The rapid expansion of the number of mobile devices has brought higher requirements for wireless applications, such as more extensive radio coverage, lower latency, higher data transmission rates and higher security, while the emergence of sixth-generation (6G) wireless technologies can perfectly meet such demands [1,2] describe the vision of future 6G communication and network architecture, which presents emerging technologies such as artificial intelligence, blockchain, quantum communications, integration of wireless information and energy transfer, integrated sensing and communication, intelligent reflecting surface and big data. Millimeter-wave (mmWave) communication has been widely regarded as a key technology of 6G, which owns ultra-wide bandwidth and high transfer speed advantages to meet the enormous capacity demand for 6G wireless communications [3,4]. An exhaustive survey of mmWave wireless communications for future 6G networks is described in [5], where the latest channel measurement, modeling and multiple input multiple output (MIMO) transceiver designs are summarized.

Also, energy harvesting (EH), as one of the main techniques to power future 6G wireless networks, has been introduced as an efficient method that allows wireless devices to capture the energy from external environments [6,7]. Radio frequency (RF) EH technique can enable an aggressive energy supply to the wireless devices (WDs) [8]. As an promising and emerging EH technology, wireless power communication networks (WPCN) are of great interest as they have the potential to provide wireless power to energy-constrained devices such as the internet of things (IoT) and wireless sensor networks (WSN) to ensure their operation [9,10]. Compared to the traditional battery-powered way, WPCN always provides more reliable and sustainable power replenishment. In WPCNs, the placement optimization of power and information access points is researched, where the WDs harvest the RF energy in the downlink and employ the harvested energy to transmit information to a base station in the uplink [11,12].

To improve the radio coverage and signal quality in a green way, an intelligent reflecting surface (IRS) has been proposed to reshape the signal propagation environment via passive reflecting arrays [13,14]. The transmitted signal can be reflected the receiver through the IRS and constitutes the received signal together with the signal transmitted by the direct link [15]. Thus, the reflected signal can be enhanced by properly adjusting the passive components of IRS [16,17].

Recently, some research on IRS-aided wireless powered sensor networks (WPSN) has been studied in [18–38]. For maximizing the sum throughput in an IRS-aided WPCN, the closed-form solution for the phase shifts of the wireless information transfer has been derived, where a low complexity scheme was proposed to reduce the computational complexity of the semidefinite programming (SDP) scheme [18]. Considering user cooperation (UC) in an IRS-assisted WPCN, the maximum throughput performance has been derived by jointly optimizing the power allocations, IRS phase shifts and the transmit time [19]. By obtaining a closed-form solution by the Lagrange dual method and the Karush-Kuhn-Tucker conditions in an IRS-aided WPSN, Chu et al. [20] optimally designed the bandwidth allocation and the transmit time schedule. Li et al. [21] have maximized the minimum throughput performance for IRS-aided UC in a WPCN. To simultaneously improve the performance of uplink information transmission and downlink energy transfer, Lyu et al. [22] investigated a hybrid-relaying self-sustainable IRS WPCN. Zhai et al. [23] has considered multicast transmissions in a game-theoretic way in an IRS-aided WPCN, where the closed-form optimal phase shifts have been derived. For a backscatter-assisted WPCN with an IRS, Ramezani et al. [24] have studied the maximization of total network throughput through a two-stage algorithm. In an IRS-assisted multiuser MISO WPCN, Zheng et al. [25] have maximized the weighted sum rate of all the energy-harvesting users by applying a block-structured optimization method. A novel dynamic IRS beamforming framework has been proposed to boost the sum throughput of an IRS-aided WPCN in [26]. In order to strictly guarantee device quality-of-service (QoS) requirements, Zeng et al. [27] have considered the non-linear energy harvesting in an IRS WPCN. In a multi-user IRS WPCN, Cao et al. [28] have maximized the system throughput by optimizing jointly the energy beamforming, phase shift and time allocation. To maximize the sum throughput by applying the Lagrange dual method, a closed-form solution has been derived for the optimal power allocation, transmission time slots and phase shift for IRS-aided wireless powered IoT system [29].

