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Stochastic Time-Scale Characterization of Nonstationary Underwater

Communication Channel

Uche A.K. Okonkwo1, Razali Ngah2, Zabih Ghassemlooy3 , Tharek Abd. Rahman4

1,2,4

Wireless Communication Center Faculty of Electrical and Electronic Engineering Universiti Teknologi Malaysia

83100 Skudai, Johor, Malaysia E-mail: [email protected], [email protected], [email protected]

3 Optical Communication Research Group,

School of Computing, Engineering & Information Sciences,University of Northumbria, Newcastle, UK.

E-mail: [email protected]

Abstract

The underwater acoustic communication channel is one of

the complex and challenging channels for communication. In most cases there is the need to provide communications

between mobile and stationary terminals. And because of

the spherical degree of freedom for the mobile terminal, such channel is characterized as highly nonstationary. In

order to account for nonstationarity, channel characterization that employs the non wide-sense

stationary uncorrelated scattering (non-WSSUS) approachis necessary. More also the inadequacy of Doppler shift in

accounting for frequency shift of the channel operator implies that the time-frequency characterization approach

is not appropriate. In this work we present the geometrical-based stochastic time-scale characterization of

the underwater channel which emphasizes on thenonstationary property of the channel. The effects of the

channel nonstationarity on the channel capacity and diversity gain are also addressed. From the simulated

example, it is inferred that channel diversity and theassumption of ergodic capacity depends on the number of

independent fades which invariably depends on theintervals of stationarity.

Keywords:

Underwater channel, nonstationarity, geometrical model,

ergodic capacity, diversity, scattering function.

1. Introduction

Over the years there is the growing need for deep sea

communication among the submerged vessels and with the

surface or on-shore transceiver stations. More also the surge

of ocean exploration activities has been steadily increasing.

The need for underwater wireless communication exists in

applications such as remote control in off-shore oil industry,

pollution monitoring in environmental systems, collection

of scientific data recorded at ocean-bottom stations, speech

transmission between divers, and mapping of the ocean

floor for the detection of objects, as well as the discovery of

new resources [1]. Coupled with this increase in ocean

exploration is the need to transmit data, collected by

sensors placed underwater, to the surface of the ocean.

From there it is possible to relay the data via a satellite to a

data collection centre.

Due to the poor propagation capability of the

electromagnetic waves in sea water which is attributed

mainly to the skin effect , acoustic signals provide the most

obvious medium to enable underwater communications.

This limits the available bandwidth for the underwater

acoustic (UWA) communication to the kilo Hertz range [2].

More also challenging issues like the refractive properties

of the UWA channel, severe fading, multipath, rapid time-

variation and large Doppler spread, impedes on the

performance of the system [1], [2], [3]. Thus a good

understanding of the UWA channel is important in the

design and simulation of the deployable component

systems.The complex UWA environment remains one of the most

challenging types of channels for information transmission.

Brady and Preisig [4] described the UWA as “quite possibly

nature’s more unforgiving wireless medium”. In general,

the physical characteristics of the UWA channel are highly

dependent on the relative distance and motion of the

terminals and the channel; the proximity and roughness of

the scattering surfaces; and the presence of ambient

interference [5]. However, the basic channel

characterization challenges can be factored into large values

of delay and Doppler spread. The discrepancy in terms of

the delay and Doppler spread (both are inversely

proportional to the velocity of propagation) between the

mobile UWA channel and the propagation in the mobile

radio channel, can be capture by 310/1, =υ τ

(underwater channel),8

10/1, =υ τ (mobile radio

channel).

More also the non-uniform Doppler shift across the

composite tones (in the case of wideband signals) makes

the evaluation of the frequency variation using Doppler

shifting as inappropriate as discussed in [6]. This issue is

even more pronounced for the underwater OFDM

communication [5], [7]. The time-scaling defined under the

time-scale domain representation is a more suitablemeasure of frequency variation in a wideband signal. For

the above reasons, the representation and characterization



of the UWA channel in the time-scale domain is more

appropriate than the Fourier domain counterpart.

