Uplink Multiple Access with Semi-Grant-Free Transmission in Integrated Satellite-Aerial-Terrestrial Networks

in Integrated Satellite-Aerial-Terrestrial Networks

Item Type Article

Authors Kong, Huaicong;Lin, Min;Han, Lve;Zhu, Wei-Ping;Ding, Zhiguo;Alouini, Mohamed-Slim

Citation Kong, H., Lin, M., Han, L., Zhu, W.-P., Ding, Z., & Alouini, M.-S. (2023). Uplink Multiple Access with Semi-Grant-Free

Transmission in Integrated Satellite-Aerial-Terrestrial Networks.

IEEE Journal on Selected Areas in Communications, 1–1. https://

doi.org/10.1109/jsac.2023.3273707 Eprint version Post-print

DOI 10.1109/jsac.2023.3273707

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Link to Item http://hdl.handle.net/10754/691639

Uplink Multiple Access with Semi-Grant-Free Transmission in Integrated

Satellite-Aerial-Terrestrial Networks

Huaicong Kong, Min Lin,Member, IEEE,Lve Han, Wei-Ping Zhu, Senior Member, IEEE, Zhiguo Ding, Fellow, IEEE, and Mohamed-Slim Alouini, Fellow, IEEE

Abstract—This paper investigates a semi-grant-free (SGF) based transmission strategy to provide a flexible connectivity for various kinds of users in an integrated satellite-aerial-terrestrial network (ISATN). Herein, a high-altitude platform (HAP) termed as a grant-based user (GBU), which serves multiple mobile terminals (MTs) through space division multiple access (SDMA), wants to access a satellite network with multiple earth stations (ESs) termed as grant-free users (GFUs) simultaneously via non- orthogonal multiple access (NOMA) assisted SGF. To this end, we first propose two SGF-based uplink transmission schemes for both perfect channel state information (CSI) and imperfect CSI cases. When perfect CSI is available, a zero-forcing based beamforming (BF) scheme is used in HAP network while an adaptive transmit power allocation (ATPA) approach is adopted for SGF transmission. When only imperfect CSI is available, BF scheme employing the derived channel correlation matrix of HAP- MT link is proposed to achieve SDMA, and a novel ATPA strategy with rate probability constraint is proposed to guarantee quality- of-service of the GBU. Next, we derive the closed-form throughput expressions to evaluate the performance of the considered ISATN with the proposed two SGF-based schemes. Finally, computer simulations are conducted to validate the theoretical performance analysis and show the superiority of the proposed schemes over the related works. Moreover, our numerical results not only demonstrate a satisfactory performance of the proposed SGF- based scheme using imperfect CSI, but also reveal the impact of CSI errors on the system performance.

Index Terms—Channel state information, integrated satellite- aerial-terrestrial network, non-orthogonal multiple access, semi- grant-free transmission

I. INTRODUCTION

N

OWADAYS, with the rollout of the commercial fifth generation (5G) systems around the world, research on

This work was supported by the Key International Cooperation Research Project under Grant 61720106003, NUPTSF under Grant NY220111, and the Postgraduate Research & Practice Innovation Program of Jiangsu Province under Grant KYCX21 0739. (Corresponding author: Min Lin.)

Huaicong Kong, Min Lin and Lve Han are with the College of Telecom- munications and Information Engineering, Nanjing University of Posts and Telecommunications, Nanjing 210003, China (e-mail: khc dream@163.com, linmin@njupt.edu.cn, hanlve@njupt.edu.cn).

Wei-Ping Zhu is with the Department of Electrical and Computer Engi- neering, Concordia University, Montreal, QC H3G 1M8, Canada (e-mail:

weiping@ece.concordia.ca).

Zhiguo Ding is with the Department of Electrical Engineering and Computer Science, Khalifa University, Abu Dhabi, UAE, and also with the Department of Electrical and Electronic Engineering, University of Manchester, Manchester, UK. (e-mail: zhiguo.ding@manchester.ac.uk).

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

the sixth generation (6G) systems has begun in full swing for providing full-dimensional wireless coverage and ubiquitous connectivity with guaranteed quality-of-service (QoS) through terrestrial and non-terrestrial networks (NTN) [1]-[3]. In the NTN, satellite communication (SatCom) network has attracted much attention since it can provide reliable user access in sparsely populated areas [4], [5]. Recently, the use of high altitude platform (HAP) systems emerges as a new wireless access platform for 6G systems, which has great potential to achieve more reliable and robust communications due to its low latency and flexible deployment [6]. To maximize the served users’ throughput while satisfying the backhaul capacity constraint in the HAP systems, SatCom networks that can offer high throughput and reliable connectivity have been envisioned to serve as backbone to connect with HAP communication networks, composing a promising paradigm called as an integrated satellite-aerial-terrestrial network (ISATN), which is essential for Internet access in remote areas and for disaster recovery and maritime communications [7].

