HAPS-enabled Downlink Coverage Enhancement in Islands and Maritime Areas
arXiv:2607.23848v1 [cs.NI] 26 Jul 2026
Hao Lin, Graduate Student Member, IEEE, Mustafa A. Kishk, Member, IEEE and Mohamed-Slim Alouini, Fellow, IEEE
Abstract—Non-terrestrial networks (NTNs) are poised to play a critical role in next-generation mobile communications, offering enhanced flexibility, improved line-of-sight (LoS) conditions, and overcoming the limitations of terrestrial networks (TNs). Among NTN platforms, high altitude platform stations (HAPSs) have emerged as a promising solution to provide Internet connectivity to underserved regions, including rural areas, islands, and maritime zones, where traditional infrastructure deployment is costly and challenging to deploy. In this paper, we investigate the feasibility of large-scale HAPS deployment to connect island and maritime users, considering real-world shadowing effects on part of HAPSs caused by the presence of island building clusters. We first analyze the coverage performance of onshore (island) and offshore (remote sea) users, in which the channels between HAPSs and the user follow the shadowed Rician distributions and Rician distributions, respectively. Next, we introduce an evaluation method for nearshore users in a hybrid channel environment with HAPSs, and propose approximations that can reduce computational complexity. Based on the simulation results, we discuss how the distance from the island boundary (i.e. the relative remoteness of maritime users) affects coverage performance under different HAPS densities. We also emphasize the importance of choosing a balanced HAPS density or an advanced HAPS deployment scheme. Index Terms—Stochastic geometry, high altitude platform stations (HAPSs), non-terrestrial networks (NTNs), coverage probability, maritime communications.
I. I NTRODUCTION In the vision of next-generation mobile communications, the rapid adoption of advanced technologies such as remote education, smart healthcare, intelligent transportation, digital twins, and extended reality, has driven unprecedented demand for global seamless connectivity [1]–[4]. Especially, with the development of marine resource utilization, ocean monitoring, and offshore facilities operation, maritime communications are becoming increasingly necessary [5], [6]. However, traditional terrestrial networks (TNs) face huge limitations in meeting these exploding demands due to high deployment costs, geographic challenges, and limited financial supports [7]. One potential solution is to build wireless mesh networks among vessels, buoys or sensors [8], but this approach still Hao Lin is with the Electrical and Computer Engineering Program, Computer, Electrical and Mathematical Sciences and Engineering Division (CEMSE), King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Saudi Arabia (e-mail: [email protected]). Mustafa A. Kishk is with the Department of Electronic Engineering, Maynooth University, Maynooth, W23 F2H6 Ireland (e-mail: [email protected]). Mohamed-Slim Alouini is with the CEMSE Division, King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Saudi Arabia (e-mail: [email protected]).
faces issues such as interference management and capacity limitation. Long-Term Evolution (LTE) technology, supported by multiple-input multiple-output (MIMO) and carrier aggregation, can realize reliable direct ship-to-shore communications [9], [10]. Since the height of the onshore base stations is limited, it is still difficult to support stable coverage for wider ocean areas. Therefore, non-terrestrial networks (NTNs), including low-altitude platforms (LAPs), high-altitude platform stations (HAPSs), low-earth orbit (LEO) satellites, and geostationary orbit (GEO) satellites, have become a viable solution to extend connectivity to these areas [11]–[13]. Different types of NTN platforms play their respective roles in future communication architectures due to their own characteristics, as shown in [14]. For example, tethered unmanned aerial vehicles (UAVs) can help extend reliable communication to nearshore users [15] and LEO satellites can be a solution to achieve Internet coverage for remote areas. HAPSs mainly operate at an altitude of 20-50 km, which can balance coverage range and propagation delay [16]. Unlike LAPs, HAPSs can support larger communication payloads and advanced antenna arrays, enabling higher data rates and precise beamforming [17]. Each HAPS has a wider coverage range of up to 140 km and an autonomous flight time of several months. Also, HAPSs have lower latency than LEO satellites because of their proximity to the earth’s surface and provide more stable connections without orbital motion [18]. Equipped with 5G and LTE access technologies, HAPSs can provide direct services to users without dedicated antennas. Although energy consumption remains a challenge, solutions such as solar panels, wind turbines, and tethering cables are being explored to ensure the sustainable operation of HAPSs [19]. Therefore, HAPSs are expected to play a pivotal role in bringing ubiquitous connectivity and supporting ultra-reliable low-latency communications (URLLC) to unconnected areas, such as islands and maritime areas [20]. HAPSs can not only provide services to areas lacking TN infrastructure, but also fill the coverage gaps of existing networks in island areas, share the traffic demand from mobile users and massive machine-type communications, and provide backup optical feeder links. Integrated sensing and communications (ISAC) has been advanced to enhance the spectrum utilization and quality of services (QoS), particularly in intelligent transportation systems, low-altitude network management and massive IoT, where HAPSs can also offer a compelling solution for implementing ISAC systems for remote scenarios [16], [19], [21]. Currently, Softbank Corp has planned to use HAPSs with a payload of up to 70 kg, to provide services in areas that
are difficult to cover by the existing mobile networks [22]. To optimize the deployment of large-scale HAPS network, a performance analysis framework of HAPS networks in a complex electromagnetic environment is urgently needed. Stochastic geometry has been widely used to study the coverage capabilities of terrestrial large-scale wireless networks (LSWN), without losing accuracy and tractability [23], [24]. It is also important to characterize the performance of future large-scale non-terrestrial networks [25]. System-level metrics including coverage probability, latency, capacity, and energy efficiency can be investigated using tools from stochastic geometry [26]. Focusing on the HAPS-based solutions in maritime applications, we establish a unified mathematical framework to model the channels between HAPSs and onshore, nearshore, and offshore users. More details on related work are shown in Sec. I-A and the contributions of this paper are summarized in Sec. I-B. A. Related Work In this paper, we investigate the potential and performance of HAPS-based solutions for island and maritime areas. Therefore, we divide the related work into: (i) HAPS-based solutions for the unconnected and (ii) stochastic geometry for HAPSbased solutions. HAPS-based solutions for the unconnected: Connecting the unconnected is an important goal of future communication systems, and HAPSs play an important role in providing services to rural and remote areas that lack broadband connectivity and infrastructure [27]. In [28], the authors investigated HAPSbased environmental monitoring applications for remote and unconnected regions, where ground Internet of Things (IoT) devices transmit information to HAPSs via long-range radio (LoRa) technique. They examined the impact of IoT device density, transmit power, and HAPS altitude on throughput and conditional success probability. Using numerical iterative methods, the authors in [29] demonstrated the potential of heterogeneous networks to achieve fair access services in both urban and rural areas. They considered a system setup where a cloud-enabled HAPS can connect with high-altitude balloons and terrestrial base stations to serve both aerial and ground users. By jointly optimizing user association and beamforming, the authors in [30] analyzed the performance of hybrid satellite-HAPS-terrestrial networks and discussed the prospects for connecting unconnected regions. They also explored the application of machine learning in such vertical heterogeneous networks (vHetNets) [31]. Due to the large payload capacity and superior LoS conditions, HAPSs can be equipped with reconfigurable intelligent surfaces (RISs) and act as relay nodes, to further connect the unconnected base stations [32]. Stochastic geometry for HAPS-based solutions: Stochastic geometry has been widely applied to the performance analysis of large-scale wireless networks, including terrestrial, nonterrestrial, and integrated networks. By analyzing system-level metrics such as coverage probability, latency, and channel capacity, operators can optimize NTNs’ performance and reduce capital expenditure (CAPEX) and operational expenditure (OPEX) [26]. For example, in [33], the authors proposed
an efficient stochastic geometry framework to evaluate the quality of service for users inside and outside disaster-affected areas. This framework utilized low-altitude platforms (LAPs) and HAPSs to address disasters of varying scales, demonstrating the effectiveness of vertical heterogeneous networks in emergency scenarios. HAPSs can not only serve as cellular or non-cellular base stations for mobile users, but also act as nodes for backhaul and computational networks [17]. For instance, in [34], the authors investigated the performance of hybrid satellite-aerial-terrestrial networks, where HAPS serves as an aerial node with file-caching capabilities. Using stochastic geometry, they derived the outage probability and the hit probability of the considered vHetNet architecture. In [35], the authors explored a heterogeneous network based on unmanned aerial vehicles (UAVs), HAPSs, and low earth orbit (LEO) satellites, analyzing the contributions of each layer to overall performance. Under constraints of practical economic costs and signal-to-noise ratio (SNR), they proposed optimization schemes for total connection probability and discussed resource allocation strategies under different constraints. Furthermore, the future communication architecture should cover users and devices on the sea. In [36], the authors studied the coverage performance of space-air-ground-sea integrated networks (SAGSIN), where onshore base stations, tethered balloons, HAPSs, and satellites provide services to surface stations at sea. Unlike the above literature, our paper aims to build a mathematical framework for downlink performance evaluation considering the relationship between users, a shadowing zone and a homogeneous HAPS network, verify the feasibility of HAPS-based solution and provide guidance for future customized HAPS deployment to realize user fairness. More details about the contributions are provided in the next subsection. B. Contributions The contributions of this paper can be summarized as follows: • We develop a more accurate evaluation approach for HAPSs-enabled solutions on islands and maritime areas, considering the shadowing effect of the island building cluster on the channels between HAPSs and users. We propose to divide the HAPS deployment area into a shadowed Rician region and a Rician region according to the geometric relationship between user, island building cluster and HAPS network. • We derive the exact expression of the coverage probability for onshore, nearshore, and offshore users using stochastic geometry. Especially for the nearshore users with a hybrid channel environment, we propose two approximations for coverage probability, whose computation complexity is less than the exact expression. • We show that onshore users operate optimally with a high HAPS density, while offshore and nearshore users perform best with a low HAPS density. A balanced HAPS density can help minimize the coverage performance gap between users at different locations, and moreover,
Shadowed Rician
Shadowed Rician 𝒓
Shadowed Rician
𝑰
Rician 𝒉𝑯
Shadowed Rician
𝒉𝑰
𝐨 Onshore 𝑰 (−𝒓𝑼 , 𝟎, 𝟎)
Shadowed Rician 𝒉𝑯
𝒓𝑰
Rician
Rician 𝒓𝑰
Nearshore (−𝒓𝑼 , 𝟎, 𝟎)
𝒉𝑯 𝒉𝑰
𝒉𝑰 𝐨𝑰
Rician
Offshore (−𝒓𝑼 , 𝟎, 𝟎)
𝐨𝑰
(a) Case 1: Onshore users with HAPS-user links (b) Case 2: Nearshore users with HAPS-user links (c) Case 3: Offshore users with HAPS-user links under shadowed Rician channels. under shadowed Rician channels and Rician chan- under Rician channels. nels. Fig. 1. Illustration of channel environment for onshore, nearshore and offshore users, respectively.
