Applied Acoustics 177 (2021) 107911

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Applied Acoustics

journal homepage: www.elsevier .com/locate /apacoust

Optimizing the number of relays for energy efficient multi-hop covertunderwater acoustic cooperative networks

https://doi.org/10.1016/j.apacoust.2021.1079110003-682X/� 2021 Elsevier Ltd. All rights reserved.

⇑ Corresponding author at: Key Laboratory of Underwater Acoustic Communica-tion and Marine Information Technology (Xiamen University), Ministry of Educa-tion, Xiamen 361005, China.

E-mail address: chenyougan@xmu.edu.cn (Y. Chen).

Yougan Chen a,b,c,d,⇑, Yuying Tang a,b,d, Junhui Liu a,b,d, Xiaokang Zhang a,b,d, Xiaomei Xu a,b,d

aKey Laboratory of Underwater Acoustic Communication and Marine Information Technology (Xiamen University), Ministry of Education, Xiamen 361005, Chinab Shenzhen Research Institute of Xiamen University, Shenzhen 518000, ChinacDongshan Swire Marine Station, Xiamen University, Xiamen 361102, ChinadCollege of Ocean and Earth Sciences, Xiamen University, Xiamen 361102, China

a r t i c l e i n f o

Article history:Received 7 September 2020Received in revised form 4 December 2020Accepted 6 January 2021Available online 1 February 2021

Keywords:Underwater acousticCovert cooperative communicationEnergy consumptionProbability of detection

a b s t r a c t

Multi-hop underwater acoustic (UWA) cooperative networks can effectively reduce the energy consump-tion of the system, yet the covertness of communication is also greatly reduced since a spatial diversitygain can be obtained by both the intended receiver and the intercept detector. In this paper, we investi-gate how to select the optimal number of relays for multi-hop UWA cooperative networks, by consideringboth the low probability of detection (PD) and energy consumption. We derive the relationship betweenPD and the number of relays and then analyze the relationship between energy consumption and thenumber of relays for the system. To address the joint PD and energy consumption optimal problem, wedefine the energy consumption rate for multi-hop UWA systems by normalizing the energy consumedin single-hop UWA systems at the same source-to-destination distance. Then, we derive the range num-ber of relays under the given PD and energy consumption rate. Moreover, we construct the objective func-tion and derive the optimal number of relays by jointly addressing PD and the energy consumption rate.Finally, we simulate and analyze the impact of several parameters on PD and energy consumption, includ-ing the signal-to-noise ratio, working distance, position angle of interceptor and data length. The simu-lation results show that the number of relays for multi-hop covert UWA cooperative networks can takethe value between 2 and 6 under a given condition, which can meet the covert communication require-ment with a PD < 0.5, while the system energy consumption is relatively low.

� 2021 Elsevier Ltd. All rights reserved.

1. Introduction balancing between covertness and multi-hop cooperation is a chal-

In recent years, multi-hop underwater acoustic (UWA) cooper-ative communication technology has become very appealingbecause of the large area coverage, the improving data transmis-sion rate, high energy efficiency and other advantages [1–3] whichis especially true with the development of internet of underwaterthings [4–6]. Meanwhile, due to special application scenarios, suchas military UWA command, control and communication, coastalmonitoring and submarine operations, the demands for covertcommunication of multi-hop UWA networks have increasedaccordingly [7–10]. However, unfortunately, as indicated in [11]the relays in multi-hop UWA systems provide spatial diversitynot only to the intended receiver but also to the intercept detector,which reduces the covertness significantly. Therefore, delicately

lenging task.A primary concern in adopting multi-hop cooperation for UWA

networks is saving the energy consumption for underwater equip-ment that is usually battery powered and expensive to replace inthe ocean. The use of a multi-hop scheme can reduce the totalpower required for transmission, since the transmission loss,related to the distance, causes an exponential reduction in receivedpower [12]. According to the results in [13] a significant amount ofpower can be saved in the multi-hop UWA networks comparedwith the direct single-hop long-distance UWA transmission. Asthe transmission loss in UWA channels is severely affected by sig-nal frequency and transmission distance, in [14] a study on thejoint optimization of node placement and carrier frequency selec-tion is presented to minimize the power consumption in multi-hop UWA networks, showing that the use of different frequencieshas an impact on the optimal relay placement. This work providesthe basis for the selection of signal frequencies according to differ-ent distances caused by the relay placement in multi-hop UWAnetworks. A power based cost function is proposed in [15] for

Y. Chen, Y. Tang, J. Liu et al. Applied Acoustics 177 (2021) 107911

multi-hop UWA networks, which can minimize the number ofhops/relays to optimize the transmission power of the entire link.Further, we investigate the trade-offs between the number of hopsand the energy consumption, the end-to-end delay, for multi-hopUWA cooperative networks in [16]. Moreover, the authors in [17]focus on the optimum choice of the number of hops, retransmis-sions, code rate, and signal-to-noise ratio (SNR), in terms of energyefficiency for multi-hop UWA networks. This study indicates thatoptimizing the number of hops/relays is fundamental to minimizethe energy consumption for medium to long links, while increasingthe number of allowed transmission trials has a smaller impact.This is an interesting result, meaning that from the energy savingsperspective, we should pay more attention to the number of hops/relays during the design of multi-hop UWA networks. However,none of the works cited above consider the covertness of communi-cation for multi-hop UWA networks.

For the stealth operations in most military underwater activi-ties, UWA communication systems are required to remain unde-tected, and the data transmission is made with low probability ofdetection (LPD) [18].

As of now, most research on covert communication has beendone for wireless electromagnetic channels [19,20]. For the UWAchannels, the study on covert communications in the existing workis usually for single-hop transmissions. A pair of energy detectorsthat are insensitive to the phase fluctuations are presented in[21] and the focus is on long code sequences for the purpose ofobtaining a high processing gain at the expense of a low data rate,so that communications can be performed at a low input SNR tominimize the probability of detection (PD) by an interceptor. In[22] a direct-sequence spread-spectrum (DSSS) scheme based ona spreading waveform is investigated for covert UWA communica-tions. Under the condition of a log-normal fading UWA channel,the covert performance of DSSS communication is investigated in[23]. Meanwhile, the directional transducers have been alsoadopted to enhance the concealment of UWA communications in[24]. A new covert underwater acoustic communication schemebased on ship-radiated noise and chaos signal has been proposedin [25] and the validity of this method is verified by sea experi-ments. In addition, orthogonal frequency-division multiplexing(OFDM), and multi-carrier spread spectrum (MC-SS) techniqueshave also been applied to covert UWA communication by manyresearchers [26–28]. Yet the signal waveforms used in the aboveschemes are easily distinguished by trained sonar operators dueto their obvious features. Furthermore, in [29–31] the authorsdemonstrate that biological mimicry method can provide covertUWA communications by using intrinsic dolphin sounds. In covertUWA communications, performance greatly depends on the intersymbol interference (ISI) caused by the channel impulse response(CIR) and on the ability of the receiver to estimate CIR. To compen-sate for these channel distortions, two methods are analyzed in[32] to improve the LPD capabilities, both using a time-updatedchannel impulse-response estimate as a matched filter to mitigatethe multipath-induced interference. However, these existing cov-ert UWA communication schemes have not considered the multi-hop transmission scenario. Recently, the authors in [11] considertwo performance metrics for LPD in multi-hop UWA cooperativenetworks. The results show that multi-hop cooperative networksexhibit worse LPD performance than single-hop systems, becauseit is more likely to be detected by the interceptor if more relaynodes are used in a cooperative network.

