Three-hop Underwater Wireless Communications

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A Novel Relay Deployment Technique

Item Type Article

Authors Xu, Jiajie;Kishk, Mustafa Abdelsalam;Zhang, Qunfei;Alouini, Mohamed-Slim

Citation Xu, J., Kishk, M. A., Zhang, Q., & Alouini, M.-S. (2023). Three- hop Underwater Wireless Communications: A Novel Relay Deployment Technique. IEEE Internet of Things Journal, 1–1.

https://doi.org/10.1109/jiot.2023.3262949 Eprint version Post-print

DOI 10.1109/jiot.2023.3262949

Publisher Institute of Electrical and Electronics Engineers (IEEE) Journal IEEE Internet of Things Journal

Rights (c) 2023 IEEE. Personal use of this material is permitted.

Permission from IEEE must be obtained for all other users, including reprinting/ republishing this material for advertising or promotional purposes, creating new collective works for resale or redistribution to servers or lists, or reuse of any copyrighted components of this work in other works.

Download date 2023-10-31 08:15:23

Link to Item http://hdl.handle.net/10754/690800

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Three-hop Underwater Wireless Communications: A Novel Relay Deployment Technique

Jiajie Xu¹, Student Member, IEEE, Mustafa A. Kishk², Member, IEEE, Qunfei Zhang³, Senior Member, IEEE, Mohamed-Slim Alouini¹, Fellow, IEEE

Abstract—Underwater long-distance wireless communication (ULWC) is a critical challenge in many applications, such as marine environmental monitoring, underwater remote control, and underwater navigation, to name a few. However, very little literature focuses on ULWC, especially where the communication distance ups to thousands of kilometers, which is urgently required in the underwater Internet of Things (UIoT) in the large-scale and deep sea. In this paper, to improve the underwater communication capacity at a level of thousands of kilometers, we propose a three-hop underwater wireless acoustic communication (3H-UWAC) structure based on the Sound Fixing and Ranging (SOFAR) channel. The proposed 3H-UWAC consists of transmitters, relay stations (RSs), and receivers. Different from the existing ULWC, 3H-ULWC can improve energy efficiency with a small vertical directivity angle (VDA). Due to the characteristics of UWAC, the straight-line communication link can be realized in the proposed three hops, and the communication distance can be increased to thousands of kilometers. Respecting the randomness of underwater devices, tools from stochastic geometry are used to model the spatial distributions of transmitters’, receivers’, and RSs’ locations. RSs are set on the SOFAR channel at a known depth. In the first hop, the transmitter sends information to the nearest first relay station (NFRS) on the SOFAR plane. In the second hop, the NFRS sends information to the nearest relay station, which is called the nearest second relay station (NSRS), to the receiver on the SOFAR plane. In the third hop, SNFS sends information to the receiver. All three communication hops can be achieved with a narrow beam width, where the energy efficiency is improved critically. With given densities of transmitters, RSs, and receivers, the coverage probabilities (CPs) of the three hops (transmitter to FRS, FRS to SRS, and SRS to receiver) are analyzed, and the final CP from a transmitter to a receiver through the 3H link is derived. Insights about the effects of VDAs at the transmitters, FNRS, and SNRS, as well as the depths of transmitters and receivers, are revealed. A rapid optimization method is proposed based on the analytical results. The accuracy of the analysis is verified by Monte-Carlo simulations.

Index Terms—Underwater Long-distance Wireless Communi- cation, Underwater Multihop Communication, Wireless Acoustic Network, UIoT, Stochastic Geometry, Coverage Probability.

I. INTRODUCTION

A. Motivation

With the exploration of undersea resources and the development of underwater technologies, short or medium-range underwater acoustic communication (UWAC) methods and

1 Computer, Electrical, and Mathematical Science and Engineering Di- vision, King Abdullah University of Science and Technology, Thuwal, Saudi Arabia;²Department of Electronic Engineering, Maynooth University, Maynooth, W23 F2H6, Ireland;³ School of Marine Science and Technol- ogy, Northwestern Polytechnical University, Xi’an, 710072, PR China. e- mail: [email protected], [email protected], [email protected], [email protected].

algorithms have been extended to achieve good performance [1]–[4]. Some technologies used in the terrestrial network can also be used in the underwater network to solve relative problems [5]–[8]. However, different from the radio frequency (RF) signal, the underwater acoustic signal propagation path is a curve (which will be discussed in more detail). When the transmitter does not know the location of the receiver, it can only use the broadcast communication method. In addition, the underwater acoustic signal attenuates quickly and consequently. So, the methods and strategies of normal UWAC can not be used in underwater long-distance wireless communication (ULWC). Furthermore, a short/medium-distance UWAC can not comprehensively promote the development of underwater applications, and ULWC is essential for some application scenarios. For example, communication between two Internet of Things (UIoT) groups in the Pacific Ocean, navigation to the submarine in the Atlantic Ocean based on the nearshore underwater navigation system, underwater navigation base station in the Arctic Ocean provides public navigation services for global underwater vehicles, to name a few. However, there are still so many problems in ULWC that need to be solved, and there is a long way to go to realize practical applications.

In some pseudo-underwater communication scenarios [9]–

[11], the UIoT devices can apply the surface buoys as relay stations and further utilize the aerial relay station to realize the communication between the transmitter and the receiver.

