1

Outage Performance of a Multiuser Distributed

Antenna System in Underwater Acoustic Channels

Zhaohui Wang†, Shengli Zhou†, Zhengdao Wang‡, Josko Catipovic§ and Peter Willett†

† Dept. of Electrical and Computer Engineering, University of Connecticut, Storrs, CT 06269, USA‡ Dept. of Electrical and Computer Engineering, Iowa State University, Ames, IA 50011, USA

§ Naval Undersea Warfare Center, Newport, RI 02841, USA

Abstract—The distributed antenna system (DAS) has beenknown for providing a large coverage and throughput in terres-trial radio environments. In this work, we investigate the benefitsof DAS relative to the centralized antenna system (CAS) in un-derwater acoustic (UWA) environments. In particular, we analyzethe outage performance in the wideband uplink transmissions.Based on theoretical derivation and numerical results, we showthat at a medium transmission SNR regime, DAS outperformsCAS in both single-user and multiuser systems. To address the co-channel interference in the multiuser DAS, we suggest a multicellstructure, which is shown having a lower outage error floorrelative to the structure which takes the operation area as asingle cell.

Index Terms—Distributed antenna system, underwater acous-tic channels, outage performance

I. INTRODUCTION

The last decade has witnessed significant progress in under-

water acoustic (UWA) communications, with typical advance-

ments for both single-carrier and multicarrier techniques in

e.g. [1]–[5]. As the point-to-point communication gradually

becomes mature, interests start growing on the underwater

acoustic networking.

Out of a myriad of network architectures, the distributed

antenna system (DAS), which was originally proposed for the

indoor wireless communications in 1980s [6], draws consider-

able attention in the wireless research community. Relative to

the centralized antenna system (CAS), DAS which is formed

by multiple distributed antenna element (DAE), provides a

larger coverage and a higher throughput for the network with

nomadic users [6]–[9].

In terms of the sytem outage performance, primary works

on DAS in terrestrial radio environments can be found in

[10]–[12]. Assuming a Rayleigh-lognormal flat fading channel,

the outage performance of a single-cell single-use DAS is

numerically studied in [10], which shows that the optimal

performance can be achieved with fully distributed antennas.

Using random matrix theory, a closed-form expression of the

uplink outage probability in the distributed multi-input multi-

output (MIMO) maximum-ratio-combining (MRC) system is

derived in [11], with an assumption that the number of transmit

antennas is no less than that of receive antennas. In [12],

the downlink outage probability is derived for DAS in both

This work by Z.-H. Wang and S. Zhou was supported by the ONR grantN00014-09-1-0704 (PECASE) and the NSF grant ECCS-1128581.

A network anchored at sea bottom A data collection network

surfaceAUV

bottom

gateway

sensors

Fig. 1. Two example underwater networks. The nodes anchored at sea bottomin the first network are connected to a control center via cables. The gatewaysin the second network can communicate with each other using radios.

single-cell and multicell environments with randomly located

antennas and mobile users. It shows that in the multicell

DAS, selection transmission is preferable to maximum ratio

transmission, while the opposite is true for the single-cell DAS.

Relative to the DAS in terrestrial environments, studies

on the underwater DAS are rare, despite the fact that the

DAS structure has been used by engineers in underwater

applications; two typical applications of the DAS are shown in

Fig. 1, where one is formed by the distributed nodes which are

anchored at sea bottom and connected via cables, and the other

is formed by the gateways which are geographically distributed

at sea surface and can communicate via radio waves. The early

testbed on the UWA networking can be found in [13] in the

Atlantic underwater test and evaluation center (AUTEC), and

an ocean technology testbed [14]. In this work, we consider the

DAS in underwater environments. In particular, we investigate

the outage performance of DAS during uplink transmissions

in the frequency-selective UWA channel, and DASs with both

single and multiple users are considered.

The UWA channel has been known different from the

terrestrial radio channel in several aspects. Particularly, the

path spreading-loss exponent in the terrestrial radio channel

varies from 3 ∼ 6, while in the UWA channel, the path

spreading-loss exponent is practically taken as 1.5, which is

much less than its terrestrial counterpart. Meanwhile, different

from the terrestrial radio channel, the UWA channel also incurs

signal absorption loss, which is highly frequency-dependent

and proportional to the transmission distance.