In general, the case with fewer relays is better for a multi-hopUWA system with respect to concealment; yet the case with morerelays is better for that with respect to energy savings. Hence it is achallenging task to select the optimal number of relays for a multi-hop UWA system, by considering both LPD and energy consump-tion. The preliminary results of our work on this topic have been

2

partly presented in [33] and further theoretical derivation andextensive numerical analysis will be reported in this paper.

The main contributions of this work are as follows:

1) We first establish the relationship between PD and the num-ber of relays based on the multi-hop UWA cooperative net-work, and then establish the relationship between energyconsumption and the number of relays for the system. Notethat the work in [11] did not consider energy consumption,and the work in [33] has not given the joint optimal objec-tive function and its solution, although they are the multi-hop covert UWA communication issues.

2) Since the value of PD is between 0 and 1, to compare PD andenergy consumption on the same scale, we define the energyconsumption rate g as the total energy consumed in a multi-hop scheme divided by that consumed in a single-hop trans-mission scheme for the same UWA system, addressing thejoint PD and energy consumption optimal problem. Then,the value of the g is also between 0 and 1, where a smallenergy consumption rate value indicates that more energyis saved in multi-hop UWA system compared with a single-hop UWA system. Then, we derive the range number of relaysunder the given PD and g. More importantly, we construct theobjective function and derive the optimal number of relaysby jointly addressing the PD and g and then present thenumerical solutions of the optimal number of relays.

Note that the work in [11,33] has not introduced the concept ofenergy consumption rate g, and has not constructed the objectivefunction considering the given PD and g for the optimizationproblem.

3) From the view of LPD, in the process of calculating distanceratio for the overall multi-hop covert UWA system, we usethe triangular area approximation method to derive statisti-cal expectation geometrically, which can easily express thedistance ratio in the multi-hop covert UWA system and bet-ter derive the upper bound of the optimal number of relays.From the view of energy consumption, we clarify the confus-ing transformation between radiated acoustic power Pt

aco

and electrical power PTel of [3,17,34] through detailed math-ematical derivation. All these have not been involved in[11,33].

4) We analyze the impact of several parameters on PD andenergy consumption, including the target SNR, working dis-tance, position angle of the interceptor and data length,which is more abundant than the existing work in [11,33].The simulation results validate the above theoretical analy-sis, and show that the number of relays for the multi-hopcovert UWA cooperative networks can be between 2 and 6under given conditions, which can meet the covert commu-nication requirement with a PD < 0.5, while the systemenergy consumption is relatively low.

The remainder of this paper is organized as follows. Section 2presents the system model and the calculation of PD. The energyconsumption is detailed in Section 3, while Section 4 presents atheoretical analysis of the optimum number of relays in terms ofPD and energy consumption. The numerical results are presentedand discussed in Section 5, and Section 6 concludes the paper.

2. System model and the probability of detection

Fig. 1(a) depicts a typical application of multi-hop covert UWAcooperative networks, where the UWA sensors are deployed in the

Fig. 1. Multi-hop covert UWA cooperative networks: (a) Application scenario; (b) Network model (K is even as an example).

Y. Chen, Y. Tang, J. Liu et al. Applied Acoustics 177 (2021) 107911

offshore area for monitoring acoustic data transmission, and theinterceptor is deployed in the open sea. As shown in Fig. 1(b), weconsider a simplified multi-hop UWA network with K relay nodes(Rk, k = 0, 1, 2,. . ., K, when k is zero, the relay node is the sourcenode) between the source node (Transmitter, Tx) and the intendeddestination node (Ds). For simplicity, all the nodes are aligned per-fectly in a straight line [11]. The interceptor/eavesdropper (Ix) isstrategically placed at the perpendicular bisector of the Tx-Rk-Ds

line.We assume that each relay node (Rk) can correctly receive the

signal from the previous relay node (Rk-1) and then amplify andforward it to the next relay node (Rk+1), which can be ensured bycontrolling the transmitting power and other measures.

In each hop, as shown in Fig. 1(b), the distance ratio is definedas follows [11]:

ck ¼rkdk

; k ¼ 0;1;2; � � � ;K ð1Þ

where rk and dk are the ranges at which Ix and Rk can successfullydetect/receive the signals from Tx. In general, a smaller distanceratio means better covertness of the communication system,because the Ix has to be closer to the Tx to detect the communica-tion signal. In an ideal case, where dk is an equal distance for allk, the average intercept range for the relay network is the smallestwhen the Ix is placed at an equal distance from both ends of the Tx-Ds line [11] as shown in Fig. 1, thus we have,

3

dk ¼ dc

kþ 1; k ¼ 0;1;2; � � � ;K ð2Þ

where dc represents the distance between Tx and Ds.

2.1. The PD for single-hop covert UWA communication

To obtain the PD of a multi-hop UWA covertness communica-tion system, we first calculate the PD of the single-hop system.

The assumption is that the same bandwidth Gaussian noiseUWA channel is experienced by both the Ds and the Ix, which hasa two-sided spectral density equal to N0/2. The signal energydetected at Ix is Ei, the signal energy required at Ds to correctlyreceive the data is Es, and the distance ratio of the single-hop sys-tem is c. Then, according to the energy consumption relationshipbetween the transmitted signal and the received signal, we canobtain [11]

Ei ¼ k/Esc�n ð3Þwhere k is the gain difference between the communication receiverand the general wideband energy detector, and u is a coefficientrepresenting the existence of the signal of interest in the receivedsignal, which is related to the signal design; n denotes the path lossexponent depending on the frequency band, relating to the soundwaves’ transmission method and transmission path in the water.According to different UWA propagation conditions, n takes differ-

Y. Chen, Y. Tang, J. Liu et al. Applied Acoustics 177 (2021) 107911

ent values. For example, n = 2 for spherical spreading, and n = 1 forcylindrical spreading. In a practical UWA communication system,we usually take the value of n = 1.5.