a customized deployment of the HAPS network has potential in the future. The rest of this paper is organized as follows: In Sec. II, we introduce the system model. In Sec. III we introduce the expression of coverage probabilities for the island (onshore) and remote maritime (offshore) users. In Sec. IV, we derive the exact expressions for nearshore users. We also propose two approximations for the coverage performance of nearshore users. In Sec. V, we conduct the Monte Carlo simulations and verify the exact expressions and approximations. Finally, we conclude this paper in Sec. VI. II. S YSTEM M ODEL In this paper, we use a circular area with radius rI to model a typical residential area (or forests) on an island, and the sea area is outside this circular area. We aim to investigate the downlink coverage performance of users at different geographic locations under the same large-scale HAPS deployment. Therefore, considering the randomness from the node failure and maintenance of the HAPS cluster, the locations of HAPSs follow a two-dimensional (2D) homogeneous Poisson point process (PPP) Φ = {xi } with density λH at altitude hH . We assume that the center of island is oI = (0, 0, 0) and the location of a typical user is zU = (−rU , 0, 0), where the distance between the typical user and the center of island is rU . We also call rU − rI the ‘relative distance to the island boundary’, where a negative value corresponds to an onshore user and a positive value corresponds to a nearshore user or an offshore user. Signal propagation is crucial for performance analysis of air-sea channels. A straightforward approach is to use the classic logarithmic path loss model and modify the parameters to account for the effects of obstacles and scatterers at sea. When the transmission antenna height is low (less than 1 km), the signal energy received by the user comes primarily from specular and diffuse reflections from the sea surface, rather than simply LoS path. However, when the transmission antenna height is higher, the majority of the signal energy received by the user comes from the LoS propagation path. This makes it particularly important to know whether the LoS signal is obstructed by random obstacles. Therefore, we model the cluster of buildings (or forests) on the island as a cylindrical area with the center of the bottom at oI , radius
rI , and altitude hI . When the LoS between a user and a HAPS crosses this cylindrical area, the power of the LoS component is randomly shadowed by the buildings, while the reflections on the buildings form the NLoS components of the signal. We assume that the shadowed LoS component follows a Gamma distribution and that the small-scale fading gain of the signal envelope is modeled using the shadowed Rician distribution. When the LoS between the user and a HAPS does not cross the cylindrical area, the small-scale fading gain of the signal envelope is modeled using the Rician distribution because clear LoS paths can be obtained, while reflections by the sea surface and evaporation ducting will form the NLoS component. In addition, as introduced in [9], [37], the ducting effect leads to coverage extension when the electromagnetic waves are emitted in certain directions, the signals are concentrated in the intended direction, which is called beyond LoS (B-LoS) transmissions. Because HAPSs have broader footprints than the onshore BSs, we ignore the B-LoS effect but consider the multipath effect. Considering different locations of typical users, we discuss three cases: • Onshore case: When rU is less than rI , the typical user is on the island and almost all LoS paths between it and HAPSs are shadowed by the island building cluster. Therefore, the typical user is in a shadowed Rician channel environment where channel gains from all HAPSs to the typical user follow the shadowed Rician distribution. • Nearshore case: When rU is larger than rI , the building cluster shadows the LoS paths between parts of the HAPSs and the typical user, and these HAPS operate in shadowed Rician fading channels. Since the building cluster does not shadow the LoS path between remaining HAPSs and the user, these HAPSs operate in Rician fading channels. Therefore, such a typical nearshore user operates in a hybrid channel environment. • Offshore case: When rU is much larger than rI , the typical user is far from the island and can obtain clear LoS paths to almost all HAPSs. Therefore, the typical user is in a Rician channel environment, where the channel gains between all HAPSs and the typical user follow the Rician distribution. It is worth noting that, when a typical user moves from the island center to the far sea area, its channel environment keeps changing. Specifically, as the user crosses the island
TABLE I TABLE OF N OTATIONS
Notation Φ = {xi }; λH ; hH oI ; rI ; hI zU ; rU ; di p t , gt , gr , λ pI,0 ; WI,i ; αI bI ; mI ; ΩI α 0 ; β0 PI,i pO,0 ; WO,i ; αO bO ; ΩO PO,i s I ; sO τ ; N0 ; Pcov
Description The set of locations of HAPSs; the density of HAPSs following a 2D PPP; the altitude of HAPSs The center of island; the radius of island; the altitude of buildings on the island The location of the typical user; the distance between the typical user and island center; the distance between the typical user at zU and HAPS at xi . Transmit power; transmitting gain; receiving gain; wave length of the carrier frequency The received power at a unit distance without channel gains; channel gain; path-loss exponent (for shadowed Rician channels); The standard deviation of NLoS components; the shape parameter; the average gain of the shadowed LoS component (for shadowed Rician channels) The parameters for approximation of shadowed Rician channel fading The signal power transmitted from HAPS at xi and received at the typical user via a shadowed Rician channel The received power at a unit distance without channel gains; channel gain; path-loss exponent (for Rician channels) The standard deviation of NLoS components; the average gain of the LoS component (for Rician channels) The signal power transmitted from HAPS at xi and received at the typical user via a Rician channel The atmospheric attenuation in shadowed Rician channels and Rician channels The decoding SINR threshold; the noise in the receiver circuit; coverage probability
boundary, over half of the HAPSs operate in the Rician channels because the building cluster no longer shadows LoS paths from the HAPSs on the sea side. When rU is larger than rI and continues to increase, more and more HAPSs can obtain clear LoS paths and operate in Rician channels. Therefore, the channel environment of the typical user gradually changes from a hybrid channel environment to a Rician channel environment. Furthermore, an increase in either the island’s size or the height of buildings results in a greater number of HAPSs experiencing shadowing effects. For ease of reading, we summarize the main notation and parameters a in Table I. We use f (z) b to represent f (a) − f (b). III. D OWNLINK A NALYSIS FOR O NSHORE AND O FFSHORE In this section, we analyze the downlink coverage performance for a typical user onshore or offshore, respectively. We specify the channel models between HAPSs and the user, and derive the expression of coverage probability for onshore users and offshore users, respectively. A. Onshore Users: Shadowed Rician Environment For users on the island, we have rU ≤ rI . As shown in Fig. 1(a), the signals from HAPSs are not only affected by the atmosphere and reflections, but also shadowed by building cluster. Therefore, we assume that the channel gains of smallscale fading between HAPSs and the onshore user follow the shadowed Rician distributions [38]. The power of signal transmitted from HAPS i and received by the typical onshore user can be modeled as: λ I )αI gt gr sI WI,i = pI,0 WI,i d−α (1) PI,i (di ) = pt ( I,i , 4πdI,i λ αI where pI,0 = pt ( 4π ) gt gr sI is the received power at a unit distance regardless of the channel gains. pt is the transmitted power, λ is the wave length, gt , gr and sI represent the transmit antenna gain, receive antenna gain and the atmospheric attenuation in shadowed Rician channels, respectively. αI is the path loss exponent for shadowed Rician channels and dI,i is the distance between the typical onshore user and HAPS located at xi . The shadowed Rician channel gain is represented using WI,i , and the PDF of WI,i is: m I I fWI,i (x)= 2b1I 2bI2bmIIm+Ω exp − 2b1I x I (2) × 1 F1 mI ; 1; 2bI [2bIΩmI I +ΩI ] x ,
where bI is the standard deviation of NLoS component, ΩI is the average of direct LoS component which follows the Gamma distribution, and mI is the shape parameter of the Gamma distribution. The CDF of WI,i can be calculated using: m I I FWI,i (x)= 2bI2bmIIm+Ω I n P∞ (mI )n (3) ΩI × n=0 (1) γ n + 1, 2bxI , n n! 2bI mI +ΩI and the CCDF of WI,i is: m I △ I F WI,i (x) = 1 − FWI,i (x) = 2bI2bmIIm+Ω I (4) n P∞ (mI )n ΩI × n=0 (1) Γ n + 1, 2bxI , n n! 2bI mI +ΩI Rx R∞ where γ(s, x) = 0 ts−1 e−t dt and Γ(s, x) = x ts−1 e−t dt represent the lower incomplete Gamma function and upper incomplete Gamma function with shape parameter s. The expectation of WI,i is E[WI,i ] = 2bI +ΩI . Since the locations of HAPSs follow a 2D homogeneous PPP with density λH and all HAPSs follow the same signal propagation model, the user selects the closest HAPS that has the strongest average received power. Therefore, we define the variable D as the distance between the user and the closest HAPS and introduce the distribution of D in Lemma 1. Lemma 1. The PDF of the distance d between the typical user and its nearest HAPS can be expressed as: fD (d) = 2λH π exp λH πh2H d exp −λH πd2 , (5) for hH < d < ∞. Proof: See Appendix A. Therefore, the closest HAPS acts as the serving HAPS to the user, and other HAPSs act as interfering ones. Based on this, we introduce the coverage probability for typical onshore users in Theorem 1. Theorem 1. When rU ≤ rI , the channel gains between the typical user and HAPSs are modeled using shadowed Rician distributions, where the coverage probability of onshore users is: △ Pcov (rU , τ )= RP{SINRI (rU ) > τ } (6) ∞ ≈ hH F WI,1 gI (d) fD (d)dd, I +ΩI )d 0 αI with gI (d) ≈ τpN d + 2πλH τ (2b αI −2 I,0 F WI,1 (x) is given in (4).
