Robust Downlink Transmission Design in IRS- Assisted Cognitive Satellite and Terrestrial Networks

Assisted Cognitive Satellite and Terrestrial Networks

Item Type Article

Authors Zhao, Bai;Lin, Min;Cheng, Ming;Wang, Jun-Bo;Cheng, Julian;Alouini, Mohamed-Slim

Citation Zhao, B., Lin, M., Cheng, M., Wang, J.-B., Cheng, J., & Alouini, M.- S. (2023). Robust Downlink Transmission Design in IRS-Assisted Cognitive Satellite and Terrestrial Networks. IEEE Journal on Selected Areas in Communications, 1–1. https://doi.org/10.1109/

jsac.2023.3288234 Eprint version Post-print

DOI 10.1109/jsac.2023.3288234

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Robust Downlink Transmission Design in IRS-Assisted Cognitive Satellite and Terrestrial

Networks

Bai Zhao, Min Lin, Member, IEEE,Ming Cheng, Member, IEEE, Jun-Bo Wang, Member, IEEE, Julian Cheng, Fellow, IEEE, and Mohamed-Slim Alouini,Fellow, IEEE

Abstract—Cognitive satellite and terrestrial network (CSTN) is considered as a promising technology to provide ubiquitous connectivity for various users within wide-coverage. This paper proposes a robust downlink transmission scheme for multi- ple intelligent reflecting surfaces (IRSs) assisted CSTN. Here, the satellite network adopts multigroup multicast transmission scheme to serve many earth stations, while the terrestrial network exploits space division multiple access and multi-IRS-enhanced non-orthogonal multiple access technology to communicate with many terrestrial users. By assuming that these two networks share the same frequency band having only the angular informa- tion based imperfect channel state information of each user, we formulate an optimization problem to minimize the total transmit power subject to the constraints of quality-of-service requirement for each user, per-antenna transmit power budgets of satellite and BS, and unit-modulus requirement for each reflecting element. To tackle this mathematically intractable problem, we then employ angular discretization together with the successive convex approx- imation method to obtain the active beamforming (BF) vectors of satellite and BS, the passive BF vector of IRS, and the power allocation coefficients. Moreover, we propose a generalized zero forcing BF and alternative optimization to obtain the suboptimal solutions of the optimization problem with low computational complexity. Finally, simulation results are given to demonstrate the effectiveness and superiority of the proposed two schemes over the benchmarks.

Index Terms—Cognitive satellite and terrestrial network, Intel- ligent reflecting surface, Non-orthogonal multiple access, Robust beamforming, Imperfect channel state information

I. INTRODUCTION

C

OMPARED with the fifth generation (5G), the sixth gener- ation (6G) networks will have more performance metrics and realize superior performance, such as achieving high

This work was supported in part by the Key International Cooperation Research Project under Grant 61720106003, in part by NUPTSF under Grant NY220111, and in part by NUPTSF under Grant NY221009.(Corresponding author: Min Lin.)

B. Zhao, M. Lin, and M. Cheng are with the College of Telecommunications and Information Engineering, Nanjing University of Posts and Telecommu- nications,Nanjing, China (e-mail:ybzb926@163.com, linmin@njupt.edu.cn, mingcheng@njupt.edu.cn).

J.-B. Wang is with the National Mobile Communications Research Laboratory, School of Information Science and Engineering, Southeast University, Nanjing 210096, China, and also with the School of Cyber Sciencer and Engineering, Southeast University, Nanjing 210096, China (e- mail:jbwang@seu.edu.cn).

J. Cheng is with the School of Engineering, University of British Columbia, Kelowna, BC V1V 1V7, Canada (e-mail:julian.cheng@ubc.ca).

M.-S. Alouini is with the Computer, Electrical, and Mathematical Science and Engineering Division, King Abdullah University of Science and Technol- ogy, Thuwal 23955, Saudi Arabia (e-mail:slim.alouini@kaust.edu.sa).

spectral efficiency particullarly by moving to higher frequency bands, green communciations, full coverage, heterogeneous services and so on. Among them, one of the most important characteristics is supporting various sevices, like content aware services and connection-centric sevices. This is because with the popularity of smart mobile devices, wireless services are not limited to the traditional connection-centric communications, but also asking for content-aware communications, like music streaming, video streaming and mobile TV. Besides, another important characteristic of the 6G future networks is to achieve global seamless coverage so that large populations, especially in isolated islands or rural areas, can enjoy ubiquitous wireless services [1], [2]. In this context, the integration of different networks, including satellite, aerial platforms and terrestrial wireless systems, has attracted significant attention [3]-[5].

