Acoustic Communication 15.4 Propagation Channel 371
Part B | 15.4
where
p D a 0 1 C k=l p ;
(15.17)
and h p is given by the expression (15.7). Note that one
can also explicitly express the salient parameters h p ; ; p
as functions of time, i. e., h p .t/; ; p .t/, if one is interested
in their variation during a long-term deployment in the
same area (deep/shallow water, etc.).
Small-Scale Effects
The above model accounts only for the variations induced by the path length displacements, i. e., it does
not include the effects of scattering. Scattering, which
causes micro-path dispersion along each propagation
path, can be modeled through an additional factor that
accompanies the path gain h p .t/. An overall channel
model is now obtained in the form
H.f ; t/ D H 0 .f /
X
p
p .f ; t/h p .t/e
i2f p.t/
:
(15.18)
In this model, the factors h p .t/ account for the slowerchanging process caused by deviations of the system
geometry from the nominal, while the factors p .f ; t/
represent the faster-changing scattering process. In
other words, during a typical communication transaction of several packets, the path gains can be considered
fixed, h p .t/ D h p , but the additional factors p .f ; t/ cannot. Assuming a relatively large number of micro-paths,
these factors can be modeled as complex-valued Gaussian processes, with some mean and variance. In general, scattering in an acoustic channel depends on the
signal frequency, making each macro-path experience
a different, possibly independent filtering effect. Note,
however, that the transfer function H.f ; t/ will in general exhibit correlation in both frequency and time (as
well as across the elements of a transmit/receive array).
Statistical Characterization
Because of the different dynamics of the two types
of processes, statistical channel analyses are typically
conducted in two forms: one that targets the locallyaveraged received signal power, and another that targets
the instantaneous channel response. Specifically, largescale statistical modeling focuses on the channel gain
defined as
G.t/ D
1
B
fcCB=2 Z
fcB=2
E
˚ˇ
ˇ H
2
.f ; t/
ˇ
ˇ
«
df ;
(15.19)
where the expectation is taken over small-scale fading.
In contrast, small-scale modeling focuses on the processes p .f ; t/, as well as the effects of random Doppler
shifting.
A complete statistical model for either large or
small-scale phenomena must specify the probability
density function (PDF) and the power spectral density (PSD), or the correlation properties of the random
process of interest. These functions can be assessed
using an analytical approach that relies on a mathematical model such as the one outlined above, a repeated
application of a propagation model such as the Bellhop ray tracer for a large number of slightly disturbed
channel conditions, or an experimental data analysis.
All approaches have been considered in the literature,
which abounds in statistical models that were found
to adequately match the channels observed in different locations, at different time scales, and in different
frequency bands. As of this time, however, the jury
is still out on establishing standard models that would
concisely describe the statistics of typical underwater
acoustic channels. Crucial to that effort is identification
of the scales of stationarity for the different processes
at hand.
As far as the large-scale modeling goes, recent experimental results offer some evidence in support of
a log-normal model for the channel gain [15.45–47].
Figure 15.12 illustrates such a model. Several of these
studies also find that the path gains, and consequently
the overall large-scale channel gain, can be well modeled as auto-regressive processes, at least on the time
scales of several seconds, which are meaningful for implementation of packet-level feedback schemes.
Experimental studies on small-scale fading have
offered evidence of Ricean [15.46, 48, 49] as well as
Rayleigh phenomena [15.50]. Figure 15.13 illustrates
an experimental data set that exhibits traits of Ricean
fading. Combining the effects of large and small-scale
fading leads to a mixture of log-normal and Ricean distributions, which can be approximated in closed form
by the compound K-distribution. Experimental evidence to this effect is provided in [15.51]. Less is known
about modeling the time-correlation functions of the
small-scale processes, but it is generally understood that
coherence times on the order of 100 ms can be assumed
for a general-purpose design. Frequency-correlation,
as well as space-correlation, as they pertain to communications systems, also remain to be addressed in
a systematic manner.
