IEEE JOURNAL OF OCEANIC ENGINEERING (TO APPEAR, 2008) 1
Multicarrier Communication over Underwater
Acoustic Channels with Nonuniform Doppler Shifts
Baosheng Li, Student Member, IEEE, Shengli Zhou, Member, IEEE, Milica Stojanovic, Member, IEEE,
Lee Freitag, Member, IEEE, and Peter Willett, Fellow, IEEE
Abstract—Underwater acoustic (UWA) channels are wideband
in nature due to the small ratio of the carrier frequency to
the signal bandwidth, which introduces frequency-dependent
Doppler shifts. In this paper, we treat the channel as having
a common Doppler scaling factor on all propagation paths, and
propose a two-step approach to mitigating the Doppler effect: (1)
non-uniform Doppler compensation via resampling that converts
a “wideband” problem into a “narrowband” problem; and (2)
high-resolution uniform compensation of the residual Doppler.
We focus on zero-padded OFDM to minimize the transmission
power. Null subcarriers are used to facilitate Doppler compensa-
tion, and pilot subcarriers are used for channel estimation. The
receiver is based on block-by-block processing, and does not rely
on channel dependence across OFDM blocks; thus, it is suitable
for fast-varying UWA channels. The data from two shallow water
experiments near Woods Hole, MA, are used to demonstrate the
receiver performance. Excellent performance results are obtained
even when the transmitter and the receiver are moving at a
relative speed of up to 10 knots, at which the Doppler shifts are
greater than the OFDM subcarrier spacing. These results suggest
that OFDM is a viable option for high-rate communications
over wideband underwater acoustic channels with nonuniform
Doppler shifts.
Index Terms—Underwater acoustic communication, multicar-
rier modulation, OFDM, wideband channels.
I. INTRODUCTION
Multicarrier modulation in the form of orthogonal frequency
division multiplexing (OFDM) has prevailed in recent broad-
band wireless radio applications due to the low complexity of
receivers required to deal with highly dispersive channels [2],
[3]. This fact motivates the use of OFDM in underwater envi-
ronments. Earlier works on OFDM focus mostly on conceptual
system analysis and simulation based studies [4], [5], [6],
Manuscript received June 1, 2007 and revised September 14, 2007. B.
Li and S. Zhou are supported by the ONR YIP grant N00014-07-1-0805
and the NSF grant ECCS-0725562. M. Stojanovic is supported by the ONR
grant N00014-07-1-0202. L. Freitag is supported by the ONR grants N00014-
02-6-0201 and N00014-07-1-0229. P. Willett is supported by the ONR
grant N00014-07-1-0055. Part of this work was presented at the IEEE/MTS
Oceans Conference, Aberdeen, Scotland, June 2007 [1]. The associate editor
coordinating the review of this paper and approving it for publication was
Prof. Urbashi Mitra.
B. Li, S. Zhou, and P. Willett are with Department of Electri-
cal and Computer Engineering, University of Connecticut, Storrs, CT
06269 (email: baosheng@engr.uconn.edu; shengli@engr.uconn.edu; wil-
lett@engr.uconn.edu).
M. Stojanovic is with Massachusetts Institute of Technology, Cambridge,
MA 02139 (email: millitsa@mit.edu).
L. Freitag is with the Woods Hole Oceanographic Institution, Woods Hole,
MA 02543 (email: lfreitag@whoi.edu).
Publisher Item Identifier
[7], while experimental results are extremely scarce [8]–[12].
Recent investigations on underwater OFDM communication
include [13] on non-coherent OFDM based on on-off-keying,
[14] on a low-complexity adaptive OFDM receiver, and [15]
on a pilot-tone based block-by-block receiver.
In this paper, we investigate the use of zero-padded OFDM
[2], [16] for UWA communications. Zero-padding is used
instead of cyclic prefix to save the transmission power spent
on the guard interval. The performance of a conventional ZP-
OFDM receiver is severely limited by the intercarrier inter-
ference (ICI) induced by fast channel variations within each
OFDM symbol. Furthermore, the UWA channel is wideband
in nature due to the small ratio of the carrier frequency to the
signal bandwidth. The resulting frequency-dependent Doppler
shifts render existing ICI reduction techniques ineffective. We
treat the channel as having a common Doppler scaling factor
on all propagation paths, and propose a two-step approach to
mitigating the frequency-dependent Doppler shifts: (1) non-
uniform Doppler compensation via resampling, which converts
a “wideband” problem into a “narrowband” one; and (2) high-
resolution uniform compensation of the residual Doppler for
best ICI reduction.