Satellite networks are suitable for multicast transmissions to provide content-aware services, owing to large coverage of the satellite footprint, multibeam technology, and broadcasting capability [6], [7]. Meanwhile, the terrestrial cellular networks, which have been evolved into a considerably mature set-up, can realize ubiquitous connection-centric communications via unicast transmissions [8], [9]. Conventionally, satellite and terrestrial networks exploit different frequency band to avoid inter-network interference. Nevertheless, to fully use the scarce satellite spectrum resource and provide seamless coverage for heterogeneous services, the cognitive satellite and terrestrial network (CSTN) has been proposed [10]-[12], which is also aligned with the development goals of the ongoing 5G and upcoming 6G communication network [13], [14]. Furthermore, intelligent reflecting surface (IRS) which is also termed as reconfigurable intelligent surface, is capable of reconstructing the wireless propagation environment and has been regarded as one of the beyond Shannon approaches to further enhance the system spectral efficiency (SE) and extend the wireless coverage [15], [16].

A. Related Works

In CSTN, to realize the spectral coexistence of two different networks, beamforming (BF) and resource allocation are often employed to manage the interference. Known BF methods include robust [17], cooperative [18], multicast [19], and hybrid analog-digital [20] BF schemes. Under the circumstances that the imperfect channel state information (CSI) is achievable, the robust BF scheme was proposed to obtain the suboptimal

solutions of BF weight vectors and guarantee the reliable communication of CSTN [17]. In a cache-eabled CSTN, the cooperative BF scheme was proposed [18]. In addition, by taking nonlinear power amplifier and large-scale CSI into account, the multicast BF scheme was proposed to, respectively, calculate the amplitude and phase of desired beams [19].

Considering that the CSTN operated at the millimeter wave band, a hybrid analog-digital BF scheme was proposed [20], where the BF weight vectors of satellite and terrestrial base station (BS) were obtained through the zero-forcing algorithm.

Although the previous literatures provide a deep insight in the cognitive satellite and terrestrial networks, their investigations are limited to the orthogonal multiple access (OMA), which suffers from the inefficient spectrum utilization. Contrary to the traditional OMA, non-orthogonal multiple access (NOMA) technique is capable of allowing many intended users to access the same time-frequency resource block, which can significantly improve the system SE [4] and connectivity [21], [22]. Therefore, the application of NOMA has been investigated in satellite network [23] and CSTN [24]-[26]. For instance, by joint optimizing the BF weight vector and power allocation coefficients, the authors of [25] proposed an iteration BF algorithm to maximize the system sum rate in a NOMA-based CSTN. Further, concerning that a single eavesdropper attempts to overhear the satellite confidential signals, the authors of [26] extended the work of [24] to the scenario of secure transmission in satellite systems. However, the limitation of previous works in [24] and [25] is that their transmission scheme design is based on the perfect CSI, which can be challenging to achieve beacuse of the large feedback delay of satellite communications and mobilty of users [26]. In this context, the robust transmission design based on the imperfect CSI is still an interesting, yet challenging topic.

On the other hand, IRS is regarded as a burgeoning technology for the 6G networks, because IRS can construct a virtual line-of-sight (LoS) link to extend coverage and strengthen signals by intelligently adjusting the phase shifts of each passive reflecting element [27]. To achieve larger coverage and higher SE of the wireless networks, combined IRS and NOMA systems have become attractive [28]- [32]. For example, concerning a typical scenario for the application of IRS, in which it is neccessary to employ the IRS to supply wider- coverage services for the celluar edge users locating in the overlapped areas of two terrestrial cellulars, the IRS-assisted NOMA technique was proposed to improve the system SE [30].

Besides, according to a practical situation that most transmitter are equipped with multiple antennas, it is inevitable to carry out the joint optimization design for the active beamforming at the transimitter side and the passive beamforming at the IRS side, which could bring exhausted pressure in the terms of computation capacity. To this end, a low-complexity algorithm based on successive convex approximation (SCA) was proposed to design the active BF of BS and the passive BF of IRS [31].

Nevertheless, this algorithm’s optimality depends on the initial point, such that an feasible initial points searching method was proposed to ensure feasibility of BF optimiation [32].

Although most works have focused on the IRS-assisted NOMA transmissiona for terrestrial communications (e.g., [28]- [32]), few studies have considered the transmission design of IRS-assisted NOMA in CSTN. To the best of the authors’

knowledge, the existing literatures related to CSTN only focused on the IRS-assisted secure transmission scheme [33]

or IRS-aided OMA transmission [34]. As for the work on the combination of IRS and NOMA techniques in CSTN, only the work in [35] proposed an IRS-assisted uplink NOMA transmission scheme to maximize the sum rate of terrestrial network, whereas the optimization design of satellite network has not been discussed. Therefore, there are still lots of works about IRS-enhanced NOMA scheme that deserve careful study in CSTN.

B. Motivations and Contributions

In wireless communications, the acquisition of the perfect channel state information (CSI) is still a challenging problem.