To satisfy the increasing demands for ubiquitous connectivity and flexible services, various multiple access technologies, such as time division multiple access (TDMA) [8], [9] and space division multiple access (SDMA) [10] have been considered for ISATN. More recently, for the capability of serving different users in the same resource block, non-orthogonal multiple access (NOMA) has been introduced in ISATN to provide higher spectral efficiency [11]. However, in the ISATN, various kinds of users are connected to satellite via different transmission strategies. Namely, the users are commonly served by the aerial platform via grant-based protocol [8]-[11], while the satellite can support direct access to advanced users, such as earth stations in a grant-free transmission manner [12]. In light of this, this paper focuses on the application of NOMA assisted semi-grant-free (SGF) transmission to the ISATN, which can achieve more spectrally efficient and meanwhile enhance system throughput.

A. Related Works

In uplink SatCom systems, the grant-free random access (RA) is becoming a prevailing scheme for its low complexity, low latency, and low signaling overhead [13], [14]. Furthermore, to achieve better performance, different techniques, such as uplink power control [15], resource allocation [16], and interference cancellation [17] have been integrated with the conventional RA schemes. More recently, the RA protocol is employed to

provide heterogeneous services for grant-free users (GFUs) in a two-hop satellite and terrestrial network [18]. Although these RA protocols have improved system performance, they lack the reliability for frequent collisions in grant-free transmissions. To alleviate the collisions, integrating NOMA with grant-free RA systems has been proposed as a potential solution to achieve reliable connectivity in the same resource with different power levels or codebooks. In this regard, a scalable massive access satellite network has been developed by combining NOMA technology with RA scheme, which includes intelligent hybrid NOMA transmission, grant-free superimposed pilot NOMA random access and multi-slot pilot allocation RA scheme [19]. Based on the NOMA assisted satellite RA system, both successive interference cancellation (SIC) and joint decoding (JD) were introduced to recover signal of each user, and it was demonstrated that the RA scheme under SIC and JD outperforms the orthogonal multiple access (OMA) scheme [20]. Furthermore, uplink NOMA based RA schemes have also been investigated for ISATN systems [21] and for satellite terrestrial relay networks [22] to serve massive numbers of users.

Many works have focused on enhancing the successful access probability and throughput by optimizing the power levels in NOMA assisted satellite RA systems, as mentioned above.

However, only few researchers have dedicated themselves to finding approaches to meet reliable access requirements for massive numbers of users under limited spectral resources.

Since the number of connected users in RA systems may be overwhelming for NOMA to decode each user’s signal successfully, the system performance especially reliability could correspondingly degrade [23]. To solve this, many researches on terrestrial networks have explored possible strategies including coordinated transmissions, massive multiple-input multiple- output parallel interference cancellation, and NOMA assisted SGF transmission schemes. In SGF transmission, it encour- ages grant-based user (GBU) to share its spectrum resource with GFUs, thus giving GFUs access to base station in an opportunistic fashion, in which way both connectivity and spectral efficiency can be improved [24]. Different from grant- free transmission, the user collision event can be effectively managed in SGF transmission. Meanwhile, relative to the grant- based transmission [25], higher reliability for each user can be achieved without bringing additional signaling overhead.

Hence, the NOMA assisted SGF transmission is considered as a good trade-off between grant-free and grant-based schemes.

The concept of SGF transmission was firstly introduced in [24], where two NOMA assisted SGF strategies, namely open-loop and distributed contention control protocols with fixed power allocation, were proposed to restrict the number of admitted GFUs and ensure the performance of the GBU without degradation from the admission of the GFUs by designing proper channel gain thresholds. To further maintain the QoS of GBU in uplink NOMA assisted SGF transmission, an adaptive power allocation scheme for the GBU was proposed in [26]. Afterwards, a new NOMA assisted SGF transmission scheme [27] leveraging hybrid SIC was proposed

to guarantee QoS of the GBU while avoiding outage error floors effectively. Furthermore, a more general SGF transmission scenario with multiple GBUs and GFUs was developed in [28], where cognitive underlay principle was exploited to manage interference from two admitted GFUs to an opportunistically selected GBU. In light of the fact that the location distribution of GFUs to base station was not considered in [24]-[28], the stochastic geometry was employed to investigate the effect of the random user locations on the NOMA assisted SGF system performance [29], [30]. Different from the threshold protocols given in [24], a dynamic threshold protocol was proposed for randomly admitted GFUs from the perspective of outage performance [29] and ergodic rate [30], respectively. More recently, advanced technologies, such as intelligent reflecting surfaces (IRS) and deep reinforcement learning, have been utilized to further improve the performance of the GFUs in IRS assisted SGF-NOMA transmission [31].

