Transmission Capacity of Underwater Sensor Networks: A Case for Fixed Distance

Mohammad Sheikh Zefreh1, Ali Mohammad Fouladgar1, Farzad Eskandari2, Pejman Khadivi1

1Department of Electrical and Computer Engineering, Isfahan University of Technology, Isfahan, IRAN 2Babol University of Technology, Babol, IRAN

{sheikhzefreh,amfooladgar, pkhadivi}@ec.iut.ac.ir , eskandari@mrc.co.ir

Abstract In this paper, upper and lower bound on the transmission capacity of underwater sensor networks are derived. Transmission capacity is defined as the maximum permissible density of simultaneous transmissions that allows a certain probability of successful reception. In order to drive these bounds it is assumed for the network model that the distance between source and destination is a fixed value. The ratio of signal to interference (SIR) for underwater acoustic channels is used as a figure of merit for the system performance. This ratio is determined and employed as a criterion for successful reception. Simulation results support the analysis.

Keywords-Underwater sensor networks; Transmission capacity; Acoustic propagation.

1. Introduction Underwater sensor networks are deemed to have applications for ocean sampling, oceanographic data collection, environmental and pollution monitoring, offshore exploration, disaster prevention, tsunami and seaquake warning, assisted navigation, distributed tactical surveillance, and mine reconnaissance. Underwater communication links are based on acoustic wireless technology, which poses unique challenges due to the harsh underwater environment, such as limited bandwidth capacity, high and variable propagation delays, high bit error rates, and temporary losses of connectivity caused by multipath and fading phenomena [1]. In addition, the path loss in underwater acoustic communication channels depends not only on the distance between the transmitter and receiver, as it is the case in many other wireless channels, but also on the signal frequency [2]. Within such an environment, designing an underwater acoustic (UWA) network that maximizes throughput and reliability while minimizing the power consumption becomes a very difficult task.

One of the questions that arise naturally at this time is that if two transmitters that their relevant receivers are in the vicinity of each other, send their data in the same time, the Signal to Interference ratio (SIR) decreases at the receivers; Hence, what are the fundamental capabilities of underwater networks in supporting multiple nodes that wish to communicate to (or through) each other over an acoustic channel in order to satisfy the QoS requirements? While research has been extremely active on assessing the capacity

of wireless radio networks, no similar analysis has been reported for underwater acoustic networks [2].

In this paper, upper and lower bound on the maximum permissible density of simultaneous transmissions in underwa- ter sensor networks are derived. In order to drive these bounds it is assumed that the distance between source and destination is a fixed value, and we postpone the problem for the case that the power control is available in the network architecture[3,4].

The rest of the paper is organized as follows: In Section 2, we discuss related works on capacity issues over wireless multi-hop networks. Underwater Acoustic Propagation model is described in Section 3. Transmission capacity of underwater sensor networks for a special case that the distance between source and its one-hop destination is a fixed value is determined in Section 4. Section 5 is dedicated to conclusions. 2. Related Works The capacity of wireless multi-hop networks is an interesting research area and current issue of many researchers. Gupta and Kumar [5] studied the capacity of ad hoc wireless networks, which is defined in the usual manner as the maximum data rate that is achievable by all source–destination pairs of nodes in the limit as the number of nodes grows to an arbitrarily large level. Under this model, stationary nodes are randomly and uniformly located (over a disk area), and each node sends data to a randomly and uniformly selected destination. Their main result indicates that as the number of nodes per unit area

)(λ increases, the throughput capacity decreases approximate -ly as λ/1 . Grossglauser and Tse [6] exploit nodal mobility to determine its effect on the network capacity. Allowing for unbounded delay and using only one-hop relaying, they show that mobility increases the capacity asymptotically. In [7], Weber et al. used a model, includes a stochastic SINR constraint along with random channel access in order to accurately model the behavior of a distributed ad hoc network. That is, under a random distribution of transmitters, the probability of the SINR ratio being inadequate for successful reception must be below some constant ε , which called the outage constraint. The results summarize the dependence of ad hoc network capacity on path loss, transmission power, target signal-to-noise ratio (SNR), spreading gain, outage probability, and other system parameters. M. Stojanovic discusses about the relationship between capacity and distance in underwater environments and the bandwidth dependency on

The Second International Conference on Sensor Technologies and Applications

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DOI 10.1109/SENSORCOMM.2008.138

706

The Second International Conference on Sensor Technologies and Applications

978-0-7695-3330-8/08 $25.00 © 2008 IEEE

DOI 10.1109/SENSORCOMM.2008.138

699

the distance is assessed using an analytical method that takes into account physical models of acoustic propagation loss and ambient noise [2].