To tackle the issue of the uplink transmission of WDs in WPCNs depending on the harvested energy from the downlink, Wang et al. [30] proposed an energy efficient optimization method in IRS-assisted WPCN by the variable-substitution approach. In order to ensure fairness, Zeng et al. [31] studied a weighted sum throughput maximization in an active IRS-aided WPCN by applying three beamforming setups. Using the semidefinite relaxation method, Lyu et al. [32] considered a sum rate maximization for the IRS-aided WPCN by jointly iteratively optimizing the time scheduling and the phase shift. In the presence of eavesdropper, Cao et al. [33] proposed three secure modes in physical layer security of an IRS-aided WPC system, where asymptotic ergodic secrecy capacity is analyzed and closed-form expression is obtained. A new transmission policy for an IRS-assisted WPSN with certain phase shifts is proposed by maximizing the system sum throughput in [34], in which IRS can collect energy to achieve its self-sustainability. In a MISO transmission form, a practical IRS can work in EH with a power budget constraint, and then utilize the harvesting energy to sustain its operations in the signal reflecting phase [35]. Considering cognitive loT devices, two transmission policies are studied with a successive interference cancellation technique [36], where the interaction between primary and secondary networks is characterized by a Stackelberg game strategy. An IRS-aided wireless powered over-the-air computation Internet-of-Things network is investigated to propose a joint design of energy beamforming at the access point, downlink and uplink phase-shift matrices, as well as transmit power, in order to minimize the mean-squared error [37]. With the aid of an IRS, in wireless powered non-orthogonal multiple access (NOMA) IoT networks, by jointly optimizing the time allocation and phase shift matrices of wireless energy transfer and wireless information transfer, a novel resource allocation approach is proposed by maximizing the sum throughput in [38].

Integrating the IRS technology with sensor networks can achieve a self-sustaining system, specifically by embedding the IRS to the existing WPSN to reflect energy and information signals to improve the wireless energy transfer (WET) and wireless information transfer (WIT) capabilities of the WPSN. Also, to achieve a better performance with the IRS-assisted scheme, we consider a mmWave WPSN UC system in the uplink, i.e., one user not only completes his own signal transmission task, but also assists other users in completing signal transmission as a relay. Different from the existing works on the IRS-aided WPCN system, the research of transmission strategies in an IRS-assisted WPSN systems with UC in the mmWave band has not been made yet. We summarize our contributions as follows:

First, an IRS has been used to improve the performance of the WPSN system with UC in the mmWave channel, where two single-antenna wireless devices can harvest energy radiated by a hybrid access point (AP) to support the separate information transmission to the AP. Here, the IRS play a role in promoting the system performance of the downlink wireless energy harvesting and the uplink wireless information transmission by energy and information reflection for the IRS, respectively.

The rest of the paper is organized as follows. Section 2 introduces the system model of an IRS-aided WPSN system with mmWave channel. In Section 3, a weighted throughput maximization problem with user cooperation is discussed carefully, and we develop a SDR-based algorithm to solve the original problem. In Section 4, our simulation results demonstrate the effectiveness of the proposed SDR algorithm. In Section 5, the paper is summarized.