Authors in [12], [13] utilize the unmanned aerial vehicle (UAV) as an aerial relay station and provide further maritime communication. Decided by the height of the UAV, the communication distance of a single UAV can reach dozens of kilometers. With a multi-relay network composed of UAVs, the communication distance may be extended to hundreds of kilometers. In pseudo-underwater communication, only the first hop (transmitter to the buoy) and the last hop (the buoy to the receiver) are underwater acoustic links, and all the other links are instead by RF communication or optical communication in the air.

However, some applications require full underwater wireless communication in a deep and large-scale ocean. One example is when the UIoT network is under the Arctic Ocean ice, a surface buoy station can not be set, and a completely underwater communication link is required. One experiment was done by the University of Washington in the Davis Strait [14]. In this experiment, sea gliders cannot access the surface and receive geolocation information with the assistance of the global navigation satellite system (GNSS). So, geolocation

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information, including range measurements from anchored acoustic navigation nodes at fixed and known locations, must be provided through long-distance UWAC. Another similar under-ice experiment for unmanned vehicle positioning and communications in the Fram Strait was presented in [15]. In this experiment, to make the best use of unmanned gliders’

capacity for long-endurance operations, a single-tube source is used for transmitting long-distance communications and navigation signals. Another motivation that cannot be ignored for ULWC is the underwater navigation station (UNS), which can provide a navigation system or an accurate reference to submarines and underwater vehicles on a large scale, for example, submarine navigation in the Atlantic ocean. For interested readers, literature [16]–[19] can be referred to. It needs to be noted that the required communication distance of ULWC spans tens to thousands of kilometers. However, as far as we know, there are no related works that can provide a stable communication link for ULWC at a distance of thousands of kilometers and verify its coverage performance.

In this paper, we propose a promising method that can be used for ULWC in the deep sea on a large scale and analyze its communication coverage probability.

A summary of notations and abbreviations is provided in Table I.

B. Related work

In this subsection, we provide a summary of the related works in two general directions of interest to this paper:

(i) long-distance UWAC, (ii) long-distance UWAC based on multi-hop.

Long-distance UWAC. As the only way to realize long- distance underwater communication, UWAC has always been a hot research topic. Long-distance communication has higher requirements for energy consumption. Japan Agency for Marine-Earth Science and Technology designed the Janus- Hammer Bell (JHB) transducer and finished some experiments with high electro-acoustic efficiency, as shown in [20]. In the experiments, the communication system can achieve a baud rate of 100 bits/s with the communication of 1000 km on the bandwidth of 450 ∼ 550 Hz. Authors in [16]

design a long-distance UWAC system. In the channel model, a rapid transition from shelf zone to deep-sea consisting of ray patterns and channel impulse responses is applied. However, in [16], the communication distance is at a level of 300 km.

Authors in [21] propose a sparse channel estimation method to improve the channel-to-reconstruction error ratio (CRER) and the bit error rate (BER) at a distance of 100 km. Other techniques for studying long-distance UWAC from coding and channel estimation can be found in [22]–[24].

Long-distance UWAC based on multi-hop. Multi-hop technology is another way to achieve long-distance communication. To reduce the effect of collision of multi-hop, authors in [25] design the structure that a long data packet can be partitioned into smaller fragments, which can confine the disruptive impact of a crash only to a few fragments. In this way, only a few crashed fragments need to be retransmit- ted. Simulation results show that data packet fragmentation

offers benefits to throughput efficiency, end-to-end latency, and energy consumption. Authors in [26] propose a practical transport-network layer scheme to achieve high throughput and low latency data transmission in multi-hop underwater networks. In [26], BATS codes are applied. A class of efficient linear network coding schemes with a matrix generalization of fountain codes is applied as the outer code. Besides that, batch-based linear network coding is used as the inner code.

Authors in [27] simulate the situation of long-distance UWAC in the high North. In the simulation, an adaptive cross-layer routing protocol is applied. Besides that, link quality, energy consumption, and topological data are used to select the best- coded modulation and relay station in the next transmission slot.

Amplify-and-Forward (AF) and Decode-and-Forward (DF) in UWAC. Forward relays, including AF and DF, are widely studied in underwater long-distance communication [28]–

[32]. In [32], an asynchronous underwater decode-interleave- forward (UDIF) cooperative protocol is proposed to shorten the end-to-end delay and improve the communication distance. Technologies, including turbo equalization, multiuser detection, and combining techniques, are applied. Authors in [33] design an asynchronous relaying strategy to enhance the reliability of underwater long-distance communication, where a method of precoded orthogonal frequency division multiplex is used.

However, as far as we know, very few works focus on underwater long-distance wireless communication (beyond a few hundred kilometers), and most of the proposed ”underwater long-distance communication” is at a level of tens or a few hundred kilometers. Even for the aforementioned papers considering the structure of multi-hop, AF/DF, etc., the maximum mentioned communication distance is 300 km.

Hence, we believe our paper is the first one to propose the coverage probability analysis at a distance level above 300 km.

In this paper, considering the shortcomings of curve propagation of the acoustic signal in terms of energy efficiency and stability, different from the existing normal multi-hop or decode/amplify and forward techniques, a new three-hop underwater acoustic communication (3H-UWAC) model is proposed, which makes the best use of the advantages of the SOFAR plane and improves the ULWC distance ups to thousands of kilometers. Taking the dynamic environment into consideration, the coverage probability (CP) of the 3H-UWAC is analyzed. Characteristics of the 3H-UWAC are clarified, and effects made by different parameters are revealed.