While travelling through the underwater channel, the

transmitted signal experiences sever distortions induced by

multipath propagation. The distortions become more sever

when either or both the transmitter and the receiver are in

motion. The resultant time-varying multipath imposes sever

limitations on the system performance [1]. Since this

channel is nonstationary, the large time-variability cannot

be ignored, thus the wide-sense stationary (WSS)

assumption is violated. On the other hand, large scale

smoothness of the scattering surfaces (as well as variations

in the mean angular spread due to motion) contributes to

correlation among the multipath components from the same

surface. Hence uncorrelated scattering (US) assumption is

also violated. Therefore channels with such rapid time-

variability and the inter-path correlations mentioned above

cannot be modelled as WSSUS [8]. Instead a non-WSSUS

approach is employed in the UWA channel characterization.

The approach in this paper follows a purely intuitive path

that hinges on the appropriation of varied statistical

intervals, other than very complex mathematical

derivations. More also instead of using the acronym ‘non-

WSSUS’ which ordinarily embodies all processes that

cannot be defined within the WSSUS assumption (grossly

nonstationarity inclusive), a ‘local sense’ statistical basis is

defined. The result of this local sense statistics is the local

sense stationary and uncorrelated scattering (LSSUS)

assumption. The parameters derived from the LSSUS

statistics are shown to be more appropriate and informative

than the WSSUS in obtaining the nonstationary

information. They also capture long-term channel properties necessary for performance analysis.

This paper is organized as follows. Section 2 presents the

stochastic time-scale characterization and presents the

concept behind LSSUS assumption. In Section 3, the

geometrical-based scattering model that typifies the

propagation in the UWA channel and considers terminal

mobility is presented. Finally examples simulations and

results are presented and discussed in Section 4.

2. Stochastic Time-Scale Characterization

The time scale representation of the time-varying channel

can be given by [3]:

2)(),()(

s

dsd

s

t xt a st y

τ τ τ ∫ ∫

∞

∞−

∞

∞−

−= W

(1)

where )()}({ t t y y= is the channel realization for a

given input )(t x , and ),( sτ W is the delay-scale

(wideband) spreading function.

Let U be the universal set of all stochastic

processes/channels, there exist subsets of whose statistical properties varies with certain degrees in respect to the

variations in some interval (acquisition interval) ℜ∈ J .

If we define a partition of B as the countable collection

of subintervals Qq B P q ,..,2,1, =⊂ , then we can

state that:

i. q ji ji P P ji ∈≠∀≠∩ ,;};0{

ii. k ji ji J J ji ∈≠∀≠∩ ,;};0{

iii.

Hence we can define the interval K k J k v ,..,2,1, = for

which some statistical properties of the associated process

under observation are assumed to be stationary. An

important process often used to characterized and simplify

slowly varying wireless channel is the wide-sense

stationary process.

Definition 1: A process is called wide-sense stationary

(WSS) if it’s first two moments, the mean and

autocorrelation are independent of time t on . Such process

is defined on if there exist some partitions for which all

intervals k J provide time independence with respect to

the mean and ACF.

Definition 2: A process is called local-sense stationary

(LSS) if there exist some partition P for which at least

one interval say i J is considered stationary. Within some

valid locally stationary interval iv J the mean and the

autocorrelation (and the associated spectral property) are

approximately independent of time and frequency, and vary

slowly in time and frequency across all other intervalsik J ≠ . Thus the autocorrelation and spectral characteristic

are WSS ativ J but vary slowly across with respect to all

other intervals }{ik

J ≠ .

For all other processes with gross time varying statistical

properties over all intervals for which no v J can be

ascertained for practical purposes, nonstationarity is

defined. From the above discussion, we can see that a little

above the strict-sense stationarity, the wide-sense stationary

channel is defined, and a little below the nonstationarity, the

local-sense stationary channel is define.

Hence for stochastic representation of the time-varyingchannel we can then define three different time instants; t

, t ′ and t ′′ . Within the quasi-stationary (WSSUS) area

for two time instants t and t ′ , the channels statistics are

constant over t t t −′=∆ . However the statistics vary

across the quasi-nonstationary (non-WSSUS) area over

t t t ′′−′=′∆ . Thus it can easily be shown that we can

define the time, scale (or frequency), scale shift (Doppler)

and delay for the LSSUS channel as:



τ τ τ τ τ ∆+∆+∆=−′′∆+∆+∆=−′′

∆+∆+∆=−′′∆+∆+∆=−′′

)(

;~)~~(~~;)(

;)(

s s s s s

s s s s s

t t t t t

(2)

The correlation function is then given by

,(),([);,,,( * s sr s s R

LSSUS ′′′′=∆∆∆ τ τ τ τ W W W, E

. It can easily be shown that:

t

s

t t t

s s s s

LSSUS

X X R

r s s R

,

)(

,)(,)(

,,.

