the PS scheme and the IRS harvests energy from the HAP by adjusting the amplitude reflection coefficients (), as illustrated in Fig. 2 (b). To be specific, only a part of the HAP’s energy signals is reflected by the IRS and the remaining part is fed into the IRS’s EH unit for harvesting.

It is assumed that all the amplitude reflection coefficients of the IRS elements have the same value, i.e. , which is a reasonable assumption for simplifying the circuit design and reducing the circuit power consumption of the IRS. The received signal at in the ET phase for the PS scheme is thus given by

(7)

The harvested energy of the IRS and for the PS scheme are denoted as and , which are respectively given by

(8)
(9)

Ii-B Information Transmission Phase

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 as . Let be the information-carrying signal of with unit power. The transmit signal of during is then expressed as

(10)

where is ’s transmit power and satisfies

(11)

with being the circuit power consumption of . The circuit power consumption of the IRS is given by , where denotes the circuit power consumption of each reflecting element [24]-[26]. To power its operations, IRS needs to harvest sufficient energy in the ET phase. Therefore, the following constraints are hold

(12)
(13)

for TS and PS schemes, respectively. Note that in the first sub-slot of the TS scheme, it is not necessary to adjust the IRS’s phase shifts to reflect signals and its circuit power consumption for only EH during is thus negligible [25].

Denote the phase shift matrix during for the IT as , where . Note that we have set the amplitude reflection coefficients to be 1 to maximize the signal reflection in the IT phase. The received signal at the HAP from , denoted by , is thus given by

(14)

where is the AWGN at the HAP. The signal-noise-ratio (SNR) at the HAP during , denoted by , is expressed as

(15)

The achievable rate from to the HAP is then formulated as

(16)

Iii Sum-rate maximization for the TS scheme

In this section, we aim to maximize the system sum-rate by jointly optimizing the phase shift design at the IRS in both ET and IT phases, time scheduling of the network for ET and IT, and power allocation at the users. The optimization problem is formulated as

(P1)
s.t.

where , , , , and .

Iii-a Optimal Solution to P1

It is obvious that P1 is a non-convex optimization problem due to the coupling of variables in the objective function and the constraints. In the following, we propose a two-step solution to solve the sum-rate maximization problem in P1. Specifically, we split P1 into two decoupled sub-problems, and solve each sub-problem separately.

Iii-A1 Optimizing the phase shift design for IT

We first present a proposition for the optimal design of phase shifts of the IRS for the IT.

Lemma 1.

The optimal IRS phase shifts for the IT during () can be found by solving the following problem

(P2)
s.t.
Proof.

Refer to Appendix A. ∎

According to Lemma 1, we proceed to solve P2 to obtain the optimal phase shifts for the IT. Denote , where . Let and . Then, can be rewritten as . Thus, P2 is reformulated as

(P2.1)
s.t.

Note that P2.1 is non-convex and difficult to be solved directly. Hence, we introduce an auxiliary matrix

and an auxiliary vector

for mathematical manipulation, which are given by

Based on and , the objective function of P2.1 is rewritten as . P2.1 can be then expressed as

(P2.2)
s.t.

Let , where and . P2.2 is then equivalent to

(P2.3)
s.t.

where denotes the -th diagonal element of . P2.3 is still a non-convex due to the rank-one constraint in C12. However, using the semidefinite relaxation (SDR) technique [28], we can relax the rank-one constraint to obtain a convex semidefinite programming (SDP) problem [29], which can be optimally solved using convex optimization toolboxes, e.g., CVX [30]. However, the solution obtained for the relaxed version of P2.3 by CVX may not satisfy the rank-one constraint. To achieve a rank-one solution, i.e., an approximate solution with satisfying accuracy for P2.3 (P2.2), the Gaussian randomization method is employed, which can construct a rank-one solution from the solution obtained by CVX.

Denote the solution to the relaxed problem as

. The singular value decomposition (SVD) of

is expressed as , where and

are the unitary matrix and diagonal matrix, respectively. Then, the approximate solution for

P2.2, denoted by , can be constructed as follows

(17)

where is a random vector with . We generate times of random vectors and compute the corresponding objective values for P2.2. The near-optimal solution to P2.2, , is the one achieving the maximum objective function value of P2.2. The near-optimal solution to P2.1, denoted by , is finally recovered by

(18)

where represents that the first elements of are taken, denotes the -th element of [20]. According to [31], the SDR technique followed by quite large number of randomizations based on the Gaussian randomization method can guarantee at least an approximation of the maximum objective function value of P2.1.

Algorithm 1 outlines the procedure for optimizing the IRS phase shift design in the IT phase.

1:  Initialize , which is a large number of generating random vectors.
2:  Use the SDR technique and solve the relaxed version of P2.3 by CVX to obtain .
3:  Compute the SVD of and obtain and .
4:  Initialize .
5:  for  do
6:     Generate by (17).
7:     Compute the objective function of P2.2 based on and denote it by .
8:     .
9:  end for
10:   Return .
11:  Compute based on (18) given , and extract .
Algorithm 1 The Algorithm for Solving P2.

According to Lemma 1, we can further obtain the following corollary.

Corollary 1.

The optimal phase shifts for the IT will align the signal of the HAP-IRS- link with the signal of the HAP- link, i.e.,

(19)

where is a positive scalar.222From Corollary 1, we find that solving P2 is equivalent to finding the optimal value of defined in (19).

Proof.

Refer to Appendix B. ∎

Remark 1.

From Corollary 1, we can observe that for a given , the received SNR at the HAP during with the assistance of the IRS can be enhanced up to compared with that of without IRS. It should be noted that is usually proportional to the number of reflecting elements. Hence, the system sum-rate can be significantly improved with the assistance of the IRS consisting of a large number of reflecting elements.