In this paper we determine maximum permissible density of simultaneous transmissions of underwater sensor networks exploiting the work by Weber et al [7].

3. Underwater Acoustic Propagation Attenuation that occurs in an underwater acoustic channel over a distance r for a signal of frequency f is given by:

rk farfrA )(),( =

where, k is the spreading factor, and a(f ) is the absorption coefficient [2]. The spreading factor k describes the geometry of propagation, and its commonly used values are k = 2 for spherical spreading, k = 1 for cylindrical spreading, and k = 1.5 for the so-called practical spreading. The absorption coefficient can be expressed empirically, using the Thorp’s formula which gives a(f ) in dB/km for f in kHz as [2]:

003.01075.24100

441

11.0)(log10 242

2

2+×+

++

+= − f

ff

fffa

4. Transmission Capacity of Underwater Sensor

Networks Transmission capacity is defined as the maximum permissible density of simultaneous transmissions that satisfies a given constraint on the probability of successful reception [7]. Our goal in this paper is that to determine this capacity for a special case that each transmitter uses a fixed transmitted power same to other transmitters. In other word, the distance between each sender and its corresponding receiver is r, which is a fixed value. Formally, let }X{ i=Π denotes a homogeneous Poisson point process on a plane, where the points Xi denote the locations of transmitting nodes [12]. We define the transmission capacity, λ , as the maximum density of points in Π such that the probability a typical receiver is unable to decode its reception is less than an outage requirement ε , 10 <<< ε .

In order to determine the capacity, it is assumed that transmitters employ a fixed transmission power ρ for receivers that are a fixed distance away. Network capacity is not independent of the modulation technique used in the physical layer. In a network that FH-CDMA is used, bandwidth W is divided into M sub-channels,

where MW is required bandwidth per channel. A receiver

attempting to decode a signal from a transmitter on sub-channel }...,,1{ Mm ∈ only sees interference from other simultaneous transmissions on that sub-channel. By contrast on a DS-CDMA network, a receiver sees interference from all transmitters, and spreading factor M is used to reduce the minimum SINR required for successful reception. If the nominal SINR requirement for FH-CDMA is β , then DS

CDMA has a reduced SINR requirement of Mβ

, assuming that the interference is treated as wideband noise.

In underwater communication, a practical system is normally designed such that the noise is negligible with respect to interference, i.e. SIRSNR >> , or equivalently,

SIRSINR ≈ . Hence, SIR is used as a figure of merit for the system performance [8]. Thus, the appropriate requirement on λ under the outage probability constraint ε for DS-CDMA is given by:

ε≤≤ β )( MSIRP

To evaluate the outage probability we consider on a typical receiver at the origin and assume that all transmitters use same transmitted power ρ , or in other word, the distance between each source and its corresponding receiver is a fixed value. Also, we ignore multi-hop routing and just consider one-hop routing. The MAC protocol adopted in this paper is also simple in the sense that all randomly distributed transmitters are presumed to transmit in an ALOHA-type fashion for a given network configuration. Based on these assumptions, upper and lower bound of the transmission capacity of underwater sensor (or ad hoc) networks when we use DS-CDMA are given in the following theorem:

Theorem : Let )1,0(∈ε , 824.0=k , 25.0=α , 2

026.0

.1r

eRr−

=β

.

Upper , DSu

,ελ , and lower, DSl

,ελ , bounds on the transmission capacity of underwater sensor networks, subject to the outage constraint ε for DS-CDMA, when transmitters employ a fixed transmission power ρ for receivers that are a fixed distance r away are:

)O(M.1 22

026.0, εε

βπλε +=

−

re r

DSu

)()21.(])21(

.).[(1.1 2222

, εεαα

απ

λ αε oMRk

DSl +−

−≥

as 0→ε .