B. Motivations and Contributions

Although the above-mentioned works for terrestrial networks [24], [26]-[31] have shed light on the characteristic and performance of NOMA assisted SGF transmission, they are only applicable to the ideal scenario with perfect channel state information (CSI) assumption. Since the acquisition of perfect CSI is still a challenging problem in wireless systems [10], [11], it is more favorable to exploit the imperfect CSI to implement the SGF transmission. Meanwhile, the ISATN is playing a more and more important role for future wireless systems to meet the rapid growth of wireless devices and huge data traffic.

However, the application of SGF-based transmission in ISATN is an open research topic and such kind of works have not been reported until now.

Motivated by this observation, we propose two SGF-based transmission schemes based on perfect CSI and imperfect CSI to provide a flexible connectivity for various kinds of users with high spectral efficiency. Furthermore, the performance of the ISATN with the proposed transmission schemes is also evaluated through deriving the closed-form system throughput expressions. Specifically, the main contributions of this paper are listed as follows:

• We present a novel uplink multiple access framework for an ISATN, where a HAP termed as GBU with higher access priority wants to access a satellite network with multiple earth stations (ESs) termed as GFUs simultane- ously via NOMA assisted SGF protocol. Here, the HAP equipped with a uniform concentric ring array (UCRA) functions as an aerial base station (BS) to serve multiple mobile terminals (MTs) through SDMA. Relative to the previous works that investigate SGF schemes in terrestrial networks [24], [26]-[31] or uplink access in the ISATN [9]-[11], the proposed framework is more general and has the advantage of providing flexible connectivity with high spectral efficiency and enhancing system throughput with satisfied user experience.

• Based on the presented framework, we propose two SGF- based uplink transmission schemes, where either perfect

CSI or imperfect CSI is available. For perfect CSI case, a zero-forcing (ZF) based beamforming (BF) is used in HAP network, and an adaptive transmit power allocation (ATPA) strategy is exploited for the GBU in SGF transmission. For imperfect CSI case, the channel correlation matrix (CCM) of the HAP-MT link with HAP employing a UCRA is first derived, based on which the BF scheme aiming at maximizing minimum average signal-to-interference-plus- noise ratio (SINR) is proposed to implement SDMA. Then, a novel ATPA strategy with rate probability constraint is proposed to guarantee the QoS of the GBU, and thus achieving reliable access for satellite network. To the best of our knowledge, this work develops for the first time such SGF-based schemes with two kinds of CSI in ISATN.

• By assuming that satellite links experience shadowed- Rician (SR) fading while the HAP links follow Nakagami- mdistribution, we derive the closed-form system through- put expressions of the considered ISATN to evaluate the performance of the proposed two SGF-based transmission schemes, which are verified through Monte Carlo simula- tions. It is worth-mentioning that the derived expressions include the existing SGF related works [24], [26]-[30] as special cases, where only the ideal case of perfect CSI and channels undergoing Rayleigh fading are considered.

Furthermore, numerical results reveal the impact of CSI errors on the throughput of the considered ISATN, and demonstrate a satisfying performance achieved by the proposed SGF-based scheme with CSI imperfection.

C. Organization and Notations

The remainder of this paper is organized as follows. The system model of the considered ISATN is described in Section II. Section III proposes two SGF-based transmission schemes by using perfect CSI and imperfect CSI, respectively. In Section IV, the total system throughput performance achieved by the proposed schemes are evaluated through closed-form expressions. Numerical results are provided in Section V, and finally Section VI concludes this paper.

Notations: Matrices and vectors are denoted by bold upper- case and lower-case letter, respectively.E[·],|·|,I_N andC^M×N stand for the expectation, the absolute value, the identityN×N matrix, and the complex space of dimensionM×N, respectively.

A^H and A⁻¹ indicate Hermitian transpose and inverse of matrixArespectively.(a)_n and

a b

are, respectively, the Pochhammer symbol and binomial coefficient. Γ(·), Γ(·,·) and exp(·) are, respectively, the Gamma function, upper incomplete Gamma function, and exponential function. In addition, Nakagami(·) andU[a,b] denotes the Nakagami-m and uniform distribution.

II. SYSTEMMODEL

The considered system model of an ISATN is illustrated in Fig. 1, where many ESs and one HAP within the coverage of a geosynchronous Earth orbit (GEO) satellite (SAT) want to access the satellite system. Unlike the existing works [9]- [11], here the NOMA assisted SGF transmission scheme is

Fig. 1. System model of the considered ISATN.

proposed to provide a flexible connectivity for various users and enhance the system throughput. In particular, the HAP is termed as GBU and expected to be served with high priority, while the multiple ESs termed as GFUs are enabled to share the spectrum resource of the GBU via uplink NOMA. Moreover, in the considered ISATN, the HAP employing a UCRA with N-antenna array is used as an aerial BS to serve multiple MTs simultaneously by SDMA, which is also different from the previous works that used the uniform linear array [8], [11] or uniform planar array [10] to implement multi-user transmission.