Proof: See Appendix B.
αI
d
z 2−αI ∞ , and
Theorem 1 describes the coverage probability of onshore users when the deployment range of HAPSs is infinite. Next, we adopt similar approaches to introduce the coverage probability of offshore users. B. Offshore Users: Rician Environment For the users in remote maritime areas, we have rU ≫ rI . As shown in Fig. 1(c), the building cluster on the island does not shadow the LoS paths between the HAPSs and the user. The reflections on the sea surface and evaporation ducting lead to multiple NLoS components. Therefore, we assume that the channel gains of small-scale fading between HAPSs and offshore users follow a Rician distribution. The power of signal transmitted from HAPS i to the typical offshore user can be modeled as: λ O )αO gt gr sO WO,i = pO,0 WO,i d−α PO,i (di ) = pt ( O,i , 4πdO,i (7) λ αO where pO,0 = pt ( 4π ) gt gr sO is the received power at a unit distance regardless of the channel gains. sO is the atmosphere attenuation in Rician channels, αO is the path loss exponent for Rician channels, and dO,i is the distance between the typical offshore user and HAPS located at xi . The Rician channel gain is represented using WO,i and the PDF of WO,i is: √ 1 ΩO x x + ΩO fWO,i (x) = I0 , (8) exp − 2bO 2bO bO where bO is the standard deviation of NLoS component, and ΩO is the strength of direct LoS component. The CDF of WO,i is: » » ΩO x FWO,i (x)= 1 − Q1 (9) bO , bO ,
IV. D OWNLINK A NALYSIS FOR N EARSHORE Different from the onshore and offshore cases, nearshore users receive the signals from HAPSs with different LoS conditions. In this section, we propose to divide the deployment area of HAPSs into a shadowed Rician region and a Rician region to evaluate the coverage probability and the approximations. A. Nearshore Users: Hybrid Channel Environment For the nearshore users, we have rU > rI but rU is not much larger than rI . As shown in Fig. 1(b), when the LoS path between one HAPS and the typical user is shadowed by the cylindrical building cluster, we assume that the channel gain follows the shadowed Rician distribution in (2) and (3), and the signal propagation model is the same as in (1). Otherwise, the channel gain follows the Rician distribution in (8) and (9), and the signal propagation model is the same as in (7). The comparison between the channel environments of onshore, nearshore and offshore user is shown in Fig. 1. We show the 3D illustration and the 2D illustration for different channel regions of HAPSs in Fig. 2 and Fig. 3, considering a typical nearshore user located at xU = (−rU , 0, 0). We project the top of the island at (−rU , 0, hI ) from the typical user to the HAPS plane at altitude hH , which is also I ), 0, hH ) with a circular area centered at o′I = (rU ( hHh−h I hH hH ′ radius rU hI : A = yl : ∥yl − oI ∥ ≤ rU hI . We consider z′U = (−rU , 0, hH ) on the HAPS plane which is directly above the typical user zU . From the perspective of z′U , the area A and the areas blocked by A form the shadowed Rician region, while other areas form the Rician region.
and the CCDF of WO,i is » » ΩO x F WO,i (x)= 1 − FWO,i (x) = Q1 (10) bO , bO , Ä 2 2ä R∞ where Q1 (a, b) = b t exp − t +a I0 (at)dt is the Marcum 2 Q-function of order ν = 1 and I0 is the modified Bessel function of first kind of order ν − 1 = 0. The expectation of WO,i is E[WO,i ] = 2bO + ΩO . Because all HAPSs operate in the Rician channels, the closest HAPS can provide the strongest average power, similar to the onshore case. Therefore, the closest HAPS acts as the serving HAPS to the user and other HAPSs act as interfering ones. Based on this, we introduce the coverage probability for typical offshore users in Theorem 2. Theorem 2. When rU ≫ rI , the channel gains between the typical user and HAPSs can be modeled using Rician distributions, where the coverage probability is: △
Pcov (rU , τ )= RP{SINRO (rU ) > τ } ∞ ≈ hH F WO,i gO (d) fD (d)dd, N0 αO O +ΩO )d with gO (d) ≈ pτO,0 d + 2πλH τ (2b αO −2 F WO,1 (x) is given in (10).
αO
(11) d
z 2−αO ∞ , and
Proof: See Appendix C. However, for nearshore users, the channel environment is more complex than in the onshore and offshore cases. Therefore, in the next section, we introduce the coverage analysis for nearshore users.
Rician Region
𝒓𝑰
𝒛′𝑼 (𝟎, 𝟎, 𝒉𝑯 )
𝒉𝑯 𝒉𝑰
Shadowed Rician ′ Region
𝒐𝑰
𝒉𝑯 (𝟎, 𝟎, 𝒉𝑰 )
Nearshore User
𝒛𝑼
𝐨𝑰
𝒓𝑰
𝒉𝑰
Fig. 2. 3D Illustration of hybrid channel environment for typical nearshore user.
To calculate the coverage probability, we assume that the locations of the HAPSs operating in the Rician channels follow a PPP with density λH and the locations of the HAPSs operating in the shadowed Rician channels follow another PPP with density λH . Because the closest HAPS does not always provide the highest signal power in the hybrid environment, we realize the selection strategy in two steps: (i) select the closest HAPS in the Rician channel region and the closest HAPS in the shadowed Rician channel region, and (ii) compare their average power and select the stronger one. We assume that the closest HAPS with a Rician channel is located at xO 1 , whose distance to the typical user is dO,1 . Similarly, the closest HAPS with a shadowed Rician channel
𝒓𝟏
Lemma 2. Assume that dO,1 is the distance between the typical nearshore user located at zU and its closest HAPS in the Rician region located at xO 1 . The CDF of dO,1 is shown in (20) on the top of the next page, and the PDF of dO,1 is f (dO,1 ) = 2λH dO,1 (1 − F (dO,1 )) ×(π − ϕ(dO,1 )1(dth,1 < dO,1 < dth,2 ) − θ1(dO,1 > dth,2 )), (17) where h2 d2 −h2 h2 +r 2 h2 −r 2 h2 ϕ(d) = arccos I √ I2 H 2 U H I H , (18)
𝒉𝑯
𝜽
Rician ′
𝒛𝑼
𝑷𝟏 𝒓𝑰 𝒉𝑰 𝑷𝟐
𝒐′𝑰
Shadowed Rician
𝒓𝟐
2
and Fig. 3. 2D Illustration of hybrid channel environment for typical nearshore user.
is located at xI1 , where the distance to the typical user is dI,1 . Therefore, we construct below two transmitter (TX) △ selection cases: HO = {Select the HAPS located at xO 1 }, △ I and HI = {Select the HAPS located at x1 }. Because the TX selection depends on the average power of these two closest HAPSs, we formulate the TX selection as: HO
−αI O pO,0 E[WO,1 ]d−α O,1 ≷ pI,0 E[WI,1 ]dI,1 .
(12)
HI
We define two functions: 1
αO
1
αI
TI (dO )=
pI,0 (ΩI +2bI ) αI αI dO , pO,0 (ΩO +2bO )
TO (dI )=
pO,0 (ΩO +2bO ) αO αO dI , pI,0 (ΩI +2bI )
and
(13) (14)
so the two TX selection cases can be rewritten using: HI
HO
HO
HI
dO,1 ≷ TO (dI,1 ) or dI,1 ≷ TI (dO,1 ),
(15)
and the coverage probability can be calculated using: Pcov (rU , τ ) = Pcov (rU , τ, HO ) + Pcov (rU , τ, HI ).