Using hypothesis testing [11] given the probability of a falsealarm (PFA), we can plug (3) into the relative formulas in [11] thenthe PD of the single-hop system can be derived as follows:

PD ¼ 12erfc erf�1 1� 2PFAð Þ � Es

N0� k/ffiffiffiffiffiffiffiffiffiffiffiffi

2T iWp � c�n

� �; T iW >> 1 ð4Þ

where Ti represents the integration time of Ix, W denotes the oper-ating bandwidth of Ix, erf and erfc are the error function and com-plementary error function, respectively. Es/N0 represents the SNRwhen Ds can receive the signal correctly. Then, the quantitySNRis ¼ Es

N0� k/ffiffiffiffiffiffiffiffiffi

2T iWp can be defined as the normalized SNR between

the Ix and Ds. When n = 1.5, PFA = 0.02, Fig. 2 shows the relationshipbetween the PD and c, and the area of PD < 0.5 is the area of LPD. Asshown in Fig. 2, the PD decreases rapidly as the c increases. Thesmaller the SNRis, the smaller the c is, meaning that the covertnessof communication system is better because the Ix needs to be closerto the Tx to intercept the signal.

In general, the complexity of the UWA channels will result indifferent SNR at the receiver. Eq. (14) in the following shows therelationship between the SNR and the transmission loss causedby the time–space-frequency varying UWA channels and the noiseat the receiver. Therefore, Es/N0 in (4) indicates that the PD of covertcommunication will be affected by the complex underwaterenvironments

2.2. The PD for multi-hop UWA cooperative covert communicationnetworks

For the communication case with K relays (Rk), due to the dif-ferent detection ranges of the UWA data transmission (rk) foreach hop, the authors in [11] propose two methods to calculatethe detection range r of the overall multi-hop system. One is totake the minimum range, that is rmin = min(rk), k = 0,1, 2,. . .,K.The second is to take the average, that is rc = E[rk], where E[ ]represents the statistical expectation for all rk. We use the sec-ond method in this paper to calculate r, i.e., r = rc(K). Thus,the distance ratio for the case with K relays can be expressedas follows [11]:

Fig. 2. The relationship between the probability of detection and detectiondistance.

4

cc Kð Þ ¼ rc Kð Þdc

ð5Þ

Taking c ¼ cc and substituting (5) into (4), we obtain the PD ofthe multi-hop UWA cooperative network with K relays, as follows:

PD Kð Þ ¼ 12erfc erf�1 1� 2PFAð Þ � Es

N0� k/ffiffiffiffiffiffiffiffi

2T iWp � 1

d�nc� r�n

c Kð Þh i

ð6Þ

Next, according to different values of K, we will calculate rc(K)by the geometric relationship shown in Fig. 1.

Assuming that the shortest distance between Ix and relay nodes(Rk) is rmin, and for any rk(k = 0,1,2,. . .,K), it is occurring with equalprobability, therefore, for an odd K,

rc Kð Þ ¼ rmin

K þ 1

XKk¼0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiK þ 12

� k� �2

� tan2hþ 1

sð7Þ

and for an even K,

rc Kð Þ ¼ rmin

K þ 1

XKk¼0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiK2� kþ 1

2

� �2

� 14

" #� tan2hþ 1

vuut ; ð8Þ

where tanh ¼ dkrmin

,and the angle h is the position parameter of Ix,

which represents the distance relationship between the Ix and theTx-Rk-Ds line. Substituting (7) and (8) into (6), we obtain the PD(K)for the multi-hop UWA cooperative network with K relays.

3. The calculation of energy consumption

In each hop, the minimum SNR required by the receiver to cor-rectly receive signals is defined as the target SNR and marked asSNR0. The energy consumed at the transmitter can be obtainedby calculating the transmission loss under the given target SNR0

at the receiver. The details will be presented next.

3.1. Transmission loss and noise

The energy attenuation of the transmitted signal in a UWAlink is related to transmission distance, including the spreadingloss and the absorption loss. Then, the path attenuation over dis-tance dk for the signal of frequency f in the k-th hop is as follows[35]:

A dk; fð Þ ¼ 1000 � dkð Þn a fð Þ½ �dk ð9Þwhere n is the spreading factor, a(f) is the coefficient of absorptionloss. Expressed in dB re lPa, the path attenuation is defined as thetransmission loss (TL),

TL ¼ 10log10A dk; fð Þ¼ n � 10log10 1000 � dkð Þ þ dk � 10log10a fð Þ ð10Þ

where the absorption coefficient given in dB re lPa /km for f in kHzcan be obtained from Thorp’s formula as follows [21]:

10log10a fð Þ ¼ 0:11f 2

1þ f 2þ 44f 2

4100þ f 2þ 2:75f 2

104 þ 0:003 ð11Þ

The ambient noise in the ocean can be modeled by Gaussianstatistics and a continuous power spectral density (p.s.d.). The totalnoise p.s.d., denoted as N0(f), is a function of frequency f in kHz. Thecalculation can refer to the Wenz Noise model in [35] usuallyincluding four different noise sources: turbulence, shipping,wind-driven waves, and thermal noise. The approximation, in dBre lPa /Hz, may be useful [35]:

10log10N0 fð Þ � 50� 18log10f ð12Þ

Y. Chen, Y. Tang, J. Liu et al. Applied Acoustics 177 (2021) 107911

Let the source power of the transmit node be Pt(dk, f), and thetotal noise be N(f), we can calculate the average SNR at the receiveras follows [35]:

K�SNR dk; fð Þ ¼ Pt dk; fð Þ

A dk; fð ÞN fð Þ ð13Þ

Expressed in dB re lPa, as follows:

SNR ¼ 10log10K�SNR dk; fð Þ

¼ 10log10Pt dk; fð Þ|fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}:¼SL

�10log10A dk; fð Þ|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}:¼TL

�W � 10log10N0 fð Þ|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}:¼NL

ð14Þ

where SL and NL, in dB re lPa, are the source level and noise level,respectively, and W is the bandwidth of receiver, in Hz.

In (13), for a given distance, choosing the optimal frequency canmake the product of A(dk, f) and N (f) smaller, which can minimizethe energy consumption under the same average SNR condition[35]. According to [14] for each hop, the relationship betweenthe optimal working frequency fopt(dk) and working distance dkcan be modeled as follows:

f opt dkð Þ ¼ a dkð Þ � d�b dkð Þk ð15Þ

where

a ¼ 23:33; b ¼ 0:16449;dk 2 ð0;1Þa ¼ 20:421; b ¼ 0:54012; dk 2 ½1;10Þa ¼ 53:197; b ¼ 0:86133; dk 2 ½10;100�

ð16Þ

In (15), the optimal frequency fopt is expressed in kHz, and theworking distance dk is in km.