Taking into account the propagation characteristics, the

following observations for underwater DAS are obtained.

• In the single-user system, the DAS enjoys lower outage

2

probabilities than CAS in the moderate transmission SNR

region, and the superiority of DAS over CAS increases

as the signal frequency increases;

• In the multiuser DAS, a multicell based uplink transmis-

sion yields better outage performance than the single-cell

based uplink transmission.

The rest of the paper is organized as follows. A system

model for the multiuser underwater DAS is developed in

Section II. The outage probabilities of the single-user DAS and

multiuser DAS are derived in Sections III and IV, respectively.

Conclusions are drawn in Section V.

II. SYSTEM MODEL FOR MULTIUSER UNDERWATER DAS

We consider an underwater distributed antenna system with

Nu users, and Nr DAEs which are geographically distributed

within an operation area, as shown in Fig. 2. To facilitate

our analysis, we consider a circular operation area of radius

Rop, and assume that users are uniformly distributed within

the disk. Let (ρµ, θµ) denote the polar coordinate of the µth

user, which are independently distributed with the probability

density function (PDF){

p(ρµ) =2ρµ

R2d

, ρµ ∈ [dmin, Rop],

p(θµ) =12π , θµ ∈ [0, 2π]

(1)

for µ = 1, · · · , Nu, where 0 < dmin ≪ Rop is the minimum

distance of each user to the DAE. Define (Dν , θν) as the

Cartesian coordinate of the νth DAE. The distance between

the µth user and the νth DAE is thus

dν,µ =√

ρ2µ +D2ν − 2ρµDν cos(θµ − θν). (2)

We consider an underwater DAS system with a center

frequency fc and bandwidth B. Let sµ(f) denote the signal

transmitted by the µth user in the frequency domain, and

define Hν,µ(f) as the channel fading coefficient between the

µth user and the νth DAE. To make the problem tractable,

we assume an identical number of independent subchannels

between each user and DAE pair, denoted by Npa. Although

in reality the numbers of paths in the channel of different

user-base-station pairs could be very different, results obtained

with this assumption shed insights into the outage performance

of the underwater DAS. We denote {f1, · · · , fNpa} as the

center frequencies of the Npa consecutive subbands within

[fc −B/2, fc +B/2], and the bandwidth of each subband is

∆f = B/Npa.

The received signal of one particular frequency fp at the

νth DAE is formulated as

yν(fp) =

Nu∑

µ=1

Hν,µ(fp)sµ(fp)e−j2πfpτν,µ + wν(fp), (3)

for ν = 1, · · · , Nr, and p = 1, · · · , Npa, where Hν,µ(fp)follows a complex Gaussian distribution CN (0, σ2(dν,µ, fp))in which σ2(dν,µ, f) depicts the signal attenuation after trans-

mitting a distance of dν,µ, and wν(fp) is the ambient noise

following a complex Gaussian distribution CN (0, N0). In

UWA channels, the signal attenuates according to

σ2(dν,µ, f) ∝ e−α(f)dν,µd−βν,µ (4)

Rd

!

"!

AUVs

Base Station

with Multiple

Antennas

(0,0)

(a) CAS

Rop

!

"!(d1,0)

(0,d2)

(0,d4)

(d3,0)

AUVs

DAEs

(0,0)

(b) DAS

Fig. 2. Illustration of a centralized antenna system and a distributed antennasystem, with users uniformly distributed within the disk of radius Rop

where α(f) and β are the frequency-dependent absorption

coefficient and the path spreading-loss exponent, respectively.

III. OUTAGE PERFORMANCE OF THE SINGLE-USER DAS

A. Outage Probability in Wideband Systems

In this section, we consider the outage performance of

single-user DAS in frequency-selective channels. The system

input-output relationship is expressed in (3) with Nu = 1. For

notational convenience, we abbreviate Hν,1(fp) as Hν,p.

Define γ̄ as the transmission signal-to-noise ratio (SNR)

γ̄ :=P

BN0(5)

where P is the transmitted signal power. For the single-user

DAS, based on the input-output relationship in (3), the outage

probability is formulated as

pout(γ̄, R|(ρ, θ)) = Pr

Npa∑

p=1

log2

(

1 + γ̄

Nr∑

ν=1

|Hν,p|2

)

< R

,

(6)

where R denotes the date rate with a unit of bits/sec/Hz.