(16)
Especially, if pI,0 E[WI,1 ] = pO,0 E[WO,1 ] and αI = αO , the selection strategy is reduced to the closest TX selection strategy. When HO is always satisfied, the typical user is in the Rician channel environment, which is the offshore case. B. Coverage Probability for Nearshore To obtain the exact expression of the coverage probability for nearshore users, we first derive the distribution of the distance dO,1 between the typical user zU and its closest HAPS in the Rician region located at xO 1 . We analyze the characteristics of the Rician region: Rician Region: As shown in Fig. 3, from the perspective of z′U , the Rician region can be divided into: (i) a sector area centered at z′U , with infinite radius and central angle 2π − 2θ where θ = arcsin rrUI , and (ii) a sector area centered at z′U , with radius r1 = hhHI (rU − rI ) and central angle 2θ, except for the regions in A. Therefore, as the search radius increases from r1 , the range of directional angle with » LoS decreases, until hH 2 − r 2 . From the the search radius is greater than r2 = hI rU I perspective of the typical » user located at zU , the » critical values of dO,1 are dth,1 = r12 + h2H and dth,2 = r22 + h2H . Based on this, we can obtain the CDF and PDF of dO,1 .
η(d) = arccos
d −hH rU hH hI
2 2 rU hH +rI2 h2H −d2 h2I +h2H h2I . 2rI rU h2H
(19)
Proof: See Appendix D. Next, we analyze the characteristics of the shadowed Rician region: Shadowed Rician Region: As shown in Fig. 3, from the perspective of z′U , the shadowed Rician region can be divided into: (i) the intersection between A»and the sector area 2 − r 2 and central centered at z′U , with radius r2 = hhHI rU I angle 2θ, and (ii) an annular sector area centered at z′U , with inner radius r2 , infinite outer radius, and central angle 2θ. From the perspective of the typical » user located at zU , the range of dI,1 is from dth,1 = r12 + h2H to infinity, where » the critical value is dth,2 = r22 + h2H . Based on this, we can obtain the CDF and PDF of dI,1 . Lemma 3. Assume that dI,1 is the distance between the typical nearshore user located at zU and its closest HAPS in the shadowed Rician region located at xI1 . The CDF of dI,1 is shown in (21) on the top of the next page, and the PDF of dI,1 is: f (dI,1 ) = 2λH dI,1 (1 − F (dI,1 )) ×(ϕ(dI,1 )1(dth,1 < dI,1 < dth,2 ) + θ1(dI,1 > dth,2 )), (22) where ϕ(dI,1 ) and η(dI,1 ) have been shown in (18) and (19). Proof: See Appendix E. Because HAPSs are separated into two independent PPPs, dO,1 and dI,1 are independent, and the joint PDF can be written as the multiplication of PDFs of dO,1 and dI,1 : f (dO,1 , dI,1 ) = f (dO,1 )f (dI,1 ),
(23)
and we can derive the coverage probability: Theorem 3. The coverage probability of nearshore users can be formulated using Pcov (rU , τ ) = Pcov (rU , τ, HO ) + Pcov (rU , τ, HI ).
(24)
The coverage probability of nearshore users with HO is: R∞ R∞ P cov (rU , τ, HO ) = hH max{dth,1 ,TI (dO,1 )} h F WO,1 gO (dO,1 , dI,1 ) f (dO,1 )f (dI,1 ) ddI,1 ddO,1 , (25) τ h O with gO (dO,1 , dI,1 ) = pO,0 N + I(d , d ) . dα 0 O,1 I,1 O,1 The coverage probability of nearshore users with HI is: R∞ R∞ P cov (rU , τ, HI ) = dth,1 max{hH ,TO (dI,1 )} (26) F WI,1 gIh (dO,1 , dI,1 ) f (dO,1 )f (dI,1 ) ddO,1 ddI,1 ,
Ä ä 2 2 −λ πd + λ πh , h ≤ dO,1 ≤ dth,1 1 − exp H H O,1 H H 2 h2H 2 2 2 hH 1 − exp −λH (π − ϕ(dO,1 ))(dO,1 − hH ) − λH sin η(dO,1 )rI rU h2 + λH η(dO,1 )rI h2 , dth,1 < dO,1 < dth,2 F (dO,1 ) = I I 1 − exp −λ (π − θ)(d2 − h2 ) − λ cot θr2 h2H + λ ( π − θ)r2 h2H ,d ≥d H
O,1
H
H
I h2I
H 2
O,1
I h2I
th,2
(20)
2 2 1 − exp −λH ϕ(dO,1 )(d2 − h2 ) + λH sin η(dI,1 )rI rU hH2 − λH η(dI,1 )r2 hH2 , dth,1 ≤ dI,1 < dth,2 I,1 H I hI hI F (dI,1 ) = 2 2 1 − exp −λH θ(d2 − h2 ) + λH cot θr2 hH2 − λH ( π − θ)r2 hH2 , dI,1 ≥ dth,2 I,1 H I h I h 2 I
τ I with gIh (dO,1 , dI,1 ) = pI,0 dα I,1 N0 + I(dO,1 , dI,1 ) . The interference is approximated using: H 2−αO dO,1 I(dO,1 , dI,1 ) ≈ pO,0 (2bO + ΩO ) (2π−2θ)λ z αO −2 ∞ R θ λH 2−αO min{dO,1 ,Td (t)} +pO,0 (2bO + ΩO ) αO −2 −θ z dt Td (t) R θ 2−α max{d ,T (t)} I,1 d λH I dt, +pI,0 (2bI + ΩI ) αI −2 −θ z ∞ (27) » 2 sin2 t). where Td (t) = hhHI (rU cos t − rI2 − rU Proof: See Appendix F. However, the computational resource requirement for this exact expression of the coverage probability is high due to the triple integral of the angle t, dO,1 and dI,1 . Therefore, we propose an approximation method by changing the geometric boundary of the Rician region and the shadowed Rician region. C. Overestimating the Shadowed Rician region To simplify the calculation of coverage probability for nearshore users, we propose an approximation by overestimating the shadowed Rician region. Approximation Method 1: As shown in Fig. 4, we overestimate the area of the shadowed Rician region. From the perspective of z′U , the shadowed Rician region is an annular sector area centered at z′U , with inner radius r1 = hhHI (rU −rI ), infinite outer radius and central angle 2θ. Therefore, the Rician region can be divided into: (i) a sector area centered at z′U , with infinite radius and central angle 2π − 2θ, and (ii) a sector area centered at z′U , with radius r1 and central angle 2θ. From the perspective»of zU , the critical value of the distance to HAPSs is dth,1 = r12 + h2H . Based on this, we can separate the set of HAPSs into two independent PPPs. The approximate CDFs and PDFs of dO,1 and dI,1 are formulated in Lemma 4. Lemma 4. Based on the approximation method 1, the CDF of the distance dO,1 between the typical nearshore user and its closest HAPS in the Rician region is: F (dO,1 ) Ä ä = 1 − exp −λH (π − θ)d2O,1 − λH θ min(dO,1 , dth,1 )2 × exp λH πh2H , (28) and the PDF of dO,1 is: f (dO,1 ) = 2λH dO,1 (π − θ) + θ1(dO,1 < dth,1 ) (29) × 1 − F (dO,1 ) , for hH < dO,1 < ∞. The CDF of the distance dI,1 between the typical nearshore user and its closest HAPS in Äthe shadowed Ricianäregion is: F (dI,1 ) = 1 − exp −λH θ(d2I,1 − d2th,1 ) , (30)
(21)
I
and the PDF of dI,1 is f (dI,1 )= 2λH θdI,1 (1 − F (dI,1 )), for dth,1 < dI,1 < ∞.
(31)
Proof: See Appendix G. Based on these, we formulate the approximation of coverage probability in Theorem 4. Theorem 4. By overestimating the shadowed Rician area, the coverage probability for nearshore users can be represented using the sum of the coverage probabilities with HO and HI , that is: Pcov (rU , τ ) = Pcov (rU , τ, HO ) + Pcov (rU , τ, HI ), (32) where dO,1 > hH and dI,1 > dth,1 . The coverage probability of nearshore users with HO is R∞ R∞ P cov (rU , τ, HO ) = hH max{dth,1 ,TI (dO,1 )} h F WO,1 gO (dO,1 , dI,1 ) f (dO,1 )f (dI,1 ) ddI,1 ddO,1 , (33) τ h O with gO (dO,1 , dI,1 ) = pO,0 dα N + I(d , d ) . 0 O,1 I,1 O,1 The coverage probability of nearshore users with HI is R∞ R∞ P cov (rU , τ, HI ) = dth,1 max{hH ,TO (dI,1 )} (34) F WI,1 gIh (dO,1 , dI,1 ) f (dO,1 )f (dI,1 ) ddO,1 ddI,1 , τ I with gIh (dO,1 , dI,1 ) = pI,0 dα I,1 N0 + I(dO,1 , dI,1 ) . The interference is approximated using: dO,1
H 2−αO I(dO,1 , dI,1 ) ≈ pO,0 (2bO + ΩO ) (2π−2θ)λ z αO −2 ∞
min{dO,1 ,dth,1 }
H +pO,0 (2bO + ΩO ) α2θλ z 2−αO d O −2
th,1
dI,1
2θλH 2−αI +pI,0 (2bI + ΩI ) α z . ∞ I −2
(35)
Proof: See Appendix H. Next, we introduce another approximation by overestimating the Rician region rather than the shadowed Rician region.
𝒓𝟏
𝒓𝑰 𝜽
Rician ′
𝒉𝑯 𝒉𝑰
𝒐′𝑰
𝒛𝑼
Shadowed Rician
𝒓𝟐
Fig. 4. Approximate method 1: overestimating the shadowed Rician region.