The proposed practical receiver algorithms rely on the
preamble and postamble of a packet consisting of multiple
OFDM blocks to estimate the resampling factor, the null
subcarriers to facilitate high-resolution residual Doppler com-
pensation, and the pilot subcarriers for channel estimation. The
receiver is based on block-by-block processing, and does not
rely on channel coherence across OFDM blocks; thus, it is
suitable for fast-varying underwater acoustic channels. To ver-
ify our approach, two experiments were conducted in shallow
water: one in the Woods Hole Harbor, MA, on December 1,
2006, and the other in Buzzards Bay, MA, on December 15,
2006. Over a bandwidth of 12 kHz, the data rates are 7.0, 8.6,
9.7 kbps with QPSK modulation and rate 2/3 convolutional
coding, when the numbers of subcarriers are 512, 1024, and
2048, respectively. Excellent performance is achieved for the
latter experiment, while reasonable performance is achieved
for the former experiment whose channel has a delay spread
much larger than the guard interval. The receiver performs
successfully even at a relative speed of up to 10 knots, resulting
in Doppler shifts that are greater than the OFDM subcarrier
spacing. These results suggest that OFDM is a viable option
for high-rate UWA communications over underwater acoustic
channels.
0000–0000/00$00.00
c
2008 IEEE
IEEE JOURNAL OF OCEANIC ENGINEERING (TO APPEAR, 2008) 2
The rest of the paper is organized as follows. In Section II,
the performance of a conventional OFDM receiver is analyzed.
In Section III, a two-step approach to mitigating the Doppler
shifts is proposed, and the practical receiver algorithms are
specified. In Sections IV and V the receiver performance is
reported. Section VI contains the conclusions.
II. ZERO-PADDED OFDM FOR UNDERWATER ACOUSTIC
CHANNELS
Let T denote the OFDM symbol duration and T
g
the guard
interval. The total OFDM block duration is T
0
= T + T
g
. The
frequency spacing is ∆f = 1/T . The kth subcarrier is at the
frequency
f
k
= f
c
+ k∆f, k = −K/2, . . . , K/2 − 1, (1)
where f
c
is the carrier frequency and K subcarriers are used
so that the bandwidth is B = K∆f.
Let us consider one ZP-OFDM block. Let d[k] denote the
information symbol to be transmitted on the kth subcarrier.
The non-overlapping sets of active subcarriers S
A
and null
subcarriers S
N
satisfy S
A
∪ S
N
= {−K/2, . . . , K/ 2 − 1}.
The transmitted signal in passband is then given by
s(t) = Re
("
X
k∈S
A
d[k]e
j2πk∆f t
g(t)
#
e
j2πf
c
t
)
,
t ∈ [0, T + T
g
], (2)
where g(t) describes the zero-padding operation, i.e., g(t) =
1, t ∈ [0, T ] and g(t) = 0 otherwise.
We consider a multipath underwater channel that has the
impulse response
c(τ, t) =
X
p
A
p
(t)δ(τ − τ
p
(t)), (3)
where A
p
(t) is the path amplitude and τ
p
(t) is the time-
varying path delay. To develop our receiver algorithms, we
adopt the following assumptions.
A1) All paths have a similar Doppler scaling factor a such
that
τ
p
(t) ≈ τ
p
− at. (4)
In general, different paths could have different Doppler scaling
factors. The method proposed in this paper is based on the
assumption that all the paths have the same Doppler scaling
factor. When this is not the case, part of useful signals are
treated as additive noise, which could increase the overall
noise variance considerably. However, we find that as long
as the dominant Doppler shift is caused by the direct trans-
mitter/receiver motion, as it is the case in our experiments,
this assumption seems to be justified.
A2) The path delays τ
p
, the gains A
p
, and the Doppler scaling
factor a are constant over the block duration T
0
.