3.2. Acoustic transmit power

From (14), we know that, given the target SNR0 at the receiver,TL, and NL, it is possible to obtain the SL and then the source powerPt(dk, f), which is expressed as follows:

Pt dk; fð Þ ¼ 10SNR0þTLþNL

10 ð17Þ

Now there is a question regarding the unit of Pt(dk, f). Theauthors in [17] indicate that it is in lPa, yet the authors in [3,34]reason that it is in dB re lPa, detailing the following:

PelT dk; fð Þ ¼ Pt dk; fð Þ � 10

�17:2

uð18Þ

where 10-17.2 is the conversion factor from acoustic power in dB relPa to electrical power PTel(dk, f) in watts, and / is the overall effi-ciency of the power amplifier and transducer [3,34]. However,many people may question why the unit of acoustic power is inlPa, the sound pressure unit, instead of in watts, the power unit.

We argue that the descriptions regarding the unit of Pt(dk, f) andthe conversion between acoustic power and electrical power in[3,17,34] are not rigorous enough. We present the detailed reason-ing as follows.

The reference intensity in underwater sound [36] is usually theintensity of a plane wave having an rms pressure equal to 1 lPa,denoted as Iref � 0.667 � 10-18 W/m2. At a large distance d, in m,let the intensity of the sound emitted by the projector be Id. Fora nondirectional projector, this intensity corresponds to a radiatedacoustic power output of Pt

aco = 4pd2Id, in watts. Combine theseaccording to the definition of SL as follows [36]:

SL ¼ 10log10IdIref

ð19Þ

5

Algorithm 1 The calculation of total energy consumption formulti-hop UWA cooperative networks

1: for each hop (k = 0,1,2,. . .,K) doStage 1: Parameter input and the basic calculation2: Given the transmission distance dk3: Calculate the optimal frequency fopt(dk) using (15) and

(16), the 3-dB bandwidth W given by (26)4: Calculate the absorption coefficient given by (11), under

the fopt(dk)Stage 2: Acoustic parameter calculation5: Calculate the transmission loss TL given by (10)6: Calculate the ambient noise using (12), and convert it into

noise level NL7: Given the target SNR0

8: Calculate the source level SL given by (14), plugging theTL and NL

9: Convert SL into acoustic power Ptaco using (20), andelectrical power PTel using (21)

Stage 3: Energy consumption calculation10: Given the data length and transmission rate, calculate

the duration of data transmission TL11: Calculate the energy consumption of transmitter Etr(k)

using (22), and the energy consumption of receiver Err(k)using (23)

12: end for13: Calculate the total energy consumption Esys(K) using

(24)

then we have the radiated acoustic power Ptaco and electrical powerPTel as follows:

Pacot

��d¼1 ¼ 4pIrefd

2|fflfflfflfflffl{zfflfflfflfflffl}10�17:2

� 10SL10|ffl{zffl}

:¼Pt

�������d¼1

ð20Þ

PelT ¼ Paco

t

uð21Þ

where we can clearly find that the conversion factor 10-17.2 is in theunit of watts, and the Pt is in the unit of 1.

Therefore, we argue that 10-17.2 is the conversion factor fromacoustic power in 1 (corresponding to source level in dB re lPa)to acoustic power in watts, and / is the conversion factor fromacoustic power in watts to electrical power in watts, which indi-cates the overall efficiency of the power amplifier and transducer.

3.3. Total energy consumption

In each hop, the receiver can correctly decode the transmittingsignal when the target SNR0 is satisfied, which means there is noretransmission for this case; hence there is no extra energy con-sumption caused by retransmission for each transmitting node.We only consider the energy consumption of encoding and decod-ing in the source node, relay nodes and destination node, andassume that the energy required to correctly receive the signal isthe same in every relay node or destination node. In the k-thhop, the energy consumption of the transmitter is as follows:

Etr dk; fð Þ ¼ Eenc dk; fð Þ þ PelT dk; fð Þ � TL ð22Þ

where Eenc represents the energy of encoding, PTel represents theelectrical power of the power amplifier, and TL is the duration ofdata transmission.

Y. Chen, Y. Tang, J. Liu et al. Applied Acoustics 177 (2021) 107911

On the other hand, the energy consumption of the receiver inthe k-th hop is as follows:

Err dk; fð Þ ¼ Edec dk; fð Þ þ PR dk; fð Þ � TL ð23Þwhere Edec is the energy required to decode the data, and PR repre-sents the electrical power of the receiver node to receive the signal,

which can be set as [37] PR ¼ 110P

elT .

Thus, the overall energy consumption to transmit a signal for amulti-hopUWAcooperative networkwithK relaynodes is as follows:

Esys Kð Þ ¼XKk¼0

Etr dk; fð Þ þ Err dk; fð Þ½ �� ð24Þ

It is normally assumed that the energy consumption is the samefor all the hops in the ideal equal distance case, and then we canrewrite (24) as follows:

Esys Kð Þ ¼ K þ 1ð Þ � Etr þ K þ 1ð Þ � Err ð25ÞAbove all, we present the process for calculating the total

energy consumption for multi-hop UWA cooperative networks,as shown in Algorithm 1.

We can use the 3-dB bandwidth around the optimum operatingfrequency as the operating bandwidth W. For simplicity, followingthe theoretical analysis in [2,3,21] the 3-dB bandwidth W, thesource power Pt, and the electrical power PT

el are function of thetransmission distance under the given target SNR0, respectively,which can be approximately expressed as follows:

W dk; fð Þ ¼ x � d�-k ð26Þ

Pt dk; fð Þ ¼ d � dvk ð27Þ

PelT dk; fð Þ ¼ Pt dk; fð Þ � 10�17:2=u ¼ w � dvk ð28Þ

where x, d and w represent the corresponding spreading factors,and w is the factor related to the target SNR0; - and v representthe corresponding exponent, which are positive values and can bereadily calculated by first-order least-squares polynomial approxi-mation on a logarithmic scale.

4. Selection of optimal number of relays

In this section, we derive the optimal number of relays based onthe above modeling of probability of detection and energy con-sumption for the multi-hop covert UWA system. Taking the LPDas the constraint, the optimal number of relays Kopt is the solutionof the objective function of minimum energy consumption, and itcan be expressed as follows:

Kopt ¼ argminK

Esys Kð Þs:t: PD Kð Þ < 0:5

ð29Þ

The following is the theoretical derivation of the optimal selec-tion of relays Kopt.