Proof :

SIR for underwater acoustic channels is determined in Appendix. Hence, the outage constrain can be written as follows:

=≤ )(M

SIRP β εβ ≤≤∑ +−−

+−−)

||.

.()(026.0

)(026.0

21

21

MXe

reP

i

ki

X

kr

i

707700

Where k is the spreading factor, r is the distance between source and its one-hop destination, and Xi represents distance between i th interferer and receiver.

Considering the outage constrain, the outage event can be defined by:

⎭⎬⎫

⎩⎨⎧

≥∑=λ +−−β

+−− )(026.0)(026.0 21

21

.||.)( krMi

ki

X reXeE i

We will define events ),( sEl λ and )s,(Eu λ such that ),()(),( sEEsE lu λ⊆λ⊂λ and the probabilities of all events )(λE , ),( sEl λ and ),( sEu λ increase in λ . Clearly, ))s,(E(P))(E(P))s,(E(P lu λλλ ≤< , therefore if we

can find the largest possible *λ such that *λλ > results ελ >)),(( sEP u then it follows that

ελ >))(E(P , i.e. the outage probability is exceeds our outage constraint ε and thus an upper bound on the contention density will be obtained. Same to this approach, we can determine the lower bound using )s,(El λ . In order to define )s,(El λ and )s,(Eu λ , we split the network in two areas, regions inside and outside a circle of radius s around the typical receiver at the origin, denoted by b(O, s) and s)O,(b respectively. The radius s is selected to be small enough such that one or more nodes within distance s would cause an outage. We consider the DS-CDMA case. The outage events associated with these two areas are defined as follows. Definition:

}0),({),( ≠∩Π= SObSEu λ

⎭⎬⎫

⎩⎨⎧ ≥∑= +−−

∩Π∈

+−− )(026.0

),(

)(026.0 21

21

.||.)( krM

SObi

ki

Xf reXeE i

βλ

),(),(),( SESESE ful λλλ ∪=

According to the definition, ),( sEu λ consists of all outcomes where there are one or more transmitters within distance s of the origin O, and the event ),( sE f λ consists of

all outcomes where the set of transmitters outside the ball b(O, s) generate enough interference power to cause an outage at the origin.

In order to find an upper bound on the transmission capacity of underwater sensor networks, we need to find the value of s. According to the definition, the radius s should be selected to be small enough such that even one node within distance s would cause an outage. Hence, s can be found from the following inequality

θ≥+−− )k(S. s.e 21

0260

Where )(026.0 21

. +−−β

=θ krM re . Using k=1.5, we have

θ≥−

2

0260

se s.

. After some mathematical calculations, bound on s

can be approximated byθ1≈≤ maxss . Substituting θ , we

have21

2

026.0 −−

⎥⎥⎦

⎤

⎢⎢⎣

⎡≤

res

rMβ . Note that for

21

2

0260 −−

⎥⎦

⎤⎢⎣

⎡≤

re

sr.

Mβ the

events have the following property:

),()(),( sEEsE lu λλλ ⊆⊂

Upper Bound: We calculate )),(( sEP u λ , set it equal toε , and solve it for λ . Consider the probability that there are no transmitters in b(O, s), which is simply the void probability for b(O, s). For a Poisson point process in the plane with intensityλ , the probability of existence of k nodes in the area A is

!k)A(e)Ak(P

kA λλ−

=∈ .