The importance of statistical channel modeling is
twofold. On the one hand, it enables the design of practical adaptive modulation methods which rely on identifying those channel parameters that can be predicted via
delayed receiver-transmitter feedback [15.52]. On the
other hand, it will enable the development standardized
simulation models, thus allowing performance analy-
Part B | 15.4
where
p D a 0 1 C k=l p ;
(15.17)
and h p is given by the expression (15.7). Note that one
can also explicitly express the salient parameters h p ; ; p
as functions of time, i. e., h p .t/; ; p .t/, if one is interested
in their variation during a long-term deployment in the
same area (deep/shallow water, etc.).
Small-Scale Effects
The above model accounts only for the variations induced by the path length displacements, i. e., it does
not include the effects of scattering. Scattering, which
causes micro-path dispersion along each propagation
path, can be modeled through an additional factor that
accompanies the path gain h p .t/. An overall channel
model is now obtained in the form
H.f ; t/ D H 0 .f /
X
p
p .f ; t/h p .t/e
i2f p.t/
:
(15.18)
In this model, the factors h p .t/ account for the slowerchanging process caused by deviations of the system
geometry from the nominal, while the factors p .f ; t/
represent the faster-changing scattering process. In
other words, during a typical communication transaction of several packets, the path gains can be considered
fixed, h p .t/ D h p , but the additional factors p .f ; t/ cannot. Assuming a relatively large number of micro-paths,
these factors can be modeled as complex-valued Gaussian processes, with some mean and variance. In general, scattering in an acoustic channel depends on the
signal frequency, making each macro-path experience
a different, possibly independent filtering effect. Note,
however, that the transfer function H.f ; t/ will in general exhibit correlation in both frequency and time (as
well as across the elements of a transmit/receive array).
Statistical Characterization
Because of the different dynamics of the two types
of processes, statistical channel analyses are typically
conducted in two forms: one that targets the locallyaveraged received signal power, and another that targets
the instantaneous channel response. Specifically, largescale statistical modeling focuses on the channel gain
defined as
G.t/ D
1
B
fcCB=2 Z
fcB=2
E
˚ˇ
ˇ H
2
.f ; t/
ˇ
ˇ
«
df ;
(15.19)
where the expectation is taken over small-scale fading.
In contrast, small-scale modeling focuses on the processes p .f ; t/, as well as the effects of random Doppler
shifting.
A complete statistical model for either large or
small-scale phenomena must specify the probability
density function (PDF) and the power spectral density (PSD), or the correlation properties of the random
process of interest. These functions can be assessed
using an analytical approach that relies on a mathematical model such as the one outlined above, a repeated
application of a propagation model such as the Bellhop ray tracer for a large number of slightly disturbed
channel conditions, or an experimental data analysis.
All approaches have been considered in the literature,
which abounds in statistical models that were found
to adequately match the channels observed in different locations, at different time scales, and in different
frequency bands. As of this time, however, the jury
is still out on establishing standard models that would
concisely describe the statistics of typical underwater
acoustic channels. Crucial to that effort is identification
of the scales of stationarity for the different processes
at hand.
As far as the large-scale modeling goes, recent experimental results offer some evidence in support of
a log-normal model for the channel gain [15.45–47].
Figure 15.12 illustrates such a model. Several of these
studies also find that the path gains, and consequently
the overall large-scale channel gain, can be well modeled as auto-regressive processes, at least on the time
scales of several seconds, which are meaningful for implementation of packet-level feedback schemes.
Experimental studies on small-scale fading have
offered evidence of Ricean [15.46, 48, 49] as well as
Rayleigh phenomena [15.50]. Figure 15.13 illustrates
an experimental data set that exhibits traits of Ricean
fading. Combining the effects of large and small-scale
fading leads to a mixture of log-normal and Ricean distributions, which can be approximated in closed form
by the compound K-distribution. Experimental evidence to this effect is provided in [15.51]. Less is known
about modeling the time-correlation functions of the
small-scale processes, but it is generally understood that
coherence times on the order of 100 ms can be assumed
for a general-purpose design. Frequency-correlation,
as well as space-correlation, as they pertain to communications systems, also remain to be addressed in
a systematic manner.
The importance of statistical channel modeling is
twofold. On the one hand, it enables the design of practical adaptive modulation methods which rely on identifying those channel parameters that can be predicted via
delayed receiver-transmitter feedback [15.52]. On the
other hand, it will enable the development standardized
simulation models, thus allowing performance analy-