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938 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 61, NO. 3, MARCH 2012

Doubly Selective Underwater Acoustic Channel

Model for a Moving Transmitter/Receiver

Chunshan Liu, Yuriy V. Zakharov, Senior Member, IEEE , and Teyan Chen

Abstract—We propose a new method of modeling the signal

transmission in underwater acoustic communications when the

transmitter and receiver are moving. The motion-induced channel

time variations can be modeled by sampling the transmitter/

receiver trajectory at the signal sampling rate and calculating, for

each position, the channel impulse response from the acoustic-ﬁeld

computation. This approach, however, would result in high com-

plexity. To reduce the complexity, the channel impulse response

is calculated for fewer (waymark) positions and then interpolated

by local splines to recover it at the signal sampling rate. To allow

higher distances between waymarks and, thus, further reduction

in the complexity, the multipath delays are appropriately adjusted

before the interpolation. Because, for every time instant, this

method only requires local information from the trajectory, the

impulse response can recursively be computed, and therefore, the

signal transmission can be modeled for arbitrarily long trajec-

tories. An approach for setting the waymark sampling interval

is suggested and investigated. The proposed method is veriﬁed

by comparing the simulated data with data from real ocean ex-

periments. For a low-frequency shallow-water experiment with a

moving source that transmits a tone set, we show that the Doppler

spectrum of the received tones is similar in the simulation and

experiment. For a higher frequency deep-water experiment with

a fast-moving source that transmits orthogonal frequency-division

multiplexing (OFDM) communication signals, we investigate the

detection performance of a receiver and show that it is similar in

the simulation and experiment.

Index Terms—B-spline, channel modeling, channel simulator,

Doppler spread, local splines, orthogonal frequency-division mul-

tiplexing (OFDM), underwater acoustic communications.

NOMENCLATURE

(·)

Matrix transpose.

(m)

mth column of a matrix C.

m Index for waymarks.

n Index for signal samples.

i Index for taps of the channel impulse response.

k Index for frequencies.

{·} Real part.

Manuscript received May 4, 2011; revised September 15, 2011 and

December 14, 2011; accepted January 18, 2012. Date of publication February 7,

2012; date of current version March 21, 2012. This paper was presented in part

at the Fourth Underwater Acoustic Measurements: Technologies, and Results

Conference, Kos, Greece, June 20–24, 2011. The review of this paper was

coordinated by Prof. T. Kuerner.

The authors are with the Department of Electronics, University of York,

YO10 5DD York, U.K. (e-mail: cl563@ohm.york.ac.uk; yz1@ohm.york.ac.uk;

tc512@ohm.york.ac.uk).

Color versions of one or more of the ﬁgures in this paper are available online

at http://ieeexplore.ieee.org.

Digital Object Identiﬁer 10.1109/TVT.2012.2187226

I. INTRODUCTION

The performance of underwater acoustic communication

systems is heavily dependent on the propagation environ-

ment. The underwater channel is considered one of the most

difﬁcult channels for communications [1]. For assessing the

communications performance, sea experiments are required.

Although sea experiments are ultimate means of assessing

the performance, they are difﬁcult to conduct and are very

expensive. In some situations, instead, the simulation of the

propagation channel can be used. The simulation also has other

advantages compared to experiments. For example, due to the

highly dynamic time and space variability of the underwater

environment, it is difﬁcult to guarantee similar experimental

conditions when comparing different systems. It is also difﬁcult

to provide reliable monitoring of the environment and thus give

valuable interpretation of experimental results. Furthermore,

ocean experiments with multiple users (such as in underwater

communications networks) are even more complicated. On the

other hand, computer simulation can provide exactly the same

propagation conditions when investigating different systems,

precise monitoring of the environment, and modeling commu-

nication networks with multiple users. However, this case is

possible only if the simulator can provide results similar to the

results observed in sea experiments. Thus, an efﬁcient approach

for simulating underwater acoustic signal transmission is highly

desirable [2]–[4].