Especially in IRS-based systems, due to the passive nature and large-scale reflecting elements of IRS, it is more difficult to accurately estimate the CSI of IRS-related links [36]. In this regard, the optimization design using angular information for IRS-assisted transmission has been considered as a more practical scheme, and received much attention recently [37], [38]. This is because the angular information can be obtained through positioning techniques. Thus, it is unnecessary to consider the impact of feedback delay. Motivated by the above observation, we propose two angular information based robust transmission schemes for an IRS-assisted CSTN to support content-aware services and connection-centric services simultaneously. The major contributions of this paper can be summarized in the following:

• We propose a downlink transmission framework for a CSTN, where the satellite offers content-aware services for many Earth stations (ESs) via multigroup multicast transmission technology. Meanwhile, the terrestrial BS provides ubiquitous connectivity for near users (NUs) through space division multiple access (SDMA), and far users (FUs) through multi-IRS-enhanced NOMA technology. In comparison with the previous works that either provide unicast services for NUs via SDMA [24], [33], or for FUs via TDMA [34], the main benefit of the framework is that it combines SDMA and IRS-assisted NOMA to facilitate the spatial diversity gain provided by multiantenna technique and extra degree-of-freedom provided by IRS, thus offering wider wireless coverage, more reliable connectivity and higher SE.

• Considering that satellite and terrestrial networks share the same radio spectrum, we formulate a joint optimization problem to minimize the total transmit power of the integrated network while satisfying the quality-of-service (QoS) requirements of all users, per-antenna transmit power limitations of satellite and terrestrial, and unit- modulus constraint of each element in IRS. To solve the mathematically intractable optimization problem, we first apply angular discretization method to deal with

the uncertainty of angular information-based CSI (AnI- CSI). Then, by jointly using the successive convex approximation (SCA),S-procedure and penalty function approaches, we propose a method to obtain the active BF weight vectors of satellite and BS, passive IRS BF weight vector and power allocation coefficients efficiently. We assume that only the angular information based imperfect CSI can be obtained, which is more practical than the related optimization design that employs perfect CSI to design the transmission scheme [24], [33], [34].

• To reduce the computational burden, we further propose a low-complexity scheme, in which the generalized zero forcing (GZF) method and alternating optimization approach are jointly employed. Specifically, we propose to adopt the GZF-based BF algorithm to attain the closed-form expression of the SAT and BS’s BF weight vectors. Then, according to the GZF-based BF weight vectors, the original optimization problem is decomposed into two simple sub-problems by using the alternating optimization approach. For the first subproblem, we employ the semidefinite program (SDP) method to obtain the transmit power of BS. For the second subproblem, we jointly use the SDP and the first-order Taylor series expansion methods to caculate the transmit power of SAT and passive BF weight vectors of IRS. The proposed low complexity scheme also exhibits a good trade-off between robustness and computational complexity.

The remainder of this paper is organized as follows. The system and channel models are presented in Section II. In Section III and Section IV, the robust transmission scheme and low-complexity scheme are, respectively, proposed in Section II and Section III. In Section V, simulation results are provided to validate the proposed schemes, and Section VI concludes this paper.

Notations: Bold lowercase and uppercase symbols represent vectors and matrices, respectively. |·| denotes the absolute value of a value;diag(·)andk·krepresents the diagonalization and Euclidean norm of a vector, respectively; (·)^T,(·)^∗,(·)^H, rank(·), Tr(·)andblkdiag(·)denote the transpose, conjugation, Hermitian transpose, rank, trace and block diagonalization of a matrix, respectively. X₋ 0 denotes that matrixXis a positive semi-definite matrix. I_N denotes an N×N identity matrix.

C^M×N denotes a M×N complex space. X⊗Y represents Kronecker product of XandY; E(·)means the expectation;

J₁(·)andJ₃(·)mean the first-order and the third-order Bessel function, respectively; log(·)means the logarithm function and hX,Yi=Tr X^HY

.C N µ,σ²

means the complex Gaussian distribution having mean µ and varianceσ².

II. SYSTEM MODEL

This paper investigates the downlink transmission in an IRS- assisted cognitive satellite-terrestrial network for supporting heterogeneous services, as depicted in Fig. 1. The satellite network termed as primary network coexists with the terrestrial network termed as secondary network by sharing the Ka band.

In particular, a geostationary orbit (GEO) satellite havingN_s

Fig. 1. System model of the considered CSTN

directional antennas generates L beams to provide content- aware services for ESs through multigroup multicast technology.

Here, the l-th cluster consists of K_l ESs. Meanwhile, the terrestrial BS, equipped with a uniform planar array (UPA) with N_b=N_b1×N_b2 antennas, serves two kinds of users, namely, NUs and FUs. Herein, the line-of-sight (LoS) link between BS and NU can be established, thus SDMA technology is employed. However, due to the existence of obstacles between BS and FUs,F IRSs withN_r=N_r1×N_r2reflecting elements are deployed, and the multi-IRS-assisted NOMA scheme is used to offer wireless service for FUs. Compared with [24] and [35], this framework can fully exploit the freedom provided by multi-antenna technology and achieve the complementary advantages of IRS and NOMA techniques, thus improving the spectral and energy efficiency of the system.

