Diversitymultiplexing tradeoff (DMT) characterizes the fundamental relationship between the diversity gain in terms of outage probability and the multiplexing gain as the normalized rate parameter
r
, where the limiting transmission rate is given by
rlog SNR
(here, SNR denote the received signaltonoise ratio). In this paper, we analyze the DMT and outage performance of an underwater network with a cooperative relay. Since over an acoustic channel, the propagation delay is commonly considerably higher than the processing delay, the existing transmission protocols need to be explained accordingly. For this underwater network, we briefly describe two wellknown relay transmissions: decodeandforward (DF) and amplifyandforward (AF). As our main result, we then show that an instantaneous DF relay scheme achieves the same DMT curve as that of multipleinput singleoutput channels and thus guarantees the DMT optimality, while using an instantaneous AF relay leads at most only to the DMT for the direct transmission with no cooperation. To validate our analysis, computer simulations are performed in terms of outage probability.
I. INTRODUCTION
Underwater networks have attracted considerable attention due to recent advances in acoustic communications technology
[1
,
2]
. However, underwater acoustic channels normally have limited bandwidth and severe signal attenuation as well as very low propagation speed, which are the main features that distinguish underwater systems from wireless radio links.
In underwater networks, a natural way to partially overcome such difficulties and to further improve the performance is the use of cooperation between terminals. Cooperative relay techniques have the advantages of extending the coverage and enhancing the endtoend quality in terms of capacity and reliability (e.g.,
[3

5]
for terrestrial radio networks). In the case of underwater networks, it was shown that cooperation gains could be achieved via simple maximum ratio combining
[6]
or distributed spacetime block coding
[7]
. To support the practical implementation of such a cooperative framework, a sparse channel estimation method
[8]
and a receiver structure including various detectors
[9]
were introduced.
In a quasistatic channel environment in which the transmitters do not have perfect channel state information (CSI), a fundamental performance measure to evaluate various cooperative strategies is the diversitymultiplexing tradeoff (DMT), originally introduced by Zheng and Tse
[10]
for pointtopoint multiple antenna systems. In the high signaltonoise ratio (SNR) regime, they defined the diversity gain as the rate of decay of the error probability (or outage probability) and the multiplexing gain as the rate of increase in the transmission rate, with increasing SNR. This work has stimulated a number of research efforts to extend the optimal DMT for wireless radio networks with cooperation
[5
,
11]
. For underwater systems, since only a small amount of CSI via delayed limited feedback may be available at the transmitters due to the low speed of sound in water (i.e., the slow propagation velocity), thus causing outages, characterizing the DMT is crucial in practice.
In this paper, we analyze the DMT and outage behavior for a threeterminal underwater network using an acoustic signal, where a single relay helps a source to better transmit its message to a destination. In the network, the construction of an optimal cooperative strategy in terms of DMT remains a challenge. Since the processing time, due to a variety of operations, at the relay node does not cause significant changes in the overall delay along the sourcerelaydestination path owing to the long propagation delay over an acoustic channel, the existing transmission protocols may operate in a fundamentally different manner from those in wireless radio channels and thus, need to be explained accordingly. For an underwater system, two relay transmissions, called decodeandforward (CD) and amplifyandforward (AF), are briefly described. Our results then indicate that a naïve instantaneous DF relay scheme achieves the same DMT curve as that of 2 × 1 multipleinput singleoutput (MISO) channels, thereby guaranteeing the DMT optimality. Meanwhile, the DMT achieved by an instantaneous AF relay is upperbounded by that of a direct transmission with no cooperation. To validate our analysis, computer simulations are performed with respect to the outage probability for a fixed target rate.
The rest of this paper is organized as follows: Section II describes the system and channel models. The DMT curves for underwater systems are derived in Section III, and the numerical evaluation is discussed in Section IV. Finally, we summarize the paper with some concluding remarks in Section V.
Throughout this paper, the superscript
H
denotes the conjugate transpose of a vector.
E
{∙} represents the expectation.
CN
(
m,σ
^{2}
) indicates the complex circular Gaussian with mean m and variance
σ
^{2}
per complex dimension. Unless otherwise stated, all logarithms are assumed to be to the base 2.
System model of threenode network, including S, R, and D.
II. SYSTEM AND CHANNEL MODELS
Consider a threeterminal relay system
[3]
, in which a source
S
aims to transmit its message to the corresponding destination
D
with the help of an intermediate relay
R
, as illustrated in
Fig. 1
.
Thus, there exists a direct link from
S
to
D
. It is assumed that
R
is located close to the direct transmission path. Each node has an average transmit power constraint
P
(constant). The relay node
R
is assumed to operate in the full duplex mode
[3]
and either to amplify what it receives (i.e., AF protocol) or to fully decode, reencode, and retransmit the source message (i.e., DF protocol). As in
[12]
, we consider slotted transmission protocols, where a cooperative block is composed of multiple time slots, each having a large number of symbols.
Now, let us turn to channel modeling. Due to the highly frequencyselective nature of underwater channels, multicarrier modulation (e.g., orthogonal frequencydivision multiplexing) is an attractive choice for reduction in receiver complexity. For analytical convenience, coding is assumed to be performed over a subchannel in a slot experiencing relatively flat fading (through channel coding across all the subchannels, full frequency diversity can be utilized, resulting in a better outage performance, which remains for further work). In this work, we focus on a subcarrier under the assumption that the same relay technique is applied to every subcarrier.
As stated earlier, suppose that the processing delay, taking place due to a variety of operations (e.g., receiving and reading a packet), at the relay is negligible as compared to the propagation delay in water (the propagation speed of an acoustic signal in water is around 1,500 m/s
[13]
, which is five orders of magnitude lower than that of a radiowave). This is because the processing delay is at most on the order of a few milliseconds, while the propagation delay can be of several seconds according to the distance between nodes. Such an assumption was similarly made in
[14]
only when the AF relay was used in the underwater system even if the AF protocol could not utilize the full spatial diversity, which will be specified in Section IIIA. In this model, the symbol generated at
R
is immediately forwarded to
D
, instead of waiting until the next time slot. That is, no idle time is assumed at
R
. Then, when the relative propagation delay between the direct and the relay paths is only a multiple of the basic symbol duration (far less than the length of each slot) under our network topology, the signal sent from
S
and the signal forwarded by
R
can be regarded as two paths in the frequency domain at a certain time by allowing a sufficiently long guard interval between the symbols. That is, synchronous cooperative communications can be possible owing to the use of multicarrier modulation (refer to
[15]
for the detailed description). Thus, unlike in the case of a wireless radio
[5
,
16]
, no additional time slot is required for cooperative transmission.
When the two instantaneous fullduplex relay schemes are used at a certain subcarrier (symbol), the output signals at the relay
R
and the destination
D
are given by
and
where
y_{R}
and
y_{D}
denote the signals received at
R
and
D
, respectively,
x_{S}
and
x_{R}
represent the transmitted symbols from
S
and
R
, respectively, and
z_{R}
and
z_{D}
refer to the independent and the identically distributed (i.i.d.) additive white Gaussian noises with variance
N
_{0}
. Here,
h_{RS}
,
h_{RD}
, and
h_{DS}
denote the i.i.d. channel coefficients of the
SR, RD
, and
SD
links, respectively, where all of them follow
CN
(0,1), i.e., Rayleigh fading (Note that Rician fading provides a good match for underwater acoustic channels
[17]
. However, since the high SNR outage behaviors of Rayleigh and Rician channels are shown to be identical
[18]
, we simply consider Rayleigh fading in this work). Moreover, we assume the quasistatic channel model, in which the channel coefficients are constant over time during one block transmission and change to a new independent value for the next block. The CSI is assumed to be available at the receivers, but not at the transmitters
For the AF transmission, the transmitted symbol at
R
is given by
where
g
represents the amplification factor and is given by
[5]
For DF transmission, the relay processes
y_{R}
by decoding an estimate of the symbol transmitted from
S
. The relay codebook is assumed to be independent of the source codebook. The relay
R
transmits the encoded symbol if it decodes the received signal successfully, i.e., the effective SNR 
h_{RS}

