uwa usys10 main
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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
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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:
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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.
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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:
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)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
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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.
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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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