As shown in Fig. 1, by performing BF with weight vector w_s,l, the multibeam satellite transmits multicast signal z_l(t)to thelth cluster, yielding the transmit signal vector as

x_s=∑^L_l=1w_s,lz_l(t). (1) At the same time, the multi-antenna BS in the terrestrial network carrys out BF with weight vectorsw_m andw_0,f, and conveys signalss_0,k(t)ands_m(t)to thekth FU andmth NU, respectively.

The the transmit signal vector of BS can be expressed as y_r,k=∑^F_f=1w_0,f∑^K_i=1√

α_is_0,i(t) +∑^M_m=1w_ms_m(t) +n_r,k(t)

=∑^F_f=1w_0,fs₀(t) +∑^Mm=1w_ms_m(t) +n_r,k(t),∀k.

Here, since NOMA is employed, the parameter αk satisfying(2)

∑^K_k=1αk=1 represents the power coefficient of the kth FU, ands₀(k) =∑^Kk=1

√α_ks_0,k(t). Besides, the transmit signals are

normalized asEh

|z_l(t)|²i

=Eh s_0,k(t)

2i

=Eh

|s_m(t)|²i

=1.

After passing through the wireless channels, the received signal at the (l,k_l)th ES, them-th NU and thekth FU can be, respectively, expressed as

yl,k_l(t) =g^H_l,k

l∑^L_l=1w_s,lz_l(t) +h^H_l,k

l∑^F_f=1w_0,fs0(t) +h^H_l,k

l∑^M_m=1w_msm(t) +nl,k_l(t),∀l, (3)

y_b,m(t) =h^H_b,m∑^F_f=1w_0,fs₀(t)

+h^H_b,m∑^Mm=1w_ms_m(t) +n_b,m(t),∀m, (4) y_r,k(t) =∑^F_f=1h_r,k,fΦΦΦfH_fw_0,f∑^Ki=1

√ αis_0,i(t)

+∑^F_f=1h_r,k,fΦΦΦfH_f∑^M_m=1w_ms_m(t) +n_r,k(t),∀k (5) where ΦΦΦ_f =diag e^jφf,1,e^jφf,2,· · ·,e^jφf,Nr

denotes the fth IRS phase shifting matrix with φ_nr ∈[0,2π),n_r=1,· · ·,N_r. Besides, n_l,kl(t)∼C N

0,σ_l,k²

l

, n_b,m(t)∼C N 0,σ_b,m²

and n_r,k(t)∼C N 0,σ_r,k²

are the additive white Gaussian noise (AWGN) with zero mean and variance σ_l,k²

l =σ_b,m² = σ_r,k² =κT B, in which κ is the Boltzmann constant, T is the noise temperature and B is the noise bandwidth. Due to the large path loss arising from high frequency band, the satellite users are often equipped with a high-gain parabolic antenna to strengthen the received satellite signal power. However, the users in terrestrial networks are always equipped with micro antennas, making it hard to receive the satellite signals. Thus, similar to the existing work [35], the interference from satellite to NUs and FUs can be ignored.

For simplicity, we define h_r,k=

h_r,k,1;h_r,k,2;· · ·;h_r,k,F

∈ C^FNr×1, ΦΦΦ = blkdiag(ΦΦΦ1,ΦΦΦ2,· · ·,ΦΦΦF) ∈ C^FNr×FNr, H₀ = blkdiag(H₁,H₂,· · ·,H_F)∈C^FNr×FNb,H= [H₁;H₂;· · ·;H_F]∈ C^FNr×Nb andw₀= [w_0,1;w_0,2;· · ·;w_0,F]∈C^FNb×1, (5) can be re-expressed as

y_r,k(t) =h^H_r,kΦΦΦH₀w₀∑^Ki=1

√α_is_0,i(t)

+h^H_r,kΦΦΦH∑^M_m=1w_ms_m(t) +n_r,k(t),∀k. (6) By taking the path loss, satellite antenna gain and rain attenuation into consideration, we model the satellite channel vector as [23]

g_l,kl =p

G_l,klC_l,klb¹/²

l,k_lr¹/²

l,k_le^{−j2πfcdl,kl}

.

c (7)

whereC_l,kl= c

4πf_cd_l,kl2

denotes the free-space loss with cbeing the light speed, f_cbeing the carrier frequency andd_l,kl being the distance between the satellite and the (l,k_l)th ES.