As shown in Fig. 1, we investigate an IRS-aided mmWave WPSN system, which contains a single-antenna AP and two single-antenna wireless users (WUs). Due to the constrained energy of devices, the wireless communication links (WCLs) of harvesting energy in the UL are established between AP and WUs. In the downlink (DL), information signals also can be transmitted through these links. Without loss of generality, we assume that WU₂ is used as a relay between WU₁ and AP in the meantime. An IRS with N dynamically adjustable elements is equipped to strengthen WCLs, and only one-time signal reflection by the IRS is considered owing to the path loss. The phase shift matrix of IRS is defined as θ=[θ1,…,θN] and Θ=diag(ejθ1,…,ejθn,…,ejθN). Denote g∈CN×1 and gRk∈CN×1 as the mmWave channel matrix for the channel of AP to IRS, IRS to WU_k, where k=1,2. hk, h21 are the channel between AP and WU_k, WU₁ and WU₂, k=1,2, and all channel gains are g=‖g‖2, gRk=‖gRk‖2, ζk=|hk|2, ζ21=|h21|2, respectively. Also, we assume that all channels between different transceiver and receiver are independent and identically distributed complex Gaussian distribution of 𝒞𝒩(0,κ2), where κ2 depending on the path loss. In this paper, our research focuses on finding a suitable optimal algorithm for IRS phase shifting, the power, the time slots allocation and deriving an upper bound on the weighted sum-rate performance of the system model under study. Therefore, we assume that CSI is perfect and we can get the CSI acquisition method under the IRS assisted WPSN system. In the paper, we further elaborate on the reasons for assuming a perfect CSI and how to obtain it.

For the mmWave channel of AP to IRS, IRS to WU_k, k=1,2, the antenna response can be written as

(1)a(ϕil,θil)=1N[1,…,ej2πλd(psin⁡ϕilsin⁡θil+qcos⁡θil),…,ej2πλd(N−1)sin⁡ϕilsin⁡θil+(N−1)cos⁡θil)]T,where ϕ and θ represent the incident signal’s angles of azimuth and elevation, λ and d are the wavelength and antenna spacing, respectively. Therefore, the mmWave channel matrix written as

(2)H=ρNtNr∑l=1Lαilar(ϕilr,θilr)at(ϕilt,θilt)Hwhere Nt and Nr represent the number of transmitting and receiving antennas, and L denotes the total number of paths, including one LOS direct path and (L−1) NLOS paths. ρ is the path loss and ρ=δd−γ, where δ, d, and γ are a constant, the geographical distance and the path-loss exponent, respectively. The WET and WIT processes are considered within the time period T that usually be assumed T=1s. In the WET, the AP charges WUs during time period τ0T(τ0∈(0,1]) in the DL. At the same time, part of the energy transmitted by the AP will be reflected to the WUs through the IRS, and the received energy contains two parts, i.e., AP → WU_k, AP → IRS → WU_k, k=1,2. Thus, the received information signals at WUs can be expressed as

(3)yk=p0(gRkHΘ1g+hk)x+nk,k=1,2,where p0 is the transmit power of the AP, and the Θ1=diag(a1,1,…,a1,N) denotes the coefficient matrix of reflected energy by IRS. Define a1,n≜ejθ1,n, θ1,n denotes the phase shift of the IRS and |a1,n|=1,n∈[1,N], x is the energy signals satisfying x∼𝒞𝒩(0,1), and n(t)∼𝒞𝒩(0,σk2) at WU_k, k=1,2. Hence, the harvested energy at WU_k was donated as

(4)Ek=ητ0Tp0|gRkHΘ1g+hk|2,k=1,2,where η represents the efficiency of energy harvesting during the WET phase.

The time of the WIT phase is (1−τ0)T, where WUs transmit the information to the AP and the IRS by using the harvested energy. In the WIT scenario, we assume that WU₁ and WU₂ can transmit information signals directly to the AP simultaneously, and WU₂ can act as a passive relay, reflecting some information signals from WU₁ to AP. As shown in Fig. 2, to further express this process, the time allocation of the WIT phase is divided into three sub-phases:

In the first sub-phase, i.e., τ1T, WU₁ transmits the information signal to the AP and WU₂ using the harvested energy (i.e., WU₁ → AP, WU₁ → WU2). Also, the signal of WU₁ is reflected to the AP and WU₂ via the IRS (i.e., WU₁ → IRS → AP, WU₁ → IRS → WU₂);

In the first sub-phase, the harvested energy obtained in WET is used in WET and the transmit power of WU₁ is

(5)p1=E1τ1T=ητ0p0|gR1HΘ1g+h1|2τ1.