The overall uplink transmission can be divided into two communication links, namely MTs-to-HAP link and HAP/ESs- to-SAT link. For the MTs-to-HAP link,K single-antenna MTs send their signalss_k(t)with transmit power p_k to HAP over sub-6 GHz links. Since each ES configured with a high-gain parabolic antenna generates a directional beam to the SAT at Ka band, the interferences to HAP can be neglected. Hence, after employing receive BF with vectors w_k∈C^N×1, the received signal of thek-th MT at the HAP can be expressed as

y_k(t) =√

p_kw^H_kh_ks_k(t) + ∑^K

j=1,j6=k

√p_jw^H_kh_js_j(t)

+w^H_kn(t),k=1,· · ·,K,

(1) wheren(t)denotes the additive white Gaussian noise (AWGN) with variance σ_h²I with variance σ_h²=κBT in which κ is the Boltzmann constant, B the bandwidth and T the noise temperature [10], [11], whileh_k∈C^N×1denotes the channel vector of the link between thek-th MT and HAP. Considering UCRA deployed at the HAP and the path loss, the channel vectorh_k can be modeled as

h_k=`_kρ_ka(θ_k,ϕ_k),k=1,· · ·,K, (2) where ρ_k is the complex channel gain satisfying

|ρ_k| ∼ Nakagami(m_k,Ω_k) [10], [11], and `_k[dB] =

1

2(20 lgλ−10α_klgd_k−20 lg 4π) with α_k being the path loss exponent corresponding to the environment, λ the wavelength, and d_k=q

x²_k+y²_k+z²_k the distance between the k-th MT and HAP with(x_k,y_k,z_k)being the three-dimensional coordinate, as depicted in Fig. 1. Meanwhile, as shown in Fig.

Fig. 2. Geometrical model of the UCRA at the HAP.

2, a(θ_k,ϕk) in (2) denotes the array steering vector of the UCRA, whose elements for the q-th ring are given by [32]

[a(θ_k,ϕ_k)]_q=

"

e^{j2πrqλ cosϕksinθk},· · ·,e^j2π

rq λcos

ϕ_k−^2π(Nq−1)

Nq

sinθ_k#

, (3) where r_q is the radius of the q-th ring, while θk = arctan

z_k.q

x²_k+y²_k

and ϕk =arccos x_k.q

x²_k+y²_k are the elevation and azimuth angles for the k-th MT, respectively.

According to (1), the output SINR of the k-th MT at the HAP is given by

γ_k= p_k w^H_kh_k

2

K

∑

j=1,j6=k

p_j w^H_kh_j

2+σ_h²w^H_kw_k

, (4)

and its corresponding achievable rate can be expressed as R_k=log₂(1+γk),k=1,· · ·,K. (5) For the HAP/ESs-to-SAT high-frequency bandwidth link, the HAP, termed GBU, maintains a backbone connection to the SAT via a directional antenna so that the received signals from MTs at the HAP can be forwarded to the SAT with high QoS requirement. Meanwhile, we assume that M ESs served as GFUs are admitted to the GBU’s channel opportunistically and transmit signals simultaneously with the GBU based on uplink NOMA. Unlike the traditional uplink NOMA systems [11], [23], the admission of the GFUs should ensure the GBU performance in NOMA assisted SGF transmission. To this end, prior to multiple access, the SAT first broadcasts the predefined thresholdτto each GFU, and then GFUs whose channel gains p_i|h_i|² are above the threshold τ will access the SAT in an opportunistic manner. The set of the number of the admitted GFUs can be expressed as D_i=n

i:p_i|h_i|²≥τ,0≤i≤Mo . Furthermore, we assume that only one GFU with the largest channel gain is scheduled for SGF implementation, which is the same as the distributed contention mechanism employed in [24]. Based on the above assumptions, the GBU transmits a composite signal x_b(t), which is multiplexed based on the decoded signals at the HAP, and the admitted GFU transmits its signalx_i(t)to the SAT concurrently. The received signal at the SAT can be written as

y(t) =√

p_bh_bx_b(t) +√

p_ih_ix_i(t) +n_s(t),i∈D_i, (6)

where p_b and p_i are the transmit power of the GBU and admitted GFU. Besides, n_s(t) denotes the AWGN while h_b andh_i are the channel responses of the SAT link for the GBU and admitted GFU, respectively. By considering the antenna gain, path loss and small-scale fading [11], [33], we express the channel response of the SAT link as

h_v=`<sub>v</sub>g<sub>v</sub>,v,b,i, (7) whereg<sub>v</sub>denotes the small-scale fading obeying SR distribution with parametersm<sub>v</sub>,Ω<sub>v</sub>andb<sub>v</sub>, while`_v includes antenna gain and path loss denoted by`_v=λp