Meanwhile,r_l,kl∈C^Ns×1=h r¹_l,k

l,r_l,k²

l,· · ·,r^N_l,k^s

l

iT

is the(l,k_l)th ES’s rain attenuation vector whose entries expressed in dB, namely, rⁿ_l,k^s,dB

l =20log₁₀ rⁿ_l,k^s

l

, follow the lognormal random distribution ln

rⁿ_l,k^s,dB

l

∼C N µ_s,σ_s²

for 1≤n_s≤N_s, in whichµ_s represents the lognormal location parameter andσ_s denotes the scale parameter. Besides,b_s,l∈C^Ns×1represents the satellite antenna gain vector, whose n_sth entry is given by

b_l,ns =b_max

J1(u_l,ns)

2u_l,ns +36^J3(u_l,ns)

u³_l,ns

2

(8) where b_max denotes the maximum beam gain, u_l,ns = 2.07123 sinθ_l,ns

sinθ_3dB withθ_l,ns representing the angle be- tween the lth ES location and then_sth beam center associated with the satellite, andθ_3dB denotes the half-power beamwidth of satellite. In (12),Gs,r denotes the antenna receive gain of satellite terminal, which is given by

G_l,kl[dB] =





 G^max_l,k

l ,0^◦<θ_l,kl<1^◦ 32−25 logθ_l,1^◦<θ_l,k

l<48^◦

−10 ,48^◦<θ_l,kl<180^◦

(9) where G^max_l,k

l means the maximum antenna gain of the satellite terminal, and θl,k_l is off-boresight angle of the(l,k_l)th ES.

Since the terrestrial network operates at Ka band featuring highly directional characteristic, the channel vector consists of a prominent LoS component and several single-bounce non-LoS (NLoS) components [25], which can be described by

h_b,m= r

G

θ_b,0^AoD,ϕ_b,0^AoD ρb,0a

θ_b,0^AoD,ϕ_b,0^AoD

+ q1

N∑^N_n=1 r

G

θ_b,n^AoD,ϕ_b,n^AoD

ρ_b,na

θ_b,n^AoD,ϕ_b,n^AoD

(10) where N denotes the number of NLoS paths, andθ_b,n^AoD(ϕ_b,n^AoD) denotes the elevation (azimuth) angle-of-departure (AoD) at then-th propagation path. Besides,ρb,0and ρb,n,n=1,· · ·,N represent, respectively, the complex channel gain of the LoS component and the nth NLoS component. Besides, G

θ_b,n^AoD,ϕ_b,n^AoD

stands for the element directional pattern at

θ_b,n^AoD,ϕ_b,n^AoD

. According to the recommendation of ITU [39], the element directional pattern in dB is given by

_

G

θ_b,n^AoD,ϕ_b,n^AoD

=G_max−minn G_x

θ_b,n^AoD,ϕ_b,n^AoD +G_y

θ_b,n^AoD,ϕ_b,n^AoD

,SLLo

(11) where Gmax denotes the maximum antenna gain, and SLL denotes the side-lobe level of the antenna pattern. In addition, G_x

θ_n^AoD,ϕ_b,n^AoD

andG_y

θ_b,n^AoD,ϕ_b,n^AoD

representing the pattern along with the X-axis and Y-axis, respectively, are given by

G_x

θ_b,n^AoD,ϕ_b,n^AoD

=min

12 arctan

cotθ_b,n^AoD .

cosϕ_b,n^AoD .

ϕ_x,3dB^AoD 2

,SLL

, and

G_y

θ_b,n^AoD,ϕ_b,n^AoD

=min

12 arctan

tanθ_b,n^AoDsinϕ_b,n^AoD .

ϕ_y,3dB^AoD 2

,SLL

(12) where ϕ_x,3dB^AoD and ϕ_y,3dB^AoD represent the 3 dB angle along with the X-axis and Y-axis, respectively. In (15),a

θ_b,n^AoD,ϕ_b,n^AoD

= a_x

θ_b,n^AoD,ϕ_b,n^AoD ⊗a_y

θ_b,n^AoD,ϕ_b,n^AoD

,n=0,1,· · ·,N stands for array steering vector of the nth propagation path where a_x

θ_b,n^AoD,ϕ_n^AoD

and a_y

θ_b,n^AoD,ϕ_b,n^AoD

are the relative array steering vector along with X-axis and Y-axis, respectively.

Then, we have a_x

θ_b,n^AoD,ϕ_n^AoD

=h

e^{−jkd1((Nu1−1)}/^2)sinθb,n^AoDcosϕ_b,n^AoD,· · ·, e^{jkd1((Nb1−1)}/^2)sinθb,n^AoDcosϕ_b,n^AoDiT

, a_y

θ_b,n^AoD,ϕ_b,n^AoD =h

e^{−jkd2((Nb2−1)}/^2)sinθb,n^AoDsinϕ_b,n^AoD,· · ·, e^{jkd2((Nub2−1)}/^2)sinθb,n^AoDsinϕ_b,n^AoDiT

(13)

where k=2π

λ denotes wavenumber with λ being the wavelength. Besides,d₁ andd₂ represent the element spacings along with the X-axis and the Y-axis respectively.