Two important phenomena that affect the performance of

underwater acoustic communications are multipath propagation

and the Doppler effect [2], [3], [5], [6]. Thus, the under-

water acoustic channel is doubly selective. Although, time-

varying multipath channel models are widely used to analyze

radio communication systems (see [7]–[9] and the references

therein), they are not directly applicable to underwater acoustic

communications [2], [10]. For radio communications, Jakes’

model combined with a set of standard power delay proﬁles

with ﬁxed multipath delays is often considered useful for study-

ing the system performance [11]. The important feature of the

underwater acoustic channel is the fast variation of multipath

delays due to a low speed of sound. As a result, the signal

distortion that is caused by the Doppler effect in the underwater

channel includes the time compression/dilation that is different

for different multipath components (see [2], [6], [12]–[15], and

the references therein). Another important consideration is that

a particular sea area can provide speciﬁc propagation conditions

that should be taken into account when studying underwater

acoustic systems. To account for the phenomena in the channel

LIU et al.: DOUBLY SELECTIVE UNDERWATER ACOUSTIC CHANNEL FOR MOVING TRANSMITTER/RECEIVER 939

model, the time-varying channel impulse response needs to be

computed for a speciﬁc sea area.

Time variations of the underwater acoustic channel are

caused by several factors such as the transmitter/receiver mo-

tion [2], internal waves [16], surface waves [17], [18], and

others. In this paper, we focus on time variations that are caused

by the transmitter/receiver motion. However, other phenomena

can also be incorporated into the proposed model.

Extensive effort has been devoted in the literature to mod-

eling underwater acoustic signal transmission. A channel sim-

ulator in [19] is based on a static channel impulse response

computed by solving the underwater acoustic wave propaga-

tion equation using the normal-mode method [20] or the ray-

tracing method [21]. This approach, however, does not account

for transmitter/receiver movement; thus, the Doppler effect is

not considered. A simulator in [22] models ﬂuctuations of

the amplitude and phase of eigenpaths (with ﬁxed delays) as

random processes. However, it is unclear how statistics of the

random processes should be deﬁned, depending on a particular

sea area. In addition, this model does not consider the delay

variations and, therefore, cannot accurately model the Doppler

effect. The simulator proposed in [10] incorporates the Doppler

effect by introducing different frequency shifts in different

eigenpaths; it then uses some statistical model for varying the

multipath amplitudes. However, it does not model the time-

varying multipath delays, and it is unclear how statistics of the

amplitude variations should be deﬁned.

One general approach for the simulation of signal transmis-

sion is to compute the channel impulse response for transmit-

ter/receiver positions that correspond to all sampling instants

of the signal. In [18], a method was developed to generate the

channel impulse response by interpolating eigenpaths that were

precomputed at a regular space grid in the area surrounding the

transmitter/receiver trajectory. However, for complicated and/or

long-time movements, where the numbers of grid points and

of signal samples are large, the complexity of this approach is

high.

In this paper, we propose a different approach for modeling

the underwater signal transmission that has a relatively low

complexity and can be used together with different ﬁeld com-

putation methods. In this approach, the trajectory is sampled

at a low rate (much lower than the signal sampling rate). At

every trajectory-sampling instant, a waymark impulse response

is computed using a ﬁeld computation method, e.g., the normal-

mode or ray-tracing method. Using the waymark impulse re-

sponses and adjusting the multipath delays, the time-varying

impulse response is recovered using local splines. The signals

are then modeled by compensating for the adjusted delays and

convolving the transmitted signal with the time-varying impulse

response. The proposed approach employs recursive (in time)

computations, thus allowing modeling the signal transmission

for arbitrary long trajectories.

This paper is organized as follows. In Section II, the

proposed approach is described. In Section III, it is ap-

plied to model signals in a shallow-water experiment. In

Section IV, the proposed approach is used for modeling sig-

nals in a deep-water experiment. Finally, Section V draws the

conclusions.

II. U

NDERWATER ACOUSTIC CHANNEL SIMULATOR

In a time-varying channel, the noise-free signal at a receiver

is described as a convolution, i.e.,

y(t)=

∞



−∞

h(t, τ )s(t − τ) dτ, t ∈ [0,T

] (1)

of the channel impulse response h(t, τ) and the source signal

s(t), where T

is the signal duration. This description of linear

time-variant systems is the most general. In underwater acoustic

communications, different variants of the description are used.