Define

Ip = log2

(

1 + γ̄

Nr∑

ν=1

|Hν,p|2

)

. (7)

For CAS, with the mathematical result in (20), the PDF of Ipis formulated as

fp(z | (ρ, θ)) =

2z ln 2

γ̄σ2CAS,p

exp

(

−2z − 1

γ̄σ2CAS,p

)[

2z − 1

γ̄σ2CAS,p

]Nr−11

(Nr − 1)!.

(8)

For DAS, with the mathematical result in (21), Ip has a PDF

formulated as

fp(z | (ρ, θ)) =Nr∑

ν=1

2z ln 2 exp(

− 2z−1γ̄σ2

ν,p

)

γ̄σ2ν,p

∏Nr

k=1,k 6=ν(1− σ2k,p/σ

2ν,p)

. (9)

The PDF of∑Npa

p=1 Ip given (ρ, θ) therefore can be obtained

as a convolution of Npa functions,

f(z | (ρ, θ)) = f1(z | (ρ, θ)) ⋆ · · · ⋆ fNpa(z | (ρ, θ)). (10)

3

−4 −2 0 2 4−4

−3

−2

−1

0

1

2

3

4

Rop

[km]

Rop

[km

]

DAE 1DAE 2DAE 3DAE 4

(a) Nr = 4

−4 −2 0 2 4−4

−3

−2

−1

0

1

2

3

4

Rop

[km]

Rop

[km

]

(b) Nr = 7

Fig. 3. Deployment of DAEs to minimize the average squared Euclideandistance from each point in a circular operation area to its nearest DAE.

The sum rate∑Npa

p=1 Ip thus follows a compound distribution

f(z) =

∫

ρ

∫

θ

f(z | (ρ, θ))p(ρ)p(θ)dρdθ. (11)

The overall outage probability can be obtained as

pout(γ̄, R) =

∫ R

0

f(z)dz. (12)

B. Numerical Results

In the single-cell single-user DAS, we employ the Lloyd’s

algorithm for DAE deployment, in which the optimal locations

of DAEs are determined by minimizing the average squared

Euclidean distance of each point in the operation area to

its nearest DAE. As an example, Fig. 3 shows the DAE

deployment strategies obtained from the Lloyd’s algorithm for

a circular operation area with different number of DAEs.

Based on (11) and (12), we next perform a semi-theoretical

calculation of the single-user outage probability in both nar-

rowband and wideband systems. We take the spreading loss

exponent as 1.5 and set dmin = 100 meters throughout the

paper.

1) Narrowband systems: We first consider a narrowband

system with Npa = 1. For Rop = 4 km and R = 1 bit/sec,

the outage probabilities of DAS and CAS at different operation

frequencies, are shown in Fig. 4. Two observations are in order.

• The performance gap between DAS and CAS in the mod-

erate transmission SNR region increases as the system

frequency increases;

• In the high transmission SNR regime, there exists a

crossing point of the outage probabilities of CAS and

DAS, meaning that CAS can catch up and supass DAS

when the transmission power is sufficiently large. The

outage probability at the crossing point decreases as the

system frequency increases.

2) Wideband systems: Fig. 5 demonstrates the outage prob-

abilities of the wideband systems with different operation

area sizes and different number of DAEs, respectively. The

advantage of DAS over CAS is observed to be pronounced

as the size of the operation area and the number of DAE

increase. Meanwhile, relative to the narrowband system, the

crossing of the outage probabilities of the wideband DAS and

CAS happens at a lower outage probability, which makes DAS

20 30 40 50 60 70 80 90 10010

−8

10−6

10−4

10−2

100

P/N0 [dB]

outa

ge p

roba

bilit

y

f = 100 Hzf = 1 kHzf = 5 kHzf = 10 kHzf = 15 kHz

Fig. 4. Outage probabilities of CAS and DAS versus transmission SNR inRayleigh fading channels, Nr = 7, β = 1.5, Rop = 4 km and R = 1 bit/sec,dashed lines: CAS; solid lines: DAS.

a more appealing choice than CAS in the practical outage

probability range.