Next, we will discuss the channel model for the IRS_f-FUs and BS-IRS_f links. By assuming that the IRS is fixed on a building’s facade in vertical direction, the IRS link can also be modeled as

h_r,k,f=ρr,0b

θ_r,0^AoD,ϕ_r,0^AoD +q

1

N∑^N_n=1ρr,nb θ_r,n^AoD,ϕ_r,n^AoD (14) where b θ_r,n^AoD,ϕ_r,n^AoD

= b_x θ_r,n^AoD,ϕ_r,n^AoD

⊗b_z θ_r,n^AoD,ϕ_r,n^AoD and the steering vectors along with X-axis and Z-axis can be, respectively, represented by [40]

b_x θ_r,n^AoD,ϕ_r,n^AoD

=h

e^{−jkd1((Nr1−1)}/^2)sinθr,n^AoDcosϕ_r,n^AoD,· · ·, e^{jkd1((Nr1−1)}/^2)sinθr,n^AoDcosϕ_r,n^AoDiT

, b_z θ_r,n^AoD,ϕ_r,n^AoD

=h

e^{−jkd2((Nr2−1)}/^2)cosθr,n^AoD,· · ·, e^{jkd2((Nr2−1)}/^2)cosθr,n^AoDiT

.

Finally, the channel matrix of BS-IRS_f link can be written as(15) H_f =q

G θ₀^AoD,ϕ₀^AoD

ρ0b θ₀^AoA,ϕ₀^AoA

a^H ν₀^AoD +

q1 N∑^N_n=1p

G(θ_n^AoD,ϕ_n^AoD)ρ_nb θ_n^AoA,ϕ_n^AoA

a^H ν_n^AoD where θ_n^AoA ϕ_n^AoA (16)

,n=0,1,· · ·,N represents the elevation (azimuth) angle-of-arrival (AoA) of thenth propagation path.

Based on (3), the output signal-to-interference-plus-noise ratio (SINR) of ES_l,kl can be written as

γl,k_l =

g^H_l,klws,l

2

∑^L_i=1,i6=l g^H_l,

klws,i

2

+∑^F_f=1 h^H_l,

klw0,f

2

+∑^M_m=1 h^H_l,

klwm

2

+σ_l,²

kl

,∀l.

(17) Then, we can express the achievable rate (AR) of ES_l,kl as

R_l,kl=log₂ 1+γ_l,kl

=log₂ 1+

g^H_l,

klw_s,l

2

∑^L_i=1,i6=l g^H_l,

klw_s,i

2

+∑^F_f=1 h^H_l,

klw_0,f

2

+∑^M_m=1 h^H_l,

klw_m

2

+σ_l,²

kl

!

(18) Similarly, the SINR of NU_m is given by

γb,m=

h^H_b,mwm

2

∑^L_i=1,i6=m h^H_b,mw_i

2

+∑^F_f=1 h^H_b,mw0,f

2

+σ_b,m²

, ∀m. (19)

As for the IRS-enhanced downlink NOMA transmission, the FUs with worse channel condition should be decoded first, and the interference from the FUs with better channel condition are treated as noise. Without loss of generality, it is assumed that the decoding order depends on the effective channel gain of BS-IRS-FUs, and the channel condition of FU_k is better than that of FU_i, wherei<k. Therefore, the signal of FU₁is decoded first, and the signal of FU_K is decoded last. According to (6) we can express the output SINR of FU_k as

γr,k= ^αk

h^H_r,kΦΦΦH₀w₀

2

∑^K_i=k+1α_i

h^H_r,kΦΦΦH₀w₀

2+∑^M_m=1

h^H_r,kΦΦΦHw_m

2+σ_r,k²

,∀k. (20)

Using the SINR expressions in (19) and (20), we can express the AR of NU_m and FU_k as

R_b,m=log₂ 1+γb,m

=log₂ 1+

h^H_b,mwm

2

∑^L_i=1,i6=m h^H_b,mw_i

2+∑^F_f=1 h^H_b,mw_0,f

2+σ_b,m²

! , (21) R_r,k=log₂ 1+γr,k

=log₂ 1+ ^αk

h^H_r,kΦΦΦH₀w₀

2

∑^K_i=1,i6=kαi

h^H_r,kΦΦΦH₀w₀

2+∑^M_m=1

h^H_r,kΦΦΦHw_m

2+σ_r,k²

! . (22) Based on the achievable rates of expressions of ESs, NUs and FUs, we will mathematically formulate a constrained optimization problem associated with the IRS-assisted downlink transmission in a CSTN in what follows.

III. PROPOSED ROBUST TRANSMISSION SCHEME

Since the positions of the BS and IRSs are fixed, the BS- IRS links are slowly time-varying and their perfect CSI can be acquired [37]. However, considering the users’ mobility, we only exploit the angular information based imperfect CSI to implement the design of transmission scheme. Here, we denote the uncertainty angle regions of ESs, NUs and FUs by ∆i = n

θ_i^AoD∈h

θ_i,L^AoD,θ_i,U^AoDi

,ϕ_i^AoD∈h

ϕ_i,L^AoD,ϕ_i,U^AoDio ,i ∈ {(g,s),(h,s),(h,b),(h,r)} withθ_i,L^AoD(ϕ_i,L^AoD) and θ_i,U^AoD(ϕ_i,U^AoD) being the lower bound of and the upper bound ofθ_i^AoD(ϕ_i^AoD), respectively. In particular,∆g,s,∆h,s,∆h,band∆h,r, respectively, results in the uncertainty of channel vectorg_l,kl,h_l,kl,h_b,mand h_r,k.