Then, the information signal received at AP and WU₂ are expressed respectively as

(6)yAP,1=p1(gR1HΘ2g+h1)s1+nAP,

(7)y2,1=p1(gR1HΘ2gR2+h21)s1+n2,where s1 is the information signal from WU₁ with E[|s1|2]=1, Θ2=diag(a2,1,…,a2,N) denotes the IRS’s reflection coefficient matrix with |a2,n|=1,n∈[1,N], nAP and n2 denote the received noise power of the AP and WU₂ with variance σAP2 and σ22, respectively. The achievable rates in this phase are expressed as

(8)RAP,1direct=τ1log⁡(1+p1|gR1HΘ2g+h1|2σAP2),

(9)R21direct=τ1log⁡(1+p1|gR1HΘ2gR2+h21|2σ22).

In the second sub-phase, the signal received at AP given by

(10)yAP,1=p2(2)(gR2HΘ3g+h2)s¯1+nAP,where p2(2) is the transmit power of WU₂ within the subphase, Θ3=diag(a3,1,…,a3,N) denotes the reflection coefficient matrix with |a3,n|=1,n∈[1,N], s¯1 represents the transmit information symbol of WU₂ associated with WU₁’s information. Then, the achievable direct rates from WU₂ to the AP are expressed as

(11)RAP,2relay=τ2log⁡(1+p2(2)|gR2HΘ3g+h2|2σAP2).

The information signals transmitted from WU₁ to AP have four different paths, where the signal of AP in the first path is the direct link from WU1 (i.e., WU₁ → AP), the signal of AP in the second path is secondarily transmitted by WU₂ with proposed UC scheme (i.e., WU₂ ⟶ID WU1 AP), the signal of AP in the third path is the direct link from WU1 via IRS (i.e., WU1→ IRS → AP), and the signal in the fourth path is secondarily transmitted by WU₂ with UC scheme via IRS to AP (i.e., WU₂ ⟶ID WU1 IRS → AP). Therefore, the achievable rate from WU₁ to AP is given by

(12)RAp,1=min(Rdirect,1,R21direct),where

(13)Rdirect,1=RAP,1direct+RAP,2relay.

In the third sub-phase, the information signal received at AP is written as

(14)yAP,3=p2(3)(gR2HΘ3g+h2)s2+nAP,where p2(3) denotes the transmit power of WU2 in this sub-phase and the achievable rate from WU2 to AP is expressed as

(15)RAP,2=τ3log⁡(1+p2(3)|gR2HΘ3g+h2|2σAP2).

The energy obtained by WU₂ in τ0T time period should be greater than that consumed in the latter two sub-phases, so that the later work can be completed normally. So, there is

(16)τ2Tp2(2)+τ3Tp2(3)≤ητ0Tp0|gR2HΘ1g+h2|2.

According to the above transmission process, the time period τ1, τ1, τ2, and τ3 in this WPSN need to satisfy the following time constraints written as

(17)τ0+τ1+τ2+τ3≤1,{τ0,τ1,τ2,τ3}≥0.

Using the derivation and definition from the previous step, a WSRM problem is formulated based on a UC scheme and then solved by jointly optimizing the phase shift matrices, time allocation, and power allocation. First, the WSRM problem is given by

(18)(P1)maxτ,p,Θ1,Θ2,Θ3αRAP,1+(1−α)RAP,2s.t.τ2p2(2)+τ3p2(3)≤ητ0p0|gR2HΘ1g+h2|2,τ0+τ1+τ2+τ3≤1{τ0,τ1,τ2,τ3}≥0,{p2(2),p2(3)}≥0,|ai,n|=1,i=1,2,3,n∈[1,N],where α and 1−α are preference constants of RAP,1 and RAP,2, respectively. Problem (P1) is not convex due to the objective function and the constant modulus constraint in the phase shift matrices Θ1,Θ2,Θ3, which leads to the time allocation τ={τ0,τ1,τ2,τ3}, and power allocation p={p2(2),p2(3)} cannot be solved efficiently.