);,,,(

τ τ

τ τ τ δ

τ τ

∆+∆+∆∆+∆+∆∆+∆+∆

=∆∆∆

y

W,

(3)

where

))((,

)(,)(t t t t a X l

s s s s

+∆+∆+∆=∆+∆+∆∆+∆+∆

τ

τ τ τ

+∆+∆+∆

+∆+∆+∆−+∆+∆+∆ s s s s

t t t t x

)(

))(())((.

τ τ τ τ

and

−=

s

t xt a X l

s τ τ )(,

. The first inner product term

in (3) is called the local-sense Scattering function (LSF):

t t t

s s s sLSSUS X R s

+∆+∆∆+∆+∆∆+∆+∆

=)(

,)(,)(,),(

τ τ τ τ

τ yP

(4)

Implicitly:

0,,),(),( →∆∆∆= st LSSUS WSSUS s s τ τ τ PP

(5)

The channel is then defined by the coherence bandwidth

c B , coherence time cT , stationarity bandwidth s B and

stationarity time sT :

WSSUS rms

c B

,,5

1

τ P= ,

cWSSUS srms

c f

T

,,

4.0

P≈

WSSUS rms

s B

,,5

1

τ P

∆= ,

cWSSUS srms

s f

T ,,

4.0P

∆≈ .

where ,.,τ rmsP and ,., srmsP are the respective delay

and scale profiles,

)()( ,,,,,, WSSUS srmsLSSUS srmsWSSUS srms PPP −=∆

. The parameters s B and sT describe the extent of

channel variation and tends to infinity in the case of

WSSUS

Hence for the flat-fading slowly varying channel, the

number of i.i.d n is approximately given by the

stationarity and coherence regions:

N k cc

c N ck nn BT

T n Bnn .

.==

(6)

Using the expression in [9], the n -dependent ergodic

capacity can be given by:

( )( ))(1log 2

1

1

qq nn

N

n

erg p+=∑−

=

C (7)

where[ ]

0

2),(.

N

f t P navn

χ =q with probability

distribution )(qn p .

It is evident that diversity performance improves

monotonically with increasing number of i.i.d [10]. In fact

as the number of i.i.d approaches infinity, the performance

of coherent diversity reception converges to the performance over a non-fading AWGN channel [12], [13].

By decoupling the stationarity region onto the time and

frequency region, the number of the i.i.d n or diversity

order can be approximately given as:

→=c

sTD

T

T n Time diversity (8)

→=

c

sTD

B

Bn Frequency diversity (9)

→=cc

s sTD BT

BT n Time-Frequency d iversity (10)

The expressions (8)-(10) imply that as sT and s B are

reduced by virtue of decrease in the correlation among

channel realizations, the diversity order reduces. Hence the

stationarity intervals set threshold and point of reference for

employing different diversity schemes.



3. Geometrical-based UWA Channel Model

A geometrical-based single bounce scattering model is

provided below in order to model and simulate the UWA

channel. Unlike the elliptical and the circular models

described for the conventional terrestrial MRC in which

mobility is restricted to the azimuthal angles and thescattering region is defined over a circular or elliptical

volume, the approach in the case of UWA channel is

slightly different. Often scatterers in UWA channels are

located at the top and bottom of the water volume, thus the

water volume can be consider as being a large rectangular

volume with the scatterers distributed on the top and bottom

lids. And the mobility of the mobile unit (MU) involves

both the azimuthal and the polar angles of movement, thus

the spherical coordinates are more appropriate for its

position descriptions. In the ensuing discussion the

geometrical-based single bounce sphero-rectangular

scattering (GBSBSRS) model for the UWA channel is

introduced as shown in Figure 1.

Figure 1- Illustration of Geometrical-based single bounce sphero-rectangular scattering (GBSBSRS) model for the

underwater channel.

Each scatterer is defined as a vector n s in a hypothetical

spatial coordinate ),,( z y x . For simplicity let 0= z ,

hence the scatterers coordinates can be specified by

),( y x sn bounded by the depth of the water and some

horizontal length determined by physical constraint or

assumed channel length. For the model above, the

following assumptions are made:

1. The temperature of the water volume is constant

over the period of simulation.