The OFDM block durations are T = 42.67, 85.33, 170.67 ms
in our experiments when the numbers of subcarriers are 512,
1024, 2048, respectively. Assumption A2) is reasonable within
these durations, as the channel coherence time is usually on
the order of seconds.
The received signal in passband is then
˜y(t) =
Re
(
X
p
A
p
X
k∈S
A
d[k]e
j2πk∆f (t+at−τ
p
)
g(t + at − τ
p
)
× e
j2πf
c
(t+at−τ
p
)
)
+ ˜n(t), (5)
where ˜n(t) is the additive noise. The baseband version y(t)
of the received signal satisfies ˜y(t) = Re
y(t)e
j2πf
c
t
, and
can be written as
y(t) =
X
k∈S
A
(
d[k]e
j2πk∆f t
e
j2πaf
k
t
×
X
p
A
p
e
−j2πf
k
τ
p
g(t + a t − τ
p
)
)
+ n(t), (6)
where n(t) is the additive noise in baseband. Based on the
expression in (6), we observe two effects:
(i) the signal from each path is scaled in duration, from T
to T /(1 + a);
(ii) each subcarrier experiences a Doppler-induced frequency
shift e
j2πaf
k
t
, which depends on the frequency of the
subcarrier. Since the bandwidth of the OFDM signal is
comparable to the center frequency, the Doppler-induced
frequency shifts on different OFDM subcarriers differ
considerably; i.e., the narrowband assumption does not
hold.
The frequency-dependent Doppler shifts introduce strong in-
tercarrier interference if an effective Doppler compensation
scheme is not performed prior OFDM demodulation.
III. RECEIVER DESIGN
We first present in Section III-A the technical approach to
mitigating the frequency-dependent Doppler shifts, and then
specify in Section III-B practical receiver algorithms that we
apply to the experimental data.
A. A Two-Step Approach to Mitigating the Doppler Effect
We propose a two-step approach to mitigating the
frequency-dependent Doppler shifts due to fast-varying un-
derwater acoustic channels:
1. Non-uniform Doppler compensation via resampling.
This step converts a “wideband” problem into a “nar-
rowband” problem.
2. High-resolution uniform compensation of residual
Doppler. This step fine-tunes the residual Doppler shift
corresponding to the “narrowband” model for best ICI
reduction.
The resampling methodology has been shown effective to
handle the time-scale change in underwater communications,
see e.g., [17], [18]. Resampling can be performed either in
passband or in baseband. For convenience, let us present these
IEEE JOURNAL OF OCEANIC ENGINEERING (TO APPEAR, 2008) 3
resamplingBPF
symbol
detection
LPFdownshifting
channel
estimation
Output
Input
synchronization partition
VA
decoding
CFO
estimation
Doppler scale
coarse estimation
Block by block processing
Fig. 1. The detailed receiver diagram on one receive-element.
steps using passband signals. In the first step, we resample the
received waveform ˜y(t) using a resampling factor b:
˜z(t) = ˜y
t
1 + b
. (7)
Resampling has two effects: (1) it rescales the waveform, and
(2) it introduces a frequency-dependent Doppler compensa-
tion. With ˜y(t) from (5) and ˜z(t) = Re{z(t)e
j2πf
c
t
}, the
baseband signal z(t) is
z(t) = e
j2π
a−b
1+b
f
c
t
X
k∈S
A
(
d[k]e
j2πk∆f
1+a
1+b
t
×
X
p
A
p
e
−j2πf
k
τ
p
g
1 + a
1 + b
t − τ
p
)
+ v(t), (8)
where v(t) is the additive noise. The target is to make
1+a
1+b
as
close to one as possible. With this in mind, we have
z(t) ≈ e
j2π
a−b
1+b
f
c
t
X
k∈S
A
(
d[k]e
j2πk∆f t
×
X
p
A
p
e
−j2πf
k
τ
p
g(t − τ
p
)
)
+ v(t). (9)
The residual Doppler effect can be viewed as the same for
all subcarriers. Hence, a wideband OFDM system is con-
verted into a narrowband OFDM system with a frequency-
independent Doppler shift
=
a − b
1 + b
f
c
. (10)
In radio applications, a carrier frequency offset (CFO) between
the transmitter and the receiver leads to an expression of the
received signal in the form (9) [19], [20]. For this reason, we
call the term in (10) as CFO when a narrowband model is
concerned.