G^r_vG^t_v/4πd_v with λ being the wavelength and dv the distance from the access user to the SAT. Here,G^r_v denotes the SAT beam pattern that can be approximated as [34]

G^r_v=G^r_v,max

J₁(u_v)

2u_v +36J₃(u_v) u³_v

2

, (8)

whereG^r_v,maxis the maximal SAT beam gain,J_n(·)the first-kind Bessel function of order n, andu_v=2.07123 sinθ_v/sinθ_{3 dB} with θ_v and θ_{3 dB} being respectively the angle between the access user and the beam center, and the 3 dB angle of the main beam. In addition, G^t_v denotes the off-boresight antenna pattern of the access user, which is given by [34]

G^t_v=











G^t_v,max−2.5×10⁻³ D_vθ¯v/λ2

,0<θ¯v<θˆ, 2+15 lg(D_v/λ),θˆ≤θ¯_v<θ,˜

32−25 lg ¯θv,θ˜≤θ¯v<48^◦,

−10,48^◦≤θ¯v≤180^◦,

(9)

whereG^t_v,max denotes the maximal gain of the access user and D_v is the antenna diameter. Besides, ¯θv is the off-boresight angle, ˆθ=_D²⁰

v

q

G^max_t − 2+15 lg^Dv

λ

and ˜θ=15.82 ^Dv

λ

^−0.6 . Similar to those related works on NOMA systems [11], [23], we also assume that the SIC decoding order is based on channel quality, i.e., a user with stronger channel quality is decoded first. Recalling that each GFU is equipped with a high-gain parabolic antenna, the GBU is assumed to have the lower channel gain, thus the signal of the GBU is decoded at the last at the SAT, indicating that the data rate of the GBU is the same as the OMA system without GFUs. As such, the output SINR of the GFU is given by

γi= p_i|h_i|²

p_b|h_b|²+σ_s², (10) and the output signal-to-noise ratio (SNR) of the GBU is written as

γb= p_b|h_b|²

σ_s² . (11)

Therefore, the achievable rate of the GFU and GBU can be, respectively, expressed as

R_i=log₂(1+γi),i∈D_i, (12) and

R_b=log₂(1+γb). (13)

III. PROPOSEDTRANSMISSIONSCHEMES

In this section, we first consider the ideal case when perfect CSI is available and propose an SGF-based transmission scheme to guarantee stable and reliable access for ESs and MTs in the considered ISATN. Then, we propose another SGF- based transmission scheme that only exploits imperfect CSI to lift the requirement of perfect CSI.

A. SGF-based transmission scheme with perfect CSI

As shown in Fig.1, to implement SDMA in the MTs-to- HAP link, we aim at maximizing the minimum SINR through solving the following BF optimization problem

w₁max,···,w_Kmin

k γ_k= p_k

w^H_kh_k

2

K

∑

j=1,j6=k

pj

w^H_kh_j

2+σ_h²

s.t. w^H_kw_k=1,∀k=1,· · ·,K.

(14)

Since receive BF weight vectors w_k are independent to each other, the optimization problem can be decomposed into K independent sub-problems as

maxw_k

p_k w^H_kh_k

2

K

∑

j=1,j6=k

p_j w^H_kh_j

2+σ_h²

s.t. w^H_kw_k=1.

(15)

Similar to [9], we here assume the availability of perfect CSI for each MT at the HAP. In this regard, the ZF method can be enabled to eliminate the inter-user interferences, namely

K

∑

j=1,j6=k

p_j w^H_kh_j

2=0. By employing ZF, the optimization problem can be simplified as follows:

maxw_k

w^H_kh_k

2

s.t.

w^H_kh_j

2=0, w^H_kw_k=1.

(16)

Then, according to (2), the optimization problem equivalent to (16) can be further expressed as

maxw_k

w^H_ka(θ_k,ϕ_k)

2

s.t. H^H_−kw_k=0_(K−1)×1, w^H_kw_k=1,

(17)

where H_−k = [a(θ₁,ϕ1),· · ·,a(θ_k−1,ϕk−1),a(θ_k+1,ϕk+1),

· · ·,a(θ_K,ϕ_K)]. With the aid of the projection theory, the closed-form expression of BF vectorw_k can be obtained as

w_k= (I_N−P_k)a(θ_k,ϕ_k)

k(I_N−P_k)a(θ_k,ϕk)k, (18) where P_k = H_−k H^H_−kH_−k−1

H^H_−k denotes the orthogonal projection matrix ontoH_−k.