For example, for a channel with discrete multipath components,

we can use the model [11], [13], [23]

h(t, τ )=



p=1

(t)δ (τ − τ

(t)) (2)

where A

(t) and τ

(t) are the time-varying amplitudes and

delays of L multipath components, and δ(t) is the Dirac delta

function. If τ

(t)=τ

− at, we arrive at a channel model

that introduces compression or dilation with a compression

factor a to the received signal; this model was adopted in

[13] to describe nonuniform Doppler shifts of subcarriers in

an orthogonal frequency-division multiplexing (OFDM) sys-

tem. If τ

(t)=τ

− a

t, we obtain a model with different

compression factors a

for different multipath components as

used in [23] to describe channel distortions that are introduced

in OFDM signals. More complicated dependencies of delays

(t) on time t allow representing time-varying compression

of signals. To describe macropaths and micropaths, as well as

the individual Doppler compression for different macropaths,

the following underwater acoustic channel model is used

[24]–[26]:

y(t)=



pa=a1

∞



−∞

s(η

t − τ )h

(τ − τ

) dτ (3)

where η

= 1 + a

and h

(τ) are impulse responses that de-

scribe different micropath structures for different discrete mul-

tipaths (macropaths). This model is equivalent to (1), with

h(t, τ )=



p=1

(τ − τ

(t)) (4)

and τ

(t)=τ

− a

t. Note that the model (4) can easily incor-

porate the frequency-dependent absorption, as opposed to the

model (2).

In discrete-time form, the received signal is given by

y(nT )=

I−1



i=0

h(nT, iT )s(nT −iT ),n= 0,...,N −1(5)

where T is the signal sampling interval, I is the number

of channel taps, and N = T

/T . To model the time-varying

channel, the impulse response h(t, τ ) should be known for

every sampling instant t = nT .

940 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 61, NO. 3, MARCH 2012

Generating the impulse response for a particular point at the

trajectory is based on the acoustic-ﬁeld computation, which is

typically of high complexity. Therefore, generating h(nT, τ)

for all n would make the simulation of long-time transmission

(i.e., for high N) complicated. In this section, one method of

modeling underwater acoustic signal transmission that allows

reduction in the complexity is proposed. For clarity, we will

concentrate on scenarios where the receiver is static and the

transmitter is moving.

A. Generating the Channel Frequency Response From

Acoustic-Field Computation

Different source/receiver positions will result in different

propagation channels. The motion-induced channel time vari-

ations can be modeled by generating the channel impulse

response for the trajectory points that correspond to signal

samples. This approach can be done by deﬁning the trajectory

by the pair [r(t),z(t)], where r(t) is the horizontal distance

between the transmitter and the receiver, and z(t) is the trans-

mitter depth at instant t, mapping the sampling instants t = nT

to positions [r(nT ),z(nT )] and ﬁnding the channel frequency

response from the acoustic-ﬁeld computation for a set of fre-

quencies of interest. The channel impulse response can then be

obtained from the channel frequency response by the inverse

Fourier transform.

At a frequency ω

, the channel frequency response

p(nT, ω

) can be computed by solving the wave equation that

governs the propagation of underwater acoustic waves. The

acoustic wave equation is deﬁned by a set of environmental

parameters such as the water density ρ, the sound speed c(z)

at depth z, the source depth z

, the receiver depth z, and the

horizontal distance to the source r [20]. With the knowledge

of the environmental parameters, different methods can be

adopted for computing the acoustic ﬁeld; the most popular

approaches are the normal-mode [20] and the ray-tracing [21]

methods.

In the normal-mode method, assuming that the environment

is range independent, the frequency response at a frequency ω

is calculated as [20]

p(nT, ω

Q(ω

)



q=1

(ω

,r(nT)) Z

(ω

,z(nT )) (6)

where Z

(ω

,z) are the normal modes, and R

(ω

,r) are

expansion coefﬁcients. The main drawback of this method is

that the normal modes should be repeated for each frequency re-

quired; this approach is computationally expensive. In addition,

the number of normal modes Q(ω

) to be taken into account

increases at higher frequencies [20].