Compensating for the CFO in z(t), we obtain
e
−j2πt
z(t) ≈
X
k∈S
A
(
d[k]e
j2πk∆f t
×
X
p
A
p
e
−j2πf
k
τ
p
g(t − τ
p
)
)
+ e
−j2πt
v(t), (11)
where the subcarriers stay orthogonal. On the output of the
demodulator in the m-th subchannel, we have [2], [16].
z
m
=
1
T
Z
T
g
+T
0
e
−j2πt
z(t)e
−j2πm∆f t
dt
≈ C(f
m
)d[m] + v
m
, (12)
where C(f) :=
P
p
A
p
e
−j2πfτ
p
and v
m
is the resulting noise.
Hence, ICI-free reception is approximately achieved. Rescal-
ing and phase-rotation of the received signal thus restore the
orthogonality of the subcarriers of ZP-OFDM. The correlation
in (12) can be performed by overlap-adding of the received
signal, followed by FFT processing [2], [16].
In practice, the scale factor b and the CFO need to be
determined from the received data. They can be estimated
either separately or jointly. Note that each estimate of b will
be associated with a resampling operation, which is costly. It
is desirable to limit the number of resampling operations to as
few as possible. At the same time, high-resolution algorithms
are needed to fine-tune the CFO term for best ICI reduction.
We next specify the practical algorithms that we apply to
the experimental data.
B. Practical Receiver Algorithms
The received signal is directly sampled and all processing is
performed on discrete-time entries. Fig. 1 depicts the receiver
processing for each element, where BPF, LPF, and VA stand
for bandpass filtering, low-pass filtering, and Viterbi algorithm,
respectively. Next, we discuss several key steps.
1) Doppler scaling factor estimation: Coarse estimation of
the Doppler scaling factor is based on the preamble and the
postamble of a data packet. (This idea was used in e.g., [17] for
single carrier transmissions.) The packet structure, containing
N
b
OFDM blocks, is shown in Fig. 2. By cross-correlating
the received signal with the known preamble and postamble,
the receiver estimates the time duration of a packet,
ˆ
T
rx
. The
time duration of this packet at the transmitter side is T
tr
. By
comparing
ˆ
T
rx
with T
tx
, the receiver infers how the received
signal has been compressed or dilated by the channel:
ˆ
T
rx
=
T
tx
1 + ˆa
⇒ ˆa =
T
tx
ˆ
T
rx
− 1. (13)
The receiver then resamples the packet with a resampling
factor b = ˆa used in (7). We use the polyphase-interpolation
based resampling method available in Matlab.
IEEE JOURNAL OF OCEANIC ENGINEERING (TO APPEAR, 2008) 4
preamble postamble
OFDM
#1
OFDM
#2
OFDM
#Nb
T
g
T
tx
T
Fig. 2. Packet structure.
2) CFO estimation: A CFO estimate is generated for each
OFDM block within a packet. We use null subcarriers to
facilitate estimation of the CFO. We collect K + L samples
after resampling for each OFDM block into a vector
1
z =
[z(0), . . . , z(K + L − 1)]
T
, assuming that the channel has
L+1 taps in discrete time. The channel length can be inferred
based on the synchronization output of the preamble, and its
estimation does not need to be very accurate. We define a
(K+L)×1 vector f
m
= [1, e
j2πm/K
, . . . , e
j2πm(K+L−1)/K
]
T
,
and a (K + L) × (K + L) diagonal matrix Γ() =
diag(1, e
j2πT
c
, ··· , e
j2πT
c
(K+L−1)
), where T
c
= T/K is
the time interval for each sample. The energy of the null
subcarriers is used as the cost function
J() =
X
m∈S
N
|f
H
m
Γ
H
()z|
2
. (14)
If the receiver compensates the data samples with the correct
CFO, the null subcarriers will not see the ICI spilled over from
neighboring data subcarriers. Hence, an estimate of can be
found through
ˆ = arg min
J(), (15)
which can be solved via one-dimensional search for . This
high-resolution algorithm corresponds to the MUSIC-like al-
gorithm proposed in [19] for cyclic-prefixed OFDM.