C. Contributions

To provide a competitive ULWC solution in a large-scale UIoT, this paper provides one of the first attempts to model the long-distance (up to thousands of kilometers) and deep- sea acoustic communication links based on the SOFAR plane in UIoT. With respect to the randomness of the underwater nodes, there are no preset requirements about the transmitter and receiver positions. The main contributions of this article are summarized below.

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Table I: Table of Notations and Abbreviations

Notation Description

UWAC; ULWC Underwater Wireless Acoustic Communication; Underwater Long-distance Wireless Communication

3H-UWAC Three-Hop Underwater Wireless Acoustic Communication

SOFAR SOund Fixing And Ranging

FRS; NFRS; SRS; NSRS First Relay Station; Nearest First Relay Station; Second Relay Station; Nearest Second Relay Station

VDA Vertical Directivity Angle

Dtf,Df s,Dsr,Dtr Distance between the transmitter and FRS, FRS and SRS, SRS and the receiver, the transmitter, and the receiver Rtf,Rsr;Rf s Projection distance ofDtf,Dsron the X-O-Y plane; Projection distance ofDf son the Y-O-Z plane

ztf;zsr Relative depth between the transmitter and FRS; Relative depth between SRS and the receiver

θtf,θf s,θsr VDA of the transmitter, FRS, SRS

• We design a 3H-UWAC link that can be applied in a scenario of large-scale and deep-sea based on the SOFAR plane. Relays in 3H-UWAC are placed on the SOFAR plane. To reflect the sparseness of underwater communication nodes, the randomness, and the unknowability of relay nodes’ locations, no additional assumptions about locations are made in 3H-UWAC, and the particularity of underwater communication networks is fully respected.

• Transducers with small VDAs can be applied, improving energy efficiency while achieving long-distance underwater wireless communication.Considering the fact that the locations of transmitters and receivers are unknown, a robust method that aligns the transmitter directly towards up to the SOFAR plane from the deep sea with a small VDA is proposed, which can significantly improve energy efficiency.

• Final CP from a random transmitter to a random receiver is derived. Tools from stochastic geometry are applied, and the locations of transmitters and receivers are modeled as two independent Poisson Point Processes (PPPs).

Monte Carlo simulations are built to identify analysis results, and some insights are proposed.

The remainder of this article is organized as follows.

Section II introduces the background of underwater long- distance wireless communication. Section III introduces the structure and the system model of 3H-UWAC and the underwater acoustic channel model. Section IV gives detailed mathematical derivation about the CP of 3H-UWAC. Section V verifies the theoretical analysis compared with the Monte Carlo simulations, reveals the insights of the 3H-UWAC, and provides solutions to maximize the CP of 3H-UWAC.

Conclusions and future work are stated in section VI.

II. BACKGROUND

A. Challenges of ULWC in deep-sea

Indeterminate sound velocity and sound profile in the underwater environment.Different from radio frequency and optical communications, where the signal speeds are almost constant in the same medium, the propagation speed of the sound is

closely related to the state of the medium, especially in the water medium. A simplified function of sound velocity, which is widely used is [34]:

c=1449.2 + 4.623T−0.0546T²

+ (1.34−0.010T)(S−35) + 0.016Z, (1) wherecis the sound velocity in m/s, T is the temperature with the unit of^◦C,−4^◦C≤T ≤30^◦C,S is the salinity with the unit of ‰, 30 ‰≤S≤37‰,Z is the depth which denotes the pressure with the unit of m. We can see that temperature is the main factor influencing sound velocity. An example of the practical sound velocity profile (SVP) of a given location is collected by Argo Float [35] in January 2020, which is shown in Fig. 1.

Surface layer

Main thermocline

Deep isothermal layer Seasonal thermocline

Figure 1: Sound velocity profile at E143N33.

As we can see that the SVP can be divided into four layers:

i) Surface layer, where the water temperature can change with time over hours as a result of sunshine, wind, waves, etc., so does the sound velocity; ii) Seasonal thermocline, where the water temperature decreases with depth and changes with the season, so do the sound velocity; iii) the Main thermocline, where the decreasing of water temperature results from the depth and almost independent of time; iv) Deep isothermal layer, where the water temperature is almost constant, but the

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sound velocity will increase with the depth (as a result of increasing pressure).

Curve propagation of the underwater acoustic signal. The underwater sound propagation path is a curve instead of a line.

Depending on the speed at which water propagates the sound signal, the curve can be expressed as:

c1

cos(θ1) = c2

cos(θ2). (2) With a grazing angle θ1 and sound velocity c1, if c1 < c2, the propagation path will be like the solid red line in Fig. 2, if c1> c2, the propagation path will be like the red dotted line in Fig. 2. Based on (2) , if θ1 is 90^◦,θ2 must be 90^◦.

!_!

!_"

"_"

"_!

Underwater sound propagation path Grazing angle

Velocity profile

"_"

Figure 2: Curve propagation of underwater sound.

B. SOFAR channel/plane in underwater

According to (2), if we assume the sound velocity at the original location is the minimum in the SVP, we can get the following grazing angle will always be 0 andc2=c1. Hence, the sound will keep at the level with the minimum velocity, which means the propagation path is a straight line instead of a curve. Curve propagation produces critical scattering and may lead to absorption and reflection at the sea surface and seabed, which will cause the signal to lose a lot of energy, reducing the signal energy efficiency. On the contrary, the straight- line propagation of the signal can effectively reduce scattering and reflection and maintain the concentration of energy. An example of the straight-line propagation and curve propagation of acoustic signal is shown in Fig. 3. Curve propagation has a longer propagation distance which will also make more transmission loss.