Furthermore, since power consumption is an important factor in CSTN [26], we utilize the minmization of the total transmit power as the design criterion. With the help of (18), (21) and (22), a constrained optimization problem can be mathematically formulated as

min

ws,l,w0,f,wm,ΦΦΦ,{α_i}^K_i=1∑^L_l=1 w_s,l

2+∑^F_f=1

w_0,f

2+∑^Mm=1kw_mk²

s.t.C1 : min

∆g,s,∆_h,sR_l,kl ≥R_s,th,∀l,∀k_l C2 : min

∆_h,b

Rb,m≥Rb,th,∀m, C3 : min

∆_h,r

R_r,k≥R_r,th,∀k,

C4 :∑^L_l=1

w_s,l

ns

2

≤P_s N_s,∀n_s, C5 :∑^F_f=1

w_0,f

n_b

2

+∑^M_m=1 [w_m]_n

b

2

6P_b

N_b,∀n_b, C6 :

[ΦΦΦ]_n

rnr

=1,∀n_r∈ {1,· · ·,FN_r}, C7 :∑^K_k=1αk=1

where C1-C3 represent the QoS constraints of ESs, NUs and(23) FUs, respectively. Meanwhile, R_s,th,R_b,th and R_r,th represent the AR threshold of ESs, NUs and FUs, respectively. C4 and C5 denote the per-antenna transmit power constraint of SAT and BS, respectively. Herein, P_s and P_b mean the maximal transmit power of SAT and BS, respectively. C6 stands for the unit-modulus constraint of IRS reflecting elements. C7 guarantees that the allocated power for FUs will not exceed the provided power.

The uncertainty of channel vectors and the strictly cou- pled optimization variables make it belong to a NP-hard problem, which is challenging to be solved. To deal with this mathematically intractable problem, we will propose an SCA-based algorithm to jointly optimize the BF weight vectors of SAT and BS, IRS phase shifting matrix and power allocation coefficients, which is referred to as a robust transmission scheme. Sequentially, a low-complexity scheme will be proposed to obtain a satisfactory system performance with low-computational load.

A. Angular Discretization Method

In this paper, we assume that only imperfect angular information of any user is available, and the achieved angular information belong to the uncertain angle regions ∆i,i∈ {(g,s),(h,s),(h,b),(h,r)}. Hence, to make the constraints C1-C3 mathematically tractable, we first use the angular discretization method to uniformly select the discrete points from elevation angle set

h

θ_i,L^AoD,θ_i,U^AoDi

and azimuth angle set h

ϕ_i,L^AoD,ϕ_i,U^AoD i

, namely θ^AoD,(qi,1)

i =θ_i,L^AoD+ (q_i,1−1)∆θ_i^AoD,qi,1=1,· · ·,Qi,1, ϕ^AoD,(qi,2)

i =ϕ_i,L^AoD+ (q_i,2−1)∆ϕ_i^AoD,q_i,2=1,· · ·,Q_i,2 where Q_i,1 and Q_i,2 denote the number of discrete(24) points for elevation angle set and azimuth angle set, respectively. In addition,∆θ_i^AoD=

θ_i,U^AoD−θ_i,L^AoD .

(Q_i,1−1) and ∆ϕ_i^AoD =

ϕ_i,U^AoD−ϕ_i,L^AoD .

(Q_i,2−1). Furthermore, θ^AoD,(qi,1)

i ,i∈ {(g,s),(h,s),(h,b),(h,r)} represents the LoS elevation angular information of discrete channel vectors h(qi,1,qi,2) and g(qi,1,qi,2), while ϕ^AoD,(q_i,2)

i represents the LoS azimuth angular information. Thus, the discrete sets of channel vector associating with imperfect CSI can be expressed as

ψ_l,k^g

l=

g(qg,1,q_g,2)

l,k_l ,· · ·,g(qg,1,q_g,2)

l,k_l

, ψ^h_j =

h(q_h,1,q_h,2)

j ,· · ·,h(q_h,1,q_h,2)

j

.