Concerning the uplink access in the HAP/ESs-to-SAT link, SGF transmission scheme is proposed for the satellite system, as described in Section II. To implement NOMA assisted SGF transmission with guaranteed QoS of the GBU, the achievable

rate of the GBU should meet the following constraint, namely, R_b=log₂(1+γb)≥R^∗_b, (19) whereR^∗_bdenotes the target rate at the GBU and is determined by the available bandwidth of the HAP-to-SAT link and required total channel capacity in the MTs-to-HAP link. Similar to most existing SGF works [24], [26]-[31], we assume the availability of perfect CSI at the GBU. In this case, by using (11) and evaluating (19), the transmit power at the GBU is given by

p_b=





 ε_b^∗σ_s²

|h_b|² ,P_max|h_b|²≥ε_b^∗σ_s²,

0 , else,

(20)

where ε_b^∗=2^R∗b−1 denotes the target SINR and P_max is the maximal transmit power at the GBU. Accordingly, the output SINR of the admitted GFU in (10) can be re-expressed as

γ_i=











p_i|h_i|²/σ_s²

ε_b^∗+1 ,P_max|h_b|²≥ε_b^∗σ_s², p_i|h_i|²

σ_s² , else.

(21)

It can be seen from (20) that if the GBU’s channel gain is strong, i.e., Pmax|h_b|²≥ε_b^∗σ_s², the GBU will achieve a fixed data rate with guaranteed QoS. Besides, by increasing pi and fixing pmax in (21), the probability for event, log₂(1+γi)<R^∗_i with R^∗_i being the target rate of the GFU, goes to zero since the interference from the GBU to the GFUε_b^∗ can be regarded as AWGN. Thus, the outage floor of the admitted GFU can be avoided.

It is noted that although perfect CSI is very difficult to obtain in practice, the above proposed scheme can be regarded as a theoretical upper bound for evaluating the performance of the practical SGF-based transmission scheme, which employs imperfect CSI and will be presented in the next subsection.

B. SGF-based transmission scheme with imperfect CSI Unlike most existing related works, in this subsection we exploit angular information-based CSI to conduct BF design so that SDMA can be implemented with low complexity in HAP network. Further, due to the disturbance of aerial platform and user mobility, we assume that only the roughly estimated angle of the MT is available. As such, the estimated angle in elevation and azimuth can be modeled as

θ˜k=θk+θ_k^e,ϕ˜k=ϕk+ϕ_k^e, (22) where θ_k^e andϕ_k^e denote the corresponding estimation errors, which follow the commonly used uniform distribution [6], namely, θ_k^e∼U[−∆θ_k,∆θ_k],ϕ_k^e∼U[−∆ϕ_k,∆ϕk]. Instead of maximizing the minimum SINR of MTs as (14), we here consider the scenario that only the estimated angle of the MT is available at the HAP, and adopt the average SINR as the design criterion. Firstly, according to (4) and using Mullen’s inequality [10], the average SINR of the k-th MT is approximated as

E[γ_k]≈

p_kE h

w^H_kh_k

2i

K

∑

j=1,j6=k

p_jEh w^H_kh_j

2i +σ_h²

. (23)

Then, the optimization problem to maximize the average SINR of the k-th MT is constructed as

maxw_k

p_kw^H_kR_kw_k

K

∑

j=1,j6=k

p_jw^H_kR_jw_k+σ_h² s.t.w^H_kw_k=1,

(24)

whereR_k=E h_kh^H_k

denotes the CCM of thek-th MT-to-HAP link. By using the estimated angle given in (22), we can derive the analytical expression of the CCM R_k through Lemma 1.

Lemma 1. The CCMR_k using the imperfect angular informa- tion of the k-th MT-to-HAP link is given by

R_k=E h_kh^H_k

=`²_kΩkR˜_k, (25) where R˜_k is the spatial correlation matrix whose elements approximated as

ωk(p,q,a,b)≈e⁻jz(p,q,a,b)sinθksinϕ0 k(p,q,a,b)

×sinc

z(p,q,a,b)∆ϕ_ksinθ_kcosϕ_k⁰(p,q,a,b)

×sinc

z(p,q,a,b)∆θ_kcosθ_ksinϕ_k⁰(p,q,a,b) ,

(26)

where p and q denote ring indexes of the steering vectors while a and b are element indexes.

Proof: See Appendix A.

By using the derived expression of the CCMR_kand resorting to the Rayleigh-Ritz theorem [10], the optimal BF weight vectors for (24) are given by

w_k=u_max





K j=1,

∑

j6=k

p_jR_j+σ_h²I_N

!−1

R_k



, (27) where u_max(·)denotes the eigenvector corresponding to the maximal eigenvalue.

Next, we focus on uplink SGF transmission in the satellite system. Apparently, due to the unavailability of CSI, it is no longer achievable to meet the instantaneous QoS constraint (19) under the ATPA strategy proposed in the previous subsection.