Hence, the channel where the acoustic can propagate similarly to a line is namedSOFAR Channel(Sound Fixing and Ranging channel). It is defined as a horizontal layer of water in the ocean at which the speed of sound is at its minimum (it will be called the SOFAR plane in this paper). According to the experimentally collected data, the SOFAR plane is a plane with a depth of around 1000 m.

In the large-scale underwater communication scenario with a critical vertical depth, since the transmitter doesn’t know where the receiver is, the transmission is similar to broadcast.

To improve energy efficiency, the broadcast-like signal is angled with a beamwidth rather than omnidirectional, and the angle is named vertical directivity angle (VDA) or vertical beamwidth. VDA is defined as the angular width between the axis of the pattern and 3dB down points at the reference distance of 1 m from the source. VDAs of two different types of transducers are shown in Fig. 4, forward transducer (FT)

and circular transducer (CT). A transducer with a large VDA can improve coverage performance.

However, due to the curve propagation, when a transmitter transmits signals with a large VDA, only a tiny part of the acoustic signal rays can be caught by the receiver, and a critical part of the signal will be scattered and reflected by the sea surface and seabed (the signal can’t be caught by the receiver), which causes low energy efficiency. Besides that, curve propagation can create reflection at the sea surface and seabed, which will cause serious multipath. Serious multipath is hard to deal with in UWAC. Furthermore, curve propagation will lead to caustics and convergence zones in long-distance underwater communication in the deep sea [36], [37].

Two long-distance underwater acoustic communication experiments were achieved by Japanese and French researchers in the deep ocean in [38], [39]. In that two experiments, long- distance acoustic communications performance is evaluated to provide a stable link for autonomous underwater vehicles (AUVs). The successful experiments prove the feasibility of long-distance communication for AUVs. However, all the transmitters and receivers were placed at a depth of around 1000 meters. This seemingly unintentional design is the key to the success of the long-distance underwater communication experiment. That’s because the SOFAR Channel is at a depth of around 1000 meters.

III. SYSTEMMODEL

A. Structure of the 3H-UWAC

The proposed 3H-UWAC and the normal ULWC structures are shown Fig. 5. The line tracks in blue denote the 3H-UWAC between the transmitter and the receiver via relay stations, and the curve track in manganese indicates the direct communication between the transmitter and the receiver without any relay.

During direct communication, the propagation path from the transmitter to the receiver is a curve, and the curve distance is noted asStr. When the straight-line distanceDtr between the transmitter and the receiver ups to thousands of kilometers, there will be a critical difference betweenStr andDtr.

In the proposed 3H-UWAC, relay stations placed at the side of the transmitter are noted as the first relay stations (FRSs), and relay stations placed at the side of the receiver are noted as the second relay stations (SRSs). Communication links are established between transmitters with a density ofγt

and receivers with a density ofγr through some relay stations (FRSs and SRSs) on the SOFAR plane. The distribution of relay stations, including FRS and SRS, is a PPP with a density of γrs. Communication links between the transmitter and the receiver can be described in three hops: (i) the transmitter transmits the signal to the FRS on the SOFAR plane, which is nearest the first relay station (NFRS); (ii) the NFRS transmits the signal to the SRS which is the nearest second relay station (NSRS) to the receiver; (iii) the NSRS transmits the signal to the receiver.

According to (2), we have the function of c1cos(θ2) = c2cos(θ1). Let c1 and c2 denote the sound speeds at the transmitter and FRS sides. Since FRS is on the SOFAR plane, we havec1 > c2. Whenθ1 is close to90^◦, according

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0 1 2 3 4 5

Distance (m) 10⁵

500

1000

1500

2000

Depth (m)

(a) UWAC using SOFAR plane

0 1 2 3 4 5

Distance (m) 10⁵

500

1000

1500

2000

Depth (m)

(b) UWAC without using SOFAR plane

Figure 3: UWAC based on SOFAR plane or not.

Vertical Vertical directivity angle, !

1 m

-3 dB

Horizontal

Axis

(a) Forward transducer

Vertical Vertical directivity angle, !

Horizontal 1 m

-3dB Axis

(b) Circular transducer

Figure 4: Vertical directivity angle of a transducer.

to the aforementioned function, θ2 must be close to 90^◦. Furthermore, if θ1= 90^◦, we must have θ2= 90^◦. Since the transmitter will transmit signals vertically up, which means the grazing angle will always be90^◦. Hence the propagation paths from the transmitter to the NFRS, from the NFRS to the NSRS, and from the NSRS to the receiver are close to Euclidean distance, and the distances can be noted as Dtf, Df s, Dsr. The coordinates of the transmitter, NFRS, NSRS, and the receiver are expressed asxt= (xt, yt, zt),xf = (xf, yf, zf), xs= (xs, ys, zs)andxr= (xr, yr, zr). Since all relay stations are on the SOFAR plane, and the depth of the SOFAR Plane zsofar can be obtained in advance, so zsofar =zf =zs. For the proposed 3H-UWAC, the summation propagation distance is

Dsum=Dtf +Df s+Dsr

≃||xt−xf||+||xf−xs||+||xs−xr||, (3)

where||·||indicates the Euclidean norm of an arbitrary vector, the approximately equal sign is because the three propagation distances during the three hops are very close to straight-line distances (depending on the sound velocity gradient), soDtf, Df s, andDsrare close Euclidean distances, and the difference between them is tiny compared with the long communication distance.