The ray-tracing method [21] is based on computing L eigen-

paths, starting from the source and hitting the receiver. The

propagation delays τ

(ω

) and attenuations a

(ω

) computed

for the lth eigenpath at the position [r(nT ),z(nT )] are then

used to compute the channel frequency response as

p(nT, ω



l=1

(ω

−jω

(ω

)

. (7)

Compared with the normal-mode method, the ray-tracing

method is more efﬁcient at dealing with high frequencies [20].

Because the channel frequency response should be calculated

for N points of the trajectory, assuming K frequencies of

interest ω

, the ﬁeld computation (e.g., the normal-mode or ray-

tracing method) should be repeated KN times. For example, to

model the deep-water signal transmission for the experiment

in Section IV, with N ≈ 10

and K ≈ 10

, about 10

ﬁeld

computations would then be required. This approach will result

in high complexity. The complexity can be reduced by assum-

ing that the multipath arrivals can approximately be treated as

frequency independent [18]. However, this step still requires

N ≈ 10

ﬁeld computations. In the following steps, we aim at

reducing this complexity.

B. Generating the Channel Impulse Response

The (real-valued) channel impulse response for the position

[r(nT ),z(nT )] can be computed as

h(nT, iT )=



K−1



k=0

p(nT, ω

jω



(8)

where ω

=(ω

−

+ kδω) ∈ [ω

−

,ω

], ω

−

> 0 and ω

> 0,

= ω

−

+(K − 1)δω, j =

√

−1, K =(ω

− ω

−

)/(δω)+

1, and i = 0,...,I − 1. In general, the frequency sampling

interval δω is determined from the maximum propagation delay

max

δω ≤

max

. (9)

In underwater acoustic channels, if the distance between the

transmitter and the receiver is high, due to the low speed of

sound, τ

max

will be high, and with the small δω, the number K

of frequencies at which the acoustic ﬁeld should be computed

will also be high; this condition will result in high complexity

of computing the frequency response.

Because the impulse response h(nT, τ ) is only nonzero for

a relatively short delay interval τ ∈ [τ

min

,τ

max

] and τ

min

usually close to τ

max

, i.e., (τ

max

− τ

min

)/τ

max

 1, we can

write

h(nT, iT )=



K−1



k=0

p(nT, ω

jω

(iT +τ

min

)



(10)

where

h(nT, iT )=h(nT, iT + τ

min

). Thus, we can replace

the ﬁltering of a signal with the impulse response h(nT, iT )

by a ﬁltering with the impulse response

h(nT, iT ) and an extra

signal delay τ

min

. Thus, the number of channel taps I can be

reduced from I ∼ τ

max

/T to I ∼ (τ

max

− τ

min

)/T . Moreover,

the frequency step δω is then calculated as

δω ≤

max

− τ

min

. (11)

Compared with (9), (11) allows higher δω and smaller K,

thus resulting in lower complexity of computing the frequency

response, particularly when the distance is long such that τ

min

is close to τ

max

, i.e., (τ

max

− τ

min

)/τ

max

 1.

LIU et al.: DOUBLY SELECTIVE UNDERWATER ACOUSTIC CHANNEL FOR MOVING TRANSMITTER/RECEIVER 941

C. Waymarks and Local-Spline Interpolation

The variation of the channel impulse response from one

signal sample to another can often be considered slow. There-

fore, the computation of the impulse response for every signal

sample will be redundant, and the trajectory sampling inter-

val T

can be made much higher than the signal sampling

interval T , i.e., T

 T . Then, an interpolation procedure

can be used for recovering the time-varying impulse response

for all signal sampling instants. In particular, we obtain

(M + 1) waymark impulse responses h

(iT )=h(mT

,iT),

where i = 0,...,I − 1, 0 ≤ mT

≤ T

, m = 0,...,M, and

= T

/M .

To approximate time-variant channels, basis expansion mod-

els (BEMs) are widely used. The most often used BEMs are

the Karhunen–Loeve (KL) functions [27], [28], discrete prolate

spheroidal (DPS) functions [29], [30], complex exponentials

(CEs) [29], [31], [32], and B-splines [33], [34]. To describe

channel time variations h(t, iT ) at a speciﬁc delay (tap) iT

using a BEM, we have the following approximation of h(t, iT ):

h(t, iT )=

b−1



m=0

i,m

(t) (12)

where b

(t) are the N

basis functions, and c

i,m

are expan-

sion coefﬁcients for the ith channel tap. The use of the KL

functions requires statistics of the path amplitude ﬂuctuations

to be known; however, we use a deterministic channel model.