D. Overestimating the Rician region We have proposed an approximation by overestimating the area of the shadowed Rician region. Next, we introduce another approximation by overestimating the area of the Rician region. Approximation Method 2: As shown in Fig. 5, we overestimate the area of the Rician region. From the perspective of z′U , the shadowed Rician region is an annular sector area » hH ′ 2 centered at zU , with inner radius r2 = hI rU − rI2 , infinite outer radius and central angle 2θ. Therefore, the Rician region can be divided into: (i) a sector area centered at z′U , with infinite radius and central angle 2π − 2θ, and (ii) a sector area centered at z′U , with radius r2 and central angle 2θ. From the perspective»of zU , the critical value of the distance to HAPSs is dth,2 = r22 + h2H .
𝒓𝟏
𝒓𝑰 𝜽
Rician ′
𝒉𝑯 𝒉𝑰
𝒐′𝑰
𝒛𝑼
Based on the approximate CDFs and PDFs of dO,1 and dI,1 , the approximation of coverage probability is provided in Theorem 5. Theorem 5. By overestimating the Rician area, the coverage probability for nearshore users can be represented using the sum of the coverage probabilities with HO and HI , that is: Pcov (rU , τ ) = Pcov (rU , τ, HO ) + Pcov (rU , τ, HI ),
(40)
where dO,1 > hH and dI,1 > dth,2 . The coverage probability of nearshore users with HO is R∞ R∞ P cov (rU , τ, HO ) = hH max{dth,2 ,TI (dO,1 )} h F WO,1 gO (dO,1 , dI,1 ) f (dO,1 )f (dI,1 ) ddI,1 ddO,1 , (41) τ h O with gO (dO,1 , dI,1 ) = pO,0 dα N + I(d , d ) . 0 O,1 I,1 O,1 The coverage probability of nearshore users with HI is R∞ R∞ P cov (rU , τ, HI ) = dth,2 max{hH ,TO (dI,1 )} F WI,1 gIh (dO,1 , dI,1 ) f (dO,1 )f (dI,1 ) ddO,1 ddI,1 , τ I with gIh (dO,1 , dI,1 ) = pI,0 dα I,1 N0 + I(dO,1 , dI,1 ) . The interference is approximated using:
Shadowed Rician
(42)
dO,1
H 2−αO z I(dO,1 , dI,1 ) ≈ pO,0 (2bO + ΩO ) (2π−2θ)λ αO −2 ∞
min{dO,1 ,dth,2 }
H +pO,0 (2bO + ΩO ) α2θλ z 2−αO d O −2
th,2
𝒓𝟐
Fig. 5. Approximate method 2: overestimating the Rician region.
In this case, we also approximate the CDFs and PDFs of dO,1 and dI,1 as formulated in Lemma 5. Lemma 5. Based on the approximation method 2, the CDF of the distance dO,1 between the typical nearshore user and its closest HAPS in the Rician region is: F (dO,1 ) ä Ä = 1 − exp −λH (π − θ)d2O,1 − λH θ min(dO,1 , dth,2 )2 × exp λH πh2H , (36) and the PDF of dO,1 is: f (dO,1 ) = 2λH dO,1 (π − θ) + θ1(dO,1 < dth,2 ) × 1 − F (dO,1 ) = 2λHÄdO,1 (π − θ) + θ1(dO,1 < dth,2 ) ä × exp −λH (π − θ)d2O,1 − λH θ min(dO,1 , dth,2 )2 × exp λH πh2H , (37) for hH < dO,1 < ∞. The CDF of the distance dI,1 between the typical nearshore user and its closest HAPS in Äthe shadowed Ricianäregion is: F (dI,1 ) = 1 − exp −λH θ(d2I,1 − d2th,2 ) , (38) and the PDF of dI,1 is f (dI,1 )= 2λH θdI,1 (1 −ÄF (dI,1 )) ä (39) = 2λH θdI,1 exp −λH θ(d2I,1 − dth,2 ) , for dth,2 < dI,1 < ∞. Proof: Similar to Lemma 4, but replace dth,1 with dth,2 .
dI,1
2θλH 2−αI +pI,0 (2bI + ΩI ) α . z ∞ I −2
(43)
Proof: Similar to Theorem 4, but replace dth,1 using dth,2 . Different from the Approximation 1, Approximation 2 is closer to the offshore coverage performance because the Rician region is overestimated. V. S IMULATION RESULTS AND DISCUSSION In this paper, we derive the exact expressions of downlink coverage probabilities for typical onshore, offshore and nearshore users, in Theorem 1, 2 and 3, respectively, where the typical users select the HAPS with strongest average received power to access. We derive the distribution of the distance between the user and its nearest HAPS in Lemma 1 for the onshore and offshore users. We also derive the distributions of the distances between the user and its nearest HAPSs operating in Rician channel or shadowed-Rician channel in Lemma 2 and Lemma 3, respectively, for the nearshore performance analysis. We also propose approximations for nearshore users in Theorem 4 and 5, by overestimating the shadowed Rician region or overestimating the Rician region, respectively, where the distance distribution for onshore and offshore users are approximated in Lemma 4 and 5. Inspired by the islands and maritime applications in the Caribbean, we assume that HAPSs are deployed in a 3000 km × 3000 km area and conduct 20,000 Monte Carlo simulations. The altitude of the typical user is 0 m and the altitude of HAPSs is hH = 20 km. We assume that the radius of an island is 20 km and the altitude of island buildings is hI = 20 m. The transmit power is pt = 45 dBm, the carrier frequency is fc = 2 GHz and the speed of light is c = 3.0 × 108 m/s. The transmit gain is gt = 41 dB and the receive gain is gr = 10 dB. The
Fig. 6. The coverage probability considering different relative distance to the island boundary rU − rI , compared to the approximations and coverage performance of remote maritime users, when HAPS density is 3 HAPSs per 105 km2 .
Fig. 7. The coverage probability considering different relative distance to the island boundary rU − rI , compared to the approximations and coverage performance of remote maritime users, when HAPS density is 1 HAPSs per 104 km2 .
atmospheric attenuation parameters are sI = sO = 2 dB and the noise power is N0 = 1 × 10−20 W. We assume that αI = 4.0, mI = 10, ΩI = 0.835, and bI = 0.126 for the shadowed Rician channels, and αO = 2.3, ΩO = 1.000, and bO = 0.200 for the Rician channels [36], [38]–[40]. In Fig. 6, we compare the simulated coverage probability with the approximations. We assume that τ = −10 dB and λH = 3 HAPSs per 105 km2 . When rU < rI , the typical user is onshore and its coverage performance can be evaluated using (6) in Theorem 1, where Pcov (rU , τ ) is 0.58. The coverage performance increases to 0.82 because more than half of HAPSs obtain the LoS condition to the user. The closest HAPS operating in the Rician channel can provide higher signal power and other HAPS operating in the Rician channel do not lead to high interference when the HAPS density is low. When rI < rU < rI +100 m and rU increases, the probability that the typical user associates to HAPSs via the Rician channel increases, so that the coverage probability increases to 0.86. When rU > rI + 100 m and rU continues to increase, more HAPSs have LoS conditions, which leads to increased interference and decreased coverage performance. When rU increases, the coverage probability Pcov (rU , τ ) approaches 0.79, where almost all HAPSs have Rician channels between the typical user. The analytical curve matches the simulation results. In addition, the simulation result is closer to the Approximation 1 in Theorem 4 than the Approximation 2 in Theorem 5, while both of them require lower computation time than the exact expression. Approximation 2 is close to the offshore case when rU > rI because we overestimate the Rician region. In Fig. 7, we show the coverage performance when the HAPS density is λH = 1 HAPSs per 104 km2 . Unlike the low HAPS density case, onshore users have better coverage probability than offshore users. When the typical user crosses the boundary of the island, the coverage probability decreases from 0.87 to 0.81 because HAPSs operating in Rician channels leads to high interference. When rI < rU < rI + 100 m and rU continues to increase, the probability that the typical user associates with the closest HAPS operating in the Rician channel increases so the coverage probability also increases. When rU > rI + 100 m, the coverage probability decreases
Fig. 8. The coverage probability considering different relative distance to the island boundary rU − rI , compared to the approximations and coverage performance of remote maritime users, when HAPS density is 1 HAPSs per 103 km2 .
due to increased interference caused by the increasing density of HAPSs. As shown in Fig. 8, when the HAPS density is λH = 1 HAPSs per 103 km2 , as the relative distance to the island boundary increases, the interference increases when more HAPSs have better LoS conditions, so that the coverage probability continues to decrease. In Fig. 9, Fig. 10 and Fig. 11, we show the relationship between coverage probability Pcov (rU , τ ) and the SINR threshold τ , considering different HAPS densities. When λH = 3 HAPSs per 105 km2 and τ < −2 dB, offshore users and nearshore users with rU = rI + 100 m have better coverage performance than onshore users. However, when τ > 2 dB, onshore users have better coverage performance than offshore users and nearshore users. This is because the increase in τ makes the coverage probability more sensitive to the value of interference, but the lack of LoS conditions and the higher path loss exponent lead to lower interference than onshore and nearshore cases. When λH = 1 HAPSs per 104 km2 , the onshore users have better coverage performance than offshore users and nearshore users. Moreover, when the HAPS density is λH = 1 HAPSs per 103 km2 , onshore users have much better coverage performance than offshore users and nearshore users. For example, when −10 dB < τ < −5 dB, onshore coverage probability is between 0.6 and 0.9, but the offshore and nearshore coverage probabilities are between 0.05 and 0.5. As shown in Fig. 12, we show the relationship between HAPS density λH and the coverage probability of onshore, nearshore, and offshore users. When λH > 8.70 HAPSs per 105 km2 , offshore users have a better coverage performance than nearshore and offshore users. Onshore users can obtain optimal coverage performance when the HAPS density is around λH = 4 HAPSs per 104 km2 , because a higher density can help the typical user connect to closer HAPSs. Unlike onshore users, offshore users have better coverage performance than onshore users when λH < 6.31 HAPSs per 105 km2 . The coverage probability can achieve 0.9 when the HAPS density is close to 1 HAPSs per 105 km2 , and the increase in the HAPS density leads to higher interference to the typical user. The increase in HAPS density also leads to a decrease in the coverage probability of nearshore users with rU = rI + 100 m.