Iii-A2 Optimizing phase shift design for ET, time scheduling, and power allocation

According to Lemma 1, P1 can be simplified as

(P3)
s.t.

where , and is obtained via Algorithm 1. P3 is still non-convex because the variables are coupled in the objective function and the constraints. We introduce auxiliary variables and . We also set . Then, we have , and . Let and , where and . Based on these new variables, the constraint C2 is recast as follows:

(20)

where

Then, P3 can be rewritten as

(P3.1)
s.t.

Due to the rank-one constraint in C16 and coupling of and in C13, P3.1 is still non-convex and difficult to be solved directly. However, it is straightforward to obtain the optimal duration of the first sub-slot in the ET phase, i.e., , as stated in the following proposition.

Proposition 1.

The optimal duration of the first sub-slot in the ET phase can be obtained as

(21)
Proof.

Refer to Appendix C. ∎

Remark 2.

From Proposition 1, we can observe that the duration of the first sub-slot in the ET phase is mainly determined by the IRS’s setting, e.g., the number of passive reflecting elements, each element’s circuit power consumption, and the channel power gain between the HAP and IRS. If each element has a higher circuit power consumption, the IRS needs more time to harvest sufficient energy, which leaves shorter time for other network operations, i.e., users’ EH with the assistance of IRS and users’ IT. However, adding the number of elements may not increase the IRS’s EH time because each element can harvest energy from the HAP individually.

We now proceed to solve P3.1 with obtained in Proposition 1. For solving P3.1, we first fix and optimize time and energy allocation in the IT phase as well as the IRS phase shift design for the ET phase. We can then find the optimal by a one-dimensional search over .

With fixed , P3.1 is reformulated as

(P3.2)
s.t.

where .

Relaxing the rank-one constraint in C16, P3.2 will be a convex optimization problem [29]. We thus use the CVX tool [30] to solve the relaxed P3.2 and obtain its optimal solution . Similar to P2.3, the obtained generally does not satisfy the rank-one constraint. Hence, the Gaussian randomization method is adopted to construct an approximate rank-one solution, which is given by

(22)

where and are the unitary matrix and diagonal matrix resulting from SVD of , and . Note that as the objective function is an increasing function of , C13 must be active at the optimal solution. Therefore, based on the generated random vectors, the energy allocation of the users is computed as

(23)

where means . Finally, is the vector achieving the maximum objective function value and is the corresponding energy allocation solution obtained from (III-A2).

The procedure for solving the sum-rate maximization problem for the TS scheme is summarized in Algorithm 2. By running Algorithm 2, we can obtain the near-optimal solution for P1. Note that the accuracy of our obtained solution is related with the number of randomizations for the Gaussian randomization method used in each sub-problem and the step-size for updating in the second sub-problem. Hence, we can achieve a solution for P1 with satisfying accuracy by increasing the number of randomizations and using a smaller step-size.

1:  Initialize and the step size . Let .
2:  Find the optimal phase shifts for the IT phase from Algorithm 1 and optimal from Proposition 1.
3:  while  do
4:     Solve the relaxed version of P3.2 with fixed and obtain its optimal solution .
5:     Compute the SVD of and obtain and .
6:     for  do
7:        Generate by (22) and find from (III-A2).
8:        Calculate the objective function value of P3.2 and denote it by .
9:     end for
10:     Set .
11:     .
12:  end while
13:  Set , , , .
14:  Set and extract from .
15:  Return , , , , and .
Algorithm 2 The Algorithm for Solving P1.

Iii-B Random phase shifts with optimized resource allocation for the TS scheme

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 [15, 32]. Letting and , we have

(24)

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

(P4)
s.t.

where . The constraint C17 is an equality at the optimal solution as we discussed earlier. Hence, we have

(25)

Substituting (25) into , we have

(26)

where , and . Proposition 1 holds here as well. Hence, P4 is modified as

(P4.1)
s.t.

It can be verified that P4.1 is a convex optimization problem [29], which can be solved by standard convex optimization techniques, e.g., Lagrange duality method. The Lagrangian of P4.1 is given by

(27)

where is the Lagrange multiplier associated with the constraint C4.

Proposition 2.

With random design of phase shifts, the optimal time scheduling for the TS scheme is given by

(28)
(29)

where is the unique solution of

(30)

and is the optimal dual variable.

Proof.

Refer to Appendix D. ∎

Using (25) and Propositions 1 and 2, the optimal energy allocation at each user can be easily obtained.

Iv Sum-rate maximization for the PS scheme

In this section, we investigate the optimal solution to the sum-rate maximization problem for the PS scheme. The problem is formulated as

(P5)
s.t.

Iv-a Optimal solution to P5

Similar to P1, P5 is a non-convex optimization problem due to the coupled variables in the objective function and the constraints. It is straightforward to observe that Lemma 1 also holds for P5. Hence, the optimal phase shifts for the IT phase can be found from Algorithm 1. Accordingly, P5 can be reformulated as

(P5.1)
s.t.
Lemma 2.

To guarantee that the value of sum-rate for the PS scheme is greater than zero, the following condition that the maximum harvested energy at the IRS from the HAP is larger than its circuit power consumption must be satisfied, i.e.,

(31)
Proof.

Refer to Appendix E. ∎

Remark 3.

From Lemma 2, we can observe that unlike the TS scheme, the PS scheme cannot always be used, i.e., if (31) is not satisfied. However, we can increase the transmit power at the HAP and/or reduce the distance between the HAP and IRS to enable the PS scheme.

In the following, we investigate P5.1 under the condition that (31) is satisfied, because otherwise, using IRS would be infeasible. Following the same steps as in Section III-A2, the sum-rate maximization problem is formulated as

(P5.2)
s.t.

where