IV. OUTAGE PERFORMANCE OF THE MULTIUSER DAS

A. Outage Probability in Wideband Systems

In a multiuser DAS, due to the low propagation speed of

acoustic waves in water, signals from different users usually

have different time-of-arrivals; the signals aligned at one DAE

could have a very large time-difference-of-arrivals at other

DAEs. Given the asynchronism of signals from multiuser at

DAEs, we consider the outage probability of a low-complexity

single-user decoding method.

Without loss of generality, we focus on the outage perfor-

mance analysis of the first user. For notional convenience, we

define ρ := [ρ1, · · · , ρNu], and θ := [θ1, · · · , θNu

]. Based

on (3), after the MRC of signals at all DAEs, the conditional

outage probability of the first user is

pout(γ̄, R|(ρ,θ)) =

Pr

Npa∑

p=1

log2

(

1 +

Nr∑

ν=1

|Hν,1,p|2

∑Nu

µ=2 |Hν,µ,p|2 + 1/γ̄

)

< R

,

(13)

where Hν,µ(fp) is abbreviated as Hν,µ,p.

Define the signal-to-interference-plus-noise ratio (SINR) of

signals from the first user at the νth DAE as

rν,p :=|Hν,1,p|

2

∑Nu

µ=2 |Hν,µ,p|2 + 1/γ̄. (14)

Based on the mathematical result in (23), the cumulative

distribution function (CDF) of rν,p is

F (rν,p | (ρ,θ)) =

1− exp

(

−rν,p

γ̄σ2ν,1,p

)

Nu∏

µ=2

1

rν,p/σ2ν,1,p + 1/σ2

ν,µ,p

(15)

4

40 50 60 70 80 90 100 110 120 13010

−6

10−5

10−4

10−3

10−2

10−1

100

γ [dB]

outa

ge p

roba

bilit

y

R

op = 4 km

Rop

= 8 km

Rop

= 12 km

(a) Nr = 7

50 60 70 80 90 100 110 120 13010

−6

10−5

10−4

10−3

10−2

10−1

100

γ [dB]

outa

ge p

roba

bilit

y

N

r = 1

Nr = 2

Nr = 4

Nr = 6

Nr = 12

(b) Rop = 8 km

Fig. 5. Outage probabilities of CAS and DAS in Rayleigh fading channelswith different operation area sizes, B/fc = 1/2, fc = 10 kHz, β = 1.5,Npa = 3, and R = 3 bits/sec/Hz. Dashed lines: CAS; Solid lines: DAS.

with the PDF formulated as

f(rν,p | (ρ,θ)) =1

σ2ν,1,p

[

1

γ̄+

Nu∑

µ=2

1

rν,p/σ2ν,1,p + 1/σ2

ν,µ,p

]

× exp

(

−rν,p

γ̄σ2ν,1,p

)

Nu∏

µ=2

1

rν,p/σ2ν,1,p + 1/σ2

ν,µ,p

. (16)

After MRC, the PDF of rp :=∑Nr

ν=1 r̄ν,p is

frp(rp | (ρ,θ)) = f(r1,p | (ρ,θ)) ⋆ · · · ⋆ f(rNr,p | (ρ,θ)).(17)

The PDF of the instantaneous mutual information zp :=log2(1 + γ̄p) can be obtained as

fzp(zp | (ρ,θ)) = 2z ln 2frp(

(2zp − 1) | (ρ,θ))

. (18)

Similar to (10), the PDF of the sum-rate z :=∑Npa

p=1 zp is

obtained as the convolution of all the PDFs of fzp(zp | (ρ,θ)),

x [meter]

y [m

eter

]

Fig. 6. An example of the large size DAS, ∗: location of DAEs; the outerDAEs within each circle are at 2/3 of the circle radius from the circle center.

2 4 6 8 10 12 14 16 1810

−8

10−6

10−4

10−2

100

number of users

outa

ge p

roba

bilit

y

f = 100 Hzf = 1 kHzf = 5 kHzf = 10 kHzf = 15 kHz

Fig. 7. Outage probabilities of CAS and DAS with different number ofusers, β = 1.5, γ̄ = 70 dB, B/fc = 1/2, Npa = 3, and R = 3 bits/sec/Hz,dashed lines: CAS; solid lines: DAS.

for p = 1, · · · , Npa. The outage probability is obtained as

pout(γ̄, R) =

∫ R

0

∫

ρ

∫

θf(z | (ρ,θ))p(ρ)p(θ)dzdρdθ.