Fig. 9. The CCDF of SINR threshold when λH = 3 HAPSs per 105 km2 , considering the onshore case (rU = rI − 100 m), nearshore case (rU = rI + 100 m), and offshore case (rU = rI + 1000 m).
Fig. 10. The CCDF of SINR threshold when λH = 1 HAPSs per 104 km2 , considering the onshore case (rU = rI − 100 m), nearshore case (rU = rI + 100 m), and offshore case (rU = rI + 1000 m).
Fig. 11. The CCDF of SINR threshold when λH = 1 HAPSs per 103 km2 , considering the onshore case (rU = rI − 100 m), nearshore case (rU = rI + 100 m), and offshore case (rU = rI + 1000 m).
Terrain Map of Jamaica
Coverage Map of Jamaica
Fig. 13. Coverage map (from GSMA) and terrain map of Jamaica. Fig. 12. The relationship between coverage probability and HAPS density, considering the onshore case (rU = rI − 100 m), nearshore case (rU = rI + 100 m), and offshore case (rU = rI + 1000 m).
As shown in the results in Fig. 6-12, Approximation 2 is close to the offshore performance because the Rician region is overestimated and the hybrid channel environment in a finite simulation area is almost reduced to a Rician environment like the offshore case. Approximation 1 is closer to the actual coverage curve because the channel conditions in fewer areas are misestimated. Two approximations can help evaluate the range of nearshore coverage performance with less waste of computational resource sensing information storage of performance analysis when the accuracy is not strict. They only require the angle range, altitude and minimum/maximum distance to the blockage clusters. Our proposed mathematical framework for coverage analysis can help provide reference to estimate coverage performance and build digital twins. As shown in Fig. 13, we took Jamaica as an example. Without HAPS deployment, the existing terrestrial network of Flow already cover most of areas. However, the deep forest areas and deep sea areas still lack of coverage. When the HAPS density in the whole Caribbean Zone is 1 HAPSs per 104 km2 , there is one or two HAPSs will be deployed upon the Jamaica. Our proposed methods in Theorem 1 and Theorem 2 can help evaluate the coverage performance, where the coverage probabilities of offshore and onshore users are higher than 0.75.
90° 35 km
692m 15 km
Fig. 14. Approximation methods for nearshore performance analysis on Jamaica.
Complex terrain make the nearshore coverage performance analysis more difficult than the ideal model. As shown in Fig. 14, through radar or camera, the considered user can sense the obstruction of forest or mountains within a range of about 90 degrees. The main terrain affecting the LoS paths between the considered user and HAPSs is on the circular area with radius rI = 35 km and altitude hI = 700 m. The distance between the user and this circular area is rU = rI + 15 km. Adopting these parameters, the coverage probability of the considered nearshore user can be approximated using Theorem 4 and Theorem 5. However, in this paper, some ideal assumptions of shadowing effect, reflections and ducting effect are adopted, to first build a mathematical framework evaluating the effect of user location, HAPSs’ location and finite shadowing zones
on the downlink performance. In the future, more complex electromagnetic environment analysis can be realized and visualized by digital twins with ray tracing technique and sensing networks. It is worth noting that, in this paper, we assume that the HAPSs interfere with each other without beamforming optimization, so that the coverage performance does not exceed 0.9. With the massive antenna array and the decoding technique, the overall performance of the HAPS network can be improved. The results shown in this paper can work as baseline or lower bound and the evaluation approaches can still work by further taking the beamforming gains into consideration. Our analysis and numerical results mainly show how to ensure user fairness by customizing the HAPS deployment. By comparing the onshore, nearshore and offshore performance, we propose to deploy less HAPSs in open areas without severe shadowing effect, to let each HAPS cover as large maritime areas as possible, but carefully select a higher HAPS density to cover onshore and nearshore users facing the shadowing effect. Based on this, the HAPS operators can design the customized HAPS deployment more easily, and realize user fairness. Furthermore, the uplink analysis and joint analysis are expected to be conducted in the future, to satisfy different applications with various user distribution and communication requirements.
By taking the derivative of Fζ (ζ), the PDF of ζ is: fζ (ζ) = 2λH πζ exp −λH πζ 2 , which R ∞satisfies: R ∞ fζ (ζ)dζ −λH πζ 2 dζ 0 R ∞ = 0 2λH πζ exp = hH 2λH π exp λH πh2H d exp −λH πd2 dd R∞ = hH fD (d)dd = 1.
A PPENDIX A P ROOF OF L EMMA 1 In this paper, we assume that HAPSs are uniformly deployed at altitude hH , where the set of their locations follows a 2D PPP with density λH . We assume that zU = (−rU , 0, 0) is the location of the typical user and z′U = (−rU , 0, hH ) is directly above zU at a height of hH . The distance between z′U and its closest HAPS to the typical user is ζ, and the distance between the typical user at zU and the closest HAPS is D. The relationship between D and ζ follows the Pythagorean theorem: D2 = ζ 2 + h2H . The CDF of ζ can be expressed as: Fζ (ζ) = 1 − exp −λH πζ 2 . (44)
(46)
Therefore, the PDF of d is fD (d) = 2λH π exp λH πh2H d exp −λH πd2 ,
(47)
for hH < d < ∞. A PPENDIX B P ROOF OF T HEOREM 1 Notice that rU is the distance from the typical user to the center of the island, and d is the distance between the user and its closest HAPS located at xI1 . We use WI,1 to represent the channel gain between them, which follows the shadowed Rician distribution shown in (2) and (3). Therefore, the coverage probability can be calculated using the conditional coverage probability and the PDF of D: △
P covR(rU , τ ) = P{SINRI (rU ) > τ } ∞ = hH P{SINRI (rU ) > τ |d}fD (d)dd R ∞ pI,0 WI,1 d−αI = hH P N0 +II (d) > τ fD (d)dd R∞ = hH F WI,1 gI (d) fD (d)dd,
(48)
mI △ I F WI,1 (x) = 2bI2bmIIm+Ω I n P∞ (mI )n ΩI × n=0 (1) Γ n + 1, 2bxI . n n! 2bI mI +ΩI
(49)
VI. C ONCLUSION In this paper, we introduced the evaluation methods of HAPS-based solutions for mobile users in islands or maritime regions. We evaluated the expressions of the coverage probabilities for onshore, nearshore, and offshore users. For nearshore users, we proposed two approximations, by overestimating the shadowed Rician region (Approximation 1) or overestimating the Rician region (Approximation 2), from which we can reduce the computation complexity. We verified that onshore users require a higher HAPS density, and offshore users require a lower HAPS density. Approximation 1 works better than Approximation 2, and both of them can help to evaluate the effect of any complex shape of islands on nearshore users. We also showed how the decoding SINR threshold and HAPS density affect the coverage performance of onshore, nearshore, and offshore users. The simulation results show that an inhomogeneous HAPS deployment is necessary in the future to realize higher and fair coverage performance for typical users at different geographical locations.