(25) Next, based on the concept of convex hull that any CSI in the uncertainty region ψ_l,k^g

l andψjh,j∈ {(l,k_l),(b,m),(r,k)}

can be substituted by a convex combination of finite discrete points [24], the convex hull of ψ according to (25) is

Ω^g_l,k

l =conv

ψ_l,k^g

l

=

∑^Q_qg,1^g,1₌₁∑^Q_qg,2^g,2₌₁µ(qg,1,qg,2)

l,k_l g(qg,1,qg,2)

l,k_l

∑^Qqg,1^g,1,q^Qg,2^g,2µ(qg,1,qg,2)

l,k_l =1

, Ω^h_j=conv

ψ^h_j

=

∑

Q_h,1 q₁=1∑

Q_h,2

q₂=1µ(qh,1,qh,2)

j h(qh,1,qh,2)

j

∑

Q_h,1,Q_h,2

q_h,1,q_h,2=1µ(qh,1,qh,2)

j =1

, (26) where µ(qg,1,q_g,2)

l,k_l ≥ 0 and µ(qh,1,q_h,2)

j ≥ 0 satisfying

∑^Q_qi,1¹₌₁∑^Q_qi,2²₌₁µ(qi,1,qi,2) = 1,i ∈ {g,h} mean the weight factors of the (q_i,1,q_i,2)th discrete channel vector.

According to the concept of convex hull, we can define ˜g_l,k

l

=∆ ∑^Q_qg,1^g,1₌₁∑^Q_qg,2^g,2₌₁µ(q_g,1,q_g,2)

l,k_l g(q_g,1,q_g,2)

l,k_l and ˜h_i =^∆

∑^Q_qh,1^h,1₌₁∑^Q_qh,2^h,2₌₁µ(q_h,1,q_h,2)

i h(q_h,1,q_h,2)

i ,i ∈ {(l,k_l),(b,m),(r,k)}.

To this end, the constraint optimization problem (23) can be reformulated as

min

w_s,l,w_0,f,w_m,ΦΦΦ,{α_i}^K_i=1

∑

^Ll=1

w_s,l

2+

∑

^Ff=1

w_0,f

2+

∑

^Mm=1kw_mk²

(27a) s.t. R˜l,k_l >Rs,th,∀l,∀k_l, (27b) R˜_b,m>R_b,th,∀m, (27c) R˜r,k>Rr,th,∀k, (27d)

C4,C5,C6,C7 (27e)

where R˜_l,kl,R˜_b,m,R˜_r,k denotes the modified version of

Rl,k_l,Rb,m,Rr,k , in which n g_l,k

l,h_io ,i ∈ {(l,k_l),(b,m),(r,k)} is replaced with n

˜g_l,k

l,˜h_io ,i ∈ {(l,k_l),(b,m),(r,k)}.

B. Decoupling and Convex Approximation

To handle the strong coupling of optimization variables in (27), we will carry out several mathematical transforma- tions, so that the problem (27) can be solved. By defining H˜_r,k=^∆diag

˜h^∗_r,k

andv=^∆

e^−jφ1,1,· · ·,e^−jφ1,Nr,· · ·,e^−jφF,NrT

, we have

˜h^H_r,kΦΦΦH₀w₀

2

=

v^HH˜_r,kH₀w₀

2

,

˜h^H_r,kΦΦΦHw_m

2

=

v^HH˜_r,kHw_m

2

,∀m.

(28)

Further, applying W_s,l =^∆ w_s,lw^H_s,l, W_0,f =^∆ w_0,fw^H_0,f, W_i =^∆ w_iw^H_i ,i = 0,1,· · ·,M, V =^∆ vv^H, G˜_l,kl =^∆ ˜g_l,k

l˜g^H_l,k

l, H˜_b,m =^∆

˜h_b,m˜h^H_b,m and H˜_l,kl =^∆ ˜h_l,k

l˜h^H_l,k

l to (27), we can express the constrained optimization problem (27) as (29) at the top of next page. Here, the rank-one constraint (29h) is added to guarantee the equivalence between (27) and (29).

It can be observed that the objective function of (29) is linear and convex, but the constraints except C7 are still non-convex.

To handle these non-convex constraints one by one, we first fo- cus on addressing the most complex constraint (29c). By defin- ing A_0,k =^∆ H˜_r,kH₀W₀H

,∀k, A_m,k =^∆ H˜_r,kHW_mH

,∀m,∀k, B_k=H^HH˜^H_r,kV,∀k, after conducting some trivial mathematical operations, (29c) can be transformed into the following form

α_kTr A^H_0,kB_k

≥ 2^Rr,th−1 _K

∑

i=k+1

α_iTr A^H_0,kB_k

M

∑

m=1

Tr

A^H_m,kB_k +σ_r,k²

,∀k.

(30)

Then, following from the equation Tr A^H_i,kB_k

=a^H_i,kb_k with a_i,k=^∆ vec A_i,k

andb_k=^∆vec(B_k), we can re-written (29) as αka^H_0,kb_k≥ 2^Rr,th−1

"

K i=k+1

∑

αia^H_0,kb_k+

M m=1

∑

a^H_m,kb_k+σ_r,k²

# ,∀k.

(31) Eq. (31) can be further expressed as

αk NbNr

∑

n=1

a_0,k,nb_k,n≥ 2^Rr,th−1 _K

∑

i=k+1

αi NbNr

∑

n=1

a_0,k,nb_k,n

+ ^M∑

m=1 N_bNr

∑

n=1

a_m,k,nb_k,n+σ_r,k²

,∀k

(32)