4.1. From the view of energy optimization

Generally, substituting (22) and (23) into (25), and taking

Eenc ¼ Edec ¼ q � PelT � TL and PR ¼ r � Pel

T , we have

Esys Kð Þ ¼ K þ 1ð Þ � Eenc þ PelT � TL

þ K þ 1ð Þ � Edec þ PR � TLð Þ

¼ K þ 1ð Þ � rþ 2qþ 1ð Þ � PelT � TL ð30Þ

where q is the difference between the power consumption ofthe encoding and that of the transmitting, and r is the differencebetween power consumption of the receiver and that of the trans-mitter. Plugging (2) and (28) into (25), we have the following:

6

Esys Kð Þ ¼ K þ 1ð Þ � rþ 2qþ 1ð Þ � w � dvk � TL

¼ rþ 2qþ 1ð Þ � dvc � w � TL|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}:¼X

� K þ 1ð Þ1�v

¼ X � K þ 1ð Þ1�v ð31ÞTaking the first-order derivative of Esys(K) with respect to K we

obtain the following:

@Esys Kð Þ@K

¼ 1� vð Þ �X � K þ 1ð Þ�v ð32Þ

Because v > 1, thus oEsys(K)/oK < 0, which means that the energyconsumption rate decreases as the number of relays K increases.For a target energy consumption e0, to meet Esys(K) < e0, plugginginto (31) we have the following:

X � K þ 1ð Þ1�v < e0 ð33Þ

which obtains K >ffiffiffiffie0X

1�vq

� 1, and this is the solution of the number

of relays from the energy consumption aspect. Taking

K0 ¼ffiffiffiffie0X

1�vq

� 1l m

as the upper bound value, K � K0 is the number

of relays that can meet the energy consumption target.However, we need to consider both Esys(K) and PD(K), and PD(K)

is a value between 0 and 1, therefore it is better to normalizeEsys(K). We define the energy consumption rate of the K-hopsystem g(K) as follows:

g Kð Þ ¼ Esys Kð ÞEsys;max

��K¼0

¼ K þ 1ð Þ1�v; ð34Þ

where Esys,max is the maximum energy consumption required and itis the case with no relay nodes, i.e., Esys,max = Esys(0). Hence g(K),taking a value between 0 and 1, represents the energy saving indexof the multi-hop UWA system. The closer the energy consumptionrate g is to 0, the better the multi-hop UWA system efficiency.

Similarly, we have og(K)/oK < 0, which means that the energyconsumption is decreasing as the number of relays K are increas-ing. For a given target energy consumption rate g0, to meetg < g0, plugging into (34) yields the following:

K þ 1ð Þ1�v < g0 ð35Þ

Then, we find the solution is K > g1

1�v0 � 1. Likewise, taking

K0 ¼ g1

1�v0 � 1

� �as the upper bound value, K � K0 is the number

of relays where the energy consumption rate can be less than thetarget energy consumption rate.

Comparing (31) with (34), we find that g(K) loses the effect ofthe target SNR0, indicated by w in (31), although it is better touse the normalized value g(K) when comparing it with PD(K). Wewant to note that both the absolute value Esys(K) and the normal-ized value g(K) are effective evaluation metrics, depending on thedifferent profiled scenarios.

4.2. From the view of probability of detection

From (4) and (6), we know that PD(K) can also be written asfollows:

PD Kð Þ ¼ 12erfc erf - 1 1� 2PFAð Þ � Es

N0� k/ffiffiffiffiffiffiffiffiffiffiffiffi

2T iWp � c�n Kð Þ

� �ð36Þ

where c Kð Þ ¼ rc Kð Þdc

¼ E rk Kð Þ½ �dc

. For simplicity, we will calculate E[rk(K)]in an approximate method according to Fig. 1(b) next. It can be seenfrom Fig. 1(b) that when K is close to infinity, the sum of rk(K) can beapproximated as the area of a triangle; then E[rk(K)] can be approx-imately expressed as follows:

Fig. 3. Optimal number of relays for selection: (a) Probability of detection vs.number of relays and energy consumption rate vs. number of relays; (b) Optimalnumber of relays, the calculation result of the objective function.

Y. Chen, Y. Tang, J. Liu et al. Applied Acoustics 177 (2021) 107911

E rk Kð Þ½ �jK!1 ¼ 1K þ 1

XKk¼0

rk Kð Þ�����K!1

� 1K þ 1

� 12rmin � dc ð37Þ

Actually, since the value of K cannot be infinity, the gain differ-ence can be expressed as 1 (apparently 1 < 1). Then, the exact valueof E[rk(K)] can be written as follows.

E rk Kð Þ½ � ¼ 1 � E rk Kð Þ½ �jK!1 ¼ 12� 1K þ 1

1 � rmin � dc ð38Þ

By tanh ¼ dkrmin

, we have rmin ¼ dktanh ¼ dc

Kþ1ð Þ�tanh, therefore,

c Kð Þ ¼ rc Kð Þdc

¼ E rk Kð Þ½ �dc

¼ 121 � rmin ¼ 1

2tanh� dc � 1

K þ 1ð Þ2ð39Þ

Thus, plugging (39) into (36), we have the following:

PD Kð Þ ¼ 12 erfc erf - 1 1� 2PFAð Þ|fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}

:¼b

� Es

N0� k/ffiffiffiffiffiffiffiffiffiffiffi

2T iWp � 1

2tanh

�n

� d�nc|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

:¼w�n

� K þ 1ð Þ2n

26664

37775

¼ 12 erfc b� w � n � K þ 1ð Þ2n

h ið40Þ

where we continue using w in (40), due to the effect of SNR, and theother effect can be synthetically expressed as n in (40). This is veryimportant because the SNR is a reflection of the dynamic changes ofmarine environments at the receiver.

In the meantime, we have oPD(K)/oK > 0, which means that theprobability of detection increases when decreasing the number ofrelays K. For a given probability of detection threshold PD0,(PD0 = 0.5 according to LPD), i.e., to meet PD(K) < PD0, we can adopt(40) to obtain the range of relays K, as follows:

K <

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

w � n b� erfcinv 2PD0ð Þ½ �2n

s� 1 ð41Þ

Taking K1 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1w�n b� erfcinv 2PD0ð Þ½ �2n

q� 1

j kas the lower bound

value, K � K1 is the number of relays that meet the PD less thanthe target threshold.

4.3. Analysis and discussion

From the discussion above, it can be found that PD(K) and Esys(K)are the two curves showing opposite performance trends withincreasing relay values once the system parameters are set andthe target SNR is given.