B. Distance model of 3H-UWAC

In 3H communication, there are three hops: the transmitter to FRS, FRS to SRS, and SRS to the receiver. In the following analysis and calculation process, transmitters, relay stations, and receivers are all considered random points and obey independent Poisson Point Processes (PPPs). To simplify the calculation without losing insights, signal-to-noise (SNR) is analyzed. Before that, three hops’ communication distances are calculated, shown in Fig. 6. In the 3H-UWAC link, the transmitters and receivers are distributed independently. So, when analyzing the first hop and the third hop, we can have two independent coordinate systems with two independent origins (the transmitter and the receiver).

The distance between the transmitter and FRS is Dtf =󰁴

(xt−xf)²+ (yt−yf)²+ (zt−zsofar)². (4) Without losing generality, we assume that the transmitter’s projection position on the SOFAR plane is directly at the origin. So the distance between the transmitter’s projection on the SOFAR plane and the NFRS is constrained to PPP, which can be modeled as Rf t = 󰁴

x²_f+y²_f. Hence, the distance between the transmitter and the NFRS can be rewritten as Dtf =󰁴

R_tf² +z_tf², wherez_tf² = (zt−zsofar)² is known. In the third hop, the distance between the receiver and the NSRS is similar to the first hop, and it can be indicated as

Dsr=󰁳

(xs−xr)²+ (ys−yr)²+ (zr−zsofar)², (5) and the distance between the receiver’s projection on the SOFAR plane and the NSRS can be modeled as Rsr =

󰁳x²_s+y²_s. Hence, the distance between the transmitter and the NFRS can be rewritten as Dsr = 󰁳

R²_sr+z²_sr, where

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!_!", thousands kilometers SOFAR

Plane X Y

Z !_!#

"_!"

!_$"

!_#$

Transmitter

Receiver

Relay station on the SOFAR plane FRS

SRS

UIoT device

Figure 5: System Model.

z_sr² = (zr−zsofar)² is known. Based on the first and third hops, the second hop can be established. In the second hop, the distance between the NFRS and the NSRS is noted as Df s, which will be analyzed in detail.Dtr is the straight line distance between the transmitter and the receiver, which is either known or has a known distribution.

As we can see now, the second hop is conditioned on the first and third hop, so, first, we will analyze the first hop and then explore the third hop, and the second hop will finally be diagnosed. As explained in Section II and Fig. 4, there is a VDA for the underwater transducer. So, if the

’receiver’ wants to be covered by the ’transmitter’, there are two conditions: 1) theSNRof the received signal is bigger than the threshold τ; 2) the ’receiver’ is covered by the VDA of the ’transmitter’. Hence, two conditions must be satisfied in each communication hop. The final CP, including three hops, can be calculated as follows.

P_total^cov =P󰀃

Successf ul transmission f rom transmitter to N F RS, Successf ul transmission f rom N SRS to receiver, Successf ul transmission f rom N F RS to N SRS󰀄

=P󰀃

SNR_tf>τ, R_tf≤z_tftan(θ_tf),SNRsr>τ, Rsr≤zsrtan(θsr), SNR_{f s}>τ, R_{f s}≤󰀃

Dtr−R_tfcos(A) +Rsrcos(B)󰀄

tan(θ_{f s})󰀄

=P󰀓Pr,tf(Dtf)

σ²_n,tf >τ, Rtf≤ztftan(θtf), Pr,sr(Dsr)

σ²_n,sr >τ, Rsr≤zsrtan(θsr),Pr,f s(Df s) σ_{n,f s}² >τ, Rf s≤󰀃

Dtr−Rtfcos(A) +Rsrcos(B)󰀄

tan(θf s)󰀔

, (6)

where Rtf and Rsr are the nearest distance in PPPs in the first and the third hops, ztf is relative depth between the transmitter and the FRS, zsr is relative depth between the receiver and the SRS, θtf, θf s and θsr are VDAs of the transmitter, FRS and SRS, Pr,tf(Dtf), Pr,sr(Dsr) and Pr,f s(Df s)are received signal powers at NFRS, NSRS, and the receiver, Dtf = 󰁴

R²_tf+z_tf² , Dsr = 󰁳

R²_sr+z_sr² and Df sare three propagation distances in three hops,Dtr is the expected distance between the transmitter and the receiver, σ_n,tf² , σ_n,sr² , and σ_{n,f s}² are three noise powers in the three hops, τ is the pre-decided decoding threshold, Rf s, A and

B are three factors which will be explained in detail in the following analysis Section.

Before we derive the total CP in detail, we will introduce the underwater acoustic channel in the following subsection.

C. Underwater acoustic channel model

The underwater acoustic channel model and the transmission loss calculation can be found in detail in [40], and here we list the channel model for simplicity as follows.

Pr(d) = 4πPtDi

1.5×10¹⁸

1

Tl(λ, f, d)×Nl, (7) wherePris the received signal power, anddis the propagation distance in km. According to the convention of underwater acoustic communication, the signal is usually measured in dB with a Reference strength, which is the plane wave sound strength with a root mean square sound pressure of 1 µpa [36]. So we can have the transmit powerPt= 10^0.1(SL⁻^170.77) whereSL is named source level in dB, the directivity index Di = 10^0.1(DI⁻^170.77) where DI is the directivity index in dB, the noise level Nl = 100.1(N L−170.77) where N Lis the noise level in dB, and the transmission lossTl(λ, f, d).