Thus, 2

101 Su e))s,O(b(P))s,(E(P λπλ −−==∩Π−=

Now, given the outage constraint ελ ≤)),(( sEP u , we obtain the parameterized upper bound

)(Lns)s(u επ

λ −−= − 11 2

for all ).,0(21

2

026.0 −−

⎥⎥⎦

⎤

⎢⎢⎣

⎡∈

reMs

r

β and all ),( 10∈ε .We

further optimize this bound over s and find that the tightest (smallest) upper bound is obtained by choosing the largest

possible 21

2

026.0

max .−−

⎥⎥⎦

⎤

⎢⎢⎣

⎡==

reMss

r

β. Thus, the final upper

bound on the optimal contention density is

)O(.1)1(.1 22

026.0

2

026.0εε

πε

πλ ββ

ε +=−⎥⎥⎦

⎤

⎢⎢⎣

⎡−=

−−

reLn

re r

Mr

Mu

Lower Bound: In order to detemine the lower bound, we

need to calculate )),(( sEP l λ . Hence, we have )),(),(()),(( sEsEPsEP ful λλλ ∪=

)),(()).,(()),(()),(( sEPsEPEPEP fufu λλελελ −+= Thus, this outage constrain can be rewritten as

708701

1)),(( ελ ≤sEP u , 2)),(( ελ ≤sEP f Now, we define the contention density imposed by

conditions in these two inequalities to be }),((|sup{)( 1

1 ελλλε ≤= sEPs uu

}),((|sup{)( 22 ελλλε ≤= sEPs ff

Thus, the lower bound of the optimal contention density can be derived as

}} , {inf{ 21

21 ),(,

εε

εε

ε λλλ fus

l Sup=

for ),0(21

2

026.0 −−

⎥⎦

⎤⎢⎣

⎡∈

reMs

r

βand ),( 21 εε satisfying

εεεεε =−+ 2121 .

Same to the upper bound, we have

)O( 1)( 21

21 εεπ

λε += −ssu for )1,0(1 ∈ε and s >0.

In order to determine )(2 sfελ , we first consider the

event fE . Let )(026.01 21

. +−−= kr reR β . Therefore, this event can

be rewritten as

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

≥= ∑∩Π∈

+−− MRXeEsObi

ki

Xf

i

),(

)(026.0 21

||.)(λ

It is not possible to compute )(2 sfελ directly; we will find a

lower bound using Chebyshev’s inequality and Campble theory [12]. Let

∑∩Π∈

+−−=),(

)(026.0 21

||.),(sObi

ki

X XesY iλ

Using Campble theory, we have

λμπλλ ).(..2)],([ )(026.0 21

srdrresYEs

kr =∫=∞ +−−

λσπλλ ).(.).(2)),(( 22)(026.0 21

srdrresYVars

kr =∫=∞ +−−

Therefore, the lower bound on )(2 sfελ is obtained as follows:

}),((|sup{ 22 ελλλε ≤= sEP ff

})||.(|sup{ 2),(

)(026.0 21

ελ ≤≥= ∑∩Π∈

+−− MRXePsObi

ki

X i

})),((|sup{ 2ελλ ≤≥= MRsYP

}1)(

|)(),(||sup{ 2ελμ

λμλλ ≤⎟⎟⎠

⎞⎜⎜⎝

⎛≥

−−≥

sMRssYP

})).((

).(|sup{ 22

2ε

λμλσλ ≤

−≥

sMRs

)(2 sfελ=

Clearly, )(2 sfελ is achieved when 22

2

)).(().( ε

λμλσ =

− sMRs .

Solving this equation for 2ελ f we have

)()(

)()( 2222

22 εε

σλε O

sMRsf += .

We now focus on obtaining the lower bound. It can be seen that if we increase )),(( sEP u λ then )),(( sEP f λ decreases,

i.e., changing s or ),( 21 εε must increase one of them and decrease another, simultaneously. Thus, a condition for the optimized lower bound is that the choice of s and ),( 21 εε such

that )()( 21 ss fuεε λλ = . Hence, we will have

1222

2 1.1)(

)( επ

εσ ssMR = .

Solving this equation is not straightforward. Consequently, we use Newton’s approach to find the root of equation and then approximate its variations. After some mathematical computations, the approximated root can be achieved which is

equal to 25.0824.0)(

AAsa = where 824.0=k and 25.0=α are

fixed parameters and1

22.)(21

εεMRA = . Substituting as in

)(1 suελ we have

.1)(1)( 12

2

12, 21 ε

πε

πλ

α

αεε

kA

Aksl == − .

Finally, we should find maximum value of 21,εελl . After some mathematical computations, it can be shown that this maximum is occurred in εαε )21(*

1 −= , and αεε 2*2 = .