2. The wind speed v is very small such that the

average height (meters) of the one-third highest

waves expressed by22

3/1 10566.0 v H −

×=

[11] is approximately zero.

3. The floor of the water volume is smooth, non-

absorptive and homogeneous.

4. The water volume is isotropic, i.e., there is no

absorption effect. Hence the sound intensity int I

falls off as the inverse of the range r , so that

within this context the transmission loss is [11]:

)(log102

10 r T loss =

(11)

5. The frequency dependent of the propagation paths

is not taken into account.

The geometric distribution y x f , of the N scatterers can

be defined using any of the appropriate known statistical

distribution functions where y x f , is independent of

frequency. To obtain the delays associated with all

multipath components (MPCs), the total path lengths have

to be obtained by considering Figure 1. Let the reference

point )0,0( be the receiver position )0,0(MU . The

path length R from )0,0(MU to BS through

),( nnn y x s is given by:

{ }nnn g f R += , N n ,..,2,1=

(12)

where:

( ) 21

22 )( nnn x X H f −+= (13)

( )22)( nn xae += (14)

and X is the distance between the MU and the BS

projected on the x-axis. The angle-of-arrival (AOA) θ is

given by:

( ) ( )( )22211 2co s nnnn g f D D f −+= −−θ

(15)

For the MU moving with a velocity v , its position at any

given time can be described as ),,( Φφ r MU . The

evolved path length R′ through the evolved scatterers’

position ),( nnn y x s ′′ with reference to its position at B is

given by:

{ }nnn g f R ′+′=′ , N n ,..,2,1=

(16)

where

2

1

2

2

2

1

22)cos())cos((

Φ+′+

−=′ r yr q f nn φ

(17)

( )22 )(nn

xae ′+′=′ (18)

and:



)cos(φ r aa −=′ , ( ) 21

222c o s2 Φ′−+′= r xr x p

nn

4. Numerical Results and Discussion

Consider an underwater communication between the

hydrophone which serves as the base station (BS) and the

mobile unit (MU) located 50 m apart on the average as

depicted in Figure 1. Assume that the operating bandwidth

is 10 KHz and the MU is moving from an initial position A

(spherical coordinate of A is )0,0,0(MU ) at a constant

velocity of 5 m/s to another position B ( ),,( Φφ r MU

). The following parameters are also defined for this

communication channel:

Water depth = 20 m; Vertical distance of hydrophone from

the surface = 10 m; Initial vertical distance of the MU from

the surface = 10 m, and the spatial extension r ,

∆+= t v

f

cr

c2

For { }0.3,0.1,8.0,5.0,2.0,1.0,08.0=∆t

sec, 0120=φ and 060=Φ , if we assume that the

speed of propagation of sound in water is 1500 m/s, the

resultant channel responses are shown in the Figure 2. The

stationarity time is obtained using vt s 2/λ =∆ , where v

is the speed of the MU.In simulating the above synthesized channel, the test signal

used is also the Mexican hat wavelet. The resultant delay-

scale scattering functions ),( sWSSUS τ P and

),( s LSSUS τ P are shown in Figure 2, and Figure 4,

respectively. From the scattering function, the power delay

profile (PDP) for the WSSUS case is derived and shown in

Figure 3. The PDP )(τ P is obtained by taking the

normalized power values at ),(),( min s s τ τ PP = over

the delay bins. To obtain the equivalent scale profile

)( sP the normalized power values is taken at

),(),( min s s τ τ PP = over the scale bins. The plots of

),( s LSSUS τ P for { }0.10,0.5,0.3,0.1=∆t are

shown in Figure 4.

Figure 2- WSSUS scattering function for the UWA channel

at st ∆ and 5 m/s.

0.033 0.034 0.035 0.036 0.037 0.038 0.039 0.040

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

N o r m a l i z e d P o w e r D

e l a y P r o f i l e

Delayτ ( sec)

Figure 3- Normalized power delay profile against the

delay for the UWA channel at 5 m/s and st ∆ .

(a) (b)

(c) (d)

Figure 4- LSSUS scattering functions at (a) 0.5 sec (b) 0.8

sec (c) 1.0 sec (d) 3.0 sec (at 5 m/s).