(45)
where
Γ(s, x) is the upper incomplete Gamma function with shape τ τ parameter s and gI (d) = pI,0 N0 dαI + pI,0 II (d)dαI . The interference can be approximated using its expectation: P I I I (d) ≈ E( pI,0 WI,k d−α I,k ) dI,k >d
(a)
R 2π R ∞ αI = pI,0 E[WI,k ] 0 √d2 −h2 λH (ζ 2 + h2H )− 2 ζdζdγ H
d
H 2−αI = pI,0 E[WI,k ] 2πλ , αI −2 z ∞
(50) where step (a) comes from Campbell theorem [23]. Because the expectation of WI,k is: E[WI,k ] = 2bI + ΩI , so that α d I +ΩI )d I 0 αI gI (d) ≈ τpN d + 2πλH τ (2b z 2−αI ∞ . αI −2 I,0 A PPENDIX C P ROOF OF T HEOREM 2 Notice that rU is the distance from the typical user to the center of the island, and d is the distance between the user and its closest HAPS located at xO 1 . We use WO,1 to represent the channel gain between them, which follows the Rician distribution in (8) and (9). Therefore, the coverage probability can be calculated using the conditional coverage probability and the PDF of D: △
P covR(rU , τ ) = P{SINRO (rU ) > τ } ∞ = hH P{SINRO (rU ) > τ |d}fD (d)dd R ∞ p WO,1 d−αO = hH P O,0 > τ fD (d)dd N0 +IO (d) R∞ = hH F WO,1 gO (d) fD (d)dd,
(51)
q » △ gO (d) ΩO . Q1 (a, b) is where F WO,1 gO (d) = Q1 , bO bO the Marcum Q-function of first kind of order ν = 1 and τ τ N0 dαO + pO,0 IO (d)dαO . The interference can gO (d) = pO,0 be approximated using its expectation: P O I O (d) ≈ E( pO,0 WO,j d−α O,j ) dO,j >d
(b)
R 2π R ∞ αO = pO,0 E[WO,j ] 0 √d2 −h2 λH (ζ 2 + h2H )− 2 ζdζdγ d
(52) where step (b) comes from Campbell theorem. Because the expectation of WO,j is: E[WO,j ] = 2bO +ΩO , so that gO (d) ≈ α τ N0 αO O +ΩO )d O 2−αO d + 2πλH τ (2b z . pO,0 d αO −2 ∞ A PPENDIX D P ROOF OF L EMMA 2 To evaluate the coverage probability for nearshore users with a hybrid channel environment, we define that HAPSs in the Rician » region follow a PPP with density λH . We define dth,1 = r12 + h2H where r1 = hhHI (rU − rI ) and dth,2 = » » 2 − r 2 . We discuss CDFs and r22 + h2H where r2 = hhHI rU I PDFs of dO,1 for the cases where hH < dO,1 < dth,1 , dth,1 < dO,1 < dth,2 and dth,2 < dO,1 < ∞, respectively. We assume that the area of the Rician region within a distance dO,1 is SO (dO,1 ). For the case where hH < dO,1 < dth,1 , we have SO (dO,1 ) = πd2O,1 − πh2H . (53) Therefore, the CDF of dO,1 can be calculated using: F (dO,1 )= 1 − exp(−λ Ä H SO (dO,1 ä )) (54) = 1 − exp −λH πd2O,1 exp λH πh2H , and the PDF of dO,1 is the derivative of F (dO,1 ), which is equal to the multiplication of HAPS density λH , angle range of Rician region at dO,1 which is 2π, the distance dO,1 , and the CCDF of dO,1 : f (dO,1 ) = 2λH πdO,1 exp −λH πd2O,1 exp λH πh2H . (55) For the case where dth,1 < dO,1 < dth,2 , we assume that ′ the » intersections of the circle centered at zU with radius ζ = d2O,1 − h2H , and the circle centered at o′I with radius rI hhHI as P1 and P2 . We can divide SO (dO,1 ) into two parts S1 (dO,1 ) and S2 (dO,1 ). S1 (dO,1 ) is a sector area centered at z′U with radius dO,1 , central angle 2π − 2ϕ(dO,1 ), where 2 h2 H −r 2 hH I h2 h2 I I h 2ζrU hH I 2 2 h2I d2O,1 −h2I h2H +rU hH −rI2 h2H 2 2 2 dO,1 −hH rU hH hI
ϕ(dO,1 ) = arccos = arccos
√
(56)
.
Especially, ϕ(dth,1 ) = 0 and ϕ(dth,2 ) = arcsin rrUI . Therefore, we have S1 (dO,1 ) = (π − ϕ(dO,1 ))(d2O,1 − h2H ). (57) We obtain S2 (dO,1 ) by subtracting the sector area centered at o′I with radius rI hhHI and central angle η(dO,1 ) from the quadrilateral z′U P1 o′I P2 , where 2 h2 2 hH rU +rI2 H −ζ 2 h2 h2 I I h h 2rI hH rU hH I I 2 2 rU hH +rI2 h2H −d2O,1 h2I +h2H h2I 2 2rI rU hH
η(dO,1 ) = arccos = arccos
h2
h2
I
I
+ sin η(dO,1 )rI rU hH2 − η(dO,1 )rI2 hH2 .
H z 2−αO ∞ , = pO,0 E[WO,j ] α2πλ O −2
2 ζ 2 +rU
Especially, η(dth,1 ) = 0 and η(dth,2 ) = arccos rrUI . Therefore, we have: h2 h2 S2 (dO,1 ) = sin η(dO,1 )rI rU H − η(dO,1 )rI2 H , (59) 2 hI h2I and SO (dO,1 )= S1 (dO,1 ) + S2 (dO,1 ) = (π − ϕ(dO,1 ))(d2O,1 − h2H ) (60)
(58)
.
The CDF can be formulated using F (dO,1 ) = 1Ä− exp(−λH SO (dO,1 )) ä = 1 − exp −λH (π − ϕ(dO,1 ))(d2O,1 − h2H ) h2 h2 × exp −λH sin η(dO,1 )rI rU hH2 + λH η(dO,1 )rI2 hH2 , I I (61) and the PDF is the derivative of F (dO,1 ), which is also equal to the multiplication of HAPS density λH , angle range of Rician region at dO,1 which is 2ϕ(dO,1 ), the distance dO,1 , and the CCDF of dO,1 , which is: f (dO,1 )Ä= 2λH (π − ϕ(dO,1 ))dO,1 ä × exp −λH (π − ϕ(dO,1 ))(d2O,1 − h2H ) h2 h2 × exp −λH sin η(dO,1 )rI rU hH2 exp λH η(dO,1 )rI2 hH2 . I I (62) For the case where dO,1 > dth,2 , S1 (dO,1 ) has a central angle θ instead of ϕ(dO,1 ) in (57): S1 (dO,1 ) = (π − θ)(d2O,1 − h2H ) and S2 (dO,1 ) can be obtained by subtracting the sector area centered at o′I with radius rI hhHI and central angle π2 − θ from the quadrilateral z′U P1 o′I P2 where ∠z′U P1 o′I = ∠z′U P2 o′I = π2 . The area S2 (dO,1 ) can be calculated by substituting dth,2 into dO,1 in (59), so that h2 S2 (dO,1 ) = rI2 hH2 cot θ − ( π2 − θ) and I
S O (dO,1 ) = S1 (dO,1 ) + S2 (dO,1 ) h2 = (π − θ)(d2O,1 − h2H ) + rI2 hH2 cot θ − ( π2 − θ) .
(63)
I
The CDF can be formulated using: F (dO,1 ) = 1Ä− exp(−λH SO (dO,1 )) ä = 1 − exp −λH (π − θ)(d2O,1 − h2H ) h2 h2 × exp −λH cot θrI2 hH2 + λH ( π2 − θ)rI2 hH2 , I
I
and the PDF can be formulated using: f (dO,1 )Ä= 2λH (π − θ)dO,1 ä × exp −λH (π − θ)(d2O,1 − h2H ) h2 h2 × exp −λH cot θrI2 hH2 + λH ( π2 − θ)rI2 hH2 . I
(64)
(65)
I
We can summarize the PDF of dO,1 as: f (dO,1 ) = 2λH dO,1 (1 − F (dO,1 )) ×(π − ϕ(dO,1 )1(dth,1 < dO,1 < dth,2 ) − θ1(dO,1 > dth,2 )). (66) A PPENDIX E P ROOF OF L EMMA 3 To evaluate the coverage probability for nearshore users within a hybrid channel environment, we also assume that HAPSs in the shadowed Rician region follow a PPP with density λH . We discuss CDFs and PDFs of dI,1 for the cases where dth,1 < dI,1 < dth,2 and dth,2 < dI,1 < ∞,
respectively. We define that the area of the shadowed Rician region within a distance dI,1 is SI (dI,1 ). For the case where dth,1 < dI,1 < dth,2 , we can calculate SI (dI,1 ) using the subtracting S2 (dI,1 ) from the sector area centered at z′U with radius dO,1 and central h2 angle 2ϕ(dI,1 ), where S2 (dI,1 ) = sin η(dI,1 )rI rU hH2 − I
h2
η(dI,1 )rI2 hH2 , and S3 (dI,1 ) = ϕ(dI,1 )(d2I,1 − h2H ), with I h2 d2 −h2 h2 +r 2 h2 −r 2 h2 ϕ(dI,1 ) = arccos I I,1√ 2I H 2 U H I H and η(dI,1 ) = 2 dI,1 −hH rU hH hI 2 2 rU hH +rI2 h2H −d2I,1 h2I +h2H h2I arccos . Therefore, SI (dI,1 ) can 2rI rU h2H be calculated using: S I (dI,1 ) = S3 (dI,1 ) − S2 (dI,1 ) = ϕ(dI,1 )(d2I,1 − h2H ) h2 h2 − sin η(dI,1 )rI rU hH2 + η(dI,1 )rI2 hH2 . I I
(67)
The CDF can be formulated as: F (dI,1 ) = 1 Ä− exp(λH SI (dI,1 )) ä = 1 − exp −λH ϕ(dI,1 )(d2I,1 − h2H ) h2 h2 × exp λH sin η(dI,1 )rI rU hH2 − λH η(dI,1 )rI2 hH2 , I