For the non-convexity of problem (P1), we can introduce some transformation forms for the phase shift matrices to tackle it. So, we have

(19)|gRkHΘ1g+hk|2=|a1g^Rk+hk|2,k=1,2,

(20)|gR1HΘ2gR2+h21|2=|a2g^21+h21|2,

(21)|gR2HΘ3g+h2|2=|a3g^R2+h2|2,where g^Rk=diag(gRkH)g and g^21=diag(gR1H)gR2. The optimal solution Θ2opt can be obtained by maximizing the signal-to-noise ratio (SNR), and it is equivalent to maximizing the expression to the right of equation in (20) with the constraint |a2,n|=1,n∈[1,N].By using the triangle inequality, we can obtain the upper bound of the objective function in (20) as

(22)|a2g^21+h21|≤∑i=1N|a2,i[g^21]i|+|h21|=∑i=1N|[g^21]i|+|h21|,where [g^21]i denotes the i-th element of g^21.The equal sign in the formula will hold when θ2,i=arg⁡(h21)−arg⁡([g^21]i) and a2,i=ejθ2,i,i∈[1,N], where arg(·) is the phase operator, and the optimal solutions a2opt and Θ2opt can be obtained. It should be noted that according to the asymptotical orthogonality of MIMO channels [39], the reflect path information can be abandoned in (6), i.e., we can reasonably consider the information that AP can only receive directly from WU₁ in the first sub-phase. Taking the similar operations for (21), we can also get the optimal solutions a3opt and Θ3opt.

Then, introducing auxiliary variables μ2=τ2p2(2) and μ3=τ3p2(3), another form of RAP,1direct, R21direct, RAP,2relay and RAP,2 can be expressed as

(23)RAP,1direct=τ1log⁡(1+λ1τ0|a1g^R1+h1|2τ1),

(24)R21direct=τ1log⁡(1+λ2τ0|a1g^R1+h1|2τ1),

(25)RAP,2relay=τ2log⁡(1+λ3μ2τ2),

(26)RAP,2=τ3log⁡(1+λ3μ3τ3),where λ1=ηp0ζ1/σAP2, λ2=ηp0|a2optg^21+h21|2/σ22 and λ3=|a3optg^R2+h2|2/σAP2. Thus, (16) can be reformed as

(27)μ2+μ3≤ητ0p0|a1g^R2+h2|2.

Define a¯1=[a1T1]T, g~Rk=[g^Rkhk]T,k=1,2, A1=a¯1a¯1H and GRk=g~Rkg~RkH. So from the modulus constraint of a1,n, n=1,…,N+1, we have [A1]n,n=1 and

(28)|a1g^Rk+hi|2=tr(GRkA1),

We further define B≜τ0A1⪰0 and the condition Bn,n=τ0,n=1,…,N+1 must hold. In the meantime, B needs to satisfy B⪰0 and rank(B)=1. Therefore, RAP,1direct, R21direct and the constraint in (25) can be reformed as

(29)RAP,1direct=τ1log⁡(1+λ1tr(GR1B)τ1),

(30)R21direct=τ1log⁡(1+λ2tr(GR1B)τ1),

(31)μ2+μ3≤ηp0tr(GR2B).

Accordingly, by introducing another auxiliary variable R^, RAP,1 can be recast as

(32)τ1log⁡(1+λ1tr(GR1B)τ1)+τ2log⁡(1+λ3μ2τ2)≥R^,

(33)τ1log⁡(1+λ2tr(GR1B)τ1)≥R^.

Due to the rank-one constraint causing the problem to be non-convex, so we use the SDR approach to remove this constraint, then (P1) can be reformulated as

(34)(P3)maxR^,μ2,μ3,B,ταR^+(1−α)RAP,2s.t.(29),(30),(31),τ0+τ1+τ2+τ3≤1{τ0,τ1,τ2,τ3}≥0,{μ2,μ3}≥0,B⪰0,Bn,n=τ0,n=1,…,N+1.

Here, the problem (34) is now a convex optimization problem. Thus, the global optimal solution can be obtained by CVX in Matlab [40,41], which is denoted as (R^opt,μ2opt,μ3opt,Bopt,τopt ). The computational complexity of the SDP relaxation scheme is n2N+9(2(N+1)3+(N+1)2+7(n+1)+n2), where n=O(2(N+1)2) [42].