The KL, DPS, CE, and many other basis functions should be

deﬁned over the entire interval [0,T

]. However, this approach

would require the following three steps: 1) computing and

keeping in memory all waymark impulse responses over [0,T

]

before expansion coefﬁcients can be computed, 2) computing

all expansion coefﬁcients at once, and 3) keeping in memory

and using all the expansion coefﬁcients for computing the

channel impulse response at any particular signal sampling

instant. With high T

, this results in high complexity and high

memory consumption.

We use local splines, which allow signiﬁcant reduction in the

complexity and memory requirement compared to the afore-

mentioned basis functions. With the same number of basis

functions, we can achieve similar performance as with an

orthogonal basis [35], [36]. When using cubic B-splines, only

four expansion coefﬁcients and four basis functions are needed

to compute the impulse response at any sampling instant.

Furthermore, to compute an expansion coefﬁcient, only a ﬁnite

number (three in our case) of waymark impulse responses are

needed. Thus, for the computation of the signal at the output of

the channel at any instant, we only need a few waymark impulse

responses. For example, for the cubic spline approximation that

we are using, for any time instant, only six waymark impulse

responses are required. In addition, all basis functions are time-

shifted versions of a single B-spline that have a ﬁnite support

(4T

for the cubic B-spline), which is usually much smaller

than T

. First, this approach signiﬁcantly reduces the modeling

complexity. Second, it allows a recursive computation of the

output signal with reduced and ﬁxed memory consumption,

independent of the signal duration.

Cubic splines provide good tradeoff between the complexity

and accuracy of approximation compared to other spline orders

[37]. We use local cubic splines for our model; however, other

local splines can also be used [33], [38], [39]. The expansion

(12) is then converted into [37]

h(t, iT )=

M+1



m=−1

i,m

b(t − mT

) (13)

whereweuseN

= M + 3, and b(t) is the cubic B-spline,

which is given by

b(t)=

⎧

⎪

⎨

⎪

⎩



2 −

|t|



−



1 −

|t|



, 0 ≤|t| <T



2 −

|t|



≤|t| < 2T

0, otherwise.

(14)

The expansion coefﬁcients c

i,m

for the local splines are found

as [33], [38]

i,m

= a

−1

m−1

(iT )+a

m+1

(iT ) (15)

where a

= a

−1

= −1/6, and a

= 4/3.

The spline interpolation can be represented in matrix form.

Let h

=[h

(0) h

(T ) ... h

(IT − T )]

and c

−1

m−1

+ a

m+1

. Then

h(T )=

M+1



m=−1

b(t − mT

Because the support of the cubic B-spline (14) is 4T

, for each

time instant t, only four spline coefﬁcients c

are required.

Therefore, for t = nT ∈ [mT

,mT

+ T

],wehave

h(nT )=Cb(nT ) (16)

where C =[c

m−1

m+1

m+2

], b(nT )=[b(ξ +

) b(ξ) b(ξ − T

) b(ξ − 2T

)]

, and ξ = nT − mT

Note that, if T

is chosen to be a multiple of T , then the

vectors b(nT ) can be precomputed for all possible ξ within

one interval T

and stored. Thus, recomputing this vector for

each signal sample is not required. It is also shown that, for

each time instant nT , the spline interpolation has a recursive

form with only six waymark impulse responses in the vicinity

of the trajectory point [r(nT ),z(nT )] involved and that no

other information is required. This condition is beneﬁcial when

modeling long-time signal transmission.

D. Delay Adjustment

Due to the low speed of sound, a small deviation of the

transmitter/receiver position may result in signiﬁcant devia-

tion of the multipath propagation delays. With a high T

this condition can result in signiﬁcant interpolation errors [4].

Fig. 1(a) shows two adjacent waymark impulse responses that

were obtained for a single-path environment. We assume that

the source moves at a constant horizontal speed of 6 m/s and

that the channel delay varies in time as a linear function.

Using T

= 125 ms, i.e., the horizontal distance between two

waymarks on the trajectory is 0.75 m, leads to an increase in the

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