I
(68)
and the PDF of dI,1 is: ä Ä f (dI,1 ) = 2λH ϕ(dI,1 )dI,1 exp −λH ϕ(dI,1 )(d2I,1 − h2H ) h2 h2 × exp λH sin η(dI,1 )rI rU hH2 − λH η(dI,1 )rI2 hH2 . I I (69) For the case where dI,1 > dth,2 , the area can be divided into two parts, the first part is the » intersection of the circular area centered at x′U with radius d2th,2 − h2H , and the circular area centered at o′I with radius rI hhHI . We have S3 (dI,1 ) = h2
h2
θ(d2th,2 − h2H ), and S2 (dI,1 ) = cot θrI2 hH2 − ( π2 − θ)rI2 hH2 , so I I that the area of the first part is S3 (dI,1 )−S2 (dI,1 ). The second part is an annular sector centered at z′U with inner radius dth,2 , outer radius dI,1 . Therefore the areas of the second part is: S4 (dI,1 ) = θ(d2I,1 − d2th,2 ). Therefore, SI (dI,1 ) can be calculated using S I (dI,1 ) = S3 (dI,1 ) + S2 (dI,1 ) + S4 (dI,1 ) (70) h2 h2 = θ(d2I,1 − h2H ) + ( π2 − θ)rI2 hH2 − cot θrI2 hH2 . I I Therefore, the CDF can be calculated using F (dI,1 ) = 1 Ä− exp(−λH SI (dI,1ä)) = 1 − exp −λH θ(d2I,1 − h2H ) (71) 2 2 π 2 hH 2 hH × exp λH cot θrI h2 − λH ( 2 − θ)rI h2 , I
I
and the PDF of dI,1 is Ä ä f (dI,1 ) = 2λH θdI,1 exp −λH θ(d2I,1 − h2H ) h2 h2 × exp λH cot θrI2 hH2 − λH ( π2 − θ)rI2 hH2 . I
(72)
I
We can summarize the PDF of dI,1 as: f (dI,1 ) = 2λH dI,1 (1 − F (dI,1 )) ×(ϕ(dI,1 )1(dth,1 < dI,1 < dth,2 ) + θ1(dI,1 > dth,2 )). (73) A PPENDIX F P ROOF OF T HEOREM 3 As shown in (16), we can calculate the coverage probability using the sum of two terms: Pcov (rU , τ ) = Pcov (rU , τ, HO ) + Pcov (rU , τ, HI ). (74)
Similar to the offshore case, the coverage probability conditioned with TX selection case HO can be calculated using: Pcov (rU , τ |HO , dO,1 , dI,1 ) = Pcov (rU , τ |dI,1 > TI (dO,1 ), dO,1 ) O pO,0 WO,1 d−α O,1 h (dO,1 , dI,1 ) , > τ = F WO,1 gO = P N0 +I(dO,1 ,d I,1 ) (75) τ h O where gO (dO,1 , dI,1 ) = pO,0 dα N + I(d , d ) 0 O,1 I,1 . We O,1 » 2 hH 2 sin t), so that the define Td (t) = hI (rU cos t − rI2 − rU interference from HAPS based on dO,1 and dI,1 is I(dO,1 , dI,1 II (dI,1 ) P P) = IO (dO,1 ) +−α I ≈ E( pO,0 WO,j dO,jO + pI,0 WI,k d−α I,k ) dO,j >dO,1
dI,k >dI,1 (2π−2θ)λH 2−αO dO,1 = pO,0 (2bO + ΩO ) αO −2 z ∞ Rθ min{dO,1 ,Td (t)} +pO,0 (2bO + ΩO ) αλOH−2 −θ z 2−αO T (t) dt d R θ 2−α max{d I,1 ,Td (t)} λH I +pI,0 (2bI + ΩI ) αI −2 −θ z dt. ∞
(76) Therefore, the coverage probability with TX selection case HO can be calculated using: P covR(rU ,Rτ, HO ) ∞ ∞ = hH dth,1 Pcov (rU , τ |HO , dO,1 , dI,1 ) ×f (dO,1 )f (dI,1 ) ddI,1 ddO,1 R∞ R∞ = hH max{dth,1 ,TI (dO,1 )} Pcov (rU , τ |dO,1 , dI,1 ) (77) ×f (d )f (d ) dd dd O,1 I,1 I,1 O,1 R∞ R∞ h (dO,1 , dI,1 ) = hH max{dth,1 ,TI (dO,1 )} F WO,1 gO ×f (dO,1 )f (dI,1 ) ddI,1 ddO,1 . Similar to the onshore case, the coverage probability with TX selection case HI can be calculated using: Pcov (rU , τ |HI , dO,1 , dI,1 ) = Pcov (rU , τ |dO,1 > TO (dI,1 ), dI,1 ) I pI,0 WI,1 d−α I,1 > τ = F WI,1 gIh (dO,1 , dI,1 ) , = P N0 +I(dO,1 ,d I,1 ) (78) τ I where gIh (dO,1 , dI,1 ) = pI,0 dα I,1 N0 + I(dO,1 , dI,1 ) . The interference is shown in (76) and the coverage probability with TX selection case HI can be calculated using: P covR(rU , τ,R HI ) ∞ ∞ = dth,1 hH Pcov (rU , τ |HI , dO,1 , dI,1 ) ×f (dI,1 )f (dO,1 ) ddO,1 ddI,1 R∞ R∞ = dth,1 max{hH ,TO (dI,1 )} Pcov (rU , τ |dO,1 , dI,1 ) (79) ×f (d )f (d ) dd dd O,1 I,1 I,1 O,1 R∞ R∞ = dth,1 max{hH ,TO (dI,1 )} F WI,1 gIh (dO,1 , dI,1 ) ×f (dO,1 )f (dI,1 ) ddO,1 ddI,1 . A PPENDIX G P ROOF OF L EMMA 4 Based on the approximation method 1, we model the HAPSs operating in the Rician channels and the HAPSs operating in the shadowed Rician channels using two independent PPPs with density λH , respectively. We discuss the cases where hH < dI,1 < dth,1 and dth,1 < dI,1 < ∞. When hH < dI,1 < dth,1 , the area of the Rician region within the distance dth,1 to ′ zU is SO (dO,1 ) = π(d2O,1 − h2H ), and when dth,1 < dI,1 < ∞ the area of the Rician region contains a sector area centered at z′U , with radius dO,1 and central angle 2π − 2θ, and a sector area centered at z′U with radius dth,1 . Therefore the area of the
′ Rician region is SO (dO,1 ) = (π − θ)(d2O,1 − h2H ) + θ(d2th,1 − 2 hH ). In short, we have ′ SO (dO,1 ) = (π − θ)d2O,1 + θ min{dO,1 , dth,1 }2 − πh2H . (80) Therefore, the CDF of the distance between the typical user and its closest HAPS in Rician region dO,1 is: ′ F (dO,1 ) = 1Ä− exp(−λH SO (dO,1 )) ä 2 = 1 − exp −λH (π − θ)dO,1 − λH θ min(dO,1 , dth,1 )2 × exp λH πh2H , (81) and the PDF of dO,1 is the derivative of F (dO,1 ): f (dO,1 )Ä= 2λH dO,1 (π − θ) + θ1(d < dth,1 ) ä × exp −λH (π − θ)d2O,1 − λH θ min(dO,1 , dth,1 )2 × exp λH πh2H , (82) for dO,1 > hH . Similarly, the Rician region within the distance dI,1 is an annular sector area centered at z′U , with inner radius r1 , infinite outer radius and central angle 2θ. Therefore, the area of the Rician region within the distance dI,1 is: SI′ (dI,1 ) = θ(d2I,1 − d2th,1 ). (83) Therefore, the CDF of dI,1 can be written as: ′ F (dI,1 )= 1 − exp(−λ ä Ä H SI (dI,1 )) (84) = 1 − exp −λH θ(d2I,1 − d2th,1 ) , and the PDF of dI,1 is the derivative of F (dI,1 ): f (dI,1 ) = 2λH θdI,1 exp −λH θ(d2I,1 − d2th,1 ) , (85) for dI,1 > dth,1 .
A PPENDIX H P ROOF OF T HEOREM 4 As shown in (16), we can calculate the coverage probability using the sum of two terms: Pcov (rU , τ ) = Pcov (rU , τ, HO ) + Pcov (rU , τ, HI ). (86) The same as the proof of Theorem 3, we can also obtain the coverage probabilityRof nearshore users with HO is ∞ R∞ P cov (rU , τ, HO ) = hH max{dth,1 ,TI (dO,1 )} h (dO,1 , dI,1 ) f (dO,1 )f (dI,1 ) ddI,1 ddO,1 , F WO,1 gO (87) τ h O with gO (dO,1 , dI,1 ) = pO,0 dα N + I(d , d ) , and the 0 O,1 I,1 O,1 coverage probability of nearshore users with HI is R∞ R∞ P cov (rU , τ, HI ) = dth,1 max{hH ,TO (dI,1 )} (88) F WI,1 gIh (dO,1 , dI,1 ) f (dO,1 )f (dI,1 ) ddO,1 ddI,1 , τ I with gIh (dO,1 , dI,1 ) = pI,0 dα I,1 N0 + I(dO,1 , dI,1 ) . Differently, based on the approximation method 1, we have approximate CDFs and PDFs of dO,1 and dI,1 formulated in Lemma 4. In this case, interference conditioned on dO,1 and dI,1 can be calculated using it expectation I(dO,1 , dI,1 II (dI,1 ) P P) = IO (dO,1 ) +−α I ≈ E( pO,0 WO,j dO,jO + pI,0 WI,k d−α I,k ) dO,j >dO,1
dI,k >dI,1 dO,1
H 2−αO = pO,0 (2bO + ΩO ) (2π−2θ)λ z αO −2 ∞
min{dO,1 ,dth,1 }
H +pO,0 (2bO + ΩO ) α2θλ z 2−αO d O −2
th,1
dI,1
2θλH 2−αI +pI,0 (2bI + ΩI ) α z , ∞ I −2
(89) which has a lower computation complexity than the interference for exact expression of coverage probability.
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