Tl(λ, f, d) =β(λ)d^λ10^0.1α(f)d, (8) where β(λ) = _1.5^4π10_×₁₀^3λ18, λ is the Geometric Spreading Loss which is set as 1.5 generally,f is the signal frequency which is known in advance, and according to the Thorp’s formula [41], the following αcan be used as a simplified model for frequencies less than 50 kHz:

α(f) = 0.11f²

1 +f² + 44f²

4100 +f² + 2.75×10⁻⁴f²+ 0.003. (9) Since λandf can be known in advance and β(λ)andα(f) will be calculated as a constant, so we will rewritePr(d)as

Pr(d) = Pt

C×T l(λ, f, d), (10) whereC= ¹⁰⁰⁰_D^λ^N^l

i is a constant,T l(λ, f, d) =d^λ10^0.1α(f)d, and we will write T l(λ, f, d)as T l(d) for simplicity sinceλ andf are given in advance.

Since, in each hop link, there are different transmit powers, noise levels, directivity indexes, and communication frequencies, received signal powers, we will denote Pt, Pr(d), C,

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SOFAR Plane

T("!, $!, %!) R("", $", %") '!#

'$"

(!"

(#$

'#$

NFRS("#, $#, %%&'())

NSRS ("$, $$, %%&'())

($"

%$"

2"!"

2"#$

%!# (!#

Figure 6: Distances in 3H-UWAC.

α(f)andT l(d) as Pt,j,Pr,j(d), Cj,αj(f), and T lj(d), j∈ {tf, f s, sr}in the three different hops.

IV. ANALYSIS OFCPIN THE3H-UWAC

In this section, we will provide the mathematical analysis of the three sub-CPs in 3H-UWAC and get the final expression of the total CP. Before calculating the CP, some critical aspects of the proposed 3H-UWAC should be clarified.

• In practical underwater acoustic communication, different types of transducers with a VDA are designed in different communication scenarios. In this paper, the forward transducer, as shown in Fig. 4(a), is used for transmitters and receivers. As shown in Fig. 4(b), circle transducers are used for FRSs and SRSs. VDAs of the transmitter, FRS and SRS are noted as θtf,θf s,θsr.

• Locations of transmitters, relay stations (including FRSs and SRSs), and receivers are all subject to homogeneous PPPs, which are independent.

• In the first hop, the transmitter needs to select its NFRS.

During the process of calculation, the density of PPP should be regarded as γrs, instead of the transmitter’s densityγt. Similarly, in the third hop, SRSs will be acti- vated by the receiver as NSRS, so during the calculation, the density of PPP should be regarded asγrs, instead of the receiver’s density γr.

• In a practical application, to reduce conflicts, the density of relay stationγrsshould be more significant than or at least equal to the densities of transmitter γtand receiver γr.

• Dtr is the straight line distance between the transmitter and the receiver, which is known or has a known distribution. Since Dtr is an independent variable with a given distribution, the expectation over it is easy to be calculated. For better tractability, we will assumeDtr is known.

Before we derive the CP of the proposed 3H-UWAC in detail, some intermediate results need to be introduced in advance.

A. Distance distributions

Lemma 1. Probability density functions (PDFs) of distances in the PPP.

In a PPP with the density of γ, the PDF of the distance from a reference location located at the origin to the i- th nearest point and the distribution of the second nearest distance conditioned on the first nearest distance in PPP are well-known results in [42], which can be written as

fRi(ri) = 2(πγ)ⁱ

(i−1)!r²ⁱ⁻¹exp(−γπr²), r >0, i= 1,2,3, ..., (11) f_R₂_|r₁(r2) = 2γπr2exp󰀃

−γπ(r₂²−r₁²)󰀄

. (12)

Lemma 2. Distributions of the nearest propagation distance in the first hop.

According to (11) in Lemma 1, we can have

fRtf(rtf) = 2γrsπrtfexp(−γrsπr_tf² ), (13) and it is easy to get FRtf(rtf) = 1−exp(−γrsπr²_tf), and then we can get the cumulative distribution function (CDF) and PDF ofDtf which is the nearest propagation distance in the first hop as follows.

FDtf(dtf) = 1−exp󰀃

−γrsπmax(d²_tf−z_tf² ,0)󰀄

, (14)

fDtf(dtf) =

󰀫2γrsπdtfexp(−γrsπ(d²_tf−z²_tf)), if dtf> ztf

0, else . (15)

Proof. See Appendix A.

Similarly, we can have the PDF of Rsr and Dsr as fRsr(rsr) = 2γrsπrsrexp(−γrsπr²_sr), (16)

fDsr(dsr) =

󰀫2γrsπdsrexp(−γrsπ(d²_sr−z_sr² )), if dsr> zsr

0, else . (17)

Regarding the transmission loss T l(d) = d^λ10^0.1α(f)d in UWAC, the following preliminaries need to be first introduced.

(9)

Lemma 3. Monotonically decreasing characteristics of received signal power in UWAC

For the function Y =d⁻^λ10⁻^0.1αd, Taking the derivative with respect to d, we get

∂Y

∂d =−10⁻^0.1αd󰀃

λd⁻^λ⁻¹+ 0.1αd⁻^λln(10)󰀄

, (18) which is always a non-positive value, so T l⁻¹(d)is a monotonically decreasing function.

Lemma 4. The inverse function of the transmission loss in UWAC.