Substituting these values, we obtain the lower bound as follows:

2 22

1 1. .[( ) . ] .(1 2 )(1 2 )l MR

kε ααλ α ε

π α= −

− where )(026.01 2

1. +−−= kr reR β and 0→ε .

5. Simulation Results In order to simulate the analytical results, we consider an area with 20 km×20 km dimensions. Thorp’s formula is used for the acoustic attenuation model. Analytical and simulation results are demonstrated in Fig.1. Due to approximations in mathematical computations, simulation carve is not between upper and lower bound for small values of spreading gain. However, it can be seen that this carve is exactly between bounds for some greater values of spreading gain. As a result, while the transmission capacity of wireless radio frequency (RF) ad hoc or sensor networks increase with spreading gain M with order α

2

M ( α is the path-loss exponent), this variation, due to different propagation model, is linear for

709702

0 10 20 30 40 50 60 7010

-4

10-3

10-2

10-1

Spreading Gain

Max

imum

Den

sity

of S

imul

tane

ous

Tra

nsm

issi

ons

Upper Bound•Lower BoundSimulation Result

Figure 1. Maximum permissible density of simultaneous transmissions versus

spreading gain

underwater ad hoc or sensor networks in the same model. 6. Conclusion and Future Works

In this paper, upper and lower bounds on the transmission capacity of underwater sensor networks are determined. It is assumed that the distance between source and destination is a fixed value. Transmission capacity is defined as the maximum permissible density of simultaneous transmissions that allows a certain probability of successful reception. As a future work, we plan to determine this capacity for a more realistic case. For example, power control can be considered where the distance between sender and receiver is variable. Also, in this paper we assumed that a simple MAC protocol in the sense that all randomly distributed transmitters are presumed to transmit in an ALOHA-type fashion for a given network configuration. However, we can extend the analytical approach to cover a more complicate MAC protocol like IEEE802.11 MAC sub-layer.

References

[1] E.Sozer, M.Stojanovic and J.Proakis, "Underwater

Acoustic Networks," IEEE Journal of Oceanic Engineering, vol.25, No.1, January 2000, pp.72-83.

[2] M.Stojanovic, “On the relationship between capacity and distance in an underwater acoustic channel,” in Proc. First ACM International Workshop on Underwater Networks (WUWNet), Sept. 2006.

[3] M. Sheikh Zefreh, P. Khadivi,”Maximum Permissible Density of Simultaneous Transmissions in Underwater Sensor Networks”, IEEE International Conference on Information & Communication Technologies, Syria, April 2008.

[4] M. Sheikh Zefreh, M.Sc. Thesis, Department of electrical and computer engineerin, Isfahan university of technology, Isfahan, IRAN, March 2008.

[5] P. Gupta and P. Kumar, “The capacity of wireless networks,” IEEE Transactions on Information Theory, vol. 46, no. 2, pp. 388–404, March 2000.

[6] M. Grossglauser and D. Tse, “Mobility increases the capacity of ad hoc wireless networks,” IEEE/ACM Trans. Netw., vol. 10, no. 4, pp. 477–486, Aug. 2002.

[7] S. P. Weber, X. Yang, J. G. Andrews, and G. d. Veciana, “TransmissionCapacity of Wireless Ad Hoc Networks With Outage Constraints”, IEEE Transactions on Information Theory, vol. 51, no. 12, December 2005.

[8] M.Stojanovic, “On the Design of Underwater Acoustic Cellular Systems,” in Proc. IEEE Oceans Conf., 2007.

[9] A. F. Harris, M. Zorzi,”Modeling the Underwater Acoustic Channel in ns2”, in Proc. of the Second ACM International Workshop on UnderWater Networks (WUWNet07), 2007.

[10] W. H. Thorp, “Analytic Description of Low Frequency Attenuation Coefficient”, J. Acoust. Soc. Am., 42, 270-271, 1967.

[11] R. J. Urick, Principles of Underwater Sound, McGraw-Hill, 1983.