In summary, to jointly consider probability of detection PD andenergy consumption e, the range of number of relays K is asfollows:

K0 6 K 6 K1;K0 ¼

ffiffiffiffie0X

1�vq

� 1l m

or K0 ¼ g1

1�v0 � 1

� �K1 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1w�n b� erfcinv 2PD0ð Þ½ �2n

q� 1

j k8>><>>: ð42Þ

where the value of K0 and K1 are related to the system parameter,target PD threshold PD0, target energy consumption e0 or targetenergy consumption rate g0, and so on.

At this time, the range of the number of relays K for the optimalsolution is determined. In the range of K, the exact value of Kdepends on the different needs, such as a lower probability ofdetection or lower power consumption.

Furthermore, to obtain the unique solution of (29), consideringboth indexes, PD and g in (42), take value between 0 and 1, theoptimal objective function can be constructed as the sum ofPD(K) and g(K), that is as follows:

PD Kð Þ þ g Kð Þ ¼ C Kð Þ;K 2 K0;K1½ � ð43Þ

7

Then, (29) can be further expressed as follows:

Kopt ¼ argminK

C Kð Þs:t: C Kð Þ ¼ PD Kð Þ þ g Kð Þ

K 2 K0;K1½ �ð44Þ

Finally, taking the first-order derivative of C(K) with respect to Kwe obtain the following:

@C Kð Þ@K ¼ @PD Kð Þ

@K þ @g Kð Þ@K

¼ 12

@@K erfc b� w � n � K þ 1ð Þ2n

h in oþ @

@K K þ 1ð Þ1�vh i

¼ � 1ffiffiffip

p � @@K b� w � n � K þ 1ð Þ2nh i

� e� b�w�n� Kþ1ð Þ2n½ �2

þ 1� vð Þ K þ 1ð Þ�v

¼ 2n�w�nffiffiffip

p � K þ 1ð Þ2n�1 � e� b�w�n� Kþ1ð Þ2n½ �2 þ 1� vð Þ K þ 1ð Þ�v

ð45ÞAs it is difficult to calculate the optimal solution by

calculating oC(K)/oK = 0, we resort to numerical solutions to findKopt for the given conditions using (45). Taking PFA = 0.02,b = 1.1987, w = 100.1SNR0-4.9040, v = 2.2047, n = 20, and SNR0 = 10 dB,the results are plotted in Fig. 3, indicating the relationship betweenthe number of relays K and performances of PD(K), g(K) and C(K),respectively.

Y. Chen, Y. Tang, J. Liu et al. Applied Acoustics 177 (2021) 107911

From Fig. 3(a), if we set PD0 = 0.5 and g0 = 0.3, then we haveK0 = 2, K1 = 6, which means that the optimal range of the numberof relays is 2 � K � 6. Hence this plot shows that there may be nosolution for certain with separately drawing the PD and g. How-ever, when adopting PD and g as the joint optimal objective func-tion, the unique solution of the optimal number of relays shouldbe Kopt = 4 from Fig. 3(b). Therefore, as shown in Fig. 3(a), wecan approximately define the number of relays according to theintersection of the two curves as the optimal number of relays Kopt,where the PD is approximately equal to g, indicating that the twoindexes are almost equal and there is no ‘‘favoritism” phenomenon.

5. Numerical analysis

In this section, we numerically evaluate the probability ofdetection/energy consumption as a function of the number ofrelays by adjusting different parameters, aiming at obtaining theoptimal number of relays for multi-hop UWA networks.

Note that, according to (31) and (34), it can be found that theeffects of factors, including target SNR, transmission distance,and transmission time, have disappeared during the calculationof the energy consumption rate. Therefore, to study the effects ofthe mentioned factors on energy consumption, we will take theabsolute value of energy consumption in (31) instead of the nor-malized value of that in (34) in the following simulation.

5.1. Simulation parameters setup

The basic parameters for simulation are as follows: the distancebetween the source node and destination node is dc = 50 km; thetarget SNR in each hop where the receiver can correctly receive asignal is SNR0 = 10 dB; the position angle of Ix is h = p/15; theenergy required for the transmitter to encode and that for receiverto decode are equal, which are set as 1% of the energy required fortransmitting; the transmitted data length LD is 256 bytes, and thetransmission bit rate Rb is 160 bps, then the time required to trans-mit the data is TL = L/Rb. Take the integration time of the Ix asTi = 1.2TL. The bandwidth of Ix is the same as that of the receiver,and the false alarm probability of Ix is PFA = 0.02. The other param-

0 10 20 30 40 5Transmission

0

5

10

15

20

25

30

Band

wid

th W

[kH

z]

Fig. 4. The relationship between the available bandwidth and trans

8

eters for calculating (26), (27) and (28) corresponding to the opti-mal frequency fopt are x = 101.4291, w = 100.1SNR0-4.9040, -=0.5392,v = 2.2074, and / = 0.25. The parameters will be adjusted accord-ing to different simulation purposes.

5.2. General covert UWA communications

First, given the distance of each hop dk, the optimal operatingfrequency fopt for each hop can be calculated by (15) and (16).Meanwhile, from (26), we can calculate the relationship betweendk and the available bandwidthW, as shown in Fig. 4. It can be seenthat W is reduced with the increase of dk. On the other hand, therelationship between dk and the transmit power is shown inFig. 5 calculated by (27) and (28) under different target SNR0 val-ues. From Fig. 4 and 5, it is clear that a shorter dk will be better,which means more relays will consume less power and offer moreavailable bandwidth, improving the transmission rate for multi-hop UWA networks.

Next, let the transmission distance be fixed at 50 km. Fig. 6shows the relationship between PD and c under different targetSNR0 values for the single-hop UWA communication system. Withthe same PD condition, it can be seen that the lower the target SNR0,the smaller the distance ratio that is required, indicating that thecovertness of the system is better. This simulation result is inaccordance with the theoretical analysis in Section 2.

Considering both covertness and energy consumption, it can beobserved from Figs. 5 and 6 that if the target SNR0 decreases from12 dB to 8 dB, the emission source power Pt will decrease from166.4425 dB to 162.4425 dB, and the distance ratio c will decreasefrom 0.875 to 0.5, aiming at PD = 0.5 over a 50-km transmission dis-tance. Therefore, designing a good transmitting signal to ensure thereceiver can correctly decode with lower target SNR0 is the key toachieve low-power consumption in covert UWA communications.In the meantime, since the complexity of uncertainty in oceanicturbulence on UWA channels will affect the target SNR0, differentUWA channel states usually need different target SNR0. Hence itis very important to design the good UWA signal and decodingscheme to overcome the shortcomings of variable UWA channelsand keep the performance of LPD at the same time in practice.

0 60 70 80 90 100 Distance [km]

Approximate curve

mission distance of a single-hop UWA communication system.