According to (10), we can have the transmission loss based on distance as

Y ≜T l⁻¹(d) =d⁻^λ10⁻^0.1αd. (19) Before we compute the distribution of Y, we need to introduce an important rule that the inverse of the function y=x⁻^a10⁻^bx can be written as

x=aW󰀃_b

a

󰀃yln^−a(10)󰀄−a¹󰀄

bln(10) , (20)

so, we can get the inverse function of Y =d⁻^λ10⁻^0.1αd as following.

d= 10λ

αln(10)W󰀃αln(10) 10λ Y⁻^λ¹󰀄

, (21)

where W(·)means Lambert function.

In the following, we will derive three sub-CPs of the three hops and then calculate the final CP.

B. CP of the first hop

The structure of the first hop that the transmitter transmits the signal to the NFRS is shown in Fig. 7. Considering the structure of VDA, the distance between the projection of the transmitter on the SOFAR plane and the NFRS, Rtf, is less than the coverage radius of the vertical directivity pattern of the transmitter with the VDA of θtf. So we can have

SOFAR Plane Relay station , !!"

T(#_#, %_#, &_#) (#$

&_#$

(#$, %$, &%&'())

)_#$

*#$

NFRS

Figure 7: From the transmitter to the first relay station.

P_tf^cov=P(Successf ul transmission f rom transmitter to N F RS󰀄

=P󰀃

SNR_tf>τ, Rtf≤ztftan(θtf)󰀄

=P󰀓P_r,tf(󰁴

R²_tf+z²_tf)

σ²_n,tf >τ, R_tf≤z_tftan(θ_tf)󰀔

=P󰀓Pt,tf

Ctf

T l⁻¹_tf(󰁴

R²_tf+z_tf² )>τ σ²_n,tf, Rtf≤ztftan(θtf)󰀔

=P󰀓 T l⁻¹_tf(󰁴

R_tf² +z_tf² )>τCtf

Pt,tf

σ²_n,tf, Rtf≤ztftan(θtf)󰀔

=P󰀓

T l⁻_tf¹(Dtf)>τC_tf Pt,tf

σ_n,tf² , Rtf≤ztftan(θtf)󰀔

. (22)

wherePt,tf is the transmit power at the transmitter,Ctf is a constant in the first hop, Dtf =󰁴

z_tf² +R²_tf, Rtf is the first nearest distance in PPP in the first hop.ztf is the relative depth between the NSRS and the receiver, which equals|zsofar−zt|, wherezsofaris the depth of the SOFAR plane which is known, θtf is the transmitter’s VDA, τ is the decoding threshold.

Lemma 5. CDF of transmission loss,T l_tf⁻¹(Dtf), in the first hop.

Based on the inverse function of transmission loss in Lemma 4, and added with the Monotonically decreasing characteristics ofY in Lemma 3, we can have Ytf ≜T l⁻_tf¹(Dtf) and ytf = ^τC_P ^tf

t,tfσ²_n,tf, and then we can get the CDF of T l_tf⁻¹(Dtf)as follows.

FYtf(ytf)

=1−P

󰀕 Rtf≤

󰁶

max󰀓󰀃 10λ

αtfln(10)W(αtfln(10) 10λ y⁻

1 λ tf )󰀄2

−z_tf² ,0󰀔󰀖

, (23)

whereW(·)means Lambert function.

Proof. See Appendix B.

Based on Lemma 5, we can have

P󰀓

T l⁻_tf¹(D_tf)> τCtf

P_t,tfσ_n,tf² 󰀔

=P

󰀕 Rtf≤

󰁶

max󰀓󰀃 10λ

αtfln(10)W(α_tfln(10) 10λ y⁻

1 λ tf )󰀄2

−z_tf² ,0󰀔󰀖

. (24)

After that, we can calculate the CP in the first hop.

Theorem 1. CP of the first hop from the transmitter to NFRS.

We can denote the CP of the first hop as the expectation of theK(Rtf)function with givenRtf. With (24), we can rewrite (22) as:

P_tf^cov=P󰀓

T l⁻_tf¹(D_tf)>τCtf

P_t,tfσ²_n,tf, R_tf≤z_tftan(θ_tf)󰀔

=P󰀓 Rtf≤

󰁶

max󰀃󰀃 10λ

1 λ tf )󰀄2

−z²_tf,0󰀄 , R_tf≤z_tftan(θ_tf)󰀔

=P󰀓

Rtf≤min󰀃󰁶

max󰀃󰀃 10λ

1 λ tf )󰀄2

−z_tf² ,0󰀄 , z_tftan(θ_tf)󰀄󰀔

(25)

=

󰁝_∞

0

fRtf(rtf) 󰀃 rtf≤A󰀄

drtf

≜ERtf[K(R_tf)], (26)

where fRtf(rtf) = 2γrsπrtfexp(−γrsπr²_tf) is the PDF of the nearest distance in PPP,K(Rtf) = 󰀃

Rtf ≤A󰀄 , and

(10)

󰀻󰁁

󰁁󰁁

󰁁󰀿

󰁁󰁁

󰀽

A= min󰀓󰁵

max󰀓󰀃 _10λ

αtfln(10)W(^α^tf_10λ^ln(10)y⁻

1 λ tf )󰀄2

−z_tf² ,0󰀔 , z_tftan(θ_tf)󰀔

y_tf=^τC_P ^tf

t,tfσ²_n,tf

.