^{2}
/
N
_{0}
at
R
exceeds a predetermined threshold. Otherwise,
x_{R}
is set to 0, i.e., no transmission at
R
.
III. DMT ANALYSIS
In this section, the DMT curves for threenode underwater acoustic systems using the AF and DF protocols are analyzed after briefly reviewing DMT
[10]
.
 A. Overview of DMT
Let
r
and
d
denote the multiplexing and diversity gains, respectively. Then,
and
where
R
_{0}
(
ρ
) represents the target rate (b/s/Hz) for a given SNR
and
P_{e}
(
ρ
) denotes the error probability assuming the maximum likelihood decoding (To simplify notation,
R
_0 (
ρ
) will be written as
R
_0 if dropping
ρ
does not cause any confusion). Here,
W
represents the bandwidth. For the sake of simplicity, the notation ≐ is used for representing the relation in (4b): particularly,
is identical to (4b), and ≐ is referred to as the exponential equality.
The optimal DMT curve represents the maximum diversity gain for a given multiplexing gain
r
and is given by
d
^{*}
(
r
). It was shown in
[10]
that the outage probability
satisfies
where
I
denotes the maximum average mutual information between the input and the output, and the error probability
P_{e}
(
ρ
) of an optimal DMTachieving scheme also satisfies
P_{e}
(
ρ
) ≐
ρ
^{d*(r)}
if the block length is sufficiently large.
 B. Achievability
In this subsection, we show that the simple instantaneous DF relay scheme achieves an optimal DMT curve. An upper bound on the DMT based on an instantaneous AF relay is also derived for the sake of comparison. We start from the following lemma:
Lemma 1.
Let F(x;k) denote the cumulative distribution function of a chisquared random variable x with k degrees of freedom.
Then, it follows that
and
The proof of this lemma is presented in
[19]
. From Lemma 1, it can be easily concluded that
F
(
x
;2)=
O
(
x
) and
F
(
x
;4)=
O
(
x
^{2}
) for small
x
—
f
(
x
)=
O
(
g
(
x
)) means that positive constants
M
and
m
exist such that
f
(
x
)≤
Mg
(
x
) for all
x
>
m
. Now, we are ready to derive the achievable DMT curve for underwater acoustic systems by using the DF relay protocol.
Theorem 1.
Suppose that the instantaneous DF relay scheme is used in threenode underwater systems.
Then,
is achievable.
Proof.
If the relay
R
fully decodes the source message, i.e., log(1+
ρ

h_{RS}

^{2}
)≥
R
_{0}
, the maximum average mutual information
I
of the DF protocol is given by