To mitigate this situation, we will propose a novel ATPA scheme to implement SGF transmission below. By taking the impact of the estimation errors and limited feedback burden into account, the estimated channel response of the GBU-to-SAT link can be modeled as [35]

hˆ_b=ηh_b+p

1−η²e, (28)

where ˆh_b denotes the estimated CSI, e the estimated error obeying complex Gaussian distribution with unit variance, and η the normalized correlation coefficient. When η =1, clearly, imperfect CSI is reduced to perfect case. In order to introduce additional degrees of freedom into our design for power allocation, we allow a target data rate probability δ ∈(0,1], as encapsulated in the following constraint:

Pr{log₂(1+γ_b)≥R^∗_b} ≥1−δ, (29)

where the probability is evaluated over the estimation error. By using (11) and substituting (28) into (29), with some algebraic manipulations, one can obtain

Pr

log₂

1+^pb

σ_s²

ηh_b+p 1−η²e

2

≥R^∗_b

=Pr

p_b σ_s²

ηh_b+p 1−η²e

2≥ε_b^∗

=Pr

|e|²≥

ε_b^∗σ_s²

p_b −η²|h_b|²

1−η²

(30)

Due to the fact that the random variable |e|² follows the exponential distribution with unit variance, (30) can be further expressed as

Pr

log₂

1+^pb

σ_s²

ηh_b+p 1−η²e

2

≥R^∗_b

=exp



−

ε∗ bσ2

s pb −η²|h_b|²

(1−η²)





(31)

Then, applying (31) to (29) yields exp



−

ε_b^∗σ_s²

p_b −η²|h_b|² (1−η²)



≥1−δ. (32) Hence, the transmit power at the GBU can be expressed as

p_b=







ε_b^∗σ_s²

ϖ+η²|h_b|², |h_b|²≥ ε_b^∗σ_s² η²P_max−ϖ

η²,

0 , else,

(33)

where ϖ=− 1−η²

ln(1−δ). Under this case, the output SINR of the admitted GFU in (10) can be accordingly re- expressed as

γ_i=









 p_i|h_i|²

,

σ_s²ε_b^∗|h_b|² ϖ+η²|h_b|²+σ_s²

!

, |h_b|²≥ ε_b^∗σ_s² η²P_max−ϖ

η²,

pi|h_i|²

σ_s² , else.

(34) It is interesting to note that when the perfect CSI of the GBU is known, namely, η=1, the proposed ATPA scheme under imperfect CSI can be simplified as (20) and (21). Or in other words, the existing SGF works in [26] can be treated as special cases of our general framework. Overall, the proposed SGF- based transmission schemes are summarized in Fig. 3.

IV. PERFORMANCEANALYSIS OF THEPROPOSEDSCHEMES

Considering the scenarios of perfect CSI and imperfect CSI, this section investigates the system throughput performance of the proposed schemes by deriving the closed-form expressions.

The total system throughput is determined by the amount of data correctly received at SAT with a fixed target rate, which can be defined as

R_tot=R_ES+R_MT=R^∗_i(1−P_out,i)+1−P_out,b^K

k=1

∑

R^∗_k 1−P_out,k

, (35)

where Pout,i denotes the outage probability (OP) of the i-th link that is defined as the probability that output SINR falls below a certain threshold valueγ_th, namely,P_out=Pr{γ<γ_th}

Fig. 3. The proposed SGF-based transmission schemes in ISATN.

with γth=2^R−1. In (35), R^∗_k is the target rate of the MT.

The difficulty in deriving (35) lies in obtaining the associated closed-form OP. To deal with this difficulty, we first focus on the outage performance analysis for the imperfect CSI case below.

According to (33) and (34), and denoting ^pi|hi|2

σ_s² =γ¯_i|g_i|² with ¯γi= ^pi`2i

σ_s² , the OP of the GBU is written as Pout,b=1−Pr{D=/0}Pr

|h_b|²≥

ε_b^∗σ_s² η²Pmax− ^ϖ

η²

−²

M−1

∑

p=1

Pr

D=D_p ∑

i∈D

Pr

i=i^∗|D=D_p

Pr

γ¯i|g_i|²

ε_b^∗|h_b|² ϖ+η²|h_b|²+1

≥ε_i^∗,|h_b|²≥

ε_b^∗σ_s² η²Pmax−^ϖ

η²

|i=i^∗

, (36) where Pr{D=/0} = ∏^M

i=1

Prn

p_i|h_i|²<τ o

= ∏^M

i=1

F_p

i|h_i|²(τ), Pr

D=D_p = ∏

i∈D_p

Prn

p_i|h_i|²≥τ o

∏

j∈D¯p

Prn p_j

h_j

2<τ o

=

∏

i∈D_p

h 1−F_p

i|h_i|²(τ)i

∏

j∈D¯p

Fpj|hj|²(τ) with Dp being the p-th nonempty set and ε_i^∗=2^R∗i −1. From (36), the closed-form OP expression for the GBU can be obtained in the following theorem.