[12] D. Stoyan, W. Kendall, and J. Mecke, Stochastic Geometry and Its Applications, 2nd ed. New York: Wiley, 1996.

APPENDIX

In an acoustic channel, the simple path loss model does not hold. Instead, the attenuation depends on the signal frequency as well as the transmission distance. As mentioned earlier, attenuation that occurs in an underwater acoustic channel over a distance r for a signal of frequency f is given by:

rk farfrA )(),( =

where, k is the spreading factor, and a(f ) is the absorption coefficient. Thorp’s formula is one of the most famous absorption models for underwater acoustic propagation and is as follows:

003.01075.24100

441

11.0)(log10 242

2

2+×+

++

+= − f

ff

fffa

where a(f ) is in dB/km, and f in kHz. (Note that although this formula is used in many recently

published papers [2],[ 8], [9], the correct one is as follows:

003.01075.24100

441

11.0)(log10 242

2

2

2+×+

++

+= − f

ff

fffa

which can be found in the original paper [10] and [11]. However, the first formula is used in this paper.) Thus, if we denote power spectral density of transmitted signal by tS , received power at a receiver at distance d away from the transmitter can be obtained as

dfrf

ff

ffrS

dffarS

dffrASrP

Wff

kt

Wff

rkt

Wff t

10ln.).003.010.75.2

41004.4

1011.0(exp(..

)(

),()(

25

2

2

2

1

minmin

minmin

minmin

++

∫ ++

+−=

∫=

∫=

−

+−

+ −−

+ −

710703

dfDCf

ffB

ffArS Wf

fk

t

))

41001(exp(..

2

2

2

2minmin

++

∫ ++

+−= +−

Where coefficients A, B, C, D are

10ln103,10ln1075.210ln4.4,10ln011.0

45 rDrCrBrA

−− ×=×===

Calculating the last integral is not straightforward; therefore we use the following approximations

11 2

2

≅+ ff

, 41004100

22 ff

f ≅+

.

The operating frequency in underwater communication systems is few kHz, thus it seems that these are fair approximations. Thus we can rewrite P(r)

∫

∫

∫

+−

+−

+−

−=

+++−≅

+++

++

−=

Wff

kt

Wff

kt

Wff

kt

dfEfKrS

dfDCffBArS

dfDCff

fBf

fArSrP

min

min

min

min

min

min

)exp(...

))4100

(exp(..

))41001

(exp(..)(

2

22

2

2

2

2

where ))(exp( DAK +−= and CBE +=4100

.

Using error function erf (x)

∫ −= x dxerf 02)exp(2)( λλ

π

we have

[ ])()(2

...

)exp(...)(

minmin

2min

min

ferfWferfE

KrS

dfEfKrSrP

kt

Wf

fk

t

−+=

−=

−

+− ∫

π

Setting ( ))()(6.17 minmin ferfWferf −+=γ and after some mathematical computations, the received power would be of the form:

)(026.0 21

...)( +−−= krt reSrP γ

Figure 2. The accuracy of the approximations

Fig. 2, represents two curves. One of them is the received

power when we do not use the approximations, and the second one is the received power when we use these approximations. The parameters that are used are as follows: r=1km k=1.5 St=1watt/kHz W=30 kHz

It seams that these two curves are matched and the difference is negligible.

Now, based on the previous calculations, we can obtain signal to interference ratio as follows

∑

∑ ∫

∫∑

+−−

+−−

+ −

+ −

=

==

i

ki

X

kr

i

Wff it

Wff t

ii

Xe

re

dffXAS

dffrASIrPSIR

i )(026.0

)(026.0

1

1

21

21

min

min

min

min

||.

.

),(

),()(

As a conclusion, although received signal power is not independent of operating frequency, the ratio of signal to interference is independent of frequency.

0 5 10 15 20 25 30 35 400

2

4

6

8

10

12

14

fmin (kHz)

Rec

eive

d P

ower

(W

att)

Real Values•Approximation

19 19.5 20 20.5 21

1.6

1.7

1.8

1.9

2

2.1

2.2

2.3

2.4

fmin (kHz)

Re

ceiv

ed

Po

wer

(W

att

)

Real Values•Approximation

711704