I=log(1+ρhDS2+ρhDR2),
which is the same as that of a 2×1 MISO system with the input covariance matrix
under the quasistatic channel assumption
[20]
. If
R
fails to decode the symbol transmitted from
S
, i.e., there is an outage at
R
, then we have
which leads to the same performance as the direct transmission case with no cooperation. Since the two aforementioned events are mutually exclusive, the outage probability
P
_{out}
(
R
_{0}
,
ρ
) in (5) becomes a sum

Pr{hRS2≥g(ρ)} Pr{hDS2+ hDR2
where
g
(
ρ
) = (2
^{R0}
 1)/
ρ
. Since 
h_{DS}

^{2}
+ 
h_{DR}

^{2}
follows the chisquared distribution with 4 degrees of freedom, the use of (6) and (7) yields
whose high SNR behavior is readily shown to be
due to the fact that
g
(
ρ
)→0 as
ρ
→∞, where the exponential equality comes from (4a), thus resulting in (8). This completes the proof. □
Further, a DMT upper bound based on an AF relay can be found as follows:
Theorem 2.
Suppose that the instantaneous AF relay scheme is used in threenode underwater systems.
Then, the DMT curve is upperbounded by
Proof.
From a genieaided removal of the noise
z_{R}
at the relay
R
, resulting in an upper bound on the performance, the output signal at the destination can be written from (1)–(3) as
Here, it is seen that
g
(
ρ
)=1/
h_{RS}
 under the condition of noise removal, and thus,
gh_{DR}
is modeled as a random variable with uniform phases distributed over [0,2
π
). Since the characteristics of the complex circular Gaussian distribution are invariant to the phase rotation, the term
gh_{DR}h_{RS}
, independent of
h_{DS}
, also follows
CN
(0,1). Hence, the performance of the DMT is bounded by the transmission case with a direct link satisfying
CN
(0,2), which completes the proof. □
On the basis of Theorems 1 and 2, we present the following interesting discussion regarding performance comparison.
Remark 1.
To verify the optimality, we consider an upper bound on the DMT in threenode underwater systems by assuming a genieaided perfect cooperation between
S
and
R
, which leads to 2×1 MISO channels. We conclude that since the 2×1 MISO DMT curve, given by
d
^{*}
(
r
)=2(1 
r
)
[10]
, exactly matches (8), the simple instantaneous DF protocol is DMToptimal, whereas for threenode wireless communications systems, the construction of an optimal DMTachieving scheme is still a challenge. On the other hand, the instantaneous AF protocol does not guarantee the optimality in underwater systems because it cannot exploit the full spatial diversity unlike the case of wireless radio systems
[5
,
11]
.
Outage probabilities for the following four schemes: direct, amplifyandforward (AF), decodeandforward (DF), and 2 × 1 multipleinput singleoutput (MISO) transmissions, where R_{0}=10. SNR: signaltonoise ratio.
IV. NUMERICAL EVALUATION
In this section, computer simulations are described to confirm our achievability results with respect to the outage performance. We compare the following four schemes: direct transmission with no relay, instantaneous
AF protocol, instantaneous DF protocol, and 2×1 MISO transmission. For
R
_{0}
= 10, that is, a fixed target rate, the simulated channels are generated 10
^{7}
times for each scheme, and the outage probability
P
_{out}
(
R
_{0}
,
ρ
) is evaluated. The results are shown in
Fig. 2
. As expected, in the case of a high SNR, the slopes, representing the maximum diversity gain, of the outage curves for DF and 2×1 MISO look identical, whereas there exists a certain SNR gap. It is also observed that the outage performance of the AF protocol is rather worse than that of direct transmission, in sharp contrast to the case of wireless radio systems.
V. CONCLUSION
The DMT and the outage probability for cooperative underwater acoustic systems have been analyzed in this study. It was shown that the use of the simple instantaneous DF protocol was indeed DMToptimal. Meanwhile, an instantaneous AF relay was shown not to provide a better DMT performance than the direct transmission with no cooperation. As a result, vital information on how to design optimal cooperative strategies in underwater systems was provided in terms of the outage performance.
Acknowledgements
This research was supported by the Basic Science ResearchProgram through the National Research Foundationof Korea (NRF) funded by the Ministry of Science, ICT &Future Planning (2012R1A1A1044151).
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