Instead of the one-dimensional search, one can also use the
standard gradient method as in [20] or a bi-sectional search.
A coarse-grid search is needed to avoid local minima before
the gradient method or the bi-sectional search is applied [21].
Remark 1: The null subcarriers can also facilitate joint re-
sampling and CFO estimation. This approach corresponds to a
two-dimensional search: when the scaling factor b and the CFO
are correct, the least signal spill-over into null subcarriers is
observed. However, the computational complexity is high for
a two-dimensional search. This algorithm can be used if no
coarse estimate of the Doppler scaling factor (e.g., from the
pre- and post-amble of a packet) is available.
3) Pilot-tone based channel estimation: After resampling
and CFO compensation, the ICI induced by CFO is greatly
reduced. Due to assumption A2, we will not consider the ICI
due to channel variations within each OFDM block. Note that
ICI analysis and suppression in the presence of fast-varying
channels have been treated extensively in the literature, see
e.g., the references listed in [22, Ch. 19]. Ignoring ICI, the
signal in the mth subchannel can be represented as [c.f. (12)]
z
m
= f
H
m
Γ
H
(ˆ)z = H(m)d[m] + v
m
, (16)
1
Bold upper case and lower case letters denote matrices and column vectors,
respectively; (·)
T
, (·)
∗
, and (·)
H
denote transpose, conjugate, and Hermitian
transpose, respectively.
where H(m) = C(f
m
) is the channel frequency response
at the mth subcarrier and v
m
is the additive noise. On a
multipath channel, the coefficient H(m) can be related to the
equivalent discrete-time baseband channel parameterized by
L + 1 complex-valued coefficients {h
l
}
L
l=0
through
H(m) =
L
X
l=0
h
l
e
−j2πlm/K
. (17)
To estimate the channel frequency response, we use K
p
pilot
tones at subcarrier indices p
1
, . . . , p
K
p
; i.e., {d[p
i
]}
K
p
i=1
are
known to the receiver.
As long as K
p
≥ L + 1, we can find the channel taps based
on a least-squares formulation
z
p
1
.
.
.
z
p
K
p
| {z }
:=z
p
=
v
p
1
.
.
.
v
p
K
p
+
d[p
1
]
.
.
.
d[p
K
p
]
| {z }
:=D
s
×
1 e
−j
2π
K
p
1
··· e
−j
2π
K
p
1
L
.
.
.
.
.
.
.
.
.
.
.
.
1 e
−j
2π
K
p
K
p
··· e
−j
2π
K
p
K
p
L
| {z }
:=V
h
0
.
.
.
h
L
|{z}
:=h
. (18)
To minimize the complexity, we will adhere to the following
two design rules:
d1) The K
p
pilot symbols are equally spaced within K
subcarriers;
d2) The pilot symbols are PSK signals with unit amplitude.
Since the pilots are equi-spaced, we have that V
H
V =
K
p
I
L+1
[23], and since they are of unit-amplitude, we have
that D
H
s
D
s
= I
K
p
. Therefore, the LS solution for (18)
simplifies to
ˆ
h
LS
=
1
K
p
V
H
D
H
s
z
p
. (19)
This solution does not involve matrix inversion, and can be
implemented by an K
p
-point IFFT. With the time-domain
channel estimate
ˆ
h
LS
, we obtain the frequency domain es-
timates using the expression (17).