0 10 20 30 40 50 60 70 80 90 100Transmission Distance [km]

120

130

140

150

160

170

180

Tran

smit

Pow

er P

t [dB

re u

Pa]

SNR0=8 dB

SNR0=10 dB

SNR0=12 dBSNR0 increase

Fig. 5. The relationship between the transmit power and transmission distance for a single-hop UWA communication system.

0 0.5 1 1.5 2 2.5 3Distance ratio

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Prob

abilit

y of

det

ectio

n P

D

SNR0=8 dB

SNR0=10 dB

SNR0=12 dB

LPD region

SNR0 increase

Fig. 6. The relationship between the probability of detection and distance ratio with different target SNRs for a single-hop UWA communication system.

Y. Chen, Y. Tang, J. Liu et al. Applied Acoustics 177 (2021) 107911

5.3. The effect of different target SNR0

To further discover the effect of the different target SNR0 valueson the covertness and energy efficiency for the multi-hop UWAnetworks, Fig. 7 shows the relationship between the probabilityof detection/energy consumption and the number of relays withdifferent target SNR0 values. Obviously, with the increase in thenumber of relays K, the energy consumption Esys of the multi-hop UWA system decreases, while the probability of detection PDincreases. On the other hand, to keep the same PD, the number ofrelays K must be reduced when the target SNR0 is increasing. This

9

is because the increased target SNR0 leads to an increased instanta-neous transmit power, which is more easily intercepted by Ix. Forexample, to maintain PD = 0.42, the number of K must be reducedfrom 6 to 3 when the target SNR0 increases from 8 dB to 12 dB,otherwise the signal is intercepted more easily.

Under the given basic simulation conditions, in the rectangularregion of Fig. 7 (i.e., the LPD region), when the target SNR0 is set at8 dB, the number of relays K should be between 2 and 6 to satisfythe requirement of the LPD, i.e., PD < 0.5. Meanwhile, the energyconsumption is relatively low. If the target SNR0 is 12 dB, the num-ber of relays K can only be 2 or 3 when taking into account the

0 1 2 3 4 5 6 7 8 9 10Number of relays K

0

0.5

1

Prob

abilit

y of

det

ectio

n P

D

0 1 2 3 4 5 6 7 8 9 100

10

20

Ener

gy c

onsu

mpt

ion

E sys [J

]

PD,SNR0=8 dB

PD,SNR0=10 dB

PD,SNR0=12 dB

Esys,SNR0=8 dB

Esys,SNR0=10 dB

Esys,SNR0=12 dB

SNR0 increase

SNR0 increase

Fig. 7. The relationship between the number of relays and probability of detection, energy consumption, respectively, with different target SNR0.

Y. Chen, Y. Tang, J. Liu et al. Applied Acoustics 177 (2021) 107911

requirements of both LPD performance and low powerconsumption.

5.4. The effect of different transmission distances

Fig. 8 shows the effect of different transmission distances dc onPD and energy consumption. It is clear that the total energy con-sumption of the system is increasing as the total transmission dis-tance increases, as shown in Fig. 8. Further, when the transmissiondistance is 20 km and the number of relays K is 3, we find thatPD = 0.3153 and the energy consumption is 0.2497 J. When thetransmission distance is 80 km and the number of relays K is 3,we find that PD = 0.1950 and the energy consumption is 5.3258 J.With the same number of relays K and the other conditionsunchanged, the distance of each hop is also increasing as the totaltransmission distance increases, which is equivalent to reducing

0 1 2 3 4 5Number of r

0

0.5

1

Prob

abilit

y of

det

ectio

n P

D

0 1 2 3 4 5

dc increase

Fig. 8. The relationship between the number of relays and the probability of detecti

10

the risk of being intercepted by a multi-hop scheme, and thusreducing the overall PD of the system. On the other hand, whenthe transmission distance is 80 km and the number of relays K isincreased to 4, the PD is only 0.3180, and the energy consumptionis reduced to 4.0679 J. Even if the number of relays K is increased to5, the PD is only 0.4740, and still meets PD < 0.5, i.e., the index of theLPD, while the energy consumption is reduced to 3.2641 J. Thismeans that, over the 80-km transmission distance, when the num-ber of relays K increases from 3 to 5, the energy consumptiondecreases from 5.3248 J to 3.2641 J, a decline of 38.7%, and it stillmeets the LPD requirements.

Therefore, based on the given basic simulation conditions, whenthe total transmission distance of the multi-hop UWA system is80 km, the number of relays K can be set between 2 and 5, whichmeets the LPD requirement, i.e., PD < 0.5, and the energy consump-tion is also relatively low. If the transmission distance reduces to

6 7 8 9 10elays K

6 7 8 9 100

20

40

Ener

gy c

onsu

mpt

ion

E sys [J

]

PD,dc=20 km

PD,dc=50 km

PD,dc=80 km

Esys,dc=20 km

Esys,dc=50 km

Esys,dc=80 km

dc increase

on and energy consumption, individually, with different transmission distances.

Y. Chen, Y. Tang, J. Liu et al. Applied Acoustics 177 (2021) 107911

20 km, the number of relays K can only be 2 or 3 when consideringthe requirements of both LPD performance and low powerconsumption.

5.5. The effect of different position angles of the interceptor

Fig. 9 presents the effect of different position angle parametersof the interceptor on PD and energy consumption. It can be seenfrom the figure, that under the same transmission distance, whenincreasing the position of angle interceptor, the interceptor is clo-ser to the transmitter and can obtain a higher PD. In particular,when the number of relays K is 3 and the angle of interceptorincreases from p/20 to p/10, the PD of the system is increased from0.1418 to 0.4982. In addition, if we only change the position of theinterceptor, it does not affect the energy consumption of the multi-

0 1 2 3 4 5Number of r

0

0.2

0.4

0.6

0.8

1

Prob

abilit

y of

det

ectio

n P

D

0 1 2 3 4 5

LD increa

Fig. 10. The relationship between the number of relays and the probability of

0 1 2 3 4 5Number of r

0

0.5

1

Prob

abilit

y of

det

ectio

n P

D

0 1 2 3 4 5

increase

Fig. 9. The relationship between the number of relays and the probability of detection, en

11

hop UWA system. Thus, if the interceptor changes position, thesystem energy consumption remains unchanged.

Therefore, given the basic simulation conditions, when the posi-tion angle of the interceptor is h = p/20, the number of relays K canbe between 2 and 6, which satisfies the requirement of the LPD,and the energy consumption stays relatively low. If the positionangle of the interceptor is h = p/10, the number of relays can onlybe 2 or 3 when considering the LPD performance and low powerrequirement of the system.