(19)

B. Numerical Results

In a large network, we consider two multiuser uplink

transmission system structures. In the first structure, all the

users are uniformly distributed within the the overall operation

area. In the second structure, we split the network into multiple

cells, and each cell has one user which is uniformly distributed

within the cell, as shown in Fig. 6. In numerical results, the

radius of each circle is set as 4 km.

1) Outage probability in the single-cell structure: Taking

the DAS in Fig. 6 as a single-cell, Fig. 7 demonstrates the

outage probabilities of DAS and CAS with different number of

users at several system center frequencies. One can see that the

outage probability increases as the number of users increases.

At high frequencies (fc ≥ 10 kHz), DAS outperforms CAS

uniformly. Meanwhile, due to multiuser interference, systems

5

40 50 60 70 80 90 100

10−8

10−6

10−4

10−2

100

γ [dB]

outa

ge p

roba

bilit

y

f = 5 kHzf = 10 kHzf = 15 kHz

Fig. 8. Outage probabilities of CAS and DAS, β = 1.5, B/fc = 1/2,Npa = 3, and R = 3 bits/sec/Hz, dashed lines: CAS; solid lines: DAS withmulticell structure; dot-dashed line: DAS with single-cell structure.

at relatively low frequencies (fc ≤ 1 kHz) are observed

having higher outage probabilities that the systems at medium

frequencies (e.g., fc = 5 kHz).

2) Single-cell versus multicell structure: Fig. 8 shows the

outage probability of both the single-cell structure and the mul-

ticell structure. Due to the multiuser interference, the outage

probability exhibits an error floor as the transmission power in-

creases. Meanwhile, comparing the outage performance of the

two DAS cellular structures with a large transmission power,

a better outage performance can be achieved in multi-cell

structure through reducing the level of multiuser interference,

especially for systems at high frequencies.

V. CONCLUSIONS

In this work, we derived and compared the outage probabil-

ities of DAS and CAS in the frequency-selective underwater

acoustic channels. With a moderate transmission SNR, DAS

was shown outperforming CAS in both single-user and mul-

tiuser uplink transmissions, and the performance gap increases

as any of the system center frequency, the number of DAEs, or

the operation area size increases. For multiuser transmissions,

an outage error floor was observed due to the multiuser

interference. A multicell network structure can reduce the

multiuser interference relative to the structure that takes the

operation area as a single cell.

APPENDIX: MATHEMATICAL RESULTS

Result 1: For n i.i.d exponential random variables Xi with pa-

rameter χ, define Y =∑n

i=1 Xi. Using the Laplace transform,

it can be shown that Y follows a gamma distribution (1, χ),with the cumulative distribution function (cdf) formulated as

FY (y) = 1− e−χy

n−1∑

i=0

(χy)i

i!. (20)

Result 2: For n independent exponentially distributed random

variables {Xi, i = 1, · · · , n} with piecewise distinct parame-

ters denoted by χ1 < χ2 < · · · < χn, respectively, the cdf of

Y =∑n

i=1 Xi is

FY (y) =n∑

i=1

1− e−χiy

∏nj=1,j 6=i(1− χi/χj)

. (21)

Result 3: Consider n independent exponentially distributed

random variables with parameter of the ith variable Xi de-

noted by χi, and define Y =∑n

i=1 Xi. Denote X0 as an

exponentially distributed random variable with parameter χ0.

Define Z := X0

Y+a, where a as a positive constant. We have

Pr(Z ≤ z) = Pr (X0 ≤ z(Y + a))

=

∫

Pr(X0 ≤ z(y + a))fY (y)dy. (22)

The PDF fY (y) can be computed with the result in (20) or

(21). Given the exponential distribution of X0, the CDF of Zcan be obtained as

FZ(z) = 1− e−aχ0z

n∏

i=1

1

χ0z + χi

. (23)

For χi = χ, ∀i = 1, · · · , n, we have

FZ(z) = 1−e−aχ0z

(χ0z + χ)n. (24)

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