The values of the corresponding coherence and stationarity

parameters are shown in Table 1. And using (8)-(10) the

available iids for the channel at different time variations are

tabulated in Table 2.

Table 1- Channel condensed parameters for the UWA

∆t (sec) Bc(Hz) F c (ms) Bs (kHz) F s (s)

st ∆ 159 152.97 ∞ ∞

08.0 159 152.97 18.165 1.887



1.0 159 152.97 12.392 1.037

2.0 159 152.97 2.3418 0.145

5.0 159 152.97 0.2025 0.186

8.0 159 152.97 0.0556 0.049

0.1 159 152.97 0.0377 0.080

0.3 159 152.97 0.0267 0.283

Table 2: Number of identically independent fading channels

t ∆ (se

c)

nTFD nTD nFD

st ∆ ∞ ∞ ∞

08.0 1409 12 114

1.0 529 6 78

2.0 14 1 15

5.0 2 1 2

8.0 1 1 1

0.1 1 1 1

0.3 1 1 1

Discussion:

In this simulation, the coherent bandwidth of the channel

defined within the WSSUS range is approximately

Hz 160 . Thus, for the operating bandwidth of kHz 20

the system is highly frequency selective. For this value, the

channel has approximately equal gain and linear phase.

This value also limits the potential data rate of a system

deployed in this environment without coding and diversity

to about Hz 160 . The coherent time is ms97.152 ,

thus the channel response is essentially invariant over this

time. The frame length or slot time at 10 kHz is not

explicitly given. But by using the Nyquist’s theorem, it can

be inferred that for BPSK modulation, up to 5 kbit/s can beobtained. Thus to transmit a single bit takes 0.1 ms. Hence

for a frame with 100 symbols the channel is essentially

slow fading. For a frame of 1000 symbols, the channel is

between the boundary of slow and fast fading, but for

frames with over 1000 symbols, the channel is essentially

fast fading.

Table 2 also shows that the ergodic durations as well as the

number of iids decreases with increase in t ∆ . The

resultant ergodic capacity (7) for flat-fading (assuming

symbol duration c s BT /1= ) is shown in Figure 5.

1 0 11 12 1 3 14 15 1 6 1 70. 2

0. 4

0. 6

0. 8

1

1. 2

1. 4

1. 6

1. 8

A v e r a ge S N R ( d B )

C h a n n e l c a p a c i t y

( b i t / s e c / H z )

W S S U S

t = 0 .8 sec

t = 0 .1 sec

t = 0 .2 sec

t = 0 .5 sec

t = 1 .0 sec

Figure 5- Ergodic channel capacity versus signal-to-noiseratio (SNR) for the UWA channel at 5 m/s for different time

scales.

From Figure 5, it can be observed that the graphs are

slightly convergent (on the WSSUS graph) up to

1.0=∆t sec. Hence ergodic assumption can be applied

over the associated distances. However, the graphs of

1.0>∆t sec are not convergent, hence the assumption

of and the use of ergodic capacity is invalid over the

corresponding distances. This implies that even at close

time displacement, the channel stationarity intervals are

small due to the high delay and Doppler variations.

As for diversity gain associated with this particular channel,

Table 2 indicates that enough diversity gain especially inwith the time-frequency diversity scheme can only be

achieved within the stationarity time over which WSSUS is

assumed.

5. Conclusion

In this work the nonstationary property of the underwater

acoustic communication channel was presented using time-

scale domain characterization. The nonstationarity is

defined using the concept of local-sense stationarity and

modeled using a geometrical-based model adequate for

UWA channel. The resultant simulation indicates that as thespatial displacement of the mobile unit increases, the

diversity gain decreases and the assumption of ergodic

capacity becomes invalid. Thus for diversity to be

achieved in most cases, either the coherent intervals are

reduced at the expense of bandwidth and channel capacity

or the mobile speed is reduced. In the latter, time-frequency

diversity will still the most viable option. In our future

work the exploitation of channel selectivity properties

instead of the coherent properties in providing capacity and

diversity estimates will be undertaken.



Acknowledgments

The authors thank the Ministry of Higher Education

(MOHE), Malaysia for providing financial support under

Grant (78368). The Grant is managed by Research

Management Center (RMC) Universiti Teknologi Malaysia

(UTM)

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