5.6. The effect of different data lengths

Fig. 10 plots the effect of different data lengths LD on PD andenergy consumption. We observe that, as the data length increases,the integration time of interceptor is also increasing, and thus the

6 7 8 9 10elays K

6 7 8 9 100

5

10

15

20

25

Ener

gy c

onsu

mpt

ion

E sys [J

]

PD,LD=128 bytes

PD,LD=256 bytes

PD,LD=512 bytes

Esys,LD=128 bytes

Esys,LD=256 bytes

Esys,LD=512 bytes

LD increase

se

detection, energy consumption, individually, with different data lengths.

6 7 8 9 10elays K

6 7 8 9 100

10

20

Ener

gy c

onsu

mpt

ion

E sys [J

]

PD, = /10

PD, = /15

PD, = /20

Esys, = /10, /15, /20

ergy consumption, individually, with the different position angles of the interceptor.

Y. Chen, Y. Tang, J. Liu et al. Applied Acoustics 177 (2021) 107911

energy is dispersed in a longer time, reducing the instantaneouspower, thereby further reducing the probability of being inter-cepted. It should be noted that although the total energy consump-tion increases as the data length increases, the increase rate is notenough to eliminate the energy dispersion due to the increase ofthe integration time. Thus, the instantaneous power is stillreduced. Specifically, taking the number of relays K as 3, if the datalength increases from 128 bytes to 512 bytes, the system energyconsumption will increase from 0.9436 J to 3.7743 J, yet the PD willdecrease from 0.3617 to 0.1528.

In summary, under the given basic simulation conditions, whenthe data length is 512 bytes, the number of relays K can be setbetween 2 and 6, which satisfies the requirement of LPD perfor-mance, and the energy consumption is relatively low. If the datalength is 128 bytes, the number of relays K can only be 2 or 3 whenconsidering the requirements of both LPD performance and lowpower consumption.

6. Conclusion

We study the optimization of the number of relays for multi-hop UWA covert communication networks. By establishing themodels for probability of detection and energy consumption forthe multi-hop system, we theoretically derive the range of thenumber of relays both in terms of probability of detection and ofenergy consumption/energy consumption rate. Furthermore, wepresent the unique solution of the optimal number of relays byconstructing an objective function as the sum of the probabilityof detection and the energy consumption rate. The impact of thetarget SNR, transmission distance, position angle of interceptor,and data length are investigated in detail by simulation, while opti-mizing the number of relays considering the above two conditions.The numerical results validate our theoretical analysis and showthat the number of relays must be reduced to maintain the samedetection probability as the target SNR increases. Under the sameprobability of detection, the number of relays should be increasedappropriately to reduce the overall energy consumption of themulti-hop system. Under the given conditions, such as a totaltransmission distance of 50 km, data length of 256 bytes, targetSNR0 of 10 dB, and interceptor position angle of p/15, the numberof relays can be between 2 and 6 to meet the requirements of cov-ert communication, i.e., a probability of detection<0.5, while theenergy consumption is relatively low.

Funding

This work was supported in part by the Basic Research Programof Science and Technology of Shenzhen, China under GrantJCYJ20190809161805508, in part by the National Key Researchand Development Program of China under Grant2016YFC1400200, and in part by the National Natural ScienceFoundation of China under Grant 41476026, Grant 41676024 andGrant 41976178, and Natural Science Foundation of Fujian Pro-vince of China under Grant 2018J05071.

CRediT authorship contribution statement

Yougan Chen: Conceptualization, Methodology, Writing -review & editing. Yuying Tang: Writing - original draft. JunhuiLiu: Writing - original draft. Xiaokang Zhang: Software. XiaomeiXu: Supervision.

12

Declaration of Competing Interest

The authors declare that they have no known competing finan-cial interests or personal relationships that could have appearedto influence the work reported in this paper.

Acknowledgment

The authors would like to thank Mr. Jianming Wu, Mr. ShenqinHuang from Xiamen University for their contributions to the earlydiscussions of this research. This work was supported in part bythe Basic Research Program of Science and Technology of Shen-zhen, China under Grant JCYJ20190809161805508, in part by theNational Key Research and Development Program of China underGrant 2016YFC1400200, and in part by the National NaturalScience Foundation of China under Grant 41476026, Grant41676024 and Grant 41976178, and Natural Science Foundationof Fujian Province of China under Grant 2018J05071.

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Yougan Chen (S’12-M’13-SM’19) received the B.S.degree from the Northwestern Polytechnical University(NPU), Xi’an, China, in 2007 and the Ph.D. degree fromXiamen University (XMU), Xiamen, China, in 2012, bothin communication engineering. He visited the Depart-ment of Electrical and Computer Engineering, Univer-sity of Connecticut (UCONN), Storrs, CT, USA, fromNovember 2010 to November 2012. Since 2013, he hasbeen with the College of Ocean and Earth Sciences,XMU, where he is currently an Associate Professor ofApplied Marine Physics and Engineering. His researchincludes the application of electrical and electronics

engineering to the oceanic environment, with recent focus on cooperative com-munication and artificial intelligence for underwater acoustic channels.

Yuying Tang received B.S., M.S. degrees in marinetechnology from Xiamen University (XMU), Xiamen,China, in 2016 and 2019, respectively. She is now pur-suing her Ph.D. degree in naval architecture and oceanengineering at Shanghai Jiao Tong University, Shanghai,China. Her research interests focus on underwateracoustic networking, signal processing, and liquidsloshing in tanks.

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Junhui Liu obtained the B.S. degree in marine technol-ogy from Xiamen University (XMU), Xiamen, China, in2018. She is currently working toward the Master’sdegree at Peking University, Beijing, China. Her researchinterests focus on signal processing for communicationsand underwater acoustic networking.

Xiaokang Zhang received the B.S. and Ph.D. degrees inmarine physics from Xiamen University (XMU), Xiamen,China, in 2005 and 2010, respectively. From 2010 to2012, he was a Postdoctoral Research Associate at theCenter of Environmental Science and Engineering, XMU.He has been a Senior Engineer with the Department ofApplied Marine Physics and Engineering, XMU, since2012. His research interests focus on underwateracoustic communications and networking.

Xiaomei Xu received B.S., M.S., and Ph.D. degrees inmarine physics from Xiamen University (XMU), Xiamen,China, in 1982, 1988, and 2002, respectively. She was aVisiting Scholar with the Department of Electrical andComputer Engineering, Oregon State University, Cor-vallis, OR, USA (1994–1995). She visited the Departmentof Electrical and Computer Engineering, University ofConnecticut (UCONN), Storrs, CT, USA, as a SeniorVisiting Scholar in 2012. She is now a Full Professor withthe Department of Applied Ocean Physics and Engi-neering, XMU. Her research interests lie in the fields ofmarine acoustics, underwater acoustic telemetry and

remote control, underwater acoustic communication, and signal processing.