Theorem 1. The OP of the GBU with the proposed SGF-based scheme in imperfect CSI case is given by (37) as shown at the top of the next page.

Proof: See Appendix B.

Similarly, the OP of the admitted GFU can be defined as Pout,i=1−²

M−1

∑

p=1

Pr

D=Dp ∑

i∈D

Pr

i=i^∗|D=Dp

Pr

γ¯i|g_i|²

ε_b^∗|h_b|² ϖ+η²|h_b|²+1

≥ε_i^∗,|h_b|²≥

ε_b^∗σ_s² η²Pmax− ^ϖ

η²

|i=i^∗ +Pr

γ¯i|g_i|²≥ε_i^∗,|h_b|²<

ε_b^∗σ_s² η²Pmax−^ϖ

η²

|i=i^∗

, (38) which can be further derived in theorem below.

Theorem 2. The OP of the admitted GFU is given by (39), shown at the top of the next page.

Proof: See Appendix C.

Next, we focus on deriving OPPout,k in (35). Specifically, we first derive the probability density function (PDF) of X_k=

p_k w^H_kh_k

2.

σ_h²wherew^H_k is given by (27). As per (2) and (4), we have X_k=γ¯_k|ρ_k|² with ¯γ_k=p_k`²_k

w^H_ka(θ_k,ϕ_k)

2. σ_h² and thus the corresponding PDF is given by

f_Xk(x) = m_k

γ¯kΩk

m_k

x^mk−1 Γ(m_k)exp

−m_kx γ¯kΩk

. (40)

Following (40) and resorting to [11, eq. (59)], the PDF of the Y_k= ∑^K

j=1,j6=k

γ¯j

w^H_khj

2can be expressed as

f_Yk(y) =







K

∏

j=1 j6=k

m_j^mj Ωjγ¯^m_j ^j







K

∑

j=1 j6=k

m_j

∑

l=1

A_j,l,k

− m_j Ωjγ¯j

y^l−1e⁻

m j Ωjγ¯jy

, (41) where A_j,l,k(s) = ¹

Γ(m_j−l+1)Γ(l) d^{m j−l} ds^{m j−l}

K

∏

t=1, t6=j, t6=k

s+ ^mt

Ωtγ¯_j

−m_t

de- notes the inverse Laplace transform coefficient. Then, with the Binomial theorem and [36, 3.381.4], we can further obtain the cumulative distribution function (CDF) ofγk=_Y^Xk

k+1 as

F_γk(z) =1−∏^K

j=1 j6=k

mjm j

Ωjγ¯^{m j}_j m_k−1

∑

q=0 K

∑

j=1 j6=k

mj

∑

l=1

e

m j Ωjγ¯jA_j,l,k

− ^mj

Ωjγ¯j

l−1

∑

i (−1)ⁱ

q!

l−1 i

mk

Ω_kγ¯_k

^q

Γ(q+l−i)z^q _m

j

Ωjγ¯j+ ^mk

Ω_kγ¯_kz−q−l+i

. (42) Let z=ε_k^∗ with ε_k^∗=2^R∗k−1 on (42), the OP expression Pout,k achieved by the proposed scheme can be obtained as

P_out,k=1−∏^K

j=1 j6=k

m_j^{m j} Ω_jγ¯^{m j}_j

mk−1

∑

q=0 K

∑

j=1 j6=k

m_j

∑

l=1

e

m j Ωjγ¯jA_j,l,k

− ^mj

Ωjγ¯j

l−1

∑

i (−1)ⁱ

q!

l−1 i

m_k Ω_kγ¯_k

q

Γ(q+l−i) ε_k^∗q _m

j

Ω_jγ¯_j + ^mk

Ω_kγ¯_kε_k^∗ −q−l+i

. (43) Finally, substituting (37), (39) and (43) into (35), the closed- form expression of the total system throughput achieved by the proposed SGF-based scheme under imperfect CSI can be easily obtained. Then, by lettingη=1 in (37), (39) and ¯γ_j=0 in (43) and resorting to (35), we can attain the throughput expression of the proposed scheme for perfect CSI.

V. NUMERICALRESULTS

In this section, we provide simulation results to verify the developed performance analysis for the proposed schemes and investigate the impact of CSI errors and various parameters on the system performance. Here, we set the QoS threshold at the GBU as δ =0.02, the target rate of the GBU, GFU, and MT are, respectively, asR^∗_b=1 bit/s/Hz,R^∗_i =0.5 bit/s/Hz andR^∗_k=0.5 bit/s/Hz. We consider two broadcast thresholds at each GFU, namely,τ₁=ε_i^∗ 1+ε_b^∗

/M andτ₂=ε_M^∗ 1+ε_b^∗ withε_M^∗ =2^MR∗i−1, and set the tranmit power of each GFU as