4) Multi-channel combining: Multi-channel reception
greatly improves the system performance through diversity;
see e.g., [24] on multi-channel combining for single-carrier
transmissions over UWA channels. In an OFDM system,
multi-channel combining can be easily performed on each
subcarrier. Suppose that we have N
r
receive elements, and
let z
r
m
, H
r
(m), and v
r
m
denote the output, the channel
IEEE JOURNAL OF OCEANIC ENGINEERING (TO APPEAR, 2008) 5
TABLE I
INPUT DATA STRUCTURE AND THE CORRESPONDING BIT RATES
# of active # of null # of blocks bit rates bit rates
K subcarriers subcarriers in a packet without coding after rate 2/3
(K
a
) (K
n
) (N
b
)
2(K
a
−K/4)
(T +T
g
)
channel coding
512 484 28 64 10.52 kbps 7.0 kbps
1024 968 56 32 12.90 kbps 8.6 kbps
2048 1936 112 16 14.55 kbps 9.7 kbps
3 packets per data burst
Stop Packet Stop StopPacket
Packet
K=512 K=1024 K=2048
Stop
Fig. 3. Each data burst consists of three packets, with K = 512, K = 1024,
and K = 2048, respectively
frequency response, and the additive noise observed at the
mth subcarrier of the rth element. We thus have:
z
1
m
.
.
.
z
N
r
m
|
{z }
:=z
m
=
H
1
(m)
.
.
.
H
N
r
(m)
| {z }
:=
˜
h
m
d[m] +
v
1
m
.
.
.
v
N
r
m
| {z }
:=v
m
. (20)
Assuming that v
m
has independent and identically distributed
entries, the optimal maximum-ratio combining (MRC) yields
ˆ
d[m] =
˜
h
H
m
˜
h
m
−1
˜
h
H
m
z
m
. (21)
Doppler scaling factor, CFO, and channel estimation are
performed independently on each receiving element according
to the procedure described in Sections III-B.1 to III-B.3. An
estimate of the channel vector
˜
h
m
is then formed, and used
to obtain the data symbol estimates in (21).
IV. PERFORMANCE RESULTS FOR THE EXPERIMENT IN
BUZZARDS BAY
The bandwidth of the OFDM signal is B = 12 kHz, and the
carrier frequency is f
c
= 27 kHz. The transmitted signal thus
occupies the frequency band between 21 kHz and 33 kHz. We
use zero-padded OFDM with a guard interval of T
g
= 25 ms
per OFDM block. The respective number of subcarriers used
in the experiment is K = 512, 1 024, and 2048. The subcarrier
spacing is ∆f = 23.44 Hz, 11.72 Hz, and 5.86 Hz, and the
OFDM block duration is T = 1/∆f = 42.67 ms, 85.33
ms, and 170.67 ms. We use rate 2/3 convolutional coding,
obtained by puncturing a rate 1/2 code with the generator
polynomial (23,35). Coding is applied within the data stream
for each OFDM block. QPSK modulation is used. For K =
512, 1024, 20 48, each packet contains N
b
= 64, 32, 16 OFDM
blocks, respectively. The total number of information bits per
packet is 30976. The signal parameters and the corresponding
data rates are summarized in Table I, where the overhead of
null subcarriers and K
p
= K/4 pilot subcarriers is accounted
for.
Fig. 3 depicts one data burst that consists of three packets
with K = 512 , K = 1024, and K = 204 8, respectively.
2.5 m
Source
ITC-6137
6 m
0.5 m
Receiver
HTI-96 Array
600 m~-110 m
Tioga
Mytilus
Fig. 4. The configuration of the experiment in Buzzards Bay.
Fig. 5. The received signal (amplitude) for the Buzzards Bay experiment.
During the experiments, the same data burst was transmitted
multiple times while the transmitter was on the move.
The WHOI acoustic communication group conducted the
experiment on Dec. 15, 2006 in Buzzards Bay, MA. The
transmitter was located at a depth of about 2.5 meters and the
receiver consisted of a four-element vertical array of length 0.5
m submerged at a depth of about 6 meters. The transmitter was
mounted on the arm of the vessel Mytilus, and the receiver
array was mounted on the arm of the vessel Tioga. OFDM
signals were transmitted while Mytilus was moving towards
Tioga, starting at 600 m away, passing by Tioga, and ending
at about 100 m away. The experiment configuration is shown
in Fig. 4.
The received signal was directly A/D converted. The signal
received on one element is shown in Fig. 5, which contains 7
data bursts or 21 packets. The following observations can be
made from Fig. 5.
1) The received power is increasing before packet 19, and
decreasing thereafter.
This observation is consistent with the fact that Mytilus
passed Tioga around that time.
2) A sudden increase in noise shows up around packet 19.