Combined with the above derivation, we can further obtain the optimal values as

(35)A1opt=Boptτ0opt,P2(2)opt=μ2optτ2opt,P2(3)opt=μ3optτ3opt,

The rank-one solution may be obtained from the relaxed WSRM (P3). So we employ the eigenvalue decomposition for A1opt to obtain the a¯1, i.e., A1opt=ΨTΨH, where Ψ∈C(N+1)×(N+1) represents a unitary matrix, T∈C(N+1)×(N+1) represents a diagonal matrix. We can obtain a¯1=ΨT12ν, where ν is complex circularly symmetric Gaussian variables with ν∼𝒞𝒩(0,IN+1). Then, the optimal a1opt is identified by θ1,i=arg⁡([a¯1]i)and a1,i=ejθ1,i,i∈[1,N].

The simulation results are presented to evaluate the performance of the user cooperation scheme in IRS-aided mmWave WPSN. The illustration of the system is shown in Fig. 3. We assume that AP and WU_k are on the same level, and the distance between the IRS and this horizontal line is 4 meters and the distance between AP and IRS is d2=5 m. Finally, the settings of other parameters are shown in Table 1. In the specific simulation, some benchmark methods are introduced as the simulation comparison to show the superiority of the proposed method:

IRS+UC: IRS helps WUs to harvest energy through reflection in the WET phase. In the WIT phase, IRS reflects signals from WUs to AP, and WU₂ can receive and secondary transmit signals from WU₁ to AP as a relay in the UC scheme.

We first evaluate the sum throughput vs. the distance d1 between the AP and WU₁ in Fig. 4. When d1 changes from 5 to 10 m, the channel between AP and WU₁ will be influenced, which will affect the sum throughput. In this simulation, p0=30 dBm, N=25. Whether equipped with IRS or not, sum throughput using UC outperform the sum throughput without the UC scheme, which shows that the UC scheme can effectively improve sum throughput in IRS-aided mmWave WPSN. When d1=10 m, the gap of sum throughput between “IRS + UC” and “IRS + Non UC” or “Non IRS + UC” and “Non IRS + Non UC” become smaller compared with d1=5 m. This is because the efficiency of the UC scheme will be affected when the distance between wireless users is large. At the same time, it can be known that the improvement of sum throughput by only using the UC scheme is greater than that only equipping with IRS.

In order to observe the sum throughput improvement with increasing p0, we second compare the performance of some different schemes when p0 increases from 30 to 40 dBm in Fig. 5. We set d1=8 m, N=25 here. It can be seen from the figure that the sum throughput of four schemes increases with the increasing p0, but the “IRS + UC” scheme outperforms that of other schemes. We can realize that the UC scheme can improve sum throughput significantly even if the WUs get less energy in WEL. This also provides a feasible way to make up for the lack of energy by introducing the UC scheme into WPSN.

Finally, the impact of the elements number N of IRS on sum throughput is shown in Fig. 6. With the increasing N, the sum throughput obtained by schemes with IRS is gradually increased due to the array gain and the sum throughput of schemes without IRS not changing. When about N<50, “IRS + UC” and “Non IRS + UC” outperforms “IRS + Non UC” and “Non IRS + Non UC”. Then “IRS + Non UC” is greater than “Non IRS + UC” as N continues to grow, and this is because the performance gain of increasing N to that only equipping with IRS is greater than that only using UC scheme in mmWave WPSN.

In this paper, we proposed a UC scheme for IRS-aided mmWave WPSN. The WSRM problem is formulated to maximize the weighted sum throughout by jointly optimizing the phase shift matrix, time slots, and power allocation. Some efficient variable substitutions and SDP relaxation are introduced to convert the original non-convex problem into a convex problem that is easy to tackle. Finally, numerical results confirm that the weighted sum throughput of our proposed UC scheme is greater than that of non UC whether equipped with IRS or not.