(27)

Remark 1. As we can see in (25), the CP of the first hop is limited by two aspects: 1) SNR > τ, which means the propagation path can not be too long; 2) the VDA of the transmitter, which is the main limitation, especially when the depth of the transmitter is not too big. If there is no limitation of VDA (which means the transmitter can broadcast the acoustic signal with high energy consumption), the depth of the transmitter, ztf, will have no effect on the CP. A bigger VDA will weaken the effect of different ztf. Besides that, we can find that the CP of the first hop is independent of the second and the third hops.

C. CP of the third hop

The structure of the third hop that the receiver receives the signal from the NSRS is shown in Fig. 8. Considering the constraints from VDA, the distance between the projection of the transmitter on the SOFAR plane and the NFRS, Rsr, should be in the coverage area of the vertical directivity pattern of the NSRS. So, the CP of the third hop can be written as follows.

!!"

R(#_", %_", &_")

&_!"

(#!, %!, &#$%&')

(!"

)!"

SOFAR Plane Relay station , *_"!

NSRS

Figure 8: From the second relay station to the receiver.

P_sr^cov=P(Successf ul transmission f rom N SRS to receiver)

=P(SNRsr>τ, Rsr≤zsrtan(θsr)󰀄

=P󰀓Pr,sr(󰁳

R²_sr+z_sr² )

σ_n,sr² >τ, Rsr≤zsrtan(θsr)󰀔

=P󰀓Pt,sr

Csr

T l_sr⁻¹󰀃󰁴

R²_sr+z_sr² 󰀄

>τ σ²_n,sr, Rsr≤zsrtan(θsr)󰀔

=P󰀓

T l⁻_sr¹(Dsr)>τCsr

Pt,sr

σ²_n,sr, Rsr≤zsrtan(θsr)󰀔

, (28)

where Pt,sr is the transmit power at SRS, Csr is a constant in the third hop, Dsr =󰁳

z_sr² +R²_sr, Rsr is the first nearest distance in PPP. zsr is the relative depth between the NSRS

and the receiver, which equals|zsofar−zr|, where θsr is the SRS’s VDA,τ is the decoding threshold.

Lemma 6. CDF of transmission loss,T l⁻¹_sr(Dsr), in the third hop.

Similar to the first hop, we can have Ysr ≜ T l⁻_sr¹(Dsr) and ysr = ^τC_P ^sr

t,srσ_n,sr² , and based on Lemma 3, Lemma 4 and Dtf = 󰁴

z²_tf+R_tf² , we can get the CDF of T l_sr⁻¹(Dsr) as follows.

F_Y_sr(ysr)

=P(Ysr≤ysr) =P(D⁻_sr^λ10⁻^0.1α^sr^D^sr≤ysr)

=P󰀓

Dsr≥ 10λ

αsrln(10)W󰀃αsrln(10) 10λ y⁻

1 srλ󰀄󰀔

= 1−P󰀓 Rsr≤

󰁶

max󰀃󰀃 10λ

αsrln(10)W(αsrln(10) 10λ y⁻

1 srλ)󰀄2

−z²sr,0󰀄󰀔

, (29)

whereW(·)means Lambert W function. So, we can have

P󰀓

T l⁻_sr¹(Dsr)> Csrτ Pt,sr

σ_n,sr² 󰀔

=P󰀓 Rsr≤

󰁶

max󰀃󰀃 10λ

1 srλ)󰀄2

−z_sr² ,0󰀄󰀔

, (30)

whereysr =^τC_P_t,sr^srσ_n,sr² , and Dsr =󰁳

R²_sr+z_sr² . After that, we can calculate the CP in the third hop.

Theorem 2. CP of the third hop from the NSRS to the receiver.

We can denote the CP of the third as the expectation of a G(Rsr) function with given Rsr. With the Lemma 6, we can rewrite (28) as follows.

P_sr^cov=P󰀓

T l⁻_sr¹(Dsr)> τCsr

Pt,srσ_n,sr² , Rsr≤zsrtan(θsr)󰀔

=P

󰀕 Rsr≤

󰁶

max󰀓󰀃 10λ

1 srλ)󰀄2

−z_sr² ,0󰀔 , Rsr≤zsrtan(θsr)

󰀖

=P󰀓

Rsr≤min󰀓󰁶

max󰀃󰀃 10λ

1 srλ)󰀄2

−zsr² ,0󰀔 , zsrtan(θsr)󰀄󰀔

(31)

=

󰁝 _∞

0

fRsr(rsr) 󰀃 rsr≤B󰀄

drsr

≜ERsr[G(Rsr)], (32)

where fRsr(Rsr) = 2γrsπrsrexp(−γrsπr²_sr) is the PDF of the nearest distance in PPP,G(Rsr) = 󰀃

Rsr ≤B󰀄, and

󰀻󰁁

󰁁󰁁

󰀿

󰁁󰁁

󰁁󰀽

B= min󰀓󰁵

max󰀓󰀃 _10λ

αsrln(10)W(^α^sr_10λ^ln(10)y⁻

1 srλ)󰀄2

−z_sr² ,0󰀔 , zsrtan(θsr)󰀔

ysr=^τ_P^C^sr

t,srσ²_n,sr

.

(33)

Remark 2. Similar to Remark 1, based on (31), we can see the CP of the third hop is limited to theSNRand the VDA of the NSRS, and the assumed value of VDA is the reason for the effect of the receiver’s depth. We can also find that the CP of the third hop is independent of the first and the second hops. Hence, the first and the third hops are independent, and